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Hydrodynamics of laser-driven ablation in planar targets Celliers, Peter Martin 1983

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HYDRODYNAMICS OF LASER-DRIVEN ABLATION IN PLANAR TARGETS by PETER MARTIN CELLIERS B.Sc. (Eng.) Queen's U n i v e r s i t y , 1980 THESIS SUBMITTED IN PARTIAL FULLFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n THE FACULTY OF GRADUATE STUDIES Department of P h y s i c s , Plasma P h y s i c s Group We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA September 1983 © Peter M a r t i n C e l l i e r s , 1983 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y a v a i l a b l e for reference and study. I further agree that permission for extensive copying of t h i s t h e s i s f o r scholarly purposes may be granted by the head of my department or by h i s or her representatives. I t i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of 1 V\ y 5 (" c S  The University of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date IZ October \^S3 DE-6 (3/81) i i ABSTRACT A comprehensive experimental and t h e o r e t i c a l study of the hydrodynamics of steady s t a t e • l a s e r - d r i v e n a b l a t i o n i n t h i n p l a n a r t a r g e t s i s g i v e n . Experiments were performed using a f r e q u e n c y - t r i p l e d Neodymium-glass l a s e r which provided a 2 ns FWHM l a s e r pulse at .355 Mm wavelength. T h i s was focussed to an 80 um spot diameter on t h i n (6 Mm to 50 Mm) aluminum t a r g e t s . Target i r r a d i a n c e s up to 2 x 1 0 1 3 W/cm2 were a c h i e v e d . F i r s t l y , t h e o r e t i c a l s c a l i n g laws f o r mass a b l a t i o n r a t e and a b l a t i o n pressure are reviewed. These are compared with experimental r e s u l t s obtained from ion c a l o r i m e t e r and Faraday cup measurements. Secondly, t a r g e t motion was s t u d i e d u s i n g these d i a g n o s t i c s and the r e s u l t s were i n t e r p r e t e d by viewing the t a r g e t as a compressible f l u i d (shock compressed) which subsequently r a r e f i e s . Target r a r e f a c t i o n was a l s o d i r e c t l y observed through measurements of l a s e r t r a n s m i s s i o n or burnthrough. The compressible f l u i d approach was • used t o c a l c u l a t e the hydrodynamic e f f i c i e n c y . T h i s c a l u l a t i o n r e q u i r e s as parameters the l a s e r i n t e n s i t y , wavelength and t a r g e t m a t e r i a l . I t p r e d i c t s a hydrodynamic e f f i c i e n c y of only 5 % at 1 0 1 3 W/cm2, .355 Mm l a s e r l i g h t on aluminum t a r g e t s , i n agreement with our experimental measurements. TABLE OF CONTENTS ABSTRACT i i TABLE OF CONTENTS i i i LIST OF FIGURES v i i ACKNOWLEDGEMENTS ix CHAPTER I INTRODUCTION 1 CHAPTER I I . THEORY OF LASER DRIVEN ABLATION: GENERAL REVIEW 4 II-1 STRUCTURE OF LASER INDUCED FLOW 5 I1-2 STEADY STATE PLANAR ABLATION MODEL 10 I I - 2 - i ISOTHERMAL RAREFACTION 13 I I - 2 - i i THE ABLATION ZONE 21 I I - 2 - i i i MATCHING AT THE ABSORPTION SURFACE 26 I I - 2 - i v ABLATION SCALING LAWS 29 II-2-v VALIDITY OF THE ABLATION MODEL ... 33 II-3 SUMMARY AND EXPECTED EXPERIMENTAL SCALINGS 45 I I - 3 - i SUMMARY OF RESULTS 45 i v I I - 3 - i i EXPECTED EXPERIMENTAL SCALINGS ... 46 CHAPTER I I I HYDRODYNAMIC EFFECTS INDUCED BY LASER DRIVEN ABLATION 47 II1-1 SHOCK PROPAGATION IN SOLIDS 48 II1-2 TARGET RAREFACTION AND DYNAMICS 52 I I I - 2 - i AXIAL RAREFACTION AND LASER BURNTHROUGH 55 I I I - 2 - i i RADIAL RAREFACTION 64 I I I - 3 HYDRODYNAMIC EFFICIENCY 69 I I I - 3 - i DEFINITION OF HYDRODYNAMIC EFFICIENCY 70 I I I - 3 - i i MEASUREMENT OF HYDRODYNAMIC EFFICIENCY 76 CHAPTER IV. EXPERIMENTAL FACILITY AND DIAGNOSTICS 80 IV- 1 LASER FACILITY 81 IV-2 THE FARADAY CUP 84 IV- 2 - i FARADAY CUP DESIGN 84 I V - 2 - i i FARADAY CUP EXPERIMENTAL TECHNIQUE 87 IV-3 THE ION CALORIMETER 92 I V - 3 - i ION CALORIMETER DESIGN 93 I V - 3 - i i ION CALORIMETER EXPERIMENTAL TECHNIQUE 98 V CHAPTER V ABLATION SCALING LAWS 101 V-1 ABLATION MEASUREMENTS 102 V - 1 - i MEASUREMENT PRINCIPLE 102 V - 1 - i i EXPERIMENTAL SETUP AND DATA REDUCTION 104 V- 2 RESULTS AND COMPARISON WITH THEORY 108 V- 2 - i EXPERIMENTAL SCALINGS 108 V- 2 - i CORRECTION FOR REFERENCE FRAME TRANSFORMATION 112 V - 2 - i i i CORRECTION DUE TO SPHERICAL DIVERGENCE 114 V-2 - i v COMPARISON INCLUDING CORRECTIONS 115 CHAPTER VI MEASUREMENT OF BURNTHROUGH TIME 117 VI- 1 EXPERIMENT 118 VI- 2 RESULTS AND DISCUSSIONS 121 CHAPTER VII DYNAMICS OF ACCELERATED THIN TARGETS 128 VII - 1 EXPERIMENT 129 V I I - 1 - i REAR SIDE SETUP 129 V I I - 1 - i i FRONT SIDE SETUP 130 V I I - 1 - i i i DATA ANALYSIS AND RESULTS 130 V I I - 2 DISCUSSION OF RESULTS 140 V I I - 2 - i ANGULAR PROFILES OF ION ENERGY ...140 V I I - 2 - i i ENERGY BALANCE 144 v i V I I - 2 - i i i EXPERIMENTAL DETERMINATION OF HYDRODYNAMIC EFFICIENCY 145 CHAPTER VIII CONCLUSIONS AND FINAL DICUSSIONS 146 VI11-1 CONCLUSIONS 147 VIII-2 FINAL DISCUSSIONS 148 V I I I - 2 - i CONTRIBUTIONS TO CURRENT DATA BASE 148 V I I I - 2 - i i MODIFICATION OF CURRENT UNDERSTANDING 149 REFERENCES 151 APPENDIX A DERIVATION OF EQUATIONS (II1-17) TO (111-22) .155 A-1 EVALUATION OF EQUATIONS (111-17), (III-18) AND (111-19) 155 A-2 EVALUATION OF EQUATIONS (111-20) , (111-21) AND (111-22) 160 v i i LIST OF FIGURES 1 STRUCTURE OF LASER INDUCED FLOW 6 2 AN X-T DIAGRAM OF LASER INDUCED FLOW 9 3 REFERENCE FRAME FOR STEADY STATE ABLATIVE FLOW 12 4 ISOTHERMAL EXPANSION PROFILES FOR A COLLISIONLESS PLASMA 18 5 STREAMLINES FOR FINITE SPOT DIVERGING FLOW AND FOR PLANAR FLOW 37 6 SCHEMATIC FLOW PROFILES FOR SHOCK UNLOADING INTO VACUUM 57 7 X-T DIAGRAM OF SHOCK RAREFACTION SEQUENCE IN FINITE THICKNESS PLANAR TARGETS 59 8 DENSITY PROFILES AT VARIOUS TIMES DURING THE SHOCK RAREFACTION SEQUENCE 60 9 SHOCK UNLOADING OF TARGET REAR AS A TWO-DIMENSIONAL SUPERSONIC AXISYMMETRIC FLOW 68 10 DIAGRAM OF EXPERIMENTAL FACILITY INCLUDING LASER BEAM LINE AND TARGET CHAMBER 82 11 FARADAY CUP MECHANICAL CONSTRUCTION 86 12 FARADAY CUPS ELECTRICAL CONNECTIONS 88 13 SAMPLE FARADAY CUP TRACE OF ION STREAM FROM STEADY STATE ABLATION 91 14 ION CALORIMETER MECHANICAL CONSTRUCTION 94 15 ION CALORIMETER ELECTRICAL SYSTEM 97 16 SAMPLE TRACES OF ION CALORIMETER SIGNALS 99 17 DETECTOR ARRAY FOR ION ABLATION MEASUREMENT 105 18 EXPERIMENTAL E(0) AND V(0) AND CORRESPONDING SPLINE FITS FOR A TYPICAL SHOT 107 19 INTENSITY DEPENDENCE OF MASS ABLATION RATE, EXPERIMENTAL AND THEORETICAL 109 v i i i 20 INTENSITY DEPENDENCE OF ABLATION PRESSURE, EXPERIMENTAL AND THEORETICAL 110 21 EXPERIMENTAL CONFIGURATION FOR BURNTHROUGH MEASUREMENTS .119 22 SAMPLE PHOTODIODE SIGNALS OBTAINED DURING BURNTHROUGH MEASUREMENTS 120 23 BURNTHROUGH TIME T g VERSUS TARGET THICKNESS D 122 24 FRACTIONAL TRANSMITTED AMPLITUDE AT PEAK TRANSMISSION VERSUS TARGET THICKNESS 123 25 CALCULATED SHOCK SPEED AND PRESSURE AS A FUNCTION OF COMPRESSION RATIO USING MEASURED T p ( D ) DATA AT 1.2 x 10' 3 W/cm2 ° 126 26 FRONT SIDE ION ENERGY ANGULAR DISTRIBUTIONS FOR VARIOUS TARGET THICKNESSES AT 1.2 x 1 0 1 3 W/cm2 132 27 REAR SIDE ION ENERGY ANGULAR DISTRIBUTIONS FOR VARIOUS TARGET THICKNESSES AT 1.2 x 1 0 1 3 W/cm2 133 28 FRONT SIDE FARADAY CUP TRACES FOR 6 Mm AND 50 nm TARGETS AT 1.2 x 1 0 1 3 W/cm2 134 29 REAR SIDE FARADAY CUP TRACES FOR 6 Mm AND 50 Mm TARGETS AT 1.2 x 1 0 1 3 W/cm2 135 30 REAR SIDE FARADAY CUP TRACES FOR A 25 Mm TARGET AT THREE ANGULAR POSITIONS AT 1.2 x 10' 3 W/cm2 136 31 REAR SIDE ION ENERGY ANGULAR DISTRIBUTIONS FOR VARIOUS LASER INTENSITIES ON 12.5 Mm TARGETS 137 32 REAR SIDE ION ENERGY ANGULAR DISTRIBUTIONS FOR VARIOUS LASER INTENSITIES ON 25 Mm TARGETS 138 33 ENERGY BALANCE FOR VARIOUS TARGET THICKNESSES AT 1.2 x 1 0 ' 3 W/cm2 139 i x ACKNOWLEDGEMENTS I would l i k e to thank my s u p e r v i s o r Dr. Andrew Ng f o r p r o v i d i n g the o p p o r t u n i t y to work with him, as w e l l as f o r h i s t i r e l e s s enthusiasm and encouragement. A great deal of a s s i s t a n c e i n running the experimental equipment was r e c e i v e d from D a n i e l P a s i n i , Andrew Ng, Dr. Joe Kwan, Dean Parfeniuk and L u i z ,DaSilva. Alan Cheuck was of g r e a t h e l p i n c o n s t r u c t i n g and p e r f e c t i n g the c a l o r i m e t e r a m p l i f i e r s . Many of the t h e o r e t i c a l i n t e r p r e t a t i o n s arose out of d i s c u s s i o n s with Andrew Ng. Both Andrew Ng and Joe Kwan were i n v o l v e d i n c o n c e i v i n g and developing the burnthrough model. Th i s r e s e a r c h was supported by NSERC Operating Grants, NSERC S t r a t e g i c Grants and B r i t i s h Columbia Hydro Fusion Grants. 1 CHAPTER I INTRODUCTION A scheme t o compress matter to high d e n s i t i e s u s i n g high i n t e n s i t y l a s e r r a d i a t i o n was f i r s t o u t l i n e d N u c k o l l s e t a l . [1] and Brueckner [ 2 ] . The o b j e c t i v e was to provide a means of i g n i t i n g a c o n t r o l l e d thermonuclear f u s i o n r e a c t i o n . T h i s was the i n i t i a l m o t i v a t i o n f o r many s c i e n t i f i c s t u d i e s of 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 matter. With t h i s impetus the f i e l d of l a s e r - m a t t e r i n t e r a c t i o n s has grown r a p i d l y over the past ten y e a r s . Formidable problems, both experimental and t h e o r e t i c a l , have been t a c k l e d w i t h i n t h i s g e neral area of r e s e a r c h . Laser-matter i n t e r a c t i o n s have a l s o opened up p r e v i o u s l y u n a t t a i n a b l e and t h e r e f o r e unexplored r e g i o n s of energy d e n s i t y and temperature. I t has r e v e a l e d i n t e r e s t i n g problems in n o n - l i n e a r plasma dynamics such as parametric i n s t a b i l i t i e s [3] and anomalous t r a n s p o r t phenomena [ 4 ] . Moreover i t can generate p r e s s u r e s i n s o l i d m a t e r i a l s i n excess of s e v e r a l tens of Mbar [ 5 ] , a pressure regime not a c h i e v a b l e with c o n v e n t i o n a l e x p l o s i v e compression methods. C e n t r a l to the compression scheme i s the uniform implosion of a s p h e r i c a l l y shaped p e l l e t . Implosion experiments have been performed with s p h e r i c a l t a r g e t s of v a r i o u s d e s i g n s and c o n s t r u c t i o n s [ 6 ] . Studies of steady s t a t e a b l a t i o n i n planar t a r g e t geometries have a l s o been conducted at v a r i o u s l a b o r a t o r i e s [ 7 ] . T h e o r e t i c a l and experimental understanding 2 has i d e n t i f i e d fundamental s c a l i n g laws which i n d i c a t e that e f f i c i e n t energy c o u p l i n g and steady s t a t e a b l a t i o n i s most f a v o u r a b l y achieved with short wavelength (sub-micron), moderate i n t e n s i t y ( < 1 0 1 S W/cm2) and long p u l s e ( > 1 ns) l a s e r i r r a d i a t i o n [8-13]. Planar t a r g e t experiments are designed to to address the hydrodynamic i s s u e s of the inward a c c e l e r a t i o n of a s p h e r i c a l p e l l e t s h e l l d u r i n g the i n i t i a l moments of the implosion. Planar geometry i s chosen as an approximation of a small s e c t i o n of the p e l l e t s h e l l which permits easy d i a g n o s t i c access of the rear s i d e of the t a r g e t . Target motions have been © measured by v a r i o u s methods- [7,8,13,27], but a l l i n t e r p r e t a t i o n s have u t i l i z e d the analogy of a p l a n a r 'rocket' [14,15,16] to d e s c r i b e t a r g e t a c c e l e r a t i o n , and to assess c o n v e r s i o n e f f i c i e n c y of l a s e r energy i n t o t a r g e t k i n e t i c energy. Shock propagation s t u d i e s have been performed [17,18,34], but c u r r e n t l i t e r a t u r e d e a l i n g with the e f f e c t s of the shock wave on t a r g e t motion i s sc a r c e . The i n v e s t i g a t i o n presented i n t h i s t h e s i s i s a l s o c e n t r e d around l a s e r i n t e r a c t i o n s with planar t a r g e t s . The l a s e r i s operated i n the sub-micron (.355 /im) low i n t e n s i t y ( 1 0 1 3 W/cm2) and long p u l s e (2 ns) regime which has been i d e n t i f i e d as favour a b l e f o r a b l a t i v e i m p l o s i o n . Steady s t a t e a b l a t i o n i s set up very e a r l y i n the p u l s e , and with t h i s i n t e n s i t y one expects strong Inverse Bremmstrahlung a b s o r p t i o n . The p r i n c i p l e o b j e c t i v e of t h i s t h e s i s r e s e a r c h i s to o b t a i n a comprehensive understanding of the hydrodynamics of 3 l a s e r d r i v e n a b l a t i o n i n planar t a r g e t s . F i r s t l y , experimental v e r i f i c a t i o n of the steady s t a t e a b l a t i o n theory i s o b tained through measurements of the i n t e n s i t y s c a l i n g of the a b l a t i o n r a t e and a b l a t i o n p r e s s u r e . T h i s c o n s t i t u t e s p a r t of a complete study of wavelength e f f e c t s on a b l a t i o n d r i v e n by sub-micron (.53, .355 and .27 Mm) l a s e r r a d i a t i o n . Secondly, the behaviour of a c c e l e r a t e d planar t a r g e t s i s examined t h e o r e t i c a l l y and e x p e r i m e n t a l l y using a compressible f l u i d approach r a t h e r than using the s i m p l i f i e d r i g i d body (rocket ) i n t e r p r e t a t i o n . The new e x p l a n a t i o n s of the t a r g e t motion l e a d to a s e l f - c o n s i s t e n t understanding of a l l the experimental o b s e r v a t i o n s . As a consequence of t h i s r e v i s e d p i c t u r e a new d e f i n i t i o n of hydrodynamic e f f i c i e n c y i s proposed using the steady s t a t e p l a n a r a b l a t i o n model. T h i s d e f i n i t i o n y i e l d s an e x p l i c i t e x p r e s s i o n which has been v e r i f i e d e x p e r i m e n t a l l y i n a manner c o n s i s t e n t with the new understanding. Chapter II d e c r i b e s i n d e t a i l the steady s t a t e a b l a t i o n p r o c e s s . T h i s i s f o l l o w e d , i n chapter I I I , by a d e s c r i p t i o n of the b a s i c hydrodynamic i n t e r a c t i o n s governing the experiment. Chapter IV c o n t a i n s d e t a i l s c o n cerning the l a s e r and t a r g e t chamber f a c i l i t y and the d i a g n o s t i c s . Experimental d e t a i l s , r e s u l t s and d i s c u s s i o n s of three d i f f e r e n t i n v e s t i g a t i o n s are then d e s c r i b e d i n c h a p t e r s V, VI and V I I . Chapter VIII summarises the main r e s u l t s and p r e s e n t s a f i n a l d i s c u s s i o n and c o n c l u s i o n s . 4 CHAPTER II THEORY OF LASER DRIVEN ABLATION: GENERAL REVIEW In t h i s chapter a c o n c e p t u a l p i c t u r e of the l a s e r - t a r g e t i n t e r a c t i o n i s presented. The chapter d e a l s mainly with c u r r e n t understanding of l a s e r d r i v e n a b l a t i o n as i t a p p l i e s to sub-micron wavelengths and multi-nanosecond l a s e r p u l s e s . The d e s c r i p t i o n given does not conform completely to any one t h e o r e t i c a l model encountered i n the l i t e r a t u r e . The models of Manheimer and Colombant,[12] and Mora [11] c o n t a i n most of the b a s i c ideas c o n c e r n i n g a b l a t i o n . However, a great d e a l of background work i s c o n t a i n e d i n many other r e f e r e n c e s [1,2,5,9,10,14,20-23,26,27]. An important purpose of t h i s review of the a b l a t i o n process i s to provide a framework f o r d i s c u s s i n g the hydrodynamic processes induced i n the t a r g e t r e g i o n s f u r t h e r ahead of the a b l a t i o n f r o n t (chapter I I I ) . S e c t i o n 11 — 1 g i v e s a broad survey of the e n t i r e l a s e r -t a r g e t i n t e r a c t i o n p r o c e s s . The main f e a t u r e s of the flow are i d e n t i f i e d to g i v e a q u a l i t a t i v e p i c t u r e . D e t a i l s of the s t r u c t u r e of the l a s e r - i n d u c e d flow are then d e s c r i b e d i n s e c t i o n I I - 2 . The p h y s i c a l processes o p e r a t i n g w i t h i n each region of the flow are d i s c u s s e d and simple a n a l y t i c s c a l i n g laws are d e r i v e d to determine the mass a b l a t i o n r a t e and a b l a t i o n p r e s s u r e . F i n a l l y , s e c t i o n II-3 g i v e s a b r i e f summary of the main p o i n t s and p r e d i c t s how these r e l a t e to 5 experimental measurements. I1-1 STRUCTURE OF LASER INDUCED FLOW Fi g u r e 1 i l l u s t r a t e s a c o n c e p t u a l p i c t u r e of the d e n s i t y , temperature and p a r t i c l e v e l o c i t y p r o f i l e s i n a t y p i c a l s o l i d t a r g e t experiment. The energy input from the l a s e r d r i v e s the flow; i t i s d e p o s i t e d i n region (1) up to the i n t e r f a c e between regions (1) and ( 2 ) . Laser l i g h t does not penetrate t h i s s u r f a c e . Region (1) extends from t h i s a b s o r p t i o n s u r f a c e i n t o the vacuum, and i s c a l l e d the corona. In t h i s r e g i o n an i s o t h e r m a l r a r e f a c t i o n wave [20-23,29] a c c e l e r a t e s t a r g e t m a t e r i a l s i n t o the vacuum. The r a r e f a c t i o n wave head j o i n s onto r e g i o n (2) and the wave propagates upstream i n t o the flow coming out of region ( 2 ) . Hence the p o s i t i o n of the boundary between regions (1) and (2) depends on the r e l a t i v e v e l o c i t i e s of the propagating r a r e f a c t i o n and the f l u i d flow. Region (2), which separates the corona from the dense t a r g e t , i s c a l l e d the a b l a t i o n zone. In t h i s r e g i o n no l a s e r energy p e n e t r a t e s ; i n s t e a d , e l e c t r o n heat conduction and X-ray t r a n s p o r t c a r r i e s energy to the a b l a t i o n s u r f a c e and i s balanced by a r e t u r n f l u x of a c c e l e r a t e d and heated t a r g e t m a t e r i a l s f l o w i n g towards the a b s o r p t i o n zone. The a b l a t i o n s u r f a c e d i v i d i n g regions (2) and (3) i s the s u r f a c e a t which dense t a r g e t m a t e r i a l s , e n e r g i s e d by the heat f l u x from the conduction zone are evaporated and a c c e l e r a t e d - or 'ablated' - i n t o the flow. 6 F i g u r e j_ S t r u c t u r e of l a s e r induced flow. P r o f i l e s of f l u i d d e n s i t y - p, v e l o c i t y - v, ion temperature - T: , and e l e c t r o n temperature - T e are shown. 1 7 The a b l a t e d t a r g e t m a t e r i a l s are a c c e l e r a t e d to la r g e v e l o c i t i e s i n the flow and i n order to balance momentum a large s t a t i c pressure e x i s t s i n the dense t a r g e t ( r e g i o n ( 3 ) ) . T h i s s t a t i c p r essure i s of the order of s e v e r a l m i l l i o n atmospheres. A complete d e s c r i p t i o n r e q u i r e s the e x i s t e n c e of a shock d i s c o n t i n u i t y propagating i n t o the unperturbed t a r g e t (region ( 4 ) ) . A l l t h e o r i e s of l a s e r - s o l i d t a r g e t i n t e r a c t i o n s are b u i l t on a s t r u c t u r e of t h i s type [1,2,5,9-12,14,26,27], The reason that such a flow s t r u c t u r e must occur i s a consequence of the e x i s t e n c e of a t h r e s h o l d e l e c t r o n d e n s i t y , n c r . (The symbol 'n' s h a l l be used to denote e l e c t r o n d e n s i t y . ) For n > n c r (CJ^) l i g h t of frequency O J ^ i s r e f l e c t e d from a plasma. Laser l i g h t i s monochromatic, hence i t has a w e l l - d e f i n e d frequency (or wavelength, X^, i n vacuum). n c r (CJ^) i s the e l e c t r o n d e n s i t y at which the e l e c t r o n plasma frequency, cjp, becomes equal to the l a s e r frequency, w^ . T h i s q u a n t i t y i s given by the r e l a t i o n , 2 2 2 n c r - * C m e / e X L II-1 n ( c m - 3 ) = (1.1 x 1 0 2 1 ) x 7 2 ( u m ) c r i* In t h i s e x p r e s s i o n X^ i s the vacuum l a s e r wavelength and i t i s evi d e n t that n c r i s i n v e r s e l y p r o p o r t i o n a l t o X,2. Hence 8 shorter-wavelength r a d i a t i o n w i l l penetrate i n t o higher d e n s i t y regions of s o l i d t a r g e t s . Once the l a r g e energy f l u x from the l a s e r i s e s t a b l i s h e d at d e n s i t y n c r then the heat wave (region ( 2 ) ) , the shock compression ( r e g i o n ( 3 ) ) , and the r a r e f a c t i o n ( r e g i o n (1)) are simple consequences of mass, momentum and energy c o n s e r v a t i o n . The l a s e r energy may a l s o be completely absorbed at a d e n s i t y n 0 l e s s than n c r due to stong c o l l i s i o n a l (Inverse Bremsstrahlung) a b s o r p t i o n [11,19]. In t h a t case the l a s e r f l u x i s e s t a b l i s h e d at d e n s i t y n 0 . The temporal e v o l u t i o n of t h i s flow s t r u c t u r e has hot yet been mentioned. The p i c t u r e i n F i g u r e 1 i s o b v i o u s l y much d i f f e r e n t from the i n i t i a l c o n d i t i o n s of zero flow v e l o c i t y , zero temperature and a step jump from vacuum to s o l i d d e n s i t y . A d e t a i l e d time h i s t o r y of the t a r g e t flow from the moment the l a s e r i s turned on i s beyond the grasp of a n a l y t i c c a l c u l a t i o n s . Most a n a l y t i c t h e o r i e s use a steady s t a t e h y p o t h e s i s assuming that the s t r u c t u r e in F i g u r e 1 i s constant i n time. In f a c t t h i s i s a good hypothesis f o r the long p u l s e , sub-micron wavelength l a s e r r a d i a t i o n used i n our experiment. In the f o l l o w i n g d e s c r i p t i o n i t w i l l be assumed t h a t the flow s t r u c t u r e of F i g u r e 1 i s q u i c k l y e s t a b l i s h e d and i s d r i v e n by a constant l a s e r i n t e n s i t y . J u s t i f i c a t i o n f o r t h i s assumption w i l l be d i s c u s s e d l a t e r . F i g u r e 2 i s an x-t diagram showing how the v a r i o u s s u r f a c e s and regions of F i g u r e 1 evolve with time a f t e r the l a s e r i s turned on. The diagram d e p i c t s the i n t e r a c t i o n i n the l a b o r a t o r y frame. I t assumes t h a t the l a s e r p u l s e i s a step F i g u r e 2 An x-t diagram of l a s e r induced flow. The l a s e r power turned on to a constant l e v e l at t = 0 . 10 f u n c t i o n of constant i n t e n s i t y . An important aspect of the flow r e v e a l e d by t h i s diagram i s that t a r g e t compression and a c c e l e r a t i o n i s achieved p r i m a r i l y by the shock wave. Further a c c e l e r a t i o n and compression may be achieved only by i n c r e a s i n g the s t a t i c a b l a t i o n p r e s s u r e at the a b l a t i o n s u r f a c e so that another shock wave i s launched i n t o the t a r g e t m a t e r i a l s . T h i s conceptual d e s c r i p t i o n i s the g e n e r a l scheme upon which hydrodynamic t h e o r i e s of s o l i d t a r g e t i n t e r a c t i o n s are based. F i g u r e s 1 and 2 are q u a l i t a t i v e r e p r e s e n t a t i o n s that are a p p r o p r i a t e f o r e i t h e r p l a n a r or s p h e r i c a l l y symmetric flows. Note, however, that the converging geometry i n s p h e r i c a l flows s t r o n g l y m o d i f i e s " the p i c t u r e when the s c a l e l e n g t h (sphere r a d i u s ) becomes s m a l l . II-2 STEADY STATE PLANAR ABLATION MODEL In t h i s s e c t i o n a d e t a i l e d d e s c r i p t i o n i s presented of the steady s t a t e planar a b l a t i o n model. The o b j e c t i v e i s to d e r i v e a number of simple s c a l i n g laws r e l a t i n g the mass ab a t i o n r a t e m, and the a b l a t i o n p r e s s u r e P a, to the l a s e r i n t e n s i t y , wavelength and t a r g e t m a t e r i a l s . In the context of F i g u r e s 1 and 2, the f o l l o w i n g d e s c r i p t i o n encompasses regions (1) and (2) of the flow which are r e s p e c t i v e l y the underdense (n < n 0) corona and overdense (n > n 0 ) a b l a t i o n zone. A number of s i m p l i f y i n g assumptions w i l l now be i n t r o d u c e d which are necessary f o r a t r a c t a b l e 11 s o l u t i o n as w e l l as f o r p h y s i c a l c l a r i t y : ( i ) the flow i s assumed to be one-dimensional i n c a r t e s i a n c o o r d i n a t e s , and t h e r e f o r e i s o t r o p i c i n planes p a r a l l e l to the t a r g e t s u r f a c e . The ch o i c e of planar geometry i s made because i t most c l o s e l y resembles the c o n d i t i o n s c r e a t e d i n the experiment. ( i i ) a s t e a d y - s t a t e flow e x i s t s i n the a b l a t i o n zone, and t h e r e f o r e a constant s e p a r a t i o n i s maintained at a l l times between the a b l a t i o n s u r f a c e and the energy absorbing s u r f a c e . ( i i i ) energy d e p o s i t i o n takes p l a c e i n the c o r o n a l region up t o a maximum e l e c t r o n d e n s i t y n 0 and w i l l be repr e s e n t e d mathematically as a d e l t a f u n c t i o n source term i n the energy e q u a t i o n . Other assumptions necessary f o r a r e a l i s t i c and t r a c t a b l e s o l u t i o n w i l l be i n t r o d u c e d i n the d i s c u s s i o n of the s p e c i f i c r e g i o n s . However, these s h a l l apply only l o c a l l y to the r e s p e c t i v e r e g i o n . F i g u r e 3 d e p i c t s the r e f e r e n c e frame i n which the f o l l o w i n g d e s c r i p t i o n w i l l apply. P o i n t s on the x-a x i s are i d e n t i f i e d w i t h planes p a r a l l e l t o the t a r g e t . The a b l a t i o n and energy a b s o r p t i o n s u r f a c e s are f i x e d to the a x i s at the p o i n t s x = 0 and x = x 0 r e s p e c t i v e l y . The flow p r o p e r t i e s of the two regions (1) and (2) w i l l now be d e s c r i b e d s e p a r a t e l y . S o l u t i o n s of the flow i n these regions are then matched at the energy a b s o r p t i o n surface x = x 0 . A l l temperatures are i n energy u n i t s (eV); the s u b s c r i p t ' 0 ' denotes the value of a q u a n t i t y at the energy 1 2 •'' ' i iL,mnwi«i» niipiinii in , nm t j C O R O N A L A S E R C O M P R E S S E D T A R G E T 0 F i g u r e 3 Reference frame f o r steady s t a t e a b l a t i v e flow. The a b l a t i o n and a b s o r p t i o n s u r f a c e s are f i x e d i n t h i s r e f e r e n c e frame. 13 a b s o r p t i o n surface x = x 0 . I I - 2 - i ISOTHERMAL RAREFACTION During the i n i t i a l moments of plasma formation there e x i s t s a dense plasma unbounded on one s i d e (vacuum). The d e n s i t y p r o f i l e i s i n i t i a l l y a st e p jump from vacuum to the unperturbed plasma. I t decays i n t o a r a r e f a c t i o n wave which propagates back i n t o the dense plasma at the i o n - a c o u s t i c speed, and a c c e l e r a t e s ions i n t o the vacuum. The l a s e r i s on du r i n g t h i s process and maintains isothermal c o n d i t i o n s f o r the e l e c t r o n d i s t r i b u t i o n i n the re g i o n of energy d e p o s i t i o n . E l e c t r o n - i o n c o l l i s i o n s are i n f r e q u e n t i n the underdense corona so t h a t the isothermal c o n d i t i o n s extend s p a t i a l l y out i n t o the wave from the a b s o r p t i o n s u r f a c e . There i s an energy f l u x c a r r i e d out to the wave by the e l e c t r o n d i s t r i b u t i o n which i s e q u i v a l e n t to a heat f l u x . The nature of the r a r e f a c t i o n s o l u t i o n as w e l l as the heat f l u x r e q u i r e d to d r i v e the wave determine the matching of t h i s type of flow t o the h i g h l y c o l l i s i o n a l flow coming from the conduction zone. The f i r s t treatment of the isothermal expansion of plasma i n t o vacuum was by Gurevich et a l . [20], The b a s i c model d i s c u s s e d there has been the b a s i s of many s i m i l a r treatments d e a l i n g with the expansion c h a r a c t e r i s t i c s of l a s e r plasmas under a number of v a r y i n g c o n d i t i o n s . These treatments have been p r i m a r i l y developed t o e x p l a i n the o b s e r v a t i o n s of high 14 energy ion streams t h a t have been d e t e c t e d with ion c o l l e c t o r s i n long wavelength experiments [21 ]. An important refinement of the b a s i c model by Mora and P e l l a t [22] i n c l u d e d the e f f e c t s of the e l e c t r o n heat f l u x , and hence i d e n t i f i e s the energy f l u x s u p p l i e d by the l a s e r that i s necessary to d r i v e the expansion wave. D e t a i l s of the d i f f e r e n t treatments can be found i n [20-23]. A new s p a t i a l c o o r d i n a t e x' i s used, where the x ' - a x i s i s f i x e d to the plasma unperturbed by the r a r e f a c t i o n wave. The r a r e f a c t i o n wave propagates i n t o the flow coming from region (2) so t h a t the unperturbed plasma used to d e f i n e the x ' - a x i s i s moving at constant v e l o c i t y (viewed from the standard r e f e r e n c e frame of F i g u r e 3). T h e r e f o r e x' must be r e l a t e d to x by a G a l i l e a n t r a n s f o r m a t i o n . We impose the c o n d i t i o n , x« (x) * ( x - x Q ) • Ct II-2 where C i s an a r b i t r a r y constant with dimensions of v e l o c i t y ; the a b s o r p t i o n s u r f a c e x = x 0 c o i n c i d e s with x' = 0 at t = 0. I n i t i a l l y the plasma i s assumed to occupy a h a l f - s p a c e x' < 0, with i n i t i a l ion d e n s i t y N and e l e c t r o n d e n s i t y n. When the s o l u t i o n i s matched t o the flow from the a b l a t i o n zone, N 0 (n 0) corresponds to the ion ( e l e c t r o n ) d e n s i t y at the matching p o i n t . At time t = 0 the plasma expands i n t o the vacuum, x' > 0. One assumes now that the plasma i s s u f f i c i e n t l y r a r e f i e d t h a t the e l e c t r o n i o n mean f r e e path i s l a r g e compared to the s i z e of the r a r e f a c t i o n wave. In t h i s case the motion i s 15 p r i m a r i l y governed by the c o l l i s i o n l e s s k i n e t i c equations f o r the e l e c t r o n s and ions, _ e • v __e - * 0 T I _ 3 i t d x m d x e ! £ i . . f i n - o S t J x Am J x P In these e x p r e s s i o n s , f e and fj represent the e l e c t r o n and ion phase space d e n s i t y d i s t r i b u t i o n s r e s p e c t i v e l y ; nip and me are the proton and e l e c t r o n p a r t i c l e masses r e s p e c t i v e l y ; e i s the e l e c t r o n charge; 0 i s a s e l f - c o n s i s t e n t e l e c t r i c p o t e n t i a l f u n c t i o n ; and, Z and A are r e s p e c t i v e l y the charge s t a t e and atomic mass number of the ion s p e c i e s . Poisson's equation i s r e q u i r e d to complete the d e s c r i p t i o n , ,2, L - l = -4tre(ZN - n) 1 1 - 5 9 X where, f i ( x , v , t ) d v 1 1 - 6 f e ( x , v , t ) d v II-7 16 g i v e the ion and e l e c t r o n p a r t i c l e d e n s i t i e s r e s p e c t i v e l y . The u s u s a l boundary c o n d i t i o n a p p l i e d to the d i s t r i b u t i o n f u n c t i o n s f e and fj , i n the l i m i t x' -> i s that they are s p a t i a l l y i s o t r o p i c with Maxwellian v e l o c i t y d i s t r i b u t i o n s c h a r a c t e r i z e d by the temperatures T e and Tj r e s p e c t i v e l y . T h i s i s the most g e n e r a l f o r m u l a t i o n of the problem. A set of approximations are u s u a l l y a p p l i e d to s i m p l i f y the system (11-3) to (11-5). The e l e c t r o n d i s t r i b u t i o n i s assumed to obey a Boltzmann d i s t r i b u t i o n i n e q u i l i b r i u m with <t> (so that (11-3) i s r e p l a c e d by n = n 0exp[e<j>/T e]); (I I - 4 ) i s r e p l a c e d by the f l u i d equations of c o n t i n u i t y and momentum ( c o l d ion approximation); and f i n a l l y ( I I - 5 ) i s r e p l a c e d by a q u a s i n e u t r a l i t y assumption N = Zn. The w e l l known s o l u t i o n i s , (C • l ) c II-8 N = exp {-( C. • 1) } II-9 T — (K * 1) 11-10 where, 1 7 11-11 and, ( Z T / A m ) v e p 1/2 c 11-12 v i s the v e l o c i t y of the ion f l u i d and c i s the i o n - a c o u s t i c speed i n a c o l l i s i o n l e s s plasma. P r o f i l e s of the dependent v a r i a b l e s v, N and <t> are i l l u s t r a t e d i n F i g u r e 4. T h i s form of s o l u t i o n i s s a i d to be ' s e l f - s i m i l a r ' because a l l dependent v a r i a b l e s are f u n c t i o n s of x' and t only i n the combination x ' / t . The c o o r d i n a t e i- i s the s e l f - s i m i l a r v a r i a b l e and a c c o r d i n g to (11-11 ) i s a s s o c i a t e d with a r e f e r e n c e frame moving at constant v e l o c i t y . I t s n a t u r a l u n i t s are i n terms of the i o n - a c o u s t i c speed, c. One might imagine a f a m i l y of i n e r t i a l (moving) r e f e r e n c e frames parametrized by £ whose o r i g i n s pass through the p o i n t x' = 0 at time t = 0, so that a l l such frames s a t i s f y c o n d i t i o n ( I I - 2 ) . The f l u i d v e l o c i t y i n the wave viewed from the o r i g i n of any such r e f e r e n c e frame i s always s o n i c . The p o i n t £ = -1 i s i d e n t i f i e d with the wave head and t h i s p o i n t propagates back i n t o the unperturbed plasma (the frame f i x e d to the x' - a x i s ) at the sound speed, c. The i s o t h e r m a l r a r e f a c t i o n i s a c o l l e c t i v e motion. Ions are a c c e l e r a t e d by the s e l f - c o n s i s t e n t p o t e n t i a l <j>. T h i s p o t e n t i a l i s l i n e a r i n |, hence the a s s o c i a t e d e l e c t r i c f i e l d decreases as l / t . E l e c t r o n s are r e p e l l e d by the p o t e n t i a l ; higher energy e l e c t r o n s are abl e to t r a v e l f u r t h e r out before 18 F i g u r e 4 Isothermal expansion p r o f i l e s f o r a c o l l i s i o n l e s s plasma. A l l flow v a r i a b l e s are f u n c t i o n s of the s e l f - s i m i l a r parameter £. D e n s i t y n i n u n i t s of n 0 ; v e l o c i t y v i n u n i t s of c; p o t e n t i a l $ i n u n i t s of T / e . 19 they are r e f l e c t e d . Note t h a t the wave p r o f i l e i s e x p o n e n t i a l i n space with a c o n s t a n t l y expanding s c a l e l e n g t h p r o p o r t i o n a l to c t . A l s o i t i m p l i e s that there are a f i n i t e ( e x p o n e n t i a l l y small) number of ions t r a v e l l i n g at a r b i t r a r i l y l a r g e v e l o c i t i e s . The approximations used t o o b t a i n the simple r e s u l t s of (II-8) to (11-12) are now examined. F i r s t l y , the Boltzmann r e l a t i o n f o r e l e c t r o n d e n s i t y ignores the d e t a i l s of the exchange of energy between the e l e c t r o n d i s t r i b u t i o n and the expanding wave. A p e r t u r b a t i o n treatment by Mora and P e l l a t [22] using the e l e c t r o n k i n e t i c equation (II-2) i n s t e a d of the Boltzmann r e l a t i o n shows that the i n c r e a s i n g k i n e t i c energy of the r a r e f a c t i o n wave i s balanced by an e q u i v a l e n t energy f l u x c a r r i e d by the e l e c t r o n s . E l e c t r o n s t r a v e l l i n g out i n t o the wave are d e c e l e r a t e d and e v e n t u a l l y r e f l e c t e d by the slowly changing s e l f - c o n s i s t e n t p o t e n t i a l <p. They r e t u r n to the unperturbed plasma with s l i g h t l y l e s s energy than they s t a r t e d with. T h i s i m p l i e s that the e l e c t r o n d i s t r i b u t i o n has a non zero mean v e l o c i t y and may be i n t e r p r e t e d as a heat f l u x ; but the energy i s not d e p o s i t e d through c o l l i s i o n s , r a t h e r by a c o l l e c t i v e i n t e r a c t i o n v i a the s e l f - c o n s i s t e n t p o t e n t i a l . The flow remains s e l f - s i m i l a r , although s l i g h t l y m o d i f i e d by t h i s e f f e c t . In the r e f e r e n c e frame f i x e d to the unperturbed plasma ( x ' - a x i s ) the e l e c t r o n heat f l u x was shown to be [22], |q I • n T c 1 n e 1 o e 11-13 20 and i t may be e a s i l y v e r i f i e d that the k i n e t i c energy of the c o l d f l u i d i n the wave measured i n t h i s frame i n c r e a s e s with time at j u s t t h i s r a t e . Next, i n s t e a d of the c o l d ion f l u i d approximation G u r e v i c h et a l . [20] have i n t e g r a t e d (II-4) n u m e r i c a l l y (although r e t a i n i n g the Boltzmann r e l a t i o n and the q u a s i n e u t r a l i t y a p p r o x i m a t i o n ) . For i n i t i a l l y Maxwellian e l e c t r o n and ion d i s t r i b u t i o n s with T e = Tj t h e i r r e s u l t s show that the ion v e l o c i t y d i s t r i b u t i o n a s y m p t o t i c a l l y approaches a spike shape ( p r a c t i c a l l y monoenergetic f o r £ > 3) with a mean v e l o c i t y c o r r e s p o n d i n g to the c o l d ion f l u i d s o l u t i o n (eq. 11-8). Hence the e f f e c t i v e ion temperature r a p i d l y decreases with i n c r e a s i n g £ and the c o l d ion f l u i d i s a reasonable approximation at l a r g e v a l u e s of £. F i n a l l y , the assumption of q u a s i n e u t r a l i t y does break down at l a r g e v a l u e s of the parameter £. Hence a r b i t r a r i l y l a r g e ion v e l o c i t i e s are not a c t u a l l y p o s s i b l e . Exact c a l c u l a t i o n s by Crow et a l . [23] show th a t the l e a d i n g edge of the wave i s c h a r a c t e r i z e d by an ion f r o n t preceeded by an e l e c t r o n c l o u d and t r a v e l s at f i n i t e v e l o c i t i e s . However the v e l o c i t y l i m i t i s l a r g e , and i s presumably r e l a t e d to the e l e c t r o n thermal v e l o c i t y . A number of a s p e c t s of the i s o t h e r m a l r a r e f a c t i o n wave merit emphasis i n the c ontext of the g l o b a l flow s i t u a t i o n . F i r s t l y , i t i s important to note that the expansion i s d e s c r i b e d with c o l l i s i o n l e s s k i n e t i c equations; t h e r e f o r e a d e s c r i p t i o n u sing the c o n v e n t i o n a l macroscopic f l u i d equations 21 (e.g ( l I - 4 3 ) - ( I I - 4 4 ) ) does not adequately r e v e a l the p h y s i c a l nature of the plasma expansion. The v a l i d i t y of the k i n e t i c d e s c r i p t i o n r e q u i r e s that the e l e c t r o n - i o n mean f r e e path be l a r g e r than the s c a l e l e n g t h , c t , of the wave. Secondly, the e l e c t r o n energy f l u x which d r i v e s the wave must couple i n t o the energy equation as a heat f l u x t r a v e l l i n g out to the wave. The magnitude of the f l u x i s dependent only on the temperature and d e n s i t y of the plasma at the wave head as evident i n equation (11-13). F i n a l l y , at remote d i s t a n c e s from the t a r g e t the expansion process a c t s to convert the thermal energy of the plasma i n t o p u r e l y d i r e c t i o n a l k i n e t i c energy of the i o n s . I I - 2 - i i THE ABLATION ZONE Region (2) i s d i f f e r e n t f rom the r a r e f a c t i o n wave because the f l u i d flow i n i t i s steady and of lower temperature and higher d e n s i t y ( c o l l i s i o n a l ) . On the le n g t h and time s c a l e s of macroscopic f l u i d motion the u s u a l hydrodynamic equations s u f f i c e f o r an adequate d e s c r i p t i o n . The assumptions made i n t h i s r e g i o n of the flow are: ( i ) the hydrodynamic macroscopic v a r i a b l e s p, v and T w i l l be used to represent the mass d e n s i t y , f l u i d v e l o c i t y , and temperature r e s p e c t i v e l y . ( i i ) C l a s s i c a l e l e c t r o n heat conduction i s the p r i n c i p l e means of energy t r a n s p o r t . ( i i i ) Steady s t a t e f l u i d equations w i l l d e s c r i b e the flow. 22 Note f i r s t l y t h a t only a s i n g l e temperature, T, was s p e c i f i e d above. I t i s assumed that the ion and e l e c t r o n d i s t r i b u t i o n s share the same temperature (T = T e = Tj ). Next, the d e n s i t y v a r i a b l e , p, r e f e r s to the mass d e n s i t y and i s r e l a t e d to the p a r t i c l e d e n s i t i e s n and N i n the c o r o n a l r a r e f a c t i o n s o l u t i o n by the r e l a t i o n , p • ANm^ « knm^/Z m To c l o s e the f l u i d e q u a tions the i d e a l gas equation of s t a t e i s assumed so that P, p and T are connected through, -.2 pc 1 1 - 1 4 where the sound speed, c, i s d e f i n e d as, c CT/H) 1/ 2 1 1 - 1 5 Am and u = i s t h e m e a n p a r t i c l e mass. 1 In the c l a s s i c a l theory of heat conduction the heat f l u x i s p r o p o r t i o n a l to a l o c a l temperature g r a d i e n t , q • vT . In an i o n i z e d plasma the constant of p r o p o r t i o n a l i t y has been determined by S p i t z e r [24] and v a r i e s as T 5 ^ 2 . Thus the heat f l u x may be w r i t t e n q u a n t i t a t i v e l y as, 1 T h i s d e f i n i t i o n of the sound speed i s d i f f e r e n t from that of equation (11-12), hence the d i f f e r e n t symbol. In a p r a c t i c a l sense, both q u a n t i t i e s c and c are n e a r l y e q u a l . The reason f o r the d i s t i n c t i o n i s due to the m i c r o s c o p i c dynamics governing the p a r t i c l e i n t e r a c t i o n s i n the r e s p e c t i v e r e g i o n s of the flow. Region (1) i s c o l l i s i o n l e s s , c i s the i o n - a c o u s t i c speed; r e g i o n (2) i s c o l l i s i o n a l , . c i s the i d e a l gas sound speed. 23 q = - < T 5 / 2 V T 1 1 - 1 6 where, ic » 2 0 ( 2 / i r ) 3 / 2 e 6 t ( i n y 2 e 4 Z l n A ) " 1 H-17 and «6 t i s a q u a n t i t y of 0(1) depending on Z approximately as [ 9 ] , e6 t « 0.095(Z • 0 . 2 4 ) / ( l * 0.24Z) 11-18 F i n a l l y , the steady s t a t e f l u i d equations f o r one-dimensional p l a n a r flow are e a s i l y d e r i v e d , ~ ( p v ) = 0 11-19 dx dv dp p V d T * "dT 11-20 2 JS[py\j • jo } * q.xj = I a « ( x - x 0 ) 11-21 (11-19) i s the c o n t i n u i t y equation and (11-20) i s the momentum equa t i o n . Here we onl y c o n s i d e r the pressure P as the d r i v i n g f o r c e of the motion. (11-21) i s the energy equation, where the 24 l a s e r absorbed f l u x I a appears as a d e l t a - f u n c t i o n source term l o c a t e d at the a b s o r p t i o n s u r f a c e x = x 0 . The a b l a t i o n f r o n t i s the s u r f a c e at which t a r g e t m a t e r i a l s , heated by the e l e c t r o n thermal f l u x , begin to a c c e l e r a t e away from the shock compressed t a r g e t . The a b l a t i o n s u r f a c e moves through the compressed t a r g e t at constant v e l o c i t y as the t a r g e t m a t e r i a l s are a b l a t e d at a constant r a t e . J u s t ahead of the a b l a t i o n s u r f a c e the pressure i s cons t a n t , equal to the a b l a t i o n pressure, P a. Behind the a b l a t i o n s u r f a c e the pressure and d e n s i t y are lower and there i s a constant mass f l u x t r a v e l l i n g towards the vacuum. The flow in the a b l a t i o n zone i s steady and i t i s n a t u r a l to choose the re f e r e n c e frame of F i g u r e 3 i n which the a b l a t i o n s u r f a c e (and a b s o r p t i o n s u r f a c e ) are f i x e d . In t h i s frame equations (11- 19) and (11-20) become simply, p v ° p 0 v o * 11-22 pv • pc » p Q V O • P Q C ( T Q ) The p h y s i c a l i n t e r p r e t a t i o n through the a b l a t i o n s u r f a c e and i s exhausted a t x = x 0 the order of the sound speed, of s t a t i c pressure p c 2 - *-a 11-23 i s c l e a r . The mass f l u x , m, enters at h i g h d e n s i t y and low v e l o c i t y at low d e n s i t y with v e l o c i t i e s of There i s a continuous c o n v e r s i o n i n t o k i n e t i c p r e s s u r e pv 2 thus 25 m a i n t a i n i n g the momentum balance throughout the r e g i o n . F i n a l l y , the energy equation (11-21) may be i n t e g r a t e d through the conduction zone from x = 0 to x = x 0 . The d e l t a f u n c t i o n source term means that no energy i s d e p o s i t e d ahead of x = x 0 ; the assumption being that there i s no preheat of the ta r g e t due to X-ray or hot e l e c t r o n p e n e t r a t i o n . A c c o r d i n g l y , ,v2 5*2» x5/2dT n p V { 7 • -c \ -cT 33- » 0 I T_ 2 4 T h i s has a simple meaning, namely that the outward enthalpy f l u x of t a r g e t m a t e r i a l s e x a c t l y balances the heat f l u x propagating inwards at a l l p o i n t s w i t h i n the a b l a t i o n zone. In t h i s r e f e r e n c e frame there i s no energy f l u x ahead of the a b l a t i o n s u r f a c e . The s o l u t i o n of equations (II-19) - (11-21) w i l l g i v e the p r o f i l e s of p, v, and T as a f u n c t i o n of x between the a b l a t i o n s u r f a c e and the a b s o r p t i o n s u r f a c e . The exact shape of the p r o f i l e over the e n t i r e a b l a t i o n zone i s r e l a t i v e l y unimportant, but a u s e f u l parameter of the flow i s the width of t h i s zone and i t s dependence on the other flow parameters. An exact s o l u t i o n r e q u i r e s numerical computation. Manheimer and Colombant [12] have made a simple a n a l y t i c approximation by n e g l e c t i n g the k i n e t i c energy term; the energy equation (11-24) then s i m p l i f i e s t o , 26 which may be i n t e g r a t e d to give T <_5/2 25 p o V o , . .2/5 T * (T • — T (X-X ) ) TT •)£ o 4 KJI o 11-26 The constant of i n t e g r a t i o n has been i n c l u d e d by r e q u i r i n g T = T 0 when p = p0 and v = v 0 at the a b s o r p t i o n s u r f a c e . The width of re g i o n (2) may be estimated simply by using the value of x 0 obtained from equation (11-26), 4 icy 5 / 2 Xo % IT p v " V ll~21 0 0 I I - 2 - i i i MATCHING AT THE ABSORPTION SURFACE The s t a t e d o b j e c t i v e of t h i s model was to develop s c a l i n g laws r e l a t i n g the a b l a t i o n c o n d i t i o n s , namely m and P a, to the l a s e r and t a r g e t parameters. In t h i s s e c t i o n i t i s shown that these flow parameters are f u l l y determined by a s u i t a b l e match of the flow from the a b l a t i o n zone to that of the r a r e f a c t i o n wave. The matching s u r f a c e i s an important region i n t h i s r e s p e c t . On a m i c r o s c o p i c s c a l e i t i s s i t u a t e d i n a complicated t r a n s i t i o n r e g ion i n which the c o l l i s i o n a l flow from the a b l a t i o n zone e v e n t u a l l y becomes c o l l i s i o n l e s s f a r out i n the corona. Yet the m i c r o s c o p i c dynamics at the matching s u r f a c e are i r r e l e v a n t i n u l t i m a t e l y determining the s c a l i n g laws. The 27 r e l a t i o n s used are simple g l o b a l c o n s i d e r a t i o n s of mass, momentum and energy c o n s e r v a t i o n combined with the requirement of steady s t a t e c o n d i t i o n s . The obvious matching technique [11,12,25,27] has been to j o i n the r a r e f a c t i o n flow s o l u t i o n onto the steady s t a t e flow at the a b s o r p t i o n s u r f a c e such t h a t the flow v e l o c i t y i s sonic at t h i s p o i n t , In f a c t t h i s i s the only way to s a t i s f y the hypothesis of steady s t a t e c o n d i t i o n s : the propagation of the r a r e f a c t i o n wave head i n t o the f l u i d coming from the a b l a t i o n zone i s s o n i c . T h e r e f o r e the flow v e l o c i t y of the a b l a t e d m a t e r i a l s must be sonic i n order balance t h i s and t h e r e f o r e to maintain the boundary between the two re g i o n s at the co n s t a n t p o s i t i o n x = x 0 . Another requirement f o r matching i s that the d e n s i t y i s continuous ac r o s s the matching s u r f a c e 1 . T h i s f a c t combined with c o n d i t i o n (11-28) s a t i s f i e s c o n s e r v a t i o n of mass and momentum a c r o s s the matching s u r f a c e . The f o l l o w i n g argument concerning the energy balance was used i n the work of Manheimer and Colombant [12] and Mora 1 Max e t . A l [9] and Fabbro [27] have d e s c r i b e d a n a l y t i c models i n which d i s c o n t i n u i t i e s i n both the d e n s i t y and v e l o c i t y p r o f i l e s may e x i s t . Such d i s c o n t i n u i t i e s are introduced t o accomodate an a r b i t r a r y e l e c t r o n heat f l u x l i m i t , or to account f o r d e n s i t y steepening e f f e c t s a t the c r i t i c a l s u r f a c e due to ponderomotive f o r c e . N e i t h e r of these e f f e c t s are co n s i d e r e d i n t h i s model. 28 [11]. Consider the energy balance that must be maintained at the a b s o r p t i o n s u r f a c e . The l a s e r energy i s c a r r i e d o f f by e l e c t r o n heat c o n d u c t i o n : there i s a f l u x , q,, f l o w i n g i n t o the r a r e f a c t i o n wave; and, there i s a f l u x , q2, p e n e t r a t i n g the a b l a t i o n zone. The f l u x q 2 i s e x a c t l y balanced by the enthalpy f l u x of heated t a r g e t m a t e r i a l s exhausted at the a b s o r p t i o n s u r f a c e so that the t o t a l energy of the a b l a t i o n zone remains constant once steady s t a t e c o n d i t i o n s are achieved. Therefore the energy of the r a r e f a c t i o n wave, f e d by q, as w e l l as the enthalpy f l u x at x =' x 0 must equal the f l u x d e p o s i t e d by the l a s e r . Keeping i n mind that the r a r e f a c t i o n wave head i s s t a t i o n a r y i n our standard r e f e r e n c e frame (Figure 3) one may i n t e g r a t e the enthalpy (pv 2 + 3pc 2)/2 over the e x p o n e n t i a l wave p r o f i l e given by (II-8) - (11-12) from the wave head to °°. The energy f l u x i n the r a r e f a c t i o n wave i s found to be 4 p 0 c 3 ( T 0 ) . Assuming s o n i c flow a t the a b s o r p t i o n s u r f a c e the t o t a l enthalpy f l u x ( p v 3 + 5pvc 2)/2 exhausted from the conduction r e g i o n i s 3 p 0 c 3 ( T 0 ) . The d i f f e r e n c e between t h i s v alue and the t o t a l energy f l u x i n the wave i s made up by the e l e c t r o n heat f l u x d r i v i n g the wave, q, = |q^| = p 0 c 3 ( T 0 ) , i n agreement with equation (11-13). A ge n e r a l r e l a t i o n between the absorbed l a s e r i n t e n s i t y , I a , n 0 f and, T 0 may now be s t a t e d by equating the l a s e r f l u x t o the energy f l u x i n the r a r e f a c t i o n wave, *a " 4 n o T o C<V a 4 n o T f / 2 C Z / A m J 1 / 2 11-29 29 The s c a l i n g laws g i v i n g the dependence of rh, and P a, on i n t e n s i t y and wavelength can then be e a s i l y determined once the a b s o r p t i o n d e n s i t y n 0 i s known. I I - 2 - i v ABLATION SCALING LAWS The d e n s i t y n 0 i s a complicated f u n c t i o n of the l a s e r i n t e n s i t y , wavelength, the plasma flow p r o f i l e and plasma temperature. For high i n t e n s i t i e s and long wavelengths a b s o r p t i o n takes p l a c e p r i m a r i l y at the c r i t i c a l s u r f a c e by the process of resonant a b s o r p t i o n [30], and a l a r g e f r a c t i o n of the i n c i d e n t energy i s r e f l e c t e d out of the plasma. In t h i s l i m i t the a b s o r p t i o n d e n s i t y i s a f u n c t i o n s o l e l y of the l a s e r wavelength ( c f . equation ( I I — 1 ) ) . For low i n t e n s i t i e s and s h o r t wavelengths the corona i s c o o l e r and n c r i s l a r g e r . Inverse Bremsstrahlung a b s o r p t i o n becomes an important process and i t i s p o s s i b l e f o r a l l of the i n c i d e n t l a s e r f l u x to be absorbed at d e n s i t i e s l e s s than n c r . These two asymptotic l i m i t s were i d e n t i f i e d by Mora [11] who used them to o b t a i n s c a l i n g laws i n both l i m i t s . 30 (A) C r i t i c a l Surface D e p o s i t i o n The d e n s i t y n 0 i s dependent only on the l a s e r wavelength as given by (11 — 1). S u b s t i t u t i n g t h i s i n t o (11-29), one o b t a i n s the dependence of T 0 on the absorbed i n t e n s i t y , r0 - 127 (*)VS ,VS X4/S n - 3 0 then u s i n g the r e l a t i o n s (11-22) and (11-23) at the matching s u r f a c e the s c a l i n g laws are, i * ( 3 3 x 1 0 3 ) ( $ , 2 / 3 , 1 / 3 , - 4 , 3 pa 1.1(4) 1 / 3 i 2 / 3 x"2/3 1 1 - 3 2 The u n i t s r e q u i r e d f o r the v a r i o u s parameters a r e : [ T 0 ] = eV, [ I a ] = TW/cm2, [ X L ] = Mm, [ft] = g c n r - s " 1 , and [ P a ] = Mbar. 31 (B) Inverse Bremsstrahlung Absorption With t h i s mechanism the a b s o r p t i o n depends on the c o l l i s i o n frequency of e l e c t r o n s and ions, and on the d e n s i t y s c a l e l e n g t h of the plasma. T h i s means a dependence on the temperature, d e n s i t y , and d e n s i t y p r o f i l e i n the region of energy d e p o s i t i o n . In h i s model Mora [11] used the WKB approximation to determine the a b s o r p t i o n f r a c t i o n of l a s e r r a d i a t i o n due to the Inverse Bremsstrahlung (IB) proc e s s . For stro n g IB a b s o r p t i o n n 0 i s l e s s than n c r and there i s t o t a l a b s o r p t i o n . In t h i s l i m i t n 0 i s roughly r e l a t e d to n c r by [11], "o - Z " c r V e l(n c rH„ where c i s the speed of l i g h t , not to be confused with the sound speed, L n i s the d e n s i t y s c a l e l e n g t h of the absorbing plasma, and i>Qi ( n c r ) i s the e l e c t r o n - i o n c o l l i s i o n frequency a t c r i t i c a l d e n s i t y [11], 4 vei(n) - (T) (7) i n lfi 11-34 m e e In the same manner as the case of c r i t i c a l s u r f a c e d e p o s i t i o n one s u b s t i t u t e s (11-33) i n t o (11-29) using the r e l a t i o n L n = c 0 t to o b t a i n the s c a l i n g f o r T 0 . Then u s i n g (11-22) and (11-23) the s c a l i n g s f o r iti and P a are a l s o o b t a i n e d , T o * 54 ( Z l n A ) 1 ' 4 ( A j l / 8 1/2 l / 2 1/4 z a * L r 11-35 A ^ (48 x 10 3}(2£nA)- 1 / 4(4) 7/4 1/2.-1/2 -1/4 2 a A L ' 1 11-36 P * 1.2 ( I l n A ) - 1 ' 8 ^ ) 7 / 1 6 I 3 / 4 X 7 1 / 4 t " 1 ' 8 H-37 U n i t s are the same as f o r (11-30) t o (11-32) with the a d d i t i o n of : [t ] = ns. The time dependences i n these s c a l i n g s are due to the expanding s c a l e l e n g t h c 0 t of the pl a n a r r a r e f a c t i o n wave. T h i s time dependence r e v e a l s a c o n f l i c t between the two i n i t i a l hypotheses of planar flow combined with steady s t a t e c o n d i t i o n s when IB i s the dominant a b s o r p t i o n mechanism. In f a c t there i s no s t r i c t l y p l a n a r steady s t a t e flow s o l u t i o n with IB a b s o r p t i o n . (C) IB Ab s o r p t i o n with F i x e d S c a l e Length The expanding s c a l e l e n g t h of planar geometry i s not r e a l i s e d f o r long p u l s e and small spot s i z e experiments s i n c e the f i n i t e spot s i z e imposes an upper l i m i t on the s i z e t h at L n can e v e n t u a l l y a t t a i n . These a s p e c t s are d i s c u s s e d i n more d e t a i l l a t e r . In t h i s case the s c a l i n g laws can be obtained i f 33 the a b s o r p t i o n s c a l e l e n g t h L n i n equation (11-33) i s taken to be a f i x e d e x t e r n a l parameter governed by the geometry of the expanding plasma. We d e f i n e the f i x e d s c a l e l e n g t h to be L x ; i t i s of the order of the spot diameter. The s c a l i n g laws obtained in t h i s case are, T 0 32 ( Z £ n A ) 2 / 9 (£) 2/9 -4/9 .4/9 .2/9 Z a A L L x 11-38 m * (125 x 1 0 3 ) ( Z t n A ) - 2 / 9 ( 4 ) 7 / 9 i 5 / 9 x - 4 / 9 -2/9 1 a L x 11-39 P a V l . l ( Z l n A ) - 1 ' 9 ( 4 ) 7 / 1 8 I 7 ' 9 x-2/9 L - l / 9 a 2 a L x Again, u n i t s are c o n s i s t e n t with the pre v i o u s s c a l i n g s ; the a d d i t i o n a l v a r i a b l e here i s : [ L x ] = Mm. I I _ 2 _ V VALIDITY OF THE ABLATION MODEL The important assumptions t h a t were made i n the model and t h e i r c o n n e c t i o n with p h y s i c a l r e a l i t y w i l l now be d i s c u s s e d . F i r s t to be d e a l t with i s the steady s t a t e h y p o t h e s i s . Secondly, the problem of steady s t a t e flow with s t r o n g IB a b s o r p t i o n i s d i s c u s s e d i n the context of r e a l i s t i c p lanar 34 geometries, i . e . , i n c l u d i n g the e f f e c t s of f i n i t e spot s i z e . T h i r d l y , the v a l i d i t y of i s o t h e r m a l r a r e f a c t i o n i s d i s c u s s e d ; t h i s i s of i n t e r e s t because the c o l l i s i o n l e s s d e s c r i p t i o n given, i n s e c t i o n I I - 2 - i i s not unique. F i n a l l y , the matching c o n d i t i o n (11-28) hides a number of important d e t a i l s c oncerning the complicated p h y s i c s i n t h i s important r e g i o n of the flow. (A) Steady State Hypothesis The steady s t a t e hypothesis s t a t e d at the beginning of s e c t i o n II-2 i s f a r from a p h y s i c a l l y r e a l i s t i c s i t u a t i o n i n v o l v i n g a temporally changing l a s e r p u l s e . The o p e r a t i n g concept i s a c t u a l l y that of s e l f - r e g u l a t e d flow i n which the flow a d j u s t s to changes i n the e x t e r n a l parameters (e.g. l a s e r i n t e n s i t y ) on a time s c a l e small compared to the c h a r a c t e r i s t i c time s c a l e s a s s o c i a t e d with the e x t e r n a l parameters [11,14,26]. Consider an . e s t a b l i s h e d steady a b l a t i v e flow. At a c e r t a i n moment the i n c i d e n t l a s e r i n t e n s i t y i n c r e a s e s s l i g h t l y . The temperature at the a b l a t i o n r e g i o n i n c r e a s e s , a l a r g e r heat f l u x p e n e t r a t e s to the a b l a t i o n s u r f a c e and m a t e r i a l i s a b l a t e d a t a l a r g e r r a t e . The i n c r e a s e d flow through the conduction region pushes back the r a r e f a c t i o n wave head so that the width of the conduction zone i n c r e a s e s thus reducing the temperature g r a d i e n t . A new steady s t a t e s i t u a t i o n i s reached where the flow has a d j u s t e d i t s e l f t o the new l a s e r i n t e n s i t y . L i kewise a drop i n i n t e n s i t y r e s u l t s i n the reverse adjustment. 35 Now d e f i n e a c h a r a c t e r i s t i c time d u r i n g which the l a s e r i n t e n s i t y changes s i g n i f i c a n t l y . A p o s s i b l e d e f i n i t i o n i s = I/(dI / d t ) f o r l a s e r i n t e n s i t y I ( t ) as a f u n c t i o n of time. (A rough estimate i s s u f f i c i e n t ; f o r a gaussian p u l s e the FWHM time i s a p p r o p r i a t e . ) I f the time i t takes f o r the flow adjustment i s , and i s small compared to , then the flow e x i s t s i n a s t a t e of near e q u i l i b r i u m with the i n c i d e n t f l u x at a l l times, i . e . , a 'quasi-steady s t a t e ' flow. The advantage of such a s i t u a t i o n i s that r e a l i s t i c experimental r e s u l t s may be compared to the simple model using the time averaged f l u x as a reasonable correspondence to the assumed i d e a l l y constant f l u x . An a p p r o p r i a t e measure of the response time of the flow [9,27] i s the t r a n s i t time f o r a f l u i d element moving from the a b l a t i o n s u r f a c e to the a b s o r p t i o n s u r f a c e , Using the mass and momentum equations (11-19) and (11-20) as w e l l as the approximated energy equation (11-26) and the e x p r e s s i o n f o r x 0 (11-27) i t can be shown t h a t , x „ 3/2_2 x * 2.7 _ . 2 . 7 ( — o) l l - 4 2 a o 25m T h i s may be combined with the s c a l i n g laws. Using (11-38) (11-40) f o r IB a b s o r p t i o n with f i x e d s c a l e l e n g t h one o b t a i n s , 36 T f * 0 . 1 2 c « t ( Z Z n A ) - 1 / 3 ( A ) - l / 3 T l / 3 x 4 / 3 L 2 / 3 II-42b L g i v i n g r f i n terms of the l a s e r parameters. For 10 TW/cm2 i n c i d e n t i n t e n s i t y at .355 urn wavelength and L x = 80 Mm t y p i c a l of our experiment ( a l s o A/Z = 2, ZlnA = 60 and e5t = .3) one can estimate r f * 7 0 p s << T ^ 2 ns. (B) Planar Flow The assumption of p l a n a r flow p r e s e n t s two problems: one i s i t s r e l a t i o n to p h y s i c a l l y r e a l i s t i c s i t u a t i o n s with f i n i t e spot s i z e ; the second i s the t h e o r e t i c a l r e s u l t mentioned above, t h a t s t r i c t l y p l a n a r flow cannot be steady s t a t e f o r the stron g IB a b s o r p t i o n s i t u a t i o n . F i g u r e 5 d e p i c t s s c h e m a t i c a l l y the flow l i n e s f o r p l a n a r flow i n comparison t o a two-dimensional p a t t e r n that might be produced by l a s e r i r r a d i a t i o n of a f i n i t e round spot on the t a r g e t . At the t a r g e t s u r f a c e the flow l i n e s are p e r p e n d i c u l a r to the t a r g e t and t h e r e f o r e p a r a l l e l to one another. E v e n t u a l l y the flow w i l l d i v e r g e so that f a r from the t a r g e t the flow l i n e s become predominantly r a d i a l c e n t r e d on the t a r g e t spot. The t r a n s i t i o n from planar to s p h e r i c a l flow must take p l a c e on a s c a l e l e n g t h t y p i c a l l y of the order of the spot diameter VSp0\ . (T h i s type of two-dimensional flow p a t t e r n i s demonstrated very c l e a r l y i n the remarkable experimental r e s u l t s of re f e r e n c e [28]) The v a l i d i t y of the p l a n a r flow assumption i s examined by r e c a l l i n g that the important a b l a t i o n parameters m and P g were 37 LASER (b) i ' X ' F i g u r e 5 St r e a m l i n e s f o r f i n i t e spot d i v e r g i n g flow and f o r pla n a r flow. (a) Flow s t r e a m l i n e s f o r f i n i t e spot s i z e p l a n a r t a r g e t experiment. S p h e r i c a l divergence s e t s i n at d i s t a n c e s of the order of the spot diameter. (b) Flow s t r e a m l i n e s i n s t r i c t l y p l a n a r geometry. 38 determined by the sonic matching c o n d i t i o n at the a b s o r p t i o n s u r f a c e . Hence an a p p r o p r i a t e c r i t e r i o n f o r us i n g the planar assumption i n e s t i m a t i n g m and P a i s that the flow i s predominantly p l a n a r from the a b l a t i o n s u r f a c e out to the a b s o r p t i o n (matching) s u r f a c e x = x 0 . Such a c r i t e r i o n r e q u i r e s t h a t the width of the conduction zone x 0 be smal l compared to the s i z e of the l a s e r spot, Dspot • I n o u r experiment D Sp 0^ i s about 80 jum while x 0 i s about 1 urn, thus s a t i s f y i n g the planar flow assumption. On the other hand one may use t h i s two-dimensional e f f e c t to address the s i t u a t i o n of steady s t a t e flow with IB a b s o r p t i o n f o r the f i n i t e spot p l a n a r geometry of F i g u r e 5 ( a ) . R e c a l l t h at the s c a l i n g laws (11-35) - (11-37) are time dependent due to the expanding s c a l e l e n g t h c 0 t of the plan a r r a r e f a c t i o n wave. When the l a s e r i s f i r s t turned on and the s c a l e l e n g t h c 0 t i s s m a l l compared t o D Sp 0^ the flow i s p l a n a r and one would expect the s c a l i n g laws (11-35) - (11-37) to h o l d . As the wave expands to s i z e s l a r g e r than the spot diameter, i t begins to di v e r g e r a d i a l l y . At t h i s - p o i n t the d e n s i t y s c a l e l e n g t h w i l l become dominated by the geometric e f f e c t s of the divergence and w i l l be l i m i t e d to a s i z e of °( Dspot 0 n e m a v t n e n u s e t n e s c a l i n g laws (11-38) - (11-40) where the f i x e d s c a l e l e n g t h L x i s taken to be E>spo\ • T h i s i s in f a c t the s i t u a t i o n t h a t most c l o s e l y corresponds to our experimental c o n d i t i o n s s i n c e the p l a n a r d e n s i t y s c a l e length c 0 t at the end of a 2 ns FWHM gaussian pulse ( i . e . roughly 3 -4 ns flow time) i s t y p i c a l l y 500 ym, compared to a spot 39 diameter of 80 win. (C) Isothermal R a r e f a c t i o n The r a r e f a c t i o n wave s o l u t i o n given i n s e c t i o n 11-2-i was based on a s e t of c o l l i s i o n l e s s k i n e t i c equations, and the p h y s i c a l c h a r a c t e r i s t i c s of that s o l u t i o n were emphasized. That p h y s i c a l d e s c r i p t i o n i s not ac c u r a t e i f the c o r o n a l plasma i s s u f f i c i e n t l y c o l l i s i o n a l . In f a c t , the iso t h e r m a l r a r e f a c t i o n wave can a l s o be obtained with c o l l i s i o n a l f l u i d e quations. In t h i s d e s c r i p t i o n the motion i s governed by the f l u i d equations of c o n t i n u i t y and momentum, I f • j n r ( p v ) = o 3t 3x 11-43 i v + v | v + lap . o T I _ 4 4 a t a x p a x T where the p r e s s u r e P i s given by P = p c 2 , and c = C-) 3 i s the iso t h e r m a l sound speed which i s assumed c o n s t a n t . For the- same i n i t i a l c o n d i t i o n s as i n s e c t i o n I I - 2 - i the r e s u l t i n g d e n s i t y and v e l o c i t y p r o f i l e s are i d e n t i c a l to equations ( I I - 8 ) , (II-9) and (11-11) [29]. But n o t i c e i n t h i s f o r m u l a t i o n the absence of the v a r i a b l e <j>. The m i c r o s c o p i c dynamics are dominated by short range c o l l i s i o n s , so that the plasma p r o p e r t i e s of the medium are unimportant. Thus iso t h e r m a l r a r e f a c t i o n i s a q u i t e general 40 r e s u l t which i s independent of the m i c r o s c o p i c d e s c r i p t i o n of the medium ( c o l l i s i o n a l or c o l l i s i o n l e s s ) . Hence the main c r i t e r i o n f o r the e x i s t e n c e of the r a r e f a c t i o n wave i s not the c o l l i s i o n a l i t y , but the f a c t that i t i s i s o t h e r m a l . For the r a r e f i e d , high c o n d u c t i v i t y corona isothermal c o n d i t i o n s are a good assumption. The d i s t i n c t i o n between c o l l i s i o n a l and c o l l i s i o n l e s s d e s c r i p t i o n s has a more s u b t l e i m p l i c a t i o n which w i l l be d i s c u s s e d next. (D) Matching C o n d i t i o n The s o n i c matching c o n d i t i o n (11-28) i s necessary to s a t i s f y the assumptions of p l a n a r flow and steady s t a t e c o n d i t i o n s . I t i s p h y s i c a l l y r e a l i s t i c and there i s no obvious reason to doubt i t s v a l i d i t y . T h e r e f o r e i t i s p o s s i b l e at t h i s p o i n t to regard the a b l a t i o n model as a reasonably s e l f -c o n s i s t e n t d e s c r i p t i o n of the a b l a t i o n p r o c e s s . Furthermore the d e t a i l e d m i c r o s c o p i c p h y s i c s of is o t h e r m a l r a r e f a c t i o n g i v e n i n s e c t i o n I I - 2 - i might appear s u p e r f l u o u s . I t was noted p r e v i o u s l y that the a b l a t i o n model i s c o n s t r u c t e d u s i n g g e n e r a l c o n s e r v a t i o n p r i n c i p l e s f o r mass, momentum and energy and y i e l d s a flow s t r u c t u r e independent of the m i c r o s c o p i c p h y s i c s i n e i t h e r of regions (1) and (2) of F i g u r e 1. In c o n t r a s t with t h i s are other approaches to t r e a t i n g the flow based on d i f f e r e n t i a l equations, which t h e r e f o r e r e q u i r e f u r t h e r assumptions about the m i c r o s c o p i c p h y s i c s . These o f t e n i n v o l v e the use of hydrodynamic computer 41 s i m u l a t i o n s . Yet the m i c r o s c o p i c dynamics are non-uniform over the e n t i r e flow r e g i o n : f a r out i n the corona the plasma i s c o l l i s i o n l e s s and t h e r e f o r e corresponds to the d e s c r i p t i o n i n s e c t i o n I I - 2 - i . On the other hand, deep i n s i d e the a b l a t i o n zone one expects the f l u i d d e s c r i p t i o n i n s e c t i o n I I - 2 - i i to apply. An examination of v a r i o u s c h a r a c t e r i s t i c s c a l e lengths of the plasma i n the matching r e g i o n shows that ' i t cannot be d e s c r i b e d by a simple set of equations such as (II-2) - (11-4) or (11-19) - (11-21). F i r s t l y , c o n s i d e r a ' c r i t e r i o n f o r c o l l i s i o n l e s s r a r e f a c t i o n . T h i s would occur i f the e l e c t r o n - i o n mean f r e e path i s somewhat l a r g e r than the s c a l e l e n g t h of the wave. The e l e c t r o n - i o n mean f r e e path X 6j may be d e f i n e d using the c o l l i s i o n frequency and the e l e c t r o n thermal v e l o c i t y , X e i " 1 1 - 4 5 The c o l l i s i o n l e s s c r i t e r i o n i s , e i ' x n'x i i 40 o o where L n i s the d e n s i t y s c a l e l e n g t h . I f L n =* L x near the a b s o r p t i o n surface x x 0 , then the c r i t e r i o n (11-46) g i v e s a r e l a t i o n between T 0, n 0 , and L x , T 2 r 1 5 . 0 > | 11-47 n Z e i n A o 42 In the case of c r i t i c a l s u r f a c e d e p o s i t i o n (11-47) may be used i n combination with the heat f l u x r e l a t i o n .(11-29) and (11 — 1 ) to g i v e a c o n d i t i o n on the l a s e r f l u x necessary to maintain a c o l l i s i o n l e s s plasma up to the d e n s i t y n c r , I a > (.53 x l 0 - 4 ) ( * ) 3 / 4 ( 2 l n A ) ! / 4 X - L 7 / 2 ^ / 4 H-48 If t h i s c r i t e r i o n i s s a t i s f i e d i t i m p l i e s that the c o r o n a l plasma i s c o l l i s i o n l e s s up to x = x 0, the su r f a c e of maximum l a s e r p e n e t r a t i o n , and that the t r a n s i t i o n to c o l l i s i o n a l flow takes p l a c e somewhere i n the a b l a t i o n zone. Ther e f o r e the f l u i d equations of S e c t i o n 11 —2 — i i are not a p p l i c a b l e , at l e a s t near x = x 0 . E x p e r i m e n t a l l y the c r i t e r i o n (11-48) i s met by long wavelength high power l a s e r f l u x e s (e.g. > 1.06 Mm, > 10 1* W/cm2). Measurements of a b s o r p t i o n and t r a n s p o r t i n t h i s regime have been i n disagreement with computer p r e d i c t i o n s u n l e s s a strong heat ' f l u x - l i m i t ' i s invoked [ 4 ] . For IB a b s o r p t i o n , the c r i t e r i o n (11-47) i s a c t u a l l y i n c o n t r a d i c t i o n with the process of a b s o r p t i o n . I t would imply a plasma that i s c o l l i s i o n a l enough to absorb a l l of the l a s e r energy, yet a l s o c o l l i s i o n l e s s so that the equations (11-3) -(II-5) are a p p r o p r i a t e to d e s c r i b e the expansion. Consider the case of IB a b s o r p t i o n with f i x e d s c a l e l e n g t h . Using (11-38), (11-33) and (11-29) one may e v a l u a t e the r a t i o of X ej to the s c a l e l e n g t h L x , 43 ex V 0.024 c £ ) 5 ' l ^ Z l n A ) - 2 ' 9 I S ' 9 A f 1 4 / 9 L 2 / 9 H-49 x o * a L For parameters t y p i c a l of our sub-micron experiment ( I a = 10 TW/cm2, X L = .355 Mm, L x =80 Mm, A/Z =2, ZlnA = 60) t h i s r a t i o i s about 0.004. Hence the r a r e f a c t i o n cannot be c o n s i d e r e d c o l l i s i o n l e s s . However one may c o n s i d e r a weaker form of the c r i t e r i o n by comparing X ej with the Debye le n g t h A T T * • 7 . The r a t i o «i| s c a l e s as, 4irne A n X o ^ • 1 , ~ 91 ($) 1/*(Zi.A)-»/*lJ/\i;i/* 11-50 A For the parameters l i s t e d above one o b t a i n s M.\ * 1 1 0 s i n c e x e j » x D o Q the plasma i s dominated by k i n e t i c e f f e c t s on time s c a l e s l e s s than l A e i and d i s t a n c e s c a l e s l e s s than X e j . However, as L x >> X e j c o l l i s i o n s w i l l dominate i n the a b s o r p t i o n process throughout the e n t i r e absorbing r e g i o n . Consider the problem of energy t r a n s p o r t . The c l a s s i c a l d e s c r i p t i o n of heat t r a n s p o r t i n a c o l l i s i o n a l plasma i s q a VT . I f the temperature g r a d i e n t ?T i s l a r g e enough |q| can not exceed a magnitude r e l a t e d to the f r e e streaming p a r t i c l e f l u x . The f r e e streaming f l u x is Iql < fn T v T e 1/2 e e e where v » (--—) i s the e l e c t r o n thermal v e l o c i t y and f i s a e parameter of 0(1) ( * 0.6) c a l l e d the f l u x l i m i t . In the matching region x = x 0 the energy f l u x t r a v e l l i n g i n any d i r e c t i o n ( e l e c t r o n 'heat' f l u x , or f l u i d enthalpy f l u x ) i s a q u a n t i t y of 0 ( 4 n 0 T 0 c 0 ) ( c f . equation (11-29)). Notice that t h i s may be regarded as a f r e e - s t r e a m i n g energy f l u x l i m i t f o r 44 the p l a n a r a b l a t i o n flow s t r u c t u r e . Furthermore, the f r e e -streaming v e l o c i t y i s not v but c 0 ! The f a c t t h a t c 0 i s the maximum fre e - s t r e a m i n g v e l o c i t y i m p l i e s an e f f e c t i v e f value of . Zm 4 c c f * —_Q \ 4 - — -v 0.07 which i s much smal l e r than the v Am e p expected value of 0(1) and a l s o c l o s e to i n f e r r e d v a l u e s of fin 0.03 [ 4 ] . T h e ' r e l a t i o n s h i p between f and / n ~ ~ ^ a s D e e n noted p r e v i o u s l y by Manheiner and Colombant [12], and by Ahlborn [31] using s i m i l a r arguments based only on the flow s t r u c t u r e . Presumably a p o s s i b l e mechanism f o r t h i s f l u x l i m i t are c o l l e c t i v e e f f e c t s that r e s t r a i n e l e c t r o n s from f r e e -streaming behaviour as i n d i c a t e d by the f a c t that X 6j >> Xp. A c o r r e c t m i c r o s c o p i c treatment of the a b l a t i v e flow t h a t takes i n t o account these f a c t s should reproduce the flow s t r u c t u r e i n c l u d i n g the inherent l i m i t s on energy t r a n s p o r t . The p o i n t of these arguments i s to i l l u s t r a t e the importance of the v a r i o u s s c a l e lengths L x , \ Q i , and Xp. With t h i s simple a n a l y t i c reasoning i t i s d i f f i c u l t to examine the problem i n any more d e t a i l . We see that, the region near x ^ x 0 i s a c o m p l i c a t e d t r a n s i t i o n zone. A l l t h r e e s c a l e l e n g t h s , X ej , Xp and L x , must be c o n s i d e r e d i n a complete treatment. They cover a range of d e s c r i p t i o n s from m i c r o s c o p i c to macroscopic, a l l e q u a l l y important i n the u l t i m a t e p i c t u r e . For t h i s reason understanding i s s t i l l l i m i t e d even with the h e l p of l a r g e s c a l e computer m o d e l l i n g . 45 II-3 SUMMARY AND EXPECTED EXPERIMENTAL SCALINGS I I - 3 - i SUMMARY OF RESULTS An a n a l y t i c model d e s c r i b i n g l a s e r - d r i v e n a b l a t i o n of s o l i d t a r g e t s has been d e s c r i b e d . The p r i n c i p l e f e a t u r e s and assumptions of the model i n c l u d e steady s t a t e ( s e l f - r e g u l a t e d ) flow, p l a n a r geometry and an i n e r t i a l ( n o n - a c c e l e r a t i n g ) r e f e r e n c e frame. The v a l i d i t y of these i n i t i a l assumptions has been examined t o i l l u s t r a t e some of the b a s i c p h y s i c a l aspects of the a b l a t i v e flow s t r u c t u r e . The model y i e l d s s c a l i n g laws f o r the mass a b l a t i o n r a t e and a b l a t i o n pressure as a f u n c t i o n of the l a s e r parameters and t a r g e t m a t e r i a l . The s c a l i n g s are not u n i v e r s a l , but depend on the ab s o r p t i o n c h a r a c t e r i s t i c s of the plasma. 46 I I - 3 - i i EXPECTED EXPERIMENTAL SCALINGS The observables m and P Q are a c c e s s i b l e t o d i r e c t measurement. The measurement technique i s o u t l i n e d i n chapter V. To compare the a b l a t i o n model to our p a r t i c u l a r experiment we must s e l e c t the a p p r o p r i a t e set of s c a l i n g laws among ( I I -31, -32), (11-36, -37) and (11-39, -40). T h i s i s done by determining which of the two asymptotic l i m i t s ( s t r o n g or weak Inverse Bremmstrahlung a b s o r p t i o n ) d e s c r i b e d i n s e c t i o n I I - 2 -i i i corresponds to our experimental c o n d i t i o n s . For a f i x e d wavelength Mora [11] presented an exp r e s s i o n f o r determining the l a s e r i n t e n s i t y at the midpoint of the two asymptotic l i m i t s . For lower i n t e n s i t i e s , a b s o r p t i o n i s dominated by str o n g IB a b s o r p t i o n ; f o r higher i n t e n s i t i e s IB a b s o r p t i o n i s weak and energy d e p o s i t i o n i s at n c r . For X^ = 0.355 Mm the t h r e s h o l d i n t e n s i t y i s approximately 3.1 x 1 0 1 6 W/cm2. Our experiment operates up t o a maximum i n t e n s i t y of about 2 x 1 0 1 3 W/cm2, hence we would expect the experimental s c a l i n g s to be c o n s i s t e n t with s t r o n g IB a b s o r p t i o n . Furthermore, s i n c e the experiment i n v o l v e s small spot s i z e and long p u l s e l e n g t h ( c 0 t >> Dgpot the u s e °f the f i x e d s c a l e l e n g t h , strong IB ab s o r p t i o n s c a l i n g laws (11-39, -40) w i l l be a p p r o p r i a t e . Comparison of experimental r e s u l t s with these s c a l i n g s i s d i s c u s s e d i n d e t a i l i n chapter V. 47 CHAPTER III HYDRODYNAMIC EFFECTS INDUCED BY LASER-DRIVEN ABLATION T h i s chapter i s concerned with the r e g i o n s ahead of the a b l a t i o n s u r f a c e . In chapter II the general flow s t r u c t u r e of l a s e r - d r i v e n a b l a t i o n was o u t l i n e d and i n that chapter i t was s t a t e d t h a t a shock wave propagates ahead of the a b l a t i o n s u r f a c e . The hydrodynamic e f f e c t s of the propagating shock are important aspects of the a b l a t i v e flow and are i n t i m a t e l y r e l a t e d to the compression and a c c e l e r a t i o n t a r g e t m a t e r i a l s . S t u d i e s of a b l a t i v e l y - d r i v e n compression of s o l i d t a r g e t s u s i n g a f u l l y hydrodynamic d e s c r i p t i o n have not, as ye t , been e x t e n s i v e l y i n v e s t i g a t e d i n the l i t e r a t u r e . For l a s e r t a r g e t s t u d i e s motivated by i n e r t i a l confinement f u s i o n the u l t i m a t e o b j e c t i v e i s the e f f i c i e n t c o n v e r s i o n of l a s e r energy i n t o hydrodynamic motion ( t a r g e t a c c e l e r a t i o n ) . However, t h e o r e t i c a l a n a l y s i s and i n t e r p r e t a t i o n of experiments [16] has been l i m i t e d t o a simple 'rocket model' to d e s c r i b e the a c c e l e r a t i o n p r o c e s s . The i n v e s t i g a t i o n s p r e s e n t e d i n c h a p t e r s VI and VII have e x p e r i m e n t a l l y r e v e a l e d s t r o n g evidence that the tar g e t behaves as a compressible f l u i d , not as a r i g i d body. A n a l y s i s of t a r g e t behaviour and l a s e r - t a r g e t energy c o u p l i n g must use a f l u i d d e s c r i p t i o n to account f o r experimental r e s u l t s . The purpose of t h i s chapter i s to d e s c r i b e the main f e a t u r e s of the shock compressed and a c c e l e r a t e d t a r g e t i n the a l t e r n a t i v e 48 hydrodynamic approach and r e l a t e them to the t h i n p l a n a r t a r g e t experiment. S e c t i o n 111 — 1 reviews the p h y s i c s of shock propa g a t i o n i n s o l i d m a t e r i a l s i n the pressure regime of s e v e r a l Mbar which i s r e l e v a n t to t h i s study. Target r a r e f a c t i o n f o l l o w i n g shock compression i s t r e a t e d i n s e c t i o n I I I - 2 . F i n a l l y , s e c t i o n III-3 examines the energy c o u p l i n g i n t o the compressed t a r g e t . 111-1 SHOCK PROPAGATION IN SOLIDS Shock compression of s o l i d m a t e r i a l s i s a complex s u b j e c t i n i t s own r i g h t and i s s u f f i c i e n t l y d i f f e r e n t from i t s analogue i n gas dynamics to warrant a d i s c u s s i o n of the main d i s t i n g u i s h i n g f e a t u r e s . For more d e t a i l s a l u c i d account can be found i n Z e l ' d o v i c h and R a i z e r [32] and r e f e r e n c e s t h e r e i n . The p r i n c i p a l d i f f e r e n c e between s o l i d s (or l i q u i d s ) and gases i s the strong i n t e r a c t i o n between the p a r t i c l e s . Atoms in condensed media are bound together by an e l a s t i c p o t e n t i a l which r e s i s t s both expansion and compression to minimize the p o t e n t i a l energy. Compression of a condensed medium must overcome t h i s e l a s t i c f o r c e as w e l l as two other c o n t r i b u t i o n s to the p r e s s u r e . These are the l a t t i c e v i b r a t i o n s of the atoms ( n u c l e i and bound e l e c t r o n s ) and thermal motions of unbound e l e c t r o n s (Fermi-Dirac e l e c t r o n g a s ) . The equation of s t a t e f o r condensed media can be expressed i n terms of the three c o n t r i b u t i o n s o u t l i n e d i n the above 49 phenomenological d e s c r i p t i o n , given by, 111 -1 c " fctl T " t e P e P c + P t l * P t e where e i s the i n t e r n a l energy of the m a t e r i a l : e_ , e., , and ° tl e^e are the c o n t r i b u t i o n s to c due to the e l a s t i c p o t e n t i a l , the v i b r a t i o n s of the l a t t i c e , and the thermal energy of the e l e c t r o n s r e s p e c t i v e l y . The pr e s s u r e P i s a l s o expressed as the sum of three c o n t r i b u t i o n s due t o : e l a s t i c compression P c, l a t t i c e thermal e x c i t a t i o n P^ , and e l e c t r o n i c thermal e x c i t a t i o n P j e • The r e l a t i v e magnitudes of these components are important f a c t o r s i n determining the s t r e n g t h and behavior of a shock. The v a r i o u s terms of equations (111 — 1 ) are e l a b o r a t e d upon i n r e f e r e n c e s [32,33,34]. T h i s form of equation of s t a t e i s used i n a n a l y s i n g shock experiments of moderate pressures (up to a few Mbar). In gas dynamics a d i s t i n c t i o n i s made between the l i m i t i n g cases of str o n g and weak shocks. T h i s can a l s o be a p p l i e d to shock compression of s o l i d s . In the weak shock l i m i t the p r i n c i p a l c o n t r i b u t i o n to the i n t e r n a l energy i s e c and the shock speed i s r e l a t e d to the c o m p r e s s i b i l i t y of the m a t e r i a l at standard c o n d i t i o n s . T h i s approaches the l i m i t of an a c o u s t i c wave, and the compression of the m a t e r i a l i s of the 50 order of a few percent. The p r e s s u r e range f o r which shocks i n s o l i d m a t e r i a l s are weak i s up to s e v e r a l hundreds of thousands of atmospheres. In the strong shock l i m i t , the energy imparted to the m a t e r i a l by the shock f a r exceeds i t s e l a s t i c p o t e n t i a l , and heats i t s u f f i c i e n t l y t h a t the p a r t i c l e s behave as a c l a s s i c a l i d e a l gas. In t h i s l i m i t the terms and e^ e are dominant, and i n f a c t the i d e a l gas equation of s t a t e may be a p p l i e d . Pressures of s e v e r a l tens, or hundreds of Mbar are necessary t o generate a s t r o n g shock i n s o l i d s . For p r e s s u r e s i n t e r m e d i a t e between these l i m i t s the three terms of equation (111 — 1 ) c o n t r i b u t e approximately e q u a l l y , hence the equation of s t a t e i s the most complicated and l e a s t w e l l understood i n t h i s regime. T h i s i s a l s o the regime (1 Mbar to tens of Mbar) where most c u r r e n t l a s e r - t a r g e t i n t e r a c t i o n experiments operate. The w e l l known shock r e l a t i o n s f o r c o n s e r v a t i o n of momentum and c o n s e r v a t i o n of mass expressed i n the l a b frame are [32], 111-2 p - P s"u III-3 where u i s the p a r t i c l e v e l o c i t y behind the shock, U i s the v e l o c i t y of the shock, and p and p are the m a t e r i a l d e n s i t i e s s c i n the unperturbed s o l i d , and the shock compressed s o l i d 51 r e s p e c t i v e l y . The parameter 0 i s the compression r a t i o . To complete the d e s c r i p t i o n of the system the energy equation a c r o s s the d i s c o n t i n u i t y i s [32], III-4 where e0 and e are the i n t e r n a l e n e r g i e s of the unperturbed and compressed m a t e r i a l s r e s p e c t i v e l y . In these equations the i n i t i a l pressure P 0 i s assumed to be n e g l i g i b l e . T h e o r e t i c a l knowledge of e ( P , P 0 , p ) can be used to s o l v e f o r p and U given the i n i t i a l s t a t e and f i n a l pressure u s i n g the shock Hugoniot. These equations form the b a s i s of a wide range of s t u d i e s , both experimental and t h e o r e t i c a l , i n the Mbar and higher p r e s s u r e regime. D e t a i l e d knowledge i n t h i s f i e l d i s scarce and q u a n t i t a t i v e p r e d i c t i o n s of experimental r e s u l t s are t h e r e f o r e d i f f i c u l t . However, hydrodynamic e f f e c t s p lay an important r o l e i n the b a s i c understanding of the experiment. These e f f e c t s are apparent and may be d i s c u s s e d q u a l i t a t i v e l y without d e t a i l e d knowledge of the e q u a t i o n - o f - s t a t e . I t i s u s e f u l to examine the r e l a t i v e magnitudes of the v e l o c i t i e s u, U and the sound speed 1, a, i n the compressed m a t e r i a l . In our experiment we are compressing Aluminum to s e v e r a l Mbar with a s i n g l e shock. The v a r i a t i o n of the q u a n t i t i e s u, U and a have been examined t h e o r e t i c a l l y by 1 In the f o l l o w i n g the symbol a denotes the sound speed i n the shock compressed t a r g e t . The symbol c continues to denote the sound speed i n the l a s e r heated corona. 52 T r a i n o r and Lee [34] f o r Aluminum i n the pr e s s u r e range 1 -100 Mbar using a G r u n e i s i n equation of s t a t e (a form of 111 - 1 ) . The r a t i o a/U v a r i e s from about 0.85 to 0.6; u/U v a r i e s from l e s s than 0.5 to 0.75; and f i n a l l y u/a v a r i e s from l e s s than 0.5 to about 1.3. The r a t i o u/a i s equal to 1 at a pressure of approximately 12 Mbar ( i . e . the p a r t i c l e v e l o c i t y i s Mach 1 i n the l a b o r a t o r y frame). The f a c t that u < a f o r p r e s s u r e s l e s s than 12 Mbar i s important f o r i n t e r p r e t a t i o n of the experimental r e s u l t s . H I - 2 TARGET RAREFACTION AND DYNAMICS In t h i s experiment the shock wave i s not d i r e c t l y observed. However, i t s e f f e c t on the t a r g e t i s to transform i t from the i n i t i a l s t a t e of a c r y s t a l l i n e s o l i d to a compressible f l u i d . Target behaviour f o l l o w i n g shock compression i s dominated by f l u i d r a r e f a c t i o n s which are then d i r e c t l y o b servable and are d i s t i n c t from r i g i d body ('rocket l i k e ' ) motion. T h i s s e c t i o n d e a l s with the r a r e f a c t i o n processes that take p l a c e i n p l a n a r t a r g e t , f i n i t e spot s i z e geometries. 53 Shock r a r e f a c t i o n sequences The forward propagating shock i s both i n i t i a t e d and s u s t a i n e d by the a b l a t i o n p r o c e s s . When the l a s e r pulse ceases the a b l a t i o n pressure d i m i n i s h e s and hydrodynamic a t t e n u a t i o n of the shock by a f o l l o w i n g r a r e f a c t i o n can occur. T h i s has been a n a l y s e d by T r a i n o r and Lee [34]. In f i n i t e t h i c k n e s s t a r g e t s i t i s a l s o p o s s i b l e f o r the shock to a r r i v e at the rear s u r f a c e of the f o i l before the end of the l a s e r p u l s e . T h i s can r e s u l t i n a sequence of events that d r a s t i c a l l y a f f e c t the subsequent p r o p e r t i e s of the t a r g e t . F i r s t l y , the shock a r r i v i n g at the rear s u r f a c e of the t a r g e t unloads i n t o the vacuum. A r e t u r n i n g r a r e f a c t i o n wave then propagates back through the compressed s o l i d towards the a b l a t i o n s u r f a c e . The m a t e r i a l s w i t h i n the r a r e f a c t i o n wave are f u r t h e r a c c e l e r a t e d i n the forward d i r e c t i o n and e j e c t e d from the rear s i d e of the t a r g e t at the expense of the compression energy s t o r e d i n the shocked s o l i d . F i n a l l y , i f the r a r e f a c t i o n wave a r r i v e s at the a b l a t i o n s u r f a c e before the end of the l a s e r p u l s e , i t w i l l s t r o n g l y p e r t u r b the steady s t a t e a b l a t i o n p r o c e s s . In f a c t i t w i l l d i s r u p t i t e n t i r e l y . The t a r g e t w i l l become underdense and the l a s e r l i g h t w i l l p e n e t r a t e . 54 Shock Unloading of the Rear Side When the shock a r r i v e s at the r e a r s i d e of the t a r g e t , t h i s s u r f a c e becomes a fr e e s u r f a c e and the compressed m a t e r i a l s begin to unload i n t o the vacuum at zero p r e s s u r e . In s t r i c t l y planar geometry t h i s u n l o a d i n g process can be t r e a t e d as a s e l f - s i m i l a r r a r e f a c t i o n . However, the compressed p o r t i o n of the t a r g e t i s l i m i t e d to the s i z e of the i n c i d e n t l a s e r spot ( f o r t a r g e t t h i c k n e s s e s small compared to the spot s i z e ) . Hence the expansion of the compressed m a t e r i a l s beyond the i n i t i a l r ear boundary of the t a r g e t cannot be t r e a t e d i n one dimension due to r a r e f a c t i o n s propagating from the edges of the expanding m a t e r i a l . To p r o p e r l y d e s c r i b e the unloading process one must c o n s i d e r both the a x i a l r a r e f a c t i o n propagating back towards the a b l a t i o n s u r f a c e through the compressed m a t e r i a l s , and the two-dimensional flow p a t t e r n of the expanding m a t e r i a l as i t emerges from the rear of the t a r g e t . The next two s e c t i o n s c o n t a i n d e t a i l s of the r a r e f a c t i o n sequence o u t l i n e d above. S e c t i o n 1 1 1 - 2 - i g i v e s a one-dimensional model d e a l i n g with a x i a l r a r e f a c t i o n and the l a s e r burnthrough mechanism. Se c t i o n I I I - 2 - i i d e s c r i b e s the two-d imensional behaviour of the u nloading flow out of the rear s u r f a c e of the t a r g e t . 55 I I I _ 2 - i AXIAL RAREFACTION AND LASER BURNTHROUGH A x i a l R a r e f a c t i o n The nature of the r a r e f a c t i o n wave propagating back towards the a b l a t i o n s u r f a c e w i l l now be examined. A f t e r the shock has passed through the t a r g e t the compressed m a t e r i a l moves at v e l o c i t y u with r e s p e c t to the o r i g i n a l t a r g e t boundaries ( l a b frame), and the unloading wave propagates backwards through t h i s moving m a t e r i a l towards the a b l a t i o n s u r f a c e . The wave head t r a v e l s at the sound speed, a, with re s p e c t to the moving m a t e r i a l . Since we are c o n s i d e r i n g u < a fo r now, i t f o l l o w s that the r a r e f a c t i o n wave head remains i n s i d e the o r i g i n a l t a r g e t boundaries. T h e r e f o r e s t r o n g edge r a r e f a c t i o n s cannot occur and i t i s reasonable to t r e a t the a x i a l r a r e f a c t i o n w i t h i n the t a r g e t boundaries as a one-dimensional flow. The a x i a l r a r e f a c t i o n , being one-dimensional, i s a s e l f -s i m i l a r wave with the same g e n e r a l p r o p e r t i e s as the c o r o n a l expansion d i s c u s s e d i n I I - 2 - i . The governing equations are those of mass and momentum c o n s e r v a t i o n coupled with an equation of s t a t e . In t h i s case the flow i s i s e n t r o p i c (compared to is o t h e r m a l c o n d i t i o n s f o r the c o r o n a l expansion). The i n i t i a l c o n d i t i o n s are a ste p d i s c o n t i n u i t y from compressed 56 d e n s i t y to vacuum. The general s o l u t i o n to t h i s type of flow i s [32,35], v - a I I I - 6 J P a v * ' — a constant I I I - 7 where v i s the flow v e l o c i t y measured i n the x ' - r e f e r e n c e frame. ( x ' , t ) = (0,0) corresponds to the i n s t a n t the shock a r r i v e s at the rear s u r f a c e ; the x ' - a x i s i s taken to be d i r e c t e d towards the vacuum and i t i s f i x e d to the compressed f l u i d (which i s moving at v e l o c i t y u i n the l a b frame). The o r i g i n c o i n c i d e s with the t a r g e t rear s u r f a c e at t = 0. The i n t e g r a t i o n i s understood to be at constant entropy. F i g u r e 6 g i v e s a schematic p r o f i l e of the flow v a r i a b l e s . Note that t h i s problem i s no d i f f e r e n t i n p r i n c i p l e from any other s e l f -s i m i l a r flow; the q u a n t i t y x ' / t p l a y s the r o l e of the s e l f -s i m i l a r parameter. A l l flow v a r i a b l e s are f u n c t i o n s of x ' / t only. T o F i g u r e 6 Schematic flow p r o f i l e s f o r shock unloading i n t o vacuum. 58 Burnthrouqh Model The propagation of the shock and r a r e f a c t i o n waves in the t a r g e t i s i l l u s t r a t e d i n F i g u r e 7. T h i s i s s i m i l a r to F i g u r e 2 w i t h the e f f e c t of f i n i t e t a r g e t t h i c k n e s s i n c l u d e d . The shock begins to propagate at time zero, at the i n s t a n t the l a s e r i s turned on. A q u a l i t a t i v e p i c t u r e of the d e n s i t y p r o f i l e s w i t h i n the t a r g e t i s shown i n F i g u r e 8. The time r Q = D/U, where D i s the o r i g i n a l t h i c k n e s s of the t a r g e t , corresponds to the a r r i v a l time of the shock wave at the r e a r s i d e of the t a r g e t . At t h i s time the t a r g e t begins to unloaded i n t o vacuum, region (3), while the r a r e f a c t i o n wave head propagates back towards the a b l a t i o n s u r f a c e r e g i o n ( 2 ) . The treatment given i n r e f e r e n c e [14] has suggested the f o l l o w i n g s c e n a r i o f o r the shock r a r e f a c t i o n sequence. When the r e t u r n i n g r a r e f a c t i o n reaches the a b l a t i o n s u r f a c e a new shock i s launched i n the forward d i r e c t i o n . A sequence of d i m i n i s h i n g shocks f o l l o w e d by r a r e f a c t i o n s takes p l a c e , e v e n t u a l l y r e a c h i n g a steady s t a t e regime i n which the t a r g e t i s a c c e l e r a t e d as a r i g i d body. The c o m p r e s s i o n - r a r e f a c t i o n events take p l a c e on a time s c a l e much smal l e r than the time i t would take to a b l a t e completely the e n t i r e t h i c k n e s s of the t a r g e t . The only way t h i s c o u l d happen i s i f the t a r g e t remains i n i t s c r y s t a l l i n e s o l i d s t a t e and i s not heated g r e a t l y by the i n i t i a l shock. T h i s can occur i f the a b l a t i o n pressure i s s u f f i c i e n t l y low that o n l y a weak shock i s generated. In t h i s l i m i t i n g case the s h o c k - r a r e f a c t i o n sequence becomes simply a 59 REFLECTED RAREFACTION WAVES F i g u r e 7 X-t diagram of s h o c k - r a r e f a c t i o n sequence i n f i n i t e t h i c k n e s s planar t a r g e t s . 60 ABLATION SHOCK LASER LASER F i g u r e 8 D e n s i t y p r o f i l e s a t v a r i o u s times ( i n d i c a t e d i n F i g u r e 7) d u r i n g the shock r a r e f a c t i o n sequence. Burnthrough occurs before the end of the l a s e r p u l s e . 61 s e r i e s of a c o u s t i c ( c o l d compression and r a r e f a c t i o n ) d i s t u r b a n c e s too weak to disassemble the l a t t i c e . In our experiment the a b l a t i o n p r e s s u r e i s i n a regime where the forward propagating shock cannot be t r e a t e d as weak. T h e r e f o r e the the s h o c k - r a r e f a c t i o n sequence and i t s e f f e c t on the a b l a t i o n e q u i l i b r i u m must be t r e a t e d w i t h i n f u l l c o ntext of hydrodynamic wave-wave i n t e r a c t i o n s . When the r e t u r n i n g r a r e f a c t i o n reaches the a b l a t i o n f r o n t the c o n d i t i o n s i n the t a r g e t ahead of the a b l a t i o n f r o n t are s t r o n g l y p e r t u r b e d from the steady, compressed s t a t e necessary f o r the a b l a t i o n e q u i l i b r i u m . The r a r e f a c t i o n i s a non-steady flow which removes and a c c e l e r a t e s m a t e r i a l from the a b l a t i o n s u r f a c e , accompanied by a decrease i n the s t a t i c p r e s s u r e . I t i s t h e r e f o r e impossible to maintain the s e l f - r e g u l a t i n g a b l a t i o n p r o c e s s , as d e s c r i b e d i n II-2-v. On the l a s e r s i d e the plasma flow takes the form of a r a r e f a c t i o n wave, and when t h i s i n t e r a c t s w i t h the wave r e t u r n i n g from the rear s u r f a c e , the two ' c o l l i d e ' to produce two r e f l e c t e d r a r e f a c t i o n waves [36] as i n d i c a t e d i n F i g u r e 8. The plasma d e n s i t y i s then q u i c k l y brought down to l e s s than n c r and the l a s e r p e n e t r a t e s the t a r g e t , r e s u l t i n g i n burnthrough. T h i s process i s e a s i l y observed by mon i t o r i n g the t r a n s m i s s i o n of l a s e r l i g h t through the t a r g e t . The burnthrough time Tg i s given by ( F i g u r e 7 ), II I - 7 6 2 where Tg i s the time f o r the shock to t r a v e r s e the t a r g e t and a r r i v e at the rear s i d e , i s the time f o r the r e t u r n i n g r a r e f a c t i o n t o reach the a b l a t i o n s u r f a c e and i s the c h a r a c t e r i s t i c time f o r the plasma to expand to d e n s i t i e s l e s s than n c r . T h e r e f o r e , D T S " U I I I - 8 R 1 " " " A a p / p c c s I I I - 9 where D i s the i n i t i a l t a r g e t t h i c k n e s s , d A i s the t h i c k n e s s of a b l a t e d m a t e r i a l , a i s the sound speed i n re g i o n (1), U i s the shock speed ( i n the l a b o r a t o r y r e f e r e n c e frame), p and p are c s the d e n s i t i e s i n the shock compressed t a r g e t ( r e g i o n ( 1 ) ) , and uncompressed s o l i d ( r e g i o n (0)) r e s p e c t i v e l y . The burnthrough time can then be. w r i t t e n as, D U 1 • a p c IT IT j 1 1 1 - 1 0 The parameter d^ depends on the t a r g e t t h i c k n e s s D. Since the a b l a t i o n f r o n t i s subsonic d^ i s much l e s s than D so that d/^ /D can be c o n s i d e r e d a small c o r r e c t i o n i n equation ( I I I —10) • The q u a n t i t y i s a l s o assumed to be s m a l l . T h i s may be j u s t i f i e d by the f o l l o w i n g arguments. When the r e t u r n i n g r a r e f a c t i o n wave meets the a b l a t i o n s u r f a c e (Figure 8 (d)) 63 p a r t i c l e s on the l a s e r s i d e stream towards the l a s e r with a v e l o c i t y of 0 ( c 0 ) , w h ile the p a r t i c l e s i n the compressed t a r g e t move i n the opposite d i r e c t i o n with a v e l o c i t y of 0 ( a ) . R e c a l l c 0 and a are the sound speeds i n the corona and compressed t a r g e t r e s p e c t i v e l y . Hence the overdense r e g i o n A disassembles with a v e l o c i t y of the order of ( c 0 + a) and we may estimate r_ E to be, T » — E c +a 1 1 1 - 1 1 o Furthermore, A = x 0+ x' x' where x 0 i s the a b l a t i o n zone t h i c k n e s s and x' i s the t h i c k n e s s of the overdense r e g i o n ahead of the a b l a t i o n zone. The a x i a l r a r e f a c t i o n p r o f i l e i s s e l f -s i m i l a r , and x' can be approximated by the time dependent 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 a r ^ which i s much l a r g e r than x 0 . Th e r e f o r e , E c +a o In g e n e r a l , c 0 i s much l a r g e r than a. T y p i c a l l y c 0 =* 10 7 cm/s, while a 10 6 cm/s. T h e r e f o r e r_ i s of the order of a few fc percent of and can be negleted i n equation ( I I I — 1 0 ) . If one assumes both d^/D and are n e g l i g i b l e equation (III—10) then i m p l i e s a l i n e a r dependence of tg on D. T h i s model and the assumptions are d i s c u s s e d and compared l a t e r along with the experimental r e s u l t s . 64 I I I - 2 - i i RADIAL RAREFACTION Consider now the motion of the unloaded m a t e r i a l once i t leaves the o r i g i n a l boundaries of the t a r g e t . The f i n i t e spot s i z e makes two-dimensional e f f e c t s a l l - i m p o r t a n t . The m a t e r i a l e j e c t e d from the rear of the i r r a d i a t e d p o r t i o n of the t a r g e t i s analogous to a supersonic gas j e t emerging from a c i r c u l a r o r i f i c e i n t o vacuum. In s i d e the t a r g e t the r a r e f a c t i o n i s a one-dimensional s e l f - s i m i l a r flow. T h i s one-dimensional flow i s transformed i n t o a two-dimensional expansion o u t s i d e the o r i f i c e from which the t a r g e t m a t e r i a l i s e j e c t e d . The t r a n s i t i o n between the two types of flow takes plac e at the o r i f i c e . R e c a l l ( s e c t i o n I I I -2 - i , ' A x i a l R a r e f a c t i o n ' ) t h a t the s e l f - s i m i l a r flow i s a f u n c t i o n only of x ' / t ( i . e . p = p ( x ' / t ) , v = v ( x ' / t ) , ... ). The x ' - a x i s i s f i x e d to the shock compressed m a t e r i a l (unperturbed by r a r e f a c t i o n s ) and moves at v e l o c i t y u i n the l a b frame. Ther e f o r e the t a r g e t rear surface corresponds to the p o i n t x ' / t = -u i n the r e f e r e n c e frame of the s e l f - s i m i l a r flow. Hence from (I I I - 6 ) the flow v e l o c i t y at the t a r g e t boundary i s v = a + x'/t = a - u, where v i s the flow v e l o c i t y measured with respect to the x ' - a x i s . A t r a n s f o r m a t i o n from the s e l f - s i m i l a r r e f e r e n c e frame t o the l a b o r a t o r y frame i s necessary to determine the c o n d i t i o n s of the flow emerging the t a r g e t r e a r suface. T h i s t r a n s f o r m a t i o n i s G a l i l e a n of r e l a t i v e v e l o c i t y u. Denote w as the flow v e l o c i t y i n the l a b frame, then at the t a r g e t rear s u r f a c e w = v + u = a. Therefore the 6 5 v e l o c i t y at the t a r g e t rear s u r f a c e i s always s o n i c . 1 A f i n a l p o i n t to be noted i s that i f the shock i s strong enough such that the v e l o c i t y u imparted to the shocked m a t e r i a l i s supersonic i n the l a b o r a t o r y frame (u > a) then an a x i a l r a r e f a c t i o n (which i s always s o n i c with r e s p e c t to the f l u i d ) cannot propagate i n t o the t a r g e t . In t h i s case there i s no soni c t r a n s i t i o n from s e l f - s i m i l a r to two-dimensional expansion; r a t h e r , the flow emerging from the t a r g e t rear s u r f a c e i s supersonic with w = u which then expands two-d i m e n s i o n a l l y . These c o n d i t i o n s are not encountered i n t h i s experiment as noted at the end of 111 — 1. The e x p l i c i t s o l u t i o n of the flow p a t t e r n beyond the t a r g e t boundary i s not attempted i n t h i s work. However the d i s c u s s i o n above has s i m p l i f i e d the problem s u f f i c i e n t l y to p o i n t towards a mathematical f o r m u l a t i o n and all o w a q u a l i t a t i v e a n a l y s i s of the flow. The problem i s s p e c i f i e d as f o l l o w s : ( i ) the a c c e l e r a t e d t a r g e t m a t e r i a l emerges as a steady sonic flow from a c i r c u l a r o r i f i c e i n t o vacuum; ( i i ) the flow i s uniform a c r o s s the diameter of the o r i f i c e and i s assumed to be axisymmetric and i r r o t a t i o n a l ; and, ( i i i ) an i s e n t r o p i c equation of s t a t e i s assumed. During the i n i t i a l moments of the unloading the flow emerging from the t a r g e t rear s u r f a c e i s t i m e - v a r y i n g . However, due t o the steady s t a t e c o n d i t i o n s a t the t a r g e t boundary i t 1 T h i s f a c t i s a s p e c i f i c case of the gene r a l statement made i n s e c t i o n I I - 2 - i , namely, that the f l u i d v e l o c i t y of a s e l f -s i m i l a r flow i s always s o n i c at the o r i g i n - o f r e f e r e n c e frames f i x e d to x'/t = cons t a n t . 66 e v e n t u a l l y becomes a steady flow. The time r e q u i r e d to set up a steady flow i s approximately = (D Sp 0t /2a). T h i s i s the time i t takes f o r and edge r a r e f a c t i o n (which i s s o n i c ) to t r a v e r s e the r a d i u s of the c i r c u l a r o r i f i c e . In our experiment a =* 10 s cm/s, and Dspot ~ ^ u m s o t n a t this time i s s e v e r a l nanoseconds. On the other hand the time r e q u i r e d to expand and e j e c t a l l of the compressed m a t e r i a l i n s i d e the t a r g e t i s approximately r D = p (D-d A)/ap (equation ( I I I - 9 ) ) where D i s H S " C the o r i g i n a l t a r g e t t h i c k n e s s , p/p i s the shock compression c s r a t i o and d^ << D i s the a b l a t e d t h i c k n e s s . In the experiment D <, 100 um, D s p o t = 80 Mm and p^/'pQ=* 2, so that r R < . T h e r e f o r e a steady two-dimensional flow i s probably never a c h i e v e d i n our experiment. Since the steady flow problem i s e a s i e r t o s o l v e we s h a l l examine i t mainly to i d e n t i f y the q u a l i t a t i v e f e a t u r e s of the two-dimensional expansion, although keeping i n mind t h a t t h i s i s the l a t e time asymptotic s o l u t i o n of the a c t u a l s i t u a t i o n . The problem i s s i m i l a r to supersonic flow i n n o z z l e s and j e t s , which has been t r e a t e d e x t e n s i v e l y i n v a r i o u s works [36,37]. The h y b e r b o l i c system i s s o l v e d u s i n g the method of c h a r a c t e r i s t i c s . The mathematical c h a r a c t e r i s t i c s a l s o have a very r e a l p h y s i c a l s i g n i f i c a n c e . In supersonic flow the c h a r a c t e r i s t i c s are the t r a j e c t o r i e s along which d i s t u r b a n c e s propagate w i t h i n the flow f i e l d . In a two-dimensional axisymmetric flow there are two c h a r a c t e r i s t i c s p a s s i n g through every p o i n t i n the flow. Both c h a r a c t e r i s t i c s i n t e r s e c t the s t r e a m l i n e s at an angle a = a r c t a n ( a / v ) where v and a are the 67 l o c a l f l u i d v e l o c i t y and the l o c a l sound speed r e s p e c t i v e l y ; a i s known as the Mach angle. The c h a r a c t e r i s t i c s C+ i n t e r s e c t s t r e a m l i n e s at a p o s i t i v e angle, and the C- c h a r a c t e r i s t i c s at a negative angle. F i g u r e 9 (a) i s a schematic r e p r e s e n t a t i o n of the c h a r a c t e r i s t i c s t h a t might occur i n t h i s problem. From the edge of the o r i f i c e , at which p o i n t the pressure drops to zero, the C+ c h a r a c t e r i s t i c s form a r a r e f a c t i o n f a n . The l e a d i n g edge of the r a r e f a c t i o n fan propagates towards the a x i s at the Mach angle a and i n t e r s e c t s i t at the p o i n t C i n F i g u r e 9. T h i s c h a r a c t e r i s t i c , and others i n t e r s e c t i n g the ax'is f u r t h e r along are r e f l e c t e d at the a x i s and become C- c h a r a c t e r i s t i c s p ropagating outwards. In 9 (b) the r a d i a l d e n s i t y p r o f i l e at v a r i o u s p o i n t s along the a x i s i s shown. The r e f l e c t i o n of the r a d i a l r a r e f a c t i o n wave at the a x i s r e s u l t s i n a d e n s i t y minimum on the a x i s , while a d e n s i t y maximum propagates r a d i a l l y outwards i n a c o n i c a l t r a j e c t o r y . In the asymptotic l i m i t of l a r g e r a d i a l d i s t a n c e s from the o r i f i c e , the Mach angle becomes small as the f l u i d c o o l s : the c h a r a c t e r i s t i c s and the s t r e a m l i n e s become c o l i n e a r and the expanded f l u i d d r i f t s along i t s e s t a b l i s h e d c o n i c a l t r a j e c t o r y . Experimental o b s e r v a t i o n s of such an expansion w i l l y i e l d the asymptotic flow v e l o c i t y and the angular l o c a t i o n of the d e n s i t y maximum i n the expanding flow. T h i s angle can be est i m a t e d roughly u s i n g the c o n d i t i o n s expected during the e a r l y phases of the unloading (when the expansion i s s t i l l t i m e - v a r y i n g ) . The r a d i a l expansion takes p l a c e at a v e l o c i t y of 0 ( a ) , while the a x i a l v e l o c i t y i s approximately u + a. ee A B C D E F i g u r e 9 Shock unloading of t a r g e t rear as a two-dimensional supersonic axisymmetric flow (a) T r a j e c t o r i e s of the s t r e a m l i n e s and c h a r a c t e r i s t i c s (C+, C-) f o r two-dimensional axisymmetric supersonic flow (b) R a d i a l d e n s i t y p r o f i l e s at v a r i o u s p o s i t i o n s i n the flow. 69 Therefore the cone angle i s * a r c t a n [ a / ( u + a ) ] . Since a/(u+a) * 1/2 t h i s angle should be roughly 30 degrees. II1-3 HYDRODYNAMIC EFFICIENCY The e f f i c i e n c y of energy c o n v e r s i o n from the l a s e r input i n t o hydrodynamic compression and a c c e l e r a t i o n i s an important parameter f o r determining the f e a s i b i l i t y of l a s e r f u s i o n schemes. I t can be roughly d e f i n e d as the f r a c t i o n a l energy d e p o s i t e d i n the forward hydrodynamic motions of the t a r g e t m a t e r i a l s a c c e l e r a t e d by the a b l a t i o n p r o c e s s . The most widely used treatment has been a rocket model. The p h y s i c a l d e s c r i p t i o n used i n t h i s model i s analogous to that of a s o l i d r ocket a c c e l e r a t e d by m a t e r i a l e j e c t e d from i t . The a c c e l e r a t e d p o r t i o n of the t a r g e t moves forward as a u n i t . T h i s d e s c r i p t i o n i s perhaps c o n s i s t e n t with the weak s h o c k - r a r e f a c t i o n sequence f o r momentum t r a n s f e r (as i m p l i e d i n r e f e r e n c e [14]) so that the t a r g e t maintains i t s r i g i d body i n t e g r i t y . For str o n g e r shocks the f u l l e f f e c t s of the hydrodynamic processes as d e s c r i b e d i n S e c t i o n I I I - 2 need to be c o n s i d e r e d . In the f o l l o w i n g d i s c u s s i o n a new d e f i n i t i o n i s proposed f o r the hydrodynamic e f f i c i e n c y t h at i s c o n s i s t e n t with the a n a l y t i c model that has been d e s c r i b e d so f a r . A c c o r d i n g l y , an ex p r e s s i o n f o r the hydrodynamic e f f i c i e n c y depending only on the t a r g e t m a t e r i a l and l a s e r parameters may be obtained. The e f f e c t s of f i n i t e geometry ( r a r e f a c t i o n s ) on energy c o u p l i n g 70 are a l s o d i s c u s s e d . T h i s i s an important c o n s i d e r a t i o n f o r p r a c t i c a l measurement of the e f f i c i e n c y . I I I - 3 - i DEFINITION OF HYDRODYNAMIC EFFICIENCY Consider the g l o b a l energy balance i n the l a b o r a t o r y r e f e r e n c e frame. In the a b l a t i o n model of chapter II the re f e r e n c e frame used i s f i x e d t o the a b l a t i o n zone ( c f . F i g u r e 3), and i n t h i s frame no energy f l u x passed through the a b l a t i o n s u r f a c e ( c f . equation 11-25). In the l a b o r a t o r y r e f e r e n c e frame the a b l a t i o n p r e s s u r e i s exer t e d on a moving s u r f a c e and i s t h e r e f o r e doing work at a constant r a t e . The r a t i o of t h i s energy f l u x t o the i n c i d e n t l a s e r energy f l u x g i v e s the f r a c t i o n of i n c i d e n t energy that i s t r a n s f e r r e d to the hydrodynamic motions of the t a r g e t . A c c o r d i n g l y one may d e f i n e the hydrodynamic e f f i c i e n c y , rj, as the r a t i o of the energy f l u x E c f l o w i n g i n t o the compressed t a r g e t ( r e g i o n (4) of F i g u r e s 1 and 2) t o the absorbed l a s e r f l u x I a , E n = ^- 111-12 a Two p o i n t s should be noted concerning t h i s d e f i n i t i o n . F i r s t l y , rj i s a time dependent q u a n t i t y s i n c e i t can change with the l a s e r i n t e n s i t y and the flow c o n d i t i o n s . Hence the arguments r e l a t i n g steady s t a t e flow and the concept of s e l f - r e g u l a t e d flow g i v e n i n II-2-v a l s o apply here. However, a p r a c t i c a l 71 measurement would g i v e a time averaged va l u e . Secondly, (111-12) i s independent of t a r g e t t h i c k n e s s or l a s e r p u l s e l e n g t h , both of which enter i n t o the rocket model. In a p r a c t i c a l sense t h i s means that f o r a given set of l a s e r parameters ( I , X^) and a given t a r g e t m a t e r i a l one should o b t a i n a s i n g l e value f o r 77. To o b t a i n a q u a n t i t a t i v e e x p r e s s i o n f o r 77 one must c o n s i d e r the usual f l u i d c o n s e r v a t i o n equations (mass, momentum and energy) w r i t t e n i n the l a b o r a t o r y r e f e r e n c e frame. Fi g u r e 2 i l l u s t r a t e s the t a r g e t motions i n t h i s frame. We c o n s i d e r only steady s t a t e c o n d i t i o n s . The a b l a t i o n and a b s o r p t i o n s u r f a c e s , as w e l l as the t a r g e t , move at constant v e l o c i t i e s i n t h i s r e f e r e n c e frame. We denote u as the f l u i d v e l o c i t y i n the shock compressed region ( s t r e a m l i n e s w i t h i n the shock compressed region of F i g u r e 2 ) , and q as the v e l o c i t y of the a b l a t i o n zone (cont a i n e d by the a b l a t i o n and a b s o r p t i o n s u r f a c e s , dashed l i n e s i n F i g u r e 2 ) . Since the a b l a t i o n f r o n t 'burns' i n t o the compressed t a r g e t a t co n s t a n t v e l o c i t y , |q| > | u j . We can r e l a t e q and u by d e f i n i n g a new parameter 0, ( 1 + ° ) U 111-13 P h y s i c a l l y a i s the burn v e l o c i t y of the a b l a t i o n of the compressed t a r g e t expressed as a f r a c t i o n of the v e l o c i t y u. The r a t i o a i s an important parameter i n t h i s c a l c u l a t i o n . I t w i l l be shown p r e s e n t l y that 77 depends s o l e l y on a. Note that the X-axis used i n F i g u r e 2 and the f o l l o w i n g equations i s d i r e c t e d o p p o s i t e t o the motions of the compressed t a r g e t , 72 a b l a t i o n and a b s o r p t i o n s u r f a c e s . T h e r e f o r e q and u have negative v a l u e s i n t h i s r e p r e s e n t a t i o n . Since the a b l a t i o n zone remains in a steady s t a t e , the p r o f i l e s of p, v and T are constant i n time. A constant mass f l u x moves through the a b l a t i o n zone but there are no net energy or momentum f l u x e s e n t e r i n g t h i s zone. However, the d e f i n i t i o n (III-12) i s concerned with steady s t a t e energy f l u x e s . T h e r e f o r e we must examine the shock compressed region and the c o r o n a l r a r e f a c t i o n r e g i o n which are l i n k e d by the a b l a t i o n zone. The shock wave and the c o r o n a l r a r e f a c t i o n wave are unsteady flows. In e i t h e r wave the t o t a l energy and t o t a l momentum c a r r i e d by the wave d i s t u r b a n c e change at a constant r a t e . Hence they i n v o l v e energy and momentum f l u x e s . When the t a r g e t i s c o n s i d e r e d as a whole the shock wave and the c o r o n a l r a r e f a c t i o n wave must s a t i s f y the general c o n s e r v a t i o n r e l a t i o n s . To e l i m i n a t e the steady s t a t e a b l a t i o n zone from our c o n s i d e r a t i o n we take the time d e r i v a t i v e of the c o n s e r v a t i o n equations, fec + fer " *a 111-14 P c • P r = 0 II I- 1 5 73 ft • ft = 0 111-16 c r (111-14) i s the energy equation where E c and E r are the time r a t e s of change of the t o t a l energy i n the shock compressed region and the c o r o n a l r a r e f a c t i o n r e s p e c t i v e l y . (III-15) i s the momentum balance; p and p are the r e s p e c t i v e r a t e s of c r change of t o t a l momentum i n the shocked and c o r o n a l r e g i o n s . F i n a l l y (111-16> i s the mass balance where m i s the ra t e at which i s mass i s a b l a t e d from the compressed t a r g e t and mr i s the r a t e at which mass i s added t o the co r o n a l r a r e f a c t i o n . Each of the s i x q u a n t i t e s i n equation can be expressed using the hydrodynamic flow d e s c r i p t i o n s of the shock compressed and c o r o n a l r a r e f a c t i o n regions given p r e v i o u s l y i n t h i s chapter and i n chapter I I 1 , K • i f / [ T P C X ) V 2 ( X ) • S p c x ) ! ] •00 T l 2 . * T1 dX 111 - 17 'qt 4 P c 3 c i • * 1 * i c3- ) 2 o o v 2 c IT v c ' o o oo 111- 18 /GOp(X)V(X)dX = 2 p o c 2 (1 • } } ) qt 1 The i n t e g r a l s i n equations (111-17) to (111-21) are sol v e d i n d e t a i l i n Appendix A. 74 r L CO * - " a f / p w d x - P c Z I 1 - 1 9 a t 1 0 0 E • -T— c dt Qt / p ( X ) P <J - f ) d x - - P u ( l - a ( B - l ) ) ^ •'ut S C /•qt Pc s dT / p ( x ) u d x = " P C 1 " o ( 6 - D ) *Mjt 111-21 *c s dT / p ^ d X * p c u o Alt 111-22 Note the r e l a t i o n E c= p c u = - P |.u. T h i s can be combined with the energy balance equation (111-16) to g i v e , E r P rU a I a 111-23 which g i v e s , ° o 111-24 o From equation (11-29), l a = 4 p 0 c 0 3 , so that the energy balance given i n equation (111-24) i m p l i e s a r e l a t i o n between q and u, or r a t h e r , u and a, namely, 75 °o a2-1 111-25 Remember that u has a negative v a l u e , and, 0 < a « 1. The mass c o n s e r v a t i o n r e l a t i o n (111-16) may be combined with (111-19) and (111-22) to give a r e l a t i o n between a and the d e n s i t y r a t i o P O / P C , u | b 1_^ o o c III-26a The hydrodynamic e f f i c i e n c y TJ i s given by, III-26b E p u n - _ £ . . l£_ . 2o • 0 ( a 2 ) J I 1 - 2 7 a a « 1 I t can be seen that rj i s improved with i n c r e a s e d l a s e r p e n e t r a t i o n d e n s i t y p 0 , as w e l l as reduced shock compression r a t i o 0 . Using the a b l a t i o n model, the s c a l i n g laws can g i v e an ex p r e s s i o n f o r p 0 i n terms of the l a s e r parameters (Po = P a/2m). Using the f i x e d s c a l e l e n g t h , strong IB a b s o r p t i o n s c a l i n g laws (11-39, -40) a p p r o p r i a t e t o our 76 experiment we o b t a i n , , * r^Hi ( z t n A ) - i / 3 ^ ) 7 / 6 i y 3 x L 2 / 5 L ; 1 / 5 ] 1 / 2 U n i t s are [ p g ] = gem* 3, [ I a ] = TW/cm2, [X^] = Mm, [ L x ] = Mm. The compression r a t i o /3 remains an unknown i n t h i s e x p r e s s i o n . Formally i t i s a f u n c t i o n of P a and the t a r g e t m a t e r i a l ( e q u a t i o n - o f - s t a t e ) and t h e r e f o r e c o u l d be expanded w i t h i n equation (111-28) to g i v e an e x p l i c i t e x p r e s s i o n . E x p e r i m e n t a l l y 0 i s t y p i c a l l y around 2 i n the few Mbar pressure regime f o r shock compression of Aluminum [33], Using these v a l u e s i n (111-28) 17 i s found to be about 5 % f o r I a = 10 TW/cm2, X L = .355 Mm, L x = 80 Mm, A/Z = 2, ZlnA = 60 and P s = 2.7 gem"3 (Aluminum). The value i s q u i t e low compared to p r e d i c t i o n s using the rocket model (maximum 77 65 % [ 1 6 ] ) . I I I - 2 - i i MEASUREMENT OF HYDRODYNAMIC EFFICIENCY The d e f i n i t i o n and t h e o r e t i c a l development of the exp r e s s i o n s (111-24, -25) f o r 77 make no re f e r e n c e to the r e a l i t i e s of f i n i t e t a r g e t s i z e or f i n i t e l a s e r pulse l e n g t h . According to the theory a measurements should y i e l d a s i n g l e value of 77 f o r a given set of l a s e r parameters ( I a , X^ ) and ta r g e t m a t e r i a l s . A proper measurement of rj r e q u i r e s a c l e a r d i s t i n c t i o n between the hydrodynamically d e p o s i t e d energy and the energies 7 7 d i s t r i b u t e d i n t o a l l other i n t e r a c t i o n s . That i s , one needs to measure only and t o t a l l y a l l of the energy d e p o s i t e d v i a the shock compression and a c c e l e r a t i o n p r o c e s s . U s u a l l y hydrodynamic e f f i c i e n c y i s obtained from ion c a l o r i m e t r y measurements of the rear s i d e t a r g e t blowoff while v a r y i n g the t a r g e t t h i c k n e s s . The r a t i o n a l e of a proper e f f i c i e n c y measurement i s o u t l i n e d as f o l l o w s . Based on the burnthrough model one can i d e n t i f y two regimes f o r which the hydrodynamic behaviour i s q u a l i t a t i v e l y d i f f e r e n t : (a) r B ( D ) < r m (b) r B ( D ) > r L H where T d ( D ) i s the burnthrough time as d e s c r i b e d by equation (111-10); TJ^ |_| i s a l a s e r time s c a l e a p p r o p r i a t e f o r e f f i c i e n c y measurements. For non-square p u l s e shapes (e.g. g a u s s i a n p u l s e s ) one must take to be the base l e n g t h of the temporal p u l s e shape (as opposed to the FWHM time or o t h e r s ) . T h i s i s the time i t takes to d e p o s i t n e a r l y a l l ( i . e . > 99% ) of the l a s e r energy i n t o the t a r g e t , and i s t h e r e f o r e the time s c a l e r e l e v a n t to the u l t i m a t e d i s t r i b u t i o n of l a s e r energy i n t o the v a r i o u s motions of t a r g e t m a t e r i a l s . In case (a) the l a s e r p e n e t r a t e s before the end of the p u l s e , and not a l l of the energy i s absorbed. The c o l l i s i o n of the r a r e f a c t i o n s at the i n s t a n t of burnthrough r e s u l t s i n two clumps of m a t e r i a l of equal momenta t r a v e l l i n g out of both s i d e s of the f o i l . R e c a l l i n g the burnthrough mechanism i t i s easy to see that the c o l l i s i o n of the r e t u r n i n g r a r e f a c t i o n 78 with the a b l a t i o n s u r f a c e separates c l e a n l y the forward going hydrodynamically induced motion from the a b l a t i v e ion f l u x d i r e c t e d back towards the l a s e r . However, the l a s e r remains on a f t e r burnthrough and c o n t i n u e s to heat the underdense plasma. T h e r e f o r e an unknown f r a c t i o n of l a s e r energy i s absorbed and 'explodes' the disassembling underdense plasma. D e t e c t o r s p l a c e d on the rear s i d e of the t a r g e t would not measure energy of hydrodynamic o r i g i n o n l y ; energy of l a s e r heated d e b r i s would be measured as w e l l . Hence a measurement of hydrodynamically d e p o s i t e d energy alone i s not p o s s i b l e i n t h i s case. For case (b) burnthrough does not occur d u r i n g the l a s e r p u l s e . However at the end of the l a s e r pule a r a r e f a c t i o n begins to propagate i n t o the compressed m a t e r i a l towards the rear of the t a r g e t . T h i s r e l e a s e s some of the hydrodynamic energy s t o r e d i n the compressed m a t e r i a l i n t o expansion towards the l a s e r s i d e (hydrodynamic shock a t t e n u a t i o n [ 3 4 ] ) . Consequently not a l l of the hydrodynamic energy w i l l be measured i n the rear s i d e p a r t i c l e e mission. T h e r e f o r e i t i s a l s o d i f f i c u l t to assess the t r u e f r a c t i o n of energy d e p o s i t e d i n t o hydrodynamic energy from measurements of ion e n e r g i e s . F i n a l l y one can c o n s i d e r the case T _ ( D ) 2 4 r . The burnthrough model as w e l l as the d i s c u s s i o n s above i n d i c a t e t h a t the t a r g e t t h i c k n e s s has to be matched to the l a s e r pulse wavelength and i n t e n s i t y so that the hydrodynamic burnthrough and a t t e n u a t i o n e f f e c t s have a minimal adverse e f f e c t on the t a r g e t dynamics. By matching the burnthrough time to the l a s e r 79 time T^ |_| one achieves the maximum t o t a l energy d e p o s i t i o n i n the t a r g e t , as w e l l as the maximum forward going energy of s o l e l y hydrodynamic o r i g i n s ( i . e . not l a s e r heated due to burnthrough). Hence a proper measurement of hydrodynamic e f f i c i e n c y can only be achieved by measuring the backward p a r t i c l e emissions f o r t a r g e t t h i c k n e s s e s such that T B ( D ) * TLH * The c h o i c e of such that > 99 % of the l a s e r energy i s dep o s i t e d before burnthrough i s a r b i t r a r y but reasonable. For example, i f burnthrough occurs l a t e i n the l a s e r p u l s e where only 5 % of the l a s e r energy remains t h i s remaining energy heats d i r e c t l y the ions on the t a r g e t rear s i d e . S ince the expected e f f i c i e n c y i s only around 5 %, such burnthrough h e a t i n g r e p r e s e n t s a l a r g e p e r t u r b a t i o n (=* 100 % of shock d e p o s i t e d energy) of the i n i t i a l hydrodynamic motions. Hence the e f f i c i e n c y can only be p r o p e r l y measured i f the percentage of the remaining l a s e r energy subsequent to burnthrough i s much l e s s than the 5 % hydrodynamic e f f i c i e n c y . 80 CHAPTER IV EXPERIMENTAL FACILITY AND DIAGNOSTICS T h i s chapter p r e s e n t s a d e s c r i p t i o n of the g e n e r a l l a s e r f a c i l i t y and the ion d i a g n o s t i c s used f o r stu d y i n g l a s e r - t a r g e t hydrodynamics. The p r i n c i p l e technique of o b t a i n i n g hydrodynamic i n f o r m a t i o n of the l a s e r - t a r g e t i n t e r a c t i o n was by remote d e t e c t i o n of the ion blowoff. Measurement of ion energy and v e l o c i t y p r o v i d e s i n f o r m a t i o n about the mass a b l a t i o n r a t e , the a b l a t i o n p r e s s u r e , as w e l l as the energy balance. These measurements were achieved with two d i a g n o s t i c d e v i c e s - the ion c a l o r i m e t e r and the Faraday cup. S e c t i o n IV-1 g i v e s a general overview of the l a s e r f a c i l i t y . Then s e c t i o n IV-2 d e s c r i b e s the Faraday cup ion c o l l e c t o r used f o r v e l o c i t y measurements. The ion c a l o r i m e t e r used f o r energy measurements i s d e s c r i b e d i n s e c t i o n IV-3. 81 IV-1 LASER FACILITY The l a s e r used i n these experiments was a Quantel neodymium-glass system, NG-34 (F i g u r e 10). I t i n c l u d e s a Nd-yag o s c i l l a t o r , a Nd-yag p r e a m p l i f i e r and two Nd-glass a m p l i f i e r s . The f i n a l rod a m p l i f i e r had an e x i t a p erture of 25 mm. The output beam can be operated at the 2nd, 3rd and 4th harmonics of the fundamental 1.06 Mm wavelength. The experiments i n t h i s work were a l l performed at the frequency t r i p l e d wavelength of .355 jum. The output l a s e r pulse i s n e a r l y gaussian with a FWHM width of about 2.0 ns (F i g u r e 10). Maximum pul s e energy at .355 Mm was approximately 1.4 J . The l a s e r was focussed onto the t a r g e t using f/10 o p t i c s . The t a r g e t normal was t i l t e d at 10 degrees with res p e c t to the a x i s of the focussed l a s e r beam to a v o i d s p e c u l a r r e f l e c t i o n back i n t o the l a s e r c h a i n . For a l l experiments the lens was p l a c e d i n the p o s i t i o n of best f o c u s . T h i s was obtained by m o n i t o r i n g B r i l l o u i n b a c k s c a t t e r [38] l e v e l s as a f u n c t i o n of lens p o s i t i o n . The lens was p l a c e d at the p o s i t i o n of maximum b a c k s c a t t e r energy. At focus 90 % of the l a s e r energy was c o n t a i n e d w i t h i n a f o c a l spot of 80 Mm diameter, and 60 % w i t h i n a 40 Mm diameter spot. T h i s gave an e f f e c t i v e spot diameter of around 80 Mm f o r the hydrodynamic p r o c e s s e s . The i n c i d e n t beam was monitored with a Gentec energy meter [46], M o n i t o r i n g of pu l s e shape and other temporal measurements (such as burnthrough) were made us i n g photodiodes [47]. Target i r r a d i a t i o n s were performed w i t h i n a vacuum chamber 82 F i g u r e 10 E x p e r i m e n t a l f a c i l i t y d e p i c t i n g l a s e r beam l i n e and t a r g e t shown?"' t e m P ° ^ l P ^ s e shape of the i n c i d e n t b e « i . 83 of approximately 2 f t i n s i d e diameter. The chamber was evacuated to about 10" 5 T o r r using a 6 i n c h d i f f u s i o n pump. Pressure was monitored with a vacuum i o n i z a t i o n gauge. Ta r g e t s were mounted near the centre of the vacuum chamber on a r o t a t a b l e d i s c d r i v e n by a stepping motor. Successive t a r g e t s c o u l d be p o s i t i o n e d at focus by e x t e r n a l o p e r a t i o n of the s t e p p i n g motor. T h i s allowed the p o s s i b i l i t y of 30 to 50 shots before exhausting the supply of t a r g e t s mounted on the d i s c . A l l t a r g e t s were A l f o i l s [39] of v a r i o u s t h i c k n e s s e s ranging from 6 Mm to 100 Mm. 84 IV-2 THE FARADAY CUP The Faraday cup ion c o l l e c t o r i s a widely used d i a g n o s t i c f o r l a s e r plasma experiments [8,21,27]. In general the Faraday cup i s used f o r i n f e r r i n g the v e l o c i t y d i s t r i b u t i o n f o r plasmas expanding i n t o vacuum, although t h i s r e q u i r e s c a r e f u l a n a l y s i s and i n t e r p r e t a t i o n of the data. The primary use i n t h i s experiment has been f o r measuring the v e l o c i t y of the near steady s t a t e a b l a t i v e flow observed i n our l o n g - p u l s e , s h o r t -wavelength experiment. I t was a l s o used to o b t a i n i n f o r m a t i o n on the ion expansion v e l o c i t i e s and v e l o c i t y d i s t r i b u t i o n s f o r the n o n - a b l a t i v e r e a r s i d e unloading, although q u a n t i t a t i v e r e s u l t s were not always p o s s i b l e i n the rear s i d e measurements. IV-2 - i FARADAY CUP DESIGN In i t s s i m p l e s t form the Faraday cup i s e s s e n t i a l l y a charge c o l l e c t o r - u s u a l l y a m e t a l l i c s u r f a c e which i s e l e c t r i c a l l y connected to a r e c o r d i n g instrument ( o s c i l l o s c o p e ) . A stream of charged p a r t i c l e s , i n t h i s case i o n s , s t r i k i n g t h i s s u r f a c e produces a c u r r e n t which i s p r o p o r t i o n a l to the c u r r e n t d e n s i t y of the ion stream. The magnitude of the d e t e c t e d ion c u r r e n t can be s t r o n g l y modified by the emission of secondary e l e c t r o n s from the c o l l e c t i n g s u r f a c e , as w e l l as by the plasma e l e c t r o n s . These introduce 85 a d d i t i o n a l components to the ion c u r r e n t s i g n a l produced at the c o l l e c t o r . To screen out plasma e l e c t r o n s and suppress the e f f e c t s of secondary e l e c t r o n emission, the Faraday cups i n t h i s experiment employ b i a s e d double g r i d and a 'deep cup' c o l l e c t o r geometry. A comparative study of v a r i o u s geometries has been made by Pearlman [40], A schematic of the d e t e c t o r used i s shown in F i g u r e 11. I t c o n s i s t s of a hollow brass c y l i n d e r with a c i r c u l a r a p e r t u r e at one end, which a c t s as the charge c o l l e c t o r . Aperture diameter of the c o l l e c t i n g cup was 7 mm. T h i s geometry serves to r e s t r i c t the escape of secondary e l e c t r o n s from the c o l l e c t o r to a small cone angle. A n e g a t i v e l y b i a s e d c o n t r o l g r i d i s p l a c e d i n f r o n t of the c o l l e c t i n g cup. The g r i d performs a dual f u n c t i o n : ( i ) e l e c t r o n s i n the plasma flow are r e j e c t e d from e n t e r i n g the cup; and, ( i i ) i t r e f l e c t s back i n t o the cup any secondary e l e c t r o n s that may escape from i t . The c o l l e c t i n g cup and c o n t r o l g r i d are mounted i n s i d e a grounded c a s i n g . Soldered to the entrance a p e r t u r e of the c a s i n g i s a second g r i d which d e f i n e s a ground plane f o r the r e j e c t i o n of plasma e l e c t r o n s . The spacing between the b i a s i n g g r i d and c o n t r o l g r i d was 1 mm. The c o n t r o l g r i d i s b i a s e d a t -215 V using dry c e l l b a t t e r i e s as a v o l t a g e source. A 10 nF ceramic c a p a c i t o r i s mounted i n s i d e the c a s i n g connected between the c o n t r o l g r i d and the c a s i n g . T h i s p r o v i d e s the c o n t r o l g r i d with a l a r g e charge c a p a c i t y which was found necessary to prevent o s c i l l a t i o n s of the g r i d v o l t a g e due to image c u r r e n t s c r e a t e d by the charge s e p a r a t i o n p r o c e s s . The use of b a t t e r i e s as the 8 6 6 R O U N D C A S I N G CONTROL G R I D COLLECTING CUP END P L A T E S I G N A L B I A S 0 1 2 c m i i 1 S C A L E F i g u r e 11 Faraday cup mechanical c o n t r u c t i o n . A l l components shown are brass (except the c a p a c i t o r and e l e c t r i c a l l e a d s ) . Nylon i n s u l a t o r s s e p a r a t i n g brass components are not shown. E l e c t r i c a l leads are connected through 50 ohm c o a x i a l c a b l e s at the end p l a t e . 87 v o l t a g e source e l i m i n a t e s ground loop problems. In a p r a c t i c a l setup more than one d e t e c t o r i s mounted i n s i d e the chamber. The b i a s l i n e of each d e t e c t o r i s connected to an RC d i s t r i b u t i o n network; t h i s allows the b i a s l i n e to occupy a s i n g l e vacuum feedthrough while e l e c t r i c a l l y i s o l a t i n g the d e t e c t o r s from one another. A schematic showing the e l e c t r i c a l c o n n e c t i o n s f o r the Faraday cups i s shown i n F i g u r e 12. The c o l l e c t i n g cup i s connected v i a a 50 ohm c o a x i a l l i n e to the 50 ohm input t e r m i n a l of an o s c i l l o s c o p e . The scope i s t r i g g e r e d e x t e r n a l l y by a s i g n a l from a photodiode monitoring the i n c i d e n t l a s e r p u l s e , thus p r o v i d i n g a zero time refere n c e f o r a n a l y s i n g the s i g n a l . I V - 2 - i i FARADAY CUP EXPERIMENTAL TECHNIQUE V e l o c i t y measurements using the Faraday cup are based on a t i m e - o f - a r r i v a l p r i n c i p l e . The d e t e c t o r i s placed at a known d i s t a n c e from the t a r g e t . The v e l o c i t y d i s t r i b u t i o n of the p a r t i c l e s s t r i k i n g the d e t e c t o r can be i n f e r r e d from the time h i s t o r y Of the c u r r e n t , with time zero d e f i n e d by the l a s e r p u l s e . Since the t r a n s i t time from t a r g e t t o d e t e c t o r (microseconds) i s much l a r g e r than the l a s e r p u l s e length (nanoseconds) i t i s reasonable to assume that the v e l o c i t y d i s t r i b u t i o n i s given by the a r r i v a l time. I t i s important to emphasize that the Faraday cup i s a charge c o l l e c t i n g d e v i c e and so cannot d i s c r i m i n a t e among 88 OSCILLOSCOPE SIGNAL 50 n BIAS I T" f PLASMA OTHER DETECTORS F i g u r e 1 2 Faraday c o a x i a l cup e l e c t r i c a l c o n n e c t i o n s . A l l leads use 50 ohm c a b l e , with the s h i e l d i n g connected to ^ 8 9 charge s t a t e s of v a r i o u s ion s p e c i e s . Since i t measures t o t a l charge d e n s i t y i t s u s e f u l n e s s f o r determining the v e l o c i t y d i s t r i b u t i o n of the ions i s compromised i f there i s a wide range of charge s t a t e s a r r i v i n g at the d e t e c t o r . T h i s may a f f e c t the p o s i t i o n of the v e l o c i t y peak, or i t s width. For the same reason i t i s not a r e l i a b l e d e v i c e f o r i n f e r r i n g energy d i s t r i b u t i o n s , or the t o t a l energy i n the ion stream, due to l a c k of knowledge of the charge s t a t e d i s t r i b u t i o n a r r i v i n g at the d e t e c t o r . For near steady s t a t e a b l a t i o n c o n d i t i o n s t h i s i s not a s e r i o u s shortcoming s i n c e the bulk of the plasma t r a v e l s at n e a r l y uniform v e l o c i t y and the s i g n a l appears as a narrow spike with a d e f i n i t e a r r i v a l time. I t should be noted a t t h i s p o i n t that a l l Faraday cup and s i m i l a r (charge s e n s i t i v e ) ion d i a g n o s t i c s r e q u i r e background p r e s s u r e s t o be approximately 10" 5 Torr or l e s s [41]. Above t h i s l e v e l the e f f e c t s of charge exchange processes can s e v e r e l y modify the charge s t a t e of the ion flow a r r i v i n g at the d e t e c t o r , and so may d i s t o r t the measured s i g n a l . A c c o r d i n g l y a l l of the Faraday cup measurements i n t h i s experiment were conducted at 10" 5 T o r r or l e s s . In our experiment the d e t e c t o r s are p l a c e d from 10 to 35 cm from the t a r g e t , depending on expected s i g n a l l e v e l s ( f u r t h e r from t a r g e t f o r high c u r r e n t d e n s i t y , c l o s e r f o r low c u r r e n t d e n s i t y ) . The a b l a t i o n peak can be viewed completely u s i n g sweep speeds ranging from 200 n s / d i v to 2 Ms/div depending on the i n c i d e n t l a s e r energy and the angle of the d e t e c t o r from t a r g e t normal. Peak c u r r e n t s range from 0 - 2 5 90 mA. F i g u r e 13 demonstrates a t y p i c a l Faraday cup s i g n a l obtained i n an a b l a t i o n measurement. The t r a c e d i s p l a y s a narrow c u r r e n t pulse which corresponds to the steady s t a t e a b l a t i o n v e l o c i t y of ion flow from the t a r g e t . 91 laaaaiii^iaaa aaaanaaaa :aaaaifaiaaan aiaarjaw^aa 28 SB 28 SS 238 ihii 38 T^siaaaaaa > |HHB F i g u r e 13 Sample Faraday cup t r a c e of i o n stream from s t e a d y s t a t e a b l a t i o n T h i s s i g n a l was measured a t t a r g e t normal on the i r o n t s i d e a t a d i s t a n c e of 34 cm from the t a r g e t s u r f a c e I n c i d e n t f l u x was 1.2 x 1 0 1 3 W/cm2. s u r t a c e . 92 IV-3 THE ION CALORIMETER The energy d e p o s i t e d i n l a s e r produced plasmas i s e v e n t u a l l y r e d i s t r i b u t e d i n t o the ion blowoff, X - r a d i a t i o n and s c a t t e r e d l a s e r l i g h t . A l l of these f l u x e s can be monitored with v a r i o u s types of d e t e c t o r s . C a l o r i m e t r i c methods f o r determining the absorbed energy are chosen f o r s e v e r a l reasons. They a r e : ( i ) i n s e n s i t i v e to the plasma charge s t a t e ; ( i i ) have l i n e a r response; ( i i i ) are capable of a b s o l u t e c a l i b r a t i o n ; and ( i v ) are adaptable to a wide v a r i e t y of c o n f i g u r a t i o n s . However these d e v i c e s are r e l a t i v e l y i n s e n s i t i v e and they have no time r e s o l u t i o n due to the slow thermal time c o n s t a n t s . As w e l l , t h e r e i s no way of s e p a r a t i n g the s i g n a l s due to the ions and X-rays from each other. The c o n f i g u r a t i o n used f o r t h i s experiment has been adapted from designs d e s c r i b e d p r e v i o u s l y by Gunn and Rupert [42] and by V i l l e n e u v e and Richardson [43], I t was chosen f o r i t s v e r s a t i l i t y i n o b t a i n i n g high angular r e s o l u t i o n of blowoff d i s t r i b u t i o n s . The i n d i v i d u a l d e t e c t o r s can be p o s i t i o n e d a r b i t r a r i l y and take up r e l a t i v e l y l i t t l e space i n the t a r g e t chamber. 93 IV-3-i ION CALORIMETER DESIGN F i g u r e 14 i s a schematic i l l u s t r a t i o n of the main components of the c a l o r i m e t e r used i n the experiment. The heat s e n s i t i v e element i s a P e l t i e r e f f e c t t h e r m o e l e c t r i c c o o l e r which was obtained commercially [44]. A temperature g r a d i e n t produced ac r o s s the de v i c e generates an EMF which can be a m p l i f i e d and recorded. The t h e r m o e l e c t r i c d e v i c e was atta c h e d on one s i d e to a massive aluminum block which serves as a heat s i n k , and on the other s i d e to a small aluminum p l a t e which serves as an absorber f o r i n c i d e n t plasma energy. The dimensions of the absorber p l a t e s were 0.75 x 0.75 x 0.083 inches. A s i l v e r - f i l l e d conducting epoxy [45] was used to bond the t h e r m o e l e c t r i c d e v i c e to both the heat s i n k and ab s o r b e r s . There are two such d e t e c t i n g elements i n each c a l o r i m e t e r ; both are connected t h e r m a l l y to the same heat s i n k , and connected e l e c t r i c a l l y i n a d i f f e r e n t i a l c o n f i g u r a t i o n . Thus the measured s i g n a l g i v e s the d i f f e r e n c e i n energie s s t r i k i n g the two s e n s i t i v e elements. The d i f f e r e n t i a l c o n f i g u r a t i o n allows one to d i s c r i m i n a t e a g a i n s t the s c a t t e r e d l i g h t s i g n a l (by p l a c i n g a g l a s s s l i d e over one d e t e c t o r ) . I t a l s o e l i m i n a t e s e f f e c t s of thermal f l u c t u a t i o n s w i t h i n the t a r g e t chamber. The absorber m a t e r i a l i s chosen to minimize the e f f e c t s of l o s s mechanisms ( s p u t t e r i n g , b a c k s c a t t e r i n g and secondary e l e c t r o n emission) t h a t reduce the energy d e p o s i t e d on the c a l o r i m e t e r . I t i s g e n e r a l l y accepted that A l i s a s u i t a b l e absorber m a t e r i a l [42], 94 HEAT SINK THERMOELECTRIC ABSORBER DEVICE CALIBRATION n RESISTORS ELECTRICAL CONNECTOR MOUNTING POST 0 1 2 c m i • « SCALE Ion c a l o r i m e t e r mechanical c o n s t r u c t i o n . For experimental measurements one of the absorbers i s covered with a f i l t e r (e.g. a g l a s s s l i d e ) . The s i g n a l i s p r o p o r t i o n a l to the d i f f e r e n c e i n energies s t r i k i n g the absorbers. 95 C a l i b r a t i o n of the c a l o r i m e t e r s i s performed i n s i t u by e l e c t r i c a l l y d e p o s i t i n g a known amount of energy i n t o a c a l i b r a t i o n r e s i s t o r mounted i n s i d e the absorber. The c a l i b r a t i o n r e s i s t o r was a 1 K 1/8 W r e s i s t o r i n s e r t e d i n t o a s m a l l hole d r i l l e d through the absorber p l a t e on one of i t s edges. The r e s i s t o r was i n s u l a t e d and f i x e d i n p o s i t i o n with s i l v e r epoxy. Energy was d e p o s i t e d by an RC d i s c h a r g e from a measured (104.2 yF) c a p a c i t o r . The e l e c t r i c a l s i g n a l produced by the c a l o r i m e t e r s are t y p i c a l l y i n the m i c r o v o l t range. A m p l i f i c a t i o n i s necessary i n order to r e c o r d the s i g n a l s . A c i r c u i t very s i m i l a r to that used i n r e f e r e n c e [43] was c o n s t r u c t e d . I t i s i l l u s t r a t e d s c h e m a t i c a l l y i n F i g u r e 15 ( a ) . The d i f f e r e n t i a l s i g n a l from the c a l o r i m e t e r enters a high g a i n , low noise h y b r i d a m p l i f i e r manufactured by Burr Brown (3626BP). The gain on t h i s a m p l i f i e r was set to 1000. F o l l o w i n g t h i s i s an a c t i v e low pass f i l t e r s e c t i o n with a c u t o f f of around 1 Hz. The f i l t e r was t h i r d order with a Butterworth c h a r a c t e r i s t i c . Another a m p l i f i e r f o l l o w i n g the f i l t e r had a gain of 100 and was used to s u b t r a c t DC o f f s e t s . T h i s was accomplished with a sample and h o l d c i r c u i t which sampled the DC l e v e l produced at the output of the a m p l i f i e r and f i l t e r c i r c u i t . The sampled s i g n a l can be switched under operator c o n t r o l i n t o the ( - ) input of the f i n a l a m p l i f i e r i n order t o zero the output s i g n a l before r e c o r d i n g on an output d e v i c e . T h i s part of the c i r c u i t was found e s s e n t i a l because l a r g e DC o f f s e t s were produced by thermal f l u c t u a t i o n s which were a m p l i f i e d by the high gain 96 c i r c u i t . The c a l o r i m e t e r s i g n a l can be modeled a c c u r a t e l y by the equation [42], g . - -k(T - T „ ) IV-1 where k i s the thermal time constant of the d e v i c e , T i s the temperature of the absorber, and T^ the temperature of the heat s i n k . For t h i s d e v i c e k i s l i m i t e d p r i m a r i l y by the c o n d u c t i v i t y of the t h e r m o e l e c t r i c sensor. The time constant i s of the order of 50 seconds. For experimental measurements the r i s e time i s l e s s than 1 second, while f o r c a l i b r a t i o n s the r i s e time was s e v e r a l seconds. The reason f o r t h i s i s that heat d e p o s i t i o n v i a the r e s i s t o r i s much l e s s uniform than i n the experimental s i t u a t i o n ; a l s o the RC time constant of the c a l i b r a t i o n process i s slower. For c a l i b r a t i o n s the e x p o n e n t i a l decay was e x t r a p o l a t e d back to zero to o b t a i n the s i g n a l magnitude (see F i g u r e 16). The minimum d e t e c t a b l e s i g n a l i s determined by the nois e l e v e l of the a m p l i f i c a t i o n system. T h i s was t y p i c a l l y 20mV at the output of the a m p l i f i e r . A t y p i c a l c a l i b r a t i o n was 2.6 mJ/V, so that the 20 mV noise l e v e l corresponds to about 50 nJ. For the 3.61 cm 2 d e t e c t o r area t h i s corresponds to a minimum d e t e c t a b l e s i g n a l of 15 nJ/cm2. Dynamic range i s l i m i t e d by the power supply l e v e l s on the a m p l i f i c a t i o n system. The power supply l e v e l s were + and - 15 V, a l l o w i n g a u s e f u l s i g n a l range of 5 - 10 V i n e i t h e r d i r e c t i o n . Compared with the 20 mV n o i s e • 1 5 V LF398H (a) -1 5 V 220 k XR084 DIFFERENTIAL CONNECTION (b) r n AMPLIFIER oOUT OTHER RESISTORS • • • ^ ^ ^ ^ ^ ^ ' * * ^ » ' • . CALIBRATION CIRCUIT F i g u r e 15 Ion c a l o r i m e t e r e l e c t r i c a l system. (a) C a l o r i m e t e r a m p l i f i e r c i r c u i t . (b) General e l e c t r i c a l connections f o r a r r a y s . x I c a l o r i m e t e r 9 8 l e v e l t h i s gives a u s e f u l range of about 2.5 decades. I V - 2 - i i ION CALORIMETER EXPERIMENTAL TECHNIQUE In the experiment the f o l l o w i n g procedure was used. The vacuum chamber was pumped to around 10" 5 T o r r . The a m p l i f i e r s were turned on s e v e r a l hours before the measurement to allow the c i r c u i t s to s t a b i l i z e t h e r m a l l y ; otherwise the DC l e v e l tended to d r i f t badly due to the h i g h gain of the c i r c u i t . Each c a l o r i m e t e r was c a l i b r a t e d j u s t before the measurement. T h i s was accomplished by d i s c h a r g i n g the c a l i b r a t i o n RC c i r c u i t at two d i f f e r e n t e n e r g i e s f o r each c a l o r i m e t e r . The average response ( i n mJ/V) was found f o r each d e v i c e . The c a l i b r a t i o n r o u t i n e was both necessary and advantageous s i n c e i t allowed a ' t o t a l ' c a l i b r a t i o n d i r e c t l y r e l a t i n g the energy d e t e c t e d to the o s c i l l o s c o p e beam d e f l e c t i o n . Hence e r r o r s i n the o s c i l l o s c o p e a m p l i f i e r gain are taken i n t o account and t h e r e f o r e do not a f f e c t the f i n a l r e s u l t . O s c i l l o s c o p e s with time bases around 5 s/ d i v were used f o r r e c o r d i n g the s i g n a l s . J u s t b e f o r e each shot the sample and hold c i r c u i - t was a c t i v a t e d to zero the DC l e v e l of a l l the c i r c u i t s . A f t e r each shot the c a l o r i m e t e r s were allowed to r e l a x t h e r m a l l y . F i g u r e 16 shows o s c i l l o g r a m s of c a l i b r a t i o n and experimental s i g n a l s obtained from two d i f f e r e n t c a l o r i m e t e r s . I t should be mentioned at t h i s p o i n t that a c a l o r i m e t e r was compared with a Gentec l a s e r energy meter i n a setup i n 9 9 F i g u r e 1 6 Sample t r a c e s of ion c a l o r i m e t e r s i g n a l s . (a) C a l i b r a t i o n t r a c e s . R i se times are longer than i n (b) because of non-uniform heat d e p o s i t i o n . S i g n a l amplitude was estimated by e x t r a p o l a t i n g the e x p o n e n t i a l back t o t = 0 . (b) Experimental t r a c e s . R i se time i s l e s s than 1 second and s i g n a l amplitude i s the peak h e i g h t . 100 which each was used to monitor a known amount of l a s e r energy. T h i s comparison r e v e a l e d a discrepancy of 20% between the c a l o r i m e t e r reading and the Gentec re a d i n g . The c a l i b r a t i o n of the Gentec energy meter was f u r t h e r compared to two other Gentec meters and was found to be i n good agreement with f a c t o r y c a l i b r a t i o n s . The observed d i s c r e p a n c y between the Gentec meter and the ion c a l o r i m e t e r was a t t r i b u t e d t o p o s s i b l e problems inherent i n the c a l o r i m e t e r c a l i b r a t i o n process; namely, energy d e p o s i t i o n d u r i n g c a l o r i m e t e r c a l i b r a t i o n s i s not uniform and the e x t r a p o l a t i o n technique may not be a c c u r a t e . Therefore the Gentec reading was accepted as the true measurement and the c a l o r i m e t e r readings were ad j u s t e d a c c o r d i n g l y to achieve s e l f - c o n s i s t e n c y . 101 CHAPTER V ABLATION SCALING LAWS The a b l a t i o n theory presented i n chapter II can be v e r i f i e d by d i r e c t measurement of the a b l a t i o n p r e s s u r e P a and the mass a b l a t i o n r a t e m. Both may be measured as a f u n c t i o n of l a s e r i n t e n s i t y to f o r comparison with the i n t e n s i t y s c a l i n g s given i n the s c a l i n g laws (11-39) and (11-40). Such measurements are important f o r two reasons. F i r s t l y , they v e r i f y the s c a l i n g laws d e r i v e d i n the t h e o r e t i c a l d e s c r i p t i o n ; secondly they y i e l d v a l u e s f o r the mass a b l a t i o n r a t e and a b l a t i o n pressure which are necessary f o r a n a l y s i s and i n t e r p r e t a t i o n of the s h o c k - r a r e f a c t i o n processes t a k i n g p l a c e i n the compressed t a r g e t r e g i o n . T h i s chapter presents the experimental measurement of m and P g. S e c t i o n V-1 d e t a i l s the measurement p r i n c i p l e and experimental setup. T h i s i s f o l l o w e d by a p r e s e n t a t i o n of the r e s u l t s and d i s c u s s i o n i n s e c t i o n V-2. 102 V-1 ABLATION MEASUREMENTS V-1 - i MEASUREMENT PRINCIPLE The s c a l i n g laws g i v i n g the mass a b l a t i o n rate and a b l a t i o n p r essure can be measured e x p e r i m e n t a l l y by remote d e t e c t i o n of the ion energy and v e l o c i t y d i s t r i b u t i o n as a f u n c t i o n of angle. The measurement p r i n c i p l e i n a p r a c t i c a l experiment can be summarised as f o l l o w s . Due to s p h e r i c a l expansion the ion f l u i d v e l o c i t y r a p i d l y approaches a constant value as i t leaves the v i c i n i t y of the t a r g e t . As w e l l , the temperature of the blowoff decreases r a p i d l y with d i s t a n c e so that a l l the energy i s c a r r i e d o f f i n k i n e t i c motion. By simultaneously measuring the energy and v e l o c i t y of the blowoff one can i n f e r the a b l a t e d mass M and momentum px c a r r i e d o f f p e r p e n d i c u l a r to the t a r g e t , 7T M 0 V-1 7T 0 V-2 103 where <V(0)> and <V 2(0)> are the mean v e l o c i t y and mean square v e l o c i t y r e s p e c t i v e l y of the ion stream a r r i v i n g at the angular p o s i t i o n 0. The values of m e x pt and P a 6xpt c a n t n e n D e i n f e r r e d assuming an e f f e c t i v e flow time Tp, and an e f f e c t i v e spot diameter D s p o t , A 4M C X P t "CD V " 3 v spot' F P ^ a. _ a.expt 777 T? v (D ^) T V-4 spot' F Both D Sp Q^ and tp are d i f f i c u l t to estimate because the l a s e r spot i s not of uniform i n t e n s i t y up to the edge, and the l a s e r p u l s e shape i s not r e c t a n g u l a r . Choosing Tp and D Sp 0f i s t h e r e f o r e somewhat a r b i t r a r y . We choose D Sp 0t t o be the diameter w i t h i n which 90 % energy d e p o s i t i o n takes p l a c e (80 Mm). Tp i s a time of the order of the FWHM of the l a s e r p u l s e , but s l i g h t l y l o n g e r . T h i s i s because rh and P a s c a l e with i n t e n s i t y at a r a t e l e s s s t r o n g l y than u n i t y . As an a r b i t r a r y estimate we l e t Tp be the f u l l width at 1/e amplitude of the gaussian pulse shape f u n c t i o n taken to the 7/9 power (pressure s c a l i n g ) . The f u l l width 1/e time of a 2 ns FWHM gaussian pulse i s 2.4 ns, so that the 1/e time of the same p u l s e exponentiated by the 7/9 s c a l i n g i s 2.7 ns (2.4 /7/9 ). We have used Tp = 2.7 ns i n equations (V-3) and (V-4). The l a s e r i n t e n s i t y i s given by, 104 4E T l a s e r „ T T _ w C D s p o t ) T L where E | a s e r i s the i n c i d e n t energy, i s the e f f e c t i v e pulse l e n g t h and na^s i s the a b s o r p t i o n f r a c t i o n . In t h i s experiment we used = 2.0 ns (the FWHM of the l a s e r pulse) and ^abs = 0*^6 (measured independently [ 4 9 ] ) . It should a l s o be noted that the these experimental measurement takes place i n the l a b o r a t o r y r e f e r e n c e frame while the s c a l i n g laws (11-39) and (11-40) are determined w i t h i n an i n e r t i a l frame moving with r e s p e c t to the l a b o r a t o r y frame ( c f . F i g u r e s 2 and 3). T h i s d i f f e r e n c e w i l l be taken i n t o account i n the d i s c u s s i o n of the r e s u l t s i n s e c t i o n V-2. V - 1 - i i EXPERIMENTAL SETUP AND DATA REDUCTION The experimental setup used i n the a b l a t i o n measurements i s shown i n F i g u r e 17. I t c o n s i s t e d of an a r r a y of ion c a l o r i m e t e r s and Faraday cups s i t u a t e d at d i f f e r e n t angles i n f r o n t of the t a r g e t . Where used, the Faraday cups were mounted d i r e c t l y above the c a l o r i m e t e r s and t h e r e f o r e at the same angular p o s i t i o n . Two separate runs with d i f f e r e n t angular p o s i t i o n s were made. One run had d e t e c t o r s (Faraday c u p / c a l o r i m e t e r p a i r s ) s i t u a t e d at 0, 15, 30, 45 and 60 degrees with r e s p e c t to t a r g e t normal; the second run used seven c a l o r i m e t e r s a t 9.5, 20, 30, 42.5, 52, 63 and 77 degrees, with Faraday cups s i t u a t e d a t the 9 .5, 20, 42.5 and 63 degree 1 05 gure 1 7 Detector a r r a y f o r ion a b l a t i o n measurement. Ion c a l o r i m e t e r / F a r a d a y cup combinations are mounted at v a r i o u s angles around the t a r g e t . 106 p o s i t i o n s . The measurement i n v o l v e d t a k i n g approximately 70 shots v a r y i n g the i n c i d e n t energy from about 0.2 J to 1.2 J . Target f o i l t h i c k n e s s was 50 Mm which exceeds the burnthrough t h i c k n e s s at a l l e n e r g i e s . The raw data g i v e s the angular d i s t r i b u t i o n of ion v e l o c i t y and energy d e n s i t y at d i s c r e t e p o i n t s . For each shot a cub i c s p l i n e was f i t through the v e l o c i t y and energy data as a f u n c t i o n of angle, thereby o b t a i n i n g continuous f u n c t i o n s r e p r e s e n t i n g E(0) and V ( 0 ) . The s p l i n e f i t t i n g was performed with the a i d of a computer g r a p h i c s t e r m i n a l so that each shot c o u l d be in s p e c t e d i n d i v i d u a l l y , and the f i t a d j u s t e d to be reasonably smooth. F i g u r e 18 demonstrates the d i s c r e t e data and f i t t e d curves f o r a t y p i c a l shot. The f u n c t i o n s <V(6)> and <V 2(0)> r e q u i r e d by equations (V-1) and (V-2) were approximated by u s i n g the s p l i n e f i t t e d curves V(6) and V 2 ( 0 ) r e s p e c t i v e l y ; these f u n c t i o n s were s u b s t i t u t e d i n t o equations (V-1) and (V-2) to o b t a i n values f o r M and p A . Equations (V-3) and (V-4) were then used to o b t a i n the v a l u e of mexp^ and P a e X p f f ° r the shot. The experimental s c a l i n g laws were c o n s t r u c t e d by a n a l y s i n g each shot i n t h i s manner and p l o t t i n g mexp^ and p a expt a s a f u n c t i o n of absorbed l a s e r f l u x . 107 in ANGLE (degrees) F i g u r e 18 Experimental E(0) and V(0) and c o r r e s p o n d i n g s p l i n e f i t s f o r a t y p i c a l shot. 108 V-2 RESULTS AND COMPARISON WITH THEORY V- 2 - i EXPERIMENTAL SCALINGS The measured i n t e n s i t y s c a l i n g s m e x p t and P a e x p t are presented i n F i g u r e s 19 and 20. They are d e s c r i b e d q u i t e w e l l by the e m p i r i c a l ( b e s t - f i t , s t r a i g h t l i n e ) r e l a t i o n s , *expt " C 7 ± . S x 1 0 5 ) i ; 5 5 V " 6 P a , e x p t " C 5 5 ± . 1 5 ) i ; 8 4 V-7 where [ P a e x p t ] " Mbar , [ A e x p t 1 = g c n r 2 s _ 1 and [ l a ] = TW/cm2. In comparison one may e x t r a c t the i n t e n s i t y dependence from the s c a l i n g laws (11-39) and (11-40). We use the parameters ZlnA = 100, A/Z = 2, Xj_ = .355 ym and L x = 80 urn, a p p r o p r i a t e to the experiment. T h i s y i e l d s , A - (.51 x l O 5 ) ! ' 5 6 V " 8 a 109 F i g u r e 19 I n t e n s i t y dependence of mass a b l a t i o n r a t e , experimental and t h e o r e t i c a l . T r i a n g u l a r data p o i n t s represent experimental mass a b l a t i o n r a t e p l o t t e d a g a i n s t absorbed l a s e r i n t e n s i t y . In comparison the t h e o r e t i c a l s c a l i n g of equation (11-39) i s shown. C o r r e c t i o n s due to s p e r i c a l divergence or r e f e r e n c e frame t r a n s f o r m a t i o n s do not modify the o r i g i n a l mass a b l a t i o n s c a l i n g (11-39). 110 F i g u r e 20 I n t e n s i t y dependence of a b l a t i o n p r e s s u r e , experimental and t h e o r e t i c a l . T r i a n g u l a r data p o i n t s represent measured a b l a t i o n p r essure p l o t t e d a g a i n s t absorbed l a s e r i n t e n s i t y . T h e o r e t i c a l s c a l i n g ( l i n e ) and c o r r e c t e d t h e o r e t i c a l p r e d i c t i o n s ( •, • ) are i n d i c a t e d . C o r r e c t i o n s are d i s c u s s e d i n V - 2 - i i , V - 2 - i i i and V- 2 - i v . .111 P - c . 7 2 ) i ; 7 8 V-9 a a with the same u n i t s as i n (V-6) and (V-7). E v i d e n t l y the measured i n t e n s i t y dependences are i n reasonable agreement with the t h e o r e t i c a l s c a l i n g s . However the magnitude of the pre s s u r e measurement i s s l i g h t l y lower than the t h e o r e t i c a l p r e d i c t i o n and the i n t e n s i t y dependence i s s l i g h t l y stonger (0.84 versus 0.78). Two e f f e c t s can account f o r t h i s d i s c r e p a n c y . The f i r s t one i s the f a c t that the t h e o r e t i c a l s c a l i n g laws (11-39) and (11-40) were obtained i n a re f e r e n c e frame f i x e d to the moving a b l a t i o n and a b s o r p t i o n s u r f a c e s , and not the l a b o r a t o r y r e f e r e n c e frame i n which the measurement was made. Secondly, the s p h e r i c a l divergence of the flow converts some of the i n t e r n a l energy of the expanding corona i n t o a l a t e r a l l y d i r e c t e d momentum f l u x which does not c o n t r i b u t e to the a x i a l momentum balance that c r e a t e s the a b l a t i o n p r e s s u r e . The m o d i f i c a t i o n s to the t h e o r e t i c a l d e s c r i p t i o n to account f o r these two e f f e c t s are now d i s c u s s e d . 1 12 V - 2 - i i CORRECTION FOR REFERENCE FRAME TRANSFORMATION A G a l i l e a n t r a n s f o r m a t i o n connects the a b l a t i o n r e f e r e n c e frame of F i g u r e 3 with the l a b frame. Transformations of t h i s type were used in the d i c u s s i o n of hydrodynamic e f f i c i e n c y i n s e c t i o n I I I - 3 - i , and some of the r e s u l t s there may be used to c o r r e c t the t h e o r e t i c a l s c a l i n g laws f o r the l a b frame measurement. Equations (111-18) and (111-19) g i v e the t o t a l momentum and mass f l u x e s r e s p e c t i v e l y , as they would be measured i n the l a b o r a t o r y r e f e r e n c e frame, 2p n< (1 o o O m P c  wo o 111 - 1 8 111 - 1 9 These may be compared with (11-22) and (11-23) which are the r e s p e c t i v e e x p r e s s i o n s i n the a b l a t i o n r e f e r e n c e frame of f i g u r e 3, P v = p c o o o o 11-22 1 13 P = p v 2 • p c 2 = 2p c 2 I : " 2 3 a Mo o *o o p o o I t i s evident that the mass f l u x i s i d e n t i c a l i n the two frames, but the momentum f l u x (pressure) measured i n the l a b frame i s reduced by the f a c t o r (1 - |q/2c 0|) from i t s magnitude seen in the a b l a t i o n frame. R e c a l l that |q| i s the v e l o c i t y of the a b l a t i o n and a b s o r p t i o n s u r f a c e s measured i n the l a b o r a t o r y frame, and c 0 i s the i o n - a c o u s t i c sound speed i n the corona. From equations (111 — 13) and (111-25) i t i s easy to show that jq/2co| - 2o + 0(o2). T h e r e f o r e one may approximate the c o r r e c t i o n f a c t o r as (1 - |q/2c 0|) (1 - rj) where a, rj << 1. A more general e x p r e s s i o n i s o b t a i n e d by s u b s t i t u t i n g the s c a l i n g law (111-25) i n t o t h i s . T h i s c o r r e c t e d pressure does not g i v e an e x p l i c i t s c a l i n g with i n t e n s i t y , P i v « I ? / 9 a,lab a: 1 - p » ( I * ' X L ' L x ' ••• }1 1 / 2 » * L » * * * * ^ 1 H o V i a a ' i . » ~ v » ' i i V 10 I t i s d i f f i c u l t to evaluate t h i s e x p r e s s i o n s i n c e i t depends on P which i n turn depends upon d e t a i l e d knowledge of the shock compression hydrodynamics and the e q u a t i o n - o f - s t a t e . However a d i r e c t measurement of 77 i s a means of determining the r e q u i r e d c o r r e c t i o n i n a manner t h a t i s s e l f - c o n s i s t e n t with the t h e o r e t i c a l d e s c r i p t i o n . 1 1 4 V - 2 - i i i CORRECTION DUE TO SPHERICAL DIVERGENCE The c o r o n a l expansion i s not planar over s c a l e lengths l a r g e r than the spot diameter. We c o n s i d e r how t h i s f a c t m o d i f i e s the t h e o r e t i c a l r e s u l t by examining f i r s t the process of f l u i d momentum t r a n s f e r i n s t r i c t l y p l a n a r geometry. In s e c t i o n III-3 i t was e s t a b l i s h e d that the flow i s reasonably planar a t l e a s t out to the a b l a t i o n s u r f a c e . T h e r e f o r e at t h i s s u r f a c e the p l a n a r model i s v a l i d . As i s e v i d e n t from equation (11-23) the a b l a t i o n pressure P& i s equal to sum of the k i n e t i c , pv 2, and thermal, p c 2 , c o n t r i b u t i o n s . At the a b s o r p t i o n s u r f a c e , x = x 0 , these two are equal because of the s o n i c matching c o n d i t i o n v 0 = c 0 (equation 11-28). Hence in the p l a n a r model the t o t a l momentum f l u x i s p a r t i t i o n e d e q u a l l y among the k i n e t i c and thermal components at t h i s s u r f a c e . Beyond the a b s o r p t i o n s u r f a c e the momentum balance i s determined by the manner i n which the thermal component of the p r e s s u r e i s e v e n t u a l l y converted to d i r e c t e d motion. At best (planar geometry) the remaining thermal pressure i s converted i n t o a x i a l l y d i r e c t e d motion and the t o t a l p e r p e n d i c u l a r momentum measured by remote d e t e c t i o n i s e x a c t l y the p l a n a r r e s u l t (modified by the G a l l i l e a n r e f e r e n c e frame c o r r e c t i o n ) . In the worst case ( s p h e r i c a l or d i v e r g i n g flow) almost a l l of the thermal pressure c o n t a i n e d i n the plasma at the a b s o r p t i o n s u r f a c e i s r e l e a s e d i n t o near i s o t r o p i c s p h e r i c a l expansion (superimposed on the e x i s t i n g a x i a l l y d i r e c t e d motion). T h i s 1 1 5 expansion c o n t r i b u t e s l i t t l e to the a x i a l (or p e r p e n d i c u l a r ) component which balances the a b l a t i o n p r e s s u r e d r i v i n g the shock. In t h i s worst case the achieved a b l a t i o n pressure i s h a l f of the planar r e s u l t due to l o s s of the thermal c o n t r i b u t i o n . V-2-iv COMPARISON INCLUDING CORRECTIONS Since the mass a b l a t i o n r a t e i s m o d i f i e d by n e i t h e r of the c o n s i d e r a t i o n s d i s c u s s e d i n V I - 2 - i i or V I - 2 - i i i , no c o r r e c t i o n s are needed i n . F i g u r e - 1 9 to f i t the d a t a . The unmodified s c a l i n g law (11-39) f i t s the measurement q u i t e w e l l . The c o r r e c t i o n s to the t h e o r e t i c a l p r e d i c t i o n s f o r the a b l a t i o n p r e s s u r e s are i l l u s t r a t e d along with the data. At I a = 1.2 x 1 0 1 3 W/cm2 (see chapter VII) our measured value f o r the e f f i c i e n c y i s p - 0.05±0.01. Hence a c o r r e c t i o n f a c t o r of (1 - .05) = 0.95 must be a p p l i e d to the t h e o r e t i c a l a b l a t i o n p r e s s u r e at t h i s i n t e n s i t y to estimate the r e s u l t that would be observed in the l a b o r a t o r y r e f e r e n c e frame. This c o r r e c t i o n i s q u i t e small i n comparison to other u n c e r t a i n t i e s such as i n Tp and Dspot * I n F ^ 9 u r e 2 ^ t w o ' c o r r e c t e d ' t h e o r e t i c a l p o i n t s are shown. The upper p o i n t ( • ) accounts only f o r the r e f e r e n c e frame c o r r e c t i o n . The lower poin t ( • ) i s p l a c e d at h a l f the magnitude of the upper one and i s the l i m i t i n g case of strong s p h e r i c a l divergence of the flow beyond the a b s o r p t i o n . I t i s obvious that the magnitude of the measured a b l a t i o n pressure i s 1 16 i n agreement with the general range of p r e s s u r e s expected by these simple arguments. The b e s t - f i t s t r a i g h t - l i n e s c a l i n g of the a b l a t i o n p r e s s u r e (equation V-7) i n d i c a t e s an i n t e n s i t y s c a l i n g of Paexpt" ^l^^compared with a t h e o r e t i c a l dependence of p a « T?J^> A d i r e c t reason f o r t h i s may be systematic v a r i a t i o n s of the l a s e r pulse and the i n t e n s i t y d i s t r i b u t i o n at the t a r g e t (spot diameter) as the i n t e n s i t y range i s scanned. However the d i s c r e p a n c y might a l s o be accounted f o r i n equation (V-10). Note from (111-27) t h a t the magnitude of the c o r r e c t i o n ( i . e . of 77) i s s e n s i t i v e to the r e l a t i v e s i z e s of p and p 0 . Both p and p 0 i n c r e a s e with i n c r e a s i n g I a , so that changes i n 77 depend on the r e l a t i v e r a t e at which each s c a l e s with I a . Hence the c o r r e c t i o n f a c t o r (1 - 77) c o u l d s c a l e p o s i t i v e l y with I g i n our measuring range i f P i n c r e a s e s more r a p i d l y than p 0 (causing 77 to decrease and (1 - TJ) to i n c r e a s e with I g ) . T h i s would e f f e c t i v e l y i n c r e a s e the measured i n t e n s i t y dependence from the 7/9 dependence. T h i s p o s s i b i l i t y cannot be t e s t e d without d i r e c t measurement of 77 at a number of i n t e n s i t i e s . 1 17 CHAPTER VI MEASUREMENT OF BURNTHROUGH TIME The purpose of t h i s measurement i s to provide evidence of the phenomenon of t a r g e t r a r e f a c t i o n and subsequent l a s e r t r a n s m i s s i o n through d e n s i t i e s l e s s than c r i t i c a l . Furthermore, an a c c u r a t e d e t e r m i n a t i o n of the burnthrough time r D as a f u n c t i o n of t a r g e t t h i c k n e s s D p r o v i d e s q u a n t i t a t i v e i n f o r m a t i o n about the s h o c k - r a r e f a c t i o n process from which one might i n f e r the shock v e l o c i t y and p r e s s u r e . S e c t i o n VI-1 p r e s e n t s the experimental method used f o r measuring Tg(D). T h i s i s followed by a q u a n t i t a t i v e a n a l y s i s of the r e s u l t s i n s e c t i o n VI-2. 118 VI-1 EXPERIMENT The measurement of was accomplished by monitoring the t r a n s m i t t e d l a s e r l i g h t as a f u n c t i o n of time. Laser energy was maintained constant while the t a r g e t f o i l t h i c k n e s s was v a r i e d . F i g u r e VI-1 shows the experimental setup. The t r a n s m i t t e d f l u x was monitored from the t a r g e t r e a r s i d e through f/10 o p t i c s (same f/number as the i n c i d e n t beam) by a photodiode as w e l l as an energy meter. Two narrow band i n t e r f e r e n c e f i l t e r s c e n t r e d at 0.35 ym were p l a c e d i n f r o n t of the photodiode t o ensure that only t r a n s m i t t e d l a s e r l i g h t was being monitored. A Tek t r o n i x 7104 o s c i l l o s c o p e was used to d i s p l a y the photodiode s i g n a l . The scope was t r i g g e r e d e x t e r n a l l y by a s i g n a l from another photodiode m o n i t o r i n g the i n c i d e n t beam. The t r i g g e r timing was t h e r e f o r e p r e s e t ( w i t h i n 100 ps) such that the t r a n s m i t t e d l a s e r pulse (no t a r g e t ) was d i s p l a y e d with the peak at approximately 2 ns from the beginning of the t r a c e . Sample o s c i l l o g r a m s of the t i m i n g measurement are shown i n F i g u r e 21. 119 PHOTODIODE F i g u r e 21 Experimental c o n f i g u r a t i o n f o r burnthrough measurements. F i g u r e 22 Sample photodiode s i g n a l s obtained d u r i n g burnthrough measurement. (a) Transmitted l a s e r p u l s e with no t a r g e t . (b) Transmitted l a s e r p u l s e through a 12.5 m^ t a r g e t at 6 x 1 0 1 2 W/cm2 i n c i d e n t i n t e n s i t y . 121 VI-2 RESULTS AND DISCUSSION The burnthrough time measurements are presented i n F i g u r e 23 f o r two d i f f e r e n t l a s e r f l u x e s . E v i d e n t l y these v e r i f y the p r e d i c t e d l i n e a r dependence of burnthrough time on t a r g e t t h i c k n e s s . The i n t e r c e p t s on the v e r t i c a l a x i s are n e g l i g i b l e which i n d i c a t e s that the c h o i c e of time zero f o r the shock ge n e r a t i o n i s reasonable and that the expansion time can indeed be ignored. F i g u r e 24 shows the f r a c t i o n a l t r a n s m i t t e d amplitude at peak t r a n s m i s s i o n as a f u n c t i o n of f o i l t h i c k n e s s . The l a s e r t r a n s m i s s i o n i s s t i l l very low even when burnthrough occurs e a r l y i n the l a s e r p u l s e . T h i s i m p l i e s t h a t the underdense plasma i s s t r o n g l y absorbing and can d i s s i p a t e much l a s e r energy in the processes of i o n i s a t i o n and expansion. T h i s f a c t i s important f o r e x p l a i n i n g the r e l a t i v e l y high a b s o r p t i o n observed (chapter VII) i n t h i n t a r g e t s which burn through e a r l y i n the l a s e r p u l s e . Thus burnthrough does not s e v e r e l y a f f e c t the d e p o s i t i o n of l a s e r energy i n t o the t a r g e t , u n l e s s the i n i t i a l t a r g e t t h i c k n e s s i s so s m a l l t h a t T d ( D ) << r. . (Refer D LH to chapter I I I , s e c t i o n I I I - 2 - i i f o r the d e f i n i t i o n of T. .) L n A q u a n t i t a t i v e e v a l u a t i o n of the model can be made at the lower i r r a d i a n c e used i n the measurement. At t h i s i r r a d i a n c e , 6x 1 0 1 2 W/cm2, the measured a b l a t i o n p r e s s u r e , from F i g u r e 20, i s P a e xp{ = 2.0±0.5 Mbar. The shock speed i n aluminum generated at t h i s p r e s s u r e has been measured independently by T r a i n o r e t . a l . [18], t o g i v e U = 1.2 x 10 s cm/s. A c c o r d i n g to a 1 22 A l T a r g e t T h i c k n e s s ( / i m ) F i g u r e 23 Burnthrough time r R versus t a r g e t t h i c k n e s s D. 123 Z I o CO I 2 4 6 10 20 40 60 100 ALUMINUM TARGET THICKNESS (/iM) F i g u r e 24 F r a c t i o n a l t r a n s m i t t e d amplitude versus t a r g e t t h i c k n e s s . 124 c a l c u l a t i o n i n T r a i n o r and Lee [34]. The sound speed i n the shock compressed region i s a = 0.82U at t h i s p r e s s u r e . Next, the compression r a t i o 0 = pc/% m a ^ ^ e e v a ^ u a t e d using the e x p r e s s i o n , 2 p s P - P I T ( 1 - -£) VI-1 s P c which may be d e r i v e d from the shock r e l a t i o n s ( 1 1 1 - 2 ) and ( I I I -3 ) . Using these v a l u e s of P, U and p = 2.7 g/cm 3, t h i s y i e l d s (S = 2. 1 ±0.4. T h i s value i s c o n s i s t e n t with v a l u e s observed i n e x p l o s i v e d r i v e n shocks i n A l [33], F i n a l l y the burnthrough depth can be i n f e r r e d from the mass a b l a t i o n measurement. For s u f f i c i e n t l y t h i c k f o i l s the measured mass a b l a t i o n r a t e of 1.6 x 10 5 g/cm 2-s y i e l d s a depth of d ^ = 1.6 urn. The t h i c k n e s s D 0 f o r which burnthrough time approximately equals the l a s e r p u l s e time T. i s about 30 um at t h i s i r r a d i a n c e , so LH the r a t i o d ^ j /D 0 =0.05 which i s indeed much l e s s than u n i t y . For t a r g e t t h i c k n e s s e s D < D 0 one expects d^/D i n equation (III—10) to remain approximately i n the same p r o p o r t i o n . The burnthrough time c a l c u l a t e d from the model (equation 1 1 1 — 10) us i n g t h i s data i s shown by the s o l i d l i n e i n F i g u r e 23, which shows e x c e l l e n t agreement with data. For the measurement at the l a r g e r i r r a d i a n c e one may use the burnthrough model to i n f e r the shock speed and pressure u s i n g the measured data. The slope of the b e s t - f i t s t r a i g h t l i n e (dotted l i n e i n F i g u r e 23) i s 7.7 x 10" 7 s/cm. For the p r e s s u r e range between 1 and 10 Mbar the sound speed t o shock 125 speed r a t i o a/U ranges from 0.84 to 0.70 [34] (where U i s measured i n the l a b frame). The compression r a t i o j3 may be l i m i t e d to a range of about 2 - 4 ( f o r a monatomic i d e a l gas the maximum compression i s l i m i t e d to 4). Without s p e c i f y i n g these parameters any f u r t h e r i t i s p o s s i b l e u s i n g equations (III—10) and (VI-1) to e x t r a c t the shock speed and pressure (Figure 25). Exact values of a and 0 are not necessary to provide a reasonable estimate of U and P. From t h i s measurement one can i n f e r U = 1.95±0.27 x 10 s cm/s and P = 6.2±0.5 Mbar. The measured a b l a t i o n p r essure at the higher i n t e n s i t y of 1.2 x 1 0 1 3 W/cm2 i s approximately 4.4±0.5 Mbar which c o n t r a s t s with the value i n f e r r e d above of 6.2±0.5 Mbar. T h i s i s c u r i o u s because of the reasonable agreement between the burnthrough model and a b l a t i o n measurement at the lower i r r a d i a n c e . The burnthrough measurements t h e r e f o r e imply a d i f f e r e n t i n t e n s i t y s c a l i n g of the pressure than the a b l a t i o n measurements. The reasons f o r the d i s c r e p a n c y are not yet understood c o n c l u s i v e l y . However, the f o l l o w i n g p o i n t s should be noted. The burnthrough and a b l a t i o n measurements d i f f e r s h a r p l y i n th a t the former i s h i g h l y s e n s i t i v e to the peak p r e s s u r e ( s p a t i a l and temporal peak) while the l a t t e r g i v e s a g l o b a l average of the p r e s s u r e . T h i s i s because the burnthrough measurement monitors the onset of l a s e r t r a n s m i s s i o n . T h i s w i l l always occur at the p o i n t where the shock propagation and subsequent r a r e f a c t i o n are f a s t e s t , hence at the p o i n t of peak p r e s s u r e . Two dimensional e f f e c t s (e.g. i n c i d e n t beam s e l f -f o c u s s i n g [48]) c o u l d e a s i l y produce l o c a l i s e d p e r t u r b a t i o n s 126 F i g u r e 25 r » ? ? « , fPeed and pressure as a f u n c t i o n of compression W/cm2 measured burnthrough data at 1.2 x 1 0 1 3 127 where the peak pressure i s s i g n i f i c a n t l y higher than the g l o b a l average. Such e f f e c t s may produce l o c a l v a r i a t i o n s i n the pressure t h a t s c a l e with i n c i d e n t i n t e n s i t y i n a d i f f e r e n t manner than the g l o b a l l y averaged p r e s s u r e . The burnthrough measurement, being h i g h l y s e n s i t i v e t o l o c a l p e r t u r b a t i o n s , w i l l tend to r e v e a l these as anomolous r e s u l t s . On the other hand, the a b l a t i o n measurement i s i n h e r e n t l y i n s e n s i t i v e to such two-dimensional p e r t u r b a t i o n s and t h e r e f o r e shows reasonable agreement with s c a l i n g s p r e d i c t e d i n the simple one-dimensional model. 128 CHAPTER VII DYNAMICS OF ACCELERATED THIN TARGETS The importance of f l u i d r a r e f a c t i o n s i n the dynamics of the t a r g e t has been emphasised in chapter I I I . Evidence f o r a x i a l r a r e f a c t i o n has a l s o been v e r i f i e d c o n c l u s i v e l y by the burnthrough measurements presented i n chapter VI. In t h i s chapter f u r t h e r experimental evidence i s presented with i n t e r p r e t a t i o n which demonstrates the f l u i d nature of the shock compressed t a r g e t . Target disassembly was s t u d i e d s y s t e m a t i c a l l y u sing the ion d i a g n o s t i c s . In these s t u d i e s the t a r g e t f o i l t h i c k n e s s as w e l l as l a s e r i n t e n s i t y were v a r i e d to demonstrate the g e n e r a l model. Experimental d e t a i l s and r e s u l t s are presented i n S e c t i o n VII-1. The r e s u l t s are d i s c u s s e d i n S e c t i o n VII-2. 129 V I 1 - 1 EXPERIMENT A hig h r e s o l u t i o n angular measurement of ion e n e r g i e s was made of both the f r o n t and rear s i d e s of the t a r g e t . The f r o n t s i d e and rear s i d e measurements are complementary and both are necessary t o determine the o v e r a l l energy balance and c o u p l i n g f o r f i n i t e t h i c k n e s s t a r g e t s . Since the number of d e t e c t o r s was l i m i t e d the f r o n t and rear s i d e measurements were performed s e p a r a t e l y with the same set of instruments. 111- 1 - i REAR SIDE SETUP The energy was monitored with an a r r a y of seven ion c a l o r i m e t e r s p o s i t i o n e d on the rear s i d e at angles ranging from n e a r l y t a r g e t normal to around 70 degrees. A l l d e t e c t o r s were p l a c e d i n a plane p a r a l l e l to the chamber f l o o r except one which was p l a c e d at an angle of 30 degrees below t h i s plane. Faraday cup s i g n a l s were monitored at three angles c e n t r e d on the c o n i c a l unloading peak at 30 degrees. Two parameters were v a r i e d to observe the q u a l i t a t i v e unloading behaviour. For one run the l a s e r energy was h e l d constant while t a r g e t t h i c k n e s s was v a r i e d . On the other the energy was v a r i e d f o r f i x e d t a r g e t t h i c k n e s s e s of 12.5 nm and 25 /urn. 130 III — 1 — i i FRONT SIDE SETUP A c a l o r i m e t e r a r r a y of s i m i l a r angular r e s o l u t i o n was p l a c e d on the f r o n t s i d e . The a b l a t e d ion energy was monitored at constant l a s e r i n t e n s i t y f o r f o i l t h i c k n e s s e s from 6 /xm and 50 /um. Faraday cup t r a c e s were a l s o taken f o r comparison with the rear s i d e measurements. The l a s e r i n t e n s i t y was s i m i l a r t o t h a t used f o r the rear s i d e constant i n t e n s i t y measurments. 1 1 1 - 1 - i i i DATA ANALYSIS AND RESULTS In p r i n c i p l e the mass d i s t r i b u t i o n s of the ions d e t e c t e d on both s i d e s of the t a r g e t can be obtained using s i m i l a r techniques as f o r the a b l a t i o n measurement. However the Faraday cup t r a c e s r e v e a l e d a l a r g e range of v e l o c i t i e s and s i g n a l amplitudes f o r the ions d e t e c t e d on the rear s i d e of the t a r g e t . Consequently one may expect that the charge s t a t e of the ions a r r i v i n g at the rear d e t e c t o r s to be non-uniform as w e l l . T h i s makes an a c c u r a t e assessment of the ion v e l o c i t y d i s t r i b u t i o n , and t h e r e f o r e <V 2(0)>, impossible to achieve with the Faraday cup data a l o n e . On the other hand, the Faraday cup t r a c e s f o r a given shot d i s p l a y e d s i m i l a r f e a t u r e s at a l l angles independent of amplitude ( F i g u r e 30). T h i s i n d i c a t e s that <V 2(0)> i s independent of ang l e . Therefore the ion energy d i s t r i b u t i o n i s an a c c u r a t e r e f l e c t i o n of the mass d i s t r i b u t i o n 131 as w e l l . The data d i s p l a y e d i n F i g u r e s 26, 27, 28, 29 and 30 show energy d i s t r i b u t i o n s and Faraday cup t r a c e s c o l l e c t e d at the constant i n t e n s i t y 1.2 x 1 0 1 3 W/cm2. The data gi v e s a comprehensive survey of t a r g e t behaviour as a f u n c t i o n of t h i c k n e s s . F i g u r e s 31 and 32 show the ion energy d i s t r i b u t i o n s measured on the rear s i d e f o r v a r i o u s l a s e r i n t e n s i t i e s , f o r the constant f o i l t h i c k n e s s e s of 12.5 Mm and 25 Mm r e s p e c t i v e l y . To o b t a i n the energy balance the ion energy d i s t r i b u t i o n s f o r both f r o n t and r e a r s i d e s were i n t e g r a t e d using c u b i c s p l i n e f i t s . An accounting of the ion energy balance i s summarised i n Fig u r e 33. 132 F i g u r e 26 Front s i d e ion energy angular d i s t r i b u t i o n s f o r v a r i o u s f o i l t h i c k n e s s e s at 1.2 x 1 0 1 3 W/cm2. The data f o r each t h i c k n e s s i s from a s i n g l e r e p r e s e n t a t i v e shot. E r r o r s were t y p i c a l l y l e s s than 0.03 J / s r . E CZ U J U J 133 F i g u r e 27 ANGLE (degrees) f«?i J 6 n e r g y a n 9 u l a r d i s t r i b u t i o n s f o r v a r i o u s f o i l t h i c k n e s s e s at 1.2 x 1 0 1 3 W/cm2. The data f o r each t h i c k n e s s was c o l l e c t e d d u r i n g a s i n g l e shot T v o i c a l e r r o r bars are i n d i c a t e d at the peak. T y p i c a l 134 T I M E (fts) ure 28 Front s i d e Faraday cup t r a c e s f o r 6 um and 50 um t a r g e t s at 1.2 x 1 0 1 3 W/cm2. The d e t e c t o r was p o s i t i o n e d at 8 degrees to t a r g e t normal, 22 cm from the t a r g e t s u r f a c e 135 T I M E (/is) F i g u r e 29 Rear s i d e Faraday cup t r a c e s f o r 6 Mm and 50 Mm t a r g e t s at 1.2 x 1 0 1 3 W/cm2. The d e t e c t o r was p o s i t i o n e d at 16 degrees to t a r g e t normal, 18 cm from the t a r g e t s u r f a c e . 1 36 T I M E ifis) F i g u r e 30 Rear s i d e Faraday cup t r a c e s angular p o s i t i o n s at 1.2 x recorded simultaneous f o r the normalized to an e f f e c t i v e t a r g e t s u r f a c e . f o r a 25 nm t a r g e t at three 1 0 1 3 W/cm2. A l l t r a c e s were same shot. The s i g n a l s were d i s t a n c e of 10 cm from the 137 0 30 60 90 ANGLE (degrees) F i g u r e 31 Rear s i d e ion energy angular d i s t r i b u t i o n s f o r v a r i o u s l a s e r i n t e n s i t i e s on 12.5 jum t a r g e t s . Data f o r each curve was c o l l e c t e d d u r i n g a s i n g l e shot. E r r o r bars are i n d i c a t e d f o r the peak. 138 1.3 « 1 0 W / c m 2 fc UJ UJ o - f 80x10 55*10 F i g u r e 32 ANGLE (degrees) Rear s i d e ion energy angular d i s t r i b u t i o n s f o r v a r i o u s l a s e r i n t e n s i t i e s on 25 nm t a r g e t s . Data f o r each curve was c o l l e c t e d d u r i n g a s i n g l e shot. E r r o r bars are i n d i c a t e d f o r the peak. >-e> QC U J u < DC 1.0 0.81 Q6 Q4 0.2 0.1 0.08 0.06 0.04 0.02 . i I r - 3 - 5 - x \ \ \ \ \ \ • FRONT SIDE IONS o REAR SIDE IONS J L_l I \ J i__L 4 6 8 10 20 40 60 80 100 ALUMINUM TARGET THICKNESS ifim) F i g u r e 33 Front s i d e and rear s i d e i n t e g r a t e d ion energy as t r a c t i o n of i n c i d e n t l a s e r energy at 1.2 x 1 0 1 3 W/cm2. 1 40 VII-2 DISCUSSION OF RESULTS V I I - 2 - i ANGULAR PROFILES OF ION ENERGY The p r o f i l e s of the ion energy measured on both s i d e s of the t a r g e t r e v e a l a number of f e a t u r e s of f i n i t e t h i c k n e s s t a r g e t behaviour d i s c u s s e d i n S e c t i o n 111-2. The r e s u l t s d i s p l a y e d i n F i g u r e s 26 to 32 can be most e a s i l y understood i n terms of the r e l a t i o n s h i p between the burnthrough depth T d ( D ) and the l a s e r pulse time We c o n s i d e r the two l i m i t s > ty_ and Tg < i j ^ a n d c a t e g o r i s e experimental r e s u l t s a c c o r d i n g to these c r i t e r i a . r B > r L H As d i s c u s s e d i n s e c t i o n I I I - 4 the burnthrough phenomenon does not occur before the end of the l a s e r pulse when T B > TLH * I n fchis regime n e a r l y a l l of the l a s e r energy i s absorbed i n the t a r g e t and reappears i n the a b l a t e d ion stream and the rear s i d e i s e n t r o p i c expansion. The c r i t e r i o n T D > T. ., D L n i s s a t i s f i e d f o r t h i c k t a r g e t s or d e c r e a s i n g l a s e r energy. The data d i s p l a y e d i n F i g u r e s 26 to 30 was ob t a i n e d at a f i x e d l a s e r i n t e n s i t y of 1.2 x 1 0 1 3 W/cm2. Accord i n g t o the 141 burnthrough data of F i g u r e 23 the burnthrough depth i s approximately 50 Mm at t h i s i n t e n s i t y . For t a r g e t s of equal or gre a t e r t h i c k n e s s than t h i s (50 Mm, 100 Mm) the rear s i d e expansion assumes the c o n i c a l d i s t r i b u t i o n ( F i g u r e 27) of the r a d i a l r a r e f a c t i o n d e s c r i b e d i n 111-2. The peak angle i n the 50 Mm case i s around 25 degrees thus demonstrating reasonable agreement with the estimated cone angle of around 30 degrees. The 100 Mm t a r g e t s unloaded at a cone angle of approximately 10 degrees. For these t h i c k f o i l s shock a t t e n u a t i o n from the f r o n t s i d e , as w e l l as two-dimensional e f f e c t s (non-planar shock at rear s i d e ) c o u l d account f o r the narrowing of the cone an g l e . The i s e n t r o p i c expansion was recorded by Faraday cup measurements ( F i g u r e 29, 50 Mm t a r g e t ) at t h i s i n t e n s i t y and these r e v e a l e d a mean v e l o c i t y of 5 x 10 s cm/s f o r the 50 Mm t a r g e t . T h i s v e l o c i t y i s s e v e r a l times the sound speed i n the shock compressed m a t e r i a l and i s comparable to the i n f e r r e d (from burnthrough) shock v e l o c i t y U of 2 x 10 6 cm/s. The c r i t e r i o n > i s a l s o s a t i s f i e d f o r t h i n n e r t a r g e t s at lower i n t e n s i t y . The same q u a l i t a t i v e behavior ( c o n i c a l d i s t r i b u t i o n i n the rear s i d e unloading) i s e x h i b i t e d f o r the 12.5 Mm and 25 Mm t a r g e t s at low i n t e n s i t i e s ( F i g u r e s 31 and 32). 142 r B < r L H The measurements conforming to t h i s c r i t e r i o n are those fo r t a r g e t s of 25 jim or l e s s at the 1.2 x 1 0 1 3 W/cm2 i n t e n s i t y ( F i g u r e s 26 to 30), and a l s o the higher i n t e n s i t y shots taken on the 12.5 urn and 25 urn t a r g e t s ( F i g u r e s 31 and 32) i n these examples the rear s i d e energy d i s t r i b u t i o n s s t i l l d i s p l a y a l o c a l peak i n the v i c i n i t y of the 30 degree c o n i c a l unloading, but t h i s i s superimposed on a smooth background d i s t r i b u t i o n s i m i l a r to the plume l i k e shape found on the f r o n t s i d e expansions. As f o i l t h i c k n e s s decreased (or r a t h e r T d / r. ,_, decreased) the c o n t r a s t r a t i o between the c o n i c a l unloading peak and the i s o t r o p i c expansion decreases. Furthermore, the rear s i d e ion Faraday cup t r a c e s r e v e a l e d a f a s t ion expansion at v e l o c i t i e s a few times 10 7 cm/s (F i g u r e s 29, 30) which i s comparable to the f r o n t s i d e a b l a t i o n v e l o c i t i e s . In a d d i t i o n a slow component i n the rear s i d e expansion i s evident f o r t h i c k e r t a r g e t s ( F i g u r e 30) with v e l o c i t i e s comparable to the i s e n t r o p i c unloading v a l u e s found f o r t h i c k ( r D > r. ._. ) D L n t a r g e t s . The amplitude of the f a s t peak i s much l a r g e r than the slow i s e n t r o p i c expansions i n d i c a t i n g that the f a s t ions may be i o n i s e d t o charge s t a t e s much higher than the slower and c o o l e r i s e n t r o p i c a l l y unloaded m a t e r i a l s . F i n a l l y , at the i n t e n s i t y 1.2 x 1 0 1 3 W/cm3 the f r o n t s i d e energy d i s t r i b u t i o n s (Figure 26) showed a c o n s i s t e n t d i m i n i s h i n g of ion energy with d e c r e a s i n g f o i l t h i c k n e s s , which i n d i c a t e s a reduced a b s o r p t i o n of t o t a l energy. The Faraday cup t r a c e s of the f r o n t s i d e 143 a b l a t i o n ( F i g u r e 28) i n d i c a t e a reduced amplitude f o r the 6 Mm t a r g e t t h i c k n e s s c o n s i s t e n t with the energy measurements. The average v e l o c i t y of a b l a t e d ions i s a l s o s l i g h t l y slower f o r the 6 Mm t a r g e t compared to the 50 Mm t a r g e t . These o b s e r v a t i o n s of the behaviour of t h i n t a r g e t s under long p u l s e i r r a d i a t i o n p o i n t to an obvious e x p l a n a t i o n i f the e f f e c t s of l a s e r burnthrough are c o n s i d e r e d . When burnthrough occurs before the end of the l a s e r p u l s e the underdense plasma i s d i r e c t l y heated by the l a s e r . A p l a u s i b l e d e s c r i p t i o n based on the e f f e c t s of t h i s d i r e c t h e a t i n g can account f o r the experimental o b s e r v a t i o n s l i s t e d above. A f t e r burnthrough occurs the l a s e r p e n e trates the underdense t a r g e t completely and i o n i s e s i t to charge s t a t e s comparable to that i n the f r o n t s i d e corona. The high degree of i o n i s a t i o n as w e l l as the d i r e c t d e p o s i t i o n of l a s e r energy i n t o the expanding plasma d i s r u p t s the slow i s e n t r o p i c expansion of the rear s i d e shock unloading process and causes a much f a s t e r and more s p a t i a l l y i s o t r o p i c i s o t h e r m a l expansion of the remaining plasma. The r a p i d h e a t i n g and expansion of the underdense t a r g e t q u i c k l y removes the absorbing plasma so that e v e n t u a l l y the l a s e r energy i s t r a n s m i t t e d with reduced a b s o r p t i o n . 144 V I I - 2 - i i ENERGY BALANCE The o v e r a l l energy balance f o r the ion measurements has been summarised i n F i g u r e 33. I t r e v e a l s the q u a l i t a t i v e l y d i f f e r e n t behaviour f o r the two regimes r < T B " LH and TB > TLH * T ^ e r e s u ^ - t s demonstrate c o n v i n c i n g l y the e f f e c t s of f i n i t e t a r g e t t h i c k n e s s on o v e r a l l l a s e r - t a r g e t energy c o u p l i n g . For t h i n t a r g e t s ( tq < ) r a t i o E J o n / E ( a s e r f o r the rear s i d e remained l e v e l at around 0.11. The d i r e c t l a s e r h e a t i n g of the underdense plasma c o n t r i b u t e s an unknown amount to the measured rear s i d e energy so that t h i s cannot be c o n s i d e r e d a measure of the hydrodynamic e f f i c i e n c y . For the t h i c k t a r g e t s ( > ) of 50 Mm and 100 Mm the decreased rear s i d e energy can be a t t r i b u t e d to the e f f e c t s of shock a t t e n u a t i o n (unloading from the f r o n t s i d e of the shock compressed t a r g e t ) so that not a l l of the energy due to hydrodynamic compression and a c c e l e r a t i o n i s measured on the rear s i d e . On the other hand the f r o n t s i d e ion energy i s observed to decrease with t h i c k n e s s f o r f o i l s l e s s 50 Mm. T h i s may be a t t r i b u t e d to the e a r l y d i s r u p t i o n of the steady s t a t e a b l a t i o n because burnthrough occured d u r i n g the l a s e r p u l s e . With t a r g e t t h i c k n e s s e s 50 Mm or g r e a t e r the f r o n s i d e ion energy remains n e a r l y c o n s t a n t . V I I - 2 - i i i EVALUATION OF HYDRODYNAMIC EFFICIENCY 145 The c o u p l i n g of l a s e r energy i n t o the hydrodynamic motions i s most f a v o u r a b l e only at one t a r g e t t h i c k n e s s f o r a given i n t e n s i t y - that t h i c k n e s s f o r which the burnthrough time Tg approximately equals the l a s e r time . The burnthrough measurements i n chapter VI i n d i c a t e that at the i n t e n s i t y 1.2 x 1 0 1 3 W/cm2 (used i n t h i s measurement) the burnthrough t h i c k n e s s i s approximately 50 ym. The measured r a t i o E j o n / E | a g e f . i s 0.05±0.01 at t h i s t h i c k n e s s . T h e r e f o r e we take t h i s to be the measured hydrodynamic e f f i c i e n c y , r\ f o r t h i s l a s e r i n t e n s i t y , wavelength, and t a r g e t m a t e r i a l (Aluminum). In comparison the p r e d i c t e d e f f i c i e n c y from (111-28) i s 0.054±0.004 using a value of 0 = 2.0±0.3 and the usual values f o r the l a s e r parameters. The agreement i s q u i t e good. 1 46 CHAPTER VIII CONCLUSIONS AND FINAL DISCUSSIONS A d e t a i l e d study of hydrodynamic processes of l a s e r - s o l i d t a r g e t i n t e r a c t i o n s i n planar geometry has been completed. The main r e s u l t s of the i n v e s t i g a t i o n are summarised i n S e c t i o n V I I I — 1 . P o i n t s that are of p a r t i c u l a r importance to c u r r e n t r e s e a r c h i n planar l a s e r - t a r g e t i n t e r a c t i o n experiments are emphasised. S e c t i o n VIII-2 then presents a f i n a l d i s c u s s i o n of the consequences of the r e s u l t s obtained i n t h i s i n v e s t i g a t i o n . T h i s d i s c u s s i o n examines how these r e s u l t s r e l a t e t o , and modify, c u r r e n t understanding of plan a r geometry hydrodynamic s t u d i e s of l a s e r - a c c e l e r a t e d s o l i d t a r g e t s . 1 47 VI11-1 CONCLUSIONS • The measurement of l a s e r d r i v e n a b l a t i o n has demonstrated that the a b l a t i o n process at .355 Mm wavelength and 1 0 1 3 W/cm2 i n t e n s i t y i s dominated by s t r o n g i n v e r s e Bremmstrahlung a b s o r p t i o n . T h i s has been v e r i f i e d by comparison of experimental i n t e n s i t y s c a l i n g s of the mass a b l a t i o n r a t e and a b l a t i o n pressure w i t h a simple theory i n c o r p o r a t i n g p l a n a r steady s t a t e flow with f i x e d a b s o r p t i o n s c a l e l e n g t h . • The f l u i d nature of the shock compressed t a r g e t has been demonstrated. S h o c k - r a r e f a c t i o n sequences dominate t a r g e t behaviour and cause t a r g e t disassembly on time s c a l e s of the order of the shock t r a n s i t time a c r o s s the t a r g e t . T h i s induces l a s e r t r a n s m i s s i o n (burnthrough) and reduces c o u p l i n g of t o t a l l a s e r energy i n t o the t a r g e t . The measurements i n c l u d e d o b s e r v a t i o n s of l a s e r t r a n s m i s s i o n through r a r e f i e d t a r g e t s (burnthrough) and a s y s t e m a t i c study of energy balance and angular d i s t r i b u t i o n s of ion blowoff on both s i d e s of the t a r g e t . Simple one- and two-dimensional models were proposed to account f o r the experimental o b s e r v a t i o n s . The burnthrough model p r o v i d e s a new way of determining shock v e l o c i t y and p r e s s u r e once the burnthrough time as a f u n c t i o n of t a r g e t t h i c k n e s s i s known. • A c a l c u l a t i o n of hydrodynamic e f f i c i e n c y based on the s i n g l e shock mechanism f o r t a r g e t a c c e l e r a t i o n and compression has been d e s c r i b e d . The model makes a q u a n t i t a t i v e p r e d i c t i o n 148 of the hydrodynamic e f f i c i e n c y which can be expressed e x p l i c i t l y as a f u n c t i o n of the l a s e r i n t e n s i t y , wavelength and t a r g e t m a t e r i a l ( e q u a t i o n - o f - s t a t e ) . Measurement of the e f f i c i e n c y at a s i n g l e i n t e n s i t y f o r the .355 Mm wavelength has shown good agreement with t h i s c a l c u l a t i o n . VI11-2 FINAL DISCUSSIONS V I I I - 2 - i CONTRIBUTIONS TO CURRENT DATA BASE The a b l a t i o n process i s fundamental to the a b l a t i v e a c c e l e r a t i o n scheme. I t i s t h e r e f o r e of great importance to measure the mass a b l a t i o n r a t e and a b l a t i o n p r e s s u r e to v e r i f y t h e o r e t i c a l s c a l i n g laws. The measurement presented i n t h i s i n v e s t i g a t i o n p r o v i d e s d e f i n i t e v e r i f i c a t i o n of the steady s t a t e , p l a n a r a b l a t i o n model o p e r a t i n g i n the long pulse strong Inverse Bremmstrahlung a b s o r p t i o n regime. T h i s s c a l i n g was p r e v i o u s l y p r e d i c t e d by Mora [11]. V I I I - 2 - i i MODIFICATION OF CURRENT UNDERSTANDING The r e s u l t s presented in t h i s i n v e s t i g a t i o n d i f f e r from the c o n v e n t i o n a l understanding of the nature of a b l a t i v e a c c e l e r a t i o n of t h i n planar t a r g e t s , namely, the rocket model which assumes r i g i d body dynamics. I t has a l r e a d y been i n d i c a t e d i n chapter III that the rocket model i n t e r p r e t a t i o n i s an i n c o r r e c t d e s c r i p t i o n of the t a r g e t a c c e l e r a t i o n p r o c e s s . T h i s r e s u l t s from simple arguments using known p r o p e r t i e s of s o l i d s compressed by s i n g l e shocks of Megabar p r e s s u r e s . I t has a l s o been demonstrated c o n c l u s i v e l y i n our experiment t h a t the t a r g e t i s not governed by r i g i d body dynamics. Furthermore, based on rocket model i n t e r p r e t a t i o n s p l a n a r t a r g e t a c c e l e r a t i o n experiments [16] have claimed that hydrodynamic e f f i c i e n c i e s as high as 20 % may be achieved. T h i s can be compared with our measurements of E j Q n / E | a s e r - .11 on the rear s i d e of very t h i n t a r g e t s . Such high values of E j o n / E | a s e r a r i s e due to burnthrough and d i r e c t l a s e r h e a t i n g of the underdense plasma. In c o n t r a s t our experiment i n d i c a t e s a hydrodynamic e f f i c i e n c y of only 5 %, i n agreement with our hydrodynamic c a l c u l a t i o n . The t h e o r e t i c a l e f f i c i e n c y c a l c u l a t i o n shows that improvements c o u l d be achieved with i n c r e a s e d l a s e r i n t e n s i t y and decreased wavelength. For s t r o n g Inverse Bremmstrahlung a b s o r p t i o n of f i x e d s c a l e l e n g t h the e f f i c i e n c y s c a l e s as ( I I I -28), 150 1.1/2 1/6 -1/3 However the s c a l i n g s of r\ with e i t h e r i n t e n s i t y or wavelength are r a t h e r weak. Coronal c o u p l i n g probems ( u n d e s i r a b l e non-l i n e a r plasma i n s t a b i l i t i e s ) l i m i t the i n t e n s i t y to l e s s than 1 0 1 5 W/cm2; the wavelength i s l i m i t e d by present technology to be g r e a t e r than 0.250 nm. A c o n s i d e r a t i o n that has been examined t h e o r e t i c a l l y [1,2], but not yet e x p e r i m e n t a l l y , i s the p o s s i b i l i t y of improving 77 by m u l t i p l e shock propagation. It should be a r e l a t i v e l y simple e x e r c i s e to extend the e f f i c i e n c y c a l c u l a t i o n t o i n c l u d e such cases. 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OSU-81-0034 1 55 APPENDIX A DETAILED DERIVATION FOR EQUATIONS (111-17) TO (111-22) A-1 EVALUATION OF (111-17), (111- 18) AND (111- 19) The s e l f - s i m i l a r r a r e f a c t i o n extends from the a b s o r p t i o n s u r f a c e out to +°°. The s o l u t i o n to the v e l o c i t y and d e n s i t y p r o f i l e s i n the wave i s given by (II-8) and (11-9), V - C o v ^ II-8 o which i s the s o l u t i o n w i t h i n a r e f e r e n c e frame (x' - a x i s ) that i s f i x e d to the moving f l u i d at the a b s o r p t i o n s u r f a c e , ( c f . s e c t i o n I I - 2 - i ) . The t r a n s f o r m a t i o n from the x' frame to the l a b frame i s a G a l i l e a n t r a n s f o r m a t i o n s i m i l a r to equation (11-2) given by, . ax _ It " I T + u r 1 56 A-1 where X i s the space c o o r d i n a t e of the l a b frame, V = dX/dt i s the f l u i d v e l o c i t y measured i n the l a b frame and w i s the r e l a t i v e v e l o c i t y of the two r e f e r e n c e frames. The f l u i d frame moves at sonic speed c 0 with respect to the a b s o r p t i o n s u r f a c e ( s o n i c matching c o n d i t i o n (11-28) ), while the a b s o r p t i o n s u r f a c e moves at v e l o c i t y q ( d e f i n e d i n I I I - 2 - i ) i n the l a b frame. Hence 1, A-2 S u b s t i t u t i n g (A-1) and (A-2) i n t o (II-8) and (II-9) g i v e s the flow p r o f i l e s f o r V and p i n the l a b o r a t o r y r e f e r e n c e frame, | R e c a l l that q has a negative value i n the l a b frame so that 1 57 V(X\ - C * \ A-3 The i n t e g r a t i o n s of (111- 17) to ( 1 1 1 - 2 0 ) are s p e c i f i e d to cover the extent of the r a r e f a c t i o n wave, which i s from the wave head X = qt to X = o c . A - 1 - i EVALUATION OF (111- 17) E, - i [l faw\xi * | fw} \ ax 00 Define a s u b s t i t u t i o n v a r i a b l e , = \ + ^ . ) c e t ^ = ax hence, I n t e g r a t i o n by p a r t s and s i m p l i f i c a t i o n g i v e s , A - 1 - i i EVALUATION OF (111-18) 00 1* Again, s u b s t i t u t e the v a r i a b l e y, ? r = fc^ « p t » + i > ^ 159 and o b t a i n , A - 1 - i i i EVALUATION OF ( i l l - 1 9 ) Oo Hence, i t l and f i n a l l y , 111 - 1 8 160 w r s | 0 4 c . 1 1 1 - 1 9 A-2 EVALUATION OF (111-20), (111-21) AND (111-22) . The shock compressed re g i o n i s bounded by the a b l a t i o n f r o n t and by the shock f r o n t . Within t h i s r e g i o n the flow v a r i a b l e s p, V and e are a l l c o n s t a n t s . The a b l a t i o n f r o n t i s l o c a t e d at X =. qt and the shock f r o n t at X = Ut. A- 2 - i EVALUATION OF (111-20) The t o t a l energy a c q u i r e d per u n i t mass of the shock compressed m a t e r i a l i s PAV where Av i s the change i n s p e c i f i c volume [32]. The energy f l u x e n t e r i n g the shock compressed reg i o n i s , R e c a l l p = P c/p hence, E0 - ?(.p-OtvU) 161 Using q = U(1 .+ o) (111 -1 3) , and the shock r e l a t i U = 110/(0 - 1) ( I I I - 2 ) , III-20 A - 2 - i i EVALUATION OF (111-21) Ut Using the r e l a t i o n u = U(0 - 1)0 ( I I I - 2 ) f * e « (*?)(£? - < M 1 62 then using the shock r e l a t i o n P = p^ Uu ( I I I - 3 ) , 1 1 1 - 2 1 A - 2 - i i i EVALUTION OF ( 1 1 1 - 2 2 ) The mass a b l a t e d from the shock compressed region i s r e l a t e d to the v e l o c i t i e s of the moving compressed m a t e r i a l and the a b l a t i o n f r o n t , u and q r e s p e c t i v e l y , ccfc 1 1 1 - 2 2 

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