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Quantitative schlieren densitometer Humphries, Christopher A. M. 1976

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A QUANT ITAT IVE SCHL IEREN DENSITOMETER < by CHRISTOPHER A . M . HUMPHRIES B . A . , O x f o r d U n i v e r s i t y , 1 9 7 3 THES I S SUBMITTED IN PART IAL FULF I LLMENT THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SC IENCE i n t h e D e p a r t m e n t o f PHYS ICS We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e , r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA September , 1976 @ Christopher A. M. Humphries In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r ee t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r ag ree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y pu rpo se s may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f The U n i v e r s i t y o f B r i t i s h Co l umb i a 20 75 Wesbrook P l a c e Vancouver, Canada V6T 1W5 ABSTRACT A s c h l i e r e n d e n s i t o m e t e r was c o n s t r u c t e d i n w h i c h a new t e c h n i q u e o f beam d e f 1 e c t i o n . m e a s u r e m e n t was e m p l o y e d . I n c o n v e n t i o n a l s y s t e m s , t h e s c h l i e r e n d e f l e c t i o n o f a p r o b i n g l i g h t beam i s m e a s u r e d b y a k n i f e e d g e t e c h n i q u e . I n t h e e x p e r i m e n t d e s c r i b e d h e r e , t h e k n i f e e d g e was r e p l a c e d by a n e u t r a l d e n s i t y w e d g e . T h i s i n n o v a t i o n b r o u g h t s e v e r a l a d v a n -t a g e s , t h e m a i n o n e b e i n g t h e e l i m i n a t i o n o f p r o b l e m s d u e t o d i f f r a c t i o n . T h e beam i n t e n s i t y , a t t e n u a t e d by t h e w e d g e , was m o n i t o r e d b y a p h o t o m u 1 t i p 1 i e r . T h e o u t p u t v o l t a g e o f t h e p h o t o m u l t i p l i e r was t i m e i n t e g r a t e d t o g i v e a l i n e a r a n a l o g u e s i g n a l f o r t h e d e n s i t y p r o f i l e o f a m o v i n g o b j e c t o r m e d i u m . T h e d e v i c e was a p p l i e d t o t h e m e a s u r e m e n t o f t h e d e n s i t y c h a n g e a c r o s s t h e s h o c k f r o n t o f a M a c h 9 p l a n e s h o c k w a v e i n a r g o n . T h e r e s u l t s w e r e i n s a t i s f a c t o r y a g r e e m e n t w i t h v a l u e s c a l c u l a t e d f r o m s h o c k w a v e t h e o r y . TABLE OF CONTENTS Page ABSTRACT '. • i i LIST OF TABLES . . . • v LIST OF ILLUSTRATIONS. . ' vi ACKNOWLEDGMENTS ix Chapter 1 INTRODUCTION . 1 1.1 General Introduction 1 1.2 Conventional Schl ieren Systems . . . . . . . 2 1.3 The Wedge Technique 4 2 OPTICS OF THE DENSITOMETER " . . . 8. 2.1 The Schl ieren Ef fect 8 2.2 E f fect of the Wedge 10 2.3 The Ca l ib ra t ion System . 16 3 THE APPARATUS : 2 0 3.1 Construction of the Apparatus 20 3.2 Use of the Apparatus . 2 8 i i i Chapter Page 4 PROPERTIES OF THE SHOCK WAVES' 32 4.1 Shock Wave Plasma Parameters . . . . . . . 32 4.2 The Refract ive Index of a Plasma 37 4.3 Numerical Ca lcu lat ions 43 5 EXPERIMENT AND RESULTS 47 5.1 Ve loc i ty Measurements. . . . 47 5.2 Shock Wave Parameters. . . 51 5.3 Density Measurements at Pi = 16 torr . . . 51 5.4 Qua l i t a t i ve Invest igat ion at Pi = 2 torr 54 6 CONCLUSIONS. . . 58 6.1 General Conclusions. 58 6 . 2 Future Work 59 BIBLIOGRAPHICAL REFERENCES 61 APPENDICES A Reference'1: Boye Ahlborn and Christopher A.M. Humphries, "A Quantitat ive Schl ieren Densitometer Employing a Neutral Wedqe," Rev. S c i . Instr . , Vo l . 47, No. 5 (1976) . . . . 63 B The Neutral Density Wedge . . . . • 68 C Optics of the Ca l ib ra t ion System 71 D The Shock Tube Apparatus 75 E Appl icat ion of the R.C'.A. 931A Photomult ip l ier 77 i v LIST OF TABLES Table Page I Shock wave parameters for Pi = 16 torr 52 v LIST OF ILLUSTRATIONS Figure Page 1 Schematic diagram of the knife edge sch l ieren apparatus . 3 2 P r inc ip le of the wedge technique '. . 5 3 Passage of a ray of l i g h t through a region with a r e f r a c t i v e index gradient 8 4 Optics of the wedge technique. . . 11 5 Beam rotat ion by the mirrors of the c a l i b r a t i o n system 17 6 Schematic diagram of the apparauts. 21 7 Horizontal section of the test sect ion 22 8 The ca l i b r a t i on system 23 9 Configuration of the photomult ip l ier , lens and wedge 24 10 C i r c u i t diagram of the integrator of the Type 0 operational ampl i f ier p lug- in unit 28 11 . Photomult ip l ier traces during one rotat ion of the mirror system of the c a l i b r a t i on system 29 v i Figure Page 12 Typical c a l i b r a t i o n curve 30 13 Plane shock wave in the frame of the shock front 33 14 (a) Var ia t ion of P with T for f ixed values of the r e f r a c t i ve index of an argon plasma . . . . . . . . . 45 (b) Refract ive index of an argon shock wave plasma as a funct ion of Pi and Mi 46 15 Smear photograph of a Mach 14 shock wave in argon . . . . '. . 48 16 Typical smear photograph for ve loc i t y measurements at the test section 49 17 Mi v s P i for shock waves in argon 50 -18 Schl ieren signal and density analogue signal for a Mach99 shock wave in argon, Pi = 16 t o r r . 53 19 Schl ieren signal for a Mach 16 shock wave in argon, Pi = 2 torr 55 20 Pa r t i c l e d i s t r i bu t i ons implied by the sch l ieren signal for the Mach 16 shock wave in argon, Pi = 2 t o r r . . . 56 B.l Typical H & D curve . 69 C l Optics of the mirror system of the ca l i b r a t i on device 72 D.l General layout of the shock tube apparatus 75 v i i Fi gure Page E.l C i r c u i t diagram for operation of the R.C.A. 931A photomult ip l ier with l imi ted dynode ampl i f i ca t ion 78 E.2 Charac ter i s t i c curve of an R.C.A. 931A photomult ip l ier with 6 dynode ampl i f i ca t ion stages in use 79 v i i i ACKNOWLEDGMENTS I should l i ke to take th is opportunity to give thanks to my supervisor, Dr. B. Ahlborn, for his suggestions and guidance during the course of the work represented by this thes i s . I should also l i k e to thank members of the technical s t a f f for the i r help and advice, p a r t i c u l a r l y Mr. D. Olson, Mr. A. Cheuck, Mr. C. Sedger and Mr. E. Wil l iams. Thanks are also due to Dr. S. Richards and Mr. B. Armstrong for many valuable consultat ions during the course of the experimental work. Chapter 1 INTRODUCTION 1 .1 General Introduction In view of the c r i t i c a l dependence of the thermodynamic behavior of plasmas on temperature and dens i ty, one of the main tasks 1 of plasma diagnost ics is the measurement of dens i ty. Since the r e f r a c t i ve index of a plasma depends upon the density of i t s component species, density can be determined by r e f r a c -t i ve index measurement. Three methods are important. The shadowgraph gives an ind ica t ion of the second spat ia l der i va -t i ve the r e f r a c t i ve index; the sch l ieren system measures i t s f i r s t de r i va t i ve , and the interferometer measures the r e f r a c -t i ve index i t s e l f . A l l three methods have the advantages over many other plasma diagnost ic methods that they neither introduce physical objects into the region of the plasma nor apply large amounts of energy to perturb i t . In the experiment to be described here, a sch l ieren densitometer was constructed to measure density steps in plane shock waves. The novel and essent ia l part of th is sch l ieren system was a neutral density wedge which replaced the usual 1 2 knife edge, giving place to several advantages over conven-t iona l methods. The instrument was tested using a detonation driven argon shock wave as a known plasma. In the present work, the descr ipt ion of the instrument is written as far as poss ible in a log i ca l sequence of parts , the content of each section fol lowing on from the development and exigences of the ones before i t . There are a few appendices in which are developed some points which seemed relevant and important, but which could not be f i t t e d into the main text without obscuring the general l ine of thought. A paper descr ib ing the instrument has been published in The Review of S c i e n t i f i c Instruments (Reference 1 ) . This is reproduced in Appendix A. 1 .2 Conventional Schl ieren Systems Successful use of the sch l ieren technique in shock tube densitometry has been reported by many authors. References 1-8 are examples . f The usual experimental technique was well descr ibed, for example, by de Boer (Reference 5 ) . The apparatus is depicted schematical ly in Figure 1 . The source s l i t Si is i l luminated by a l i gh t source and is s i tuated at the focus of the lens L i . Light selected by the beam def ining s l i t S 2 traverses the test section of the shock tube and is brought to a focus in the plane of the knife edge by the lens l2- The 3 knife edge l i e s in such a plane that a part of the l i gh t beam from L 2 is blocked of f from the photomult ip l ier . The photo-m u l t i p l i e r signal is proportional to the amount of l i gh t escaping the knife edge. i TEST SECTION PHOTOMULT IPL IER D E T E C T O R KNIFE EDGE Figure 1. Schematic diagram of the knife edge sch l ieren apparatus. If the test section is assumed to be th in , the func-t ioning of the instrument is simply understood. Through each part of the test section passes a pencil of l i gh t from S 2 . This penci l forms an image in the plane of the knife edge, the dimensions of which are fz/f\ times the dimensions of the source 4 s l i t , f i and f 2 are the focal lengths of L i and L 2 respec-t i v e l y . When a density gradient is present at the point under cons iderat ion, the penci l i s deviated by the sch l ieren ef fect , by an angle proportional to the gradient. The correspon-ing displacement of the contr ibut ion to the image from th is penci l causes a change in the amount of l i gh t escaping the knife edge, and th is change is followed by the photomul t ip l ier . The tota l change in the photomult iplter output voltage is the sum of contr ibut ions from the penci l s of l i gh t from a l l points on S 2 . Hence the instrument integrates the density gradient to g ive, at any ins tant , the density change over the width of the probing beam. A method of improving the s e n s i t i v i t y of the instrument by a factor of about 100 was suggested by Hall (Reference 9). It was proposed that the de f lec t ion of the l i gh t penci l be e f f e c t i v e l y increased by an etalon system. In the experiments of K iefer and Lutz (References 6, 7 & 8) an expanded l i gh t beam was not used, but a c lo se ly pa ra l l e l and highly intense He Ne laser beam reduced to a small diameter by a simple te lescope. This system -determined density gradient with a spat ia l reso lut ion of approximately 1 mm. 1 . 3 The Wedge Technique The knife edge method, common to a l l conventional sch l ieren densitometers, was replaced in this experiment by 5 the wedge technique. This technique was previously employed by Potter (References 11 & 12) in the measurement of small sh i f t s of wide spectral l i n e s . A neutral wedge is a neutral density f i l t e r whose l i gh t transmission varies in one d i rec t i on across i t s surface (Figure 2). The wedge in use in this app l i ca t ion was produced D E T E C T O R Figure 2. P r inc ip le of the wedge technique. on a photographic p la te . Its production and cha rac te r i s t i c s are described in Appendix B. It w i l l be shown in Section 2.2 6 that, in the present app l i ca t i on , i t . is not necessary that the var ia t ion of transmission with distance be l i n e a r ; in some app l i ca t ions , for example in the experiment of Pot ter , .a l inear wedge is required. probing beam is t rans lated into an in tens i ty change by the wedge. The instantaneous de f lec t ion a is proportional to the transverse r e f r a c t i ve index gradient encountered by the beam, assuming this to vary l i t t l e along the path of the beam. Accordingly, the factor by which the wedge attenuates the in tens i ty of the def lected beam increases with a, and so the change in photomult ip l ier signal AV is a measure of a. For a 1i near wedge AW dn A V = * dT where x 1 S a constant. If the re f r ac t i ve material moves per-pendicular ly to the beam'with ve loc i ty v in the d i rec t i on of the re f rac t i ve index gradient, a varies with time. The re f r ac t i ve index change An as the material passes through the beam is found by in tegra t ion : Figure 2 i l l u s t r a te s how a de f lec t ion of a narrow (2) n (3) (4) 7 Now the re f r a c t i ve index of a gas or of the neut ra l , ion or e lectron species of a plasma is proport ional to p a r t i c l e dens i ty. If the sch l ieren de f lec t ion is caused by a gas or plasma in motion as a whole, measurement of JAVdt as a funct ion of time y ie lds an analogue signal for the density p r o f i l e . This p r inc ip le underl ies the present densitometer. The opt ics of the device is treated f u l l y and general ly in the fol lowing chapter. It is shown that accurate resu l t s are promised by a simple design not requir ing l i n e a r i t y of the wedge over the whole in tersect ing beam width, but only over the small sch l ieren displacements of ind iv idua l rays. And problems with d i f f r a c t i o n are not encountered, a customary d i f f i c u l t y with conventional sch l ieren densitometers (References 5 & 10) . Chapter 2 OPTICS OF THE DENSITOMETER 2.1 The Schl ieren E f fect The sch l ieren e f fec t is the de f lec t i on of a l i gh t beam due to gradients in the re f r ac t i ve index of the transmitt ing medium. Consider a ray of l i gh t t ravers ing a medium in which the re f r ac t i ve index varies in the x d i r e c t i o n . Let £ be the angle between the ray and the y d i rec t i on as shown in Figure 3. Figure 3. Passage of a ray of l.ight through a region with a r e f r a c t i v e index gradient. 8 9 S n e l l ' s law may be wr i t ten : n cos <{> = constant (5) D i f f e ren t i a t i n g equation (5) gives Now dn dn dx dy ~ dx dy (6) (7) = t a n $ dn dx (8) Subst i tut ing equation (8) into equation (6) gives cos o> tan op ^ = n sin o) ^ (9) Hence dx dy (10) For a ray entering the medium pa ra l l e l to the y axis and t r a -versing a region of thickness 6, the total deviat ion a follows from integrat ion of equation (10). ret n d c> (11) (12) 10 If n and dn/dx vary neg l i g ib ly over the path of the ray, equation (12) becomes dn dx 6 = na (13) so that a = n dx (14) 2.2 Ef fect of the Wedge Figure 4 shows the geometry of the test sect ion of the.shock tube, the wedge and the probing laser beam. The long i tud ina l axis of the shock tube, the x ax i s , is perpendicular to the axis of the laser beam. The in ter sect ion of these axes is taken as the o r i g in of x. The d i rec t i on X of the t rans-mission gradient of the neutral wedge is pa ra l l e l to x. The o r i g i n of X is at the in ter sec t ion of the X axis and the laser beam ax i s . The transmission funct ion of the wedge is written T(X). D is defined as the distance from the test sect ion to the wedge, the separation of the x and X.axes. i t s cross section is Gaussian; i t w i l l be assumed to be con-f ined within a disc whose radius at the x axis is r. Since the beam has a natural divergence, this radius is not preserved. In general, a ray cross ing the axis of the test sect ion at x crosses the X axis at L(x) , say. The in tens i ty d i s t r i b u t i o n of the laser beam over Figure 4. Optics of the wedge techni que. 12 Thus far, the undef 1 ected beam has been considered. During the passage of a shock wave whose re f r ac t i ve index may be described by the funct ion n ( x j t ) , the instantaneous de f l e c -t ion of a ray at x by the sch l ieren e f fec t is a ( x , t ) . This d isplaces the ray a distance across the wedge which is small compared with the width of the disc formed by the in te r sec t ion of the undisturbed beam with the wedge. It w i l l be assumed throughout that the wedge is l i near over these small d i sp l ace -ments, but not necessar i ly l i near over the whole beam width. Consider a laminar sect ion of the undisplaced beam, the c o l l e c t i o n of rays passing between x and x + dx as shown in Figure 4. The displacement of th is section by the sch l ieren e f fec t is indicated by broken l i n e s . The l i gh t f lux cj>(x)dx is the area integra l of beam in tens i t y across the sec t ion . I f n(x,t) and 9/9x n(x,t) vary neg l i g ib l y over the paths of the rays within the sect ion, a (x,t ) is given by equation (14). For a shock wave plasma,the n of the denominator may be set at unity with neg l i g ib le loss of accuracy, n being considered unequal to 1 only in terms involv ing (n - 1). Then a(x,t ) = 6 n(x,t) (15) The re f r ac t i ve index p r o f i l e n(x,t) of the plasma propagates in the x d i rec t ion at the ve loc i t y v of the shock wave, so that 1 3 n(x,t) = f(x - vt) (16) Accordingly, equation (15) becomes a(x,t ) = 6 f ( x - vt) (17) The change in transmitted l i gh t f lux due to the sch l ieren de f lec t ion of the beam is monitored by a photomult ip l ier . A change AV in signal corresponds to a proportional change in l i ght f lux A$. Then AV = k A $ (18) The elementary contr ibut ion x d(AV) = k <J)(x) dx Subst i tut ing equation (17) into equation (19) gives d(AV) = kD o)(x) T ' (L (x ) ) 6 f ' ( x - vt) dx (20) Integrating equation (20), from the section of l i gh t through '9T(X)' 8X X=L(x) a(x,t ) D (19) AV = kD 6 J dp(x) T ' (L (x ) ) f ' ( x - r - vt) dx (21) 14 AV is integrated through time from before the a r r i v a l of the shock to time r. AVdt = kDS <J>(x) T' (L(x)) f ' (x - vt) dt dx . 0 0 - y ^ jr <j>(x) T'(L(x)) « - r f (X - Vt) - f ( - 0 0 ) I dx (22) (23) f(-°°) is the r e f r a c t i v e index of the undisturbed gas. If r represents a time a f ter the shock front has passed the beam completely, then f(x - vr) is evaluated as the step in refrac-t i ve index gradient at the fr-ont, plus a slowly varying func-t ion representing the decay in r e f r a c t i ve index of the wave behind the f ront . Defining r in th i s way, - f(x - vr) - f(-») varies neg l i g ib ly over - r < x < r, and so equation (23) becomes AVdt kDS j —00 f(x - vr) - f(-»)}• T <|>(x) T'(L(x)) dx (24) The integral on the r .h.s . is a constant of the apparatus. If the whole beam is rotated by a small angle 0 1 , the change AV1 in the signal i s , from equation (19), AVi = kD 0 1 x) T' (L(x)) dx (25) Subst i tut ing equation (25) into equation (24) gives 15 AVdt Av 6 a i v f ( x - vr) - f ( - o o ) } (26) From equation (16), equation (26) becomes AVdt = AV_i _5 a i v n(x - vr) - n(-°°) (27) This may be rewritten AVdt dccL v 4-n (x - vt) - n ( (28) for t > r The quantity fdV da is the rate of change of voltage with c t , equal to AV 1/ai'; the subscr ipt B denotes that the quantity in the brackets must be evaluated for a rotat ion of the whole beam, and at the pos i t ion at which the beam axis coincides with i t s undeflected pos i t i on . Equation (28) is the fundamental equation of the instrument. The condit ion t > r ar i ses because of the large var ia t ion of n in the shock front over the distance 2r, the width of the laser beam. The integrated signal AVdt does — oo not minutely describe the spat ia l var ia t ion of n over the shock f ront , but i t does track the jump in n with accuracy as the shock front passes the beam. Evidently the spat ia l resolu-t ion of the device is the width 2r of the laser beam. This was approximately 2mm. 16 It should be noted that equation (28) is the same as i t would have been had the whole laser beam in tens i ty been concentrated long the beam ax i s . The transverse spread of the laser beam, character ized here by the function 4>(x), a n d the var ia t ion in the transmission gradient of the wedge over the area on which the beam f a l l s do not a f fec t the resu l t of t h e c a l c u l a t i o n . The quantity dV must be measured in order to B dc^ ca l i b ra te the system. An opt ica l system to do this is described in the next sec t ion . 2 . 3 The Ca l ib ra t ion System The laser beam is swept across the wedge at a constant known angular ve loc i t y by a system of three mirrors , two of which are mounted on a revolving tab le . Measurement of the var ia t ion of signal with time leads d i r e c t l y to . ' l d a J B The geometry of the mirror system is i l l u s t r a t e d in Figure 5. Ful l l ines show the system as i t passes the beam undef1ected; broken l ines indicate the e f fec t of a small rotat ion 0 of the rotat ing mirror assembly, and dotted l ines are l ines of construct ion. The centre of rotat ion 0 of the mirror assembly l i e s at the point of in ter sect ion of l ines drawn perpendicular to the rotat ing mirror surfaces and through the points of incidence of the laser beam on the mirrors when the mirrors are in the symmetric pos i t i on , the pos i t ion for 1 7 which there is no resultant beam displacement. The included angle of the two rotat ing mirrors is 9 5 ° . Figure 5. rotat ion by the mirrors of the c a l i b r a t i on system. 18 Consider a rotat ion of the mirror assembly by a small angle 8. The laser beam is twice re f l ec ted from rotated surfaces and is consequently i t s e l f rotated by an angle 46, appearing to diverge from i t s o r i g ina l axis at a point A. The distance AD i s , for p rac t i ca l purposes, equal to the distance CD. The opt ics of the mirror system is discussed more f u l l y in Appendix C. In view of the equal i ty of thedis tancesAD and CD, the ca l i b ra t i on system was constructed and placed in such a way that the distance CD was equal to the distance between D and the centre of the test sect ion. The system then appeared to make the laser beam rotate from the point at which the sch l ieren de f lec t ion arose. When the mirror assembly was rotated by 180° from the pos i t ion in which i t is drawn in Figure 5, the laser beam was not incident on e i ther mirror . The mirrors were placed thus when the system was out of use, conveniently avoiding the necessity of removing the c a l i b r a t i o n system from the rest of . the apparatus of the densitometer. Writing co for the angular ve loc i t y d9 dt of the mir rors , dV da 1 dV 4 d0 (29) = 1 d V 1 T dt u (30) Whereupon 19 dV_ da 4OJ dV dt (31) Measurement of fdVl dt from the output of the photomult ip l ier gives dvl d i rec t l y from equation (31). Allowance must be made for l i gh t absorption by the mirrors . Chapter 3 THE APPARATUS 3.1 Construction of the Apparatus A schematic diagram of the apparatus is found in Figure 6. The observed plasma was a shock wave produced in argon by a chemical detonation. The shock wave propagated along a pyrex shock tube of 2.5 cm internal diamter, and was produced by i gn i t i ng a mixture of equal parts by pressure of oxygen and acetylene in a detonation chamber separated from the shock tube by a mylar diaphragm. The i n i t i a l pressure of the explosive mixture was always 350 t o r r ; the i n i t i a l pressure of the argon test gas was varied between 1 torr and 160 to r r . The detonation dr iver has been described by Huni (Reference 13) and Redfern (Reference 14). Detai l s of the shock tube apparatus are given in Appendix D. The test section consisted of a piece of brass tube of the same interna l diameter as that o,f the shock tube. Two para l l e l quartz windows were mounted on opposite sides of the tube in order to transmit the probing laser beam. A hor izontal section of . the test section is shown in Figure 7. 20 21 \ / / * / > / \ / \ / \ HLM. I«.ier \ C a l i k r o Via r\ f . M . -10OV -vIS V -IS V p Koboi^ iAlhpli + aal"t6 E D ouhpul- O-~IB"1 I Ttkfcroni* S5I d u a l o « - 4 m O S C i ' l l o i C o f t kio|k p^ s s F i g u r e 6 . S c h e m a t i c d i a g r a m o f t h e a p p a r a t u s . 22 P R O B I N G L A S E R B E A M Q U A R T Z W INDOWS B R A S S ^ _ _ ^ T U B E Figure 7. Horizontal section of the test sec t ion . A smear camera was used to photograph the shock wave over a length of the shock tube inc luding the test sec t ion . The shock wave ve loc i t y was measured in this way. It was necessary to remove the laser beam in order to take these photographs. The smear camera is described in deta i l in Reference 13. A l l the opt ica l components of the densitometer were mounted on opt ica l benches bolted to a piece of steel U channel. This was f ixed to a table which was phys i ca l l y separate from the table on which the shock tube was mounted, and which was braced and heavi ly weighted in order to minimize v ib ra t i ons . The probing beam was provided by a Spectra 115 helium o neon laser of 1/2 mW beam power and 6328 A beam wavelength. 23 The c a l i b r a t i o n system is drawn in Figure 8. The rotat ing mirrors were mounted on the two symmetrical faces of an isosceles aluminium prism, the included angle being 95° . To allow the laser beam to pass unobstructed when the prism was rotated by 180° from the pos i t ion shown, a small part of the prism was cut away as shown in Figure 8. The prism was rotated at 1200 r.p.m. by a Globe SC B1702 synchronous motor. Dynamic balancing of the prism assembly was necessary in order to prevent overheating of the motor. Optical alignment was achieved by adjust ing the three screws supporting the base plate of the motor mount, and the micrometer screws of the f ixed mirror mount. F I X E D M I R R O R M I R R O R M O U N T C U T - A W A Y S E C T I O N R O T A T I N G P R I S M M O T O R M O T O R M O U N T B A S E P L A T E A D J U S T I N G S C R E W S Figure 8. The c a l i b r a t i o n system. 24 A cer ta in f r ac t i on of the laser l i gh t was los t by r e f l e c t i o n at the three mirrors . When the c a l i b r a t i o n system was not being used, a neutral density f i l t e r was placed in the laser beam path. A f i l t e r was chosen with such a density that the attenuation of the beam by the f i l t e r was as nearly as possible the same as the attenuation of the beam by the mirrors . In th is way, the only overa l l e f fec t of the c a l i b r a -t ion system was to rotate the beam. The wedge was mounted on a r i g i d frame d i r e c t l y in front of a 50 mm focal length convex lens (Figure 9). The lens served to focus the def lected l i gh t beam on to a roughly f ixed region of the photosensit ive surface of the photomul-t i p l i e r . Because of t h i s , the photomult ip l ier signal varied with beam de f lec t i on only on account of the varying transmission of the wedge, not being appreciably af fected by s e n s i t i v i t y var ia t ions over the surface of the photocathode. D E F L E C T E D B E A M B E A M A X I S P H O T O M U L T I P L I E R T U B E Figure 9. Conf igurat ion of photomul t ip l ier , lens and wedge. \ 25 An R.C.A. 931A photomult ip l ier was used. In order to improve the signal to noise r a t i o of this device, a l im i ted number of i t s dynodes were used. This modi f icat ion of the c usual c i r c u i t r y is described in Appendix E. The laser l i gh t f a l l i n g on the photocathode was reduced to an appropriate in tens i ty range by a neutral, density f i l t e r placed over the window of the photomult ip l ier housing. The photomult ip l ier output was appl ied across a lk load re s i s t o r to the input of a National LH 0063 buffer amp l i f i e r . This produced a voltage ampl i f i ca t ion of 1, working into 50 fi. The output was taken by an RG 58 co-ax ia l cable terminated in a 50 fi r e s i s t o r . The terminator matched the impedance of the cable, so pre-venting r e f l e c t i o n s of the s i gna l . The minimum 10% to 90% signal r i set ime was estimated from the s ingle photon pulses to be 9 ns. In addit ion to the def lected l i gh t of the laser beam, the photomult ip l ier behind the wedge also received l i gh t emitted by the plasma. An attempt was f i r s t made to compensate for th is l i gh t by construct ing a monitor comprising a photo-m u l t i p l i e r system s imi la r to the one already descr ibed, and a semi - si 1vered mirror . It proved f u t i l e , however, to try to balance the two channels'. This was attempted by s imulating the plasma l i gh t with a l i gh t f lash from a Solatron 122 e lec -tronic f lash uni t , and balancing the l i gh t f lux received by the two photomult ip l iers by adjustment of the i r i s diaphragms 26 placed before the lenses of the two channels. The f l a s h l i g h t could be balanced approximately, but the plasma l i gh t was i n t r a c t ab l e . Instead of this measure, a narrow band inter ference f i l t e r was placed between the test sect ion and the c a l i b r a t i o n o system. This f i l t e r allowed the 6328 A laser rad iat ion to pass scarcely attenuated. Its band pass proved to be narrow enough to reduce the plasma l i g h t , even of the most luminous shock waves, to a level that caused neg l i g i b le response from the photomult ip l ier . The e lec t ron ic s used to detect and process the photomult ip l ier s ignals is depicted schematical ly in Figure 6. Both the sch l ieren signal and i t s time integra l were displayed on the dual beam Tektronix 551 cathode ray o sc i l l o s cope . A Tektronix 7704 C.R.O. was used for timing and ampl i f i ca t ion purposes . A timing pulse was derived from an R.C.A. 931A photomult ip l ier placed next to the shock tube and close to the test sect ion. The luminous fronts of the shock waves created timing pulses of approximately 40 V amplitude. The timing pulse was applied to the t r i g ge r , input of time base A (p lug- in unit 7B71) of the 7704 C.R.O. The delay generator of th is time base was adjusted to t r i gger time base B (plug- in unit 7B70) af ter a delay equal to the time elapsed between the production of the t r igger pulse and the a r r i v a l of the shock front at the test sect ion. In th i s way, time base B was \ 27 synchronized with the sch l ieren signal from the photomul t ip l ie r . This time base was used to t r i gger the 7704 C.R.O. and, by means of the "+ gate" s i gna l , to t r i gger the 551 C.R.O. as wel 1 . Though a l l measurements were car r ied out using the 551 C.R.O., i t was convenient to apply the photomult ip l ier sch l ieren signal f i r s t to the 7704 C.R.O. A 7A12 v e r t i c a l signal p lug- in unit was used, with an input s e n s i t i v i t y of 50 mV/div. The osc i l l o scope output then provided a ten times ampl i f ied version of the input s i gna l . The output signal was appl ied to the Type K p lug- in unit control 1ing the upper beam of the 551 C.R.O. The same signal was passed through a high pass f i l t e r and then i n te -grated by the operational ampl i f ie r Type 0 p lug- in unit which contro l led the lower beam. The high pass f i l t e r served to reduce the amplitude of a 13 kHz saw tooth r i pp le produced by the buffer amp l i f i e r c i r c u i t . Measurements of the integrated signal could not be made i f th is waveform was not f i l t e r e d . The cu t - o f f frequency of the f i l t e r was approximately 10 kHz. The c i r c u i t diagram of the integrator is shown in Figure 10. A 300 mV x 1 y s square pulse from a Bradley E lec t ron ics Pulse Generator 233 was used to c a l i b r a te the integrator . For the spec i f i ed impedance values shown in F i g u r e 1 0, e 0 = -1.22 where t is measured in ys. e i dt (32) 28 O-o-oi M n A A A V W O O - O O O l jiF F i g u r e 10. C i r c u i t d i ag ram of the i n t e g r a t o r o f the Type 0 o p e r a t i o n a l a m p l i f i e r p l u g - i n u n i t . 3.2 Use o f the Appa r a t u s The f a c t o r by wh ich the m i r r o r s o f the c a l i b r a t i o n sys tem reduced the beam i n t e n s i t y was found to be 0 . 62 . The c l o s e s t e q u i v a l e n t n e u t r a l d e n s i t y f i l t e r a v a i l a b l e i n the l a b o r a t o r y was o f d e n s i t y 0 . 2 ; i t a t t e n u a t e d the beam i n t e n s i t y by a f a c t o r 0 . 63 . These f a c t o r s were c l o s e enough t o g e t h e r t h a t i t c o u l d be assumed t h a t , i n both c a s e s , the photomul-. t i p l i e r was o p e r a t i n g a t p o i n t s o f the same s l o p e on i t s c h a r a c -t e r i s t i c . To c o r r e c t f o r the d i s c r e p a n c y , t h e n , a l l t h a t was needed was the sma l l c o r r e c t i o n f a c t o r 1.02 (~ 0 .63/0.62) wh ich m u l t i p l i e d the c a l i b r a t i o n c o n s t a n t i n o r d e r to s t a n d a r d i z e the v o l t a g e s c a l e s of the c a l i b r a t i o n and of the e x p e r i m e n t . F i g u r e 11 i s a t r a c i n g o f a pho tog raph o f two p h o t o -m u l t i p l i e r t r a c e s t aken d u r i n g one r o t a t i o n of the m i r r o r \ 29 PHOTOMULTIPLIER OUTPUT V O L T A G E 1 TRACE I 5 0 mV TRACE 2 T w = 126 RAD IAN/S —H h— TRACE I. Syus TRACE 2, 5ms •» T I M E -Figure 11 . Photomult ip l ier traces during one ro ta t ion of the mirror assembly of the c a l i b r a t i o n system. assembly. The r i s i n g slope of trace 1 is the response of the photomult ip l ier as the mirrors swept the beam across the wedge. Trace 2 is the signal produced during the time that the cut-away sect ion of the rotat ing prism (Figure 8) allowed the rotat ing mirror assembly to not in terrupt the beam. Escaping attenuation by r e f l e c t i o n , the beam in tens i t y was enhanced by a factor 1/0.62 over the re f l ec ted beam i n ten s i t y . The unref lected beam was attenuated on passing through the centre of the wedge, roughly by 1/2. It would be expected, then, that trace 2 would be a f l a t bottomed pulse with a voltage of about 0.8 times the greatest (negative) voltage of trace 1. The actual voltage is ev ident ly of about this magnitude, but the 30 bottom of the pulse is not f l a t . This is because the pulse is of such an area that a quantity of charge was drawn from the photomult ip l ier which was not small in comparison with the charge on the l a s t speed-up capac i tor . Fortunately, th is trace is never required for any measurement; the pulse of trace 1 is too small to arouse the same d i f f i c u l t y . Figure 12 shows a photograph of a t yp i ca l c a l i b r a t i o n curve. The r i s i n g slope is i s o l a t e d . The gradient of the slope is measured at the centre of the l i near por t ion . The gradient in th is case is 98 mV/ys, and the centre of the l i near region occurs around the centre of the time scale at a signal of about 260 mV. The gradient of the c a l i b r a t i o n curve must be mu l t ip l i ed by the factor 1.02, and also by a fac tor 10 because i t was always measured from a trace of the 7704 C.R.O. The signal received by the 551 C.R.O. was ampl i f ied by a fac tor 10 by the 7704 C.R.O. as described in Section 3.1. P H O T O M U L T I P L I E R O U T P U T V O L T A G E Figure 12. Typical c a l i b r a t i o n curve. \ 31 In genera l , once the c a l i b r a t i o n curve had been determined, the pos i t ion of the wedge was adjusted in such a way that the undeflected beam' axis f e l l on the centre of the l i near port ion of the wedge, the point for which the gradient had been measured. In order to accomplish t h i s , the fo l lowing procedure was adopted. The rotat ing mirror assembly was turned out of play, and the 0 .2 neutral density f i l t e r was emplaced. The wedge was then adjusted to such a pos i t ion that the d.c. output voltage V i of the photomult ip l ier was 1.02 times the voltage V 2 at the centre of the l i nea r region of the c a l i b r a t i o n curve. The d.c. component V i of the photomult ip l ier output was removed during the experiment proper by use of the a.c. coupling f a c i l i t y of the input to the 7A12 p lug- in unit of the 7704 C.R.O. The a.c. components were ampl i f ied by 10 and passed to the 551 C.R.O., The sett ing of the delay generator of the 7B71 p lug- in unit was adjusted to a value for which the timing pulse caused both osc i l l o scopes to tr igger a few microseconds before the shock front crossed the beam. In th is way, the V and jAVdt signals were displayed simultaneously on the 551 dual beam osc i l lo scope on a time base of 1 us /d iv . The performance of the densitometer was invest igated by making measurements on a shock wave whose propert ies were ca lculated from the i n i t i a l condit ions of the test gas, and the measured shock wave v e l o c i t y . These ca l cu l a t i ons and the theory behind them are presented next. Chapter 4 PROPERTIES OF THE SHOCK WAVES 4.1 Shock Wave Plasma Parameters A shock wave is a discontinuous r i se in the pressure, dens i ty, temperature and entropy of a f l u i d . The d i s cont inu i t y propagates supersonica l ly with respect to the undistrubed gas ahead of i t . The f l u i d behind the shock wave moves along in the same d i rec t ion , in such a way that the ve loc i t y of the i shock wave with respect to the f l u i d behind i t is subsonic. The flow takes place a d i a b a t i c a l l y but is i r r e v e r s i b l e and hence not i sen t rop i c . A real f l u i d cannot have an actual d i s -cont inu i ty ; this is an i d e a l i z a t i o n of what is r e a l l y a thin region with very high gradients in the thermodynamic quan t i t i e s . Consider a plane shock wave in a frame in which the shock front is at rest (Figure 13). Let the subscr ipt 1 . denote quant i t ies before the shock front and le t the subscr ipt 2 denote quant i t ies behind i t . P is pressure; p is dens i ty ; T is temperature; H is enthalpy per unit mass, and u is f l u i d ve loc i ty r e l a t i ve to the shock f r on t . 32 \ 33 S H O C K F R O N T -K -UcH P, T , H -u, S H O C K T U B E D I R E C T I O N O F M O T I O N O F S H O C K W A V E W I T H R E S P E C T T O T U B E F i g u r e 1 3 . P l a n e s h o c k wave i n t h e f r a m e o f t h e s h o c k f r o n t . The c o n s e r v a t i o n e q u a t i o n s o f m a s s , momentum and e n e r g y may be w r i t t e n ( R e f e r e n c e 15) P i U i = p 2 u 2 ( 3 3 ) P i + P i U i 2 = P 2 + P 2 u 2 2 ( 3 4 ) H x + 1 U ! 2 = H 2 + \ u 2 2 ( 3 5 ) F o r an i d e a l gas , t h e e q u a t i o n o f s t a t e m a y b e w r i t t e n H = — — ( 3 6 ) Y - l p Y i s t h e r a t i o o f t h e p r i n c i p a l s p e c i f i c h e a t s o f t h e gas and i s a c o n s t a n t . I t i s c o n v e n i e n t t o i n t r o d u c e t h e Mach number 34 M. This is the ra t io r e l a t i n g the ve loc i t y of a d i s • turbance or flow in a gas to the loca l ve loc i t y of sound a. Hence M i a i (37) The ve loc i ty of sound is re la ted to the thermodynamic parameters of an ideal gas by a = XP P (38) Solut ion of equations (33) through (38) y ie ld s the fo l lowing equations: Pi 2 Y M t 2 - (y-1) Y + 1 (39) £2. - ( Y + 1 ) M I 2  P L " ( Y - 1 ) + 2 Ti Y M Y - 1 ^ 1 M l 2 + 1 ( 4 0 ) ( 4 1 ) Mi where the temperature, fol1ows from the ideal gas law. Since the condit ions ahead of the shock are known, these equations determine P2> P2 and T 2 once the shock wave ve loc i t y has been meas ured. \ 35 In monatomic gases the only departures from ideal gas behavior are caused by i on iza t ion and by a contr ibut ion to the s p e c i f i c heat from e lec t ron i c e x c i t a t i o n . The popula-t ion of an excited e lec t ron ic energy level at equ i l ibr ium is - E /kT proport ional to e , where E is the exc i t a t i on energy. e x For most atoms and molecules, even at comparatively high temperatures, E is large enough to render the energy r e s i d -e x ing in the excited e lec t ron ic leve l s neg l i g i b le in comparison with the c l a s s i c a l energy -^ RT of the other energy modes. Ionizat ion may be taken into account in e i ther of two ways. Most commonly (e.g. Reference 16) the energy con-servation equation (35) is modified to include an energy term AE, th is being the energy taken to produce the i o n i z a t i o n . Equation (35) then becomes H 2 - Hi = 1 U j 2 - \ u 2 2 - AE (42) In the present work, the a l te rna t i ve procedure is adopted. An e f f ec t i ve ad iabat ic exponent g is introduced, defined by 9-1 P H, P and p re fer to the ionized gas, and so g is a function of temperature and pressure. This funct ion tends to the con stant y in the lower temperature regime of zero i o n i z a t i o n . The l a t t e r approach has the advantage of separating the (43) 36 hydrodynamic ca l cu la t ions from the thermodynamic c a l c u l a t i o n s , i the l a t t e r amounting to the evaluat ion of the funct ion g(P,T). The so lut ions of equations (33), (34), (35) and (43) are £ i = 92 + I n (44) g i Mi Subst i tut ing equations (37) and (38) into equation (45) with the approximation g x Mj 2 >> g 2 - l y i e l d s Pz - \V+U\* (46) Equations(43), (44) and (46) then give (92 + 1) (47) Equations (44) through (47) a l l contain g 2 on the r .h . s . , and g 2 is an unknown. It may be obtained by a simple i t e r a t i o n (Reference 17) involv ing equations (46) and (47) and a table giving g as a funct ion of P and H. The compression r a t i o p 2 / p i then follows from equation (44). Since H is a known function of P and T (Reference 18), T 2 may be obtained from values of P 2 and H 2 folowing from equations (46) and (47) \ 37 re spec t i ve l y . F i n a l l y , the gas composition is derived from the Saha equation: N. N Q. _ j e _ o i N a Q0 2TT m kT 3/2 e E .j / kT (48) with N . = N i e ( 4 9 ) and P 2 - ( N - + N + N a ) k T :  v i e * ( 5 0 ) N.j , N g and N a are the pa r t i c l e dens i t ies of ions, e lectrons and neutral atoms re spec t i ve l y . E^  is the i on i za t i on po ten t i a l . Q. and QQ are the internal p a r t i t i o n functions of ions and neutral atoms respect ive ly (Reference 18). In order to ca lcu la te the r e f r a c t i v e index i t is necessary to know i t s re l a t i on to the plasma parameters. This is the subject of the next sec t ion . 4•2 The Refract ive Index of a Plasma The re f r a c t i ve index of an ionized gas includes con-t r ibut ions from neutral atoms, from ions and from the ion-e lectron plasma. As w i l l be seen later , ' under the physical 38 condit ions encountered in th is experiment, the plasma contr i bution is that of a c l a s s i c a l e lectron gas. This j u s t i f i e s the use of the Gladstone-Dale expression (Reference 19) for the r e f r a c t i ve index n: n-1 = I k. N i (51) k. is the s p e c i f i c r e f r a c t i v i t y of the i - t h component of the mixture, and is i t s concentrat ion. For the. argon shock wave plasma in question th is may be written n-1 plasma • • n-1 + n-1 + n-1 I J a i (52) The subscripts a, i and e denote atoms, ions and electrons re spec t i ve ly . It is the phase index of re f rac t i on that is referred to here. It is t h i s , rather than the group index of r e f r a c t i o n , that is of i n te res t in r e f r a c t i v e phenomena such as the sch l ieren e f f e c t . 4.2.1 Neutral Atoms In the c l a s s i c a l theory (Reference 20), the atom is regarded as a set o f ' e l e c t r on s of charge e and mass m . Each electron is harmonically and isotropica11y bound to i t s i n d i -vidual equi l ibr ium pos i t i on . When the atom is placed in a beam of l i g h t , the a l te rnat ing e l e c t r i c f i e l d causes forced \ 39 o s c i l l a t i o n of the p a r t i c l e s . The induced e l e c t r i c d ipole moment is ca lcu lated as a function of time. From th is fol lows the po lar i zab i 1 i ty a. a = I — ? — " 2 (53) m v oj. 2 - OJ 2 e i i 0J.J is the resonant angular frequency of the i - t h e l ec t ron ; OJ is the angular frequency of the appl ied r a d i a t i o n . The o s c i l l a t o r strengths f.. re fer to e lec t ron ic t rans i t i ons with resonant frequencies OJ . . The quantum mechanical treatment (Reference 20) y ie ld s an equation of the same form. The o s c i l l a t o r strengths f ind in terpreta t ion as cer ta in matrix elements re-lated to the dipole moments. The re f r ac t i ve index is re lated to the polar i z -a b i l i t y by n-1 = 2 TT N a (54) so that n-1 2 TT N e :  a m_ i w i O J ' (55) Resonance l ines of neutral atoms l i e almost exc lu s i ve l y in t h e u l t r a v i o l e t region and in shorter wavelength parts of the spectrum. Equation (55) may, then, be expanded and written in the form: 40 n-1 / a k l + X k 2 (56) For argon, kx = 1.03 x 1 0 " 2 3 cm3 and k 2 = 0.58 x 1 0 - 3 3 cm5 (Reference 19). 4.2.2 Ions Again the po l a r i z ab i1 i t y must be ca l cu la ted . Several methods are ava i l ab le , and a l l are approximate. The best and most commonly used involves S l a t e r ' s screening constants. These are used to evaluate mean square r ad i i of e lectron orb i t s (References 21 and 22). 4 9 ' B a = £ r . " 1 I < r 2 i > z (57) 2 r„ is the Bohr radius and <r .> is the mean square value of B e i the distance of the i - t h e lectron from the nucleus. This leads to a s p e c i f i c r e f r a c t i v i t y for the argon ion of 0.67 times that of the neutral atom. Since the e lectron contribu-t ion to the re f r ac t i ve index dominates the ion cont r ibu t i on , the probable error in the l a t t e r is not s i g n i f i c a n t . 4.2.3 Electrons From Maxwell's equations the e l e c t r i c f i e l d wave equation is 41 V 2 E - 4irc 2Va J_ i l l + 4-rr 1^ (58) If j is p a r a l l e l to the wave f ron t , V«j vanishes, no charges accumulate, and so a vanishes a t must be found from the macroscopic equations of motion of the plasma, derived from the Bolzmann equation (Reference 23): m m c e Z p e 3t E + v x B n J eZp m. VP i e Zm VP . e i m . - Zm J x B l e J > (59) 'm. and mg are the masses of ion and e lectron re spec t i ve l y ; P. and P are the i r respect ive pressures. p is the plasma i e z dens i ty. ~ is the charge of the ions, n is a constant defined by the assumption n e N + P • = J ei c J (60) where P • is the tota l momentum transferred to the ions per ei unit time by c o l l i s i o n s with the e lec t rons . Where V«j^ = a = 0, the pressure does not change during the o s c i l l a t i o n , and so VP = V P . = 0 e l . (61) 42 B may be set at zero a l so , since there is no external magnetic f i e l d and the force due to the magnetic f i e l d vector of the wave is small compared with the e l e c t r i c force. Then equation (59) becomes m i m e c 8 l _ * Z p e 2 3t (62) For a wave in which E z is propagated along the z ax i s , the wave equation i s , from equations (58) and (62), d2 E 4TTp Ze 2 E z _ z — § x"2" m. rn c 2 i e 1 3 2 E z 8 t z (63) The dispers ion re l a t i on for so lut ions to equation (63) of the f orm exp i ( kz - cot) i s 2 1 - O ) 2 / O J Z p (64) v is the phase ve loc i t y of the wave, and ojp is the plasma frequency defined by 4TT N e 2  e m m 1 + Z — m i (65) The re f r ac t i ve index (66) now follows from equation (64): 43 W n 2 n 2 = 1 - (67) For an argon plasma with N g = TO 1 7 c m - 3 , w ~ 1.5 x 1 0 1 3 s " 1 . o For the 6328 A laser r ad i a t i on , to ~ 3 x 1 0 1 5 s - 1 . In these circumstances, ojp/w ~ 5 x 1 0 - 3 . E lectron dens i t ies for the shock waves of th is experiment did not exceed 1 0 1 7 c m - 3 , and so the approximation oo 2 << to 2 was always v a l i d . This being the case, and taking ~ as neg l i g i b l e , equations (65) and i (67) render 2 TT e 2 N n-1 = -A (68) me OJ 2 This is the same resu l t as that which would fol low from taking the d ispers ion equation of the neutral gas and evaluating i t o for the case of no resonances. Again for 6328 A r ad i a t i on , the constants of equation (68) may be evaluated. Then, with N in cm e - 3 n - 1 = -1 .796 x l O " 2 2 N e (69) e 4.3 Numerical Ca lcu lat ions The resu l t s of th is chapter were used to evaluate the r e f r a c t i v e index var ia t ion with temperature and pressure of a-general argon plasma and of a shock wave produced argon plasma . 44 \ In performing the thermodynamic c a l cu l a t i on s , g and T were obtained as functions of P and H not from external data but from f i r s t p r i nc ip l e s using H = 1 P f P + N. E i (70) Equation (70) follows from the d e f i n i t i o n of enthalpy and simple k inet i c theory. The pa r t i t i on funct ions were ca lcu lated fol lowing the method of Dre l l i shak et al. (Reference 18). The tempera-tures considered here were not high enough for second and further ion izat ions to be of s i g n i f i c a n c e , and so the assump-tion made in equation (49) remained a va l id one, given charge neut ra l i t y of the o r i g ina l test gas. Computer programmes were written in Fortran for an I.B.M. 370 computer. P vs T curves for constant values of r e f r ac t i ve index were determined. These are drawn in Figure 14(a). Also,(n - 1) was determined for a shock wave plasma as a function of the Mach number Mi of the shock wave and the i n i t i a l pressure Pi of the test gas. The curves are drawn in Figure 14(b). The ca lcu la t ions were performed throughout for argon o and for l i gh t of wavelength 6328 A. \ 45 15-0 * T E M P E R A T U R E ( D E G R E E S K x l O 3 ) Figure 14(a). Var iat ion of P with re f r ac t i ve index of T for f ixed values of the an argon plasma . 46 F i g u r e 1 4 ( b ) . R e f r a c t i v e i n d e x o f an a r g o n s h o c k wave p l a s m a as a f u n c t i o n o f P x and Mi. \ Chapter 5 EXPERIMENT AND RESULTS 5.1 Ve loc i ty Measurements The shock wave v e l o c i t y , necessary for the c a l c u l a -t ion of the shock wave parameters, was measured with a smear camera. The construct ion of the smear camera is described in Reference 13. The shock wave was f i r s t photographed in the part of the pyrex shock tube through which the shock wave passed just before entering the test sec t ion . Thin black v e r t i c a l markers, spaced by 5 cm, were attached to the shock tube in order to measure the spat ia l progress of the shock wave. The wri t ing speed of the camera was 46.96 cm/s, th is f i gure fo l low-ing from a rotat ion frequency measurement on the drum bearing the mirror . A Monsanto Programmable Counter-Timer was used to make th is measurement. Figure 15 shows a colour smear photograph of an argon shock wave for which P i = 4 torr and M = 14.3. 47 CM Figure 1 5 . Smear photograph of a Mach 1 4 shock wave in argon. 49 It was found that the shock wave v e l o c i t y measured over the test section by the smear camera f e l l from i t s value in the e a r l i e r section of the shock tube. A typ ica l smear camera photograph for the measurement at the test sect ion is shown in Figure 16. The thin v e r t i c a l black l i ne is the image of a marker on the shock tube; i t shows the end of the test section at which the shock wave entered. R=4 T O R R D I S T A N C E <~ T E S T S E C T I O N T H I N M A R K E R P Y R E X S H O C K T U B E 11 u Figure 16. Typical smear photograph at the test sect ion. for ve l oc i t y measurement \ 50 The shock wave ve loc i t y was measured from the pos i t ions on the time axis of the leading edge of the shock wave as i t just entered and just l e f t the test sect ion. Five measurements were made for each pressure Pi that was invest igated. Mach numbers were ca lcu lated using the fact that the ve loc i t y of sound in argon at 20°C is 0.318 km/s. The resu l t s are shown in Figure 17. SHOCK WAVE MACH NUMBER 4 0 B O ! 6 0 >p, ( I N T O R R ) Figure 17. Mi vs Pi for shock waves in argon. \ 51 5.2 Shock Wave Parameters Using the resu l t s of Sections 4.1 and 4.2, and 5.1 for the shock wave v e l o c i t i e s , the thermodynamic parameters and the r e f r a c t i ve indexes of the shock wave plasmas were ca lcu la ted for various values of P x . The r e f r a c t i v e index of the i n i t i a l test gas is important, and this was also ca l cu l a ted . Table I shows the shock wave parameters for Pi = 16 t o r r ; th is was the f i r s t shock wave to be inves t i ga ted. An i n i t i a l temperature of 20°C was assumed. The atomic weight of argon is 39.944 for the i sotopic mixture found in the environment. 5.3 Density Measurements at Pi = 16 Torr It remains to treat of the app l i ca t i on of the den-sitometer to check that the measured density jump across the shock front was in accordance with the value expected from shock wave theory and the ve loc i t y measurements. In order to carry th is out, the re f r ac t i ve index jump across the shock front was measured for a Pi = 16 torr shock wave. The resu l t was compared with the expected value of 1.51 x 1 0 " 5 . It was observed during the course of the experiment that the quartz windows of the test sect ion darkened with successive shock wave detonations. This darkening was caused by the deposit of thin layers of carbon by the dr iver gas. In consequence, i t was necessary to ca l i b ra te the densitometer before every measurement. 52 I N I T I A L T E S T G A S S H O C K W A V E P L A S M A S H O C K W A V E V E L O C I T Y ( K ^ / S ) 3 -15 M A C H N U M B E R 9 - 9 0 P R E S S U R E ( D Y N E / c r Y v v ) 2 - 1 3 x 1 0 ^ 2-64 x l O b D E N S I T Y (3/ c^ ) 3 - 5 0 x 1 0 " * 1-27 x l O " " C O M P R E S S I O N R A T I O 3 ^ 6 2 T E M P E R A T U R E ( D E G R E E S K ) 2 9 3 8 ' 9 6 X I 0 3 E F F E C T I V E A D I A B A T I C E X P O N E N T 1-6 7 1-63 N A (ch," 3) 5-2 8 xlO n 2-1 2 x l d ' N, ( c h ^ ) 8-33 x l d S Ne ( c m - 3 ) 8-33 x 1 C f 5 - 4 9 x IO" k 2 - 2 0 x l O " 5 5-8 K lO* l - 5 | x I O " 5 - l - 5 0 x l O " b T a b l e I. S h o c k Wave P a r a m e t e r s f o r P i = 16 T o r r . T h e l e n g t h 6 o f t h e p a t h o f t h e l a s e r beam i n t h e t e s t s e c t i o n was m e a s u r e d a s 2.45 cm. ' A t y p i c a l o s c i l l o g r a m i s shown i n F i g u r e 1 8 . T h e l o w e r t r a c e i s t h e s c h l i e r e n s i g n a l ; t h e u p p e r t r a c e i s t h e i n t e g r a t e d s i g n a l , w h i c h i s a l i n e a r a n a l o g u e f o r t h e d e n s i t y \ 53 p r o f i l e . The voltage scale of the upper trace is ca l ib ra ted also for dens i ty . S im i l a r l y , the time axis is ca l i b ra ted also for distance along the shock tube. SIGNAL VOLTAG E(-VE) t 2<OOmv(2-36x 1 0 ' e m f 1 0 0 m V ^ T DENSITY ANALOGUE SIGNAL SCHL I E REN SIGNAL l/JS ( 0 . 2 9 c r * ) C A L I BRAT I O N (J~\ = 1-08 x 1 0 b m V / RADIAN ^TIME (SHOCK WAVE DISPLACEMENT) Figure 18. Schl ieren signal and density analooue signal for a Mach9-9shock wave in argon, Pi = 16 torr Four osci l lograms were taken for Pj = 16 t o r r . The average value of An was, from equation (28), An = 1.45 x 1 0 " 5 . This d i f f e r s from the expected value, An = 1.51 x 1 0 - 5 , by about 4%. There were three major contr ibut ions to this d i s -crepancy, each one d i f f i c u l t to estimate q u a n t i t a t i v e l y . There were f i r s t l y the inaccuracies introduced by the e l e c t r o n i c s . In p a r t i c u l a r , the high pass f i l t e r may have had an e f f ec t on the lower frequency Fourier components of the s ch l i e ren s i gna l , 54 thereby sh i f t i n g the e f f e c t i v e c a l i b r a t i o n constant of the integrator for that s i gna l . Secondly, the flow f i e l d of the shock wave may not have been uniform across the whole of the diameter of the shock tube traversed by the laser beam. Laminar flow e f fec t s would serve to e f f e c t i v e l y decrease 6. Las t ly , the measured value of Pi may have been in e r ro r , causing an error in the ca l cu l a t i on of the expected value of An. In view of these p o s s i b i l i t i e s , the agreement of p red ic -tion and measurement seems to be within acceptable l i m i t s . 5.4 Qua l i ta t i ve Invest igat ion at Pi = 2 Torr As Pi is decreased, the shock wave ve loc i t y increases. The compression r a t i o , temperature and s p e c i f i c enthalpy increase, and ion iza t ion becomes appreciable. For Pi = 2 t o r r , the negative contr ibut ion to (n - 1) from the e lectrons is greater than the pos i t i ve contr ibut ion from the neutral atoms. The approximate e lectron r e f r a c t i v e index is (n - 1) ~ - 1 0 - 5 . For the neutral atoms, (n - 1), ~ 5 x 1 0 " 6 . For the ions, a (n - 1). ~ 0.5 x 1 0 " 6 . For the undisturbed test gas, (n - 1) ~ 0.7 x 10~ 6 . The change in r e f r a c t i v e index between the assembly of neutral p a r t i c l e s in the shock wave and the o r i g ina l test gas is then approximately 4 x 1 0 " 6 . For the e lectron assembly, the change in r e f r a c t i v e index is approximately - 1 0 " 5 . The e f fec t of the ions on the r e f r a c t i v e index change is smal l . Figure 19 shows an osci l logram of the sch l ieren signal for a Pi = 2 torr shock wave. No attempt was made to integrate 55 the s ignal e l e c t r o n i c a l l y . The c a l i b r a t i o n curve gradient was 49 mV/us (7704 C.R.O. t race) . The shock wave v e l o c i t y , from Section 5.1, was 5.01 mm/us, Mach 15.7. S I G N A L V O L T A G E ( - V E ) I • 1 T -X k-> T I M E ( S H O C K W A V E \ D I S P L A C E M E N T ) C A L I B R A T I O N : ( 4 j - \ = 0 - 9 9 5 mV/ R A D I A N Figure 19. Sch l ieren signal for a Mach 16 shock wave in argon, Pi = 2 t o r r . A pos i t i ve voltage de f l ec t i on ind icates a sch l ieren de f l ec t i on due to the passage in the d i r ec t i on of the shock wave of an increase in r e f r ac t i ve index. That is to say, the neutral atom contr ibut ion def lec t s the signal p o s i t i v e l y , while the change in r e f r ac t i ve index due to the e lectron assembly of the shock wave plasma de f lec t s the signal negat ively. In the osc i l logram of Figure 19, the e a r l i e s t s ignal de f l ec t i on above the noise can be seen in the second g ra t i cu le square through which the trace runs. This is a pos i t i ve d e f l e c t i o n . It is almost immediately followed by a negative s ignal d e f l e c -t i on . Making an estimate of the area of these s i gna l s , the r e f r a c t i v e index jumps corresponding to the f i r s t and second \ 56 def lec t ions respect ive ly are about + 2 x 10~ 6 and -6 x 1 0 - 6 . These are roughly half the values predicted above, which is probably accounted for part ly by the l i ke l i hood of some cance l l a t i on of the opposing e f fect s of the e lectrons and the neutral atoms. density causes a sch l ieren signal which fol lows the signal due to the neutral atoms by about 1 ys . This corresponds to a length of 5 mm on the shock wave ax i s . The pos i t i ve and nega-t i ve r e f r a c t i ve index gradients almost c e r t a i n l y overlap to a cer ta in extent, and so a deta i led d i s t i n c t i o n of e lectron and neutral pa r t i c l e density p ro f i l e s is not poss ib le . The form S I G N A L i V O L T A G E (-ve) / \ A The trace of Figure 19 also shows that the e lectron * T I M E P A R T I C L E D E N S I T I E S 1- i • • + I ~J~ ! T | % t N 1 =4'3K I O M + " j " J \ 1 4- N e = N j - 1 x io' t H 5 rntn k - t -• N E U T R A L A T O M S - E L E C T R O N S - ^ D I S T A N C E + I O N S Figure 20. Pa r t i c l e d i s t r i bu t i on s implied by the sch l ieren signal for the Mach 16 shock wave in argon, Pi = 2 t o r r . of the pa r t i c l e d i s t r i b u t i o n s implied by the osc i l logram indicated in Figure 20. The reso lu t ion obtained supports the expectation (Section 2.2) of a spat ia l reso lut ion of approximately 2 mm. Chapter 6 CONCLUSIONS 6.1 General Conclusions The attempt to construct a sch l ieren densitometer u t i l i z i n g the wedge technique was success fu l , agreement having been demonstrated within the l im i t s of accuracy between the density measured and the density predicted by theory. Refrac-t i ve index changes of the order of 10~ 5 were measurable by the device without d i f f i c u l t y . A spat ia l reso lut ion of approximately 2 mm was expected, and found to be achieved. The a b i l i t y of the densitometer to track both pos i t i ve and negative r e f r a c t i v e index changes was also demonstrated. The pr inc ipa l advantage of the wedge technique over the conventional knife edge techniques l i e s in the s imp l i c i t y and f i d e l i t y of the opt ica l system i t employs. Loss of accuracy due to geometric complexit ies is n e g l i g i b l e , and the operating equation (28) of the device is based on very few assumptions. The wedge need not even be l inear over the whole region of i n ter sec t ion of the laser beam; i t needs to 58 \ 59 be l i near only over the sch l ieren de f l ec t i on of ind iv idua l rays. D i f f r a c t i on problems are not encountered as they are with knife edge techniques. 6.2 Future Work There are several ways in which the device might be improved. A double wedge technique s im i l a r to the one used by Potter (References 11 & 12) could be introduced. Here the beam is divided into two channels which are ident i ca l but for opposite senses of the wedge gradient. As long as the two channels are exactly balanced, when the i r s ignals are subtracted the plasma l i gh t contr ibut ions cancel . The remain-ing sch l ieren signal measures the r e f r a c t i v e index gradient with twice the s e n s i t i v i t y that would be obtained using a s ing le wedge. The s e n s i t i v i t y might also be improved by increasing the distance from the test section to the wedge. However, the advantage gained by going th is is severely o f f se t by the expansion of the beam's cross sect ion due to i t s natural divergence. Distances of between 1 and 2 metres are appropriate. The appropriate wedge gradients are then not great enough to create d i f f r a c t i o n e f f e c t s . It should be noted that th is arrangement of fers a more compact system than most conventional techniques al1ow. The spat ia l reso lut ion could be improved by reduc-tion of the diameter of the beam by opt i ca l means, such as \ \ 60 is described in Reference 6. However, an attempt to reduce the beam diameter was made and proved unsuccessful . This indicated that a great deal of e f f o r t would be necessary to make a s i g n i f i c a n t improvement. The instrument could be adapted to measure the change in r e f r ac t i ve index across s tat ionary objects by sweeping the beam through the object in quest ion. A possible opt ica l arrangement to perform this funct ion is suggested in Reference 1. \ BIBLIOGRAPHICAL REFERENCES 1. Boye Ahlborn and Christopher A.M. Humphries, Rev. S c i . , Instr. , Vol . 4J7_, No. 5 ( 1 976 ) . 2. Michiru Yasuhara, Kenji Yoneda and Susumu Sato, J . Phys. Soc . Japan , 3_6 , 555 ( 1 974) . 3. E'.L. Res ler, J r . and M. Scheibe, J . Acoust, Soc. Am., 27, 932 (1955). 4. Jerome Daen and P.C.T. de Boer, J . Chem. Phys., 3_6, 1222 ( 1 962 ) . 5. P.C.T. de Boer, Rev. S c i . Instr . , 3_6 , 1 1 35 ( 1 965 ). 6. John H. Kiefer and Robert W. Lutz, Phys. F lu id s , 8, 1393 ( 1 965) . 7. John H. Kiefer and Robert W. Lutz, J . Chem. Phys., 44, 658 (1966). 8. Robert W. Lutz and John H. K ie fe r , Phys. F lu id s , 9, 1638 ( 1 966). 9. Laurence S. H a l l , Rev. Sc i . Ins t r . , 37, 1735 (1966). 10. G. Dodel and W. Kunz, Applied Opt ics , 1_4, 2537 ( 1 975). 11. Michael U. Potter, M. Sc. Thes is , Department of Physics, * The Univers i ty of B r i t i s h Columbia, 1967. 12. M.U. Potter and B. Ahlborn, AIAA Journa l , 6, 2227 (1968). 61 \ 62 13. Jean-Paul R. Hum', Ph.D. Thes i s , Department of Physics, The Univers i ty of B r i t i s h Columbia, 1970. 14. P. Redfern and B. Ahlborn, Can. J . Phys., 5_0_, 1 771 (1 972 ) 15. Ya. B. Ze l 'dov ich and Yu. P. Ra izer, Elements of Gas-dynamics and the C l a s s i c a l Theory of Shock Waves, Academic Press Inc., 1968. 16. A.G. Gaydon and I.R. Hurle, The Shock Tube in High-Temperature Chemical Physics, Chapman and Hall Ltd. London, 1963. 17. Boye Ahlborn, Can. J . Phys., 5_3, 976 (1 975). 18. K.S. Dre l l i shak, C F . Knopp and A l i Bulnet Campbell, Phys. F lu id s , 6, 1280 (1963). 19. V. Hermoch, Czech. J . Phys., 1_2 , 939 (1 970). 20. Joseph 0. H i r sch fe lder , Charles F. Curt i ss and R. Byron B i rd , Molecular Theory of Gases and Liquids, John Wiley and Sons Inc., 1954. 21.. Ralph A. Alpher and Donald R. White, Phys. F lu id s , 2, 153 (1959). 22. Ralph A. Alpher and Donald R. White, Phys. F lu id s , 2, 1 62 ( 1 959 ) . 23. Lyman Sp i tzer , J r . , Physics of F u l l y Ionized Gases, John Wiley and Sons Inc., 1962. \ APPENDIX A Reference 1: Boye AhlOTrn and Christopher A.M. Humphries, "A Quant itat ive Schl ieren Densitometer Employing a Neutral Wedge," Rev. S c i . Ins t r . , Vol . 47, No. 5 (1976). 63 APPENDIX A (LEAVES 64-67) NOT MICROFILMED FOR REASONS OF COPYRIGHT. PLEASE CONTACT THE UNIVERSITY FOR FURTHER INFORMATION. UNIVERSITY OF BRITISH COLUMBIA ATTENTION: LAURENDA DANIELLS SPECIAL COLLECTIONS DIVISION THE LIBRARY 2075 WESBROOK PLACE VANCOUVER, B.C., CANADA V6T 1W5 /_\V ^ A A hectic Sfr . Quantitative schlieren densitometer employing a neutral Of^'^ • density wedge Boye Ahlborn and Chr is topher A . M. Humphr ies Department of Physics, University of British Columbia, Vancouver, British Columbia V6T 1W5, Canada (Received 29 September 1975; in final form, 19 January 1976) In this schlieren system the probing laser beam is attenuated by a neutral density wedge (instead of the usual knife edge) so that the intensity is proportional to the deflection angle, which in turn is proportional to the density gradient. Time integration of the attenuated probe beam intensity yields the absolute refractive index variation across moving objects. The device is tested by measuring the density jump across a Mach 9 shock wave. INTRODUCTION Schl ieren techniques have been used in shock wave studies for a long time to detect density steps and to measure density variat ions. 1 " " ' l n these diagnostic sys-tems one uses the fact lhat an opt ica l density gradient (schliere) deflects a probing light beam and the deflection angle a is direct ly proport ional to the gradient of the refractive index. In the standard arrangement the prob-ing beam is init ial ly b locked by a knife edge or slit. When the density gradient is introduced, the beam is deflected to pass the knife edge and reach a detector. If the beam deflection is small compared to the beam width at the knife edge, the deflection angle a wi l l be proport ional to the amount of flux wh ich passes the edge. T o obtain a signal wel l above the noise, one would l ike to have as large an angle a as possible, so that the measuring beam wou ld have to be quite wide in turn. A wide signal beam, however, implies a low spatial resolution. In conven-tional schl ieren systems one can find an opt imum trade-off between spatial resolution and angular deflection a . 5 The uncertainty of the angular measurement is essen-tially due to diffraction of the slit or knife edge cm-ployed in these schl ieren systems. Therefore, to min i -mize the dif fract ion effects, one ought to el iminate the slit or knife edge. W e have developed a quantitative schlieren system in wh ich the knife edge is replaced by a neutral density wedge wh ich essentially el iminates the diffraction ef-fects. T he flux behind the wedge is proportional to the location of the center of gravity of the probing beam, and varies therefore with the deflection angle a. T h e beam shape is unimportant, as was discussed in a previous appl ication o f neutral density wedges to the measure-ment of small shifts of wide spectral l ines . 8 , 7 APPARATUS A sketch of the apparatus is given in F ig . 1. T h e probing beam is provided by a Spectra Physics 115 H e -N e laser of 0.5-mW beam power and 6328-A wavelength. The neutral density wedge N with transmission vary ing in the X d irect ion is produced photographical ly, as re-ported prev ious ly . 0 The lens L focuses the deflected beam to roughly the same area on the photomult ip l ier ( P M ) independently of the deflection a, so that the signal varies with deflection only on account o f the neutral wedge. L im i ted use was made o f the dynode cha in , the seventh dynode being used as the anode in order to re-duce the signal-to-noise ratio. A narrow-band inter-ference filter I is interposed to el iminate plasma light emitted during the measuring process. DENSITY STEPS ACROSS MOVING OBJECTS The schl ieren deflection a o f a ray of light passing through a length 5 of a material wi th transverse refrac-tive index gradient dn/dx is given by _ S dn n dx which f c r laboratory plasmas (n — 1) may be taken as dn (1) a ~ 8 -dx (2) If the deflection is caused by material mov ing at uni form velocity v — dx/dt, one has 8 dn v dt (3) Hence, one finds the change of refractive index A/i be-tween t = -so and tl s imply by integrating Eq . (3): a(t) dt = —Anitt). (4) Th i s time integration may be carr ied out electronical ly so that the variation of the index of refract ion and, hence, the density can be measured immediately. CALIBRATION Our apparatus is designed to measure density steps 'across shock waves. The observat ion region P (F ig . 1) is a section of a 2.5-cm-diam shock tube with paral lel windows. Shock waves propagate in the ^ d i r e c t i o n . The 5 7 0 R e v . Sci. Insirum., V o l . 4 7 , N o . 5, M a y 1 9 7 6 C o p y r i g h t © I976 A m e r i c a n I n s t i t u t e o f P h y s i c s 5 7 0 6k I H e N e shock M 2 FIG. 1. Schlieren system with neutral wedge N. strong density gradient of a nonionizing shock front deflects the beam as shown by an angle a, which varies as the shock moves through the test section at P. To calibrate the device, the laser beam is artificially deflected (with no object at P) by a known angle «, and the variation of intensity is recorded. The calibrated deflection is accomplished with a rotating prism carrying mirrors M, and M3, and the stationary mirror M2: The angle between M, and M 2 (95°), the distance between rotation axis 0, signal beam axis, and the position of M 2 are carefully chosen so that the beam appears to be ro-tating about P at 4 times the angular frequency v„ of the prism. Figure 2(a) shows the resulting photomultiplier signal as depicted on an oscilloscope with a sweep fre-quency <VQ. V, corresponds to the voltage signal for the light path shown in Fig. 1 (solid line). The narrow u— 1 0 m s e c (a) 2 p sec FIG. 2. (a) Photomultiplier signal obtained when the signal beam is chopped by rotating prism (V,) and deflected by mirrors M„ M2. and M3 (V2). (b) Calibration signal obtained with rotating prism. Time scale ( is also calibrated in deflection angle a for / = 7.17 cm. spike Vo is the calibration signal and is shown on an expanded time scale in Fig. 2(b). The time scale of Fig. 2(b) can be converted into an angle scale if v0 and the distance / between P and the wedge are known. For the measurements we used the shaded region, where the transmission varies approximately linearly with angle a. This gives a range of about 1.5 x 10-3 rad for the schlieren deflection. During measurements the prism is stopped and the signal beam has the path shown by the solid line in Fig. I. The wedge is laterally adjusted so that the undeflected signal produces a voltage V0> indi-cating that the beam lies in the middle of the linear range. Note that the rotating calibration beam producing the voltage in Fig. 2(a) is less intense than the interrupted direct beam, which produces the voltage V,. This arises from reflection losses at the surfaces M,, M2, and M3. Therefore, the actual deflection measurement must be carried out with a lower intensity. This is accomplished by interposing the neutral filter F, which reduces the laser beam by the ratio V,JVt. With this precaution the variation of intensity of the measuring' beam V - V0 = AV can be transformed into a schlieren deflection angle, a - BAV, fi = ^ = 1.2 x 10-dV mV (5) so that the change of refractive index is found with Eqs. (4) and (5): An{t) =-B \AVdt. (6) Note that the sensitivity of the device, expressed by B, can be changed by varying the distance /. The integral (4) yields the increase of the refractive index up to the time /, which can be correlated to a position within the moving density gradient (shock front) as AX = vAt. Hence the local refractive index profiles can be found in the optically inhomogeneous region with a resolution A A' of the width of the probing laser beam. In our case AX is typically 1 mm. EXPERIMENTAL TEST OF THE DEVICE To test the schlieren wedge system the density varia-tion across shock waves was measured. The shocks were generated with a gaseous detonation driver de-5 7 1 R e v . S c i . I n s t r u m . , V o l . 4 7 , N o . 5, M a y 1 9 7 6 S c h l i e r e n d e n s i t o m e t e r 5 7 1 65 scribed previously.8 The initial pressure of the driver gas, an equimolar mixture of C 2 H 2 plus 0 2 , was 350Torr. The shock wave velocity was measured for various pres-sures of the argon test gas by using a smear camera. The shock tube reaches Mach 17.9 in 1 Torr of argon test gas, Mach 14.3 in 4 Torr argon, and Mach 9.2 in 16 Torr argon. The dependence of the refractive index upon the plasma composition is well understood. The subject is treated by Alpher and White. 9 For argon gas the relationships at the wavelength of 6328 A are Na= 1 + 1.04 x 10- 2 3«„-, N, = 1 + 0.70 x 1 0 - 2 3 « ( , Ne = 1 - 18.0 x 1 0 - 2 3 « e , where «„, nt, and ne are the refractive indexes, and N„, Nf, and Ne are the number densities in c m - 3 of argon atoms, ions, and electrons, respectively. The partial refractive indexes are summed up and related to the total refractive index of the gas mixture n by n - 1 = £ (« , - 1). (7) j In the test experiment, shocks of Mach 9.2 in argon of Po = 16 Torr were studied. Figure 3(a) is the schlieren wedge signal recorded by P M , and calibrated in units of the deflection angle. Figure 3(b) is the integrated signal calibrated to give the jump of the index of refraction. The time is also calibrated as distance within the shock given by AA" = u s l l 0 c : k Af. With standard shock wave theory one then finds for the shock-compreSsed gas a pressure p2 - 2.0 x 10" dyn/cm 2, an equilibrium tem-perature of T 2 = 7500 K, and the composition Na2 2 3 t FIG. 3. Schlieren wedge signal for Mach 9.2 shock in 16 Torr argon, (a) Deflection angle a as a function of time I or position \X within the shock, (b) Increase of index of refraction. 5 7 2 R e v . S c i . I n s t r u m . , V o l . 4 7 , N o . 5 , M a y 1 9 7 6 FIG. 4. Arrangement to sweep the probing beam through a sta-tionary object ftP. The glass blocks G, and C 2 rotate synchronously in opposite directions, thus producing a constant sweep speed for small angles y. = 2.2 x 1018 c m " 3 and Na = N i 2 = 9.7 x 1014 cm" 1 . The contribution of the electrons to the refractive index is negligible in this case; that is, for practical purposes the shock is nonionizing. The refractive index for the shock-compressed gas is nt = 1 + 2.25 x 10 - 5. Sub-tracting the refractive index for the gas in the initial stage,/i, = 1 + 5.5 x 10_G, one finds that the theoretical increase across the shock is An = 1.7 x 10"5, (8) which causes the schlieren deflection and is measured in the experiment. Entering this result into Eq. (3), written in the form aAt = (S/v)An, and using S = 2.5 cm and v = 2.94 x 105 cm/sec, we have aAt = 1.4 x i d " ' 0 rad sec. (9) This value should be compared with the area under the experimental schlieren deflection curve [Fig. 3(a)], which was measured to be 1.2 x 10 - 1 0 rad sec ( ± 1 0 % ) . The agreement is reasonably good considering that the calculations depend critically on the initial filling pres-sure. Since no absolute calibration of the pressure manometer was available, no error margin for the ex-pected value in Eq. (8) could be quoted. MODIFICATION FOR STATIONARY OBJECTS In order to obtain the jump of the index of refraction, it was necessary to transform the spatial variation into a temporal variation so that the integration could be carried out. This is quite simple for a moving object travelling with constant velocity v = clX/dt through the observation region. For stationary objects one can sweep the measuring beam through the object rather than sweep the object through the measuring station P. If the sweep velocity vs is known, one can again replace dX by vs dt in Eq. (3) and find the refractive index through the integration of Eq. (4). A possible arrange-ment for sweeping the observation beam through the stationary object (such as an arc) is shown in Fig. 4. In analyzing cylindrical (or nonplanar) objects, one must account1" for the fact that the probing beam passes through layers of different density gradients which may cause sections of the probing beam to converge or diverge. S c h l i e r e n d e n s i t o m e t e r 5 7 2 66 ACKNOWLEDGMENTS We thank B. Armstrong for valuable comments on the manuscript. This work was supported by a grant from the AECB of Canada. 1 E. L. Resler and M. Schcibe, J. Acousl. Soc. Am. 27, 932 (1955). ! J . H. Kiefer and R. W. Lutz. Phys. Fluids 8, 1393 (1965). 3 A. G. Gaydon and I. R. Hurie, The Sliock Tube in High Temperature ChemicalI'hysics (Chapman and Hail, London, 1961). 4 M. Yasuhara, K. Yorieda, and S. Sato, J. Phys. Soc. Jpn. 36, 555 (1974). * C. Dodel and W. Kunz. Appl. Opt. 1 4 , 2537 (1975). ° B. Ahlborn and A. J. Barnard, AIAA J. (Am. Inst. Aeronaut. Astronaut.) 4 , 1136 (196b). 7 B. Ahlborn and R. Morris, J. Quant. Spectrosc. Radiat. Transfer 9 , 1519 (1969). • P. Redfern and B. Ahlborn. Can. J. Pliys. 5 0 , 1771 (1972). » R. A. Alpherand D. R. White, Phys. Fluids 2, 162 (1959). " P. C T. de Boer. Ph.D. thesis. University of Maryland, 1962 (University Microfilm, 313 First St., Ann Arbor, Ml, Order No. 63-7137). 5 7 3 R e v . S c i . I n s t r u m . , V o l . 4 7 , N o . 5 , M a y 1 9 7 6 S c h l i e r e n d e n s i t o m e t e r 67 5 7 3 \ APPENDIX B THE NEUTRAL DENSITY WEDGE The important c h a r a c t e r i s t i c of a photographic emulsion is i t s H & D curve. This is a plot of the density D of the negative as a funct ion of the logarithm of the exposure. The exposure E is re lated to the in tens i t y I and the time t of exposure by E = It (B.l ) The transmission T of the negative is re lated to i t s density by D = - l o g 1 0 T (B.2) A typ ica l H & D curve is shown in Figure B. l . The shape of the curve depends upon the type of developer used and the time of development. The range of l i n e a r i t y of the H & D curve is re ferred to as the gamma of the f i l m . It covers the useful range of exposures for ordinary f i lms . In th is region 68 69 D = k + ylog ! o E (B. 3) Subst i tut ing equation (B.2) into equation (B.3) gives: T = k' E " Y (B.4) where l o g 1 0 k1 - -k (B.5) The wedge was produced on a Kodak photographic p la te , type 649, developed 5 minutes in Kodak D 19 developer. A \ 70 one-dimensional l i nea r gradient of l i gh t i n tens i t y was pro-jected on to the plate by a half shadow technique. As can be seen from equation (B . l ) , th i s brought about a l i nea r var i a t ion of E. Equation (B.4) shows, then,that the va r i a t i on of transmission T of the plate with distance should not be l i near but should vary as distance raised to the negative power of y• However y is a small number and so the trans-miss ion-distance curve should be approximately l i n e a r . This was demonstrated by the c a l i b r a t i o n curves (e.g. Figure 12), which indicated that the wedge was approximately l i near between transmissions of 25%'and 75%, with a gradient of 64%/mm. APPENDIX C OPTICS OF THE CALIBRATION SYSTEM The geometry of the mirror system is i l l u s t r a t e d in Figure C l . The rotat ing mirror assembly is drawn in two pos i t ions : in the pos i t ion drawn with f u l l l i n e s , the mirrors pass the beam in such a way that i t emerges undeflected from i t s o r i g ina l path; in the pos i t ion drawn with dotted l i n e s , the mirror assembly is rotated by a small angle 6 from the above pos i t i on . Rays indicated by f u l l l ines show the r e f l e c -t ion of l i gh t by the system when the mirrors are in the pos i t ion drawn with f u l l l i n e s . Real and v i r t u a l rays indicated by dashed l ines show the e f fec t on the beam of the rotat ion of the f i r s t mirror by 6. Here i t is hypo thetica11y assumed that the second mirror of the prism assembly has not rotated. The path of rays re f l ec ted by both mirrors when rotated by 6 is shown by dotted l i n e s . The dashed ray is deviated through an angle 26 by i t s r e f l e c t i o n at the f i r s t mirror. It appears to diverge from the undeflected ray from a point X a f ter further re f l e c t i on s at the second and th i rd mirrors . The real point of divergence 71 Figure C.l Optics of the mirror system of the c a l i b r a t i o n device. 73 is Y. This is s l i g h t l y displaced from B because the mirror rotates about 0 and not B. It is easy to see that C'Y = C'Z ( C l ) It fol lows from equation ( C l ) that D'Z = D ' C '. + C'Y (C.2) Applying the law of r e f l e c t i o n to the r e f l e c t i o n of the dashed ray at D', i t fol lows that D'X = D'Z (C.3) Hence, subst i tut ing equation (C.2) in equation (C.3), D'X = D'C' + C'Y' (C.4) However, the actual r e f l e c t i o n of the rotated beam is at D", as indicated by the dotted ray. The rota t ion of the mirror surface by 8 causes the dotted and dashed rays to diverge from each other by an angle 26. Hence, the dotted and f u l l rays diverge by 46. Now i f 6 i s a small angle, the distance between D' and D" may be neglected. Then, again for small 8, D"A = iD'X ( C 5 ) 74 From equation (C.4), this becomes D" A = i D 'C + C'Y ( C 6 ) F i n a l l y , i f DD1 and BY are both taken as smal l , equation (C,6) becomes DA = i(DC + CB) = DC (C.7) (C.8) This is the resu l t quoted in Section 2.3. The quant i t ies that have been taken as small are a l l of f i r s t and higher order in 6. The de f lec t ions encountered in th is experiment are of the order of 10~ 4 radian, and so the approximations are j u s t i f i e d . \ APPENDIX D THE SHOCK TUBE APPARATUS The general layout of the apparatus in which the shock wave plasma was produced is shown in Figure D.l. S P A R K GAP V A C U U M P U M P 0 0-2400 T O R R -0 A I R A C E T Y L E N E ] O X Y G E N D E N S I T O M E T E R - 0 O - 4 O T O R R ^ 0 O I 6 O T O R R - ® — - A R G O N -0O-2OOO/; (PI R A N Y l ) - ® A IR V A C U U M ( ) < — P U M P Figure D. l . General layout of shock tube apparatus. 75 \ 76 The detonation chamber and the mixing tank were both about three l i t r e s in volume. The 0 - 2400 torr pressure gauge monitored the pressure in the mixing tank and the f i l l i n g pressure when the detonation chamber was f i l l e d ; the 0 - 4 0 torr pressure gauge was used to monitor the pressure in the detonation chamber when i t was being pumped down. S im i l a r l y the 0 - 160 torr and the 0 - 4 0 torr pressure gauges measured the pressure of the argon test gas, while the Piranyi gauge monitored the evacuation of the shock tube. In order to minimize the movement of the test sect ion due to the detonation, the detonation chamber and the shock tube were mounted on separate tab les . The shock tube was f i rmly f ixed to i t s tab le , but the detonation chamber and i t s support structure were allowed to s l i de on the surface for a cer ta in distance (beyond which the bellows would be over-extended). A lead weight of 50 lbs wt was appended to the support structure of the detonation chamber in order to increase f r i c t i o n and to reduce the reco i l v e l o c i t y . The tables were laden with bricks and sandbags for reduction of v ib ra t i ons . The oxygen-acetylene mixture was ignited by an e l e c t r i c spark produced at the spark gap by the discharge of a 1.72 yF capac i tor . A charging voltage of 18 kV was s u f f i c i e n t to pro-vide a spark that would set o f f the detonation. The c i r c u i t diagram of the i gn i t i on system is given in Reference 13. \ \ APPENDIX E APPLICATION OF THE R.C.A. 931A PHOTOMULTIPLIER In normal use, the ten dynodes of the photomult ip l ier tube serve as ampl i f i ca t ion stages for the e lectron current, which or ig inates at the photocathode. The ampl i f ied signal i s picked up by the anode and dr ives a load r e s i s t o r , thus producing a voltage output. Each photoelectron produced at the photocathode eventual ly produces a pulse of output voltage, and the output voltage waveform is the sum of the voltage pulses of a l l the photons of the inc ident l i gh t beam. Consider a l i gh t beam of constant in tens i ty and wave-length. Electrons are l i bera ted from the photocathode at a constant ra te , and with the same energies. This leads to the expectation of a constant voltage s i gna l . But there is a s t a t i s t i c a l var ia t ion in the number of e lectrons in the cascade produced for each photoelectron by amp l i f i ca t i on at the dynodes. Because of th is f l u c t u a t i o n , the output voltage has a noise component superimposed on the constant voltage. And the signal to noise ra t io becomes less favourable with each successive stage of dynode amp l i f i c a t i on . 77 78 In the sch l ieren densitometer, the laser beam intens i ty had to be reduced by a neutral density f i l t e r to a level which was su i tab le for the photomul t ip l ier . The excess of ava i lab le l i g h t made poss ible a modi f icat ion of the normal use of the photomul t ip l ie r . A lesser number of dynode stages were employed, leading to an improved signal to noise r a t i o . The loss of amp l i f i ca t i on was compensated by changing the neutral density f i l t e r to allow an increase in the f r a c t i o n of incident l i g h t f a l l i n g on the photocathode. Using 6 dynode stages, the signal to noise ra t io was improved from 4.7 to 95. The seventh dynode was used as the anode. The c i r c u i t diagram is shown in Figure E . l . O -700 V I O O K I O O K I O O K Figure E.l C i r c u i t diagram for operation of the R.C.A. 931A photomult ip l ier with l imited dynode a m p l i f i c a t i o n . \ 79 The dynode re s i s t o r chain current is approximately 1 ma. The photomult ip l ier c h a r a c t e r i s t i c curve (output voltage vs l i gh t in tens i ty ) should be l i nea r i f the tube current never exceeds one tenth of t h i s . In these circumstances, the output voltage must not exceed 100 mV. The photomult ip l ier c h a r a c t e r i s t i c curve was measured. It is shown in Figure E.2. O U T P U T ' V O L T A G E ( m y ) . 6 0 0 -5 0 0 H 4 0 0 -3 0 0 -2 0 0 -I O O -O o Figure E . 2 O 0-2 0-3 0 - 4 0 - 5 0 6 0 7 > L I G H T I N T E N S I T Y ( A R B I T R A R Y U N I T S ) Charac te r i s t i c cut-ve of an R.C.A. 931A photomul-t i p l i e r with 6 dynode amp l i f i ca t i on stages in use. During the experiment, the photomult ip l ier wa~s oper-ated on a non- l inear part of i t s c h a r a c t e r i s t i c . But i t was 80 the var ia t ion in signal due to the sch l ieren e f f ec t that was being measured, and this was small enough that the charac-t e r i s t i c was e f f e c t i v e l y l inear over i t s range. The noise was increased by the buffer ampl i f ie r c i r c u i t . Considering the voltage va r i a t i on due to the sch l ieren e f f ec t as the .s ignal , experimental signal to noise ra t io s were about 20. 

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