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Energy balance and temperature in a CO₂ laser produced plasma Popil, Roman 1984

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ENERGY BALANCE AND TEMPERATURE IN A C 0 2 LASER PRODUCED PLASMA by ROMAN EWHEN POPIL B . S c , U n i v e r s i t y of R e g i n a , 1977. M . S c , U n i v e r s i t y of B r i t i s h C o l u m b i a , 1979. A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES (Department of P h y s i c s ) We accept t h i s t h e s i s as c o n f i r m i n g to the r e q u i r e d s t andard THE UNIVERSITY OF BRITISH COLUMBIA September 1984 © Roman Ewhen P o p i l , 1984 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r equ i rement s f o r an advanced degree a t 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 agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g ran ted by the head o f my department o r by h i s o r her r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l no t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f P h y s i c s  The U n i v e r s i t y o f B r i t i s h Columbia 1956 Main Mall V a n c o u v e r , Canada V6T 1Y3 Date October 1,19 84. A b s t r a c t The a b s o r p t i o n of i n t e n s e j< 1 0 1 ** W/cm 2 C 0 2 l a s e r r a d i a t i o n by an underdense plasma i s i n v e s t i g a t e d e x p e r i m e n t a l l y . The plasma i s produced by the C 0 2 l a s e r beam focus sed onto a s t a b i l i z e d l aminar gas j e t emanating from a L a v a l n o z z l e . The e l e c t r o n temperature of the l a s e r produced plasma i s measured by s o f t x - r a y d i a g n o s t i c s which y i e l d a 300 eV thermal and a 2000 eV supra thermal t empera ture . Time r e s o l v e d s t u d i e s are made of the r a d i a l expans ion of the plasma to determine the absorbed energy and temperature u s i n g a m o d i f i e d b l a s t wave a n a l y s i s tha t i n v o l v e s the i d e n t i f i c a t i o n of the Chapman-Jouguet d e t o n a t i o n p o i n t i n the e x p a n s i o n . The absorbed energy i s a l s o de te rmined from U l b r i c h t sphere photometry . I n f o r m a t i o n on the plasma d imens ions arid e l e c t r o n d e n s i t y i s o b t a i n e d by time r e s o l v e d ruby l a s e r i n t e r f e r o m e t r y . The e l e c t r o n temperature and absorbed energy measurements are. used to c o r r o b o r a t e one another and- to p r o v i d e a s e l f -c o n s i s t e n t p i c t u r e of the l a s e r plasma c o u p l i n g . A l t h o u g h vacuum l a s e r i n t e n s i t i e s are s u f f i c i e n t l y h i g h where s a t u r a t i o n of c o l l i s i o n a l ( i n v e r s e b r e m s s t r a h l u n g ) , a b s o r p t i o n i s expec ted to o c c u r , the e x p e r i m e n t a l ev idence i n d i c a t e s tha t l i n e a r i n v e r s e b r e m s s t r a h l u n g account s for the observed thermal e l e c t r o n temperature and measured l e v e l of a b s o r p t i o n . The energy ba lance shows tha t t h e r e i s a s u f f i c i e n t l e v e l of a b s o r p t i o n to accommodate the measured thermal e l e c t r o n temperature as w e l l as < 15% of the e l e c t r o n s at a 2 keV supra thermal t e m p e r a t u r e . i i i Tab le of Contents A b s t r a c t i i T a b l e of Content s i i i L i s t of T a b l e s v L i s t of F i g u r e s ' v i Acknowledgements ix Chapter I : I n t r o d u c t i o n 1 Chapter I I : A b s o r p t i o n of La ser R a d i a t i o n 6 11 — 1 Inverse b remss t r ah lung a b s o r p t i o n . 6 11 - 2 S a t u r a t i o n of C o l l i s i o n a l A b s o r p t i o n 10 11 -3 Resonance A b s o r p t i o n 12 11 - 4 I o n - A c o u s t i c T u r b u l e n c e 15 II - 5 Thermal C o n d u c t i v i t y 17 I I -6 E l e c t r o n H e a t i n g Model 21 I I - 7 R e s u l t s from Hydrodynamic Computer Codes 27 Chapter I I I : The Apparatus and i t s A p p l i c a t i o n 32 III — 1 Laminar Gas J e t T a r g e t . . . 34 I I I - 2 The C 0 2 Laser S y s t e m . . 42 I I I -3 X - r a y D i a g n o s t i c s 48 111 -4 U l b r i c h t I n t e g r a t i n g Sphere 58 I I I - 5 Image C o n v e r t e r S t reak Camera 63 Chapter I V : R e s u l t s from the Exper iments 67 IV- 1 E l e c t r o n Temperature Measurements 67 IV-1a I n i t i a l O b s e r v a t i o n s at Low I n t e n s i t i e s 68 IV-1b Change i n X - r a y E m i s s i o n wi th Targe t C o n d i t i o n s 72 IV-1c The E f f e c t of Hot E l e c t r o n s 74 V - l d X - r a y Measurements at High Laser I n t e n s i t y . . 7 8 IV-2 O p t i c a l I n v e s t i g a t i o n of the Plasma Expans ion 83 IV-3 B l a s t waves w i t h heat t r a n s f e r 93 IV-4 T i m e - r e s o l v e d i n t e r f e r o m e t r i c study of the plasma expans ion 103 IV-5 C - J A n a l y s i s of the F r o n t Plasma I s l a n d . . . . . . 1 0 7 IV-6 Energy A b s o r p t i o n Measurements 111 Chapter V : A n a l y s i s and D i s c u s s i o n 115 V - 1 Overview 115 V-2. Inverse Bremss t rah lung A b s o r p t i o n 117 V-3 Energy Ba lance and E l e c t r o n Temperature 124 V-4 The Nature of the Supra thermal T e m p e r a t u r e . . . 130 V-4a Smoothing E f f e c t s of the X - r a y D i a g n o s t i e s . . 1 3 0 V-5 D e c o n v o l u t i o n of the X - r a y Data 140 V-6 Energy C o n t a i n e d i n the Plasma Waves 146 Chapter V I : C o n c l u s i o n s 148 VI-1 F u r t h e r Work 1 52 B i b l i o g r a p h y 155 Appendix A : N i t r o g e n l i n e e m i s s i o n e f f e c t on T d e t e r m i n a t i o n ' 165 e Appendix B: C a l c u l a t i o n of R a d i a t i v e Power Loss from the Plasma 170 V L i s t of T a b l e s V-1 E l e c t r o n temperatures c a l c u l a t e d by u s i n g i n v e r s e bremss t rah lung 119 V-2 Energy c o n t a i n e d i n i o n i z a t i o n u s i n g i n t e r f e r o m e t r i c data 126 V-3 E l e c t r o n temperatures c a l c u l a t e d from the energy ba lance 129 V-4 R e s u l t s from the c o n v o l u t i o n c a l c u l a t i o n s w i t h T e = 2000 eV 134 V-5 Hot e l e c t r o n f r a c t i o n c a l c u l a t e d by energy b a l a n c e . . . 145 A-1 E x c i t a t i o n e n e r g i e s and o s c i l l a t o r s t r e n g t h s for N ion l i n e s 1 68 B-1 Re l avent q u a n t i t i e s for r e c o m b i n a t i o n r a d i a t i o n from a n i t r o g e n plasma 171 v i L i s t of F i g u r e s II — 1 Form of the assumed r a d i a l temperature p r o f i l e 23 II-2 R e s u l t s from the n u m e r i c a l s o l u t i o n of the energy t r a n s p o r t e q u a t i o n 25 I I - 3 Temperature s c a l i n g w i t h i n c i d e n t l a s e r energy from MEDUSA computer runs 29 III — 1 O v e r a l l d e p i c t i o n of the e x p e r i m e n t a l geometry . . . . 33 I I I - 2 D e t a i l s of the L a v a l N o z z l e 36 111-3 E l e c t r i c a l and gas systems of the t a r g e t chamber 37 I I I - 4 Ruby l a s e r shadowgraphs of the gas j e t t a r g e t 39 I I I - 5 S t a b i l i z e d j e t o p e r a t i n g p r e s s u r e s de te rmined from ruby l a s e r shadowgraphy 40 111 -6 R a y l e i g h s c a t t e r i n g on the gas j e t 42 111-7 Schematic diagram of the o p t i c a l l a y - o u t of the C 0 2 l a s e r system 44 I I I -8 X - r a y t r a n s m i s s i o n through aluminum and b e r y l l i u m f o i l s 50 I I I - 9 X - r a y d i a g n o s t i c s s e t - u p for the z - p i n c h 52 111 — 10 X - r a y t r a n s m i s s i o n data fo r the z - p i n c h plasma 54 111-11 X - r a y d i a g n o s t i c s s e t - u p for the gas j e t t a r g e t 55 111 - 1 2 P h o t o m u l t i p l i e r c h a r a c t e r i z a t i o n 56 111 - 1 3 Schematic of an U l b r i c h t sphere 59 111 - 1 4 U l b r i c h t sphere p h o t o m e t r i c s e t - u p 61 111 — 15 Arrangement for the i n f r a r e d energy i n v e n t o r y . . . . 62 111 — 16 C i r c u i t diagram for the image c o n v e r t e r s t r e a k camera 64 111- 17 Arrangement for time s y n c h r o n i z a t i o n of the spark gap l i g h t source fo r shadow s t r e a k photography 66 111- 18 Se t -up for time r e s o l v e d s c h l i e r e n photography 66 v i i IV-1 R a t i o of t r a n s m i t t e d x - r a y i n t e n s i t i e s 9 and 1 8 ym f o i l v s . T g 69 IV-2 E l e c t r o n temperature s c a l i n g w i t h i n c i d e n t l a s e r e n e r g y . Laser i n t e n s i t i e s < 1 0 1 3 W/cm 70 IV-3 Hard x - r a y i n t e n s i t y as a f u n c t i o n of t a r g e t p r e s s u r e 73 IV-4 E l e c t r o n temperature ver sus l a s e r energy . La se r i n t e n s i t i e s <_ l 0 1 1 + W / c m 2 74 IV-5 X - r a y t r a n s m i s s i o n data from aluminum and copper f o i l s 79 IV-6 R e l a t i v e t r a n s m i t t e d x - r a y i n t e n s i t i e s from b e r y l l i u m f o i l s 78 IV-7 Summary of x - ray t r a n s m i s s i o n data 82 IV-8 Luminous f r o n t s t reak photograph 85 IV-9 R-t curve s from the theory of Sakura i for v a r i o u s absorbed e n e r g i e s 88 IV-10 Absorbed l a s e r e n e r g i e s as o b t a i n e d from the compar i son of luminous f r o n t s t reak photographs w i t h S a k u r a i B l a s t wave curve s 88 IV-11 Time r e s o l v e d shadowgraphy 91 IV-12 Absorbed e n e r g i e s o b t a i n e d by a n a l y s i s of shadow s t reak photographs 92 IV-13 Regions of a thermal b l a s t wave 96 IV-14 Space-t ime h i s t o r y of a b l a s t wave w i t h heat t r a n s f e r 97 IV-15 Chapman-Jouguet a n a l y s i s of s t r e a k photographs 100 IV-16 Absorbed energy o b t a i n e d by C - J a n a l y s i s of s t r e a k photographs 101 IV-17 E l e c t r o n temperatures de termined from C - J a n a l y s i s of shadowgraph s t r e a k s . . . . 1 0 3 IV-18 Geometry of the Mach-Zehnder i n t e r f e r o m e t e r 104 IV-19 Sample l a t e - t i m e i n t e r f e r o g r a m 105 IV-20 R a d i a l plasma d e n s i t y p r o f i l e s 106 IV-21 L o g - l o g p l o t s of the plasma r a d i u s 107 v i i i IV-22 Shadow s t reak photograph 108 IV-23 S c h l i e r e n s t reak photography of the f r o n t He-N i n t e r f a c e r e g i o n 109 IV-24 E l e c t r o n temperatures o b t a i n e d by C- J a n a l y s i s of the f r o n t He-N i n t e r f a c e r e g i o n 110 IV-25 S c a t t e r e d - r e f r a c t e d l a s e r energy measured by the U l b r i c h t photometer 112 IV- 26 Absorbed l a s e r energy determined by the U l b r i c h t sphere 113 V - 1 E l e c t r o n q u i v e r v e l o c i t y e f f e c t on the e l e c t r o n temperature measurement 123 V-2 D e c o n v o l u t i o n of the x-ray t r a n s m i s s i o n data 143 V - 3 E l e c t r o n energy d i s t r i b u t i o n from x- ray data 144 V I - 1 Summary of the measured and c a l c u l a t e d e l e c t r o n temperatures , 150' A-1 L i n e e m i s s i o n i n t e n s i t y as a f u n c t i o n of T 169 e A-2 Change i n r e l a t i v e t r a n s m i t t e d x - r a y i n t e n s i t i e s due to x - ray l i n e e m i s s i o n 169 ix Acknowledgement I am i n d e b t e d to D r . J . Meyer fo r h i s i n v i t a t i o n to work on l a s e r - p l a s m a i n t e r a c t i o n s and for the o p p o r t u n i t y to do so under h i s a u s p i c e s . The C 0 2 l a s e r plasma p r o j e c t i s a l a r g e c o l l a b o r a t i v e e f f o r t i n v o l v i n g the work of s e v e r a l p e o p l e . John B e r n a r d , Grant M c i n t o s h and Hubert Houtman were d i r e c t l y i n v o l v e d i n many of the exper iments a s s i s t i n g w i t h the c o l l e c t i o n and a n a l y s i s of the d a t a . The C 0 2 l a s e r system e v o l v e d over s e v e r a l y e a r s l a r g e l y through the l a b o u r s of Hubert Houtman, D r . C . J . W a l s h , John Bernard and D r . B. H i l k o . I t i s a l s o a p l e a s u r e to acknowledge many u s e f u l d i s c u s s i o n s w i t h D r . B. A h l b o r n , D r . A . Ng, Dean P a r f e n i u k , L u i z D a S i l v a , D r . E . H i o b , D r . W. L i e s e , D r . S. K a s t n e r , and D a n i e l P a s i n i . Jack Bosma and A l Cheuck p r o v i d e d much of the r e q u i r e d t e c h n i c a l e x p e r t i s e throughout the cour se of t h i s work. 1 Chapter I : I n t r o d u c t i o n S h o r t l y a f t e r the i n v e n t i o n of the ruby l a s e r i n 1960 by Maiman, exper iments began to produce plasmas by f o c u s s i n g l a s e r r a d i a t i o n onto m a t e r i a l s . An e x t e n s i v e review of these numerous s t u d i e s can be found i n H u g h e s 1 . These plasmas were c o m p a r a t i v e l y much h o t t e r and denser than those t y p i c a l l y found i n e l e c t r i c a l d i s c h a r g e d e v i c e s . The p o s s i b i l i t y of a c h i e v i n g s u f f i c i e n t l y h i g h temperatures and d e n s i t i e s to o b t a i n c o n t r o l l e d thermonuclear f u s i o n has been the impetus for much of the r e s e a r c h i n plasma p h y s i c s s i n c e the 1950 ' s . A c c o r d i n g l y , the e x t r a p o l a t i o n of l a s e r produced plasmas to c o n d i t i o n s where thermonuc lear f u s i o n can occur remains to t h i s day a t o p i c of i n t e n s i v e r e s e a r c h e f f o r t s throughout the w o r l d . The main idea beh ind the l a s e r f u s i o n program i s the i s e n t r o p i c i m p l o s i o n of a d e u t e r i u m - t r i t i u m m i x t u r e . The i n c r e a s e i n d e n s i t y produced by such an i m p l o s i o n can reduce the energy r e q u i r e d to produce a g i v e n thermonuc lear output energy by a f a c t o r of as much as 10" below that needed fo r h e a t i n g an uncompressed p l a s m a 2 . Most of the e a r l y i m p l o s i o n exper iments i n the 1970's used s imple t h i n s h e l l g l a s s m i c r o b a l l o o n s f i l l e d w i t h D-T gas at h i g h p r e s s u r e which a c t e d l i k e " e x p l o d i n g p u s h e r s " . These exper iments were c h a r a c t e r i z e d by shor t plasma s c a l e l e n g t h s and s h o r t (<1 ns) l a s e r p u l s e s of s e v e r a l kJ energy . A l t h o u g h these exper iments y i e l d e d l a r g e amounts of neutrons and h i g h plasma t empera ture s , thermonuclear i g n i t i o n cannot be a c h i e v e d w i t h any f e a s i b l e l a s e r system based on an e x p l o d i n g pusher scheme. S i n c e 2 1 980 3 the emphasis has t u r n e d towards a b l a t i v e compress ion s t u d i e s i n which the D-T gas i s a d i a b a t i c a l l y compressed . These exper iments are done w i t h t a r g e t s that have a c o m p a r a t i v e l y t h i c k p l a s t i c ou ter s h e l l which i s a b l a t e d by a mul t inanosecond l a s e r p u l s e r e s u l t i n g i n an e x t e n s i v e tenuous underdense c o r o n a . (The e l e c t r o n d e n s i t y of the corona plasma i s s a i d to be underdense when the l a s e r l i g h t f requency i s g r e a t e r than the plasma f requency and can propagate through the p la sma. Laser l i g h t i s r e f l e c t e d at the c r i t i c a l d e n s i t y ( n c r ) a n ( 3 plasma d e n s i t i e s g r e a t e r than t h i s are s a i d to be o v e r d e n s e . ) The i n t e r a c t i o n between the l a s e r and t h i s underdense c o r o n a l plasma i s important i n these a b l a t i v e pusher exper iments where phenomena such as s t i m u l a t e d B r i l l o u i n (SBS) and Raman s c a t t e r i n g (SRS) can have s e r i o u s d e t r i m e n t a l consequences on the e f f i c i e n c y of l a s e r energy c o u p l i n g to the t a r g e t . In SBS and SRS, the i n c i d e n t l a s e r l i g h t i n t e r a c t s w i t h d e n s i t y p e r t u r b a t i o n s i n the plasma r e s u l t i n g in a s c a t t e r e d l a s e r beam and subsequent enhancement of these p e r t u r b a t i o n s . These phenomena can cause a s u b s t a n t i a l r e f l e c t i o n of the i n c i d e n t l a s e r energy . The e f f e c t i v e a b s o r p t i o n of l a s e r l i g h t i s perhaps the most important i s s u e i n a l l l a s e r f u s i o n schemes. The o b j e c t i v e i s to heat the body of the e l e c t r o n d i s t r i b u t i o n a v o i d i n g the g e n e r a t i o n of supra thermal e l e c t r o n s which can p e n e t r a t e the c o l d dense plasma above the c r i t i c a l d e n s i t y l a y e r c a u s i n g unwanted p r e h e a t i n g of the t a r g e t . S e v e r a l a b s o r p t i o n mechanisms are known to occur i n the underdense c o r o n a . The most important mechanism i s c o l l i s i o n a l i n v e r s e b remss t r ah lung a b s o r p t i o n which 3 a r i s e s from the i n t e r a c t i o n of the e l e c t r o n s i n the Coulomb f i e l d of the i o n s . E l e c t r o n s o s c i l l a t e under the i n f l u e n c e of the l a s e r e l e c t r i c f i e l d and subsequent ly t r a n s f e r t h e i r l a s e r f i e l d induced o s c i l l a t o r y mot ion to thermal k i n e t i c e n e r g y . A l t h o u g h the v a l i d i t y of the theory for i n v e r s e b r e m s s t r a h l u n g a b s o r p t i o n been t e s t e d and v e r i f i e d for low l a s e r i n t e n s i t i e s of < 1 0 1 0 W / c m 2 , at h i g h e r i n t e n s i t i e s where the o s c i l l a t o r y v e l o c i t y of the e l e c t r o n s v , becomes comparable to the os thermal v e l o c i t y v , t h i s t h e o r y i s expected to f a i l as th e l e c t r o n - i o n c o l l i s i o n s become i n c r e a s i n g l y l e s s e f f e c t i v e fo r energy t r a n s f e r of the o s c i l l a t o r y mot ion to thermal e n e r g y . To d a t e , most l a s e r plasma exper iments a re concerned w i t h the a b s o r p t i o n of l a s e r r a d i a t i o n by s o l i d t a r g e t s . In these c a s e s , the presence of a c r i t i c a l d e n s i t y l a y e r c o m p l i c a t e s the o v e r a l l p i c t u r e so t h a t c o n c l u s i o n s c o n c e r n i n g the r o l e of i n v e r s e b remss t r ah lung a b s o r p t i o n are somewhat i n s u b s t a n t i a l . T h i s t h e s i s e x p e r i m e n t a l l y examines the a b s o r p t i o n of C 0 2 l a s e r l i g h t by a l a s e r produced underdense p la sma. In order to determine the absorbed l a s e r energy , the plasma i s d i agnosed by s o f t x - ray measurements, U l b r i c h t sphere photometry , i n t e r f e r o m e t r y and time r e s o l v e d photography . The main q u e s t i o n i s how the absorbed l a s e r energy i s c o r r e l a t e d w i t h the measured e l e c t r o n t e m p e r a t u r e . T h i s enab le s a compar i son between the expected a b s o r p t i o n due to i n v e r s e b remss t rah lung and measured v a l u e s fo r an underdense p la sma. The energy ba lance s tudy a l l o w s the i n v e s t i g a t i o n of the d i s t r i b u t i o n of the absorbed energy among the v a r i o u s p roce s se s which take p l a c e i n the l a s e r - p l a s m a i n t e r a c t i o n , e . g . hot e l e c t r o n p r o d u c t i o n , r a d i a t i o n l o s s e s , 4 e t c . The o r g a n i z a t i o n of t h i s t h e s i s i s as f o l l o w s . The theory of a b s o r p t i o n of l a s e r r a d i a t i o n by a plasma i s d i s c u s s e d semi-q u a n t i t a t i v e l y i n the f o l l o w i n g chap te r to p r o v i d e some p h y s i c a l i n s i g h t to the p roce s se s as w e l l as to i n t r o d u c e some of the c o m p l i c a t i o n s and e f f e c t s t h a t are expec ted to a r i s e for the e x p e r i m e n t a l c o n d i t i o n s e n c o u n t e r e d . A s imple e l e c t r o n h e a t i n g model i s p r e s e n t e d a l o n g w i t h e s t i m a t e s of the expected e l e c t r o n t e m p e r a t u r e . These e s t i m a t e s are then supplemented by the r e s u l t s of s e v e r a l computer s i m u l a t i o n s t h a t were performed u s i n g a v a i l a b l e hydrodynamic computer c o d e s . S i n c e t h i s t h e s i s i s m a i n l y an e x p e r i m e n t a l work d e s c r i b i n g the i n t e r a c t i o n of a C 0 2 l a s e r beam and the plasma i t p r o d u c e s , b r i e f d e s c r i p t i o n s of the appara tus used f o l l o w i n Chapter I I I . To ensure p r o d u c t i o n of a s t r i c t l y underdense p lasma, the C 0 2 l a s e r beam i s d i r e c t e d and focus sed upon a v a r i a b l e d e n s i t y gas j e t t a r g e t . The d e v i c e for p r o d u c i n g the l aminar gas t a r g e t and i t s c h a r a c t e r i z a t i o n fo r use as a t a r g e t i s d e s c r i b e d i n t h i s c h a p t e r . In a d d i t i o n to the hardware of the l a s e r and i t s t a r g e t , the d i a g n o s t i c i n s t r u m e n t a t i o n deve loped and implemented i n the cour se of the exper iments i s a l s o p r e s e n t e d h e r e . The e x p e r i m e n t a l r e s u l t s f o r the e l e c t r o n temperature and energy a b s o p t i o n are c o n t a i n e d i n Chapter I V . E r r o r estimates are c o n s i d e r e d and the r e s u l t s are compared w i t h the data o b t a i n e d from o t h e r l a b o r a t o r i e s fo r comparable e x p e r i m e n t a l c o n d i t i o n s . A u n i f y i n g s e l f - c o n s i s t e n t . p i c t u r e of the a b s o r p t i o n proce s s of the l a s e r - p l a s m a i n t e r a c t i o n i s deve loped i n Chapter 5 V . The r e s u l t s of v a r i o u s c a l c u l a t i o n s are t a b u l a t e d and the energy i n v e n t o r y i s p r e s e n t e d . The nature of the supra thermal temperature measured by so f t x - r a y d i a g n o s t i c s r e c e i v e s c o n s i d e r a t i o n and s p e c u l a t i o n r e g a r d i n g i t s p o s s i b l e o r i g i n . The r e s u l t s are summarized and c o n c l u d i n g s tatements c o n c e r n i n g the na ture of the a b s o r p t i o n of l a s e r r a d i a t i o n for C 0 2 l a s e r i n t e n s i t i e s <10 1 " W/cm 2 i n underdense plasma are made i n Chapter V I . The most p r o b a b l e mechanism for the g e n e r a t i o n of the supra thermal e l e c t r o n s i s p r o f e r r e d . S i n c e no i n v e s t i g a t i o n can ever produce a l l the answers , a few s u g g e s t i o n s for f u r t h e r work are i n c l u d e d as w e l l . The t h e s i s and the new s i g n i f i c a n t r e s u l t s i t c o n t a i n s can be summarized as f o l l o w s . The t h e s i s d e s c r i b e s the a b s o r p t i o n and energy ba lance of a unique l a s e r - u n d e r d e n s e plasma i n t e r a c t i o n based on x - r a y d i a g n o s t i c s , i n t e r f e r o m e t r y , s t r e a k photography and U l b r i c h t sphere measurements. A new method of a n a l y s i s of the s t r e a k r e c o r d s of the r a d i a l plasma expans ion i s i n t r o d u c e d and implemented to p r o v i d e an independent e s t imate of the plasma tempera ture , p r e s s u r e and absorbed l a s e r energy . The U l b r i c h t sphere and x - ray d i a g n o s t i c s measurements, a l o n g w i t h i n t e r f e r o m e t r i c e l e c t r o n d e n s i t y i n f o r m a t i o n , y i e l d the absorbed energy and e l e c t r o n temperature as w e l l , e n a b l i n g a comparison of the r e s u l t s for i n v e r s e bremss t rah lung a b s o r p t i o n and to make an energy i n v e n t o r y for the i n t e r a c t i o n . A l l of the r e s u l t s can be e x p l a i n e d on the b a s i s of i n v e r s e b remss t r ah lung a b s o r p t i o n , a l t h o u g h s a t u r a t i o n of t h i s a b s o r p t i o n mechanism i s p r e d i c t e d by theory fo r the e x p e r i m e n t a l c o n d i t i o n s e n c o u n t e r e d . 6 Chapter 1 1 : A b s o r p t i o n of Laser Rad ia t ion The s tudy of the energy ba l ance r e q u i r e s a d e t a i l e d account of the energy a b s o r p t i o n mechanisms and of the energy l o s s p r o c e s s e s . S e v e r a l d i f f e r e n t a b s o r p t i o n p roce s se s may i n whole or i n p a r t , account fo r the a b s o r p t i o n observed i n l a s e r - p l a s m a e x p e r i m e n t s . I t i s t h e r e f o r e a p p r o p r i a t e to p r e s e n t l y review some of these v a r i o u s a b s o r p t i o n mechanisms. The purpose of t h i s c h a p t e r i s to p r o v i d e some i n s i g h t i n t o the r e l e v a n t p h y s i c s by which a l a s e r beam i s absorbed by a p la sma . The d i s c u s s i o n w i l l be l i m i t e d to s e m i - q u a n t i t a t i v e rev iews of the p h y s i c a l p r o c e s s e s of a b s o r p t i o n mechanisms s i n c e a thorough t h e o r e t i c a l t rea tment of t h i s s u b j e c t i s beyond the scope of t h i s t h e s i s . The review of the p h y s i c a l p roce s se s i n v o l v e d i n the c o l l i s i o n a l a b s o r p t i o n of l a s e r l i g h t serves as a p r e l u d e to a d i s c u s s i o n of some of the anomalous mechanisms t h a t can o c c u r . In a d d i t i o n , some n u m e r i c a l e s t i m a t e s of the expected e x p e r i m e n t a l r e s u l t s w i l l be p re sen ted a l o n g w i t h the r e s u l t s of some computer s i m u l a t i o n s tha t were per fo rmed . I I -1 Inver se b r e m s s t r a h l u n g a b s o r p t i o n Inver se b remss t r ah lung a b s o r p t i o n i s c o n s i d e r e d to be the most important mechanism f o r the a b s o r p t i o n of l a s e r l i g h t by a p la sma. I t i s the p r o c e s s by which the e l e c t r o n p o p u l a t i o n i n c r e a s e s i t s thermal energy through the a b s o r p t i o n of photons i n the Coulomb f i e l d of the i o n s . Other a b s o r p t i o n proce s se s to 7 be d i s c u s s e d l a t e r c r e a t e an e n e r g e t i c component of hot e l e c t r o n s which u s u a l l y c o n s t i t u t e s a m i n o r i t y of the e l e c t r o n p o p u l a t i o n . The i n v e r s e b remss t r ah lung p r o c e s s a r i s e s from the o s c i l l a t i o n of the e l e c t r o n s i n the e l e c t r i c f i e l d of the i n c i d e n t l a s e r l i g h t . T h i s d i r e c t e d energy of e l e c t r o n motion i s randomized and hence c o n v e r t e d i n t o heat energy by e l e c t r o n c o l l i s i o n s w i t h i o n s . The e q u a t i o n of motion of the e l e c t r o n under these c i r c u m s t a n c e s i s approx imate ly -»• • m(dv/dt ) + mv v = - e E , e i here the c o l l i s i o n a l damping term mp v c o n t a i n s the i o n ei. c o l l i s i o n f requency v . There i s an induced c u r r e n t J , due to e i the e l e c t r o n mot ion a l o n g the d i r e c t i o n of the e l e c t r i c f i e l d v e c t o r J = n ev , e and J i s i n t u r n governed by M a x w e l l ' s e q u a t i o n s . The s i m u l t a n e o u s s o l u t i o n of M a x w e l l ' s e q u a t i o n s a l o n g w i t h the above e q u a t i o n s , w i t h the as sumpt ion of p l ane wave o s c i l l a t o r y dependences of the form • f e ( r , t ) v ( r * , t ) e x p ( i ("K-T - at) ' produces the d i s p e r s i o n r e l a t i o n f o r t r a n s v e r s e ( k * E = 0) e l e c t r o m a g n e t i c waves: 8 k 2 c 2 / c o 2 = 1 - C L > 2 / C O t (coT + iv .) , ^ p L L ex = 1 - C J 2 / W 2 + (iv /u ) (o)2/o)2) , p L e i L p L where the expans ion i s done assuming " e i < < 1 » w p * s t n e plasma f requency co p = ^ n ^ e V m . S i n c e the energy of a wave i s p r o p o r t i o n a l to i t s a m p l i t u d e squared , the s p a t i a l damping r a t e of wave energy , k i s then twice the imag inary p a r t of k as i b o b t a i n e d from the above d i s p e r s i o n r e l a t i o n : V 6 J 2/CCJ 2 e i p L k * , i b (1- U2/L>2)^ p. L The s c a l i n g of the e l e c t r o n - i o n c o l l i s i o n f requency v ^ i s o b t a i n e d by the f o l l o w i n g r e a s o n i n g . The c o l l i s i o n f requency i s v = n.o- v t o r an e l e c t r o n of t h e r m a l v e l o c i t y v , c o l l i d i n g e i i tn th w i t h background ions of number d e n s i t y n 1 . The c o l l i s i o n c r o s s s e c t i o n i s o =irb2 where the d i s t a n c e of c l o s e s t approach b i s g i v e n by the ba l ance of p o t e n t i a l and k i n e t i c e n e r g y : Z e 2 / b = 0 .5m e v t 2 h , where v t h = ( k T e / m e ) 3 5 The c o l l i s i o n f requency s c a l i n g i s o b t a i n e d by s u b s t i t u t i n g the above e x p r e s s i o n fo r b and i s 2 2 1.5 v . « n . ( 2 Z e /m v )v K n Z / T , 6 1 1 e th t h e e so t h a t the i n v e r s e b r e m s s t r a h l u n g a b s o r p t i o n c o e f f i c i e n t i s 9 k c c | z n 2 / T - n / n ) 1 e e e c r where n C T = 1. x l O 1 9 cm" 3 f o r C 0 2 l a s e r l i g h t wavelength X = 10.6 ym. From the above r e l a t i o n , i t i s apparent tha t i n v e r s e b remss t r ah lung a b s o r p t i o n i s g r e a t e s t for low tempera ture , h i g h d e n s i t y , h i g h Z plasmas . A much more r i g o r o u s c a l c u l a t i o n of the i n v e r s e b remss t r ah lung c o e f f i c i e n t i n v o l v e s the use of k i n e t i c theory which takes i n t o account the a c t u a l d i s t r i b u t i o n f u n c t i o n of the e l e c t r o n s and a c t u a l p o s i t i o n of the ions under the i n f l u e n c e of the l a s e r e l e c t r i c f i e l d " . In a d d i t i o n , quantum mechan ica l c o r r e c t i o n s to the c l a s s i c a l r e s u l t s are i n c o r p o r a t e d i n t o a Guant f a c t o r . N u m e r i c a l l y , to s u f f i c i e n t a c c u r a c y , the a b s o r p t i o n c o e f f i c i e n t o b t a i n e d i s the Johns ton and Dawson v a l u e 5 : 2 -35 Z n e 1 3 k « 9.74 x 10 cm" ( T ^ e V , n ^ c m " ) i b T 3 / 2 (1 - n / n ) ~ h e e' c r B i l l m a n and S t a l l c o p 6 have c o n s i d e r e d o ther low temperature mechanisms such as e l e c t r o n - n e u t r a l atom c o l l i s i o n a l a b s o r p t i o n , p h o t o i o n i z a t i o n , e t c . T h e i r a b s o r p t i o n c o e f f i c i e n t i s h i g h e r than t h a t of the Johnston and Dawson v a l u e by as much as a f a c t o r of 5 fo r e l e c t r o n tempera tures of the o r d e r of a few eV. However, f o r T £ > 20 eV the d i f f e r e n c e between the two a b s o r p t i o n c o e f f i c i e n t s i s n e g l i g i b l e . 10 11-2 Sa tura t ion of C o l l i s i o n a l A b s o r p t i o n At very h i g h l a s e r i n t e n s i t i e s the i n v e r s e b remss t rah lung c o e f f i c i e n t as c a l c u l a t e d by Johnston and Dawson i s expected to s a t u r a t e w i t h i n c r e a s i n g i n t e n s i t y so t h a t the a b s o r p t i o n becomes i n e f f i c i e n t . T h i s i s because the c a l c u l a t i o n of the a b s o r p t i o n c o e f f i c i e n t assumes the e l e c t r o n d i s t r i b u t i o n to remain M a x w e l l i a n whereas a c t u a l l y , the e l e c t r o n - i o n Coulomb c o l l i s i o n f requency i s m o d i f i e d by the e l e c t r o n ' s coherent motion i n the l a s e r e l e c t r i c f i e l d . S c h l e s s i n g e r and W r i g h t 7 review and compare the d i f f e r e n t approaches to q u a n t i f y t h e e f f e c t of i n t e n s e f i e l d s on the i n v e r s e b remss t r ah lung a b s o r p t i o n r a t e . The r i g o r o u s t h e o r e t i c a l t r ea tment s produce a c o r r e c t i o n term to the a b s o r p t i o n r a t e t h a t i s i n agreement w i t h the h e u r i s t i c t reatment of F a e h l and R o d e r i c k 8 to w i t h i n 10%. i 2 In s tead of h a v i n g a t y p i c a l mean square v e l o c i t y v t n ' the e l e c t r o n now has an e f f e c t i v e average v e l o c i t y <v2> = v 2 + <v 2 > th os where v Q S i s the o s c i l l a t o r y v e l o c i t y of the e l e c t r o n v = eE(t ) /mcj 0 , E ( t ) = E 0 c o s ( w 0 t + A ) , os The a b s o r p t i o n c o e f f i c i e n t i s p r o p o r t i o n a l to the e l e c t r o n i o n c o l l i s i o n f requency which i n t u r n has the v e l o c i t y dependence - 1 . 5 v * <v2> e i 11 T h i s suggests tha t the i n v e r s e b remss t r ah lung c o e f f i c i e n t must be reduced by the f a c t o r : "3/2 k -»• k / ( I + <v 2 >/3v 2 J i b i b o s t h T h i s s imple theory agrees w i t h o ther c a l c u l a t i o n s to w i t h i n 10% up to at l e a s t v / v = 10. For the e x p e r i m e n t a l c o n d i t i o n s i n os th t h i s t h e s i s v / v , < 3. os th — A second n o n - l i n e a r m o d i f i c a t i o n to i n v e r s e b remss t r ah lung r, was d i s c u s s e d by L a n g d o n 9 . I t a r i s e s from the f ac t that the e l e c t r o n d i s t r i b u t i o n can be far from M a x w e l l i a n when the l a s e r i n t e n s i t y i s h i g h . For a M a x w e l l i a n d i s t r i b u t i o n , the h e a t i n g r a t e fo r an e l e c t r o n i n the e l e c t r i c f i e l d of the l a s e r i s 0.5m v 2 v e os e i However, the r a t e at which t h i s e l e c t r o n can share t h i s energy w i t h o ther e l e c t r o n s to form a M a x w e l l i a n d i s t r i b u t i o n i s p r o p o r t i o n a l to v v 2 , ee t h where v i s the e l e c t r o n - e l e c t r o n c o l l i s i o n f requency v p p ee =v / Z . T h e r e f o r e , the h e a t i n g r a t e can exceed the r a t e at which e i e l e c t r o n s can form a M a x w e l l i a n d i s t r i b u t i o n , e s t a b l i s h i n g the c o n d i t i o n f o r the f o r m a t i o n of a non-Maxwel l i an d i s t r i b u t i o n : Z v 2 / v 2 > 1 os t h 12 I t i s the low v e l o c i t y e l e c t r o n s , v e < < v ^ , t h a t are the most s t r o n g l y p e r t u r b e d by c o l l i s i o n s . When the above c o n d i t i o n i s s a t i s f i e d , the number of these low energy e l e c t r o n s i s r e d u c e d . T h i s has the e f f e c t t h a t the amount of c o l l i s i o n a l a b s o r p t i o n decrea se s below the v a l u e c a l c u l a t e d fo r a Maxwel l i an d i s t r i b u t i o n . The r e d u c t i o n as c a l c u l a t e d by Langdon takes the form of k' + k exp ( - v 2 / 2 v 2 ) , i b i b CJ t h where v u i s the e l e c t r o n v e l o c i t y for which y e . = 0 ^ , s i n c e the i n v e r s e b remss t r ah lung a b s o r p t i o n r a t e i s p r o p o r t i o n a l to the number of e l e c t r o n s i n the d i s t r i b u t i o n which have v . =CJ T . e i L G e n e r a l l y , t h i s r e d u c t i o n i n the a b s o r p t i o n r a t e i s g r e a t e r than t h a t c a l c u l a t e d by the" h e u r i s t i c model and i s the same at v / v - 7. os t h 11-3 Resonance A b s o r p t i o n Another important p o s s i b l e mechanism of a b s o r p t i o n of l a s e r l i g h t t h a t i s c o n s i d e r e d i n t h i s t h e s i s i s t h a t of resonance a b s o r p t i o n . T h i s i s a l s o known as c o l l i s i o n l e s s a b s o r p t i o n s i n c e the c o l l i s i o n frequency does not e n t e r i n t o the a b s o r p t i o n r a t e f o r t h i s mechanism. For the case of l i g h t i n c i d e n t on a plasma d e n s i t y g r a d i e n t at ang le to the g r a d i e n t d i r e c t i o n S n e l l ' s law and the d i s p e r s i o n r e l a t i o n 1 3 u2 = CJ 2 + c 2 k 2 , L p L can be combined to show tha t the t u r n i n g p o i n t fo r the l i g h t wave o c c u r s at the d e n s i t y n e = n c r c o s 2 e . For p p o l a r i z e d l i g h t , the e l e c t r i c f i e l d E , i s i n the p l ane formed by k and Vn . At the t u r n i n g p o i n t , the e l e c t r i c f i e l d i s i n the d i r e c t i o n Vn . T h i s component of the e l e c t r i c f i e l d can t u n n e l i n t o the c r i t i c a l d e n s i t y l a y e r becoming very l a r g e and r e s o n a n t l y . e x c i t e e l e c t r o n plasma waves. The p o l a r i z a t i o n dependence of resonance a b s o r p t i o n i s apparent from M a x w e l l ' s e q u a t i o n s fo r e l e c t r o m a g n e t i c waves i n a p la sma: V * ( e E ) = 0 = ( E « V e + eV-E) , where e — 1 - u 2 / u 2 = 1 - n / n p L e c r The e l e c t r o n d e n s i t y p e r t u r b a t i o n $"n of the e l e c t r o n plasma wave f o l l o w s : V E = -47rerjn — e = - E . V e / e <* E ' Vn /e e ->-T h e r e f o r e , i f E *Vn = 0 , t h e n §ne = 0 and t h e r e are no d r i v e n e l e c t r o n waves. But i f there i s a component of the e l e c t r i c f i e l d a l o n g the d e n s i t y g r a d i e n t , E ' V n h 0, then 6 n 0 0 s i n c e e e ^ 0 as the e l e c t r o n d e n s i t y approaches c r i t i c a l d e n s i t y n c r . For a l i n e a r d e n s i t y g r a d i e n t the power absorbed from the l a s e r f i e l d may be shown to b e 1 0 P a b s " " o L E 2 /8* , 14 where L i s the d e n s i t y g r a d i e n t s c a l e l e n g t h and E d i s the component of the e l e c t r i c f i e l d tha t d r i v e s the waves and has been c a l c u l a t e d by K r u e r 1 1 . The f r a c t i o n of i n c i d e n t l i g h t tha t can be absorbed by the resonance p r o c e s s i s l i m i t e d to 50%, and v a l u e s of t h i s a b s o r p t i o n as h i g h as t h i s v a l u e have been observed for C 0 2 l a s e r - s o l i d t a r g e t e x p e r i m e n t s . A second important consequence of resonance a b s o r p t i o n i s the g e n e r a t i o n of h o t - e n e r g e t i c e l e c t r o n s . The l a s e r energy c o u p l e d i n t o the plasma o s c i l l a t i o n s can d i s s i p a t e through Landau damping appear ing then as k i n e t i c energy of heated e l e c t r o n s . Two d i m e n s i o n a l computer s i m u l a t i o n s have shown tha t the r e s o n a n t l y d r i v e n wave f i e l d can grow to s u f f i c i e n t i n t e n s i t y t h a t the e l e c t r o n s are t r apped and a c c e l e r a t e d i n one wave p e r i o d . The hot e l e c t r o n s are found to be c h a r a c t e r i z e d by a s u p r a t h e r m a l Maxwel l i an d i s t r i b u t i o n whose temperature i s a p p r o x i m a t e l y 1 2 0.33 0.33 T = 14 ( IX 2 ) T (keV) , H c where I i s the l a s e r i n t e n s i t y i n u n i t s of 1 0 1 6 W/cm 2, X i s the l a s e r wavelength i n um and T c i s the thermal t e m p e r a t u r e . For the e x p e r i m e n t a l c o n d i t i o n s encountered i n t h i s t h e s i s , the hot e l e c t r o n temperature t h a t may be expected i s 6 - 10 keV. 1 5 11-4 Ion-Acous t i c T u r b u l e n c e Another l i k e l y mechanism for a b s o r p t i o n i s i o n a c o u s t i c t u r b u l e n c e , a phenomenon which has been invoked to account for the h i g h l e v e l s of a b s o r p t i o n i n exper iments which cannot be e x p l a i n e d i n terms of i n v e r s e b remss t r ah lung or resonance a b s o r p t i o n . The idea i s tha t i f by some means a c o n t i n u o u s n o i s e spectrum of ion a c o u s t i c waves i s e x c i t e d , then t h i s r e s u l t i n g t u r b u l e n c e can s c a t t e r e l e c t r o n s e f f e c t i v e l y , r e d u c i n g the heat f l o w , thus c r e a t i n g a h i g h e r temperature than can be expected from c o l l i s i o n a l ( i n v e r s e bremss t rah lung) a b s o r p t i o n a l o n e . A l s o the e l e c t r o n ion c o l l i s i o n f requency i s i n c r e a s e d by the l e v e l of t u r b u l e n c e and hence t h e r e i s a h i g h amount of a b s o r p t i o n of l a s e r r a d i a t i o n . There are s e v e r a l models of ion wave t u r b u l e n c e (see e . g . M o n c h i c o u r t 1 3 ) . One common cause i s thought to be the f o l l o w i n g . The energy f l u x \ = KVT , where K i s the thermal c o n d u c t i v i t y , produces a d i s t o r t i o n of the e l e c t r o n d i s t r i b u t i o n f u n c t i o n f ( r , v ) . I t s maximum i s s h i f t e d by ~v' = -e^/m v = a/c/ ( e V T / W ) , e e e i e i s i n c e j = a E + « VT = 0 , where cr i s the e l e c t r i c a l c o n d u c t i v i t y and a i s the Seebeck c o e f f i c i e n t . There i s a r e t u r n c u r r e n t of c o l d e l e c t r o n s n e e v e -y -v which compensates the hot e l e c t r o n s of f ( r , v ) which c a r r y the heat f l u x . I t was shown by Gray and K i l k e n n y 1 4 i s t h a t the heat f l u x i s c a r r i e d by e l e c t r o n s w i t h v £ =2.3 to 3 t imes v t h . 1 6 The ion a c o u s t i c i n s t a b i l i t y appears when the e l e c t r o n d r i f t v e l o c i t y exceeds the ion a c o u s t i c speed 0.5 v >_ (ZT /m ) e e i When the above c o n d i t i o n i s f u l f i l l e d » a c o n t i n u o u s t u r b u l e n t spectrum of ion a c o u s t i c modes i s genera ted whose e f f e c t i s an i n c r e a s e i n the momentum t r a n s f e r between ions and e l e c t r o n s s i n c e the e l e c t r o n s can now s c a t t e r from ion charge bunches r a t h e r than i n d i v i d u a l i o n s . In the presence of ion t u r b u l e n c e the energy damping r a t e i s 1 5 where e i s the plasma d i e l e c t r i c f u n c t i o n and < I m ( l / e ( k , u )> = 0.3 for n e = 0 . 2 5 n c r f o r f l u c t u a t i o n s w i t h kXD = 0.5 , XD b e i n g the Debye wave leng th . The term ( k ' E ) 2 i s u s u a l l y averaged over a l l k and taken to be 0 . 5 . The f l u c t u a t i o n spectrum for a gas j e t experiment s i m i l a r to the one i n t h i s t h e s i s has been measured by a Thomson s c a t t e r i n g t e c h n i q u e at the U n i v e r s i t y of A l b e r t a 1 6 where (Z<$n,-(t)/n ) 2 = 0 .04 . Should the same l e v e l of k c r t u r b u l e n c e be p re sen t i n the p r e s e n t exper iment , t h i s would r e s u l t i n v . /v . = 1.20 p r o d u c i n g a 20% i n c r e a s e i n the e i e i c 3 a b s o r p t i o n l e v e l . Because of a lower temperature and h i g h e r e l e c t r o n d e n s i t y , the A l b e r t a exper iments demonstrate a 4X enhancement i n the a b s o r p t i o n over t h a t which can be a t t r i b u t e d to i n v e r s e b remss t r ah lung a l o n e . E s t a b r o o k 1 7 has shown, i n one d i m e n s i o n a l computer 17 s i m u l a t i o n s , tha t ion a c o u s t i c t u r b u l e n c e can produce a supra therma l temperature g i v e n a p p r o x i m a t e l y by T ' = T ( 1 . 5 ) / ( l - n / n ) . e e e c r For the e l e c t r o n d e n s i t i e s of 0.25n as encountered i n the cr exper iments of t h i s t h e s i s , the hot temperature due to ion a c o u s t i c t u r b u l e n c e can be expected to remain below one keV. 11-5 Thermal C o n d u c t i v i t y The measured e l e c t r o n temperature i n t h i s t h e s i s i s used to determine the energy c o n t e n t of the p la sma. The e l e c t r o n temperature t h a t i s u l t i m a t e l y a t t a i n e d by the plasma depends on the thermal c o n d u c t i v i t y s i n c e fo r shor t l i v e d l a s e r p lasmas , e l e c t r o n therma l c o n d u c t i o n i s the dominant energy t r a n s p o r t mechanism. In t h i s s e c t i o n , the thermal c o n d u c t i v i t y of a l a s e r produced plasma i s examined i n view of some of the v a r i o u s phenomena t h a t can e f f e c t i t . S i n c e the e l e c t r o n thermal speed i s always much l a r g e r than the ion speed , i t i s the e l e c t r o n thermal c o n d u c t i v i t y tha t i s dominant . In the c l a s s i c a l t h e o r y of e l e c t r o n thermal c o n d u c t i o n , the heat f l u x i s assumed to be g i v e n by F o u r i e r ' s law, q = -/cVT e . The most commonly used va lue fo r the thermal c o n d u c t i v i t y *c, i s t h a t of S p i t z e r and H a r m 1 8 where a p e r t u r b a t i o n approach was used t o s o l v e the Boltzmann e q u a t i o n w i t h a F o k k e r - P l a n c k c o l l i s i o n t e r m . The p e r t u r b a t i o n parameter used i s Xa/L.j, , the r a t i o of the e l e c t r o n mean f ree pa th X e , t o 18 the temperature g r a d i e n t s c a l e l e n g t h L T = | T e/VT e|. When X e < < Lrr and the p a r t i a l time d e r i v a t i v e s <5/<5t < < v . the heat f l u x T ^ e i i s q| - ( X p / L )q T max w h i c h i s e q u i v a l e n t t o F o u r i e r ' s law w i t h t h e t h e r m a l c o n d u c t i v i t y 1 .5 K = 20(2/TT) k. ( eg ) B (kT ) e 5/2 1/2 4 (m ) e Z In A where e , 6 , are the Z dependent S p i t z e r f a c t o r s f o r a non-L o r e n t z i a n gas with no net c u r r e n t flow. The q u a n t i t y q x i s the f l u x of energy a c r o s s a plane i n a c l a s s i c a l gas which i s 0.5 q = 2k T n /4 (8k T A m ) max B e e B e e = 0.8 (n v )k T e th B e S e v e r a l e f f e c t s can occur in a l a s e r produced plasma that i n v a l i d a t e the theory of S p i t z e r and Harm. F i r s t , the p e r t u r b a t i o n approach to s o l v i n g the Fokker-Planck equation f a i l s f o r s u f f i c i e n t l y l a r g e v a l u e s of X e / L T • Gray and K i l k e n n y 1 0 examined S p i t z e r ' s 1 8 c a l c u l a t i o n with the c r i t e r i o n that the p e r t u r b e d e l e c t r o n d i s t r i b u t i o n f u n c t i o n become negative with v i n the range 2.3 v < v < 3 v . A negative e th e th d i s t r i b u t i o n f u n c t i o n i s p h y s i c a l l y u n r e a l and i s a consequence 19 of the o m i s s i o n of h i g h e r order terms i n the s o l u t i o n of the F o k k e r - P l a n c k e q u a t i o n . The c r i t e r i o n de termined by Gray and • K i l k e n n y se t s the l i m i t : q / q = 2.4X / L < 0.036 , max T ( f o r Z = 1 ) . For the plasmas encountered i n the exper iments o f t h i s t h e s i s , the t y p i c a l r a d i a l temperature g r a d i e n t s c a l e l e n g t h i s L = |T / V T I = 300 eV/(250 eV/ .05cm) = 0.06 cm . T e e The temperature g r a d i e n t c o n s i d e r e d i s tha t i n the r a d i a l d i r e c t i o n p e r p e n d i c u l a r to the l a s e r a x i s . In the a x i a l d i r e c t i o n , the plasma i s heated u n i f o r m l y by the l a s e r and the g r a d i e n t s t h e r e are s m a l l as measured i n a s i m i l a r experiment by Wyndham et a l . 1 9 The e l e c t r o n mean f r e e p a t h i s X = T v e ee t h T 3 / 2 = 1.44 x 10 ^—— cm, (T i n eV, n i n cm - ) n Z InA e • e where Z i s the charge s t a t e and InA i s the Coulomb l o g a r i t h m -10. For an e l e c t r o n temperature of 300 eV and n e = 2.5 x 1 0 1 8 cm , Z = 7, X e = 74/ixm. T h i s makes \ > / L T = 0.12 so t h a t the c r i t e r i o n of Gray and K i l k e n n y i s not s a t i s f i e d . B e l l et a l . 2 0 extended the work of S p i t z e r and Harm by n u m e r i c a l s i m u l a t i o n s of the F o k k e r - P l a n c k e q u a t i o n which examine the heat flow i n temperature s c a l e l e n g t h s a few t imes the e l e c t r o n mean f ree p a t h . For the case of \ / L T = 0.12 t h e i r 20 r e s u l t i s that the thermal c o n d u c t i v i t y i s reduced from that of the c l a s s i c a l value by a f a c t o r of 3.5. The experiments of Wyndham et a l . suggest that f o r X g / L j . = 0.12 the r e d u c t i o n of the c o n d u c t i v i t y i s a f a c t o r of 2. In a d d i t i o n to the inadequacy of the Spitzer-Harm c o n d u c t i v i t y f o r s i t u a t i o n s of l a r g e thermal g r a d i e n t s , two other e f f e c t s can f u r t h e r decrease the thermal c o n d u c t i v i t y . -3 Since the Coulomb c o l l i s i o n r a t e i s p r o p o r t i o n a l to v , when vo s / v t h > 1 the c o l l i s i o n a l t r a n s p o r t can be c o n s i d e r a b l y m o d i f i e d . Secondly, i f T e > > T i as i s u s u a l l y the case i n l a s e r produced plasmas, ion a c o u s t i c t u r b u l e n c e can occur, reducing the e l e c t r o n mean f r e e path and the thermal c o n d u c t i o n . Manheimer 2 1 c a l c u l a t e s the thermal c o n d u c t i v i t y i n the presence of ion a c o u s t i c t u r b u l e n c e . Gray and Kilkenny present t h i s r e s u l t i n the form of a r a t i o between the anomalous and S p i t z e r c o n d u c t i v i t y as K an S p i t z e r Assuming a An/n = 0.02 as measured by A l - S h i r a i d a et a l . 1 6 and a -3 T = 300 eV, n„ = 2.5 x 1 0 1 8 cm t h i s r a t i o i s 0.04. e e In summary, there are s e v e r a l e f f e c t s a l l of which decrease the thermal c o n d u c t i v i t y from the c l a s s i c a l S p i t z e r v a l u e . T h i s has s e r i o u s consequences i n computer code s i m u l a t i o n s s i n c e the thermal c o n d u c t i v i t y i s p r o p o r t i o n a l to T^ 2, l a r g e temperatures can p r e d i c t a p h y s i c a l l y u n r e a l i s t i c heat f l u x . In p r a c t i c e , the thermal f l u x used i n s i m u l a t i o n s i s an i n t e r p o l a t i o n between Y 2 n A 3 ' e D l a n I [320 TTSE 21 F o u r i e r ' s law and the f ree s t reaming l i m i t -K VT i q max where ^ i s a f a c t o r v a r i e d , to f i t the observed e l e c t r o n temperatures w i t h the r e s u l t s from s i m u l a t i o n s . Many exper iments i n d i c a t e & = 0 .03 , but v a l u e s as h i g h as 0.15 have been o b t a i n e d 1 9 . 11-6 E l e c t r o n Heat i n g Model An o v e r s i m p l i f i e d model for the h e a t i n g of e l e c t r o n s under the a c t i o n of l a s e r r a d i a t i o n i s p r e s e n t e d i n t h i s s e c t i o n . The purpose i s to a r r i v e at an e s t i m a t e of p o s s i b l e e l e c t r o n temperatures for t y p i c a l e x p e r i m e n t a l c o n d i t i o n s . The c a l c u l a t i o n s shown below were i n i t i a l l y per formed f o r the case of l a s e r i n t e r a c t i o n w i t h a pre formed z - p i n c h plasma co lumn. However, w i t h some i n t e r p r e t a t i o n the r e s u l t s can be a p p l i e d to the i n t e r a c t i o n of i n t e n s e l a s e r r a d i a t i o n w i t h a low p r e s s u r e gas j e t t a r g e t as w e l l . In t h i s mode l , a focus sed l a s e r beam impinges upon a u n i f o r m d e n s i t y plasma t h a t i s i n i t i a l l y at 50 eV . T h i s removes c o m p l i c a t e d i n i t i a l e f f e c t s of plasma f o r m a t i o n from a n e u t r a l gas t a r g e t , and a l s o v a l i d a t e s the use of the Johns ton and Dawson i n v e r s e b remss t r ah lung c o e f f i c i e n t to c a l c u l a t e the l a s e r a b s o r p t i o n . For mathemat ica l e x p e d i e n c y , the energy l o s s e s due to volume expans ion of the plasma are n e g l e c t e d , an as sumpt ion q = 22 p a r t l y j u s t i f i a b l e on t h e b a s i s t h a t t h e r e i s l i t t l e e x p a n s i o n i n t h e d u r a t i o n o f a n a n o s e c o n d l a s e r p u l s e . T h e e n e r g y l o s s e s c o n s i d e r e d i n t h i s m o d e l a r e t h e r m a l c o n d u c t i o n , r a d i a t i o n a n d e n e r g y t r a n s f e r t o t h e i o n s . T h e t h e r m a l c o n d u c t i o n i s t h a t a s c a l c u l a t e d by S p i t z e r a n d i n i t i a l l y no s a t u r a t i o n e f f e c t s a r e i n c l u d e d i n t h e a b s o r p t i o n c o e f f i c i e n t . T h e u s e o f t h e S p i t z e r c o n d u c t i v i t y w o u l d t e n d t o u n d e r e s t i m a t e t h e e l e c t r o n t e m p e r a t u r e w h i l e t h e e x c l u s i o n o f b r e m s s t r a h l u n g s a t u r a t i o n r e s u l t s i n an o v e r e s t i m a t e . H o w e v e r , i t i s t h e g e n e r a l t r e n d i n s u c h m o d e l s t o n e g l e c t t h e s e e f f e c t s e . g . H a n d k e e t a l . 2 2 B u r n e t t a n d O f f e n b e r g e r 2 3 . T h e m o d e l d e t e r m i n e s t h e c h a n g e i n T e w i t h t i m e by means o f t h e e n e r g y t r a n s p o r t e q u a t i o n cTt(lnekBTe) - I / * <1 - exp(-k. b£)) - V-q - C P • P ) - 1^  f f f b r ( i n i kB T i ) ' w h e r e k i b i s t h e i n v e r s e b r e m s s t r a h l u n g a b s o r p t i o n c o e f f i c i e n t , I i s t h e a b s o r p t i o n l e n g t h a n d P f f a n d P f b a r e t h e c o n t i n u u m r a d i a t i v e p o w e r s d i s c u s s e d a n d e v a l u a t e d i n A p p e n d i x B. E n e r g y t r a n s f e r t o t h e i o n s i s c a l c u l a t e d by S p i t z e r ' s 2 * e q u a t i o n : dT. i = ( T - T ) / t d t e l e q w h e r e t i s t h e e q u i p a r t i t i o n t i m e b e t w e e n e l e c t r o n s o f mas s m e a n d i o n s o f mas s nt^: 23 3m m k e i 3/2 eq 8(2ir ^ n zVlnA ( T e / m e + T . / m . ) 3/2 Thermal c o n d u c t i o n l o s s e s are g iven by the d i v e r g e n c e of F o u r i e r ' s law ( for Z=2) V » q = V ' ( 3 . 6 V T / T 0.8n kT v ) e e th A x i a l g r a d i e n t s i n temperature can be c o n s i d e r e d n e g l i g i b l e i n compar i son to r a d i a l g r a d i e n t s ( t r a n s v e r s e to the beam a x i s ) . T h e r e f o r e , the temperature g r a d i e n t s are e s t i m a t e d from e i t h e r the geometry of the f o c a l r e g i o n or the s i z e of the l a s e r produced p la sma . The temperature p r o f i l e assumed has the form shown i n F i g u r e 11 — 1 where the d i s t a n c e d i s the d iameter of the Fipure I I - l Form of the assumed r a d i a l temperature p r o f i l e . f o c a l r e g i o n or the plasma r a d i u s < 1 mm. The background temperature T 0 i s set to 50 eV and T e i s the temperature 24 a t t a i n e d , a t t ime t i n t o the l a s e r p u l s e . W i t h the temperature p r o f i l e shown, the d i v e r g e n c e of the heat f l u x may be approximated to f i r s t order a s : + 2.5 2.5 V ' q a V ( T e - VT) = T e * ( ( T e - T 0 ) / 3gd2) F u r t h e r a p p r o x i m a t i o n s i n c l u d e l i n e a r i z i n g the e x p o n e n t i a l term (1 - e x p - ( k i b ) ~ k ^ which i s j u s t i f i e d on the b a s i s tha t the s c a l e l e n g t h s of the i n t e r a c t i o n are 3 1mm so t h a t k i b < < 1. For s i m p l i c i t y , the l a s e r p u l s e i s approx imated by a l i n e a r ramp pu l se I = l e f t , where I 0 i s the maximum i n t e n s i t y i n the p u l s e . The e l e c t r o n temperature i s expected to f o l l o w the l a s e r p u l s e i n t e n s i t y and so c a l c u l a t i o n s are c a r r i e d out to the t ime t = 1/ct , s p e c i f i c a l l y to 1.3 n s . The e q u a t i o n s fo r e l e c t r o n h e a t i n g are s o l v e d n u m e r i c a l l y u s i n g a second order Runge-Kutta method fo r v a r i o u s i n c i d e n t l a s e r i n t e n s i t i e s . Thermal c o n d u c t i o n beg ins to have a s i g n i f i c a n t e f f e c t o n l y at i n t e n s i t i e s g r e a t e r than 1 0 1 1 W / c m 2 . Due to the r e l a t i v e l y long e l e c t r o n - i o n e q u i p a r t i t i o n t imes of > 40 ns , there i s l i t t l e energy t r a n s f e r to the i o n s and t h e i r temperature i s seen to i n c r e a s e o n l y a few eV. The c o r r e s p o n d i n g r e d u c t i o n i n e l e c t r o n temperature due to energy t r a n s f e r to the ions i s thus o n l y a few p e r c e n t . S i m i l a r l y , the i n c l u s i o n of r a d i a t i o n l o s s terms was found to produce a' n e g l i g i b l e e f f e c t on the r e s u l t a n t e l e c t r o n t empera ture . The r e s u l t s of the c a l c u l a t i o n are shown i n F i g u r e I I - 2 . The uppermost curve i s c a l c u l a t e d assuming no thermal c o n d u c t i o n and o ther l o s s e s and i s o b t a i n e d by a s t r a i g h t f o r w a r d 25 Figure I I - 2 Results from the numerical s o l u t i o n of the energy transport equation. i n t e g r a t i o n of the energy t r a n s p o r t equation, y i e l d i n g : 2.5 -6 0.4 T = (5/2(2/3T 0 + 1 . 1 8 x 1 0 I 0 ) ) » a t t = 1.3 e with T e , T 0 i n eV and I 0 i n W/cm2. The c a l c u l a t i o n f o r S p i t z e r c o n d u c t i v i t y i s shown as the c e n t e r curve through the c r o s s hatched r e g i o n . The upper curve r e p r e s e n t s the r e s u l t s o b t a i n e d w i t h the thermal c o n d u c t i v i t y lowered from the S p i t z e r value by a f a c t o r of two as i s suggested by the experimental r e s u l t s of Wyndham et a l . 1 9 The lower curve was c a l c u l a t e d by a l l o w i n g the i n v e r s e bremsstrahlung a b s o r p t i o n c o e f f i c i e n t k±t, , to s a t u r a t e at the higher i n t e n s i t i e s a c c o r d i n g to the model of Faehl and K r u e r 8 26 k i b k . + , i b 3/2 ( 1 + 7.02 x 1 0 , 2 I 0 / T ) e w i t h I 0 i n W/cm 2 and T e i n eV. A l t h o u g h the r e d u c t i o n i n the c o l l i s i o n a l a b s o r p t i o n i s as h i g h as a f a c t o r of 3 at i n t e n s i t i e s approach ing 10 1 * W / c m 2 , the r e s u l t i n g e l e c t r o n temperature i s not d r a s t i c a l l y reduced by the s a t u r a t i o n e f f e c t s because most of the h e a t i n g o c c u r s at e a r l y t imes at low i n t e n s i t i e s , when the a b s o r p t i o n i s much h i g h e r due to a lower e l e c t r o n t e m p e r a t u r e . S i m i l a r c a l c u l a t i o n s were repea ted fo r the case of a f u l l y i o n i z e d n i t r o g e n gas j e t t a r g e t which produced p r e d i c t e d e l e c t r o n temperatures i n the range of 400 - 600 eV for l a s e r i n t e n s i t i e s 1 0 1 3 - 1 0 1 " W / c m 2 . The i n c r e a s e i n temperature i n a n i t r o g e n t a r g e t compared to the h e l i u m i s due to the l i n e a r dependence on Z i n the a b s o r p t i o n c o e f f i c i e n t , whereas the thermal c o n d u c t i v i t y decrea se s w i t h i n c r e a s i n g Z . The c a l c u l a t i o n s p r e s e n t e d here are crude i n the sense that no t ime dependent i o n i z a t i o n or hydrodynamic e f f e c t s are c o n s i d e r e d . For the case of a preformed 50 eV t a r g e t plasma the i o n i z a t i o n i s s u e can be i g n o r e d . However, for a l a s e r produced plasma t h i s may not n e c e s s a r i l y be so . The hydrodynamics of the phenomenon may a l s o i n f l u e n c e the s t a t e of i o n i z a t i o n which i n t u r n de te rmines the l e v e l of a b s o r p t i o n and e l e c t r o n temperature and v i c e v e r s a . T h e r e f o r e , to model a l a s e r produced plasma w i t h c o n s i d e r a b l e more f i n e s s e r e q u i r e s the s imul taneous s o l u t i o n of a system of c o u p l e d hydrodynamic d i f f e r e n t i a l e q u a t i o n s which 27 d e s c r i b e the c o n s e r v a t i o n of mass, energy and momentum toge ther w i t h a system of e q u a t i o n s d e s c r i b i n g the t ime dependent s t a t e of i o n i z a t i o n . The c o m p l e x i t y of the problem r e q u i r e s the use of l a r g e computer codes . For compari son w i t h the p r e d i c t e d e l e c t r o n temperatures c a l c u l a t e d i n t h i s s e c t i o n , the r e s u l t s of s e v e r a l computer code s i m u l a t i o n s are p r e s e n t e d i n the f o l l o w i n g sec t i o n . 11-7 R e s u l t s from Hydrodynamic Computer Codes The implementa t ion and i n t e r p r e t a t i o n of computer codes to l a s e r - p l a s m a exper iments i s a t h e s i s t o p i c i n i t s e l f . Only a b r i e f mention of the r e s u l t s of the s e v e r a l computer codes used w i l l be p r e s e n t e d here to o b t a i n e s t i m a t e s of the expected e l e c t r o n temperature for the e x p e r i m e n t a l c o n d i t i o n s encountered i n t h i s work. The one d i m e n s i o n a l Lagrang ian code MEDUSA has been a v a i l a b l e s i n c e 1 974 2 5 and has been i n use a t UBC s i n c e 1 980 2 6 . A complete d e s c r i p t i o n of the code and i t s method of s o l u t i o n of the hydrodymnamic e q u a t i o n s i s g i v e n i n the above r e f e r e n c e s . Of r e l e v a n c e to t h i s t h e s i s i s tha t the a b s o r p t i o n of l a s e r r a d i a t i o n i s by i n v e r s e b remss t r ah lung and the thermal c o n d u c t i v i t y i s the c l a s s i c a l v a l u e of S p i t z e r . Unique to the UBC v e r s i o n of MEDUSA are the n u m e r i c a l changes e f f e c t e d by J . K w a n 2 7 to the i n v e r s e b remss t r ah lung a b s o r p t i o n c o e f f i c i e n t and thermal c o n d u c t i v i t y . These changes brought the i n v e r s e b remss t r ah lung c o e f f i c i e n t i n t o agreement w i t h tha t of Johnston and Dawson and the thermal c o n d u c t i v i t y was c o r r e c t e d fo r 28 v a r y i n g v a l u e s of t a r g e t Z . S i n c e MEDUSA i s a one d i m e n s i o n a l c o d e , a l l the hydrodynamic phenomena can o n l y take p l a c e i n one d imens ion a l o n g the l a s e r a x i s . O b v i o u s l y , i f r a d i a l plasma expans ion i s ah impor tant f a c t o r i n the a c t u a l exper iment s , the output of the code w i l l o v e r e s t i m a t e T £ and i s then sub jec t to i n t e r p r e t a t i o n . That plasma expans ion cannot be n e g l e c t e d i s e v i d e n c e d by the f a c t t h a t i f the a c t u a l gas j e t m o l e c u l a r d e n s i t y of 4 x 10~ 5 g/cm 3 i s used as an i n i t i a l t a r g e t d e n s i t y , the code almost i n s t a n t a n e o u s l y produces f u l l i o n i z a t i o n and an o v e r c r i t i c a l d e n s i t y t a r g e t so tha t t h e r e i s no t r a n s m i s s i o n of the l a s e r beam. S i n c e the i n t e n t i o n i s to s tudy the i n t e r a c t i o n of a l a s e r beam w i t h an underdense p lasma, the i n i t i a l t a r g e t d e n s i t y was lowered by a f a c t o r of 10 to 3 to get r e s u l t i n g e l e c t r o n d e n s i t i e s of 0.1 and 0.45 n c r r e s p e c t i v e l y . G e n e r a l l y , the e l e c t r o n temperatures o b t a i n e d from the MEDUSA runs are s u b s t a n t i a l l y over 1 keV as F i g u r e I I -3 shows. These h i g h e l e c t r o n temperatures are a r e s u l t of the f a c t t h a t fo r an underdense p lasma, there are n e g l i g i b l y s m a l l thermal g r a d i e n t s a x i a l l y a l o n g the l a s e r beam a x i s . T h e r e f o r e , the therma l c o n d u c t i o n l o s s e s are q u i t e s m a l l r e s u l t i n g i n h i g h e l e c t r o n t e m p e r a t u r e s . I t was observed tha t i n o r d e r to reduce the r e s u l t i n g e l e c t r o n temperatures by a f a c t o r of 2, i t was nece s s a ry to a r t i f i c i a l l y i n c r e a s e the thermal c o n d u c t i v i t y by a f a c t o r of 100, thus i n d i c a t i n g the s m a l l r o l e of thermal c o n d u c t i o n i n a one d i m e n s i o n a l model . The a n a l y t i c s o l u t i o n for no therma l c o n d u c t i o n from the p r e v i o u s s e c t i o n i s a l s o p l o t t e d i n F i g u r e I I -3 for the case of 0.2 n e l e c t r o n d e n s i t y . 29 ^ ' • 1 1 • l i 1 1 1 — ^ 3 5 7 Joules Figure II-3 Temperature s c a l i n g wi th i n c i d e n t l a s e r energy from MEDUSA computer runs . O b v i o u s l y , t o model the i n t e r a c t i o n of a l a s e r beam with an underdense plasma a two dimensional code should be used. N. H. Burnett of NRC has simulated the i n t e r a c t i o n u s i n g a 2-D E u l e r i a n code f o r a n i t r o g e n t a r g e t with d e n s i t y of 9 x 1 d~6 g/cm 3 and a l a s e r i n t e n s i t y of 1 0 1 3 W/cm2. The r e s u l t of i n t e r e s t i s that at the peak of the pul s e the e l e c t r o n temperature i s 540 eV f o r a t a r g e t of 0.25 n i n i t i a l e l e c t r o n cr d e n s i t y . Another s i m u l a t i o n was performed by E. Hiob u s i n g the w e l l documented 2-D E u l e r i a n code CASTOR 2 8. The i n i t i a l c o n d i t i o n s f o r the CASTOR s i m u l a t i o n was a t a r g e t d e n s i t y of 3.7 x 10" 5 g/cm3 and a 6 J o u l e i n c i d e n t l a s e r p u l s e of t r i a n g u l a r temporal shape with a r i s e time of 2 ns. The beam f o c a l r a d i u s was 100 Mm. The i n i t i a l e l e c t r o n temperature was 1 eV and a time dependent atomic p h y s i c s p a c k a g e 2 9 was implemented. The e l e c t r o n 30 temperatures a l o n g the l a s e r a x i s at the time of the peak of the p u l s e were i n the range of 500-700 eV, the a x i a l d e n s i t y be ing 0.26 n c r . I f the n e u t r a l gas d e n s i t y were to be i n s t a n t a n e o u s l y f u l l y i o n i z e d , the mean e l e c t r o n d e n s i t y would be 1.1 n c r however, the e l e c t r o n d e n s i t y remains underdense throughout the s i m u l a t i o n . I n i t i a l l y , the l e v e l of i o n i z a t i o n i s low, the average Z b e i n g 4-5 at t imes up to 0.5 n s , a f t e r which plasma expans ion i s s u f f i c i e n t to keep the a x i a l e l e c t r o n d e n s i t y s u b s t a n t i a l l y below the c r i t i c a l d e n s i t y even a f t e r near f u l l i o n i z a t i o n i s a c h e i v e d . T h e r e f o r e , the 2-D codes are i n approximate agreement w i t h each o ther i n p r e d i c t i n g an e l e c t r o n temperature of the order of 500 - 700 eV. The n u m e r i c a l s o l u t i o n of the energy t r a n s p o r t e q u a t i o n w i t h r a d i a l heat c o n d u c t i o n a l s o p r e d i c t s e l e c t r o n temperatures i n t h i s range for l a s e r i n t e n s i t i e s > 1 0 1 3 W / c m 2 . A shor tcoming of a l l these p r e d i c t i o n s for the e l e c t r o n temperature i s tha t they must assume some i n i t i a l e l e c t r o n temperature o t h e r w i s e , the i n v e r s e b remss t r ah lung c o e f f i c i e n t becomes i n f i n i t e . The assumption of an i n i t i a l e l e c t r o n temperature makes an i m p e r c e p t i b l e d i f f e r e n c e to the maximum e l e c t r o n t empera ture , but the i n i t i a l i o n i z a t i o n that a r i s e s from t h i s i n i t i a l temperature does c r e a t e a preformed e l e c t r o n d e n s i t y p r o f i l e which does not resemble the c o m p l i c a t e d mult icomponent s t r u c t u r e a c t u a l l y observed i n the i n t e r f e r o g r a m s (Chapter I V ) . S e c o n d l y , the computer s i m u l a t i o n s do not s e l f -c o n s i s t e n t l y model the pa th of the l a s e r through the e v e r -chang ing d e n s i t y c o n t o u r s . A l l of the i n c i d e n t l a s e r l i g h t i s sent through the plasma w i t h no r e f r a c t i o n taken i n t o a c c o u n t . A 31 s i g n i f i c a n t amount of the i n c i d e n t l a s e r beam can i n p r i n c i p l e be r e f r a c t e d by a tenuous p r e c u r s o r upstream plasma r e d u c i n g the l a s e r h e a t i n g of the denser core of the p la sma. 32 Chapter I I I ; The Apparatus and i t s A p p l i c a t i o n The main o b j e c t i v e of t h i s t h e s i s i s to o b t a i n a complete energy ba l ance of the l a s e r - p l a s m a i n t e r a c t i o n by e x p e r i m e n t a l methods. The energy ba lance i s used to e x p l a i n the e l e c t r o n temperature o b t a i n e d by an independent measurement. The means by which these measurements are made are i n d i c a t e d i n t h i s c h a p t e r . B r i e f d e s c r i p t i o n s of the appara tus used i n the exper iments are p r e s e n t e d i n t h i s c h a p t e r . The t e c h n i c a l d e t a i l s are l e f t to a minimum and i n s t e a d the u n d e r l y i n g p h y s i c a l p r i n c i p l e s and t h e i r r e l a t i o n to the d e s i g n of the appara tus w i l l be emphas ized . In s imple terms , the e x p e r i m e n t a l appara tus c o n s i s t s of a l a s e r , a t a r g e t and the a s s o c i a t e d d i a g n o s t i c s f o r the i n v e s t i g a t i o n of the l a s e r produced p la sma. The o v e r a l l d e p i c t i o n of the e x p e r i m e n t a l geometry i s shown i n F i g u r e 111 — 1. Each of the components of the exper iment w i l l be t r e a t e d i n d e t a i l subsequent ly i n t h i s c h a p t e r . Some p r e l i m i n a r y work i n v o l v e d a z - p i n c h d e v i c e d e s c r i b e d by H o u t m a n 3 0 and c h a r a c t e r i z e d by H i l k o 3 1 . A t t e n t i o n q u i c k l y t u r n e d to a v a r i a b l e i n i t i a l d e n s i t y gas j e t t a r g e t f i r s t d e s c r i b e d by B u r n e t t 3 2 and l a t e r u t i l i z e d e x t e n s i v e l y by Of fenberger et a l . 3 3 The gas j e t t a r g e t w i l l be d e s c r i b e d i n s e c t i o n 111 — 1 a l o n g w i t h i t s c h a r a c t e r i z a t i o n f o r use as a s u i t a b l e t a r g e t . A few p e r t i n e n t r e s u l t s from the theory of s u p e r s o n i c n o z z l e f low are a l s o mentioned s i n c e they determine the j e t o p e r a t i n g c o n d i t i o n s . The d e s c r i p t i o n of the C 0 2 l a s e r system w i l l be 3 3 Figure I I I - l Overall depiction of the experimental geometry. a p p r o p r i a t e l y s u c c i n c t as many of the t e c h n i c a l d e t a i l s of the components are c o n t a i n e d i n a UBC l a b o r a t o r y r e p o r t 3 * and a concurrent UBC t h e s i s 3 5 . The o p e r a t i o n of the l a s e r system i n c o n j u n c t i o n with the other components of the experiments i s d i s c u s s e d i n s e c t i o n I I I - 2 . The l a s e r produced plasma i s i n v e s t i g a t e d using s e v e r a l d i a g n o s t i c s . Apparatus was designed and developed by the author f o r the measurement of e l e c t r o n temperature by s o f t x-ray d i a g n o s t i c s , energy a b s o r p t i o n by an U l b r i c h t i n t e g r a t i n g sphere, and plasma expansion by an image c o n v e r t e r s t r e a k camera. These d e v i c e s and t h e i r a p p l i c a t i o n i n the experiments are t r e a t e d s e c t i o n s I I I - 3 , 4 and 5. 34 111 — 1 Laminar Gas J e t T a r g e t . The v a r i a b l e d e n s i t y gas j e t t a r g e t was s e l e c t e d as the optimum t a r g e t for the energy ba lance i n v e s t i g a t i o n . I t p r o v i d e s a f a i r l y un i fo rm h i g h - t e m p e r a t u r e , h i g h - d e n s i t y ; underdense plasma which i s c o n d u c i v e towards a b s o r p t i o n . s t u d i e s . In c o n t r a s t , the m a j o r i t y of l a s e r - p l a s m a i n t e r a c t i o n exper iments i n l a b o r a t o r i e s throughout the w o r l d are performed on v a r i o u s s o l i d t a r g e t s , l a r g e l y i n the i n t e r e s t of a s c e r t a i n i n g the hydrodynamic e f f i c i e n c y i . e . , the r a t i o of a c c e l e r a t e d t a r g e t energy to absorbed l a s e r energy , (see e . g . , N i s h i m u r a et a l . 3 6 ) In such t a r g e t s , the l a s e r - p r o d u c e d - p l a s m a d e n s i t y decays e x p o n e n t i a l l y a l o n g the l a s e r a x i s i n t o the s u r r o u n d i n g vacuum. S i n c e d i f f e r e n t a b s o r p t i o n and p a r a m e t r i c p r o c e s s e s occur i n d i f f e r e n t d e n s i t y r e g i o n s , i t i s d e s i r a b l e i n s t e a d , to have a t a r g e t of a s i n g l e u n i f o r m d e n s i t y to s i m p l i f y the l a s e r - p l a s m a i n v e s t i g a t i o n . To t h i s end , gas t a r g e t s have s e v e r a l advantages over s o l i d t a r g e t s . F i r s t l y , a t l a s e r i n t e n s i t i e s of 1 0 1 3 W / c m 2 , the o s c i l l a t o r y e l e c t r o n energy 0.5m v o s , (where v o s i s the q u i v e r v e l o c i t y of the e l e c t r o n i n the l a s e r f i e l d ) , exceeds the i o n i z a t i o n energy of most low Z gases so t h a t 100% i o n i z a t i o n can be a n t i c i p a t e d i n t imes l e s s than the p u l s e d u r a t i o n . Thus the plasma d e n s i t y i s r o u g h l y equa l t o the d e n s i t y of the n e u t r a l gas i n which i t i s formed so t h a t the plasma d e n s i t y can be c o n t r o l l e d by v a r y i n g the gas p r e s s u r e . Gaseous t a r g e t s c o n s i s t i n g of l a r g e volume gas c e l l s i n t o which l a s e r l i g h t i s focussed were i n i t i a l l y used by Y a b l o n o v i t c h 3 7 to e l u c i d a t e resonant a b s o r p t i o n phenomena i n a 35 u n i f o r m l y overdense p la sma. B u r n e t t 3 8 s t u d i e d s t i m u l a t e d B r i l l o u i n b a c k s c a t t e r i n He gas m a i n t a i n e d at p r e s u r e s such tha t at f u l l i o n i z a t i o n , the e l e c t r o n d e n s i t y was 0.1 - 0.5 n . A major d i s advantage of t h i s method, however, i s tha t i n the c r i t i c a l d e n s i t y c a s e , the l a s e r i n t e n s i t y can never exceed that r e q u i r e d to a c h i e v e i o n i z a t i o n s i n c e any f u r t h e r i n c r e a s e i n the i n t e n s i t y w i l l cause the i o n i z a t i o n f r o n t to move upstream w i t h no i n t e n s i t y i n c r e a s e at the plasma f r o n t . S i m i l a r l y , i n the underdense t a r g e t c a s e , B u r n e t t 3 8 found a complete absence of i o n i z a t i o n downstream of the f o c a l p o i n t and " e x p l o s i v e " r e f r a c t i o n around the f o c a l volume due to the cone-shaped i o n i z a t i o n r e g i o n ahead of the f o c u s , t e r m i n a t i n g the i n t e r a c t i o n . To overcome the d i f f i c u l t i e s of plasma genera ted upstream towards the i n c i d e n t l a s e r beam, the s t a b i l i z e d ^ s u p e r s o n i c gas j e t i s used by B u r n e t t 3 2 , O f fenberger et a l . 3 3 In t h i s t a r g e t , s t a b l e t w o - d i m e n s i o n a l f low i s o b t a i n e d by d i s c h a r g i n g h i g h p r e s s u r e n i t r o g e n gas through a c o n v e r g e n t -d i v e r g e n t ( L a v a l ) n o z z l e i n t o ambient He gas a t a p r e s s u r e of a few T o r r . The 1 mm t h i c k 1 cm wide j e t of v a r i a b l e d e n s i t y s i m u l a t e s p l ane s o l i d t a r g e t s . He l ium gas i s used to s t a b i l i z e the f low thus e n s u r i n g a low d e n s i t y plasma compared to tha t formed from n i t r o g e n i n the event of upstream breakdown. The d e t a i l s of the p a r t i c u l a r L a v a l n o z z l e are shown i n F i g u r e I I I -2 . The p a r t i c u l a r d e s i g n i s due to N . H . Burne t t of NRC. For such a n o z z l e , the c r i t i c a l d e s i g n f e a t u r e i s the r a d i u s of c u r v a t u r e i n the d i v e r g i n g p a r t and t h i s i s r e l a t e d to the d e s i r e d Mach number as shown i n S h a p i r o 3 9 . The L a v a l n o z z l e c o n s i s t s of a p a i r of " j aws" set i n a L u c i t e c o n e . T h i s s i t s 36 gas flow 0.197" 0.033" Figure III-2 Details of the Laval nozzle. atop a short brass pipe connected to a s o l e n o i d v a l v e . A high p r e s s u r e n i t r o g e n r e s e r v o i r of 0.5 l i t e r volume p r o v i d e s the n i t r o g e n gas. To f a c i l i t a t e p r e c i s e f i l l i n g of the r e q u i r e d ambient He gas and r e s e r v o i r pressures, an automatic gas h a n d l i n g system was i n c o r p o r a t e d F i g u r e I I I - 3 where e l e c t r i c s o l e n o i d v a l v e s are operated i n a p r e s e t sequence by the e l e c t r o n i c s . The e l e c t r o n i c s senses the output from c a p a c i t a t i v e p r e s s u r e t r a n s d u c e r s and a l l o w s the t a r g e t chamber and n i t r o g e n r e s e r v o i r t o be a u t o m a t i c a l l y r e f i l l e d t o the d e s i r e d p r e s s u r e s whenever r e q u i r e d . A manual push button i s used to a c t i v a t e the s o l e n o i d v a l v e atop the n i t r o g e n r e s e r v o i r and e s t a b l i s h e s laminar flow. A p i e z o probe l o c a t e d near the n o z z l e senses the change i n gas pr e s s u r e and i s used as a t i m i n g s i g n a t u r e . The output p u l s e from the p i e z o probe e n t e r s a t r i g g e r e d v a r i a b l e delay p u l s e g e n e r a t o r . The p u l s e from t h i s d e l a y u n i t i s used to a c t i v a t e other time delay u n i t s f o r the o p e r a t i o n of ruby or C0 2 l a s e r s 37 automated pressure control — manual control piezo I pressuref) T&f probe Y pressure _ _ j ^ ^ ^ J - - transducer comparator and delay ruby laser -timing unit CO2 laser timing unit inlet Figure III-3 E l e c t r i c a l and gas systems of the target chamber. as c i r c u m s t a n c e s d i c t a t e . The o p e r a t i n g c o n d i t i o n s of the j e t are de te rmined by the geometry of the j e t n o z z l e and a few r e s u l t s from the gas dynamics of n o z z l e flow (see e . g . Landau and L i f s h i t z " 0 ) . In the x convergent p a r t of the n o z z l e the maximum amount of mass f l u x a t the c o n s t r i c t i o n i s g i v e n by Q = p v S , max * * min where S . i s the c r o s s s e c t i o n a l a rea at the c o n s t r i c t i o n . From mm the theory f o r s t r e a m l i n e f l o w , the maximum f l u x o c c u r s when the gas v e l o c i t y i s equa l to the l o c a l v e l o c i t y of sound such t h a t the c r i t i c a l d e n s i t y and v e l o c i t y r e s p e c t i v e l y a r e : 38 .7/(7-1 ) P = Po * \ ( 7 + 1 )J (1 ) and v = c 0 ( 2 / ( 7 + 1)) * 0.5 (2) where y i s the r a t i o of s p e c i f i c h e a t s . The flow becomes s u p e r s o n i c i n the d i v e r g e n t p a r t of the n o z z l e s i n c e Q = pvS = c o n s t a n t , thoughout the n o z z l e . At the wides t p a r t , c o n s e r v a t i o n of mass f l u x i s w r i t t e n a s : S p v = S p v (3) min * * t h th t h where v = c 0 t h 2 7-1 (Y-1 ) (1 " (p/Po) ) ; (4) a g e n e r a l r e s u l t f o r s t r e a m l i n e f l o w . The s u b s c r i p t s * , 0 , and t h a p p l y to c o n d i t i o n s a t the c o n s t r i c t i o n , the r e s e r v o i r and the e x i t p l a n e of the n o z z l e . The s u b s t i t u t i o n of e q u a t i o n s ( 4 ) , ( 2 ) , and (1) i n t o (3) produces the r e l a t i o n between the c r o s s - s e c t i o n a l a rea s S .^ and S h and the r e s e r v o i r and e x i t i n g gas d e n s i t i e s : S r ( 7 + 1 ) / 2 ( 7 - 1 ) - * min 2 2 39 P /Po 1 " <P /Po) t h |_ t h For n i t r o g e n , 7 = 1 . 4 , so that to a good approximation 1/7 p / p 0 = S /S (0.259) = (P /P 0) . (5) th min th th The r e q u i r e d r e s e r v o i r p r essure to reach a c e r t a i n e x i t p r e s s u r e f o r a giv e n area r a t i o S . /S , , i s determined by the min tn above e q u a t i o n . When the ambient He gas p r e s s u r e i s l e s s than P then the j e t p r e s s u r e f a l l s to the lower He p r e s s u r e as the j e t e x i t s i n an expansion fan that o r i g i n a t e s at the jaw edges. T h i s phenomenon, c a l l e d "underexpansion", m a n i f e s t s i t s e l f v i s u a l l y i n the "candle flame" appearance of the j e t boundaries as made v i s i b l e by shadowgraphy i n F i g u r e I I I - 4 . T h i s a l l o w s a (a) (b) Figure I I I - 4 Ruby laser shadowgraphs o f the gas .let target. (a) P th He <b> P t h > PHe t e s t of the steady s t a t e n o z z l e flow equation ( 5 ) as w e l l as a v e r i f i c a t i o n of c o r r e c t pressure s e t t i n g s t o o b t a i n an optimum ta r g e t with p a r a l l e l boundaries and subsequent uniform d e n s i t y . 40 C a l c u l a t i o n s based on the method of hodograph c h a r a c t e r i s t i c s ( S h a p i r o , 3 9 c h a p t e r 15) show t h a t i f the background p r e s s u r e i s lower than the e x i t p r e s s u r e by 30% then the j e t w i d t h w i l l be r o u g h l y double i t s wid th at the e x i t a t a d i s t a n c e 5 mm from the e x i t and thus be c l e a r l y v i s i b l e . The background He p r e s s u r e s neces sa ry to o b t a i n p a r a l l e l t a r g e t b o u n d a r i e s for a g i v e n r e s e r v o i r p r e s s u r e fo r a j e t w i t h S m i n / S t h = 0.0846 (jaw s e p a r a t i o n = 0.0111cm) are shown i n F i g u r e 111-5 where i t i s seen t h a t t h e r e i s good agreement between the e x p e r i m e n t a l 20 V) IT) ,6121 o> | 81 15 o 1 1" I — I r — Torr -< reservoir pressure i x . psi 20 40 60 Figure I I I - 5 S t a b i l i z e d j e t operat ing pressures determined from ruby l a se r shadowgraphy. v a l u e s and those p r e d i c t e d by e q u a t i o n (5) ( s o l i d l i n e ) . F u r t h e r shadowgraph s e r i e s were taken f o r v a r i o u s S m i n / S t h r a t i o s by B. H i l k o and agreement w i t h t h e o r y was w i t h i n 20%. More r e c e n t l y , Houtman and M e y e r " 1 have u t i l i z e d a m u l t i p a s s s c h l i e r e n 41 t e c h n i q u e to enhance the image of the d e n s i t y g r a d i e n t s which i s neces sa ry for low He p r e s s u r e s . T h e i r r e s u l t s on the gas j e t t a r g e t as o p e r a t e d i n the exper iments v e r i f i e d tha t the j e t t a r g e t c o n s i s t e d of p a r a l l e l l aminar boundar ie s as e x p e c t e d . Ruby l a s e r R a y l e i g h s c a t t e r i n g o f f the gas j e t was a l s o used to c o r r o b o r a t e the above o b s e r v a t i o n s of the gas j e t p r e s s u r e . The i n t e n s i t y W of s c a t t e r e d l a s e r l i g h t of i n c i d e n t power P i s W = CTRn0P dz d n where crR i s the R a y l e i g h s c a t t e r i n g c r o s s s e c t i o n , n 0 the d e n s i t y of m o l e c u l e s , dz the l e n g t h of gas the l i g h t i s s c a t t e r e d from and dfi the s o l i d ang le subtended by the c o l l e c t i o n o p t i c s . The R a y l e i g h s c a t t e r i n g system shown i n F i g u r e I I I - 6 a i s c a l i b r a t e d by f i l l i n g the t a r g e t chamber w i t h a known p r e s s u r e of N 2 gas and o b s e r v i n g the s c a t t e r e d ruby l a s e r s i g n a l l e v e l as d e t e c t e d by the c o l l e c t i o n o p t i c s and p h o t o m u l t i p l i e r t u b e . The ruby l a s e r s c a t t e r e d i n t e n s i t i e s from the gas j e t t a r g e t are then compared w i t h the c a l i b r a t i o n data t a k i n g i n t o account t h a t the n i t r o g e n emanating from the L a v a l n o z z l e i s a t 69 degrees K e l v i n whereas the c a l i b r a t i o n data i s for n i t r o g e n at room t e m p e r a t u r e . The s e t - u p for R a y l e i g h s c a t t e r i n g and the r e s u l t s are shown i n F i g u r e I I I - 6 b . Once a g a i n , the e x p e r i m e n t a l v a l u e of 10+4 T o r r for the gas j e t i s i n good agreement w i t h the e x i t p r e s s u r e p r e d i c t e d by e q u a t i o n (5) for a n i t r o g e n r e s e r v o i r p r e s s u r e of 2280 T o r r . Thus the m o l e c u l a r d e n s i t y of the gas j e t can be de termined by the r e s e r v o i r p r e s s u r e and jaw s e p a r a t i o n w i t h c o n s i d e r a b l e c o n f i d e n c e . (b) 42 ( a ) 0 10 30 50 70 Figure I I I - 6 Rayleigh s c a t t e r i n g on the gas j e t . (a) experimental set-up (b) r e s u l t s , c a l i b r a t i o n and target shots I The CO_2 La ser System L a s e r - p l a s m a i n t e r a c t i o n phenomena c o n s i s t of two b a s i c t y p e s , f l u i d mot ion and m i c r o s c o p i c i n t e r a c t i o n s t h a t occur i n two d i f f e r e n t t ime s c a l e s . Hydrodynamic e f f e c t s , such as heat and mass f low occur i n t ime s c a l e s t y p i c a l l y much longer then those i n which the m i c r o s c o p i c p h y s i c s , a b s o r p t i o n , p a r a m e t r i c i n s t a b i l i t i e s , e t c . , t akes p l a c e . In o r d e r to i n v e s t i g a t e the l a s e r plasma i n t e r a c t i o n i n a u n i f o r m plasma i t i s d e s i r a b l e t o m i n i m i z e the c o m p l i c a t i n g hydrodynamic phenomena. T h e r e f o r e a s h o r t s i n g l e l a s e r p u l s e o f a few ns d u r a t i o n i s a u s e f u l t o o l fo r l a s e r plasma s t u d i e s . The p r o d u c t i o n of a h i g h i n t e n s i t y , s h o r t - d u r a t i o n l a s e r p u l s e i s a c h i e v e d i n our system by u s i n g a f a s t e l e c t r o n i c a l l y 43 c o n t r o l l e d o p t i c a l gate to " s l i c e " a shor t p u l s e from a l onger l a s e r p u l s e and subsequent ly a m p l i f y i t u s i n g o ther a m p l i f i e r u n i t s . The system i s d e s i g n e d w i t h s e v e r a l c r i t e r i a : e f f i c i e n t use of the g a i n medium, avo idance of exceed ing damage t h r e s h o l d s f o r o p t i c a l components, avo idance of p a r a s i t i c o s c i l l a t i o n s , c o n t r o l of the b a c k s c a t t e r beam and c o n t r o l of the beam q u a l i t y . The f i r s t and second c r i t e r i a are s a t i s f i e d by the o p t i c a l system which a l l o w s the l a s e r beam to pass through each a m p l i f i e r c a v i t y twice w i t h a p p r o p r i a t e beam expans ion and c o l l i m a t i o n upon each p a s s . P a r a s i t i c o s c i l l a t i o n s , i . e . s e l f -l a s i n g of the a m p l i f i e r s , i s suppres sed by t i l t i n g the o p t i c a l f l a t s used as windows fo r the a m p l i f i e r s and by i n s e r t i n g s a t u r a b l e absorber gas c e l l s i n the beam p a t h . S a t u r a b l e a b s o r b e r s , e . g . S F 6 gas , suppress s m a l l ampl i tude o s c i l l a t i o n s but become b l e a c h e d at l a r g e i n t e n s i t i e s a l l o w i n g r e l a t i v e l y u n a f f e c t e d t r a n s m i s s i o n of the h i g h e r i n t e n s i t i e s . S p a t i a l f i l t e r s , each c o n s i s t i n g of a p i n h o l e a p e r t u r e at the f o c a l p l a n e s a l o n g the beam p a t h , precede each a m p l i f i e r s t a g e . These p i n h o l e a p e r t u r e s ensure a smooth r a d i a l i n t e n s i t y p r o f i l e of the l a s e r beam. The C 0 2 l a s e r system i n c o r p o r a t e s the t e c h n o l o g y of t r a n s v e r s e l y e x c i t e d a tmospher ic (TEA) systems known s i n c e 1 9 7 0 " 2 . The system c l o s e l y resembles the 60 J NRC system d e s c r i b e d by Tan et a l . " 3 i n tha t i t c o n s i s t s of s e v e r a l s t a g e s : p u l s e s e l e c t i o n , p r e a m p l i f i c a t i o n and main a m p l i f i e r s . A schemat ic r e p r e s e n t a t i o n of the l a s e r system i s shown i n F i g u r e I I I - 7 where the o p t i c a l beam pa th through the v a r i o u s components i s d e p i c t e d . 44 SPATIAL LUMONICS K103 F i g u r e I I I - 7 S c h e m a t i c d i a g r a m o f t h e o p t i c a l l a y - o u t o f t h e CC>2 l a s e r s y s t e m . The p u l s e switched out to the a m p l i f i e r modules o r i g i n a t e s at the h y b r i d l a s e r s e c t i o n . The h y b r i d l a s e r 0 * c o n s i s t s of a small TEA s e c t i o n with a low pr e s s u r e CW s e c t i o n i n the same l a s e r c a v i t y . The low pr e s s u r e s e c t i o n i s operated c o n t i n u o u s l y so that there i s a s i g n i f i c a n t l e v e l of r a d i a t i o n of a s i n g l e l o n g i t u d i n a l mode at the time when the hig h gain pressure s e c t i o n i s p u l s e d . T h i s mode then grows f i r s t and d e p l e t e s the a v a i l a b l e g a i n i n the TEA s e c t i o n before other modes have b u i l t up a p p r e c i a b l y . Thus the output from the h y b r i d l a s e r i s a h i g h -power s i n g l e l o n g i t u d i n a l mode pul s e superimposed on a low-power CW output. In g e n e r a l , a C0 2 l a s e r can be made to emit at l e a s t 400 i n d i v i d u a l v i b r a t i o n - r o t a t i o n l i n e s i n the 8.7 - 11.8 m wavelength r e g i o n . Without any attempt to s e l e c t the emitted wavelength, o s c i l l a t i o n may occur on e i t h e r one of s e v e r a l 45 l i n e s , of which the P ( 1 8 ) , P(20) and P(22) l i n e s have the h i g h e s t g a i n . For t h i s r e a s o n , a temperature c o n t r o l l e d F a b r y -Pero t e t a l o n i s used as the output c o u p l e r a t the end of the CW s e c t i o n . By c a r e f u l c o n t r o l of the temperature of the e t a l o n and i t s a l i g n m e n t , the P(20) l i n e (10.58 um) i s m a i n t a i n e d as the e m i t t e d wavelength d u r i n g cw o p e r a t i o n . The P(20) wavelength has the optimum g a i n c h a r a c t e r i s t i c s f o r subsequent a m p l i f i c a t i o n i n the TEA a m p l i f i e r s . N o n e t h e l e s s , the s t a b i l i t y of the e m i t t e d wavelength has a tendency to be somewhat t r a n s i e n t so that the output of the CW s e c t i o n must be c o n t i n u o u s l y m o n i t o r e d by a spectrum a n a l y z e r and the temperature and a l i gnment of the e t a l o n a d j u s t e d a c c o r d i n g l y . The p o l a r i z a t i o n of the output beam of the h y b r i d l a s e r i s set by the i n t r a c a v i t y Brewster windows and by the germanium p o l a r i z e r immedia te ly p r e c e d i n g a Pocke l s c e l l . P r i o r to the t ime of peak output o f . t h e h y b r i d l a s e r g a i n - s w i t c h e d p u l s e , a c o - a x i a l c a b l e spark gap i s d i s c h a r g e d . T h i s p r o v i d e s a 2 ns FWHM h i g h v o l t a g e p u l s e which i s a p p l i e d to the P o c k e l s c e l l t h a t c o n s i s t s of a s l a b of GaAs mounted i n a 50 t r a n s m i s s i o n l i n e arrangement . The h a l f wave v o l t a g e of 13 kV a c r o s s the GaAs c r y s t a l causes the p lane of p o l a r i z a t i o n of the l a s e r beam to be r o t a t e d by 90 d e g r e e s . As a r e s u l t , the Germanium p o l a r i z e r f o l l o w i n g the GaAs c r y s t a l becomes p a r t i a l l y r e f l e c t i n g and s w i t c h e s out a p u l s e whose d u r a t i o n i s equal t o t h a t of the v o l t a g e p u l s e a p p l i e d to the P o c k e l s c e l l . The s w i t c h e d out 2 ns p u l s e i s then d i r e c t e d i n t o the Lumonics K103 p r e a m p l i f i e r which has been d i s c h a r g e d some few hundred nanoseconds p r i o r to the t ime of e n t r y of the p u l s e . The 46 swi tched out pu l se of 100 uJ energy double passes through the a c t i v e g a i n medium of the K103 a m p l i f i e r , so tha t i t s energy i s augmented to about 100 mJ. Some of the 100 ns d u r a t i o n g a i n swi tched p u l s e from the h y b r i d l a s e r passes through the P o c k e l s c e l l p o l a r i z e r arrangement , and so an SF G s a t u r a b l e absorber c e l l i s p l a c e d a f t e r the K103 a m p l i f i e r to e l i m i n a t e the low l e v e l " l e a k t h r o u g h " r a d i a t i o n . The 100 mJ beam from the K103 passes through t h i s S F 6 c e l l and i n t o a second s p a t i a l f i l t e r p l a c e d i n s i d e a m u l t i p l e s e c t i o n TEA a m p l i f i e r u n i t d e s i g n a t e d as the 3-stage a m p l i f i e r i n F i g u r e I I I - 6 . The s t r a t e g i c l o c a t i o n of the s p a t i a l f i l t e r s i n the a m p l i f i e r c h a i n a l s o p r o v i d e s nece s s a ry overdense breakdown plasma i s o l a t o r s f o r any r e t r o - r e f l e c t e d p u l s e which becomes a m p l i f i e d i n t r a v e r s i n g backwards down the l a s e r c h a i n . The rea r m i r r o r of the t h r e e s tage a m p l i f i e r i s c o n t a i n e d i n a gas c e l l which i s used t o prevent s e l f - l a s i n g of t h i s a m p l i f i e r and to p r o v i d e p r e f e r e n t i a l wavelength s e l e c t i o n f o r the P(20) l i n e . T h i s gas c e l l i s f i l l e d w i t h 1 - 2 T o r r S F 6 , 20 T o r r e t h a n o l , 100 T o r r F reon 502 and 640 T o r r of He . The l a s e r beam i s double passed through both the three s tage and Lumonix 600 modules . The e x i t beam r a d i i of the 3-stage and Lumonix 600 a m p l i f i e r s a re 2 cm and 5 cm r e s p e c t i v e l y . The input i n t e n s i t y to these u n i t s i s s u f f i c i e n t l y h i g h so tha t on the r e t u r n pa th they are o p e r a t e d i n the s a t u r a t e d regime where a l i n e a r growth i n energy d e n s i t y w i t h d i s t a n c e t r a v e r s e d i s o b t a i n e d . A l l the a v a i l a b l e energy d e n s i t y per u n i t l e n g t h i s thus e x t r a c t e d . From the geometry and e x p e r i m e n t a l g a in s measured by Houtman and W a l s h 3 4 the a v a i l a b l e e x t r a c t a b l e energy from the t h r e e s tage 47 a m p l i f i e r opera ted at 1 atmosphere p r e s s u r e i s 3 - 5 J o u l e s , w h i l e tha t from the Lumonix 600 i s about 15 J o u l e s . Thus i n t h e o r y , a t o t a l a v a i l a b l e energy of some 20 J o u l e s can be e x t r a c t e d . In a c t u a l p r a c t i c e , the t y p i c a l observed e n e r g i e s are about 10 J o u l e s . The r e l a t i v e l y low y i e l d can be a t t r i b u t e d to the i n h e r e n t problem of TEA l a s e r s , which i s a r c i n g i n the d i s c h a r g e . N o r m a l l y , a glow d i s c h a r g e at a tmospher ic p r e s s u r e i s u n s t a b l e and r a p i d l y deve lops i n t o an a r c . However, the t ime to d e v e l o p and arc i s f i n i t e and hence arc fo rmat ion can be i n h i b i t e d by e x c i t a t i o n of the gas mix w i t h p u l s e s whose d u r a t i o n i s l e s s than the arc f o r m a t i o n t i m e . A l s o e f f e c t i v e and neces sa ry i s un i form p r e i o n i z a t i o n of the gas volume by UV r a d i a t i o n from an a u x i l l i a r y low energy d i s c h a r g e p r i o r to the main d i s c h a r g e as f i r s t demonstrated by Seguin and T u l i p " 5 . A l t h o u g h these f e a t u r e s a re i n c o r p o r a t e d i n t o the l a s e r modules , a r c i n g remains a s p o r a d i c phenomenon. To remedy t h i s s i t u a t i o n , a sma l l c o n t r o l l e d c o n c e n t r a t i o n of t r i p r o p y l a m i n e vapour i s i n t r o d u c e d i n t o the c o n t i n u o u s f l o w i n g gas mix . T h i s i m p u r i t y has low i o n i z a t i o n p o t e n t i a l and a quantum e f f i c i e n c y fo r p h o t o i o n i z a t i o n of n e a r l y u n i t y . Seguin et a l . " 6 found tha t a t r a c e amount of t r i p r o p y l a m i n e i s s u f f i c i e n t to r a i s e the p h o t o e l e c t r o n d e n s i t y by up to four o r d e r s of magni tude . A d d i t i o n of t r i p r o p y l a m i n e i n t o our l a s e r system i s found to be an e f f e c t i v e a r c suppres sant a t the c o s t of some a b s o r p t i o n of the C 0 2 l a s e r beam by the v a p o u r . 48 X - r a y D i a g n o s t i c s So f t x - r a y d i a g n o s t i c s are used e x t e n s i v e l y fo r e l e c t r o n temperature d e t e r m i n a t i o n of l a s e r produced plasmas s i n c e o ther methods such as Thomson s c a t t e r i n g and s p e c t r o s c o p y are c o m p a r a t i v e l y more cumbersome and u n s u i t a b l e fo r s h o r t - l i v e d s m a l l s c a l e h i g h temperature p la smas . The method used to measure the e l e c t r o n temperature of the plasma i n the exper iments i s the f o i l ab sorber t echn ique p i o n e e r e d by Jahoda et a l . " 7 on a t h e t a p i n c h p la sma. The e l e c t r o n temperature i s i n f e r r e d from measurements of the t r a n s m i t t e d x - r a y i n t e n s i t y through v a r i o u s meta l f o i l s . The t e c h n i q u e r e l i e s upon the f a c t t h a t a hot plasma of 100 eV e l e c t r o n temperature emit s p r i m a r i l y cont inuum r a d i a t i o n . The peak of the continuum spectrum i s a t the wavelength X = 6200/T , A , (T i n eV) p e and i s i n the so f t x - r a y range . S i n c e the wavelength dependence of the cont inuum e m i s s i v i t y v a r i e s as e x p ( - h c / X k T e ) , i t i s a s e n s i t i v e f u n c t i o n of the temperature i n the s o f t x - r a y wavelength r ange . A b s o r p t i o n measurements y i e l d i n f o r m a t i o n about the spectrum of i n c i d e n t x - r a y s and hence the e l e c t r o n temperature by the shape of the p l o t of the t r a n s m i t t e d i n t e n s i t y ver sus absorber t h i c k n e s s . The a b s o r p t i o n c o e f f i c i e n t s of m a t e r i a l s as a f u n c t i o n of wavelength a re very w e l l known and documented (see e .g .Henke e t a l . " 8 ) and have the g e n e r a l form of 49 n *i(X) = CX , w h e r e n i s o f t h e o r d e r o f 3 a n d C i s a c o n s t a n t p r o p o r t i o n a l t o Z , a b e i n g d e t e r m i n e d e m p i r i c a l l y . T h e t r a n s m i s s i o n o f x - r a y s e m a n a t i n g f r o m a p l a s m a e m i t t i n g b r e m s s t r a h l u n g a n d r e c o m b i n a t i o n r a d i a t i o n t h r o u g h a f o i l t h i c k n e s s x i s e x p r e s s e d a s T ( x ) = J l ( X ) e x p ( - * i ( X ) x ) d X = / A e x p ( - h c / X k T ) e x p ( - * i ( X ) x ) d X , w h e r e A i s ( s e e e . g . D o n a l d s o n " 9 ) : - 1 8 2 7 . 5 8 x 10 n V A = Z f Z + Z f P P P P P H T L. e. x I l i z p , n ( ! n n) x (In H P » n e x p H e r e n i s t h e e l e c t r o n d e n s i t y i n c m 3 , V i s t h e v o l u m e o f t h e e m i s s i o n s o u r c e , Z i s t h e c h a r g e , n_L d e n s i t y o f i o n s p e c i e s p , f t h e f r a c t i o n a l a b u n d a n c e (=n / n ) , n a t h e d e n s i t y o f i o n p p e ' e •* s p e c i e s p , X H i s t h e i o n i z a t i o n p o t e n t i a l ( i n e V ) o f h y d r o g e n , X i s t h e i o n i z a t i o n p o t e n t i a l f r o m t h e n t h s h e l l o f i o n p , n s p e c i e s p a n d J[n i s t h e n u m b e r o f s t a t e s a v a i l a b l e i n t h e n t h s h e l l f o r a r e c o m b i n i n g e l e c t r o n . T h e f o r m o f t h e i n t e g r a n d I ( X ) e x p ( - j i ( X ) x ) i s s h o w n i n F i g u r e I I I - 8 ( E l t o n 5 0 ) f o r v a r i o u s t h i c k n e s s e s o f Be a n d A l f o r b r e m s s t r a h l u n g e m i s s i o n f r o m a p l a s m a o f T = 500 e V . T h e x - r a y t r a n s m i s s i o n ( i n a r b i t r a r y u n i t s ) , i n F i g u r e 1 1 1 - 8 i s shown a s a 50 f u n c t i o n o f x - r a y w a v e l e n g t h w i t h f o i l t h i c k n e s s a s a p a r a m e t e r , ou 108 10 6 ' 102 10 BERYLLIUM Te 500 eV ou 10< 10* 10: 10 A loo ALUMINUM Te = 500eV \\\ * las 1 1 ib A Figure III-8 X-ray transmission through aluminum and^beryllium f o i l s of thicknesses i n d i c a t e d i n mg/cm . Two f e a t u r e s i m m e d i a t e l y become a p p a r e n t u p o n e x a m i n a t i o n o f F i g u r e I I I - 8 . F i r s t l y , t h e o v e r a l l t r a n s m i t t e d i n t e n s i t y d r o p s many o r d e r s o f m a g n i t u d e i n c o m p a r i s o n t o t h e i n c r e a s e o f a b s o r b e r t h i c k n e s s . T h i s a l l o w s t h e d e t e r m i n a t i o n o f t h e e l e c t r o n t e m p e r a t u r e by s t r a i g h t f o r w a r d c o m p a r i s o n o f t h e f a l l o f t r a n s m i t t e d x - r a y i n t e n s i t y a s a f u n c t i o n o f a b s o r b e r t h i c k n e s s . S e c o n d l y , f o r c o m p a r a t i v e l y t h i n f o i l s , m o s t o f t h e t r a n s m i t t e d x - r a y i n t e n s i t y l i e s i n t h e k - e d g e w a v e l e n g t h o f t h e a b s o r b e r ( e . g . f o r A l t h e k e d g e i s 7 . 9 A ) . T h u s u s i n g s u i t a b l y t h i n f o i l a b s o r b e r s o f v a r i o u s d i f f e r i n g Z a l l o w s a d e t e r m i n a t i o n o f t h e x - r a y s p e c t r u m d i r e c t l y . I n c r e a s i n g t h e f o i l t h i c k n e s s e s d i m i n i s h e s t h e t r a n s m i t t e d i n t e n s i t y b u t a l s o s h i f t s t h e p e a k o f t h e t r a n s m i t t e d s p e c t r u m t o w a r d s s h o r t e r w a v e l e n g t h s . T h u s , when a w i d e v a r i e t y o f f o i l a b s o r b e r s a r e u s e d , t h e w a v e l e n g t h r e s p o n s e o f t h e x - r a y d e t e c t o r m u s t b e c o n s i d e r e d a s w e l l . O f a l l c o m m o n l y a v a i l a b l e x - r a y d e t e c t o r s , t h e 51 p h o t o m u l t i p l i e r - s c i n t i l l a t o r system has the most s e n s i t i v i t y per x - r a y photon absorbed . P l a s t i c s c i n t i l l a t o r s (NE 110) were used to d e t e c t the t r a n s m i t t e d x - r a y s and t h e i r v i s i b l e l i g h t output i n t u r n i s d e t e c t e d by a p h o t o m u l t i p l i e r c o u p l e d to the s c i n t i l l a t o r v i a a s u i t a b l e l i g h t g u i d e . I m p l i c i t i n t h i s t e c h n i q u e i s the assumption t h a t the v i s i b l e l i g h t output from the s c i n t i l l a t o r s i s p r o p o r t i o n a l to the x - r a y energy ab sorbed . Indeed , the response of p l a s t i c s c i n t i l l a t o r s have been v e r i f i e d by s e v e r a l i n v e s t i g a t o r s 5 1 ' 5 2 t o be l i n e a r i n the x - r a y range of 0.6 - 15 keV. The t r a n s m i t t a n c e p r o p e r t i e s of the f o i l s t o g e t h e r w i t h the a b s o r p t i o n p r o p e r t i e s of the s c i n t i l l a t o r de termine the response R of the x - r a y d e t e c t i o n system R(X) = exp(-M (X)x ) • (1 - exp(-M (X)x ) , a a sc sc where the s u b s c r i p t s " a " and " s c " r e f e r t o absorber and s c i n t i l l a t o r r e s p e c t i v e l y . The s i g n i f i c a n c e of t h i s r e s u l t i s tha t the x - r a y d e t e c t o r has a f i n i t e wavelength range of s e n s i t i v i t y . T h i s can be put i n t o p r a c t i c a l use by j u d i c i o u s s e l e c t i o n of s c i n t i l l a t o r t h i c k n e s s x and absorber t h i c k n e s s sc x a such t h a t R(X) i s a maximum i n a narrow wavelength r a n g e , ( B e r n s t e i n 5 3 ) . For l a s e r produced plasmas which are commonly c h a r a c t e r i z e d by a two component t e m p e r a t u r e , the s c i n t i l l a t o r must be t h i c k enough to absorb the x - r a y photons b e i n g t r a n s m i t t e d through the f o i l s , ye t t h i n enough so tha t the hard x - r a y components from the supra therma l e l e c t r o n s do not i n t e r f e r e s i g n i f i c a n t l y when the s o f t e r p a r t of the x - r a y spectrum i s b e i n g sampled. In the exper iments d e s c r i b e d i n t h i s t h e s i s , the t h i c k n e s s of the s c i n t i l l a t o r s used was 0.6 cm so 52 t h a t x - r a y p h o t o n s o f e n e r g i e s l e s s t h a n 15 k e V a r e e f f i c i e n t l y a b s o r b e d . V a r i o u s x - r a y d e t e c t o r s w e r e c o n s t r u c t e d a n d u t i l i z e d i n t h e c o u r s e o f t h e e x p e r i m e n t s . T h e i m p o r t a n t d e s i g n f e a t u r e s common t o a l l t h e a r r a n g e m e n t s u s e d s u c c e s s f u l l y i n t h e e x p e r i m e n t s a r e c l e a r l y i l l u s t r a t e d i n t h e d e t e c t o r u s e d f o r m e a s u r e m e n t s o n t h e z - p i n c h p l a s m a c o l u m n 5 • shown i n F i g u r e I I I -9 . T h e z - p i n c h p l a s m a s e r v e d a s a u s e f u l t e s t - b e d f o r t h e Figure I I I - 9 X-ray diagnostics set-up f o r the z-pinch. (a) o v e r a l l set-up (b) detector head d e t a i l d e v e l o p m e n t o f x - r a y d i a g n o s t i c s o f h i g h s e n s i t i v i t y . S i n c e t h e z - p i n c h p l a s m a a n d t h e g a s j e t p l a s m a a r e b o t h n o t p a r t i c u l a r l y c o p i o u s s o u r c e s o f x - r a y s , a l a r g e d e t e c t o r a r e a = 5 c m 2 i s e x p o s e d t o t h e x - r a y s o u r c e f o r o p t i m u m s e n s i t i v i t y . S i n c e d e l i c a t e f o i l s o f t h i c k n e s s e s a s l o w a s 5 jim a r e u s e d , p r o v i s i o n m u s t be made f o r e v a c u a t i o n o f a i r t r a p p e d b e h i n d t h e f o i l a n d t h e s c i n t i l l a t o r t o p r e v e n t f o i l r u p t u r e u p o n e v a c u a t i o n o f t h e 53 t a r g e t c h a m b e r . T h e f o i l s m u s t be d i r e c t l y p l a c e d a g a i n s t t h e s c i n t i l l a t o r s o t h a t t h e y a r e s u p p o r t e d a g a i n s t t h e b l a s t f r o m t h e c o l l a p s i n g p i n c h o r l a s e r p r o d u c e d p l a s m a . A 0 . 5 mm v a c u u m c h a n n e l d r i l l e d i n t o t h e s c i n t i l l a t o r a l l o w s e s c a p e o f a n y a i r e n t r a p p e d b e t w e e n f o i l a n d s c i n t i l l a t o r . T h e s c i n t i l l a t o r i s c e m e n t e d w i t h i n d e x m a t c h i n g e p o x y o n t o a L u c i t e l i g h t g u i d e 60 cm i n l e n g t h w h i c h i n t u r n i s o p t i c a l l y c o u p l e d w i t h Dow S i l i c o n e 200 f l u i d o n t o t h e f a c e o f a n RCA p h o t o m u l t i p l i e r t u b e . T h e m e a s u r e d l i g h t a t t e n u a t i o n f r o m t h e l i g h t g u i d e was 50 %. To r e d u c e e l e c t r i c a l n o i s e . f r o m t h e p l a s m a a n d l a s e r t r i g g e r i n g K r y t r o n d i s c h a r g e u n i t s f r o m i n t e r f e r i n g w i t h t h e photomultiplier s i g n a l s , t h e photomultiplier i s e n c a s e d i n a d o u b l y s h i e l d e d e n c l o s u r e . T h e o u t e r m o s t p a r t o f t h e e l e c t r i c a l s h i e l d i s a s e x t e n s i o n o f t h e s c r e e n e d r o o m h o u s i n g t h e e x p e r i m e n t ' s e l e c t r o n i c i n s t r u m e n t a t i o n . T h e s i g n a l g r o u n d a n d t h e s c r e e n e d r o o m e x t e n s i o n a r e e l e c t r i c a l l y i s o l a t e d a n d c o m m o n l y g r o u n d e d a t t h e s c r e e n e d r o o m . T h e e l e c t r i c a l n o i s e l e v e l i s t h u s r e d u c e d f r o m s e v e r a l v o l t s t o a t o l e r a b l e l e v e l o f 10 mV. F l u o r e s c e n c e o f t h e d e t e c t o r f r o m i r r a d i t i o n by p l a s m a b r e m s s t r a h l u n g was k e p t t o a m i n i m u m b y t h e e x c l u s i v e u s e o f l o w Z m a t e r i a l s i n t h e c o n s t r u c t i o n o f t h e x - r a y d e t e c t o r . F o r e x a m p l e , t h e f l u o r e s c e n c e c o n v e r s i o n e f f i c i e n c y f o r b r a s s c o m p a r e d t o a l u m i n u m i s t w e n t y - f i v e 5 * t i m e s l a r g e r . T h e x - r a y d e t e c t i o n a p p a r a t u s a n d t h e p r i n c i p l e o f t h i s t e m p e r a t u r e m e a s u r i n g t e c h n i q u e was c o n f i r m e d b y d e t e r m i n i n g t h e e l e c t r o n t e m p e r a t u r e o f t h e z - p i n c h p l a s m a . T h e r e s u l t s o b t a i n e d a r e s h o w n i n F i g u r e 1 1 1 - 1 0 w h e r e t h e r e l a t i v e t r a n s m i t t e d 54 01 QO-1 5 6 7 e 9 microns F i ^ 111-10 X-ray transmission data f o r the z-pinch plasma. i n t e n s i t i e s t h r o u g h v a r i o u s a l u m i n u m f o i l s a r e p l o t t e d a g a i n s t t h e t h e o r e t i c a l r e l a t i v e t r a n s m i s s i o n s f o r v a r i o u s e l e c t r o n t e m p e r a t u r e s . I t i s s e e n i n F i g u r e I I I - 9 t h a t t h e r e s u l t s s u g g e s t a o f 40 e V w h i c h i s i n a g r e e m e n t w i t h t h e t e m p e r a t u r e s p r e v i o u s l y o b t a i n e d b y T h o m s o n s c a t t e r i n g , s p e c t r o s c o p y a n d o t h e r m e a n s 5 6 . T h e d e s i g n o f t h e x - r a y d e t e c t o r f o r t h e l a s e r i n c o r p o r a t e s t h e same d e s i g n f e a t u r e s a s f o r t h e z - p i n c h d e t e c t o r b u t i n v o l v e s a f e w m o r e c o n s i d e r a t i o n s m a i n l y d u e t o t h e p r o b l e m s t h a t a r i s e f r o m s u p r a t h e r m a l e l e c t r o n s . T h e e n e r g e t i c e l e c t r o n s w h i c h a r e a t t r i b u t e d t o v a r i o u s p a r a m e t r i c p r o c e s s e s g e n e r a t e d b y t h e l a s e r p l a s m a i n t e r a c t i o n , c a n h a v e mean f r e e p a t h s e x c e e d i n g t h e d i m e n s i o n s o f t h e v a c u u m c h a m b e r , e n a b l i n g t h e m t o s t r i k e t h e s u r f a c e s p r o d u c i n g c h a r a c t e r i s t i c K a r a d i a t i o n . T h e s e e l e c t r o n s m u s t a l s o n o t be a l l o w e d t o r e a c h t h e x - r a y d e t e c t o r s w h e r e t h e y c o u l d p e n e t r a t e t h e a b s o r b e r f o i l s . B o t h o f t h e s e e f f e c t s u n l e s s c o n t r o l l e d , c a n p r o d u c e a a d d i t i v e 55 background l e v e l to the observed s i g n a l s . When d e t e r m i n i n g the e l e c t r o n temperature by t a k i n g s i g n a l r a t i o s , the presence of a background l e v e l would l e a d to an e r r o n e o u s l y h i g h f i g u r e f o r the e l e c t r o n tempera ture (see e . g . Lachambre e t a l . 5 7 and M a r t i n e a u et a l . 5 8 ) . The x - r a y d e t e c t o r system used f o r the e l e c t r o n temperature measurement of the l a s e r produced plasma from the gas j e t t a r g e t i s shown i n F i g u r e 111 — 11 . To f a c i l i t a t e an e l e c t r o n tempera ture PM HOUSING « RCA UTS'S 0.5 FIBtR OPTICS LIGHT GUIDES • «« • ^•6AS JET NOZZLE SOLENOID VALVE 2 RESERVOIR Figure I I I - l l X-ray d iagnos t i c s set-up for the gas j e t target . m e a s u r e m e n t o n a s i n g l e s h o t b a s i s , t h e r e a r e now f o u r x - r a y c h a n n e l s e a c h w i t h a s c i n t i l l a t o r , 1 cm d i a m e t e r f i b e r o p t i c s l i g h t g u i d e a n d a n RCA 8 5 7 5 p h o t o m u l t i p l i e r t u b e . T h e s c i n t i l l a t o r - f o i l a s s e m b l i e s a r e c o n t a i n e d i n a b l a c k L u c i t e 8 cm d i a m e t e r d i s k w h i c h i s a l s o e n c l o s e d i n a t u b u l a r L u c i t e 56 e n c l o s u r e t h a t s e r v e s a s s h i e l d i n g o f t h e d e t e c t o r s f r o m t h e e l e c t r o n s . T h e t u b u l a r h o u s i n g a l s o c o n t a i n s a p a i r o f p e r m a n e n t b a r m a g n e t s b e h i n d a n a p e r t u r e . T h e s e m a g n e t s p r o v i d e a f i e l d o f 700 g a u s s o v e r a d i s t a n c e o f 5 cm s o t h a t i n t h e g e o m e t r y u s e d i n t h e e x p e r i m e n t s , e l e c t r o n s o f e n e r g i e s l e s s t h a n 10 MeV a r e s w e p t away f r o m t h e d e t e c t o r s . L i n e a r i t y o f t h e p h o t o m u l t i p l i e r s was e s t a b l i s h e d u s i n g a c o - a x i a l s p a r k g a p l i g h t s o u r c e w h o s e d u r a t i o n was l i m i t e d t o 20 n s by t h e c a b l e l e n g t h . T h e s a t u r a t i o n s i g n a l l e v e l o f t h e p h o t o m u l t i p l i e r s was d e t e r m i n e d by t h e o b s e r v a t i o n o f t h e p e a k s i g n a l o u t p u t v o l t a g e a s a f u n c t i o n o f t h e l i g h t i n t e n s i t y v a r i e d b y t h e i n c l u s i o n o f n e u t r a l d e n s i t y f i l t e r s a s shown i n F i g u r e 111-12. A l l f o u r s i g n a l o u t p u t s f r o m t h e p h o t o m u l t i p l i e r s (a) (b) 1-2 kV variable length cable pm tube 2 3 Figure III-l2 Photomultiplier Characterization (a) set-up (b) p.m. response scaling with light intensity a r e d e l a y e d w i t h r e s p e c t t o o n e a n o t h e r u s i n g c a b l e d e l a y s , t e r m i n a t e d by 50 Ohms a n d t h e n d i s p l a y e d o n a T e k t r o n i x 7904 57 o s c i l l o s c o p e w i t h 7A12 and 7A24 d u a l t r a c e p l u g - i n a m p l i f i e r s opera ted i n the add ing mode. The t ime response of the system was checked by i n j e c t i n g a 680 ps p u l s e from an N 2 TEA l a s e r 5 9 of UV wavelength 337.1 nm onto the s c i n t i l l a t o r s u r f a c e . The observed p e n e t r a t i o n depth of the beam i s about 2 mm i n t o the s c i n t i l l a t o r c a u s i n g i t to f l u o r e s c e . The r e s u l t i n g s i g n a l s as measured on the same o s c i l l o s c o p e as the x - r a y s i g n a l s shows t h a t the F W H M of the response i s 10 n s . S i n c e the l a s e r p u l s e i s o n l y 2 ns t h e r e w i l l be c o n s i d e r a b l e tempora l smoothing of the d e t e c t e d x - r a y s . In a d d i t i o n , t h e r e i s s p a t i a l smoothing as w e l l s i n c e the o n l y s p a t i a l d i s c r i m i n a t i o n i s t h a t p r o v i d e d by the a p e r t u r e and the s o l i d ang le subtended by the x - r a y d e t e c t o r s . Both t empora l and s p a t i a l smoothing p r o p e r t i e s of x - r a y d i a g n o s t i c s f o r shor t l a s e r p u l s e plasmas have been c o n s i d e r e d by P u e l l e t a l . 6 0 who c o n c l u d e d t h a t a temperature e q u a l t o the maximum temperature i s o b t a i n e d by the r a t i o method i f the t r a n s m i s s i o n of the f o i l s used i s not g r e a t l y d i f f e r e n t from one a n o t h e r . A s i m i l a r c o n c l u s i o n was reached f o r the s p a t i a l i n t e g r a t i o n assuming rea sonab le d e n s i t y g r a d i e n t s f o r the plasma and c o n s i d e r i n g tha t b r e m s s t r a h l u n g power i s p r o p o r t i o n a l to the e l e c t r o n d e n s i t y s q u a r e d . However, the f o i l a b s o r b e r s used i n the exper iment s i n t h i s t h e s i s have a range of d i f f e r e n t t r a n s m i s s i o n c h a r a c t e r i s t i c s so t h a t the t empora l and s p a t i a l smoothing e f f e c t s as they a f f e c t the e x p e r i m e n t a l r e s u l t s w i l l be d i s c u s s e d a t some l e n g t h i n Chapter V . 5 8 U l b r i c h t I n t e g r a t i n g Sphere As p a r t of the energy b a l a n c e , the s c a t t e r e d f r a c t i o n of l a s e r l i g h t t h a t does not c o n t r i b u t e to plasma h e a t i n g must be known. O p t i c a l energy ba l ance measurements are the most d i r e c t means of g a i n i n g t h i s i n f o r m a t i o n . In l a s e r produced plasma e x p e r i m e n t s , a l a r g e f r a c t i o n of the non-absorbed l a s e r l i g h t may be s c a t t e r e d i n a d i f f u s e manner. Hence, a r e f l e c t o m e t e r t o be used i n such exper iments s h o u l d have a 4^ c o l l e c t i o n a n g l e . An U l b r i c h t s p h e r i c a l p h o t o m e t e r 6 1 i s i d e a l f o r such a measurement and has been f i r s t used by Godwin e t a l . 6 2 f o r a l a s e r produced p la sma . I t b a s i c a l l y c o n s i s t s of a h o l l o w sphere w i t h i t s i n t e r i o r s u r f a c e f a s h i o n e d to be a d i f f u s e r e f l e c t o r . The a t t r a c t i v e f e a t u r e of t h i s d e v i c e i s t h a t the measured i n t e n s i t y , which i s p r o p o r t i o n a l to the r e f l e c t e d l a s e r l i g h t , i s independent of the a n g u l a r d i s t r i b u t i o n of the r a d i a t i o n . The p h o t o m e t r i c p r o p e r t i e s of the U l b r i c h t sphere a re immedia te ly apparent from the g e n e r a l laws of r a d i a t i o n , namely t h a t the i n t e n s i t y of i l l u m i n a t i o n of a s u r f a c e i s i n v e r s e l y p r o p o r t i o n a l to the square of i t s d i s t a n c e from the source and t h a t the luminous i n t e n s i t y of an element of a d i f f u s i n g s u r f a c e i s d i r e c t l y p r o p o r t i o n a l to the c o s i n e of the ang le between the normal to the s u r f a c e and the l i n e of e m i s s i o n 6 3 . In the schemat ic d iagram of F i g u r e I I I - 1 3 , the f l u x per u n i t s o l i d ang le f a l l i n g n o r m a l l y onto the s u r f a c e a t p o i n t x ' from the s u r f a c e element dS at x w i t h normal luminance b i s b dS c o s 0 / d 2 . The a rea element a t x ' i s a t an ang le Q t o the l i n e of 59 Figure 111-13 Schematic of an U l b r i c h t sphere. s i g h t x x ' s o t h a t t h e i l l u m i n a t i o n a t x ' , I d u e t o t h e e l e m e n t d S , i s I = b dS c o s 0 c o s f l / d 2 = b d S / 4 R 2 s i n c e d =2R c o s 0 by g e o m e t r y o f t h e s p h e r e . A s x i s a n a r b i t r a r y p o i n t on t h e s p h e r e , t h e n i t f o l l o w s t h a t a n y e l e m e n t o f t h e s u r f a c e i s i l l u m i n a t e d by a l l o t h e r s e q u a l l y . T h e r e f o r e , t h e i l l u m i n a t i o n o f t h e s u r f a c e by r e f l e c t e d l i g h t o n l y i s e q u a l a t a l l p o i n t s r e g a r d l e s s o f a n y a s y m m e t r y o f t h e l i g h t e m i t t e d by t h e s o u r c e . I f I d i s t h e a v e r a g e i l l u m i n a t i o n o f t h e s p h e r e by d i r e c t l i g h t a n d I i s t h e i l l u m i n a t i o n by d i f f u s e r e f l e c t e d l i g h t t h e n t h e a v e r a g e t o t a l i l l u m i n a t i o n i s 1 = 1 + 1 d m 2 3 = 1 + m l + m l + m l + . . . , d d d d w h e r e m i s t h e r e f l e c t i o n f a c t o r o f t h e s p h e r e ' s s u r f a c e . S i n c e t h e n u m b e r o f r e f l e c t i o n s i s i n f i n i t e a n d m < 1, t h e n I = I d m / ( 1 - m ) . A m o r e t h o r o u g h t r e a t m e n t 6 ' i n c l u d e s c o n s i d e r a t i o n o f l i g h t l o s t t h r o u g h t h e n e c e s s a r y p o r t s o f t h e s p h e r e . T h e m a i n r e s u l t h e r e i s t h a t t h e t h r o u g h p u t o f t h e s p h e r e , i . e . t h e r a t i o o f 60 e x i t i n g f l u x t o t h a t e n t e r i n g t h e s p h e r e , i s T = f m / d - m d - I f ) ) e i i w h e r e t h e f a c t o r f e i s t h e r a t i o o f t h e e x i t p o r t a r e a t o t h e a r e a o f t h e s p h e r e s u r f a c e , a n d Lf^ i s t h e r a t i o o f t h e s u m m a t i o n o f a l l t h e p o r t a r e a s t o t h a t o f t h e s p h e r e s u r f a c e . T h e u s e o f U l b r i c h t s p h e r e s f o r C 0 2 l a s e r p r o d u c e d p l a s m a s h a s b e e n c o m p a r a t i v e l y r e s t r i c t e d t o a few e x p e r i m e n t s i n t h e p u b l i s h e d l i t e r a t u r e : G a r b a n - L a b a u n e e t a l . 6 5 , K r i s t a l 6 6 , N i s h i m u r a e t a l 6 7 . F o r C 0 2 l a s e r p r o d u c e d p l a s m a s t h e r e f l e c t e d l i g h t i s i n f r a r e d s o t h a t t h e i n t e r i o r s u r f a c e o f t h e s p h e r e m u s t be s a n d - b l a s t e d a n d g o l d c o a t e d t o p r o v i d e a d i f f u s e l y r e f l e c t i n g s u r f a c e . By i t s v e r y n a t u r e , t h e U l b r i c h t s p h e r e i s a t i m e i n t e g r a t i n g d e v i c e . K r i s t a l 6 6 h a s m e a s u r e d t h e d e c a y p u l s e s h a p e a n d f o u n d i t t o be o f t h e f o r m I ( t ) = I 0 e x p ( - t / T > w h e r e r i s t h e d e c a y c o n s t a n t a p p r o x i m a t e d b y T / a , T b e i n g t h e r e p r e s e n t a t i v e t r a n s i t t i m e o f t h e s p h e r e , a i s t h e l o s s p e r b o u n c e . T h e t i m e i n t e g r a t i n g f e a t u r e o f t h e U l b r i c h t s p h e r e c a n i n p r i n c i p l e l e a d t o a n o v e r e s t i m a t e o f t h e r e f l e c t e d l a s e r e n e r g y s i n c e t h e e n e r g y r a d i a t e d i n b r e m s s t r a h l u n g c a n c o n t r i b u t e t o t h a t a t t r i b u t e d t o r e f l e c t i o n a l o n e . H o w e v e r , a s s h o w n i n A p p e n d i x B , t h e p o w e r r a d i a t i o n l o s s e s f r o m t h e o b s e r v e d p l a s m a a r e 3 o r d e r s o f m a g n i t u d e s m a l l e r t h a n t h e l a s e r p o w e r i n p u t a n d t h e r e f l e c t e d ( r e f r a c t e d a n d o r s c a t t e r e d ) i n f r a r e d r a d i a t i o n i s o f t h e same o r d e r a s t h e i n p u t p o w e r . T h e U l b r i c h t s p h e r e u s e d i n t h e e x p e r i m e n t s i s d e p i c t e d i n F i g u r e I I I - 1 4 . I t i s c o m p r i s e d o f t w o 18 cm d i a m e t e r a l u m i n u m h e m i s p h e r e s t h e i n t e r i o r o f w h i c h was s a n d b l a s t e d a n d u n i f o r m l y 61 Figure III-14 U l b r i c h t sphere photometric se t-up. g o l d - c o a t e d b y v a c u u m e v a p o r a t i o n . T h e i n c i d e n t l a s e r beam e n t e r s a n d e x i t s t h e s p h e r e t h r o u g h t w o 2 cm p o r t s . T h e e x i t p o r t i s n e c e s s a r y t o a v o i d h i g h i n t e n s i t y t r a n s m i t t e d l a s e r r a d i a t i o n f r o m d a m a g i n g t h e g o l d s u r f a c e . T h e i n c i d e n t l a s e r p o w e r a n d t h e t r a n s m i t t e d e n e r g y a r e m e a s u r e d f o r e a c h s h o t b y t h e a r r a n g e m e n t s h o w n i n F i g u r e 111 — 1 5 . A beam s p l i t t e r i n f r o n t o f t h e f o c u s s i n g l e n s s e n d s a s m a l l f r a c t i o n o f t h e i n c i d e n t l a s e r b e a m , a s w e l l a s t h e r e t r o - r e f l e c t e d l i g h t c o l l e c t e d b y t h e f o c u s s i n g l e n s , o n t o a p h o t o n d r a g d e t e c t o r t h r o u g h s u i t a b l e I R a t t e n u a t o r s . T h e p h o t o n d r a g d e t e c t o r f o r m o n i t o r i n g t h e i n c i d e n t l a s e r e n e r g y i s c a l i b r a t e d p e r i o d i c a l l y r e l a t i v e t o t h e A p o l l o e n e r g y m e t e r w h i c h i s u s e d f o r t h e t r a n s m i t t e d e n e r g y m e a s u r e m e n t . C a l i b r a t i o n i s a c h i e v e d b y d i r e c t c o m p a r i s o n o f t h e p h o t o n d r a g d e t e c t o r s i g n a l r e l a t i v e t o t h e A p o l l o e n e r g y m e t e r when t h e l a t t e r i s p l a c e d i n f r o n t o f t h e t a r g e t c h a m b e r . T h e a c t u a l d e t a i l s o f t h e I R i n c i d e n t a n d b a c k s c a t t e r r a d i a t i o n a r e 62 INCIDENT ENERGY DETECTOR ULBRICT SPHERE incident \ Position laser beam D-0 b e a m x splitter /VACUUM CHAMBER o • APOLLO ENERGY METER (transmitted energy) LAVAL NOZZLE BACKSCATTERED ENERGY DETECTOR Figure III-15 Arrangement for the infrared energy inventory. i n t h e t h e s i s o f B e r n a r d 3 5 . T h e r e f l e c t e d l a s e r e n e r g y f r o m t h e p l a s m a i s m e a s u r e d b y a G e n - T e c ED - 200 e n e r g y m e t e r p l a c e d d i r e c t l y a b o v e t h e g a s j e t a s s h o w n i n F i g u r e 111 — 14. A N a C l s a l t f l a t p o s i t i o n e d i n f r o n t o f t h e G e n - T e c m e t e r s e r v e d a s a f i l t e r p r e v e n t i n g d i r e c t h e a t t r a n s f e r f r o m t h e p l a s m a t o t h e e n e r g y m e t e r . T o c h e c k t h a t v i s i b l e r a d i a t i o n f r o m t h e p l a s m a m a k e s n o c o n t r i b u t i o n t o t h e m e a s u r e m e n t s , t h e f i l t e r i n g s a l t f l a t was r e p l a c e d b y a P l e x i g l a s d i s k . No d e t e c t a b l e s i g n a l was o b s e r v e d w i t h t h e p l a s m a p r e s e n t a n d w i t h t h e i n c l u s i o n o f t h e P l e x i g l a s d i s k . C a l i b r a t i n g t h e U l b r i c h t s p h e r e e s s e n t i a l l y i n v o l v e s f i r i n g t h e C 0 2 l a s e r beam i n t o t h e e v a c u a t e d s p h e r e w i t h t h e r e a r p o r t c o v e r e d b y a m a t c h i n g c a p , t h e i n s i d e c o n c a v e s u r f a c e o f w h i c h was a l s o s a n d b l a s t e d a n d g o l d c o a t e d . C o m p a r i s o n o f t h e G e n - T e c s i g n a l t o t h e i n c i d e n t beam p h o t o n d r a g s i g n a l u n d e r t h e s e 63 c i r c u m s t a n c e s e s t a b l i s h e s the c a l i b r a t i o n of the U l b r i c h t s p h e r e . The r a t i o of i n c i d e n t l a s e r energy to t h a t measured by the Gen-Tec meter was de te rmined to be 74 + 2. I s o t r o p y of the sphere was checked by t i l t i n g the back p o r t cap at v a r i o u s degrees of i n c i d e n c e to the i n c i d e n t l a s e r beam and o b s e r v i n g i f t h e r e was any change i n the c a l i b r a t i o n of the Gen-Tec s i g n a l s . No b a c k s c a t t e r s i g n a l s were obse rved d u r i n g the c a l i b r a t i o n ( < 0 . 0 1 1 0 ) t o w i t h i n the a c c u r a c y of the measurements. The i n t e g r a t i n g sphere as used appeared to be e n t i r e l y symmetric i n i t s r a d i o m e t r i c p r o p e r t i e s . Image C o n v e r t e r S t reak Camera A l t h o u g h the t ime d u r a t i o n of the l a s e r - p l a s m a i n t e r a c t i o n i s s h o r t - l i v e d (< 4 n s ) , the plasma p e r s i s t s f o r a l o n g t ime a f t e r w a r d . O b s e r v a t i o n s of the l i g h t output from the plasma by a Hamamatsu vacuum photod iode i n d i c a t e s t h a t the v i s i b l e e m i s s i o n has a 1/e decay t ime of about 60 n s . I t was t h e r e f o r e d e s i r a b l e to o b t a i n a t ime r e s o l v e d r e c o r d of the growth of the p la sma. In p a r t i c u l a r , the amount of absorbed l a s e r energy can i n p r i n c i p l e be d e t e r m i n e d by compar i son of the r a d i a l growth of the plasma w i t h t h e o r e t i c a l b l a s t wave space- t ime c u r v e s , (see e . g . Leonard and M a y e r 6 8 ) . For t h i s purpose a s t r e a k camera u n i t was c o n s t r u c t e d which u t i l i z e d an RCA C73435 image c o n v e r t e r tube w i t h a c o - a x i a l c a b l e d i s c h a r g e c i r c u i t which p r o v i d e s the gate and d e f l e c t i o n p l a t e v o l t a g e s . A schemat ic d iagram of the s t r e a k camera u n i t i s shown i n F i g u r e I I I - 1 6 . A s i m i l a r c i r c u i t of t h i s d e s i g n was f i r s t used by Cavenor and M e y e r 6 9 i n the s tudy of 64 s p a r k d i s c h a r g e s . T h e u s e o f t h e RCA i m a g e c o n v e r t e r t u b e t o 20 p s r e s o l u t i o n h a s b e e n d e m o n s t r a t e d b y S c h e l e v e t a l . 7 0 One c a b l e L 2 i s c h a r g e d t o 6 k V by a n e x t e r n a l p o w e r s u p p l y . When t h e s p a r k g a p i s d i s c h a r g e d t h e v o l t a g e d r o p s t o 3 k V w h i l e t h e v o l t a g e on L j r i s e s 3 k V . T h e s e v o l t a g e p u l s e s p r o p a g a t e a l o n g t h e c a b l e s t o t h e d e f l e c t i o n p l a t e s w h i c h t h e n become c h a r g e d w i t h a r i s e t i m e g i v e n b y 2 R S C w h e r e C i s t h e c a p a c i t a n c e o f t h e d e f l e c t i o n p l a t e s (= 6 p F ) a n d R g i s t h e v a l u e o f t h e c h a r g i n g r e s i s t a n c e . T h e s t r e a k s p e e d i s t h u s d e t e r m i n e d by t h e v a l u e o f R s . T h e t e m p o r a l r e s o l u t i o n o f t h e s t r e a k s y s t e m was c a l i b r a t e d b y t h e u s e o f a m o d e - l o c k e d r u b y l a s e r s y s t e m ( H o u t m a n e t 65 a l . 7 ' ) . The mode- locked output t r a i n of p u l s e s a re s epara ted i n time by about 2l/c, where I i s the c a v i t y l e n g t h and c the speed of l i g h t . The s t reak speed of the camera i s e s t a b l i s h e d once a s t r e a k r e c o r d of the mode- l o c k e d t r a i n of p u l s e s i s o b t a i n e d . For the c a l i b r a t i o n , the s t r e a k camera spark gap c i r c u i t was t r i g g e r e d by a K r y t r o n c i r c u i t a c t i v a t e d by a v a r i a b l e t ime d e l a y p u l s e u n i t which i t s e l f was a c t i v a t e d by the time d e l a y u n i t used to opera te the ruby l a s e r . A b r i g h t l i g h t source of t ime d u r a t i o n > 100 ns was n e c e s s a r y f o r b a c k l i g h t i n g the plasma for t ime r e s o l u t i o n of the r a d i a l growth of i t s shadow. S i n c e Q s w i t c h e d l a s e r p u l s e s a re l i m i t e d to 60 ns at best and have as much t empora l j i t t e r , a t r i g g e r e d h i g h p r e s s u r e spark gap l i g h t source was c o n s t r u c t e d . The spark gap l i g h t source was powered by a low impedance 1 Ohm t r a n s m i s s i o n l i n e as shown i n F i g u r e I I I - 1 7 . The background l i g h t source was measured to have a decay t ime of 500 n s . Both the l i g h t source and the s t r e a k camera are t r i g g e r e d by a 2 ns 13 kV p u l s e from the h y b r i d l a s e r P o c k e l s c e l l d r i v e r system v i a s u i t a b l e c a b l e d e l a y s to compensate fo r the t ime of a r r i v a l of the C 0 2 l a s e r p u l s e and l a s e r t r i g g e r i n g . S t u d i e s of the plasma expans ion on f a s t e r t ime s c a l e s were per formed u s i n g a Hamamatsu M1763 tempora l d i s p e r s e r u n i t . The h i g h g a i n of t h i s u n i t as p r o v i d e d by the i n c l u s i o n of a m i c r o -c h a n n e l p l a t e i n the s t r e a k t u b e , a l l o w e d s u b - m i l l i m e t e r s p a t i a l r e s o l u t i o n . Plasma expans ion was r e c o r d e d up t o 5 ns a f t e r the onset of the p la sma , which i s the upper l i m i t of the time base of t h i s u n i t . A s c h l i e r e n system was a l s o set up and used i n these s t u d i e s . The s c h l i e r e n system e s s e n t i a l l y c o n s i s t e d of a 66 CABLE DELAY TO STREAK CAMERA TRIGGER PRESSURIZED TRIGGERED SPARK GAP 5.2nF X-RAY DIAGNOSTIC EXTENSION LENS I APERTURE / SLIT RCA C7305 IMAGE CONVERTER STREAK CAMERA 2ns.27kV trigger pulse from hybrid loser spark gap Figure 111-17 Arrangement f o r time synchronization of the spark-gap l i g h t source f o r shadow streak photography. s u b m i l l i m e t e r f o i l s p e c k p l a c e d i n t h e f o c a l p l a n e o f t h e i m a g i n g l e n s a s s h o w n i n F i g u r e I I I - 1 8 . T h e t u r n i n g m i r r o r f o r dove ^>rism Q o foil speck mounted on glass L-p r - ruby laser beam r trigger pulse from hybrid laser spark gap Hamamatsu streak camera Figure I I I - l 8 Set-up f o r time-resolved s c h l i e r e n photography. t h e r u b y l a s e r beam was m o u n t e d o n a g i m b a l m o u n t a n d a l l o w e d a v e r t i c a l d i s p l a c e m e n t o f t h e i m a g e o n t h e s t r e a k c a m e r a s l i t s s o t h a t t h e r a d i a l e x p a n s i o n o f t h e p l a s m a a t d i f f e r e n t p o i n t s a l o n g t h e f o c a l r e g i o n c o u l d be s t u d i e d . 67 Chapter I V : R e s u l t s from the Exper iment s In o r d e r to examine how the l a s e r energy i s c o u p l e d to the p l a sma , s e v e r a l of the p h y s i c a l parameters of the plasma i n a d d i t i o n to the energy a b s o r p t i o n must be measured. The q u a n t i t i e s of p r imary importance are the e l e c t r o n t empera ture , the d e n s i t y and s i z e of the p la sma . T h i s c h a p t e r r e p o r t s the r e s u l t s of the exper iments u s i n g the appara tus and t e c h n i q u e s d e s c r i b e d i n Chapter I I I . The d i s c u s s i o n of the r e s u l t s here w i l l be r e s t r i c t e d to the p o s s i b i l i t i e s of s y s t e m a t i c e r r o r and the v a l i d i t y of the r e s u l t s . The p l a u s i b i l t y of the e x p e r i m e n t a l r e s u l t s can be a s c e r t a i n e d to some degree by compar i son w i t h the a v a i l a b l e p u b l i s h e d data from o t h e r l a s e r l a b o r a t o r i e s . A more d e t a i l e d d i s c u s s i o n of the s i g n i f i c a n c e of the e x p e r i m e n t a l r e s u l t s a re l e f t to Chapter V . I V - 1 : E l e c t r o n Temperature Measurements T h i s s e c t i o n d e s c r i b e s the e l e c t r o n temperature measurements o b t a i n e d by the x - r a y a b s o r p t i o n t e c h n i q u e d e s c r i b e d i n Chapter I I I . I n i t i a l o b s e r v a t i o n s a re r e p o r t e d t h a t were made at c o m p a r a t i v e l y low l a s e r i n t e n s i t i e s i m p i n g i n g on a low d e n s i t y t a r g e t such t h a t no hot e l e c t r o n g e n e r a t i o n was o b s e r v e d . The i n t e r a c t i o n i s thus s i m p l i f i e d by the absence of the c o m p l i c a t i n g f e a t u r e of hot e l e c t r o n g e n e r a t i n g p a r a m e t r i c i n s t a b i l i t i e s . A s c a l i n g of e l e c t r o n temperature w i t h i n c i d e n t l a s e r energy i s o b t a i n e d f o r the i n t e r a c t i o n o f . C 0 2 l a s e r l i g h t 68 w i t h an underdense p l a sma . The e x p e r i m e n t a l c o n d i t i o n s were then changed so t h a t p r o d u c t i o n of hot e l e c t r o n s i s observed s i n c e t h i s t h e s i s work i s p a r t of an o v e r a l l program i n i t i a t e d to s tudy hot e l e c t r o n g e n e r a t i n g i n s t a b i l i t i e s . The c o r r e s p o n d i n g changes i n the x - r a y e m i s s i o n c h a r a c t e r under these c i r c u m s t a n c e s are noted a c c o r d i n g l y . The presence of hot e l e c t r o n s i n t r o d u c e s e v e r a l problems t h a t are c o n s i d e r e d and e v a l u a t e d i n terms of t h e i r consequences on the measurements. X - r a y a b s o r p t i o n da t a o b t a i n e d w i t h a v a r i e t y of x - r a y a b s o r b e r s p r o v i d e x - r a y s p e c t r a l i n f o r m a t i o n which i s p r e s e n t e d and summarized. IV-1 a : I n i t i a l O b s e r v a t i o n s a t Low I n t e n s i t i e s As ment ioned i n Chapter I I , the e l e c t r o n temperature of a l a s e r produced plasma i s expec ted to depend on the i n c i d e n t l a s e r i n t e n s i t y . As rev iewed by Donaldson et a l . 7 2 , the dependence of e l e c t r o n temperature on the i n c i d e n t l a s e r f l u x i n t e n s i t y can be o b t a i n e d from r e l a t i v e l y s imple hydrodynamic a n a l y t i c a l mode l s . These r e s u l t i n s c a l i n g laws of the form Tg* I n where I i s the l a s e r f l u x , T^ the t e m p e r a t u r e , and n v a r i e s from 4/9 to 2/3 depending upon the a s sumpt ions employed . The models have been used w i t h some succe s s to e x p l a i n the observed e l e c t r o n temperature s c a l i n g o b t a i n e d i n p l a n a r s o l i d t a r g e t e x p e r i m e n t s . The gas j e t s i m u l a t e s a s o l i d p l a n a r t a r g e t i n geometry, but w i t h the impor tant d i f f e r e n c e t h a t t h e r e i s no c r i t i c a l d e n s i t y l a y e r and no consequent shock h e a t i n g of the t a r g e t m a t e r i a l b e h i n d i t . T h u s , i t i s of i n t e r e s t t o compare 69 t h e s c a l i n g w i t h i n c i d e n t l a s e r f l u x f o r t h e u n d e r d e n s e g a s j e t t a r g e t . F o r t h e s c a l i n g s t u d y o f e l e c t r o n t e m p e r a t u r e , a 2 c h a n n e l v e r s i o n o f t h e x - r a y d e t e c t o r d e s c r i b e d i n C h a p t e r I I I was u s e d , t h e e l e c t r o n t e m p e r a t u r e s i m p l y b e i n g d e t e r m i n e d by t h e r a t i o o f t h e x - r a y i n t e n s i t i e s t r a n s m i t t e d t h r o u g h t w o s u i t a b l y d i f f e r e n t a b s o r b e r f o i l s . T h e g a s j e t t a r g e t was o p e r a t e d a t l o w p r e s s u r e c o n d i t i o n s s u c h t h a t i f f u l l i o n i z a t i o n w e r e t o o c c u r a s e x p e c t e d , t h e maximum e l e c t r o n d e n s i t y w o u l d be 5 x 1 0 1 B c m - 3 . S i n c e t h e r a n g e o f a c c u r a c y o f t e m p e r a t u r e d e t e r m i n a t i o n f r o m a n y p a i r o f f o i l s i s l i m i t e d a s s h o w n i n F i g u r e I V - 1 , v a r i o u s a l u m i n u m f o i l p a i r s i n t h e r a n g e o f t h i c k n e s s f r o m 6 18 um w e r e s e l e c t e d f o r d i f f e r e n t t e m p e r a t u r e r a n g e s . T h e r e s u l t s a r e 50 AO 30 20 10 signal ratio theoretical calculated curve typical experimental datum u 100 200 300 A00 eV Figure IV-1 Ratio of transmitted x-ray i n t e n s i t i e s through 9 and 18 ym f o i l vs. T g s u m m a r i z e d i n t h e p l o t o f F i g u r e I V - 2 w h e r e t h e s o l i d l i n e i s t h e f i t o f T = 4 1 E ° - 6 0 , E b e i n g t h e i n c i d e n t l a s e r e n e g y . T h e i n t e n s i t y s c a l i n g o f 0 . 6 c a n be c o m p a r e d w i t h t h a t f r o m o t h e r l a b o r a t o r i e s . D i r e c t c o m p a r i s o n s h o u l d be a p p r o a c h e d w i t h some Figure IV-2 E l e c t r o n temperature s c a l i n g wi th i n c i d e n t l a s e r energy. Laser i n t e n s i t i e s <^  1 0 1 3 W/cm 2 . c a u t i o n s i n c e d i f f e r e n t e x p e r i m e n t s u s e d i f f e r e n t c o n d i t i o n s s u c h a s f o c a l s p o t s i z e a n d t i m e d u r a t i o n o f p u l s e . N o n e t h e l e s s , i t i s o f i n t e r e s t t o m e n t i o n a f e w o f t h e s e r e s u l t s . T h e r e a r e t w o p a r a m e t e r s t o be c o m p a r e d h e r e , e l e c t r o n t e m p e r a t u r e a n d t h e s c a l i n g w i t h l a s e r i n t e n s i t y . I n t e r m s o f p u l s e d u r a t i o n a n d i n t e n s i t y t h e r e s u l t s o b t a i n e d h e r e c a n be c o m p a r e d t o t h o s e o f P e p i n e t a l . 7 3 w h e r e a 4 J , 1 .8 n s C 0 2 l a s e r p u l s e was f o c u s s e d o n t o a p o l y e t h y l e n e t a r g e t . T h e r e p o r t e d t e m p e r a t u r e s r a n g e d f r o m 1 5 0 - 3 0 0 e V a n d t h e i n t e n s i t y s c a l i n g i s 0 . 3 - 0 . 4 . Y a m a n a k a 7 * s u m m a r i z e s t h e r e s u l t s o f t h e E c o l e P o l y t e c h n i q u e a n d L i m e i l l a b o r a t o r i e s a s w e l l a s t h a t o f O s a k a , a l l o f w h i c h show a s c a l i n g o f I V 2 S a n d a t e m p e r a t u r e r a n g e f r o m 100 e V a t 1 0 9 W / c m 2 t o 5 0 0 e V a t 1 0 1 * W / c m 2 . M o r e r e c e n t l y , N i s h i m u r a e t a l . 6 7 h a v e r e p o r t e d a 0 . 6 p o w e r s c a l i n g f o r t h e b l a c k b o d y t e m p e r a t u r e o f C 0 2 l a s e r i r r a d i a t e d g o l d t a r g e t s . T h e o b s e r v e d t e m p e r a t u r e s f r o m t h e g a s j e t a r e t h e r e f o r e 71 w i t h i n t h e same r a n g e o b t a i n e d f o r s i m i l a r i n t e n s i t i e s on s o l i d t a r g e t s . I n a d d i t i o n t o t h e a b o v e m e n t i o n e d d a t a , E n r i g h t e t a l . 7 5 r e p o r t s a t e m p e r a t u r e T e o f 250 e V f r o m a C 0 2 l a s e r p r o d u c e d p o l y e t h y l e n e p l a s m a a t i n t e n s i t i e s o f 1 0 1 * W / c m 2 . S i m i l a r l y , V i o l e t a n d P e t r u z z i 7 6 f i n d T £ = 300 - 500 eV f o r 10 -40 J o u l e C 0 2 l a s e r p u l s e s o f 1 - 3 n s d u r a t i o n on p l a n a r p o l y e t h y l e n e t a r g e t s . R e c e n t r e s u l t s i n t h e S o v i e t l i t e r a t u r e 7 7 i n d i c a t e a t h e r m a l t e m p e r a t u r e o f 220 eV o b t a i n e d u n d e r s i m i l a r c o n d i t i o n s a s t h o s e a b o v e . I t s h o u l d be m e n t i o n e d t h a t t h e m a j o r i t y o f x - r a y e m i s s i o n i n s o l i d t a r g e t s c o m e s f r o m t h e c o n d u c t i v e l y h e a t e d r e g i o n b e h i n d t h e c r i t i c a l d e n s i t y l a y e r . T h e c o m p u t e r s i m u l a t i o n s o f M a r s h a n d T h o m p s o n 7 8 f o r l a s e r i r r a d i a t i o n o f a c a r b o n s l a b r e v e a l t h a t t h e s u p e r c r i t i c a l c o n d u c t i v e l y h e a t e d r e g i o n i s c h a r a c t e r i z e d by a P l a n c k i a n s p e c t r a l d i s t r i b u t i o n c o r r e s p o n d i n g t o t e m p e r a t u r e s o f < 200 e V . P r e c e d i n g t h e c r i t i c a l d e n s i t y l a y e r i s t h e t e n u o u s o p t i c a l l y t h i n c o r o n a p l a s m a w h i c h i s l a s e r h e a t e d t o s e v e r a l k e V p r o d u c i n g a b r e m m s t r a h l u n g s p e c t r u m i n t h e p h o t o n e n e r g y r a n g e o f 2 k e V a n d b e y o n d . O n l y a f e w r e s u l t s o f e l e c t r o n t e m p e r a t u r e m e a s u r e m e n t s o n g a s e o u s t a r g e t s i r r a d i a t e d by C 0 2 l a s e r s h a v e b e e n p u b l i s h e d . G i l e s a n d O f f e n b e r g e r 7 9 m e a s u r e 150 e V i n a n o x y g e n g a s j e t t a r g e t i r r a d i a t e d by a 40 n s p u l s e o f i r r a d i a n c e 6 x 1 0 1 2 W / c m 2 . B a l d i s a n d W a l s h 8 0 m e a s u r e a B r i l l o u i n b a c k s c a t t e r r e d s h i f t c o r r e s p o n d i n g t o a 200 e V p l a s m a o f e l e c t r o n d e n s i t y 0 . 1 - 0 . 2 n c r . I n c o n t r a s t , Y a b l o n o v i t c h 3 7 c l a i m s a n e l e c t r o n t e m p e r a t u r e o f 1 k e V f o r a n i t r o g e n p l a s m a u s i n g o n l y 0 . 1 J o f i n p u t e n e r g y . Y a b l o n o v i t c h ' s r e s u l t s a r e p a r t i c u l a r l y i n t e r e s t i n g i n t h a t a 72 r e s o n a n t n a t u r e i n t h e x - r a y p r o d u c t i o n i s o b s e r v e d . A s t h e g a s j e t t a r g e t p r e s s u r e i s i n c r e a s e d t o a d e n s i t y w h e r e a c r i t i c a l e l e c t r o n d e n s i t y i s e x p e c t e d f o r f u l l i o n i z a t i o n , a s u d d e n o n s e t o f x - r a y a n d e l e c t r o n p r o d u c t i o n i s o b s e r v e d ( K o l o d n e r a n d Y a b l o n o v i t c h 8 1 ) a n d i s a t t r i b u t e d t o t h e r e s o n a n c e a b s o r p t i o n m e c h a n i s m . I V - 1 b : C h a n g e i n X - r a y E m i s s i o n w i t h T a r g e t C o n d i t i o n s S i m i l a r b e h a v i o u r a s r e p o r t e d b y K o l o d n e r a n d Y a b l o n o v i t c h was o b s e r v e d w i t h t h e g a s j e t t a r g e t w i t h a n i n c r e a s e o f t a r g e t d e n s i t y a n d a s u b s e q u e n t i m p r o v e m e n t i n t h e f o c a l s p o t r a d i u s . T h e f o c a l s p o t r a d i u s was m e a s u r e d by t h e m u l t i p l e s l i t F r a u n h o f e r d i f f r a c t i o n b u r n m a r k t e c h n i q u e a s d e s c r i b e d b y O ' N e i l e t a l . 8 2 a n d B e r n a r d 3 5 . T h e f o c a l s p o t was r e d u c e d f r o m 250/im u s e d i n t h e x - r a y a b s o r p t i o n m e a s u r e m e n t s d e s c r i b e d e a r l i e r t o 50 nm t h u s i n c r e a s i n g t h e f o c a l i n t e n s i t y t o < 1 0 1 " W / c m 2 . A c r i t i c a l d e n s i t y i n p r i n c i p l e c a n be r e a c h e d f o r f u l l i o n i z a t i o n i f t h e p r e s s u r e o f t h e n i t r o g e n e m a n a t i n g f r o m t h e n o z z l e i s 4 T o r r o r g r e a t e r . I n d e e d , when t h e j e t p r e s s u r e was i n c r e a s e d t o 4 T o r r , t h e x - r a y s i g n a l l e v e l s t h r o u g h a 9 ixm f o i l i n c r e a s e d b y a f a c t o r o f 45 f r o m t h a t o b s e r v e d f r o m t h e 2 T o r r j e t t a r g e t . A t t h i s h i g h e r i n i t i a l t a r g e t d e n s i t y , t h e x - r a y s i g n a l s h a d a t e n d e n c y t o f l u c t u a t e c o n s i d e r a b l y f r o m s h o t t o s h o t r e s u l t i n g i n a l a r g e s c a t t e r o f t h e d a t a . T h e r e a l s o a p p e a r e d t o be no a p p a r e n t s c a l i n g o f t h e e l e c t r o n t e m p e r a t u r e w i t h i n c i d e n t l a s e r e n e r g y . T h e x - r a y s i g n a l s o b s e r v e d t h r o u g h 300 *xm o f A l w i t h a 0 . 3 N . D . f i l t e r f o r v a r i o u s j e t t a r g e t 73 p r e s s u r e s a r e shown i n F i g u r e I V - 3 . E n e r g e t i c e l e c t r o n e m i s s i o n was o b s e r v e d s i m u l t a n e o u s l y w i t h a n e l e c t r o n s p e c t r o m e t e r b y M c i n t o s h 8 3 . T h e s e r e s u l t s s h o w e d a s i m i l a r l y s t r i k i n g i n c r e a s e a s t h e x - r a y s w i t h t a r g e t p r e s s u r e . T h e s u b s e q u e n t i n v e s t i g a t i o n s w e r e a l l p e r f o r m e d on a 5 T o r r j e t t a r g e t a s t h e u p p e r l i m i t s i n c e h i g h e r t a r g e t p r e s s u r e s r e s u l t e d i n a b r e a k d o w n He p l a s m a w i t h o u t t h e p r e s e n c e o f N 2 . F o r t h e 5 T o r r j e t t a r g e t , t h e t w o c h a n n e l x - r a y d e t e c t o r m e a s u r e m e n t s c o n s i s t e n t l y i m p l i e d a c o n s t a n t t e m p e r a t u r e o f 2 k e V i r r e s p e c t i v e o f t h e i n c i d e n t l a s e r e n e r g y . Some r e p r e s e n t a t i v e d a t a i s s h o w n i n F i g u r e I V - 4 w h i c h i s t h e t e m p e r a t u r e o b t a i n e d b y means o f a 100 nm a n d 3 0 0 nm a l u m i n u m f o i l p a i r . T h e s u d d e n i n c r e a s e i n e l e c t r o n p r o d u c t i o n a n d x - r a y i n t e n s i t y w i t h t a r g e t p r e s s u r e a l o n g w i t h t h e o b s e r v e d h i g h e l e c t r o n t e m p e r a t u r e s u g g e s t s t h e o n s e t o f p a r a m e t r i c V 2 1 N2 Torr Figure IV-3 Hard x-ray i n t e n s i t y as a function of target pressure 74 Figure IV-4 Electron temperature versus laser energy. Laser intensit ies <_ 1011* W / c m 2 i n s t a b i l i t i e s c a u s i n g t h e p r o d u c t i o n o f e n e r g e t i c e l e c t r o n s . I t h a s b e e n s h o w n by o t h e r i n v e s t i g a t o r s , M a r t i n e a u e t a l . 5 8 , L a c h a m b r e a n d N e u f e l d 5 7 , P e p i n a n d B a l d i s 8 * , t h a t i n t h e p r e s e n c e o f e n e r g e t i c e l e c t r o n s t h e t w o f o i l r a t i o m e t h o d o f e l e c t r o n t e m p e r a t u r e d e t e r m i n a t i o n c a n be g r e a t l y e r r o n e o u s a s d i f f e r e n t e l e c t r o n d i s t r i b u t i o n s p r o d u c e d i f f e r e n t s p e c t r a o f x -r a y r a d i a t i o n a c c o r d i n g t o t h e i r e n e r g y d i s t r i b u t i o n s . T h i s f a c t i s d r a m a t i c a l l y d e m o n s t r a t e d by t h e r e s u l t s o f D y e r e t a l . 8 5 who o b t a i n e d 100 e V t o 2 6 0 0 e V f r o m t h e same l a s e r p l a s m a u s i n g v a r i o u s p a i r s o f f o i l s . I V - l c : T h e E f f e c t o f H o t E l e c t r o n s O f p r i m a r y c o n c e r n i s t h e e f f e c t o f e l e c t r o n s b o m b a r d i n g m a t e r i a l s i n t h e t a r g e t c h a m b e r . E i d m a n n a n d S i g e l 8 6 d e m o n s t r a t e d t h i s e f f e c t a s b e i n g i m p o r t a n t s i n c e p r o p e r s h i e l d i n g o f t h e i r x - r a y d e t e c t o r s f r o m t h e r a d i a t i o n e m i t t e d b y 75 t h e t a r g e t c h a m b e r w a l l s r e d u c e d t h e s i g n a l y i e l d b y a f a c t o r o f 3 - 1 0 . I n t h e p h y s i c s o f b r e m s s t r a h l u n g e m i s s i o n , t h e r e a r e t w o r e l e v a n t mean f r e e p a t h s : t h e mean f r e e p a t h f o r b i n a r y c o l l i s i o n s g i v i n g b r e m s s t r a h l u n g a n d t h e mean f r e e p a t h f o r s c a t t e r i n g t h r o u g h 90 d e g r e e s by m u l t i p l e s m a l l a n g l e C o u l o m b c o l l i s i o n s . T h i s s e c o n d mean f r e e p a t h i s t y p i c a l l y l a r g e r t h a n t h e f i r s t a n d d e t e r m i n e s w h e t h e r a n e l e c t r o n c a n be a r u n a w a y e l e c t r o n ( S p i t z e r 2 " ) . T h e v a l u e o f t h i s mean f r e e p a t h c a n be c a l c u l a t e d f r o m S p i t z e r ' s e q u a t i o n X = E 2 / 2 i r n e * 1 / l n A e = 7 . 6 8 x l 0 1 , E 2 / n e , c r r i / ( E i n e V f n ^ i n c m ~ ^ ) f o r e l e c t r o n s o f d e n s i t y n e , e n e r g y E a n d C o u l o m b l o g a r i t h m InA-F o r e l e c t r o n s o f e n e r g y 2 k e V , t h e mean f r e e p a t h i s o v e r a cm f o r a t y p i c a l d e n s i t y o f 2 . 5 x 1 0 1 8 cm . T y p i c a l l a s e r p r o d u c e d p l a s m a s a r e m u c h s m a l l e r t h a n t h i s s o t h a t e l e c t r o n s o f e n e r g y o f a f ew k e V p r o d u c e no b r e m s s t r a h l u n g i n t h e i r p a s s a g e t h r o u g h t h e p l a s m a . P h o t o n s w i t h e n e r g i e s i n t h i s r a n g e a r e n o t p r o d u c e d i n t h e p l a s m a . T h e r e m a i n i n g o r i g i n s a r e t h e n e i t h e r e l e c t r o n c o l l i s i o n s w i t h t h e N 2 g a s s u r r o u n d i n g t h e plasma' o r m o r e l i k e l y , w i t h m a t e r i a l s i n t h e t a r g e t v a c u u m c h a m b e r . I t i s w e l l known t h a t t h e i n t e r a c t i o n o f f a s t e l e c t r o n s w i t h t h e t a r g e t m a t e r i a l i s t h e same a s t h a t o f f a s t e l e c t r o n s s t o p p i n g i n n o n - i o n i z e d m a t e r i a l a s f a r a s b r e m s s t r a h l u n g a n d K a e m i s s i o n a r e c o n c e r n e d f o r r a d i a t i o n o f p h o t o n s o f h i g h e n e r g y i . e . , h X » k T e ( M c C a l l 8 7 ) . T h u s e l e c t r o n s b o m b a r d i n g n e u t r a l n i t r o g e n y i e l d t h e same r a d i a t i o n a s i f t h e n i t r o g e n w e r e i o n i z e d . A c c o r d i n g l y , t o m a i n t a i n t h i s p h y s i c a l e q u i v a l e n c e o f r a d i a t i o n 76 throughout the t a r g e t chamber, a l l meta l s u r f a c e s i n the v i c i n i t y of the f o c a l volume were c o v e r e d by a l a y e r of P l e x i g l a s . E n e r g e t i c e l e c t r o n s have been observed to l e a v e the f o c a l v o l u m e . 8 8 ' 8 9 These e l e c t r o n s a re c h a r a c t e r i z e d by an energy of 100 keV and were e s t i m a t e d to c o n s t i t u t e 1% of the t o t a l number of e l e c t r o n s i n the f o c a l vo lume. The r a d i a t i o n c o n t r i b u t i o n of these e l e c t r o n s i m p i n g i n g upon m a t e r i a l s i n the t a r g e t chamber was c o n s i d e r e d as f o l l o w s . The x - r a y p r o d u c t i o n from a t a r g e t w i t h a tomic number 2 i s g i v e n by 9 0 x - r a y power/beam power = 7 x 10* ZE , where E i s the e l e c t r o n energy i n MeV. W i t h the a s sumpt ion t h a t the 100 keV e l e c t r o n s are e m i t t e d i n a p u l s e d u r a t i o n of 2 n s , the x - r a y power from these e l e c t r o n s bombarding the P l e x i g l a s i n the chamber i s e s t i m a t e d to b e . 1 0 2 - 1 0 3 W a t t s . In c o m p a r i s o n , the t o t a l i n t e g r a t e d b r e m s s t r a h l u n g (Jahoda * 7 ) i s -32 0.5 JE dX = 1.5 x 10 T g N N W a t t s , (T f i i n e V ) , X e e i which f o r T e of the o r d e r of s e v e r a l hundred eV and t o t a l e l e c t r o n number of 1 0 1 6 y i e l d s an e s t i m a t e of j u s t a few wat t s , an i n s i g n i f i c a n t power compared to t h a t p roduced by e l e c t r o n bombardment of P l e x i g l a s . The Gaunt f a c t o r g above, i s = 1. The s h o r t e s t wave length t h a t can be e m i t t e d by an e l e c t r o n of energy E i s the Duane-Hunt l i m i t : 77 X = h c / E , min which i s 0.01 nm for 100 keV e l e c t r o n s . As t h i s wavelength i s c o n s i d e r a b l y s h o r t e r than t h a t e m i t t e d by thermal e l e c t r o n s i n the p la sma , the shape of the spectrum of x - r a y s e m i t t e d by the e n e r g e t i c e l e c t r o n s becomes an i s s u e . S e v e r a l r e s u l t s from the theory of p r o d u c t i o n of x - r a y s from x - r a y tubes become u s e f u l (Green and C o s s l e t t 9 1 ) . The number of photons N e m i t t e d as cont inuum r a d i a t i o n i n the wavelength range X m i n to \ i s g iven by the i n t e g r a t i o n of Kramers ' f o r m u l a : "x "x -14 Z b min N = m 1.8 x 10 In — + c X X X min - min- L b J where m i s the number of bombarding e l e c t r o n s per s econd . C a l c u l a t i o n s r e v e a l 10 8 p h o t o n s / s e c w i t h the peak of the cont inuum spectrum at 2 X m i n . Green and C o s s l e t t o b t a i n a r e l a t i o n fo r the number of p h o t o n s / s e c N f c of K a l i n e r a d i a t i o n which has the form: N = u mA x E / E l n ^ E / E \ - ( E / E - 1) , k k k 1 k' k where C J k i s the f l u o r e s c e n c e e f f i c i e n c y and A i s a c o n s t a n t . Both w k and A are t a r g e t dependent and are t a b u l a t e d i n Green and C o s s l e t t . For a carbon t a r g e t the c a l c u l a t i o n s show tha t 1 0 1 8 p h o t o n s / s e c are produced at the carbon K Q l i n e of 4.4 nm. T h e r e f o r e , the hard cont inuum r a d i a t i o n produced by the bombardment of e n e r g e t i c e l e c t r o n s i s not a s i g n i f i c a n t 78 c o n t r i b u t i o n to the d e t e c t a b l e r a d i a t i o n . The s o f t x - ray r a d i a t i o n of the K l i n e s of low Z m a t e r i a l need not be c o n s i d e r e d s i n c e the t r a n s m i s s i o n of the f o i l s used f a l l s o f f s h a r p l y at 1.0 nm. For example, the f r a c t i o n of 4.4 nm K r a d i a t i o n t r a n s m i t t e d through a 5 Mm A l f o i l i s exp-(M(X)x) = 1 0 ~ 1 9 . The e f f e c t of h i g h Z m a t e r i a l i n the v i c i n i t y of the L a v a l n o z z l e has been observed i n the e x p e r i m e n t s . A f a c t o r of 2 decrease i n the x - ray s i g n a l i n t e n s i t y was seen to occur when the s t a i n l e s s s t e e l jaws of the n o z z l e were c o v e r e d w i t h P l e x i g l a s . I V - 1 d : X - r a y Measurements at H i g h Laser I n t e n s i t y An e x t e n s i v e s e r i e s of o b s e r v a t i o n s of the 5 T o r r N 2 j e t x-ray e m i s s i o n were made u s i n g aluminum f o i l f i l t e r s w i t h the four c h a n n e l d e t e c t o r . The e f f e c t i v e n e s s of the e l e c t r o n s c r e e n i n g by the magnets was checked by o b s e r v i n g the r e l a t i v e x - r a y s i g n a l s be fore and a f t e r r o t a t i n g the magnet assembly by 90 degrees . Upon removal of the magnets from the d e t e c t o r the p h o t o m u l t i p l i e r tube s i g n a l s were observed to s a t u r a t e , i n d i c a t i n g t h a t the i n c l u s i o n of the magnets i s n e c e s s a r y . L i g h t l e a k s through the f o i l s were checked by the replacement of the magnet assembly by a 6 cm t h i c k c l e a r P l e x i g l a s c y l i n d e r which e f f e c t i v e l y a t t e n u a t e s e l e c t r o n s and x - r a y s but i s t r a n s p a r e n t to the v i s i b l e l i g h t . The dynamic range of the p h o t o m u l t i p l i e r s was extended by the i n c l u s i o n of g e l a t i n n e u t r a l d e n s i t y f i l t e r s a t the ends of the o p t i c a l f i b e r s l e a d i n g to the 79 p h o t o m u l t i p l i e r s . C a l i b r a t i o n of the d e t e c t o r s r e l a t i v e to one another was performed whenever the n e u t r a l d e n s i t y ( N.D.) a t t e n u a t i o n was changed. T h i s was done by having the same t h i c k n e s s of f o i l on each channel and o b s e r v i n g the x-ray p u l s e s from the plasma. Because of the f l u c t u a t i o n i n x-ray i n t e n s i t y from shot to shot, many x-ray s i g n a l s were recorded and analyzed f o r a p a r t i c u l a r l a s e r energy. A l l the aluminum f o i l data c o n s i s t e n t l y y i e l d e d an e l e c t r o n temperature of 2 + 0.4 keV i r r e s p e c t i v e of i n c i d e n t l a s e r energy. The o b s e r v a t i o n s were extended to i n c l u d e the t r a n s m i s s i o n s through copper f o i l s as w e l l . The data are summarized i n F i g u r e IV-5, where the e r r o r Figure IV-5 X-ray transmission data from aluminum (o) and copper (A) f o i l s . bars denote the standard d e v i a t i o n of the mean of ten or more sh o t s . The copper f o i l data are p l o t t e d on the same graph by 80 c a l c u l a t i n g the c u t - o f f t h i c k n e s s of the copper f o i l and p l a c i n g each copper datum on the a b s c i s s a c o r r e s p o n d i n g to an e q u i v a l e n t t h i c k n e s s of aluminum f o i l r e q u i r e d to a c h i e v e the same c u t - o f f e n e r g y . The c u t - o f f energy i s d e f i n e d as tha t photon energy at which the a t t e n u a t i o n of t r a n s m i t t e d x - r a y s i s 1/e i . e . , M(E) • x = 1, where x i s the f o i l t h i c k n e s s and M(E ) i s the photon energy dependent mass a b s o r p t i o n c o e f f i c i e n t of the f o i l m a t e r i a l . The observed temperature of 2 keV i s much too h i g h to be the therma l temperature of the bu lk of the e l e c t r o n s i n compar i son w i t h o t h e r C 0 2 l a s e r - p l a s m a exper iments u s i n g s i m i l a r i r r a d i a n c e . The r e s u l t s from o ther i n v e s t i g a t o r s 7 " ' 7 2 ' 9 2 i n d i c a t e t h a t a lower temperature i s o b t a i n e d fo r f o i l s w i t h c u t - o f f e n e r g i e s i n the range 2 keV and l o w e r . T h u s , a s e r i e s of measurements were taken u s i n g t h i n b e r y l l i u m f o i l s from 15 to 90 Mm t h i c k n e s s . The data are d i s p l a y e d i n F i g u r e IV-6 where i t i s seen t h a t the temperature now i s about 300 eV. I f the temperature were 1 keV or more, then the t r a n s m i t t e d x - r a y i n t e n s i t i e s through the f o i l s i n t h i s range would be r o u g h l y i d e n t i c a l . The data shown i n F i g u r e i v - 6 are averaged from 26 l a s e r shots where the i n c i d e n t energy ranged from 5 to 8 J o u l e s and no d e t e c t a b l e s c a l i n g of the temperature w i t h l a s e r energy was o b s e r v e d . T h i s i s p a r t l y due t o the f a c t tha t below 5 J o u l e s i n c i d e n t l a s e r energy no o b s e r v a b l e s i g n a l s are t r a n s m i t t e d through the t h i n n e s t Be f o i l s . S i m i l a r behav iour i n x - r a y r a d i a t i o n was observed by B u c h l 9 2 u s i n g a neodymium g l a s s l a s e r on a d e u t e r i u m i c e t a r g e t . A s t r i k i n g f e a t u r e of the b e r y l l i u m f o i l da ta i s t h a t t h e i r s i g n a l i n t e n s i t i e s are not r a d i c a l l y d i f f e r e n t from those seen through the t h i n n e s t A l f o i l s . T h i s 81 Figure IV-6 R e l a t i v e transmitted x-ray i n t e n s i t i e s from b e r y l l i u m f o i l s . s u g g e s t s t h a t t h e r e i s a s u b s t a n t i a l f r a c t i o n o f s u p r a t h e r m a l e l e c t r o n s c h a r a c t e r i z e d b y a 2 k e V t e m p e r a t u r e . T h e d a t a f r o m a l l t h e f o i l s a r e s u m m a r i z e d i n F i g u r e l v - 7 w h e r e t h e r e l a t i v e i n t e n s i t i e s a r e a l l p l o t t e d v e r s u s t h e c u t -o f f e n e r g y o f t h e v a r i o u s f o i l a b s o r b e r s . S u c h a p l o t o f r e l a t i v e t r a n s m i t t e d i n t e n s i t i e s v e r s u s t h e c u t - o f f e n e r g y a s s u m e s a s i n g l e t e m p e r a t u r e M a x w e l l i a n d i s t r i b u t i o n o f e l e c t r o n s . T h e d a t a p r e s e n t e d i n F i g u r e I V - 7 a r e v e r y s i m i l a r i n f o r m t o t h a t o b t a i n e d by P e p i n e t a l . 7 3 a n d M i t c h e l l e t a l . 9 3 f o r C 0 2 l a s e r p r o d u c e d p l a s m a s . T h e q u e s t i o n a r i s e s w h e t h e r l i n e e m i s s i o n f r o m h i g h l y i o n i z e d n i t r o g e n may c a u s e a s e r i o u s e r r o r i n t h e m e a s u r e m e n t o f t h e e l e c t r o n t e m p e r a t u r e t h r o u g h t h e b e r y l l i u m f o i l s . C a l c u l a t i o n s s h o w n e x p l i c i t l y i n A p p e n d i x A r e v e a l t h a t t h e e m i s s i o n o f t h e Lyman s e r i e s o f N V I a n d N V I I i o n s h a s t o be i n c r e a s e d b y a f a c t o r o f 1 0 3 i n o r d e r f o r t h e o b s e r v e d t r a n s m i t t e d i n t e n s i t i e s t o g i v e a n u n d e r e s t i m a t e o f t h e 82 temperature by a f a c t o r of two. T h e r e f o r e , x - r a y l i n e e m i s s i o n from n i t r o g e n i s c o n s i d e r e d not t o a f f e c t the temperature o b t a i n e d from the b e r y l l i u m f o i l s . In summary, the x - ray data sugges t s t h a t the plasma can be c h a r a c t e r i z e d by two t e m p e r a t u r e s . The lower temperature of 300 eV i s measured i n the so f t x - r a y range of 1 - 2 keV w h i l e the 2 keV tempera ture i s measured i n the range of 3 - 18 keV. The measured i n t e n s i t i e s through the b e r y l l i u m f o i l s a re m a i n l y due t o the s o f t photons e m i t t e d from the c o l d e r t h e r m a l e l e c t r o n s which s h o u l d be c o n s i d e r a b l y more abundant than any supra thermal e l e c t r o n s . The aluminum and copper f o i l s t r a n s m i t o n l y the harder photons from the h o t t e r plasma and thus s i g n a l i n t e n s i t i e s from these f o i l s s h o u l d be c o r r e s p o n d i n g l y s m a l l e r than those obse rved from the b e r y l l i u m f o i l s . The shape of the t r a n s m i t t e d i n t e n s i t i e s v e r s u s c u t - o f f energy thus c o n t a i n s 83 i n f o r m a t i o n of the r e l a t i v e p o p u l a t i o n s of e l e c t r o n s of d i f f e r e n t t e m p e r a t u r e s . T h i s t o p i c w i l l be f u r t h e r t r e a t e d i n Chapter V . The p l a u s i b i l i t y of the observed e l e c t r o n tempera tures can be a s c e r t a i n e d by the knowledge of the thermal energy content of the p l a sma . Thus , s e v e r a l t e c h n i q u e s were used to o b t a i n a f i g u r e for the energy a b s o r p t i o n of the plasma and these r e s u l t s a re r e p o r t e d i n the f o l l o w i n g s e c t i o n s of t h i s c h a p t e r . I V - 2 : O p t i c a l I n v e s t i g a t i o n of the Plasma Expans ion T h i s s e c t i o n d e s c r i b e s the r e s u l t s o b t a i n e d from time r e s o l v e d s t u d i e s of the plasma expans ion u s i n g the image c o n v e r t e r s t r e a k camera d e s c r i b e d i n Chapter I I I . O b s e r v a t i o n s of the c h r o n o l o g y of the luminous f r o n t i s r e p o r t e d and the r e s u l t s a re compared w i t h s t a n d a r d b l a s t wave t h e o r y to o b t a i n a f i g u r e fo r the absorbed e n e r g y . To i n v e s t i g a t e the expanding plasma d e n s i t y s t e p wi thout a m b i g u i t y , the shadowgraph method i s subsequent ly a p p l i e d . An encompassing s e r i e s of o p t i c a l i n v e s t i g a t i o n s were i n s p i r e d by the f e a s i b i l i t y of e x t r a c t i n g i n f o r m a t i o n on the p e r t i n e n t p h y s i c a l parameters of the plasma namely tempera ture , energy a b s o r p t i o n and e l e c t r o n d e n s i t y . Comparison of the plasma expans ion h i s t o r y w i t h r e s u l t s p r e d i c t e d by the t h e o r y of s e l f -s i m i l a r f low l eads to an e s t i m a t i o n of the absorbed l a s e r energy i n the plasma ( A r i g a and S i g e l 9 * , Leonard and M a y e r 9 5 , Basov et a l . 9 6 , Hughes and E l - F e r r a 9 7 ) . Once the absorbed energy i s de te rmined and the number of e l e c t r o n s i n the plasma are known 84 by i n t e r f e r o m e t r i c means, the e l e c t r o n temperature may be c a l c u l a t e d through an energy ba l ance (Geunther and P e n d l e t o n 9 8 ) . The expans ion i n the a x i a l d i r e c t i o n of a l a s e r - p r o d u c e d plasmas can be c h a r a c t e r i z e d , depending upon the e x p e r i m e n t a l c o n d i t i o n s , by a l a s e r d r i v e n d e t o n a t i o n wave 1 w i t h a f r o n t v e l o c i t y p r o p o r t i o n a l to ( J 0 / P o ) 0 , 3 3 > where J 0 i s the i n s t a n t a n e o u s absorbed f l u x d e n s i t y and p 0 i s the i n i t i a l gas d e n s i t y . The t r a n s v e r s e t h e r m a l l y d r i v e n r a d i a l l y symmetric expans ion of a l a s e r produced plasma i s expected to be d e s c r i b a b l e i n terms of a s imple b l a s t wave theory which r e l a t e s the f r o n t v e l o c i t y to the absorbed e n e r g y . I n i t i a l o b s e r v a t i o n s were t ime r e s o l v e d s t u d i e s of the r a d i a l expans ion of the luminous e m i s s i o n f r o n t of the plasma u s i n g the RCA image c o n v e r t e r s t r e a k camera mentioned i n Chapter I I I . S ince t h i s u n i t has a s m a l l o p t i c a l g a i n of 4, a f a i r l y wide s l i t of 2 .0 mm had to be u sed . T h i s r e s t r i c t s the tempora l r e s o l u t i o n of the s t r e a k r e c o r d s to about 1 n s . P r e l i m i n a r y t ime r e s o l v e d i n t e r f e r o g r a m s and shadowgraphs taken by H i l k o for the 110 um t h r o a t j e t r e v e a l e d a r o u g h l y s p h e r i c a l plasma c o n f i n e d w i t h i n the gas j e t boundar ie s and e v o l v i n g i n t o an e l l i p s o i d . A c c o r d i n g l y , a 0.5 mm s l i c e of the c e n t e r of the j e t t a r g e t was imaged onto the s t reak camera s l i t s . The sweep r a t e used for the v i s i b l e e m i s s i o n f r o n t h i s t o r i e s was 6.9 ns/cm a l l o w i n g the luminous f r o n t to be f o l l o w e d fo r a p p r o x i m a t e l y 50 n s . A r e p r e s e n t a t i v e s t reak photograph of the luminous f r o n t and i t s accompanying r - t t r a c e i s shown i n F i g u r e I V - 8 . A l l s t r eak photographs taken of the luminous f r o n t show tha t when the plasma f i r s t becomes luminous i t a l r e a d y has a f i n i t e i n i t i a l 85 (a) (b) (mm) | 1 1 ' 7 Figure IV-8 Luminous front streak photograph (a) photograph (b) r-t plot for (a) s i z e o f a b o u t 400 * i m . The p e r i o d i c i n t e n s i t y m o d u l a t i o n s i n t h e s t r e a k p h o t o g r a p h a r e a n a r t i f a c t o f t h e s t r e a k c a m e r a a p p a r a t u s a n d a r e t h o u g h t t o be d u e t o be some " r i n g i n g " n o i s e s u p e r i m p o s e d u p o n t h e a p p l i e d v o l t a g e p u l s e s . T h e o b s e r v e d i n i t i a l v e l o c i t i e s a r e c o n s t a n t f o r t h e f i r s t 10 n s o r s o a n d a r e n o t s t r o n g l y c o r r e l a t e d w i t h t h e i n c i d e n t l a s e r e n e r g y . T h e a v e r a g e v a l u e f o r t h e i n i t i a l v e l o c i t y i s 6 x 1 0 s c m / s e c . By 20 n s o r e a r l i e r a f t e r t h e s t a r t o f t h e v i s i b l e s t r e a k , t h e r a d i a l g r o w t h o f t h e l u m i n o u s f r o n t s l o w s d r a m a t i c a l l y , l i m i t i n g t h e e x p a n s i o n t o 2 mm r a d i u s o r l e s s . Ahmed e t a l . " o b t a i n e d s i m i l a r s t r e a k p h o t o g r a p h s f o r r u b y l a s e r p r o d u c e d s p a r k s i n 4 a t m o s p h e r e s o f He g a s . T h e r a d i a l g r o w t h t h e r e a l s o s t o p p e d c o m p l e t e l y a f t e r 50 n s o f i n i t i a l b r e a k d o w n . I n c o r p o r a t i o n o f t h e e f f e c t s o f i o n i z a t i o n i n t o t h e t h e o r y f o r t h e t e m p o r a l r a d i u s o f c y l i n d r i c a l b l a s t w a v e s r e s u l t e d i n t h e o r e t i c a l p r o f i l e s t h a t a l s o s h o w e d t h e l i m i t t o t h e r a d i a l g r o w t h . I n t h e s t r o n g s h o c k r e g i o n o f p l a s m a e x p a n s i o n , t h e w e l l 86 known b l a s t wave s c a l i n g a p p l i e s ( Z e l ' d o v i c h and R a i z e r 1 0 0 ) 0.2 0.4 R( t ) = A ( 7 ) ( E 0 / p o ) t where R( t ) i s the r a d i u s of the b l a s t w a v e , E 0 i s the energy d e p o s i t e d i n t o a p o i n t at R = 0 and t = 0, p 0 i s the ambient gas d e n s i t y and 0.2 75 A ( 7 ) = 1 67T (7+1) (7-1)(37+1) where 7 i s a c o n s t a n t = 1.4, an a p p r o x i m a t i o n v a l i d for s m a l l Mach numbers. T h i s r e l a t i o n s h i p can be used to o b t a i n space- t ime c u r v e s for v a r i o u s d e p o s i t e d l a s e r e n e r g i e s to which the e x p e r i m e n t a l s t r e a k r e c o r d s of the plasma expans ion can be compared. However, most i n v e s t i g a t i o n s of l a s e r produced plasma expans ions have shown b e t t e r agreement w i t h the theory of S a k u r a i 1 0 1 . T h i s theory t r e a t s the d i f f e r e n t i a l hydrodynamic mass, energy and momentum c o n s e r v a t i o n e q u a t i o n s and f i n d s s o l u t i o n s i n terms of a second o r d e r power s e r i e s i n the r e c i p r o c a l Mach number. T h i s theory i s e s p e c i a l l y a p p l i c a b l e for low Mach number shocks M < 4. A l s o , the d i m e n s i o n a l i t y of the b l a s t waves has been c o n s i d e r e d , s p h e r i c a l b l a s t waves s t a r t i n g from an i n i t i a l p o i n t s o u r c e , and c y l i n d r i c a l b l a s t waves from a l i n e s o u r c e . T h i s l a t t e r p r o p e r t y makes the theory a p p r o p r i a t e to l a s e r plasmas formed i n gases s i n c e they i n g e n e r a l , t end to be e l l i p s o i d a l i n shape. S a k u r a i g i v e s the g e n e r a l s o l u t i o n fo r b l a s t waves i n the form: 87 2 c (C/U) ( R „ / R ) a +1 J 11 + X (C/U) + 0.5X (C/U) where a = 0 , 1 , 2 , c o r r e s p o n d i n g to p l a n e , c y l i n d r i c a l and s p h e r i c a l geometry r e s p e c t i v e l y , U i s the shock f r o n t v e l o c i t y , C i s the speed of sound i n the u n d i s t u r b e d f l u i d , R 0 i s a c h a r a c t e r i s t i c l e n g t h r e l a t e d to the energy of the e x p l o s i o n and J 0 and X are c o n s t a n t s . For the 5 T o r r N 2 j e t the above e q u a t i o n reduces to the c o n v e n i e n t form: where E 0 i s the energy d e p o s i t e d per u n i t l e n g t h . With the assumption tha t the l a s e r energy i s d e p o s i t e d i n a l e n g t h e q u a l to the depth of focus * 1 mm, the above f u n c t i o n was p l o t t e d as a f u n c t i o n of the d e p o s i t e d l a s e r energy and t ime i n F i g u r e IV-9. For c o m p a r i s o n , a s p h e r i c a l b l a s t wave curve i s superposed, i n F i g u r e I V - 9 . P l o t t i n g l o g - l o g p l o t s of the r - t t r a c e s of the luminous f r o n t s t r eak photographs i n d i c a t e s tha t the time dependence of the r a d i a l growth i s t 0 ' 5 r a t h e r than t ° * 4 . The r a d i u s t ime h i s t o r i e s of the plasma fo r v a r i o u s l a s e r input e n e r g i e s were compared w i t h the t h e o r e t i c a l b l a s t wave curves to y i e l d a r e l a t i o n s h i p between the i n p u t l a s e r energy and absorbed l a s e r e n e r g y . The r e s u l t s a re shown i n F i g u r e I V - 1 0 . The f i t of the t h e o r e t i c a l curves to the a c t u a l t ime h i s t o r i e s i s q u i t e rough but the g e n e r a l t r e n d i s t h a t the absorbed energy s c a l e s l i n e a r l y w i t h l a s e r energy and the f r a c t i o n of absorbed energy appears to be about 12%. A l t h o u g h cont inuum e m i s s i o n i s p r o p o r t i o n a l to the square 0.25 0.25 0.5 t R ( t ) = ( 7 / O . 8 8 ) (E / p ) o o 88 Figure IV-9 R-t curves from the Sakurai theory f o r various absorbed energies. Figure IV-10 Absorbed l a s e r energies as obtained from the comparison of luminous front streak photographs with Figure IV-9 . of the e l e c t r o n d e n s i t y , the luminous f r o n t does not n e c e s s a r i l y r e p r e s e n t the boundary of the p l a sma . The b l a s t wave theory a p p l i e s to the shock f r o n t t h e r m a l l y d r i v e n by the hot co re p la sma . T h u s , the shock f r o n t would be c h a r a c t e r i z e d by a r e g i o n 8 9 of h i g h d e n s i t y i n comparison to the r e g i o n of ambient gas ahead of i t and plasma behind i t . The luminous f r o n t may very w e l l be j u s t an i o n i z a t i o n f r o n t and not be r e p r e s e n t a t i v e of the t r u e shock wave boundary . T h i s i s indeed the case for many l a s e r plasmas which are i n i t i a l l y bounded by a s u p e r s o n i c heat wave or thermal wave, which l a t e r as a consequence of the decrease i n energy f l u x d e n s i t y , change to a subson ic heat wave w i t h the a s s o c i a t e d l e a d i n g shock wave. That the luminous f r o n t may m i s r e p r e s e n t the shock f r o n t of the b l a s t wave has been observed by o ther i n v e s t i g a t o r s (see e . g . Da iber and T h o m s o n 1 0 2 , A l c o c k et a l . 1 0 3 ) . A l c o c k et a l . per formed t ime r e s o l v e d s t u d i e s of l a s e r sparks by s c h l i e r e n photography , s c a t t e r e d ruby l a s e r l i g h t from the p lasma, and by luminous e m i s s i o n from the plasma as above . A l l t h r e e methods y i e l d the same i n i t i a l t ime dependence of the r a d i a l g rowth , but a f t e r the peak of the l a s e r p u l s e , the luminous f r o n t has a t ime dependence of t ° a s p r e d i c t e d fo r a s p h e r i c a l b l a s t wave, whereas the shock f r o n t as v i s u a l i z e d by s c h l i e r e n time r e s o l v e d photography has a t ime dependence of t 0 ' 2 1 . Da iber and Thomson on the o ther hand f i n d tha t the luminous f r o n t lagged b e h i n d the shock f r o n t fo r t h e i r e x p e r i m e n t a l c o n d i t i o n s . Thus , c o n s i d e r a b l e i n f o r m a t i o n can be o b t a i n e d i n t ime r e s o l v e d shadography of the p la sma. The shadowgraph method works on the p r i n c i p l e tha t l i g h t p a s s i n g through a g r a d i e n t i n an index of r e f r a c t i o n i s d e f l e c t e d to a s u f f i c i e n t ex tent to make d e n s i t y changes o b s e r v a b l e . The shadowgraph method measures the second d e r i v a t i v e of the r e f r a c t i v e index and t h e r e f o r e makes v i s i b l e o n l y those p a r t s of the medium where the d e n s i t y 90 g r a d i e n t s through which the l i g h t t r a v e r s e s change very r a p i d l y as i n the case of shock waves. The arrangement fo r t ime r e s o l v e d shadowgraphy has been shown i n the p r e v i o u s c h a p t e r . S ince the l u m i n o s i t y of the spark gap l i g h t source i s comparable to tha t of the p lasma, the spark l i g h t source i s focus sed through a s m a l l a p e r t u r e whereas the plasma l i g h t i s de focus sed as i m p l i e d i n the rays of F i g u r e 1 1 1 - 1 . Such a method has been used s u c c e s s f u l l y by Cheng et a l . 1 0 4 i n the s tudy of e x p l o d i n g w i r e s . The f e a s i b i l i t y of the method was t e s t e d by u s i n g a f r e e l y expanding s o n i c He j e t i n a i r . When the o v e r p r e s s u r e i n t h i s j e t n o z z l e i s j u s t s l i g h t l y above a t m o s p h e r i c , a l ong s teady stream of He gas emanates from the e x i t . Such a j e t i n a i r was observed to produce a comparable f r i n g e s h i f t i n an i n t e r f e r o m e t e r s e t - u p as the plasma i t s e l f . The s t r e a k shadowgraph of t h i s j e t showed i t s e l f as a dark band, so t h a t t h i s method was then known to be u s e f u l to v i s u a l i z e the plasma e x p a n s i o n . Demonstrated g r a p h i c a l l y by H i l k o 3 1 i s the f a c t tha t the s i z e of the shadowgraph images a re somewhat dependent on the p o s i t i o n of the o b j e c t p l ane of the imaging o p t i c s . The f u r t h e r the o b j e c t p l ane i s away from the r e f r a c t i n g o b j e c t , then the l a r g e r i s the r e s u l t i n g shadow. T h i s i s because s m a l l ray d e f l e c t i o n s due to r e g i o n s of c o m p a r a t i v e l y low e l e c t r o n d e n s i t y become apparent i n a much more d i s t a n t o b j e c t p l a n e . However, these e f f e c t s are o n l y p r e v a l e n t fo r those phase o b j e c t s which have a smoothly v a r y i n g d e n s i t y w i t h r a d i u s . For the case of an expanding l a s e r plasma bounded by a dense shock f r o n t t h i s problem s h o u l d not be s i g n i f i c a n t . The o b j e c t p l ane for the p a r t i c u l a r s e t - u p used was 10 cm beh ind the p la sma. The 91 s e n s i t i v i t y of the arrangement was e s t i m a t e d by g e o m e t r i c a l o p t i c s c o n s i d e r a t i o n s 3 1 and the i n d i c a t i o n i s tha t e l e c t r o n d e n s i t i e s g r e a t e r than 0.11 n s h o u l d be v i s i b l e . cr A sample s t r e a k shadow photograph and i t s accompanying r - t d iagram are shown i n F i g u r e I V - 1 1 . The two dark bands a c r o s s the Figure IV-11 Time-resolved shadowgraohy (a) Sample shadow streak photograph (b) R-t plot for (a) s t r e a k photograph are taken to r e p r e s e n t the boundary of the expanding shock wave. The v e r t i c a l s t r i a t i o n s a re an a r t i f a c t of the appara tus and are to be i g n o r e d . The fundamental d i f f e r e n c e between the shadowgraphs and the luminous f r o n t s t r e a k photographs i s t h a t i n the shadowgraphs, the plasma boundary i s seen t o grow from a p o i n t . The shadowgraphs a l s o show a r a p i d l y approached l i m i t t o the growth of the plasma at about 20 n s . The shadowgraphs i n d i c a t e a maximum r a d i a l ex tent about 40% s m a l l e r than those observed for luminous f r o n t s t r e a k s fo r a s i m i l a r l a s e r input e n e r g y . 92 The compar i son of the b l a s t wave c u r v e s w i t h the shadowgraph r - t t r a c e s r e v e a l the absorbed energy as a f u n c t i o n of i n c i d e n t l a s e r energy shown i n F i g u r e I V - 1 2 . The e r r o r bars Figure IV-12 Absorbed energies obtained by ana lys i s of the shadow streak photographs of the plasma expansion. come from the judgement of the q u a l i t y of the f i t of the e x p e r i m e n t a l to t h e o r e t i c a l curve s and measurement e r r o r from the s t r e a k p h o t o g r a p h s . The c o n c l u s i o n of t h i s shadowgraph compar i son w i t h the b l a s t wave c u r v e s i s t h a t the energy a b s o r p t i o n i s l i n e a r w i t h i n c i d e n t l a s e r energy but t h a t the energy a b s o r p t i o n now i s about 5% of the i n p u t l a s e r e n e r g y . 93 IV-3 : B l a s t waves w i t h heat t r a n s f e r A l t h o u g h the a p p l i c a t i o n of s t a n d a r d b l a s t wave theory i s r e l a t i v e l y common p r a c t i c e i n many a n a l y s e s of l a s e r spark e x p e r i m e n t s , i t i s r a t h e r l i m i t e d i n i t s u s e f u l n e s s i n p r o v i d i n g i n f o r m a t i o n on plasma parameter s . In a d d i t i o n , i t s a p p l i c a b i l i t y to l a s e r produced plasmas i s somewhat q u e s t i o n a b l e for reasons d i s c u s s e d below. A c c o r d i n g l y , a d i g r e s s i o n i s made i n t h i s s e c t i o n where the theory of b l a s t waves w i t h heat t r a n s f e r i s i n t r o d u c e d . T h i s p r o v i d e s the background fo r a more u s e f u l a n a l y s i s of the s t reak photographs of the r a d i a l expans ion tha t a l l o w s the d e t e r m i n a t i o n of the e l e c t r o n temperature as w e l l as the absorbed energy . Comparison of e x p e r i m e n t a l t ime h i s t o r i e s of plasma expans ion w i t h b l a s t wave curve s must be approached w i t h some c a u t i o n , s i n c e the b l a s t wave theory makes s e v e r a l a s sumpt ions which may not p e r t a i n a t a l l to the case of l a s e r produced p la smas . B l a s t wave t h e o r y e v o l v e s from the s e l f s i m i l a r f low as sumpt ion which r e q u i r e s t h a t the flow v a r i a b l e s are a l l f u n c t i o n s of ( r , t ) thus e n a b l i n g the s o l u t i o n of the se t of hydrodynamic d i f f e r e n t i a l e q u a t i o n s of energy , mass and momentum c o n s e r v a t i o n . T h i s a s sumpt ion c o r r e s p o n d s to r e q u i r i n g t h a t the p r o f i l e s of the p h y s i c a l q u a n t i t i e s r e t a i n t h e i r i n i t i a l form and so remain s e l f - s i m i l a r to t h e m s e l v e s . S e c o n d l y , a d i a b a t i c f low i s assumed, i m p l y i n g t h a t the energy t r a n s f e r i s o n l y due to shock h e a t i n g and the medium i s not supposed to l o s e , t r a n s m i t or r e t a i n energy by c o n d u c t i o n or r a d i a t i o n . T h e r e f o r e , t h i s a s sumpt ion l i m i t s the a p p l i c a b i l i t y of b l a s t wave theory to 9 4 those cases where the c e n t e r temperature i s r e l a t i v e l y low and much m a t e r i a l i s i n i t i a l l y a c c e l e r a t e d . T h i r d l y , the b l a s t wave i s bounded by a shock wave i n the u s u a l t h e o r y . However, as mentioned p r e v i o u s l y , many i n v e s t i g a t i o n s have shown tha t the luminous f r o n t has no resemblance to a shock f r o n t i n i t i a l l y and does not c o i n c i d e w i t h the shock f r o n t l a t e r . Thus , the t r a n s p o r t p roce s se s i n the gas are n e g l e c t e d , and the s p e c i f i c heat r a t i o 7 and the degree of i o n i z a t i o n are assumed to be c o n s t a n t . N o n e t h e l e s s , s e l f - s i m i l a r b l a s t wave theory a d e q u a t e l y d e s c r i b e s the l a t e s tages of c e r t a i n l a s e r spark exper iments (Guenther and P e n d l e t o n 9 8 , Buch l et a l . 1 0 5 , Lampis and B r o w n 1 0 6 ) . However, i t i s not a p p l i c a b l e a t the space or t ime o r i g i n of a r e a l l a s e r - p r o d u c e d b l a s t wave s i n c e i t assumes an i n f i n i t e co re temperature and z e r o c o r e d e n s i t y . For l a s e r sparks i n low p r e s s u r e gas where a l a r g e amount of energy i s d e p o s i t e d i n a shor t span of t i m e , the c e n t e r c o r e has a h i g h i n i t i a l temperature so t h a t r a d i a t i v e and c o n d u c t i v e heat t r a n s f e r i n v a l i d a t e the above as sumpt ions of s e l f - s i m i l a r flow b l a s t wave t h e o r y . In such c a s e s , the heat t r a n s f e r dominates r e s u l t i n g i n a s u p e r s o n i c heat wave which at a l a t e r t ime i n the e v o l u t i o n of the l a s e r spark changes to a subson ic heat wave d r i v i n g a shock wave ahead of i t . T h i s sequence i n the e v o l u t i o n of a l a s e r spark i s the consequence of a r e d u c t i o n of the energy f l u x d e n s i t y i n the plasma due to the i n c r e a s e of i t s s u r f a c e a rea w i t h t i m e . The t h e o r y fo r b l a s t waves w i t h heat t r a n s f e r has been deve loped by A h l b o r n and A r i g a 1 0 7 . In t h i s mode l , the b l a s t wave expands due t o heat t r a n s f e r d e s c r i b e d by a s u r f a c e r a d i a t i o n 95 i n t e n s i t y . Step wave p r o f i l e s for a l l p h y s i c a l q u a n t i t i e s by narrow p lane f r o n t s are assumed so that i n t e g r a l " jump" c o n s e r v a t i o n e q u a t i o n s of mass, momentum and energy app ly a c r o s s a l l the f r o n t s and determine the type of f l ow. A subson ic heat f r o n t w i l l be preceded by a compress ion shock f r o n t and a s u p e r s o n i c heat f r o n t w i l l be f o l l o w e d by a r a r e f a c t i o n wave. T h i s r e s u l t i s r e a l i z e d from the p h y s i c a l r e a s o n i n g tha t for heat waves, the f r o n t v e l o c i t y i s g i v e n by v = w/h p a b a where W i s the energy f l u x , h i s the e n t h a l p y of the r e g i o n b beh ind the heat wave f r o n t and p i s the d e n s i t y of the medium a i n t o which the heat wave t r a v e l s . The v e l o c i t y of a s t r o n g shock f r o n t ahead of a heat f r o n t i s (Ahlborn and L i e s e 1 0 8 ) 0.5 v = (const h ) , s b so t h a t no shock can precede the heat wavefront i f v > v or i f a s 2/3 h < cons t (W/p ) b a T h i s means t h a t once the e n t h a l p y of the plasma has decrea sed to a c e r t a i n l e v e l due to expans ion and a consequent decrease i n W, the shock wave w i l l over take the heat wave f r o n t . In the subsonic heat wave's frame of r e f e r e n c e , p a r t i c l e s en te r the h e a t i n g zone where they are heated and are s i m u l t a n e o u s l y a c c e l e r a t e d towards the energy s o u r c e , l e a v i n g the f r o n t w i t h 96 i n c r e a s e d v e l o c i t y as d e p i c t e d i n F i g u r e I V - 1 3 . T h i s causes a heot wove shock wove I T / T Q (b) (2) (a) P/PQ | w0 V f J P/Pa 0 r Fieure IV-13 Regions of a thermal b l a s t wave. t h r u s t which i s r e s p o n s i b l e fo r d r i v i n g a shock wave ahead of the heat wave. The p o i n t i n the plasma e v o l u t i o n where the shock compres s ion f r o n t and the heat expans ion f r o n t are c o i n c i d e n t i s known as a Chapman-Jouguet d e t o n a t i o n . T h e r e f o r e , the theory of b l a s t waves w i t h heat t r a n s f e r models a space- t ime h i s t o r y as shown i n F i g u r e I V - 1 4 . I n i t i a l l y , when the energy f l u x i s h i g h compared t o the e n t h a l p y h b t h e r e i s a s u p e r s o n i c heat wave ( i . e . , the i n t a k e v e l o c i t y v a i s s u p e r s o n i c w i t h r e s p e c t to the gas b e h i n d i t . When the r a d i u s has i n c r e a s e d to the p o i n t where h, = c o n s t (W/p ) 2 ^ 3 , a Chapman-Jouguet d e t o n a t i o n momentar i ly b a e x i s t s where the r a r e f a c t i o n wave head i n s i d e the hot c o r e has caught up t o the f r o n t and the heated p a r t i c l e s are l e a v i n g the f r o n t w i t h the speed of sound. As W decrea se s f u r t h e r , the subson ic mode of the heat wave then p e r s i s t s and the shock p e e l s 97 o f f . The c l a s s i c a l b l a s t wave t h e o r y i s expected t o d e s c r i b e the space- t ime c h a r a c t e r i s t i c s of t h i s heat wave i n the l a t e s tages p r o v i d e d t h a t the heat t r a n s f e r i s f l u x l i m i t e d . By f l u x l i m i t e d t r a n s f e r i t i s meant t h a t each p a r t i c l e t r a v e l i n g at i t s thermal speed T 1 ^ 2 c a r r i e s i t s i n t e r n a l energy through the hot s u r f a c e t o p r o v i d e the f l u x W ^ T 3 ^ 2 . Many exper iments f i n d the t ime dependence of the r a d i a l expans ion t o be t ° ' 2 i n s t e a d of the s t andard p r e d i c t e d v a l u e of t 0 ' * (see e . g . A l c o c k e t a l . 1 0 3 ) . T h i s can be e x p l a i n e d by a b l a s t wave t h a t i s d r i v e n by heat t r a n s f e r due to b lackbody r a d i a t i o n W « T * ( A h l b o r n 1 0 8 ) . The a n a l y s i s by A h l b o r n and A r i g a shows t h a t f o r the e a r l y s tages of therma l b l a s t waves, the r a d i u s r w i l l v a r y as 1/(3a-2) r a t , and at l a t e r t imes a f t e r the C - J p o i n t i t s h o u l d vary as 9 8 1/(a+1) r « t , where a i s the exponent t h a t c h a r a c t e r i z e s the heat " r e s p o n s e " of the plasma i . e . , W = c o n s t h a . Thus a s t r e a k r e c o r d of the r a d i a l expans ion c o n v e r t e d i n t o a l o g r - l o g t p l o t s h o u l d r e v e a l r e g i o n s w i t h two d i f f e r e n t s l o p e s c o r r e s p o n d i n g to d i f f e r e n t modes of plasma e x p a n s i o n . The i n s t a n t of t ime tha t the s l o p e changes t=t . , i n d i c a t e s a Chapman-Jouguet d e t o n a t i o n . c j The i d e n t i f i c a t i o n of t h i s p o i n t i n t ime i n the h i s t o r y of the plasma has a p a r t i c u l a r s i g n i f i c a n c e s i n c e the l o c a l p r e s s u r e P c j , sound speed a and energy c o n t e n t E 0 can be o b t a i n e d from a measurement of the f r o n t v e l o c i t y V , and r a d i u s r c j c j the f o l l o w i n g r e l a t i o n s a p p l y : At t = t c j ^0.5 g P b b g V b c j g +1 b P = p V / ( g +1) b a c j b 87T 3 9 2 b / ( g - 1) b 3 2 2 3 r p V = C p V r c j a c j a c j c j The s u b s c r i p t b denotes the r e g i o n behind the f r o n t , the s u b s c r i p t a denotes the r e g i o n ahead of the f r o n t , p i s the d e n s i t y and g i s the e n t h a l p y c o e f f i c i e n t d e f i n e d by 99 h P g = = , +  h - P / p 3 / 2 P + I n E i . i i O n c e a l o g r - l o g t p l o t o f a s h a d o w s t r e a k r e c o r d i d e n t i f i e s t h e C h a p m a n - J o u g u e t p o i n t , t h e v e l o c i t y a t t=tQ^, V a n d t h e r a d i u s o f t h e p l a s m a a t t h a t t i m e r . c a n be u s e d t o f i n d t h e d e p o s i t e d l a s e r e n e r g y a n d p r e s s u r e o f t h e p l a s m a a t t h i s t i m e a n d c o n s e q u e n t l y t h e t e m p e r a t u r e . T h e s t r e a k p h o t o g r a p h s ( s u c h a s shown i n F i g u r e I V - 1 1 ) , w e r e e n l a r g e d s u f f i c i e n t l y t o m i n i m i z e t h e m e a s u r e m e n t e r r o r a n d r - t a n d l o g l o g p l o t s w e r e made a e x a m p l e o f w h i c h i s s h o w n i n F i g I V - 1 5 . T h e a n a l y z e d s t r e a k p h o t o g r a p h s a l l s h o w e d t h e same g e n e r a l f e a t u r e s i n d i c a t i n g a b r e a k i n t h e s l o p e a t t = 3 n s a f t e r t h e i n i t i a l b r e a k d o w n . T h e i n i t i a l a v e r a g e s l o p e u p t o 3 n s was 0 . 8 2 a n d a f t e r t h e C J p o i n t t h e a v e r a g e s l o p e was 0 . 5 4 , i n g o o d a g r e e m e n t w i t h t h e b l a s t wave t h e o r y o f S a k u r a i . T h e r ^ a n d V ^ v a l u e s f o r e a c h s h o t w e r e f o u n d b y t h e g r a p h i c a l m e t h o d i n d i c a t e d a n d t h e a b s o r b e d e n e r g y was c a l c u l a t e d u s i n g t h e e x p r e s s i o n f o r t h e C - J p o i n t . I t s h o u l d b e n o t e d t h a t t h i s c a l c u l a t i o n i s s u b j e c t t o c o n s i d e r a b l e i n a c c u r a c y s i n c e i t c o n t a i n s t h e s q u a r e o f t h e v e l o c i t y a n d t h e c u b e o f t h e r a d i u s . I n a d d i t i o n , t h e c o n s t a n t C c o n t a i n s t h e e n t h a l p y c o e f f i c i e n t g . F o r s h o r t t i m e s c a l e s w h e r e f u l l i o n i z a t i o n i s n o t r e a c h e d e v e n when t h e s h o c k v e l o c i t y i s h i g h , t h e n g = y = 1 . 4 , t h e a d i a b a t i c e x p o n e n t f o r a c o l d g a s . I f t i m e s c a l e s a r e l o n g e r s o t h a t t h e s h o c k e d g a s w i l l h a v e a t i m e t o r e a c h a n e q u i l i b r i u m d e g r e e o f i o n i z a t i o n t h a t i s c o n s i s t e n t w i t h t h e e n t h a l p y j u m p 100 ( a ) 0 2 A 6 ns Figure IV-15 Chapman-Jouguet a n a l y s i s of streak photographs (a) r - t p l o t from a shadowgraph streak (b) l o g - l o g p l o t f o r (a) showing the change i n slope a c r o s s t h e s h o c k , t h e n t h e v a l u e f o r g c a n be g i v e n a p p r o x i m a t e l y . C a l c u l a t i o n s w e r e p e r f o r m e d by A h l b o r n a n d A r i g a a n d show t h a t f o r s h o c k v e l o c i t i e s g r e a t e r t h a n 1 0 s c m / s e c , t h e v a l u e o f t h e c o n s t a n t C i s a b o u t 3 0 . I t i s n o t e d t h a t i f f u l l i o n i z a t i o n r i g h t a t t h e s h o c k f r o n t i s a s s u m e d , t h e n g = 1 . 6 7 a n d C = 9> w h e r e a s f o r a c o l d n e u t r a l g a s C = 1 2 . T h u s , t h e g r e a t e r p a r t o f t h e u n c e r t a i n t y l i e s i n t h e d e g r e e o f i o n i z a t i o n a t t a i n e d a t t h e s h o c k f r o n t . T h e a b s o r b e d e n e r g y E 0 a s o b t a i n e d u s i n g t h i s a n a l y s i s i s s h o w n i n F i g u r e I V - 1 6 t a k i n g t h e o p t i m i s t i c v a l u e o f 30 f o r C . T h e e r r o r b a r s i n t h i s g r a p h a r i s e f r o m t h e u n c e r t a i n t y i n t h e r c j a n d V c j v a l u e s w h i c h s t e m f r o m t h e u n c e r t a i n t y i n d e t e r m i n i n g t . b y g r a p h i c a l a n a l y s i s . 101 Figure IV-16 Absorbed energy obtained by an a l y s i s of shadow streak photographs. N o n e t h e l e s s , the i n d i c a t i o n i s t h a t the upper bound fo r the a b s o r p t i o n by t h i s method of a n a l y s i s i s 17% of the i n c i d e n t l a s e r e n e r g y . The p r e s s u r e de te rmined at the C - J p o i n t i s a more c o n f i d e n t e s t i m a t e s i n c e the e x p r e s s i o n used o n l y depends upon the f a c t o r l / ( g +1) so t h a t the u n c e r t a i n t y of the degree o f i o n i z a t i o n i s unimportant compared t o the a c c u r a c y of the measurements t h e m s e l v e s . Of i n t e r e s t here i s t o c a l c u l a t e the temperature and compare the r e s u l t i n g v a l u e s w i t h t h a t o b t a i n e d u s i n g the x - r a y d i a g n o s t i c s . Assuming the i d e a l gas law we have : P = nkT = n kT + n kT e e i i n k(Z+1)T e - f o r T = T = T Z e i ^ n kT f o r Z >3 e 102 For hydrodynamic motion one u s u a l l y expec t s t h a t Tg = , however, the e l e c t r o n - i o n e q u i l i b r a t i o n time tg^ as c a l c u l a t e d i n c h a p t e r II i s of the order of 100 ns so t h a t i n the shor t time s c a l e s of the l a s e r p u l s e and c e r t a i n l y up to the t ime of the C - J p o i n t , T g > T . T h e r e f o r e , the e l e c t r o n temperature fo r the plasma can be o b t a i n e d from the C - J p o i n t v e l o c i t i e s by the r e l a t ion P 2 T c j p V e = = 1 c j n k e B n k (g + 1 ) e B b The a n a l y s i s does not i n c o r p o r a t e the p o s s i b l e e f f e c t of hot e l e c t r o n s on the hydrodynamic m o t i o n . I t i s assumed t h a t the hot e l e c t r o n number i s s m a l l compared to the c o l d e l e c t r o n number d e n s i t y . The C - J v e l o c i t i e s V . . , a re de termined by the g r a p h i c a l method d e s c r i b e d p r e v i o u s l y , p^ i s the n i t r o g e n gas j e t d e n s i t y i n t o which the b l a s t wave expands and i t i s known from the i s e n t r o p i c flow r e l a t i o n s f o r a L a v a l n o z z l e as shown i n Chapter I I I . The e l e c t r o n d e n s i t y beh ind the b l a s t wave i s measured u s i n g a Mach-Zehnder i n t e r f e r o m e t e r t o be d e s c r i b e d s h o r t l y . The e n t h a l p y c o e f f i c i e n t g fo r n i t r o g e n of a few hundred eV i s 1.5 based on LTE c a l c u l a t i o n s 1 1 0 . The r e s u l t s fo r the measured V, v e l o c i t i e s are shown i n F i g u r e I V - 1 7 , where the e r r o r bar s r e p r e s e n t the u n c e r t a i n t y i n d e t e r m i n i n g t . The C J r e s u l t s thus i n d i c a t e tha t tempera tures of a few hundred eV are p l a u s i b l e therma l temperatures fo r the p lasma. A s c a l i n g of e l e c t r o n temperature w i t h i n c i d e n t l a s e r energy i s suggested by 103 o 5 I 5 I ib J Figure IV-17 El e c t r o n temperatures determined from C-J an a l y s i s of shadowgraph streaks. t h e d a t a o f F i g u r e I V - 1 7 a l t h o u g h t h e r e i s t o o m u c h s c a t t e r t o make a g o o d q u a n t i t a t i v e s t a t e m e n t f o r t h i s s c a l i n g . I V - 4 ; T i m e - r e s o l v e d i n t e r f e r o m e t r i c s t u d y o f p l a s m a e x p a n s i o n A M a c h - Z e h n d e r i n t e r f e r o m e t e r , t h e g e o m e t r y o f w h i c h i s s h o w n i n F i g u r e I V - 1 8 was s e t - u p t o o b t a i n a s e r i e s o f t i m e r e o l v e d i n t e r f e r o g r a m s f r o m w h i c h t h e d e n s i t y p r o f i l e s o f t h e p l a s m a c o u l d be o b t a i n e d . I n t e r f e r o g r a m s w e r e d i g i t i z e d a n d A b e l u n f o l d e d b y a c o m p u t e r r o u t i n e u t i l i z e d b y M c i n t o s h 8 3 . T i m e r e s o l u t i o n was p r o v i d e d b y u s i n g a n 80 p s d u r a t i o n r u b y l a s e r p r o b e beam t h a t c a n be a r b i t r a r i l y d e l a y e d w i t h r e s p e c t t o t h e C 0 2 l a s e r p u l s e , P e r h a p s t h e m o s t s t r i k i n g f e a t u r e o f t h e i n t e r f e r o g r a m s i s t h e r e s u l t t h a t t h e p l a s m a f i r s t f o r m s a t t h e t w o H e - N 2 i n t e r f a c e s . T h e s e t w o i n i t i a l l y s e p a r a t e p l a s m a i s l a n d s q u i c k l y m e r g e b y t h e t i m e o f t h e p e a k o f t h e l a s e r p u l s e , t h e f r o n t p l a s m a e x p a n d i n g m u c h m o r e r a p i d l y a n d t o 1 04 beamsplitter > mirror FigureIV-18 Geometry of the Mach-Zehnder in te r fe rometer . g r e a t e r d imens ions than the r e a r plasma i s l a n d . The a x i a l l e n g t h of the plasma by the end of the l a s e r p u l s e reaches s e v e r a l mm w i t h the f r o n t plasma expanding i n t o the ambient He . T y p i c a l a x i a l e l e c t r o n d e n s i t i e s at the t ime of the l a s e r p u l s e are around 2 x 1 0 1 8 cm" 3 . The reason why two plasma i s l a n d s form i s a t o p i c f o r some s p e c u l a t i o n . The obv ious s u g g e s t i o n i s f o c u s s i n g l e n s s p h e r i c a l a b e r r a t i o n e f f e c t s c a u s i n g the f o c a l r e g i o n t o have s e v e r a l r e g i o n s of h i g h i n t e n s i t y near the f o c a l p l ane (Evans and G r e y -M o r g a n 1 1 1 ) . However, l a s e r produced burnmarks on a l u m i n i z e d mylar f i l m s i n the v i c i n i t y of the f o c a l p l a n e showed no i n d i c a t i o n of such a b e r r a t i o n . Other p o s s i b l e e x p l a n a t i o n s may l i e i n the cascade theory of gas breakdown by l a s e r r a d i a t i o n where the i o n i z a t i o n t h r e s h o l d s a re c r i t i c a l l y dependent on the c o l l i s i o n f requency for momentum t r a n s f e r between e l e c t r o n s and atoms. T h i s might imply a Penning e f f e c t phenomena i . e . , there i s a l o w e r i n g of the breakdown t h r e s h o l d a t the H e - N 2 boundary . An i n t e r f e r o g r a m of the plasma taken s e v e r a l nanoseconds 105 a f t e r the s t a r t of the l a s e r p u l s e i s shown i n F i g u r e IV-19 toge ther w i t h the c a l c u l a t e d d e n s i t y p r o f i l e s from Abe l (Q) (b) mm 1.0-0.6-02-0 J 1 R 1(1 i jet boundary i i laser O05n. Figure IV-19 A sample late-time interferogram (a) Interferogram at 9.3 ns a f t e r the s t a r t of the l a s e r . (b) Abel unfolded density contours of Ca) i n v e r s i o n . From the A b e l u n f o l d e d c o n t o u r s the s i g n i f i c a n t r e s u l t i s t h a t an o f f a x i s h i g h d e n s i t y annulus forms which s t ay s c l o s e t o the expanding plasma boundary . T h i s i s a c l e a r i n d i c a t i o n of shock wave f o r m a t i o n . S e v e r a l s e r i e s of u n f o l d e d i n t e r f e r o g r a m s were s e l e c t e d and the e l e c t r o n d e n s i t i e s a t v a r i o u s a x i a l p o s i t i o n s were p l o t t e d as a f u n c t i o n of t ime as shown i n F i g u r e I V - 2 0 . The two a x i a l p o s i t i o n s used fo r these p l o t s are the f r o n t H e - N 2 i n t e r f a c e toward the f o c u s s i n g l e n s and the m i d d l e of the gas j e t t a r g e t . T h i s i n t e r f e r o m e t r i c s tudy r e v e a l s t h a t s t r o n g shock f o r m a t i o n i s l a r g e l y p r e v a l e n t and r e s t r i c t e d to the r e g i o n of the f r o n t H e - N 2 boundary , where the 106 expans ion of the plasma i s e v i d e n t l y more r a p i d than i n the middle of the i n t e r a c t i o n volume. To c o n f i r m the two mode na ture of plasma e x p a n s i o n , the outer edge of the plasma o b t a i n e d from the i n t e r f e r o g r a m s was p l o t t e d on l o g - l o g p l o t s shown i n F i g u r e I V - 2 1 . A l t h o u g h t h e r e i s some s c a t t e r i n the p o i n t s due t o da ta b e i n g taken from d i f f e r e n t l a s e r shot s of s l i g h t l y v a r y i n g i n c i d e n t l a s e r energy , the break i n the s l o p e i n d i c a t i n g a Chapman-Jouguet c o n d i t i o n i s q u i t e d r a m a t i c . The s l o p e a f t e r the C - J p o i n t i s 0.21 as expected fo r r a d i a t i v e t r a n s f e r . The a p p l i c a t i o n of the same procedure t o the midd le j e t r e g i o n c o r r o b o r a t e s the o b s e r v a t i o n s of the shadow s t r e a k photographs , namely t h a t the s lope a f t e r the C - J p o i n t i s 0.4 and the break i n the s l o p e o c c u r s a t about 3 n s . 107 Figure IV-21 Log-log r - t p l o t s of the plasma r a d i u s . (a) at the front He-^ boundary region (b) at the j e t target center I V - 5 : C - J A n a l y s i s of the F r o n t Plasma I s l a n d The r a d i a l expans ion of the f r o n t H e - N 2 i n t e r f a c e r e g i o n was i n v e s t i g a t e d more c l o s e l y u s i n g the Hamamatsu s t r e a k camera sys tem. Due to the h i g h g a i n of t h i s d e v i c e the s p a t i a l r e s o l u t i o n c o u l d be made as low as 5 Mm of the j e t w i d t h and the f i r s t 3-4 ns of the plasma expans ion s t r e a k e d . In t h i s c a s e , the plasma was b a c k l i g h t e d by the 7 ns output from a c a v i t y dumped ruby l a s e r . A sample shadow s t r e a k taken of the plasma a t the f r o n t H e - N 2 i n t e r f a c e i s shown i n F i g u r e I V - 2 2 . The i n i t i a l v e l o c i t i e s a t t h i s i n t e r f a c e are h i g h e r than those observed at the j e t t a r g e t c e n t e r , the average b e i n g 2 . 2 x 1 0 7 c m / s e c . S c h l i e r e n s t r e a k photography was a l s o performed u s i n g the arrangement d e s c r i b e d i n the p r e v i o u s c h a p t e r . The s c h l i e r e n 108 image was s c a n n e d a c r o s s t h e s t r e a k c a m e r a s l i t b y c a l i b r a t e d a d j u s t m e n t o f t h e t u r n i n g m i r r o r f o r t h e p r o b e r u b y l a s e r b e a m . O n l y a t t h e r e g i o n o f t h e f r o n t H e - N 2 i n t e r f a c e d i d a s t r i k i n g t w o f r o n t s t r u c t u r e a p p e a r c o n s i s t i n g o f t w o s e p a r a t i n g l u m i n o u s b a n d s t h a t c o r r e s p o n d t o t h e e x p a n d i n g c y l i n d r i c a l s h o c k wave d e n s i t y f r o n t . T h e l o g - l o g s p a c e - t i m e p l o t s o f s h o t s t a k e n i n t h i s r e g i o n r e v e a l a s h a r p C - J b r e a k i n t h e s l o p e w i t h t h e s l o p e b e i n g 0 . 2 a f t e r t h e C - J p o i n t a s shown i n F i g u r e I V - 2 3 . C - J wave a n a l y s i s m u s t be a p p l i e d somewhat d i f f e r e n t l y f o r t h i s d a t a s i n c e t h e e x a c t d e n s i t y i n t o w h i c h t h e e x p a n s i o n o c c u r s i n t h e r e g i o n o f t h e f r o n t H e - N 2 b o u n d a r y i s u n c e r t a i n . T h e e l e c t r o n t e m p e r a t u r e c a n s t i l l be o b t a i n e d by t h e f o l l o w i n g m e a n s . A t t h e t i m e o f t h e C - J p o i n t , t h e r a r e f a c t i o n wave h e a d i n s i d e t h e h o t p l a s m a h a s c a u g h t u p t o t h e f r o n t a n d t h e h e a t e d p a r t i c l e s a r e l e a v i n g t h e f r o n t w i t h t h e s p e e d o f s o u n d 2 2 P [v 2 1 a • V • g b = g c j b b b b . 9' +1. 109 Figure IV-23 Schlieren streak photography of the front He-^ interface region. (a) R-t plot of a sample schlieren photo (inset) (b) Log-log r-t plot for (a) The p r e s s u r e / d e n s i t y term i n the above e x p r e s s i o n can be approximated a s : a = g b b P L b Zn kT + n kT i e i i n m + n m i i e e = g (ZkT + kT )/m b e i i • g kT /2m f o r Z > 1 , b e p where the l a s t s t e p a r i s e s from the f a c t tha t the mass of the 1 10 ion rrij i s 2Zm_, fo r f u l l i o n i z a t i o n , m be ing the mass of the 1 P p p r o t o n . The combina t ion of t h i s l a s t e x p r e s s i o n w i t h the p r e v i o u s one r e s u l t s i n a s imple e x p r e s s i o n for the t e m p e r a t u r e : 2 2 kT = (Zm g / ( g +1 ) V ) e p b b c j The above e q u a t i o n , thus o n l y depends on one e x p e r i m e n t a l parameter 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 d i s c r e p a n c i e s i n g b . The temperature as a f u n c t i o n of the i n c i d e n t l a s e r energy i s p l o t t e d i n F i g u r e IV-24 u s i n g t h i s e q u a t i o n and the v e l o c i t i e s o b t a i n e d from the g r a p h i c a l a n a l y s i s of the Hamamatsu shadow and s c h l i e r e n p h o t o g r a p h s . The average observed e l e c t r o n eV 600 600-40CH 200 6 _ ft A 1 ' 1 0 2 4 6 8 Figure IV-24 E l e c t r o n temperatures obtained by C-J a n a l y s i s of the streak data for the front F e - ^ i n t e r f a c e r e g i o n . tempera ture of 360 + 80 ev i s i n good agreement w i t h t h a t o b t a i n e d f o r expans ion i n the midd le of the j e t t a r g e t . In summary, i t has been shown t h a t an e l e c t r o n temperature of 3 0 0 -111 500 eV i s o b t a i n e d from a C- J wave a n a l y s i s of s t r eak photographs which i s based on the i d e n t i f i c a t i o n of the i n s t a n t of t ime the plasma expans ion changes i t s t ime dependence. S tandard b l a s t wave a n a l y s i s of the s t r e a k photographs y i e l d s a r a t h e r poor f i t of the observed space- t ime h i s t o r i e s of the plasma and a low energy a b s o r p t i o n . The C - J a n a l y s i s p r e s e n t e d here suggests a h i g h e r energy a b s o r p t i o n then i s sugges ted by the b l a s t wave a n a l y s i s and i s about 17% of the i n c i d e n t l a s e r e n e r g y . Which v a l u e of a b s o r p t i o n i s more a c c e p t a b l e can be d e c i d e d by measurement of the energy a b s o r p t i o n , by an U l b r i c h t i n t e g r a t i n g s p h e r e , the r e s u l t s from which are r e p o r t e d s u b s e q u e n t l y . I V - 6 : Energy A b s o r p t i o n Measurements The a b s o r p t i o n of i n c i d e n t l a s e r r a d i a t i o n was i n v e s t i g a t e d by t ime i n t e g r a t e d measurements of the s c a t t e r e d , b a c k s c a t t e r e d , t r a n s m i t t e d and r e f l e c t e d i n f r a r e d r a d i a t i o n from the plasma u s i n g the e x p e r i m e n t a l arrangement d e s c r i b e d i n Chapter I I I . The b a c k s c a t t e r e d , t r a n s m i t t e d and i n c i d e n t e n e r g i e s are r o u t i n e l y moni tored on every l a s e r s h o t . For a b s o r p t i o n measurements, the s c a t t e r e d - r e f r a c t e d l a s e r r a d i a t i o n i s measured by the U l b r i c h t s p h e r e . The s i g n a l l e v e l s from the p h o t o m e t e r ' s Gen-Tec energy meter were q u i t e s t r o n g , r ang ing from 0.2 - 0.015 v o l t s and f ree from e l e c t r i c a l n o i s e . The s c a t t e r e d energy d e t e c t e d by the U l b r i c h t sphere was found to be l i n e a r as a f u n c t i o n of the i n c i d e n t l a s e r energy and i s shown i n F i g u r e IV-25 where i t i s seen t h a t h a l f of the l a s e r energy i s thus decoup led from the 1 1 2 pla sma . J 3 2 1 0 2 4 6 J F igure IV-25 Sca t te red- re f rac ted l a s e r energy measured by the U l b r i c h t photometer. The f r a c t i o n of l a s e r energy t h a t i s b a c k s c a t t e r e d and c o l l e c t e d back through the f o c u s i n g o p t i c s i s measured to s c a l e l i n e a r l y w i t h i n c i d e n t l a s e r energy and s a t u r a t e s at 10% for 10 J o u l e s i n c i d e n t e n e r g y . The t r a n s m i t t e d l a s e r energy f a l l s e x p o n e n t i a l l y w i t h i n c r e a s i n g l a s e r e n e r g y . S u b t r a c t i n g the b a c k s c a t t e r e d energy E b t o g e t h e r w i t h the r e f r a c t e d - s c a t t e r e d E g and t r a n s m i t t e d energy F from the i n c i d e n t l a s e r energy E l n f o r each shot produces a f i g u r e f o r the absorbed l a s e r energy E 0 E 0 = E - ( E + E + E i n b s t ) The r e s u l t s of the measurements u s i n g the above e q u a t i o n are shown i n F i g u r e I V - 2 6 . The absorbed energy i s (20 + 3)%, which i s i n good agreement w i t h average a b s o r p t i o n o b t a i n e d by 1 13 C - J a n a l y s i s o f t h e p l a s m a e x p a n s i o n . The a b s c i s s a i n t e r c e p t i s Figure IV-26 Absorbed l a s e r energy determined by TJlbricht sphere. a c o n s e q u e n c e o f t h e f a c t t h a t t h e b r e a k d o w n t h r e s h o l d t o f o r m a p l a s m a r e q u i r e s a b o u t o n e J o u l e o f l a s e r e n e r g y . C o m p a r i s o n o f t h e a b o v e a b s o r p t i o n r e s u l t s t o t h o s e o b t a i n e d i n o t h e r l a b o r a t o r i e s i s d i f f i c u l t s i n c e r e l a t i v e l y l i t t l e h a s b e e n p u b l i s h e d on t h e a b s o r p t i o n o f h i g h p o w e r C 0 2 l a s e r r a d i a t i o n . T h e e x p e r i m e n t s o f V i l l e n e u v e e t a l . 1 1 2 , G a r b a n - L a b a u n e e t a l . 6 5 , L i n d m a n 1 1 3 , N i s h i m u r a 6 7 a r e c o n c e r n e d w i t h t h e a b s o r p t i o n o f C 0 2 l a s e r r a d i a t i o n on s o l i d t a r g e t s a s a f u n c t i o n o f t h e a n g l e o f i n c i d e n c e o f p o l a r i z a t i o n t o v e r i f y t h e p r e s e n c e o f r e s o n a n c e a b s o r p t i o n . T h e y f i n d t y p i c a l a b s o r p t i o n f i g u r e s o f 30-45% f o r l a s e r i n t e n s i t i e s b e t w e e n 5 x 1 0 1 1 a n d 3 x 1 0 1 * W / c m 2 . T h e a m o u n t o f r a d i a t i o n s c a t t e r e d i n t o a 4TT s o l i d a n g l e i s a b o u t 60%. I n t h e s e e x p e r i m e n t s t h e s c a l e l e n g t h s a n d e l e c t r o n t e m p e r a t u r e s a r e s u c h t h a t i n v e r s e b r e m s s t r a h l u n g 1 14 a b s o r p t i o n account s for l e s s tha t 5% of the observed a b s o r p t i o n i n these c a s e s . Fabre and S t e n z 1 1 " i n v e s t i g a t e d an underdense preformed plasma i n the i n t e n s i t y range of 10 8 to 1 0 1 1 W/cm 2 and found the a b s o r p t i o n to be anomalous ly h i g h at 50% at 1 0 1 1 W / c m 2 . Inver se b remss t r ah lung a b s o r p t i o n becomes l e s s e f f i c i e n t w i t h h i g h e r i n t e n s i t i e s as the e l e c t r o n temperature i n c r e a s e s . O f f e n b e r g e r 3 3 at i n t e n s i t i e s of 1 0 1 2 W/cm 2 p u r p o r t s a 90% a b s o r p t i o n of which o n l y 40% can be accounted for by i n v e r s e b r e m s s t r a h l u n g . T h e r e f o r e , the observed a b s o p t i o n of 20% i s low compared to o ther exper iments e l s e w h e r e , but i t i s a l s o by the same measure an e n t i r e l y p l a u s i b l e f i g u r e . The i s s u e of how the measured degree of a b s o r p t i o n can be accounted for i n terms of the e l e c t r o n temperature w i l l be addres sed i n the f o l l o w i n g c h a p t e r . 115 Chapter V : A n a l y s i s and P i s c u s s i o n The r e s u l t s p r e s e n t e d i n the p r e v i o u s chap te r w i l l now be d i s c u s s e d and i n t e r p r e t e d i n o r d e r to o b t a i n a coherent s e l f -c o n s i s t e n t p i c t u r e of the l a s e r plasma i n t e r a c t i o n i n v e s t i g a t e d i n t h i s work. By u n i f y i n g a l l the r e s u l t s o b t a i n e d , some degree of c o n f i d e n c e can then be a t t a c h e d to the v a l u e s for the measured plasma parameter s . The measurements w i l l be i n t e r p r e t e d on the b a s i s of an energy i n v e n t o r y to determine how the l a s e r l i g h t i s absorbed by the p la sma . V - 1 : Overview I t i s u s e f u l to summarize the important r e s u l t s o f . the measurements r e p o r t e d t h u s f a r . P r e l i m i n a r y x - r a y measurements on a 2 - T o r r j e t t a r g e t i n the i n t e n s i t y range of 0.3 - 2 x 1 0 1 2 W/cm 2 produced low e l e c t r o n tempera tures under 200 eV t h a t s c a l e d w i t h i n c i d e n t l a s e r energy as expected by comparison w i t h o ther s i m i l a r e x p e r i m e n t s . However, when the C 0 2 l a s e r beam was focus sed onto a 5 T o r r gas j e t t a r g e t at h i g h e r i n t e n s i t i e s , two e l e c t r o n temperatures were found t h a t were c h a r a c t e r i z e d by two d i f f e r e n t ranges of x - r a y e m i s s i o n s p e c t r a . A supra thermal temperature of about 2 keV was found to be independent of the i n c i d e n t l a s e r energy w h i l e a low 300 eV temperature can be measured o n l y w i t h t h i n b e r y l l i u m f o i l s . The two temperatures as p r e s e n t e d by the x - ray data of F i g u r e IV-5 suggest tha t 300 eV i s a thermal temperature which c h a r a c t e r i z e s the m a j o r i t y of the 116 e l e c t r o n s w h i l e the 2 keV temperature i s a s c r i b e d to a supra therma l component c o m p r i s i n g of a m i n o r i t y of the e l e c t r o n s . Which temperature c h a r a c t e r i z e s the plasma as a bu lk temperature i s a s c e r t a i n e d by a n a l y s i s of s t r e a k photographs of the r a d i a l plasma e x p a n s i o n . The Chapman-Jouguet d e t o n a t i o n p o i n t i n the plasma expans ion h i s t o r y i s i d e n t i f i e d by the g r a p h i c a l method d e s c r i b e d i n the p r e v i o u s c h a p t e r . R e l a t i o n s for the l o c a l sound speed and p r e s s u r e that a p p l y to the C - J p o i n t then g i v e the e l e c t r o n temperature of the plasma near the t ime of the peak of the l a s e r p u l s e . A l t h o u g h there a re d i f f e r e n t r a t e s of expans ion at d i f f e r e n t p o i n t s a l o n g the f o c a l a x i s , the average temperatures i n the i n c i d e n t l a s e r energy range a v a i l a b l e are 460 + 200 eV for the midd le of the i n t e r a c t i o n r e g i o n and 360 + 160 eV for the r e g i o n of the f r o n t He~N 2 i n t e r f a c e neare s t the f o c u s s i n g l e n s . These are the averages of a l l the s t reak photographs a n a l y z e d for both r e g i o n s of the t a r g e t , and t h e . q u o t e d d e v i a t i o n i s the s t andard mean d e v i a t i o n of 10 or more s h o t s . These two measurements a f f i r m the statement t h a t the bulk temperature of the plasma as c h a r a c t e r i z e d by i t s hydrodynamic motion i s of the order of 300 - 500 eV. A plasma of a few hundred eV i s a l s o p l a u s i b l e based on the c a l c u l a t i o n s and computer code r e s u l t s p r e s e n t e d i n Chapter 11. A c h r o n o l o g i c a l s e r i e s of i n t e r f e r o g r a m s c l e a r l y show the expected change i n the tempora l behav iour of the plasma expans ion as i t e v o l v e s from the s u p e r s o n i c to the subsonic mode. D e n s i t y p r o f i l e s o b t a i n e d from A b e l i n v e r s i o n of the 1 17 i n t e r f e r o g r a m s show the fo rmat ion of a d e n s i t y s t ep at the edge of the plasma as expected fo r a b l a s t wave. The i n t e r f e r o m e t r i c r e s u l t s p r o v i d e j u s t i f i c a t i o n of the a p p l i c a t i o n of the C - J wave a n a l y s i s to the s t r e a k photographs . An e s t imate of the absorbed energy i s a l s o p r o v i d e d by the C - J wave a n a l y s i s , the f r a c t i o n of absorbed energy be ing (17% ± 9)%. T h i s f i g u r e for the a b s o r p t i o n i s h i g h e r than that e s t i m a t e d by the compari son of the s t r e a k photographs w i t h h y p o t h e t i c a l b l a s t wave space- t ime curve s o b t a i n e d from the theory of S a k u r a i or T a y l o r and Sedov. However , -an U l b r i c h t photomet r i c sphere used to measure the s c a t t e r e d and r e f r a c t e d r a d i a t i o n from the plasma showed tha t the amount of l a s e r energy decoupled from the plasma i s about 50% of the i n c i d e n t l a s e r e n e r g y . Upon c a l c u l a t i n g the energy i n v e n t o r y by s u b t r a c t i n g a l l i n f r a r e d l o s s e s from the plasma the f r a c t i o n of absorbed energy i s de termined to be (20 + 3)%, i n agreement w i t h the e s t i m a t e o b t a i n e d by C - J wave a n a l y s i s . V - 2 : Inver se Bremss t rah lung A b s o r p t i o n The degree of a b s o r p t i o n as deduced from the U l b r i c h t sphere and s t r e a k photographs can be used to determine the e l e c t r o n temperature of the p l a s m a , i f i t i s assumed that the a b s o r p t i o n i s by i n v e r s e b r e m s s t r a h l u n g . The measured degree of a b s o r p t i o n r?, i s then r e l a t e d to the e l e c t r o n temperature t h r o u g h : 1 - exp(-k L) = TJ , i b 118 where k i b i s the temperature dependent i n v e r s e b remss t rah lung c o e f f i c i e n t ( Johnston and Dawson 5) as g i v e n i n Chapter I I . The plasma i s f a i r l y u n i f o r m i n d e n s i t y by the time of the peak of the l a s e r p u l s e so tha t the a b s o r p t i o n d e s c r i b e d by the above e q u a t i o n i s t h a t which occur s for a l a s e r beam t r a v e r s i n g a u n i f o r m plasma of average ion charge Z, average e l e c t r o n d e n s i t y n of l e n g t h L . The d e n s i t y c o n t o u r s from the i n t e r f e r o g r a m s e s e r i e s p r o v i d e i n f o r m a t i o n on the pa th l e n g t h L . A ray t r a c i n g computer r o u t i n e c o n c e i v e d and implemented by B e r n a r d 3 5 shows that the c e n t r a l rays of the l a s e r beam propagate through t y p i c a l l y about 2 mm of plasma at a d e n s i t y of 2 - 3 x 10 i e cm -3 S o l v i n g fo r the e l e c t r o n temperature r e s u l t s i n the s i n g l e e x p r e s s i o n -35 2 9.74 x 10 Zn L 2/3 1 -n \0.b e n 1 c r (-In (1-r?)) The above e x p r e s s i o n was e v a l u a t e d for s e v e r a l i n t e r f e r o g r a m s w i t h L and n g de termined by the ray t r a c i n g r o u t i n e assuming f u l l i o n i z a t i o n Z=7 and the measured a b s o r p t i o n of t) = 0 . 2 . The r e s u l t s of t h i s c a l c u l a t i o n are g iven i n T a b l e V - 1 . The i n t e r f e r o m e t r i c data r e v e a l why the a b s o r p t i o n i s l i n e a r w i t h i n c i d e n t energy s i n c e the l e n g t h of the plasma 119 INTERF. NUMBER INCIDENT ENERGY (J) TIME (NS) INTO PULSE n / n e c r L(cm) T (eV) | 36 2 . 5 2 . 0 0 . 2 5 0 . 1 8 2 5 0 38 2 . 5 2 . 5 0 . 2 0 0 . 2 9 2 5 0 45 2 . 7 3 . 3 0 . 2 0 0 . 2 3 2 2 0 39 2 . 9 1 . 9 0 . 3 0 0 . 1 7 3 2 0 35 2 . 9 3 . 8 0 . 2 0 0 . 2 2 2 1 0 18 3 . 8 1 . 5 0 . 2 5 0 . 2 0 2 7 0 40 3 . 6 2 . 1 0 . 2 5 0 . 2 3 300 9 4 . 6 3 . 0 0 . 2 0 0 . 5 7 390 31 6 . 3 4 . 3 0 . 2 0 0 . 3 5 290 16 6 . 5 1 . 6 0 . 2 0 0 . 4 5 6 1 0 6 6 . 5 2 . 4 0 . 2 2 0 . 5 3 4 3 0 1 1 7 . 8 3 . 2 0 . 1 6 0 . 5 6 290 E l e c t r o n temperatures c a l c u l a t e d by u s i n g i n v e r s e b r e m s s t r a h l u n g . i n c r e a s e s w i t h i n c i d e n t l a s e r e n e r g y . The i n c r e a s e i n l e n g t h of plasma wi th h i g h e r input l a s e r energy i s compensated by a c o r r e s p o n d i n g l y h i g h e r e l e c t r o n temperature and so the a b s o r p t i o n remains l i n e a r . The o n - a x i s e l e c t r o n d e n s i t y remains f a i r l y c o n s t a n t throughout the d u r a t i o n of the l a s e r p u l s e . The average e l e c t r o n temperature c a l c u l a t e d i n t h i s manner i s 280 + 40 eV, which i s i n good agreement w i t h the average temperature o b t a i n e d from the x - r a y d i a g n o s t i c s and s t r e a k photograph a n a l y s i s . The above c a l c u l a t i o n shows t h a t l i n e a r i n v e r s e b remss t r ah lung account s fo r the observed e l e c t r o n t e m p e r a t u r e . However, the l a s e r i n t e n s i t i e s on the t a r g e t a re i n the regime where n o n - l i n e a r a b s o r p t i o n can be expected to o c c u r . The r a t i o of the e l e c t r o n o s c i l l a t o r y to thermal v e l o c i t y i s : 0 . 5 [eE/m w_}/ (kT /m ) 1 20 -6 0.5 2 , T i n eV) ' e = 6.49 x 10 ( I / T ) (I i n W/cm e = 1 - 3, f o r 300 eV, 1 -10 J o u l e s = 0.4 - 1.2, f o r 2 keV, 1 - 10 J T h e r e f o r e , s i n c e the o s c i l l a t o r y v e l o c i t y i s comparable or g r e a t e r than the thermal v e l o c i t y throughout the range of l a s e r i n t e n s i t i e s on t a r g e t , the a b s o r p t i o n s h o u l d be l e s s than the l i n e a r r e s u l t by the f a c t o r the p h y s i c a l reasons for t h i s a re p r e s e n t e d i n Chapter I I . For the range of vacuum i n t e n s i t i e s i n c i d e n t on the p lasma, the i n v e r s e b r e m s s t r a h l u n g a b s o r p t i o n c o e f f i c i e n t i s lowered by a f a c t o r r a n g i n g from 0.79 to 0.25 which i n the temperature c a l c u l a t i o n shown above r e s u l t s i n a lower temperature of about 100 eV at moderate i n t e n s i t i e s of 3 x 10 1 3 W / c m 2 . A more d e t a i l e d c a l c u l a t i o n a l l o w s the e l e c t r o n temperature to vary w i t h t ime and thus the a b s o r p t i o n becomes a . t i m e v a r y i n g f u n c t i o n . Recent e v i d e n c e 1 1 5 of Thomson s c a t t e r i n g from e x c i t e d TPD modes i n d i c a t e s that the temperature at the peak of the pu l se i s a round 600 eV. In t h i s c a l c u l a t i o n , the temperature was a l l o w e d to f o l l o w the l a s e r p u l s e shape t e m p o r a l l y w i t h the peak temperature b e i n g 600 eV. The r e s u l t i n g c a l c u l a t e d a b s o r p t i o n was found t o have the l i n e a r dependence 2 -3 /2 ( 1 + (<v > / 3v ) ) os t h r?(t) = 0 .14t - 0.12 121 wi th t i n n s . The r e s u l t i n g a b s o r p t i o n i n t e g r a t e d over the p e r i o d of the l a s e r p u l s e i s / r?(t) E ( t ) dt = 0.21 / E ( t ) dt T h e r e f o r e , t h i s s c e n a r i o of a time dependent temperature and a b s o r p t i o n f i t s the measured time i n t e g r a t e d a b s o r p t i o n p r o v i d e d by the U l b r i c h t photometer . Due to the l a r g e amount of beam r e f r a c t i o n measured i t i s apparent t h a t the o n - t a r g e t i n t e n s i t y i s a f u n c t i o n of t ime and not equa l to the vacuum f o c a l i n t e n s i t y of 1 0 1 " W / c m 2 . T h i s e f f e c t may very w e l l be a mechanism t h a t l i m i t s the i n t e n s i t y to l e v e l s such that c o l l i s i o n a l a b s o r p t i o n remains l i n e a r . For a 300 eV p lasma, t h i s s tatement i m p l i e s t h a t the o n - t a r g e t i n t e n s i t i e s are < 1 0 1 3 W / c m 2 . However, at the t ime of the peak of the l a s e r p u l s e , the d imens ions of the plasma observed from the i n t e r f e r o g r a m s are on a s u b - m i l l i m e t e r s c a l e which are i n s u f f i c i e n t to cause severe beam r e f r a c t i o n . Thus the i n t e n s i t i e s e a r l y i n the p u l s e must be near to 10 1 * W / c m 2 . The s a t u r a t i o n of i n v e r s e b remss t rah lung a b s o r p t i o n has not yet to date been c o n c l u s i v e l y c o n f i r m e d . For example, Gray and K i l k e n n y 1 " i n c o r p o r a t e d t h i s e f f e c t into t h e i r one d i m e n s i o n a l computer code fo r s i m u l a t i n g a C 0 2 l a s e r i n t e r a c t i o n w i t h an underdense Z - p i n c h plasma at the modest i n t e n s i t i e s of 1 0 1 1 W / c m 2 . They found t h a t the a b s o r p t i o n had to be i n c r e a s e d by f a c t o r of 1.1 to get a good f i t of the s i m u l a t i o n to the e x p e r i m e n t a l r e s u l t s . 122 Langdon 9 a p p l i e s h i s c o r r e c t i o n to the a b s o r p t i o n i n the exper iments of Shay et a l . 1 1 6 which i n v o l v e d i n t e n s i t i e s of 10 1 * W/cm 2 at a wavelength of 1.06 Mm, and e l e c t r o n tempera tures of 400 eV. The c a l c u l a t e d r e d u c t i o n i n a b s o r p t i o n i s 40% which i s comparable to o ther r e f inement s invoked i n the computer model ing of the exper iment . More r e c e n t l y , Herbs t et a l . 1 1 7 f o r s i m i l a r e x p e r i m e n t a l c o n d i t i o n s to those of Shay et a l . used a two d i m e n s i o n a l c y l i n d r i c a l hydrodynamic code w i t h c o l l i s i o n a l a b s o r p t i o n and c l a s s i c a l thermal c o n d u c t i o n and found the e x p e r i m e n t a l v a l u e s of the a b s o r p t i o n to be c o n s i s t e n t or s l i g h t l y g r e a t e r than the c a l c u l a t e d v a l u e s , but no more than by 10% of the i n c i d e n t energy . Thus the presence of s a t u r a t i o n of i n v e r s e bremss t rah lung at h i g h l a s e r i n t e n s i t i e s i s a t o p i c of some c o n t r o v e r s y . S i n c e the bremss t rah lung e m i s s i v i t y and a b s o r p t i o n are d i r e c t l y r e l a t e d to one another i t i s a p p r o p r i a t e to c o n s i d e r what the e f f e c t of the h i g h i n t e n s i t y n o n - l i n e a r i t y i s on the b remss t r ah lung e m i s s i o n . T h i s e f f e c t may i n t r o d u c e an e r r o r i n t o the d e t e r m i n a t i o n of e l e c t r o n temperature by the method of x - r a y a b s o r p t i o n as used i n t h i s t h e s i s . T h i s e f f e c t has been c o n s i d e r e d by P e r t 1 1 8 . The bremss t r ah lung e m i s s i o n c r o s s - s e c t i o n for x - r a y s i s a <* v 2 and the e l e c t r o n s possess a v e l o c i t y which i s the sum of the thermal and o s c i l l a t o r y components v = v o s + v t h . The time averaged e m i s s i o n r a t e for a M a x w e l l i a n d i s t r i b u t i o n of e l e c t r o n s i s then R = < 1/(V + v )>f(v., ) os th t h P r o v i d e d t h a t the o s c i l l a t i o n energy 0.5mvo^ i s much l e s s than the e m i t t e d photon energy hp then the r e s u l t i n g e m i t t e d i n t e n s i t y i s c a l c u l a t e d to be augmented by the f a c t o r of 123 ( 1 + ( 2 W k T - 3 ) m < v 2 > / 6 k T ) . 6 os e To i n v e s t i g a t e the s i g n i f i c a n c e of t h i s e f f e c t , the above e x p r e s s i o n was i n c o r p o r a t e d i n t o the c a l c u l a t i o n of x - r a y i n t e n s i t y t r a n s m i t t e d through f o i l f i l t e r s . The r e s u l t s are shown i n F i g u r e V-1 where i t i s seen tha t the e l e c t r o n temperature measurements s h o u l d not be g r e a t l y i n e r r o r . ' ' ' I I I 15 40 65 pm Be Figure V-1 E l e c t r o n q u i v e r v e l o c i t y e f f e c t on the e l e c t r o n temperature measurement. The dashed curves are the x-ray f o i l t r a n s m i s s i o n s w i t h the i n c o r p o r a t i o n of the e f f e c t . A l t h o u g h the n o n - l i n e a r b r e m s s t r a h l u n g e m i s s i o n i s s u b s t a n t i a l l y i n c r e a s e d over the l i n e a r e m i s s i o n , the r e l a t i v e t r a n s m i t t e d i n t e n s i t i e s t h r o u g h the f o i l s i s not changed s i g n i f i c a n t l y . The reason fo r t h i s i s tha t the s p e c t r a l d i s t r i b u t i o n of the x - r a y r a d i a t i o n i s l i t t l e changed by the l a s e r f i e l d and the f o i l s 124 t r a n s m i t a l i m i t e d wavelength band. I t i s of i n t e r e s t to note tha t a l t h o u g h the e l e c t r o n temperature i n c r e a s e s due to b remss t r ah lung a b s o r p t i o n at an energy g a i n of 0.5m<v^ s> = 1300 eV per c o l l i s i o n , the r a p i d expans ion a l o n g w i t h thermal c o n d u c t i o n c o o l s the plasma s u f f i c i e n t l y q u i c k l y so tha t the a c t u a l temperatures are t y p i c a l l y no g r e a t e r than a few hundred eV i n l a s e r p lasmas . T h e r e f o r e , l i n e a r i n v e r s e b remss t r ah lung a b s o r p t i o n can account for the e l e c t r o n temperature o f 300 eV determined by two independent t e c h n i q u e s . A l t h o u g h the vacuum f o c a l i n t e n s i t i e s suggest that s a t u r a t i o n of i n v e r s e b remss t r ah lung can o c c u r , beam r e f r a c t i o n may l i m i t the o n - t a r g e t i n t e n s i t i e s to the l i n e a r regime on a t ime i n t e g r a t e d b a s i s . The p l a u s i b i l i t y of the e l e c t r o n temperature can be checked f u r t h e r by a d e t a i l e d energy ba l ance c a l c u l a t i o n and t h i s f o l l o w s i n the next s e c t i o n . V - 3 : Energy Ba lance and E l e c t r o n Temperature The r e s u l t s p r e s e n t e d t h u s f a r now enable the energy ba lance to be c a l c u l a t e d . Once the energy ba l ance i s d e t e r m i n e d , the thermal e l e c t r o n temperature of the plasma can a l s o be e s t i m a t e d and compared w i t h the v a l u e s o b t a i n e d by the so f t x - r a y d i a g n o s t i c s and the C - J a n a l y s i s of the s t r e a k photographs . By means of a d e t a i l e d energy i n v e n t o r y , the e f f i c i e n c y of energy c o u p l i n g of the i n c i d e n t l a s e r beam to the plasma i t produces can be a s s e s s e d . The e l e c t r o n temperature can be de termined once the thermal energy conten t of the plasma i s known and thus serves as a check of the p l a u s i b i l t y of the 1 25 r e s u l t s p r e s e n t e d t h u s f a r . Both s t r e a k photograph a n a l y s i s and U l b r i c h t photomet r i c measurements i n d i c a t e an energy a b s o r p t i o n of 20% of the i n c i d e n t l a s e r energy . T h i s absorbed energy i s c o n v e r t e d i n t o o ther forms i n s i d e the p la sma. P a r t s of i t a re used to i o n i z e the p la sma , and to do work i n expans ion of the p la sma . Some of the energy appears i n the k i n e t i c energy of hot e l e c t r o n s , and i n the p o t e n t i a l energy of the e x c i t e d modes t h a t occur i n the observed p a r a m e t r i c i n s t a b i l i t i e s . Some absorbed energy i s a l s o l o s t i n r a d i a t i o n . In t h i s s e c t i o n , each of these energy s i n k s w i l l be e v a l u a t e d to determine what remains as the thermal energy c o n t e n t of the p la sma. The energy ba lance i s w r i t t e n a s : E = 0 .20E = | k B T e N e + I k B T i N i o i n + IN E + / PdV + IP dt + E N i i r ad h h where P r a d i s the r a d i a t i o n power, E h N h i s the energy c o n t a i n e d i n the hot e l e c r o n s , the thermal and hot e l e c t r o n number are r e p r e s e n t e d by N e , N h and N ± i s the ion number. Measurements of the hot e l e c t r o n e m i s s i o n from the plasma (Meyer et a l . 8 9 ) i n d i c a t e tha t about 1% of a l l the e l e c t r o n s i n the i n t e r a c t i o n volume are e j e c t e d at the mean energy of 85 keV. The f o c a l volume i s 7 r r 2 l = t x 50Mm2 x 300 um. The energy c o n t a i n e d i n the Langmuir wave a c c e l e r a t e d e l e c t r o n s i s 7 r r 2 l n e x 0.01 x ^ B T e ' ' ? B T e ^ e i n g 85 keV and i s of the order of a m i l l i j o u l e . T h e r e f o r e the 85 keV e l e c t r o n s are an i n s i g n i f i c a n t s i n k fo r the absorbed l a s e r energy . The energy expended i n i o n i z i n g the plasma can be c r u d e l y 126 e s t i m a t e d by assuming t h a t a l l the e l e c t r o n s p re sen t are from f u l l y s t r i p p e d i o n s . T h i s w i l l be an upper bound to the i o n i z a t i o n energy c o n t e n t of the p l a s m a . To f u l l y s t r i p n i t r o g e n of e l e c t r o n s r e q u i r e s a t o t a l of about 1500 e V / i o n . The c a l c u l a t i o n s fo r the v a r i o u s i n t e r f e r o g r a m r e s u l t s a re p r e s e n t e d i n T a b l e V - 2 . INTERF. INPUT TIME INTO TOT AT, // ,c % ENERGY NO. ENERGY PULSE (NS) ELECTRONS 1 0 l 5 IN IONIZATION 39 2 . 9 1 . 9 0 . 6 0 . 7 3 6 2 . 5 2 . 0 1 .3 1 .7 8 2 . 5 2 . 5 2 . 1 2 . 9 4 5 2 . 7 3 . 3 3 . 2 4 . 0 3 5 2 . 9 3 . 6 3 . 1 3 . 6 28 4 . 2 1 .0 0 . 0 1 3 0 . 0 1 19 4 . 2 1 . 2 0 . 2 3 0 . 0 2 18 3 . 8 1 . 5 0 . 6 5 0 . 5 8 40 3 . 6 2 . 1 1 .6 0 . 0 2 9 4 . 6 3 . 0 5 . 1 3 . 8 26 6 . 5 0 . 9 0 . 0 2 8 0 . 0 2 16 6 . 5 1 .6 1 .6 0 . 8 4 6 6 . 5 2 . 4 5 . 5 2 . 6 31 6 . 3 4 . 3 6 . 3 3 . 4 Figure V -2 Energy contained i n i o n i z a t i o n using i n t e r f e r o m e t r i c data. From the t a b l e i t i s seen t h a t by the end of the l a s e r p u l s e < 4 n s , the maximum amount of i o n i z a t i o n energy the plasma can c o n t a i n i s about 4% of the i n c i d e n t l a s e r e n e r g y . At the peak of the l a s e r p u l s e , there are fewer e l e c t r o n s so t h a t the energy i n i o n i z a t i o n i s s m a l l , l e s s than 1%. The energy consumed i n expans ion work of the plasma a g a i n s t i t s ambient s u r r o u n d i n g s i s now e s t i m a t e d as f o l l o w s . From the C - J wave a n a l y s i s of the s t reak photographs , the p r e s s u r e of the 127 plasma at the t ime of the C- J p o i n t i s known. In a d d i t i o n , the volume of plasma as a f u n c t i o n of t ime i s known by contour i n t e g r a t i o n of the i n t e r f e r o g r a m d a t a . T h e r e f o r e the expans ion work i s c a l c u l a t e d by j*PdV = P AV , . c j where AV i s the volume change of the plasma d u r i n g the l a s e r p u l s e . S ince s t r e a k photographs and i n t e r f e r o g r a m s are not taken s i m u l t a n e o u s l y the data fo r l a s e r shot s for s i m i l a r i n p u t l a s e r e n e r g i e s are combined . A t y p i c a l plasma volume at the t ime of the C - J p o i n t i s 2 x 10~ 3 c m 3 ( w h i l e the p r e s s u r e at t h i s t ime i s about 1 x 10 9 d y n e s / c m 2 . T h i s r e s u l t s i n expans ion work of the order of s e v e r a l hundred m i l l i j o u l e s or 3 - 5 % of the i n c i d e n t l a s e r e n e r g y . R a d i a t i o n powers i n c l u d e the sum of f r e e - f r e e , f r ee -bound and l i n e r a d i a t i o n . The d e t a i l s of t h i s c a l c u l a t i o n are shown i n Appendix B where i t i s seen t h a t r a d i a t i o n l o s s e s can be n e g l e c t e d i n the d u r a t i o n of the l a s e r p u l s e . S ince the e l e c t r o n - i o n e q u i l i b r a t i o n t ime fo r a plasma i s ( S p i t z e r 2 " ) 3 / 2 / r 4 T = { m A/m ei >/^/m | /3/STT ^ T e I n A ] 8 3 / 2 = 2 . 8 4 x 10 Z T / j n I n A j l e 300 ns , (n i n c m - 3 , T i n e V ) , for our plasma c o n d i t i o n s , the c o l l i s i o n a l t r a n s f e r between 128 e l e c t r o n s and ions i s i n s u f f i c i e n t to impart an a p p r e c i a b l e amount of energy to the ions and so ion h e a t i n g can be n e g l e c t e d in t h i s e s t i m a t e . The energy ba l ance can now be w r i t t e n as E = 0 .20E = i N k T + 4 - N . k T . + £ N E + J*PdV + E N 0 i n 2 e B e 2 i B i n h h = f N i k B T e + 0 .04E + 0.04E i n i n The e l e c t r o n temperature can be c a l c u l a t e d from T ( t ) = 2/3 e (0.12 + .04)E ( t ) i n N ( t ) e where N e i s the t o t a l e l e c t r o n number which i s o b t a i n e d from the i n t e g r a t i o n of the A b e l u n f o l d e d d e n s i t y c o n t o u r s from the i n t e f e r o g r a m s . I t i s emphasized i n the above e q u a t i o n that the energy input E i n and e l e c t r o n number N g are both f u n c t i o n s of t i m e . The l a s e r p u l s e i s a c t u a l l y d e s c r i b e d by a t r i a n g l u l a r p u l s e shape of the form I ( t ) = I o 0 . 8 3 t , f o r 0 < t < 1.2 ns = I 0 | l - (t-1 . 2 ) 0 . 3 8 5 } , for 1.2 < t < 3.8 ns . The f r a c t i o n a l energy input at a t ime t i n t o the l a s e r p u l s e d u r a t i o n i s o b t a i n e d by i n t e g r a t i o n and n o r m a l i z a t i o n of the above e x p r e s s i o n fo r the p u l s e shape. The temperature as de termined from the energy ba l ance i s c a l c u l a t e d and p r e s e n t e d i n T a b l e V-3 u s i n g the i n t e r f e r o m e t r i c d a t a . Temperatures c a l c u l a t e d i n t h i s manner are q u i t e h i g h for 129 INTERF. TOTAL TIME (NS) TOTAL TOTAL # T ( e V ) SHOT * INPUT INTO ENERGY ELECTRONS e ENERGY PULSE AT TIME t X 1 0 1 5 39 2 . 9 1 . 9 1 .8 0 . 6 1500 36 2 . 5 2 . 0 1 . 7 1 . 3 760 6 2 . 5 2 . 5 2 . 1 2 . 1 500 45 2 . 7 3 . 3 2 . 6 3 . 2 410 28 4 . 2 1 . 0 0 . 9 2 0 . 0 1 3 3510 19 4 . 2 1 .2 1 . 3 3 0 . 2 3 2 9 0 0 18 3 . 8 1 .5 1 . 8 0 0 . 6 5 1620 40 3 . 6 2 . 1 2 . 5 0 1 . 6 910 9 4 . 6 3 . 0 4 . 3 5 . 1 420 26 6 . 5 0 . 9 1 .2 0 . 0 2 8 2 1 4 0 16 6 . 5 1 .6 3 . 3 1 .6 1030 6 6 . 5 2 . 4 5 . 2 5 . 5 550 31 6 . 3 4 . 3 6 . 3 6 . 3 580 Table V-3 E l e c t r o n temperatures c a l c u l a t e d from the energy ba lance . those i n t e r f e r o g r a m s taken at t imes near the peak of the p u l s e . T h i s a r i s e s because the plasma f i r s t appears near the peak of the p u l s e and t h e r e i s too much energy for too few e l e c t r o n s . The h i g h tempera tures at e a r l y t imes are a l s o a consequence of u s i n g time i n t e g r a t e d measurements of a b s o r p t i o n on a t ime dependent phenomenon. N o n e t h e l e s s , shots 4 5 , 3 5 , 9 , and 31 a l l demonstrate t h a t there i s s u f f i c i e n t a b s o r p t i o n to a t t a i n tempera tures from 4 0 0 - 5 0 0 eV s i n c e these are a l l i n t e r f e r o g r a m s taken at the end of the l a s e r p u l s e . T h e r e f o r e , the 300 eV temperature measured by the s o f t x - r a y d i a g n o s t i c s as i n f e r r e d from the s t r e a k p h o t o g r a p h i c s tudy i s f u r t h e r c o r r o b o r a t e d by the s imple energy ba lance c a l c u l a t i o n shown above . 1 30 V - 4 ; The Nature of the supra thermal temperature As shown i n the p r e c e d i n g s e c t i o n s , the 300 eV thermal temperature de termined by the x - ray d i a g n o s t i c s i s c o r r o b o r a t e d by the temperatures d e r i v e d from a m o d i f i e d b l a s t wave a n a l y s i s of the s t reak photographs and a l s o from an energy ba lance c a l c u l a t i o n . However, the x - ray d i a g n o s t i c s a l s o show the consp icuous presence of a 2 keV component i n the p la sma . In t h i s s e c t i o n , the nature of t h i s supra thermal temperature w i l l be d i s c u s s e d i n terms of i t s p o s s i b l e o r i g i n and meaning . Smoothing E f f e c t s of the X - r a y D i a q n o s t i e s In Chapter III i t i s mentioned t h a t due to the s m a l l tempora l and s p a t i a l ex tent of the l a s e r produced p lasma, the x-ray d i a g n o s t i c s are both t ime and s p a t i a l l y i n t e g r a t i n g measurements. S i n c e the f o i l f i l t e r s used sample a wide x - r a y spectrum r a n g i n g from 1 - 18 keV, the s p a t i a l and tempora l smoothing of the x - r a y d e t e c t i o n system has important consequences for the i n t e r p r e t a t i o n of the d a t a . The i n f o r m a t i o n of where and when the 2 keV e l e c t r o n s occur remains obscured by the smoothing e f f e c t s . However, s e v e r a l s c e n a r i o s may be p o s t u l a t e d and t h e i r v a l i d i t y t e s t e d by c o n s i d e r i n g the t r a n s m i s s i o n of x - r a y s through v a r i o u s f o i l s r e s u l t i n g from a time v a r y i n g e l e c t r o n . t e m p e r a t u r e and d e n s i t y . In g e n e r a l terms , any input s i g n a l S i ( t ) can be c o n s i d e r e d as a l i n e a r combina t ion of d e l t a f u n c t i o n impulses ( e . g . 131 H e l s t r o m 1 1 9 ) S ( t ) = J S ( t ' )6 " ( t-t ' ) d t ' . i i The output of a l i n e a r d e v i c e to a D i r a c d e l t a f u n c t i o n input i s the impulse response K ( t ) . For an a r b i t r a r y input s i g n a l , the output s i g n a l must be a l i n e a r c o m b i n a t i o n of d e l a y e d input re sponses S ( t ) = J S ( f ) K ( t - t ' ) d t ' o i = / K (T )S ( t - r ) d r . i For s i m p l i c i t y and e x p e d i e n c y , the impulse response f u n c t i o n i s taken to be the H e a v i s i d e u n i t s t ep f u n c t i o n K (T ) = 0 , r < 0 = 1 , 0 < T < 10 ns , where the l i m i t of 10 ns i s the FWHM of the measured x - r a y d e t e c t o r impulse response f u n c t i o n . The input s i g n a l s S^ft) for which the c o n v o l u t i o n s were performed were the t r a n s m i t t e d i n t e n s i t i e s of two p a i r s of f o i l s as f u n c t i o n s of a t ime v a r y i n g t e m p e r a t u r e . The t r a n s m i s s i o n of x - r a y s through any g i v e n f o i l i s i n g e n e r a l a m o n o t o n i c a l l y i n c r e a s i n g f u n c t i o n of the e l e c t r o n temperature and i s thereby r e a d i l y amenable to a c o n v e n i e n t p o l y n o m i a l f i t s u i t a b l e fo r subsequent i n t e g r a t i o n fo r the temperature as a f u n c t i o n of t i m e . For these c a l c u l a t o n s the t r a n s m i t t e d i n t e n s i t i e s through a 15 (im and 30 Mm Be f o i l 132 p a i r as w e l l as tha t for a 30 nm and 55 am A l f o i l p a i r were c o n v o l u t e d w i t h the s tep f u n c t i o n to determine what d i f f e r e n c e i n measured temperature o c c u r s fo r v a r i o u s t ime dependent t e m p e r a t u r e s . These p a r t i c u l a r f o i l p a i r s were s e l e c t e d s i n c e the b e r y l l i u m p a i r t r a n s m i t s the s o f t e r x - ray r a d i a t i o n i n the v i c i n i t y of 1 - 2 keV w h i l e the aluminum p a i r t r a n s m i t s x - r a y r a d i a t i o n i n the v i c i n i t y of 6 -7 keV. The aluminum f o i l p a i r i s thus s e l e c t i v e i n the sense tha t i t t r a n s m i t s o n l y the harder r a d i a t i o n from a c o m p a r a t i v e l y h o t t e r p lasma. In a d d i t i o n to the hard r a d i a t i o n , the b e r y l l i u m f o i l s w i l l d e t e c t the s o f t e r r a d i a t i o n from c o o l e r p a r t s of the plasma which may p e r s i s t for a l onger p e r i o d of t ime than the h o t t e r components . The s i m p l e s t p o s s i b i l i t y of tempora l temperature v a r i a t i o n i s tha t the plasma i s heated to a 2 keV temperature and then s l o w l y c o o l s . S i n c e t e m p o r a l l y the c o o l e r plasma predomina te s , i t may be p o s s i b l e t h i s would r e s u l t i n the observed two t e m p e r a t u r e s . Exper iments where t ime r e s o l v e d measurements were made on l a s e r heated plasmas ( e . g . Gray and K i l k e n n y 1 " ) show t h a t the temperature tempora l p r o f i l e f o l l o w s the l a s e r p u l s e and tha t a f t e r the maximum temperature i s r eached , the temperature f a l l s e x p o n e n t i a l l y w i t h t i m e . A slow e x p o n e n t i a l decay of the temperature can be somewhat j u s t i f i e d on the b a s i s tha t the v i s i b l e e m i s s i o n from the plasma has been observed w i t h a Hamamatsu vacuum photod iode to have an 1/e f o l d i n g t ime of 50 n s . The temperature tempora l p r o f i l e was assumed to be of the form: T ( t ) = 1667t 0 < t < 1.2 ns e 133 = 2000 j e x p - ( t - 1 . 2 ) / T j f o r t > 1.2 ns , where T , the 1/e f o l d i n g t i m e , v a r i e d from 1 t o 20 n s . I t i s a l s o neces sa ry to c o n s i d e r the t ime v a r i a t i o n of the x - ray e m i s s i o n due to the growth of the e m i t t i n g volume of p lasma, p a r t i c u l a r l y s i n c e there i s r e l a t i v e l y l i t t l e plasma at the time of the peak of the l a s e r p u l s e . A study of the i n t e r f e r o g r a m s e r i e s shows tha t the average d e n s i t y of the plasma i s cons t an t but the volume i s a r a p i d l y v a r y i n g f u n c t i o n : V ( t ) = 0 , f o r t < 0.5 ns 4 = At , 0.5 < t < 2.5 ns 4 = A ( 2 . 5 ) , for t > 2.5 ns T h i s f u n c t i o n for the plasma volume was i n c o r p o r a t e d i n t o the c o n v o l u t i o n . The i n t e g r a t i o n s were n u m e r i c a l l y performed by means of the UBC l i b r a r y r o u t i n e of DCADRE. The r e s u l t s for the temperature h i s t o r y d e s c r i b e d are shown i n T a b l e V - 4 . The r e s u l t s i n Tab le V-4 i n d i c a t e that the b e r y l l i u m f o i l s shou ld measure a temperature t h a t i s 60% w i t h i n the peak temperature and the aluminum f o i l s w i t h i n 70% p r o v i d e d that T, the f a l l t ime i s around a few nanoseconds as i s e x p e c t e d . T h e r e f o r e , the temperature measured by the b e r y l l i u m f o i l would always be about 1 keV i f the plasma peak temperature i s 2 keV. The o ther p o s s i b i l i t y i s t h a t the supra therma l temperature p e r s i s t s o n l y fo r a shor t t ime i n comparison w i t h the l a s e r p u l s e d u r a t i o n and i t c h a r a c t e r i z e s o n l y a f r a c t i o n of the t o t a l e l e c t r o n d i s t r i b u t i o n . T h i s i s suggested by the f a c t tha t time 134 T Be F O I L % OF M A X . A l F O I L % OF M A X . (NS) T E M P . ( e V ) T e T E M P . ( e V ) T e 1 6 4 0 32 1240 62 2 7 8 0 39 1280 64 5 920 46 1300 65 10 1 160 58 1420 71 15 1300 65 1520 74 20 1440 72 1580 79 Table V-A Results from the convolution c a l c u l a t i o n s with T max = 2000 eV r e s o l v e d Thomson s c a t t e r i n g o b s e r v a t i o n s ( B e r n a r d 3 5 ) of the e x c i t e d TPD (two-plasmon decay p a r a m e t r i c i n s t a b l i l i t y ) and SBS modes i n d i c a t e t h a t these phenomena e x i s t f o r t imes 0.7 t o 1.5 ns a f t e r the s t a r t of the l a s e r p u l s e . T h i s h y p o t h e s i s was t e s t e d by super impos ing upon the low temperature h i s t o r y , a hot temperature " s p i k e " whose d u r a t i o n was 0.8 ns as i n d i c a t e d below: T ( t ) = 250(t ) f o r 0 <_ t <_ 1.2 ns e = 300 exp-{(t-1 .2)/ !0} for 0 < t < 10 ns = T ( t ) + T 0.7 < t < 1.5 ns e hot The c o n t r i b u t i o n of t r a n s m i t t e d x - r a y i n t e n s i t y from T was not weighted by the f a c t o r 2 ( n / n ) hot t o t a l 1 35 s i n c e the cont inuum e m i s s i o n i s p r o p o r t i o n a l to the square of the e l e c t r o n d e n s i t y . The use of a s t ep f u n c t i o n for the tempora l p r o f i l e for T b o t ^ s j u s t i f i e d on the b a s i s t h a t for the plasma d e n s i t i e s observed, the mean f ree pa th for a k i l o -e l e c t r o n v o l t e l e c t r o n exceeds the d imens ions of the p la sma. S e v e r a l po s s ib l e values of T^^. were c o n s i d e r d . Immediately apparent i s the e f f e c t of the 100 keV e l e c t r o n s tha t a r i s e as a r e s u l t of the SRS and TPD i n s t a b i l i t i e s . For the hot e l e c t r o n f r a c t i o n of 15% the r e s u l t of 2500 eV was o b t a i n e d u s i n g the A l f o i l p a i r . A r e s u l t of 1500 eV was o b t a i n e d when the f r a c t i o n was lowered to 10%. The t i m e - smoothed t r a n s m i t t e d i n t e n s i t y through the b e r y l l i u m f o i l s i s u n a f f e c t e d by the shor t T h o t p u l s e , the temperature o b t a i n e d be ing around 270 eV. Resonance a b s o r p t i o n i f p r e s e n t , i s p r e d i c t e d to produce a hot temperature of around 10 keV ( F o r s l u n d 1 2 ) for n = 0.25 n c • e c r a c o l d temperature of 300 eV, and l a s e r i n t e n s i t i e s of 10 W / c m 2 . The c a l c u l a t i o n s fo r t h i s T hot s ^ o w tha t i f the hot e l e c t r o n f r a c t i o n i s 30% then the temperature i n f e r r e d from the aluminum f o i l t r a n s m i t t e d i n t e n s i t i e s i s 1600 eV, w h i l e the Be f o i l s as be fore imply a temperature of 250 eV. Ion a c o u s t i c t u r b u l e n c e can be expected to produce a supra therma l temperature of a few keV maximum. Only by a l l o w i n g a l l the e l e c t r o n s to have a 3 keV temperature for the i n t e r v a l 0.7 to 1.5 ns c o u l d a temperature of 1600 eV be o b t a i n e d from the aluminum p a i r w h i l e a temperature of 300 eV i s o b t a i n e d from the b e r y l l i u m f o i l s . F i n a l l y , a 2.5 keV temperature was a l l o w e d to have an 136 e x p o n e n t i a l f a l l t ime a l o n g w i t h the c o l d temperature to t e s t the e f f e c t of a c o n t i n u o u s hot component i n s t e a d of a shor t sp ike as d e s c r i b e d above. A temperature of 1600 eV i s o b t a i n e d from the aluminum f o i l s i f the hot e l e c t r o n f r a c t i o n i s 40 % and both hot and c o l d temperatures have a decay t ime of 10 n s . However, i n t h i s c a se , the temperature o b t a i n e d from the b e r y l l i u m f o i l s i s 800 eV. A r e d u c t i o n i n the decay time of the hot temperature to 2 ns lowered the b e r y l l i u m f o i l temperature to 420 eV but then the aluminum f o i l measured temperature i s o n l y 1 keV. The e f f e c t of s p a t i a l smoothing of the x - r a y d e t e c t i o n system was a l s o c o n s i d e r e d . The i n t e r f e r o g r a m s i n g e n e r a l , r e v e a l t h a t the plasma has a c o n s t a n t average d e n s i t y up to a r a d i u s of 300 - 200 nm for t imes up to about 2 ns a f t e r the s t a r t of the l a s e r p u l s e . Thus a s p a t i a l r a d i a l temperature p r o f i l e was assumed of .the form: T ( r ) = T e max where R i s 300 um. T h i s i s mode l ing the plasma by a hot c e n t r a l a x i a l core surrounded by p r o g r e s s i v e l y c o o l e r c y l i n d r i c a l s h e l l s a l l of the same e l e c t r o n d e n s i t y . I f T and T . were set to •* max mm 500 and 100 eV r e s p e c t i v e l y then both f o i l p a i r s measure a temperature t h a t i s w i t h i n 10% of the maximum. For the extreme case of a 2 keV core temperature f a l l i n g o f f r a d i a l l y to 100 eV at R both f o i l p a i r s measure 2 keV to w i t h i n 40%. T h e r e f o r e , on the b a s i s of t h i s c a l c u l a t i o n , the f o i l temperature measurement " T - T max min R f 137 s p a t i a l l y o b t a i n s the maximum temperature from a uni form d e n s i t y r e g i o n p r o v i d e d the temperature g r a d i e n t s are not too s t e e p . In summary, the o b s e r v a t i o n of a two component e l e c t r o n temperature by the x- ray absorber method can be best e x p l a i n e d by the h y p o t h e s i s tha t a hot e l e c t r o n f r a c t i o n p e r s i s t s f o r a shor t t ime i n t e r v a l . The three unknown parameters are the hot e l e c t r o n f r a c t i o n , the hot temperature and i t s t ime d u r a t i o n . The s c e n a r i o s tha t y i e l d the e x p e r i m e n t a l measurements a l l have the bulk of the e l e c t r o n s c h a r a c t e r i z e d by a temperature of 300 eV which peaks at the l a s e r p u l s e and decays e x p o n e n t i a l l y w i t h a f a l l t ime of s e v e r a l nanoseconds . T h i s produces the c o l d temperature measured through the b e r y l l i u m f o i l s . The hot e l e c t r o n t e m p e r a t u r e , because i t o c c u r s o n l y fo r a c o m p a r a t i v e l y shor t t ime c o n t r i b u t e s a s m a l l background s i g n a l t o the b e r y l l i u m f o i l s but a much more s i g n i f i c a n t s i g n a l through the aluminum and copper f o i l ab sorber s due to t h e i r b i a s toward t r a n s m i s s i o n of the harder x - r a y r a d i a t i o n . There a re s e v e r a l combina t ions of hot e l e c t r o n temperature and f r a c t i o n t h a t produce a 2 keV temperature measurement from the aluminum f o i l p a i r . The presence of 100 keV e l e c t r o n s i n the time i n t e r v a l s p e c i f i e d produced 2 - 2.5 keV when the f r a c t i o n of hot e l e c t r o n s was between 10 and 15%. T h i s i s however g r e a t l y i n excess of the f r a c t i o n of hot e l e c t r o n s (1%) e s t i m a t e d by the exper iments d e s c r i b e d by Meyer et a l . 8 9 A temperature of 10 keV, which may a r i s e i f resonance a b s o r p t i o n i s p r e s e n t , r e q u i r e s 30% of the e l e c t r o n s to be at t h i s temperature to o b t a i n a measurement of 2 keV. Resonance a b s o r p t i o n r e q u i r e s that a c r i t i c a l d e n s i t y l a y e r be p r e s e n t . An 138 o v e r c r i t i c a l d e n s i t y can be a c h i e v e d i f the j e t t a r g e t were to be i n s t a n t a n e o u s l y f u l l y i o n i z e d . However, a l l i n t e r f e r o g r a m s taken i n d i c a t e an e l e c t r o n d e n s i t y < 0.4 n , so t h a t resonance cr a b s o r p t i o n as a mechanism fo r p r o d u c i n g the hot e l e c t r o n s i s u n a c c e p t a b l e . Ion a c o u s t i c t u r b u l e n c e and f i l a m e n t a t i o n (Ng et a l . 1 2 0 ) are known to produce tempera tures of a few keV. I f a 3 keV temperature i s a l l o w e d to p e r s i s t for the 0.8 ns i n t e r v a l , then o n l y by a l l o w i n g a l l the e l e c t r o n s to be at t h i s temperature can the e x p e r i m e n t a l t ime and space i n t e g r a t e d x - r a y data be d u p l i c a t e d . As i n d i c a t e d i n Chapter II and v e r i f i e d by Of fenberger et a l . 3 3 , the presence of s i g n i f i c a n t l e v e l s of ion a c o u s t i c t u r b u l e n c e r e q u i r e s tha t the a b s o r p t i o n of the l a s e r r a d i a t i o n be q u i t e h i g h . With the measured a b s o r p t i o n l i m i t e d to 20%, i o n -a c o u s t i c t u r b u l e n c e i s t h e r e b y an u n l i k e l y c a n d i d a t e for the o r i g i n of the observed supra therma l t empera ture . T h i s l e a v e s f i l a m e n t a t i o n as the r e m a i n i n g o r i g i n of the supra therma l t e m p e r a t u r e . Ng et a l . 1 2 0 demonstrated e x p e r i m e n t a l l y a c o r r e l a t i o n of the i n t e n s i t y of hard x - r a y s a t t r i b u t e d to f i l a m e n t a t i o n and s a t u r a t i o n of the s t i m u l a t e d B r i l l o u i n b a c k s c a t t e r . In t h i s gas j e t exper iment , s a t u r a t i o n of SBS and SRS were both observed wi th a peak of SRS a c t i v i t y o c c u r r i n g at about 6 J o u l e s of i n c i d e n t e n e r g y . A s i m i l a r behav iour • of the t r a n s m i s s i o n of x - r a y s through t h i c k aluminum f o i l s was o b s e r v e d . The d e f i n i t i v e proof of f i l a m e n t a t i o n would be x - ray p i n h o l e photographs ( F o n g 1 2 1 ) of the f o c a l volume showing s m a l l r e g i o n s of i n t e n s e x - r a y e m i s s i o n . U n f o r t u n a t e l y , 139 the c o m b i n a t i o n of plasma d e n s i t y and temperature d i d not a l l o w s u f f i c i e n t x - r a y i n t e n s i t i e s to produce c o n v i n c i n g s i n g l e shot x - ray photographs . In the few i n s t a n c e s where p i n h o l e photographs were o b t a i n e d by s e v e r a l i n t e g r a t e d exposure s , the x - ray e m i t t i n g volume was comparable i f not g r e a t e r than the f o c a l volume i t s e l f . However, the s a t u r a t i o n of the SRS e l e c t r o n s and 3/2 CJ0 r e f l e c t i v i t y would suggest tha t wavebreaking of the SRS e l e c t r o n waves p r o v i d e s the source of suprathermal e l e c t r o n s . -Wavebreaking, i n the sense used i n t h i s t h e s i s , r e f e r s to the phenomenon observed i n the computer s i m u l a t i o n s of F o r s l u n d et a l . 1 2 2 . The s i m u l a t i o n s demonstra ted the d i s r u p t i o n of the s t i m u l a t e d B r i l l o u i n and Raman waves for i n c i d e n t l a s e r powers where v / v , >1. In these c i r c u m s t a n c e s , the plasma becomes os th r compressed i n t o a wave form by the ponderomotive f o r c e p a t t e r n of the pump and backward l i g h t waves. The d e n s i t y p e r t u r b a t i o n s may be as much as an o r d e r of magnitude above the background d e n s i t y . However, the plasma e v e n t u a l l y r e a c t s a g a i n s t t h i s squeez ing and t h i s r e a c t i o n - m a n i f e s t s i t s e l f i n the computer s i m u l a t i o n s as a wave b r e a k i n g i n the e l e c t r o n or ion phase space . At the p o i n t of b r e a k i n g , the p a r t i c l e energy d e n s i t y i s e q u i l i b r a t e d w i t h the r a d i a t i o n p r e s s u r e . At wavebreak ing , the p r e s s u r e ba l ance per u n i t volume i s 2 E 0 /Sir = n kT , e e where E 0 i s the e l e c t r i c f i e l d of the l a s e r beam. For 5 j o u l e s of l a s e r energy d e l i v e r e d i n 2 ns i n t o a plasma of d e n s i t y 2.5 x 1 40 1 0 1 8 cm 3 , the above c o n d i t i o n y i e l d s a T g = 2 .5 keV. V - 5 : D e c o n v o l u t i o n of the X - r a y Data Thusfar , the hot e l e c t r o n p o p u l a t i o n , as e v i d e n c e d by the supra therma l e l e c t r o n t empera ture , has not been c o n s i d e r e d i n the energy b a l a n c e . T h i s s e c t i o n a t tempts to e s t i m a t e the number of these hot e l e c t r o n s by a d e c o n v o l u t i o n procedure a p p l i e d to the x- ray t r a n s m i s s i o n d a t a , and by a s imple energy ba lance c a l c u l a t i o n . A common procedure i n x - r a y d i a g n o s t i c s of l a s e r produced plasmas i s to o b t a i n a spectrum of the x - r a y e m i s s i o n from the t r amsmis s ion d a t a . The i n t e n s i t y spectrum p r o v i d e s i n f o r m a t i o n of the temperature of the plasma from the s lope of the f i t of the data much the same as the methods used i n Chapter I V . F u r t h e r m o r e , the i n t e n s i t y spectrum i s used' to o b t a i n an e s t i m a t e of the e l e c t r o n d i s t r i b u t i o n , e . g . B r u e c k n e r ' 2 3 . D i r e c t l y a p p l i c a b l e to the data p r e s e n t e d i n t h i s t h e s i s are the a n a l y t i c a l methods p r e s e n t e d by M . H . K e y 1 2 ' , where i t i s shown tha t s u b j e c t to some a p p r o x i m a t i o n s , knowledge of the continuum e m i s s i o n spectrum i n a g i v e n photon range as c o n t a i n e d i n x - ray t r a n s m i s s i o n da ta a l l o w s d e c o n v o l u t i o n of the e l e c t r o n energy d i s t r i b u t i o n i n the same energy range . The r e a s o n i n g beh ind the c o n v o l u t i o n p rocedure i s o u t l i n e d be low. The s p e c t r a l i n t e n s i t y per u n i t f requency i n t e r v a l due to e l e c t r o n s of number d e n s i t y n e (v)dv, i n the v e l o c i t y range v to v+dv, i n t e r a c t i n g w i t h ions of d e n s i t y N^ and charge Z i s d l U ) = N, v n ( v ) p U ) d V , 141 w h e r e p ( »> ) i s t h e c r o s s s e c t i o n f o r b r e m s s t r a h l u n g e m i s s i o n a n d i s oc 1 / v 2 . T h e t o t a l e m i s s i o n s p e c t r u m d u e t o a n y d i s t r i b u t i o n o f e l e c t r o n v e l o c i t i e s i s t h e n t h e i n t e g r a l : -1/2 1 ( f ) = AN J n ( E ) E d E w h e r e t h e v a r i a b l e change v E was made a n d E 0 = h p . I f n £ (E) d e c r e a s e s r a p i d l y w i t h i n c r e a s i n g e n e r g y i . e . , n ( E ) i e x p ( - E / k T ) e -1/2 -1/2 t h e n E ' = E 0 m t h e i n t e g r a t i o n s o t h a t n ( E 0 ) = _ ° . d I ( v ) AhN^ dv s u b j e c t t o t h e c o n d i t i o n t h a t k T / 2 E 0 < < 1. T h e a b o v e e q u a t i o n r e p r e s e n t s t h e s i m p l e s t f o r m o f d e c o n v o l u t i o n . R e f i n e m e n t s i n v o l v e t h e i n c l u s i o n o f G a u n t f a c t o r s ( K a r s a s a n d L a t t e r 1 2 5 ) t o a d j u s t t h e c l a s s i c a l r e s u l t s t o q u a n t u m m e c h a n i c a l l y c o r r e c t e d v a l u e s . H o w e v e r , f o r t h e p h o t o n e n e r g i e s a n d e l e c t r o n t e m p e r a t u r e s c o n s i d e r e d , t h e G a u n t f a c t o r s a r e s l o w l y v a r y i n g a n d f o r f r e e - f r e e r a d i a t i o n a r e e q u a l t o 1 w i t h i n 15% t h r o u g h o u t t h e r a n g e o f p a r a m e t e r s c o n s i d e r e d . T h e r e f o r e , f o r e x p e d i e n c y t h e G a u n t f a c t o r s a r e s e t e q u a l t o o n e . T o d e t e r m i n e t h e e l e c t r o n e n e r g y d i s t r i b u t i o n , t h e c o n t i n u u m i n t e n s i t y 1(f) m u s t be d e d u c e d f r o m t h e t r a n s m i t t e d x -r a y i n t e n s i t y a s a f u n c t i o n o f a b s o r b e r t h i c k n e s s T ( t ) . T h e a b s o r p t i o n c o e f f i c i e n t o f a n x - r a y a b s o r b e r i s t a k e n a s u(v) = ki>~3 s o t h a t t h e t r a n s m i t t e d f l u x i s 142 -3 T ( t ) = J l U ) e x p (-fetM )6v . The i n t r o d u c t i o n of the c u t - o f f frequency by = ( f e t ) 1 / 3 and subsequent d i f f e r e n t i a t i o n of the above e x p r e s s i o n g i v e s : d T ( t ) / d t = v / t J Kv) U A ) 3 exp ( -U A ) 3 ) dU A ) . Key c o n s i d e r s t h i s i n t e g r a l f o r the case of the s p e c t r a encountered i n x-ray continuum emission from low atomic number l a s e r produced plasmas. The i n t e g r a n d above shows a maximum c l o s e to v = v^ with a s p e c t r a l spread of the order of Lv/v^ = 0.3. T h i s i n t e g r a l i s approximated i n terms of the s p e c t r a l i n t e n s i t y I(v ) i n terms of a decaying e x p o n e n t i a l and the c r e s u l t i s that T ( „ ) _ i d T j v c ) . 1 I UC 1 ~ 3 d v c x where X r e p r e s e n t s the value of the integ r a n d c o n v o l u t e d with the approximation f o r 1 ( f ) i n terms of I(v ) and i s a slowly v a r y i n g f u n c t i o n which to a f i r s t approximation may be assumed to be c o n s t a n t . Key a p p l i e d t h i s method of d e c o n v o l u t i o n to the experimental r e s u l t s of Buchl et a l . 1 0 5 " who observed a two temperature plasma i n the x-ray photon range of 1 - 15 keV. The d e c o n v o l u t i o n method d e s c r i b e d i s c o n s i d e r e d t o have a 25% accuracy which i s s u f f i c i e n t f o r an estimate of the hot e l e c t r o n f r a c t i o n as d e s i r e d here. X-ray t r a n s m i s s i o n data such as presented i n F i g u r e IV-6 g e n e r a l l y show m o n o t o n i c a l l y d e c r e a s i n g f u n c t i o n s with i n c r e a s i n g c u t - o f f energy. However, the experimental data 143 obt a i n e d i n d i c a t e that dT/dE-> 0 f o r the energy range 5 - 10 keV. T h i s i s a n o n p h y s i c a l r e s u l t s i n c e dT/dE«E/kT i . e . , the slope of the x-ray spectrum i s i n v e r s e l y p r o p o r t i o n a l to the e l e c t r o n temperature. A c c o r d i n g l y , the best f i t h y p e r b o l i c curve through the data was used as the s t a r t i n g p o i n t f o r the d e c o n v o l u t i o n procedure. The accuracy of the f i t may be judged from F i g u r e V-1. The h y p e r b o l i c f i t to the data was subsequently (a) (b) Figure V-2 Deconvolution of the x-ray transmission data. (a) Curve f i t (dashed) of the data. (b) I n t e n s i t y Spectrum obtained from (a) d i f f e r e n t i a t e d and the r e s u l t being i n t h e second graph of F i g u r e V-1. D i f f e r e n t i a t i o n of the i n t e n s i t y spectrum curve m u l t i p l i e d by E 1 ^ 2 leads to the e l e c t r o n d i s t r i b u t i o n curve shown i n F i g u r e V-2. Maxwellian energy d i s t r i b u t i o n s of the form: 0.5 f ( E ) = A E exp (-E/kT ) , e 1 44 w e r e f i t t e d t o t h e d i s t r i b u t i o n w i t h T b e i n g 300 e V a n d 2000 e e V . T h e r a t i o o f t h e a r e a s u n d e r n e a t h t h e t w o M a x w e l l i a n c u r v e s t h e n g i v e s t h e h o t e l e c t r o n f r a c t i o n a s b e i n g (12 + 5 ) % . 1 1 1 1 1 1 r \ Figure *~ 3 Electron energy distribution from x-ray data. A n o t h e r e s t i m a t e o f t h e h o t e l e c t r o n p o p u l a t i o n i s o b t a i n e d by r e e x a m i n a t i o n o f t h e e n e r g y b a l a n c e . A s p r e v i o u s l y c a l c u l a t e d i n T a b l e V - 3 t h e r e i s s u f f i c i e n t e n e r g y a b s o r p t i o n o f t h e i n p u t l a s e r e n e r g y t o h e a t a l l t h e e l e c t r o n s i n t h e p l a s m a t o s e v e r a l h u n d r e d e V . A s s u m i n g t h e t h e r m a l p l a s m a t e m p e r a t u r e i s 300 eV a n d t h e s u p r a t h e r m a l t e m p e r a t u r e i s 2 0 0 0 e V , t h e f r a c t i o n o f h o t e l e c t r o n s c a n be s i m p l y e s t i m a t e d b y : E = ( 0 . 1 2 + . 0 3 ) E > = | N e c k B T e c + | N e H k B T e R t h i n T h e t o t a l n u m b e r o f e l e c t r o n s i s 1 45 N + N = N e c eH T o t a l w h e r e N T o t a l i s m e a s u r e d by i n t e r f e r o m e t r y . T h e h o t e l e c t r o n f r a c t i o n was c a l c u l a t e d by t h e s o l u t i o n o f t h e p r e v i o u s t w o e q u a t i o n s u s i n g d a t a f r o m l a t e t i m e i n t e r f e r o g r a m s a n d t h e r e s u l t s a r e p r e s e n t e d i n T a b l e V-5. T h e a v e r a g e h o t e l e c t r o n I N T E R F . TOTAL T I M E T O T A L T O T A L HOT SHOT # I N P U T I N T O ENERGY E L E C T R O N # X 1 0 1 5 E L E C T R O N ENERGY P U L S E AT T I M E t F R A C T I O N 36 2 . 5 2 . 0 1 .7 1 .3 0 . 2 1 B 2 . 5 2 . 5 2 . 1 2 . 1 0 . 1 2 45 2 . 7 3 . 3 2 . 6 3 . 2 0 . 0 6 3 35 2 . 9 3 . 8 2 . 9 3 . 1 0 . 1 0 40 3 . 6 2 . 1 2 . 5 1 .6 0 . 2 8 9 4 . 6 3 . 0 4 . 3 5 . 1 0 . 1 1 6 6 . 5 2 . 4 5 . 2 5 . 5 0 . 1 0 31 6 . 3 4 . 3 6 . 3 6 . 3 0 . 1 2 Table V-5 Hot electron f r a c t i o n c a l c u l a t e d b y energy balance. f r a c t i o n c a l c u l a t e d i n t h i s m a n n e r a p p e a r s t o be ( 1 2 _+ 3)% i n a g r e e m e n t w i t h t h e e s t i m a t e o b t a i n e d b y t h e d e c o n v o l u t i o n p r o c e d u r e d e s c r i b e d p r e v i o u s l y . I t s h o u l d be n o t e d t h a t t h e s c e n a r i o t o w h i c h t h e s e e s t i m a t e s a p p l y a r e t h o s e w h e r e t h e h o t e l e c t r o n f r a c t i o n i s a l w a y s p r e s e n t t h r o u g h o u t t h e l a s e r i r r a d i a t i o n o f t h e p l a s m a . 146 V-61 Energy C o n t a i n e d i n the Plasma Waves The p r e v i o u s c a l c u l a t i o n s i n v o l v i n g the energy ba lance do not c o n s i d e r the energy t r a n s f e r r e d to the i o n and e l e c t r o n waves that a r i s e from the SBS and SRS p a r a m e t r i c i n s t a b i l i t i e s . T h i s a d d i t i o n a l p o s s i b l e energy s ink w i l l be c o n s i d e r e d i n t h i s s ec t i o n . Computer s i m u l a t i o n s of SBS by C . E . M a x 1 2 6 i n d i c a t e the c r e a t i o n of i o n heated " t a i l s " i n the M a x w e l l i a n d i s t r i b u t i o n . P h y s i c a l l y , the energy source for ion h e a t i n g i s from the momentum ba l ance as expres sed by the Manley-Rowe r e l a t i o n s 2 2 E /CJ E / CJ , L L w w where E T , CJ t and E and a> are the e l e c t r i c f i e l d s and L Li W W f r e q u e n c i e s of the l a s e r and plasma wave r e s p e c t i v e l y . S i n c e energy i s p r o p o r t i o n a l to E 2 , the f r a c t i o n of i n c i d e n t l i g h t energy g i v e n to the plasma wave i s u /u^ . For the case of SBS t h i s f r a c t i o n i s CJ / CJ = 2c / c , w L S here c g i s the i o n sound speed v e l o c i t y and c i s the speed of l i g h t . The assumption of an e l e c t r o n temperature of 300 eV fo r the c a l c u l a t i o n of the ion sound speed shows t h a t o n l y 0.04% of the i n c i d e n t l a s e r l i g h t can be c o n t a i n e d i n the ion a c o u s t i c wave. The same r e a s o n i n g as above l eads to 50% of the l a s e r 147 energy b e i n g absorbed i n t o the e l e c t r o n plasma waves at 0.25 n . However, as demonstrated by the t h e o r e t i c a l work of cr ' J C o f f e y 1 2 7 , e l e c t r o n plasma waves s a t u r a t e when the e l e c t r o s t a t i c f i e l d of the wave becomes l a r g e enough to t r a p and a c c e l e r a t e e l e c t r o n s . The parameter of r e l e v a n c e i s /3 = 3v f c h / v p h where v i s the thermal v e l o c i t y of the e l e c t r o n s and v ^ i s the phase v e l o c i t y of the e l e c t r o n plasma waves which for Raman s c a t t e r i n g at 0.25 n c r i s c / / 3 (Hiob and B a r n a r d 1 2 8 ) . The r e s u l t from C o f f e y ' s work i s the s a t u r a t i o n of the e l e c t r i c f i e l d of the e l e c t r o n plasma wave by the f a c t o r : 0.25 0.5 0.5 E ' -»• E ( 1 - 1 / 3 / 3 - 8 / 3 / 3 +2/3 ) w w For a s u b - k i l o e l e c t r o n - v o l t p la sma, the r e d u c t i o n i n the e l e c t r i c f i e l d of the e l e c t r o n plasma wave due to e l e c t r o n t r a p p i n g i s about 50%. T h i s means tha t by the Manley-Rowe r e l a t i o n s , the maximum f r a c t i o n of l a s e r energy tha t can be c o n v e r t e d i n t o the e l e c t r o n wave energy i s ( 0 . 5 ) 2 x 0.5 or about 13% of the input pump energy . From the energy ba lance c o n s i d e r a t i o n s p r e s e n t e d e a r l i e r i n t h i s c h a p t e r , i t i s obv ious tha t the energy c o n t a i n e d i n the e l e c t r o n plasma waves can o n l y be at most , a few p e r c e n t of the i n c i d e n t l a s e r e n e r g y . 1 48 Chapter V I : C o n c l u s i o n s The i n t e r a c t i o n of a C 0 2 l a s e r beam w i t h an underdense plasma has been i n v e s t i g a t e d i n t h i s work. The d e t a i l e d energy ba l ance of the l a s e r - p l a s m a i n t e r a c t i o n was o b t a i n e d i n order to e x p l a i n the observed plasma e l e c t r o n t e m p e r a t u r e s . The d i a g n o s t i c s used by the author are s o f t x - ray a b s o r p t i o n , s t r eak photography , U l b r i c h t sphere photometry and supplemented by time r e s o l v e d ruby l a s e r i n t e r f e r o m e t r y . The i n f o r m a t i o n o b t a i n e d from these d i a g n o s t i c s was used to determine the energy c o u p l i n g of the l a s e r beam to the plasma i t produces in a low p r e s s u r e n i t r o g e n gas j e t t a r g e t . E l e c t r o n temperatures of a few hundred eV were o b t a i n e d , c o n s i s t e n t w i t h n u m e r i c a l e s t i m a t e s based on i n v e r s e b remss t r ah lung a b s o r p t i o n and thermal c o n d u c t i o n . The observed e l e c t r o n temperatures do not i n d i c a t e a s a t u r a t i o n of c o l l i s i o n a l a b s o r p t i o n as i s t h e o r e t i c a l l y p r e d i c t e d for f o c a l i n t e n s i t i e s where v o s / v t h > 1 . An energy i n v e n t o r y of the i n f r a -red r a d i a t i o n l eads to the f i g u r e of a b s o r p t i o n of the i n c i d e n t l a s e r energy of 20% which i s a c c o u n t a b l e by c o l l i s i o n a l a b s o r p t i o n . Of t h i s f r a c t i o n , a few percent has been c a l c u l a t e d to go i n t o i o n i z a t i o n of the plasma and expans ion work. The s m a l l number of e l e c t r o n plasma wave a c c e l e r a t e d e l e c t r o n s c o n s i t u t e a n e g l i g i b l e energy s i n k . The energy i n v e n t o r y a l l o w s an e s t imate of the thermal energy content of the plasma which upon assuming tha t a l l the observed e l e c t r o n s can be c h a r a c t e r i z e d by the same tempera ture , y i e l d s temperature e s t i m a t e s t h a t account for h e a t i n g to s e v e r a l hundred eV. 1 49 Shadow and s c h l i e r e n t echn iques were used i n the time r e s o l v e d s t u d i e s of the r a d i a l plasma e x p a n s i o n . These i n v e s t i g a t i o n s were used i n i t i a l l y to de termine the energy d e p o s i t i o n by compari son of the space- t ime c h r o n o l o g i e s of the plasma r a d i u s w i t h tha t p r e d i c t e d by the c o n v e n t i o n a l b l a s t wave t h e o r y . T h i s method of a n a l y s i s of the s t r e a k r e c o r d s showed t h a t the f r a c t i o n a l a b s o r p t i o n of the input energy i s a c o m p a r a t i v e l y s m a l l 5% to 12% depending on the p a r t i c u l a r o p t i c a l method u sed . C l o s e r i n s p e c t i o n of the t ime dependence of the r a d i a l growth of the plasma i n d i c a t e s that t h e r e i s a sudden change d u r i n g the l a s e r i r r a d i a t i o n . T h i s i s s u g g e s t i v e of the two mode expans ion observed i n o ther l a s e r spark exper iments and e l e c t r o t h e r m a l shock t u b e s . A s c r i b i n g Chapman-Jouguet wave d e t o n a t i o n c o n d i t i o n s at the time of change of the tempora l behav iour of the plasma expans ion a l l o w s the s t r a i g h t f o r w a r d c a l c u l a t i o n of the e l e c t r o n temperature and absorbed energy s u b j e c t t o a few j u s t i f i a b l e a s sumpt ions . The e l e c t r o n temperature c a l c u l a t e d by t h i s method y i e l d s s e v e r a l hundred eV i n good agreement w i t h t h a t o b t a i n e d by the x - r a y d i a g n o s t i c s . The absorbed energy i s found to be about 20% u s i n g the C- J wave a n a l y s i s . The thermal temperatures measured and c a l c u l a t e d i n the cour se of t h i s work are a l l summarized i n the p l o t of F i g u r e V I - 1 . The o p t i c a l study a l s o r e v e a l e d the complex s t u c t u r e of the l a s e r produced p la sma. The plasma i n i t i a l l y c o n s i s t s of s e p a r a t e d h i g h d e n s i t y i s l a n d s which have d i s t i n c t l y d i f f e r e n t growth r a t e s due to expans ion i n d i f f e r e n t ambient c o n d i t i o n s . T h i s behav iour was v e r i f i e d by time r e s o l v e d i n t e r f e r o m e t r y 150 I I I I I 1010 1011 IO'2 1013 W/cm2 Figure VI-1 Summary of measured and calculated electron temperatures. w h i c h n o n e t h e l e s s i n d i c a t e d t h e u n m i s t a k a b l e p r e s e n c e o f t h e t w o mode t e m p o r a l e x p a n s i o n o f t h e p l a s m a . P a r t i c u l a r l y s t r i k i n g i s t h e o b s e r v a t i o n o f t h e r a d i a l p r o p a g a t i o n o f s h o c k f r o n t s by a s c h l i e r e n t e c h n i q u e w h i c h s h o w e d a t i m e d e p e n d e n c e o f t ° * 2 . T h i s i s i n d i c a t i v e o f r a d i a t i v e h e a t t r a n s f e r d r i v i n g t h e e x p a n s i o n . The a p p l i c a t i o n o f t h e C h a p m a n - J o u g u e t wave c o n d i t i o n s t o t h e a n a l y s i s o f s t r e a k r e c o r d s i s a n o v e l t e c h n i q u e u s e d s u c c e s s f u l l y t o o b t a i n t h e e l e c t r o n t e m p e r a t u r e a n d a b s o r b e d l a s e r e n e r g y . T h e a g r e e m e n t o f . t h e c a l c u l a t e d v a l u e s w i t h e x p e r i m e n t a l l y m e a s u r e d q u a n t i t i e s i l l u s t r a t e s t h e u s e f u l n e s s o f t h i s m e t h o d t o d e t e r m i n e t h e p a r a m e t e r s o f a p l a s m a f r o m s t r e a k p h o t o g r a p h s a l o n e . T h e t e m p e r a t u r e s o b t a i n e d f r o m t h i s a n a l y s i s a l s o e s t a b l i s h t h e x - r a y d i a g n o s t i c s m e a s u r e d t e m p e r a t u r e o f 300 eV a s b e i n g t h e t h e r m a l t e m p e r a t u r e o f t h e p l a s m a . The x - r a y d i a g n o s t i c s a l s o m e a s u r e c o n s i s t e n t l y a 2 k e V 151 e l e c t r o n temperature i n a harder x - r a y energy range than i n which the 300 eV was o b s e r v e d . A c c o u n t i n g for t h i s temperature i n terms of an energy ba lance i s not s t r a i g h t f o r w a r d s i n c e the x - r a y and energy a b s o r p t i o n measurements are time i n t e g r a t e d . T h i s l e ads to the p o s t u l a t i o n of s e v e r a l s c e n a r i o s of t ime dependent hot e l e c t r o n temperatures a t t r i b u t e d to v a r i o u s f r a c t i o n s of the e l e c t r o n d i s t r i b u t i o n be ing compr i sed of hot e l e c t r o n s . C o n v o l u t i o n c a l c u l a t i o n s were performed to determine what plasma c o n d i t i o n s would s i m u l a t e the o b t a i n e d r e s u l t s from the x - r a y d i a g n o s t i c s . From the many p o s s i b l e combina t ions of hot e l e c t r o n temperature and f r a c t i o n a l d e n s i t y , one p l a u s i b l e c o m b i n a t i o n i s t h a t the plasma i s at 3 keV for a shor t i n t e r v a l of 0.8 ns d u r i n g which SRS and TPD modes have been observed to e x i s t by t ime r e s o l v e d Thomson s c a t t e r i n g . T h i s sudden temperature jump i s superimposed upon a temperature h i s t o r y t h a t f o l l o w s the l a s e r p u l s e and has a peak temperature of a few hundred eV. P o s s i b l e mechanisms for t h i s 2 keV supra thermal temperature are ion a c o u s t i c t u r b u l e n c e , f i l a m e n t a t i o n and wavebreaking of the e x c i t e d e l e c t r o n plasma waves. The s imple ba lance between r a d i a t i o n and plasma p r e s s u r e t h a t i s thought to occur i n wavebreaking does l e a d to a c a l c u l a t e d e l e c t r o n temperature of 2 .5 keV. I f the hot e l e c t r o n temperature were p re sen t throughout the l a s e r - p l a s m a i n t e r a c t i o n , then s imple energy ba lance c a l c u l a t i o n s i n d i c a t e t h a t about 12% of the e l e c t r o n s are at 2 keV the rema in ing b e i n g at 300 eV. A g r a p h i c a l d e c o n v o l u t i o n p rocedure of the x - r a y data a l s o suggests tha t 12% of the 152 e l e c t r o n s can be a s c r i b e d to be s u p r a t h e r m a l . The a m b i g u i t y tha t a r i s e s from the s p a t i a l and temporal smoothing of the x- ray d i a g n o s t i c s does not a l l o w a c o n c l u s i v e statement to be made c o n c e r n i n g the nature of the observed supra thermal t empera ture . However, s e v e r a l rea sonab le s p e c u l a t i o n s can be p r o f f e r e d as i l l u s t r a t e d above . In c o n c l u s i o n , l i n e a r i n v e r s e b remss t r ah lung a b s o r p t i o n i s shown to account for the observed thermal t e m p e r a t u r e . The o b s e r v a t i o n s are shown to be s e l f - c o n s i s t e n t and are c o r r o b o r a t e d by the a n a l y s i s of s t reak photographs and from the energy ba lance de termined by i n t e g r a t i n g sphere photometry . The supra thermal e l e c t r o n d i s t r i b u t i o n de termined by the x - ray d i a g n o s t i c s i s a s c r i b e d to be due to the wavebreaking of the observed p a r a m e t r i c i n s t a b i l i t i e s . F u r t h e r Work C o n s i d e r a b l e more i n s i g h t may be ga ined about the l a s e r plasma i n t e r a c t i o n i f t ime r e s o l v e d x - ray d i a g n o s t i c s were to be made a v a i l a b l e . The time i n t e g r a t e d measurements a l t h o u g h u s e f u l i n p r o v i d i n g some i n f o r m a t i o n cannot g ive a d e t a i l e d account o f the c h r o n o l o g y of the e l e c t r o n t empera ture . For example, the exper iments of Lee and R o s e n 1 2 9 u s i n g an x - r a y s t r e a k camera have shown t h a t w h i l e the thermal temperature of the plasma from a l a s e r i r r a d i a t e d g o l d d i s k t a r g e t remains f a i r l y cons t an t throughout the l a s e r p u l s e , the hot e l e c t r o n temperature has a tempora l behav iour which f o l l o w s the l a s e r p u l s e i n t e n s i t y . A problem w i t h C 0 2 l a s e r plasmas i s tha t because of t h e i r 153 low d e n s i t y , the x - ray f l u e n c e i s s e v e r a l o r d e r s of magnitude s m a l l e r than t h a t from Nd l a s e r plasmas thus h i g h g a i n x- ray measurement d e v i c e s are n e c e s s a r y . Commercial x - r a y s t reak cameras w i t h m i c r o c h a n n e l p l a t e a m p l i f i c a t i o n have become r e c e n t l y a v a i l a b l e . The use of such a camera may p r o v i d e tempora l r e s o l u t i o n on the e v o l u t i o n of the e l e c t r o n t e m p e r a t u r e . As t h i s work i s p a r t of an o v e r a l l program d e d i c a t e d to the study of p a r a m e t r i c i n s t a b i l i t i e s , knowledge of the s p a t i a l and tempora l d i s t r i b u t i o n of the thermal e l e c t r o n temperature i s e s s e n t i a l f o r the c a l c u l a t i o n of i n s t a b i l i t y t h r e s h o l d s fo r the v a r i o u s p a r a m e t r i c p r o c e s s e s . Time r e s o l v e d ruby l a s e r Thomson s c a t t e r i n g can be i n p r i n c i p l e be implemented fo r t h i s purpose . The i n h e r e n t d i f f i c u l t y here i s the f a c t tha t o n l y a f a c t o r of 1 0 ~ 1 2 of the i n c i d e n t l i g h t i s s c a t t e r e d and e l i m i n a t i o n of s t r a y l i g h t from the c o l l e c t i o n o p t i c s becomes a major t e c h n i c a l p rob lem. A p o s s i b l e c i r c u m v e n t i o n to the problem of the low l e v e l s of Thomson s c a t t e r e d l i g h t from thermal f l u c t u a t i o n s i s to observe the s c a t t e r i n g from the enhanced d e n s i t y f l u c t u a t i o n s a r i s i n g from the pumped i n s t a b i l i t i e s . E l e c t r o n plasma waves are s u b j e c t to the d i s p e r s i o n r e l a t i o n : 2 2 2 2 co = co + 3/2 k v 7 e p e t h where cog and cop a re the e l e c t r o n wave and plasma f requency r e s p e c t i v e l y . Two plasmon decay modes occur at the d e n s i t y of 0.25 n so t h a t co i s known. The s c a t t e r e d spectrum shou ld show cr P 1 54 a peak at a f r e q u e n c y c o r r e s p o n d i n g to the e l e c t r o n wave f r e q u e n c y , by the f requency matching c o n d i t i o n : CJ = CJ + CJ . L e sca t By the measurement of the s c a t t e r e d spectrum and subsequent i d e n t i f i c a t i o n of CJ the e l e c t r o n temperature may then be S C e l t c a l c u l a t e d from the d i s p e r s i o n r e l a t i o n . Once a g a i n , t ime r e s o l v e d s p e c t r o s c o p y i s d e s i r a b l e to d i s c r i m i n a t e the s c a t t e r e d ruby l a s e r spectrum from n i t r o g e n l i n e s from recombin ing i o n s . The rema in ing q u e s t i o n s of the presence of beam f i l a m e n t a t i o n or ion a c o u s t i c t u r b u l e n c e a l s o remain t o p i c s t h a t warrant f u r t h e r i n v e s t i g a t i o n . 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S p e c t r o s . Transf . J_9 , 591 (1977) . 131) M. G a l a n t i and N . J . Peacock, J . Phys . B 1_4, 2427 (1975) . 164 132) R. P . M c W h i r t e r , "Plasma D i a g n o s t i c T e c h n i q u e s " e d i t e d by R. H . H u d d l e s t o n e , and S. L . Leonard (Academic P r e s s , New York 1965). 133) C . W. A l l e n , " A s t r o p h y s i c a l Q u a n t i t i e s " ( U n i v e r s i t y of London, A t h l o n e Pre s s 1955). 165 Appendix A N i t r o g e n l i n e e m i s s i o n e f f e c t on T D e t e r m i n a t i o n For a n i t r o g e n plasma of a few hundred eV tempera ture , the dominant r a d i a t i o n i s l i n e r a d i a t i o n from the h i g h l y i o n i z e d , i o n s . T h i s exceeds the r e c o m b i n a t i o n r a d i a t i o n and b r e m s s t r a h l u n g r a d i a t i o n combined by as much as four o r d e r s of magnitude ( Jacobs et a l . 1 3 0 ) at 10 eV. I t i s a p p r o p r i a t e to c o n s i d e r the e f f e c t of the l i n e r a d i a t i o n on the x - r a y measurements p a r t i c u l a r l y when t h i n f o i l s of low c u t - o f f energy are b e i n g u sed . The c o m p a r a t i v e l y s o f t l i n e r a d i a t i o n may be a p p r e c i a b l y t r a n s m i t t e d through these f o i l s c a u s i n g a background s i g n a l which upon t a k i n g r a t i o s of t r a n s m i t t e d x - r a y i n t e n s i t i e s may l e a d to an er roneous f i g u r e fo r the t empera ture . The c a l c u l a t i o n s performed to c o n s i d e r t h i s e f f e c t of the l i n e e m i s s i o n are p r e s e n t e d i n t h i s a p p e n d i x . For s i m p l i c i t y , the s teady s t a t e corona model i s used fo r the ion p o p u l a t i o n s . In a l a s e r produced p lasma, the p o p u l a t i o n d e n s i t i e s of v a r i o u s ion s p e c i e s i s a t ime dependent problem and r e q u i r e s the s o l u t i o n of a system of Z+1 c o u p l e d r a t e e q u a t i o n s ( M a g i l l 2 9 ) . H e r e , the e l e c t r o n d e n s i t y , temperature and hence c o l l i s i o n a l i o n i z a t i o n c o e f f i c i e n t s are a l l t ime dependent q u a n t i t i e s of the e l e c t r o n temperature and d e n s i t y de termined by the p h y s i c s of the l a s e r plasma i n t e r a c t i o n . N o n e t h e l e s s , the s teady s t a t e corona model i s an o f t e n used s i m p l i f i c a t i o n a p p l i e d to l a s e r plasmas (see e . g . Pep in et a l . 7 3 , Donaldson et a l . 7 2 ) . As shown by G a l a n t i and P e a c o c k 1 3 1 , the s teady s t a t e corona model 166 o v e r e s t i m a t e s the p o p u l a t i o n d e n s i t i e s of the h i g h e r charge s t a t e s but much l e s s so t h a t the LTE Saha e q u i l i b r i u m model . The l i n e e m i s s i o n , b remss t r ah lung and r e c o m b i n a t i o n c o n t i n u a and t h e i r t r a n s m i t t e d i n t e n s i t i e s through v a r i o u s f o i l s are c a l c u l a t e d u s i n g the computer program of F o n g 1 2 1 w i t h minor m o d i f i c a t i o n s as f o l l o w s . The a b s o r p t i o n p r o p e r t i e s of the s c i n t i l l a t o r have been taken i n t o account by c o n v o l u t i n g the b remss t r ah lung and r e c o m b i n a t i o n e m i s s i v i t i e s w i t h the f a c t o r (1 exp (X)x ) x b e i n g the t h i c k n e s s of the s c i n t i l l a t o r i n s c u n i t s of mg/cm 2 and n (X) i s the wavelength dependent mass a b s o r p t i o n c o e f f i c i e n t of the s c i n t i l l a t o r . The a n a l y t i c e x p r e s s i o n s fo r the mass a b s o r p t i o n c o e f f i c i e n t s of the s c i n t i l l a t o r and the b e r y l l i u m f o i l s had to be i n c o r p o r a t e d i n t o the program. These e x p r e s s i o n s a r e : for the s c i n t i l l a t o r : 2.88 2.38 M (X) x = 0.929 X + 0.120 X , sc sc for b e r y l l i u m f o i l s : 3.063 *i(X) = 0.269 X , f o r copper f o i l s : 2.68 M(X) = 118 X , where X i s i n Angstroms . An e x p r e s s i o n f o r the i n t e n s i t i e s of the l i n e e m i s s i o n i n the x - r a y r e g i o n was o b t a i n e d and summed t o the v a l u e of the c o n t i n u u a i n t e n s i t y i n the wavelength i n t e r v a l of 0.1 - 50 A . In g e n e r a l terms , the i n t e n s i t y of a l i n e t r a n s i t i o n p + q 167 of an i o n w i t h charge Z i s ( M c W h i r t e r 1 3 2 ) : e = hc/47rX A ( p , q ) N ( z , p ) , X where A ( p , q ) i s the t r a n s i t i o n p r o b a b l i t y and N ( z , p ) ' i s the p o p u l a t i o n of an ion of charge Z i n s t a t e p . In the corona model , the p o p u l a t i o n d e n s i t i e s of the e x c i t e d l e v e l s are de termined by a ba lance between the r a t e of c o l l i s i o n a l e x c i t a t i o n from the ground l e v e l ba l anced by the r a t e of spontaneous r a d i a t i v e decay : N N ( Z , g ) X(T , g , p ) e e N ( Z , p ) Z A ( p , q ) q<p where X ( p , q ) i s the c o l l i s i o n a l e x c i t a t i o n r a t e of l e v e l q to l e v e l p . In the c a l c u l a t i o n s , o n l y the t r a n s i t i o n s to the ground l e v e l are c o n s i d e r e d to be s u f f i c i e n t l y i n t e n s e so tha t the sum over q d i s a p p e a r s . T h e r e f o r e , the l i n e i n t e n s i t i e s are g i v e n b y : e = hc/47rX N N ( Z , g ) X(T , g , p ) and X(T , g , p ) e -4 n 6.5 x 10 E ( p , g ) T 1/2 f ( g , p ) e x p - ( E ( p , g ) / k T ) , where E ( p , g ) i s the e x c i t a t i o n p o t e n t i a l from the ground s t a t e ( i n e V ) , f ( g , p ) i s the o s c i l l a t o r s t r e n g t h . In u n i t s a p p r o p r i a t e to the i n c o r p o r a t i o n of the e x p r e s s i o n i n t o the computer program, the e x p r e s s i o n fo r the l i n e i n t e n s i t i e s i s w r i t t e n a s : 168 8 _ e = 6.38 x 10 X f ( g , p ) N(Z) X(A) E T p e J exp ( -E / k T ) P e The r e l e v a n t parameters ( A l l e n 1 3 3 ) a re g i v e n i n the T a b l e where o n l y the f i r s t few Lyman s e r i e s l i n e s a re c o n s i d e r e d : N V I N V I I WAVELEGTH E (eV) P f 23.771 521.55 0. 052 24.898 497.94 0. 144 28.787 430.67 0. 674 WAVELENGTH E (eV) f P 19.826 625.3 0.0290 20.910 592.9 0.0791 24.781 500.3 0.4162 Table A - l E x c i t a t i o n energies and o s c i l l a t o r strengths for N ion l i n e s . The c o n t r i b u t i o n s of the l i n e i n t e n s i t i e s t o the t o t a l r a d i a t e d i n t e n s i t y as a f u n c t i o n of the e l e c t r o n temperature i s shown i n F i g u r e A-1. S i n c e the s h o r t e s t wavelength of the l i n e s e r i e s i s 19 A the t r a n s m i t t e d i n t e n s i t y even through the t h i n n e s t f o i l s i s not a s i g n i f i c a n t e f f e c t . C a l c u l a t i o n s f o r the case of x - r a y t r a n s m i s s i o n through b e r y l l i u m f o i l s a re p r e s e n t e d i n F i g u r e A-3 were i t i s e v i d e n t t h a t the l i n e i n t e n s i t y must be i n c r e a s e d by a f a c t o r of 103 f o r the e l e c t r o n temperature measurement to be i n e r r o r by 50%. Figure A - 2 Change i n r e l a t i v e transmitted x-ray i n t e n s i t i e s due to x-ray l i n e emission. 170 Appendix B C a l c u l a t i o n of R a d i a t i v e Power Loss from the Plasma The r a d i a t i o n from the plasma i s r a d i a t e d as b r e m s s t r a h l u n g , r e c o m b i n a t i o n and l i n e r a d i a t i o n . In t h i s append ix , the r a d i a t i v e power l o s s i s c a l c u l a t e d e x p l i c i t l y for a plasma c h a r a c t e r i z e d by a 300 eV e l e c t r o n t e m p e r a t u r e . The e q u a t i o n s used here may be found i n P u e l l 6 0 . B remss t rah lung a r i s e s from the d e c e l e r a t i o n of e l e c t r o n s i n the Coulomb f i e l d of i o n s , and photons are e m i t t e d c o r r e s p o n d i n g to the change i n energy . The t o t a l b remss t r ah lung power P e m i t t e d per u n i t volume i s -32 1/2 2 3 P = 1 . 5 x 1 0 T n Z n Z , (W/cm ) f f e e Z i where the Gaunt f a c t o r s are taken to be one and T i s i n eV. The e sum i s over the charge s t a t e s . A s teady s t a t e c o r o n a l e q u i l i b r i u m c a l c u l a t i o n shows t h a t at 300 eV, 91% of the N atoms are f u l l y i o n i z e d , the rema in ing b e i n g N VII i o n s . Recombinat ion r a d i a t i o n a r i s e s from f ree e l e c t r o n s b e i n g c a p t u r e d by i o n s . For the n i t r o g e n p lasma, r e c o m b i n a t i o n r a d i a t i o n exceeds b remss t r ah lung (Appendix A ) . In the f requency range hv > X where X i s the i o n i z a t i o n p o t e n t i a l of an i o n w i t h charge Z-1 for e l e c t r o n s i n the n th s h e l l , r e c o m b i n a t i o n r a d i a t i o n and bremss t rah lung show the same s p e c t r a l b e h a v i o u r . At hv = X the r e c o m b i n a t i o n r a d i a t i o n spectrum shows a d i s c o n t i n u i t y , s i n c e t h i s case c o r r e s p o n d s to the c a p t u r e of a 171 r e s t i n g f r ee e l e c t r o n by an i o n . The t o t a l power e m i t t e d per u n i t volume as r e c o m b i n a t i o n r a d i a t i o n i s -31 1/2 2 P = 2 x 1 0 n T I n (Z) I (X / X ) _ / n , fb e e Z i n n H n where T i s i n eV, n ^ Z ) i s the d e n s i t y of ions of charge s t a t e Z, n i s the p r i n c i p a l quantum number and _ £ n i s the number of unoccup ied s i t e s i n the n th s h e l l . The r e l e v a n t parameters are g i v e n i n T a b l e B-1 Z + 1 NVIII NVII NVI NV NIV N i l I N i l z 7 6 5 4 3 2 1 X 6 6 7 5 5 2 9 7 . 9 7 7 . 5 4 7 . 4 2 9 . 6 1 4 . 5 n 1 1 2 2 2 2 2 2 1 6 7 6 5 4 Table B-1 Relevant q u a n t i t i e s f o r recombination r a d i a t i o n from a nitrogen plasma. When t h e r e are ions w i t h one or more bound e l e c t r o n s p r e s e n t i n the p la sma , the e x c i t a t i o n of these e l e c t r o n s and the c o r r e s p o n d i n g l i n e r a d i a t i o n must be c o n s i d e r e d . Shown i n Appendix A i s t h a t the l i n e r a d i a t i o n exceeds the b remss t r ah lung and r e c o m b i n a t i o n r a d i a t i o n combined f o r a plasma of a few hundred eV t e m p e r a t u r e . As i n Appendix A , o n l y the c o n t r i b u t i o n of Lyman s e r i e s l i n e s of the N VI and N VII a r e c o n s i d e r e d as b e i n g s u f f i c i e n t l y i n t e n s e . The power e m i t t e d as l i n e r a d i a t i o n 1 72 per u n i t volume i s -25 -1/2 P = 3.5 x 10 T n L n (Z) exp ( - E ( Z ) / k T ) , 1 e e Z i e where the e x c i t a t i o n e n e r g i e s are g i v e n i n Appendix A . Assuming an e l e c t r o n d e n s i t y of 2 x 1 0 1 8 c m - 3 and a temperature of 300 eV, the r a d i a t i o n power l o s s from the plasma i s P + P + P = ( 7 X 1 0 6 + 2 X 1 0 7 + 5 X 1 0 8 ) « W/cm 3 f f fb 1 The volume of the plasma a t the end of the l a s e r p u l s e i s t y p i c a l l y 4 x 10~ 3 cm 3 so tha t the power l o s s through r a d i a t i o n i s 2 x 10^ W a t t s . In c o n t r a s t , the i n p u t l a s e r power i s t y p i c a l l y 5 x 10 9 W a t t s . T h e r e f o r e , the r a d i a t i o n power from the plasma i s up to 3 o r d e r s of magnitude s m a l l e r than the l a s e r input power. T h i s has important consequences i n the U l b r i c h t sphere measurements and i n the c a l c u l a t i o n of the energy ba lance as d e s c r i b e d i n the t e x t . 

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