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Quantitative schlieren densitometer Humphries, Christopher A. M. 1976

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A  QUANTITATIVE  SCHLIEREN  DENSITOMETER  <  by  CHRISTOPHER B.A., THESIS THE  A.M.  Oxford  SUBMITTED  University, 1973  IN  REQUIREMENTS MASTER in  the  HUMPHRIES  PARTIAL  FULFILLMENT  FOR THE DEGREE OF OF  SCIENCE  Department of  PHYSICS  We a c c e p t to  this  thesis  the, r e q u i r e d  THE UNIVERSITY  conforming  standard  OF BRITISH  September, @  as  COLUMBIA  1976  C h r i s t o p h e r A. M. Humphries  In  presenting  this  an a d v a n c e d  degree  the L i b r a r y  shall  I  f u r t h e r agree  for  scholarly  by h i s of  this  written  thesis at  the U n i v e r s i t y  make  it  It  The U n i v e r s i t y o f  British  20 75 W e s b r o o k P l a c e V a n c o u v e r , Canada V6T 1W5  Columbia  requirements  for  I agree  r e f e r e n c e and copying of  this  that  not  copying or  for  that  study. thesis  by t h e Head o f my D e p a r t m e n t  financial gain shall  of  the  B r i t i s h Columbia,  is understood  permission.  Department  of  for extensive  p u r p o s e s may be g r a n t e d  for  fulfilment of  freely available  that permission  representatives. thesis  in p a r t i a l  or  publication  be a l l o w e d w i t h o u t my  ABSTRACT  A schlieren densitometer 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  In c o n v e n t i o n a l light  beam  experiment neutral tages,  i s measured described  here,  d e n s i t y wedge.  t h e k n i f e e d g e was  This  t h e main one being  innovation  by a p h o t o m u 1 t i p 1 i e r .  photomultiplier  f o r the density  several  profile  due t o  voltage  of the  a l i n e a r analogue  of a moving object  or medium.  t h e s h o c k f r o n t o f a Mach 9 p l a n e  s h o c k wave i n  The r e s u l t s were i n s a t i s f a c t o r y agreement with  calculated  advan-  by t h e wedge,  The output  integrated to give  by a  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  change across argon.  was t i m e  replaced  brought  intensity, attenuated  probing  In t h e  the e l i m i n a t i o n o f problems  was m o n i t o r e d  device  employed.  by a k n i f e edge t e c h n i q u e .  T h e beam  The  i n which a  systems, the s c h l i e r e n d e f l e c t i o n of a  diffraction.  signal  was c o n s t r u c t e d  from  s h o c k wave  theory.  values  TABLE OF CONTENTS  Page ABSTRACT  '.  LIST OF TABLES LIST OF ILLUSTRATIONS.  •  ii  .. . •  v  .'  vi  ACKNOWLEDGMENTS  ix  Chapter 1  2  3  INTRODUCTION  .  1.1  General  1.2  Conventional  1.3  The Wedge Technique  1  Introduction  1  S c h l i e r e n Systems  The S c h l i e r e n E f f e c t  2.2  E f f e c t of the Wedge  2.3  The C a l i b r a t i o n  System  THE APPARATUS  2 4  OPTICS OF THE DENSITOMETER 2.1  . . . . . . .  "...  8. 8 10  .  16 2  0  :  3.1  C o n s t r u c t i o n of the Apparatus  3.2  Use of the Apparatus  iii  20 .  2  8  Chapter 4  5  Page PROPERTIES OF THE SHOCK WAVES' 4.1  Shock Wave Plasma  4.2  The R e f r a c t i v e  4.3  Numerical  32  Parameters  . . . . . . .  32  Index of a Plasma  37  Calculations  43  EXPERIMENT AND RESULTS  47  5.1  V e l o c i t y Measurements.  . . .  47  5.2  Shock Wave Parameters.  . .  51  5.3  Density  5.4  Qualitative  Measurements  at P i = 16 t o r r  Investigation  . . .  51  at  Pi = 2 t o r r 6  CONCLUSIONS.  54  .  .  6.1  General  Conclusions.  6 .2  Future Work  58 59  BIBLIOGRAPHICAL REFERENCES APPENDICES A  58  61  Reference'1: Boye Ahlborn and C h r i s t o p h e r A.M. Humphries, "A Q u a n t i t a t i v e S c h l i e r e n Densitometer Employing a Neutral Wedqe," Rev. S c i . I n s t r . , V o l . 47, No. 5 (1976) . . . .  63  B  The Neutral  . . . . •  68  C  Optics  System  71  D  The Shock Tube Apparatus  75  E  A p p l i c a t i o n of the R.C'.A. 931A Photomultiplier  77  Density  Wedge  of the C a l i b r a t i o n  iv  LIST OF TABLES  Table I  Page Shock wave parameters  for  v  Pi = 16 t o r r  52  LIST OF ILLUSTRATIONS  Figure 1  Page Schematic diagram of the k n i f e edge s c h l i e r e n apparatus  2  P r i n c i p l e of the wedge technique  3  Passage of a ray of l i g h t  3  .  5  through a region  with a r e f r a c t i v e index g r a d i e n t 4  Optics  of the wedge t e c h n i q u e . . .  5  Beam r o t a t i o n by the m i r r o r s of the calibration  '.  .  system  8 11  17  6  Schematic diagram of the apparauts.  21  7  H o r i z o n t a l s e c t i o n of the t e s t s e c t i o n  22  8  The c a l i b r a t i o n system  23  9 10 11  C o n f i g u r a t i o n of the p h o t o m u l t i p l i e r , lens and wedge C i r c u i t diagram of the i n t e g r a t o r of 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 . P h o t o m u l t i p l i e r t r a c e s during one r o t a t i o n of the m i r r o r system of the c a l i b r a t i o n system  vi  24 28 29  Figure  Page  12  T y p i c a l c a l i b r a t i o n curve  30  13  Plane shock wave in the frame of the shock f r o n t  33  14  (a)  (b)  V a r i a t i o n of P with T f o r f i x e d values of the r e f r a c t i v e index of an argon plasma . . . . . . . . .  45  R e f r a c t i v e index of an argon shock wave plasma as a f u n c t i o n of Pi and Mi  46  15  Smear photograph of a Mach 14 shock wave in argon . . . . '. .  16  T y p i c a l smear photograph f o r v e l o c i t y measurements at the t e s t  17 -18  19 20  Mi v s P i  f o r shock waves  48  section  49  in argon  50  S c h l i e r e n s i g n a l and d e n s i t y analogue s i g n a l f o r a Mach99 shock wave in argon, Pi = 16 t o r r . S c h l i e r e n s i g n a l f o r a Mach 16 shock wave in argon, Pi = 2 t o r r  55  P a r t i c l e d i s t r i b u t i o n s i m p l i e d by the s c h l i e r e n s i g n a l f o r the Mach 16 shock wave in argon, Pi = 2 t o r r . . .  56  53  B.l  T y p i c a l H & D curve  .  Cl  Optics of the m i r r o r system of the c a l i b r a t i o n device  72  D.l  General  75  layout of the shock tube apparatus  vii  69  Fi gure E.l  E.2  Page C i r c u i t diagram f o r o p e r a t i o n of the R.C.A. 931A p h o t o m u l t i p l i e r with l i m i t e d dynode amplification  78  C h a r a c t e r i s t i c curve of an R.C.A. 931A p h o t o m u l t i p l i e r with 6 dynode a m p l i f i c a t i o n stages i n use  79  vii i  ACKNOWLEDGMENTS  I  should l i k e  to my s u p e r v i s o r ,  Dr.  to take t h i s  o p p o r t u n i t y to give  B. A h l b o r n , f o r his  thanks  suggestions and  guidance during the course of the work r e p r e s e n t e d by  this  thesis. I staff  should a l s o  l i k e to thank members of the t e c h n i c a l  f o r t h e i r help and a d v i c e , p a r t i c u l a r l y Mr.  Mr. A. Cheuck, Mr.  C. Sedger and Mr.  Thanks are a l s o  due to Dr.  E.  Williams.  S.  Richards  Armstrong f o r many v a l u a b l e c o n s u l t a t i o n s of the experimental work.  D.  Olson,  and Mr.  B.  during the course  Chapter 1  INTRODUCTION  1 .1  General In  Introduction view of the c r i t i c a l  dependence of the  behavior of plasmas on temperature and d e n s i t y , main tasks of plasma d i a g n o s t i c s 1  is  its  tive  component s p e c i e s , d e n s i t y  index measurement.  one of  first tive  itself.  All  over many other plasma introduce physical apply  large In  three methods diagnostic  objects  amounts  derivaits  the r e f r a c -  have the advantages  methods  that  they n e i t h e r  i n t o the region of the plasma  of energy to perturb  nor  it.  the experiment to be d e s c r i b e d h e r e , a s c h l i e r e n  densitometer was c o n s t r u c t e d to measure d e n s i t y shock waves.  The  the s c h l i e r e n system measures  d e r i v a t i v e , and the i n t e r f e r o m e t e r measures index  density  can be determined by r e f r a c -  Three methods are important.  the r e f r a c t i v e index;  density.  depends upon the  shadowgraph gives an i n d i c a t i o n of the second s p a t i a l tive  the  the measurement of  Since the r e f r a c t i v e index of a plasma of  thermodynamic  The novel  system was a n e u t r a l  and e s s e n t i a l  steps  part of t h i s  plane  schlieren  d e n s i t y wedge which r e p l a c e d the  1  in  usual  2  k n i f e edge, g i v i n g tional  methods.  place to several  advantages over conven-  The instrument was t e s t e d  d r i v e n argon shock wave as a known In  plasma.  in a l o g i c a l  content of each s e c t i o n f o l l o w i n g  and exigences  a detonation  the present work, the d e s c r i p t i o n of the  i s w r i t t e n as f a r as p o s s i b l e the  using  of the ones before i t .  in which are developed some points  instrument  sequence of  parts,  on from the development There are a few appendices  which seemed r e l e v a n t and  i m p o r t a n t , but which could not be f i t t e d i n t o the main without  obscuring  the general  A paper d e s c r i b i n g i n The Review is  of S c i e n t i f i c  l i n e of  text  thought.  the instrument  has  been p u b l i s h e d  (Reference 1 ) .  Instruments  This  reproduced in Appendix A.  1 .2  Conventional  S c h l i e r e n Systems  Successful  use of the s c h l i e r e n technique i n  tube densitometry has been r e p o r t e d by many a u t h o r s .  shock References  1-8 are examples . f  The for  usual  experimental  example, by de Boer  technique was well  (Reference 5 ) .  The apparatus  depicted schematically  in F i g u r e 1 .  The source s l i t  i l l u m i n a t e d by a l i g h t  source and is  situated  the  lens L i .  traverses to a focus  Light  the t e s t  is Si is  at the focus  s e l e c t e d by the beam d e f i n i n g s e c t i o n of the shock  described,  slit  tube and is  S  of  2  brought  in the plane of the k n i f e edge by the lens  l2  The  3  k n i f e edge l i e s from L  2  is  multiplier escaping  in such a plane that a part of the l i g h t  blocked o f f from the p h o t o m u l t i p l i e r . signal  is  proportional  beam  The photo-  to the amount of  light  the k n i f e edge.  PHOTOMULTIPLIER DETECTOR  i  KNIFE EDGE  TEST SECTION  Figure 1.  If  Schematic diagram of the k n i f e edge s c h l i e r e n apparatus.  the t e s t  s e c t i o n is  t i o n i n g of the instrument part of the t e s t  simply  understood.  the f u n c -  Through each  s e c t i o n passes a p e n c i l of l i g h t  This p e n c i l forms an image dimensions  is  assumed to be t h i n ,  from  S . 2  in the plane of the k n i f e edge, the  of which are fz/f\  times the dimensions  of the source  4  slit,  f i and f  tively.  2  are the f o c a l  lengths  When a d e n s i t y g r a d i e n t  is  of L i and L  present at the  under c o n s i d e r a t i o n , the p e n c i l  is  e f f e c t , by an angle  to the g r a d i e n t .  ing  proportional  2  respecpoint  d e v i a t e d by the s c h l i e r e n The c o r r e s p o n -  displacement of the c o n t r i b u t i o n to the image from  pencil  causes  a change  k n i f e edge, and t h i s The t o t a l  change  2  to g i v e ,  change  is  escaping  from the p e n c i l s  Hence the instrument at any i n s t a n t ,  the  f o l l o w e d by the p h o t o m u l t i p l i e r .  in the p h o t o m u l t i p l t e r output v o l t a g e  sum of c o n t r i b u t i o n s on S .  in the amount of l i g h t  this  of l i g h t  integrates  the d e n s i t y  from a l l  the d e n s i t y  is  the  points  gradient  change over the width  of the probing beam. A method of improving by a f a c t o r of about 100 was It  the s e n s i t i v i t y  suggested by Hall  of the  (Reference  was proposed that the d e f l e c t i o n of the l i g h t  effectively In  i n c r e a s e d by an e t a l o n  parallel  and h i g h l y  i n t e n s e He Ne l a s e r  1 .3  with a s p a t i a l  p e n c i l be  (References  beam was not used, but a c l o s e l y  diameter by a simple t e l e s c o p e . gradient  9).  system.  the experiments of K i e f e r and Lutz  6, 7 & 8) an expanded l i g h t  instrument  This  beam reduced to a small  system -determined d e n s i t y  r e s o l u t i o n of approximately  1 mm.  The Wedge Technique The k n i f e edge method, common to a l l  s c h l i e r e n d e n s i t o m e t e r s , was r e p l a c e d in t h i s  conventional experiment by  5  the wedge t e c h n i q u e .  This  by P o t t e r  11 & 12) in the measurement of  shifts  (References  of wide s p e c t r a l  technique was p r e v i o u s l y  transmission  (Figure  2).  small  lines.  A n e u t r a l wedge i s light  employed  varies  a neutral  density  filter  in one d i r e c t i o n across  The wedge in use in t h i s  its  whose surface  a p p l i c a t i o n was produced  DETECTO R  Figure 2.  on a photographic  P r i n c i p l e of the wedge t e c h n i q u e .  plate.  Its  are d e s c r i b e d in Appendix B.  p r o d u c t i o n and It  will  characteristics  be shown in S e c t i o n  2.2  6  that,  in the present a p p l i c a t i o n ,  variation  of t r a n s m i s s i o n  applications,  i t . is  with d i s t a n c e  not necessary be l i n e a r ;  f o r example in the experiment of  that  the  in some  Potter,.a  l i n e a r wedge is r e q u i r e d . Figure  2 i llustrates  probing  beam is  translated  wedge.  The instantaneous  how a d e f l e c t i o n of a narrow  i n t o an i n t e n s i t y d e f l e c t i o n a is  t r a n s v e r s e r e f r a c t i v e index g r a d i e n t assuming t h i s Accordingly,  to vary l i t t l e along  by the  proportional  the path of the beam.  the f a c t o r by which the wedge attenuates  in p h o t o m u l t i p l i e r  to the  encountered by the beam,  i n t e n s i t y of the d e f l e c t e d beam i n c r e a s e s change  change  signal  AV i s  AW  dn  the  with a, and so the  a measure of a.  For a  1i near wedge  A V  where x  1 S  a  constant.  If  =  * dT  the r e f r a c t i v e m a t e r i a l  p e n d i c u l a r l y to the beam'with  beam i s  index change An as  found by  per-  v e l o c i t y v in the d i r e c t i o n  of the r e f r a c t i v e index g r a d i e n t , refractive  moves  a v a r i e s with time.  the m a t e r i a l  The  passes through  the  integration:  (2)  n  (3)  (4)  7  Now the r e f r a c t i v e or e l e c t r o n species density. plasma  If  of a plasma  is  or of the n e u t r a l ,  proportional  the s c h l i e r e n d e f l e c t i o n is  to  an analogue  p r i n c i p l e underlies The o p t i c s  generally results  signal  f o r the d e n s i t y  the present  chapter.  as a f u n c t i o n profile.  treated f u l l y It  is  and  shown that  are promised by a simple design not r e q u i r i n g  of the wedge over the whole i n t e r s e c t i n g over the small  or  densitometer.  of the device i s  in the f o l l o w i n g  ion  particle  caused by a gas  in motion as a whole, measurement of J A V d t  of time y i e l d s This  index of a gas  s c h l i e r e n displacements  of  accurate linearity  beam w i d t h , but individual  only  rays.  And problems with d i f f r a c t i o n are not encountered, a customary difficulty 5 & 10) .  with conventional  s c h l i e r e n densitometers  (References  Chapter 2  OPTICS OF THE DENSITOMETER  The S c h l i e r e n  2.1  Effect  The s c h l i e r e n e f f e c t i s due  to g r a d i e n t s  medium.  Consider  in the r e f r a c t i v e a ray of l i g h t  the r e f r a c t i v e index v a r i e s angle  the d e f l e c t i o n of a l i g h t index of the  traversing  Figure  3.  transmitting  a medium in which  in the x d i r e c t i o n .  between the ray and the y d i r e c t i o n as  Let £ be the  shown in Figure  Passage of a ray of l.ight through a r e g i o n with a r e f r a c t i v e index g r a d i e n t . 8  beam  3.  9  Snell's  law may be w r i t t e n :  n cos  Differentiating  equation  <{> = constant  (5)  (5) gives  (6)  Now dn dn dx dy ~ dx dy  (7)  dn dx  (8)  = tan $  Substituting  equation  (8) i n t o equation  cos o> tan op ^  = n sin  (6) gives  o)  (9)  ^  Hence  dx  For a ray e n t e r i n g the medium p a r a l l e l versing  a region of thickness  from i n t e g r a t i o n  of equation  (10)  dy  to the y axis  6, the t o t a l  and  deviation a  tra-  follows  (10).  (11) ret  n d c>  (12)  10  If  n and dn/dx vary n e g l i g i b l y  equation (12)  over the path of the r a y ,  becomes  dn 6 = na dx so  (13)  that (14)  a = n dx  2.2  E f f e c t of the Wedge Figure 4 shows the geometry of the t e s t  the.shock t u b e , the wedge and the probing l a s e r longitudinal to the axis is  axis  of the l a s e r  beam.  is  parallel  at the i n t e r s e c t i o n of the X a x i s  T(X).  d e f i n e d as the d i s t a n c e from the t e s t  the wedge, the s e p a r a t i o n of the x and  section is  Gaussian;  it will  f i n e d w i t h i n a d i s c whose radius beam has a n a t u r a l In  g e n e r a l , a ray c r o s s i n g  crosses  the X a x i s  at L ( x ) ,  the axis say.  laser  written  s e c t i o n to  beam over  be assumed to be con-  at the x a x i s  divergence, this  The  X.axes.  The i n t e n s i t y d i s t r i b u t i o n of the l a s e r cross  to x.  f u n c t i o n of the wedge is  axes  trans-  and the  The t r a n s m i s s i o n  its  perpendicular  The d i r e c t i o n X of the  beam a x i s . D is  The  The i n t e r s e c t i o n of these  g r a d i e n t of the n e u t r a l wedge is  o r i g i n of X i s  beam.  of the shock tube, the x a x i s ,  taken as the o r i g i n of x.  mission  s e c t i o n of  radius  is  is  r.  Since  not p r e s e r v e d .  of the t e s t  s e c t i o n at x  the  Figure  4.  Optics of the wedge techni que.  12  Thus far, the undef 1 ected beam has During  been c o n s i d e r e d .  the passage of a shock wave whose r e f r a c t i v e index may  be d e s c r i b e d by the f u n c t i o n n ( x j t ) ,  the instantaneous  t i o n of a ray at x by the s c h l i e r e n e f f e c t i s displaces  the ray a d i s t a n c e across  compared with the width of the d i s c  t h a t the wedge i s  ments, but not n e c e s s a r i l y  This  the wedge which i s  small  formed by the i n t e r s e c t i o n  of the undisturbed beam with the wedge. throughout  a(x,t).  deflec-  It  will  be assumed  l i n e a r over these small  displace-  l i n e a r over the whole beam w i d t h .  Consider a laminar s e c t i o n of the u n d i s p l a c e d beam, the c o l l e c t i o n of rays i n Figure 4. effect  is  i n d i c a t e d by broken l i n e s .  the area i n t e g r a l  If  n(x,t)  and 9/9x n ( x , t )  The l i g h t  vary n e g l i g i b l y  w i t h i n the s e c t i o n , a ( x , t )  at u n i t y with n e g l i g i b l e  is  flux  cj>(x)dx  the s e c t i o n .  over the paths  given  by equation  to 1 only in terms i n v o l v i n g  = 6  (14).  (n - 1).  considered  Then  n(x,t)  The r e f r a c t i v e index p r o f i l e n ( x , t )  of the plasma  (15)  propagates  in the x d i r e c t i o n at the v e l o c i t y v of the shock wave, that  of  n of the denominator may be set  l o s s of a c c u r a c y , n being  a(x,t)  shown  s e c t i o n by the s c h l i e r e n  of beam i n t e n s i t y across  For a shock wave plasma,the  unequal  between x and x + dx as  The displacement of t h i s  is  the rays  passing  so  1 3  Accordingly,  The change  n(x,t)  = f(x  (15)  becomes  a(x,t)  = 6 f(x  equation  in t r a n s m i t t e d  light  d e f l e c t i o n of the beam is A change AV in s i g n a l in l i g h t  f l u x A$.  - vt)  (16)  - vt)  (17)  f l u x due to the s c h l i e r e n  monitored by a p h o t o m u l t i p l i e r .  corresponds  to a p r o p o r t i o n a l  change  Then  AV = k A $  (18)  The elementary c o n t r i b u t i o n from the s e c t i o n of l i g h t  through  x d(AV)  Substituting  equation  d(AV)  Integrating  = k <J)(x) dx  (17)  '9T(X)' 8X  i n t o equation  = kD o)(x) T ' ( L ( x ) )  equation  a(x,t) D  X=L(x)  (19)  6 f'(x  (19)  gives  - vt)  dx  (20)  (20),  AV = kD 6 J -r  dp(x) T ' ( L ( x ) )  f ' ( x - vt)  dx  (21)  14  AV i s  i n t e g r a t e d through time from before the a r r i v a l  shock to time  r.  .00  j  r  (22)  <J>(x) T ' (L(x)) f ' (x - vt) dt dx  AVdt = kDS  ^  of the  - y  <j>(x) T'(L(x)) « f (X - V t ) - f  (-00)  dx  I  (23)  -r  f(-°°) i s the r e f r a c t i v e index of the undisturbed gas. represents a time a f t e r the shock f r o n t has passed completely, tive  then f ( x  index g r a d i e n t  - vr)  is  If  r  the beam  evaluated as the step in r e f r a c -  at the fr-ont, plus a slowly  varying  func-  t i o n r e p r e s e n t i n g the decay i n r e f r a c t i v e index of the wave behind the f r o n t . varies  negligibly  kDS  AVdt  r in t h i s way, - f(x - vr) - f(-»)  Defining  over - r < x < r,  and so equation  f(x - vr) - f(-»)}• T  (23)  becomes  <|>(x) T'(L(x)) dx  (24)  j —00  The  integral If  the  on the r . h . s .  is  the whole beam i s  change AV1 in the s i g n a l  AVi = kD  Substituting  a constant  equation  01  (25)  of the  apparatus.  r o t a t e d by a small  is,  from equation  angle 0 1 ,  (19),  x) T ' (L(x)) dx  into equation  (24)  (25)  gives  15  From equation  AVdt  Av 6 ai v  (16),  equation  f(x  (26)  AV_i _5 ai v  AVdt =  -  n(x  vr)  -  f ( - o o )  }  (26)  becomes  - vr)  - n(-°°)  (27)  - n(  (28)  T h i s may be r e w r i t t e n  AVdt  for  t > r  The q u a n t i t y  with c t , equal quantity  dccL v  4-n (x - vt)  fdV  is  da  the r a t e of change  to AV /ai'; the s u b s c r i p t 1  in the brackets  the fundamental  its  equation of the  of n in the shock  width of the l a s e r  beam.  the of  at which the beam a x i s  undeflected p o s i t i o n .  Equation  (28)  is  instrument.  The c o n d i t i o n t > r a r i s e s variation  that  voltage  must be evaluated f o r a r o t a t i o n  the whole beam, and at the p o s i t i o n c o i n c i d e s with  B denotes  of  because of the  f r o n t over the d i s t a n c e  The i n t e g r a t e d  signal  large 2r,  the  AVdt does — oo  not minutely d e s c r i b e shock  f r o n t , but i t  the shock  the s p a t i a l  does  track  was approximately 2mm.  the  the jump in n with accuracy  f r o n t passes the beam.  t i o n of the device is  v a r i a t i o n of n over  Evidently  the s p a t i a l  the width 2r of the l a s e r  beam.  as  resoluThis  16  It as  should be noted that equation  i t would have been had the whole l a s e r  concentrated long the beam a x i s . the l a s e r  (28)  is  the same  beam i n t e n s i t y been  The t r a n s v e r s e  spread of  beam, c h a r a c t e r i z e d here by the f u n c t i o n 4>(x),  the v a r i a t i o n  in the t r a n s m i s s i o n  the area on which the beam f a l l s  gradient  a  n  d  of the wedge over  do not a f f e c t the r e s u l t  of  thecalculation. The q u a n t i t y calibrate  the system.  dV dc^  must be measured in order to  B  An o p t i c a l  system to do t h i s  is  described  in the next s e c t i o n .  2.3  The C a l i b r a t i o n System The l a s e r  known angular  beam i s  swept across  the wedge at a constant  v e l o c i t y by a system of three m i r r o r s ,  which are mounted on a r e v o l v i n g v a r i a t i o n of s i g n a l  table.  with time leads '  Measurement of the  d i r e c t l y to  The geometry of the m i r r o r system i s Figure 5.  Full  lines  show the system as  undef1ected; broken l i n e s  it  m i r r o r assembly  lies  l  d  a  J  . B  illustrated  in  passes the beam  i n d i c a t e the e f f e c t of a small  r o t a t i o n 0 of the r o t a t i n g m i r r o r assembly, are l i n e s of c o n s t r u c t i o n .  two of  and dotted  lines  The c e n t r e of r o t a t i o n 0 of the  at the point of i n t e r s e c t i o n of  drawn p e r p e n d i c u l a r to the r o t a t i n g m i r r o r s u r f a c e s  lines and  through  the points of i n c i d e n c e of the l a s e r beam on the m i r r o r s when the m i r r o r s are in the symmetric p o s i t i o n , the p o s i t i o n  for  1 7  which there is  no r e s u l t a n t  angle of the two r o t a t i n g  Figure 5.  beam displacement.  mirrors  The i n c l u d e d  is 9 5 ° .  r o t a t i o n by the m i r r o r s  of the c a l i b r a t i o n  system.  18  Consider a r o t a t i o n of the m i r r o r assembly angle  8.  surfaces  The l a s e r and i s  appearing  twice r e f l e c t e d from r o t a t e d  consequently  itself  to d i v e r g e from i t s  The d i s t a n c e AD i s , d i s t a n c e CD. more f u l l y  beam i s  r o t a t e d by an angle  original  for practical  The o p t i c s  by a small  46,  a x i s at a point A.  purposes,  equal  of the m i r r o r system  is  to the discussed  in Appendix C. In  view of the e q u a l i t y of t h e d i s t a n c e s A D  the c a l i b r a t i o n  and CD,  system was c o n s t r u c t e d and placed in such a  way that the d i s t a n c e  CD was equal  and the centre of the t e s t to make the l a s e r  section.  to the d i s t a n c e  between D  The system then appeared  beam r o t a t e from the point at which the  s c h l i e r e n d e f l e c t i o n arose. When the m i r r o r assembly the p o s i t i o n  in which i t  is  was r o t a t e d by 180° from  drawn in Figure 5,  was not i n c i d e n t on e i t h e r m i r r o r .  the l a s e r  beam  The m i r r o r s were placed  thus when the system was out of use, c o n v e n i e n t l y a v o i d i n g  the  n e c e s s i t y of removing the c a l i b r a t i o n system from the r e s t of . the apparatus  of the densitometer.  Writing  co f o r the angular  1 dV  dV da  =  Whereupon  4 d0  1  d V  T dt  1 u  d9 v e l o c i t y dt of the m i r r o r s ,  (29)  (30)  19  dV_ da  Measurement  of  fdVl dt  dV 4OJ  (31)  dt  from the output of the p h o t o m u l t i p l i e r  dvl directly for  light  from equation  absorption  (31).  by the m i r r o r s .  Allowance must be made  gives  Chapter 3  THE APPARATUS  3.1  Construction  of the  Apparatus  A schematic diagram of the apparatus Figure 6.  is  found  in  The observed plasma was a shock wave produced  argon  by a chemical  detonation.  along  a pyrex shock  tube of 2.5 cm i n t e r n a l  produced by i g n i t i n g  The shock wave  a mixture of equal  in  propagated  d i a m t e r , and was  parts  by pressure  of  oxygen and a c e t y l e n e in a d e t o n a t i o n chamber separated  from the  shock  of  tube by a mylar diaphragm.  e x p l o s i v e mixture was always of the argon  test  gas  are given  The t e s t  parallel  the  pressure  (Reference  of the shock  tube  13)  apparatus  D.  diameter as  quartz windows  that  of a piece of brass o,f the shock  were mounted on o p p o s i t e  tube in order to t r a n s m i t section of.the test  Details  section consisted  of the same i n t e r n a l  the i n i t i a l  been d e s c r i b e d by Huni  (Reference 14).  in Appendix  350 t o r r ;  pressure  was v a r i e d between 1 t o r r and 160 t o r r .  The detonation d r i v e r has and Redfern  The i n i t i a l  the probing  section  is  shown  20  laser  beam.  in Figure  7.  tube.  sides  of  tube Two the  A horizontal  21  \  / / / / /  /  * > \ \ \  HLM.  I«.ier \  Cal  i k r o Via r\  + aal"t6  ED Ttkfcroni* S5I ouhpul- Of.  M.  dual  -10OV  V  p Koboi^iAlhpli ~IB"1  Figure  oft  kio|k  -vIS V -IS  o«-4m  OSCi'lloiC  6.  Schematic  I  diagram  of the apparatus.  p^  s s  22  PROBING LASER  BEAM  BRASS ^__^TUBE  QUARTZ WINDOWS  Figure 7.  Horizontal  s e c t i o n of the t e s t  section.  A smear camera was used to photograph over a length of the shock  tube i n c l u d i n g  The shock wave v e l o c i t y was measured necessary  to remove the l a s e r  photographs.  the shock wave  the t e s t  in this  way.  section. It was  beam in order to take  these  The smear camera i s d e s c r i b e d in d e t a i l  in  Reference 13. All  the o p t i c a l  mounted on o p t i c a l  components  of the densitometer were  benches b o l t e d to a piece of s t e e l  This was f i x e d to a table which was p h y s i c a l l y the t a b l e on which the shock braced and h e a v i l y weighted The probing neon l a s e r  U channel.  separate  from  tube was mounted, and which was in order to minimize  vibrations.  beam was provided by a Spectra 115 helium o 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 i s  drawn in Figure 8.  r o t a t i n g m i r r o r s were mounted on the two symmetrical an i s o s c e l e s  aluminium p r i s m , the i n c l u d e d angle  To allow the l a s e r  the  prism was cut away as  r o t a t e d at 1200 r.p.m. Dynamic b a l a n c i n g  faces  shown, a small  shown in Figure 8.  to prevent o v e r h e a t i n g of the motor. achieved by a d j u s t i n g  prism part  was necessary  Optical  the three screws  of  The prism was  by a Globe SC B1702 synchronous  of the prism assembly  of  being 9 5 ° .  beam to pass unobstructed when the  was r o t a t e d by 180° from the p o s i t i o n  The  motor. in order  alignment  supporting  the  p l a t e of the motor mount, and the micrometer screws  was  base  of  the  f i x e d m i r r o r mount.  FIXED  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  ROTATING  P R I S M  M O T O R M O T O R  BASE  Figure  8.  The  calibration  M O U N T  PLATE  system.  ADJUSTING  SCREWS  24  A c e r t a i n f r a c t i o n of the l a s e r r e f l e c t i o n at the three m i r r o r s . was not being the  laser  used, a n e u t r a l  beam path.  A filter  light  When the c a l i b r a t i o n  density f i l t e r  possible  mirrors.  by  system  was placed in  was chosen with such a d e n s i t y  that the a t t e n u a t i o n of the beam by the f i l t e r as  was l o s t  was as  nearly  the same as the a t t e n u a t i o n of the beam by the  In  this  way, the only o v e r a l l  e f f e c t of the c a l i b r a -  t i o n system was to r o t a t e the beam. The  wedge was mounted on a r i g i d  f r o n t of a 50 mm f o c a l lens  served to focus  length convex lens  the d e f l e c t e d l i g h t  f i x e d region of the p h o t o s e n s i t i v e tiplier.  Because of t h i s ,  frame d i r e c t l y in (Figure  9).  beam on to a roughly  s u r f a c e of the photomul-  the p h o t o m u l t i p l i e r s i g n a l  with beam d e f l e c t i o n only on account of the v a r y i n g of the wedge, not being a p p r e c i a b l y a f f e c t e d by variations  DEFLECTED  BEAM  The  varied transmission  sensitivity  over the s u r f a c e of the photocathode.  BEAM  AXIS  P H O T O M U L T IPLIER TUBE  Figure 9.  Configuration  of p h o t o m u l t i p l i e r , lens and wedge.  \  25  An R.C.A. 931A p h o t o m u l t i p l i e r was to improve the s i g n a l number of i t s usual  dynodes were used.  c i r c u i t r y is  falling  to noise r a t i o  used.  of t h i s  In  device, a limited  This modification  d e s c r i b e d in Appendix E.  of the  The l a s e r  c  light  on the photocathode was reduced to an a p p r o p r i a t e  intensity  range  by a n e u t r a l , d e n s i t y  window of the p h o t o m u l t i p l i e r output was a p p l i e d across a National  filter  housing.  LH 0063 b u f f e r a m p l i f i e r .  by an RG 58 c o - a x i a l  placed over the  The p h o t o m u l t i p l i e r  a lk load r e s i s t o r  a m p l i f i c a t i o n of 1, working  into  50  This fi.  cable terminated  to the input  produced a  venting  r e f l e c t i o n s of the s i g n a l . r i s e t i m e was  estimated  of  voltage  The output was  taken  in a 50 fi r e s i s t o r .  The terminator matched the impedance of the c a b l e ,  signal  order  so p r e -  The minimum 10% to 90%  from the s i n g l e  photon  pulses  to be 9 ns. In a d d i t i o n  to the d e f l e c t e d l i g h t  of the l a s e r  the p h o t o m u l t i p l i e r behind the wedge a l s o r e c e i v e d emitted by the plasma. for this  light  multiplier  An attempt was f i r s t  by c o n s t r u c t i n g  system s i m i l a r  semi - si 1vered m i r r o r .  It  balance the two channels'. the plasma tronic  light  flash  a monitor  compensate a photo-  to the one a l r e a d y d e s c r i b e d , and a proved f u t i l e , however, to t r y This was attempted by  with a l i g h t  flash  u n i t , and balancing  the two p h o t o m u l t i p l i e r s  light  made to  comprising  beam,  simulating  from a S o l a t r o n  the l i g h t  by adjustment  flux  to  122  elec-  r e c e i v e d by  of the i r i s  diaphragms  26  placed before the lenses  of the two c h a n n e l s .  could be balanced a p p r o x i m a t e l y ,  The  but the plasma  flashlight  light  was  intractable. Instead of t h i s filter  was  system.  measure,  placed between the t e s t  This  filter  allowed  pass s c a r c e l y a t t e n u a t e d .  shock waves, the  to a l e v e l  s e c t i o n and the o  the 6328 A l a s e r  Its  enough to reduce the plasma  a narrow band i n t e r f e r e n c e calibration  radiation  to  band pass proved to be narrow  light,  even of the most  that caused n e g l i g i b l e  luminous  response  from  photomultiplier. The e l e c t r o n i c s used to d e t e c t and process  photomultiplier signals  is  Both the s c h l i e r e n s i g n a l on the dual Tektronix  depicted schematically and i t s  time i n t e g r a l  the  in Figure  were  displayed  beam T e k t r o n i x 551 cathode ray o s c i l l o s c o p e .  7704 C.R.O. was used f o r  timing  and  6.  A  amplification  purposes . A timing photomultiplier the t e s t  timing  placed next to the shock  section.  c r e a t e d timing  pulse was d e r i v e d from an R.C.A. 931A  The luminous  pulses  of approximately  of the shock  to  waves  40 V amplitude.  The  pulse was a p p l i e d to the t r i g g e r , input of time base A  (plug-in  u n i t 7B71)  of the 7704 C.R.O.  this  time base was adjusted  unit  7B70)  to t r i g g e r  a f t e r a delay equal  the p r o d u c t i o n of the t r i g g e r shock  fronts  tube and c l o s e  f r o n t at the t e s t  The delay generator time base B  to the time elapsed pulse and the a r r i v a l  section.  In  this  of  (plug-in between of  the  way, time base B was  \  27  synchronized with the s c h l i e r e n s i g n a l This  time base was used to t r i g g e r  means of the "+ gate"  signal,  from the  photomultiplier.  the 7704 C.R.O. and, by  to t r i g g e r  the 551 C.R.O.  as  wel 1 . Though a l l 551 C.R.O., schlieren signal mV/div.  measurements  were c a r r i e d out using  i t was convenient to apply  signal  plug-in  first  to the 7704 C.R.O.  u n i t was  output  a m p l i f i e d v e r s i o n of the input The output s i g n a l  passed  photomultiplier A 7A12 v e r t i c a l  used, with an input  The o s c i l l o s c o p e  u n i t c o n t r o l 1ing  the  sensitivity  then provided a ten  was  through  grated  by the o p e r a t i o n a l  was a p p l i e d  a high  The same  pass f i l t e r and then  The high  pass f i l t e r  the b u f f e r a m p l i f i e r c i r c u i t . could not be made i f  The c u t - o f f  this  of the  waveform was  u n i t which  not  10.  of the i n t e g r a t o r  A 300 mV x 1 y s square  Electronics  Pulse  integrator.  pulse  Generator 233 was  is  to  produced by integrated filtered.  frequency of the f i l t e r was approximately  The c i r c u i t diagram Figure  Measurements  inte-  served  reduce the amplitude of a 13 kHz saw tooth r i p p l e  signal  times  to the Type K p l u g - i n  a m p l i f i e r Type 0 p l u g - i n  c o n t r o l l e d the lower beam.  of 50  signal.  the upper beam of the 551 C.R.O.  signal  the  10 kHz.  shown  in  from a Bradley  used to c a l i b r a t e  For the s p e c i f i e d impedance values  shown  the in  F i g u r e 1 0, e where t i s  measured in  0  = -1.22 ys.  e  i  dt  (32)  28  o-oi M n A A A V W  O-  O  O - O O O l jiF  Figure  3.2  10.  C i r c u i t diagram of the i n t e g r a t o r of operational amplifier plug-in unit.  Use o f  the  The system reduced closest  Apparatus  f a c t o r by w h i c h  was  of  density  by a f a c t o r 0 . 6 3 . that  it  tiplier  density 0.2;  operating  To c o r r e c t f o r  needed was  the small  found  filter  in  both  scales  the  Figure multiplier  traces  11  is  enough  1.02  taken during  of  one  (~  the intensity  together photomul-.  on i t s  charac-  that  was  0.63/0.62)  in order the  a photograph r o t a t i o n of  The  in  then, a l l  t h e c a l i b r a t i o n and o f a tracing  the  same s l o p e  the d i s c r e p a n c y ,  correction factor  of  be 0 . 6 2 .  available  cases,  which 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 the v o l t a g e  the c a l i b r a t i o n to  were c l o s e  at pointsof  teristic.  of  i t a t t e n u a t e d t h e beam  These f a c t o r s  c o u l d be assumed t h a t , was  the m i r r o r s  t h e beam i n t e n s i t y was  equivalent neutral  laboratory  t h e Type 0  to  standardize  experiment. of  the  two  photo-  mirror  \  29  PHOTOMULTIPLIER OUTPUT VOLTAGE  5 0 mV  TRACE I  1  TRACE 2 T  —H h— w  =  126  T R A C E I. Syus T R A C E 2 , 5ms  RADIAN/S  •» T I M E  -Figure 1 1 .  assembly.  P h o t o m u l t i p l i e r t r a c e s during one r o t a t i o n of the m i r r o r assembly of the c a l i b r a t i o n system.  The r i s i n g  slope of t r a c e 1 is  p h o t o m u l t i p l i e r as  the m i r r o r s  wedge.  the s i g n a l  Trace 2 i s  the  cut-away  the  r o t a t i n g m i r r o r assembly  Escaping  the response of the  swept the beam across produced during  s e c t i o n of the r o t a t i n g  prism  the  the time  (Figure  8)  that allowed  to not i n t e r r u p t the beam.  a t t e n u a t i o n by r e f l e c t i o n , the beam i n t e n s i t y was  enhanced by a f a c t o r  1/0.62 over the r e f l e c t e d beam i n t e n s i t y .  The u n r e f l e c t e d beam was attenuated on passing  the  centre of the wedge, roughly  by 1/2.  that t r a c e 2 would be a f l a t  bottomed pulse with a v o l t a g e  about 0.8 times  (negative)  actual  voltage  the g r e a t e s t is  It  through  would be e x p e c t e d , t h e n ,  voltage  e v i d e n t l y of about t h i s  of  of t r a c e 1.  magnitude,  but the  The  30  bottom of the pulse  is  not f l a t .  of such an area t h a t a q u a n t i t y  This  trace  is  trace  1 is  because the pulse  in comparison with the  speed-up c a p a c i t o r .  Fortunately, this  never r e q u i r e d f o r any measurement; the pulse too small  to arouse  The r i s i n g  slope  is  isolated.  this  case  is  calibration  The g r a d i e n t of the  i s measured at the centre of the l i n e a r p o r t i o n . in  98 mV/ys, and the c e n t r e of the l i n e a r  The g r a d i e n t  by the f a c t o r 1.02,  of the c a l i b r a t i o n curve must  and a l s o  slope  The g r a d i e n t  occurs around the centre of the time s c a l e at a s i g n a l 260 mV.  of  the same d i f f i c u l t y .  Figure 12 shows a photograph of a t y p i c a l curve.  is  of charge was drawn from the  p h o t o m u l t i p l i e r which was not small charge on the l a s t  is  region of  about  be m u l t i p l i e d  by a f a c t o r 10 because i t was  always measured from a t r a c e of the 7704 C.R.O.  The  signal  r e c e i v e d by the 551 C.R.O. was a m p l i f i e d by a f a c t o r 10 by the 7704 C.R.O. as d e s c r i b e d in S e c t i o n  3.1.  PHO T O M ULTIPLIER O U T P U T VOLT AG E  Figure  12.  Typical c a l i b r a t i o n curve.  \  31  In  g e n e r a l , once the c a l i b r a t i o n curve had been  determined,  the p o s i t i o n of the wedge was a d j u s t e d  way that the u n d e f l e c t e d beam' axis l i n e a r p o r t i o n of the wedge, had been measured.  turned out of p l a y ,  the p o i n t f o r which the  The r o t a t i n g  and the 0 . 2 n e u t r a l  1 . 0 2 times  output voltage  the v o l t a g e  V  2  this,  the  m i r r o r assembly  The wedge was then a d j u s t e d  that the d . c .  on the c e n t r e of  In order to accomplish  procedure was adopted.  emplaced.  fell  in such a  density  gradient following was  filter  to such a  the  was  position  V i of the p h o t o m u l t i p l i e r was  at the c e n t r e of the l i n e a r  region  of the c a l i b r a t i o n c u r v e . The d . c .  component V i of the p h o t o m u l t i p l i e r  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 7 A 1 2 p l u g - i n  7 7 0 4 C.R.O.  The a . c .  u n i t of the  components were a m p l i f i e d by 10 and  passed to the 551 C.R.O., The s e t t i n g u n i t was a d j u s t e d  of the 7B71  to a value f o r which the timing  both o s c i l l o s c o p e s shock f r o n t crossed  to t r i g g e r the beam.  s i g n a l s were d i s p l a y e d oscilloscope  of the delay generator  a few microseconds In  this  simultaneously  way,  pulse  caused  before the  the V and jAVdt  on the 551 dual  beam  on a time base of 1 u s / d i v .  The performance of the densitometer was by making measurements calculated  plug-in  investigated  on a shock wave whose p r o p e r t i e s were  from the i n i t i a l  conditions  the measured shock wave v e l o c i t y . theory behind them are presented  of the t e s t  gas,  These c a l c u l a t i o n s next.  and  and the  Chapter 4  PROPERTIES OF THE SHOCK WAVES  4.1  Shock Wave Plasma  Parameters  A shock wave i s density,  rise  temperature and entropy of a f l u i d .  propagates gas  a discontinuous  supersonically  ahead of  it.  with r e s p e c t to the  i shock wave with r e s p e c t to the f l u i d place a d i a b a t i c a l l y  The d i s c o n t i n u i t y undistrubed  behind i t  but is  continuity;  i d e a l i z a t i o n of what i s  an  f l u i d cannot  region with very high g r a d i e n t s  is  the  subsonic.  i r r e v e r s i b l e and  A real  is  along  the v e l o c i t y of  hence not i s e n t r o p i c . this  pressure,  The f l u i d behind the shock wave moves  in the same d i r e c t i o n , in such a way that  The flow takes  in the  have an a c t u a l  dis-  r e a l l y a thin  in the thermodynamic  quantities.  Consider a plane shock wave in a frame in which the shock  front  is  at r e s t  . denote q u a n t i t i e s  behind i t .  temperature; H is  velocity  13).  before the shock  2 denote q u a n t i t i e s T is  (Figure  f r o n t and l e t  P is  enthalpy  r e l a t i v e to the shock  Let the s u b s c r i p t 1  pressure;  the  p is  subscript density;  per u n i t mass, and u is front.  32  fluid  \  33  SHOCK  FRONTSHOCK  P,  K  TUBE  T,  H - U c H -u, DIRECTION WAVE  Figure  The energy  may  13.  written  For  an  ideal  x  gas, t h e  equations  (Reference  Pi  H  +  +  1  Pi  U i  P i  U i  U !  2  equation  H =  Y is  is  the  ratio  a constant.  of It  the is  MOTION  RESPECT  P l a n e s h o c k wave i n of the shock f r o n t .  conservation  be  OF  WITH  = H  convenient  mass,  frame  momentum  and  —  (33)  2  +  \  state  Y-l  principal  2  +  2  of  u  2  = P  —  the  SHOCK TUBE  15)  = p  2  of  OF  TO  P  u  2  u  2  (34)  2 2  (35)  2  maybe  written  (36)  p  specific to  heats  introduce  of the  the Mach  gas  and  number  34  M.  This  is  the r a t i o  turbance or flow  the v e l o c i t y of a d i s •  r e l a t i n g  in a gas  to  the l o c a l  v e l o c i t y of  sound  a.  Hence  Mi  The v e l o c i t y of sound of an i d e a l  gas  is  a  r e l a t e d to the thermodynamic  parameters  by  XP  a =  Solution  (37)  i  of equations  (33)  (38)  P  through  (38)  yields  the  following  equations:  2Y  Pi  "  L  M  Y  - (y-1)  2 t  Y  £2. P  M  +  (39)  1  (Y+1)MI (Y-1) + 2 2  Y-1  ^1  (40)  2 M  l  + 1 (41)  Ti  Mi  where the temperature, fol1ows from the i d e a l the c o n d i t i o n s determine P > 2  meas ured.  ahead of the shock P2 and T  2  gas  law.  are known, these  Since  equations  once the shock wave v e l o c i t y  has  been  \ 35  In monatomic gases the only departures gas  from i d e a l  behavior are caused by i o n i z a t i o n and by a c o n t r i b u t i o n  to the s p e c i f i c  heat from e l e c t r o n i c e x c i t a t i o n .  The popula-  t i o n of an e x c i t e d e l e c t r o n i c energy l e v e l at e q u i l i b r i u m i s - E /kT p r o p o r t i o n a l to e , where E is the e x c i t a t i o n energy. ex For most atoms and m o l e c u l e s , even at c o m p a r a t i v e l y high temperatures, ing  E  is  ex  large  enough to render the energy  in the e x c i t e d e l e c t r o n i c l e v e l s  with the c l a s s i c a l  Most commonly  servation  equation  AE,  being  this  Equation  (35)  comparison  (35)  (e.g. is  i n t o account  Reference 16)  modes.  in e i t h e r of  the energy  m o d i f i e d to i n c l u d e an energy  conterm  the energy taken to produce the i o n i z a t i o n .  then becomes H  In  in  energy ^- RT of the other energy  I o n i z a t i o n may be taken two ways.  negligible  resid-  2  - Hi = 1  the present work,  U j  2  -  \  the a l t e r n a t i v e  An e f f e c t i v e a d i a b a t i c  exponent g is  9-1  This  2 2  (42)  - AE  procedure i s  adopted.  i n t r o d u c e d , d e f i n e d by  (43)  P  H, P and p r e f e r to the i o n i z e d gas, of temperature and p r e s s u r e .  u  and so g i s  f u n c t i o n tends  a function to the con  s t a n t 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 s e p a r a t i n g  the  36  hydrodynamic c a l c u l a t i o n s  from the thermodynamic  calculations,  i  the  l a t t e r amounting The  to the e v a l u a t i o n of the f u n c t i o n  solutions  of equations  (33),  (34),  (35)  g(P,T). and  (43)  are  £i  92 +  =  I  (44)  n  Mi  gi  Substituting the  equations  approximation g  (37)  Mj  x  2  >> g - l  (44)  i n t o equation  (45)  with  yields  2  - \V+ \*  Pz  Equations(43),  and (38)  U  and (46)  then  (46)  give  (47) (92 Equations g  2  is  (44)  an unknown.  (Reference giving p  2  / p i  through  17)  It  (47)  + 1)  all  contain g  equations  g as a f u n c t i o n of P and H. from equation  (46)  of P  2  and H  2  folowing  and (47)  (44). 2  and  iteration and a t a b l e  The compression  f u n c t i o n of P and T (Reference 18), T values  on the r . h . s . ,  may be o b t a i n e d by a simple  involving  then f o l l o w s  2  Since H i s  ratio a known  may be obtained from  from equations  (46)  and  (47)  \  37  respectively.  Finally,  the gas  composition  is  d e r i v e d from  the Saha e q u a t i o n : N. N _j e _ N  Q. o  2TT m  e  i  Q  a  3/2  kT  E .j / kT  (48)  0  with  N. = N i e  (49)  and  P  N.j , N  g  and N  and n e u t r a l Q. and Q neutral  Q  +N )kT * a  are the p a r t i c l e d e n s i t i e s  are the i n t e r n a l  atoms r e s p e c t i v e l y  necessary  4•2  v  atoms r e s p e c t i v e l y .  In  is  a  -(N-+N i e  2  E^ is  partition (Reference  (50)  :  of  ions,  the i o n i z a t i o n  functions  to know i t s  The R e f r a c t i v e  and  18).  r e l a t i o n to the plasma  it  is  parameters.  This  section.  Index of a Plasma  The r e f r a c t i v e index of an i o n i z e d gas from n e u t r a l  e l e c t r o n plasma.  potential.  of ions  order to c a l c u l a t e the r e f r a c t i v e index  the s u b j e c t of the next  tributions  electrons  atoms,  As w i l l  from ions  includes  con-  and from the i o n -  be seen l a t e r , ' under the  physical  38  conditions bution  is  encountered in t h i s that of a c l a s s i c a l  e l e c t r o n gas.  the use of the G l a d s t o n e - D a l e the r e f r a c t i v e index  is  expression  This  wave plasma  is  its  I  =  k.  N  (51)  this  component of  For the. argon  plasma  I  n-1  Ja  +  a,  i and e denote atoms,  respectively.  It  is  refraction,  that  the phase  It is  is  shock  •  n-1  The s u b s c r i p t s  to here.  the  may be w r i t t e n  •  referred  for  i  concentration.  in question  n-1  justifies  (Reference 19)  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  m i x t u r e , and  contri  n:  n-1  k.  experiment, the plasma  i  (52)  n-1  +  ions  and e l e c t r o n s  index of r e f r a c t i o n that  this,  r a t h e r than the group  of i n t e r e s t  is  index  in r e f r a c t i v e phenomena  of  such  as the s c h l i e r e n e f f e c t .  4.2.1  Neutral In  Atoms  the c l a s s i c a l  theory  (Reference 20),  the atom  regarded as a set o f ' e l e c t r o n s of charge e and mass m . electron vidual  is  harmonically  and i s o t r o p i c a 1 1 y  equilibrium position.  beam of l i g h t ,  the a l t e r n a t i n g  bound to i t s  When the atom is  Each indi-  placed in a  e l e c t r i c f i e l d causes  is  forced  \  39  oscillation is  of  the p a r t i c l e s .  The induced e l e c t r i c d i p o l e  c a l c u l a t e d as a f u n c t i o n of time.  p o l a r i zab i 1 i ty  I  =  is  the angular  the resonant  strengths  20) y i e l d s strengths lated  angular  (53)  2  frequency of the i - t h  The quantum mechanical  an equation find  electron; OJ  radiation. with  treatment  of the same form.  The  The  oscillator  resonant (Reference  oscillator  i n t e r p r e t a t i o n as c e r t a i n matrix  elements  re-  moments.  The r e f r a c t i v e  index  is  related  to the p o l a r i z -  by  n-1  so  OJ  -  frequency of the a p p l i e d  to the d i p o l e  ability  2  f.. r e f e r to e l e c t r o n i c t r a n s i t i o n s  f r e q u e n c i e s OJ . .  the  2  — ? — "  m v oj. e i i is  follows  a.  a  0J.J  From t h i s  moment  = 2 TT N a  (54)  that 2 TT N a m_  n-1  Resonance  lines  theultraviolet spectrum.  of n e u t r a l region  Equation  in the form:  e  :  i  atoms l i e  w  i  (55)  OJ'  almost  exclusively  and in s h o r t e r wavelength  (55)  may,  parts  then, be expanded and  in of  the  written  40  k  n-1  k l  +  /a For argon,  k  x  = 1.03 x 1 0 "  (56)  2  X  cm and k  2 3  3  2  = 0.58 x 1 0  - 3 3  cm  5  (Reference 19).  4.2.2  Ions Again the p o l a r i z a b i 1 i t y must be c a l c u l a t e d .  methods are a v a i l a b l e ,  and a l l are approximate.  most commonly used i n v o l v e s  Slater's  screening  These are used to e v a l u a t e mean square r a d i i (References  Several  The best and constants.  of e l e c t r o n o r b i t s  21 and 22).  4 a = £ 9 r' .B"  r „ i s the Bohr radius B  and <r  1  I <r > 2  (57)  z  i  2  .> i s the mean square value of ei  the d i s t a n c e of the i - t h e l e c t r o n from the n u c l e u s .  This  leads  to a s p e c i f i c r e f r a c t i v i t y f o r the argon ion of 0.67  times  that of the n e u t r a l atom.  Since the e l e c t r o n c o n t r i b u -  t i o n to the r e f r a c t i v e index dominates  the ion c o n t r i b u t i o n ,  the probable e r r o r in the l a t t e r i s not s i g n i f i c a n t .  4.2.3  Electrons From Maxwell's equations  equation i s  the e l e c t r i c f i e l d wave  41  J_ i l l  V E - 4irc Va 2  If  j  is  parallel  2  4-rr  +  ^1  (58)  to the wave f r o n t , V«j v a n i s h e s ,  accumulate, and so a vanishes  at  must be found from the  macroscopic equations  of motion of the plasma,  the Bolzmann equation  (Reference  m c e Z p e  m  3t  E + v x B  no charges  d e r i v e d from  23):  n J  eZp  m . - Zm J x B l e  m. VP i e  Zm VP . e i  (59) >  J  'm. and m P. and P i  density. defined  g  e  a r e the masses of ion and e l e c t r o n r e s p e c t i v e l y ; are t h e i r r e s p e c t i v e p r e s s u r e s .  p is  the  plasma  z  ~  is  by the  the charge of the i o n s ,  n  the t o t a l  u n i t time by c o l l i s i o n s the pressure  a constant  assumption  P • = ei where P • i s ei  n is  e  N  c  + J  (60)  J  momentum t r a n s f e r r e d to the ions with the e l e c t r o n s .  does not change during  VP  e  per  Where V«j^ = a = 0,  the o s c i l l a t i o n , and so  = VP. = 0 l.  (61)  42  B may be set at zero a l s o , field  and the f o r c e due to the magnetic  wave is (59)  s i n c e there i s  small  no e x t e r n a l  magnetic  f i e l d v e c t o r of the  compared with the e l e c t r i c  force.  Then equation  becomes  m  For a wave in which E equation i s ,  z  i e Z p e m  is  —  2  §  E  (58)  4TT  z _ x" "  p  Ze  and E  2  m. rn c i e  2  (62)  3t  2  propagated along  from equations d  8l _ *  c  the z a x i s ,  the wave  (62),  z  1  3  2  E  z  8 t  2  The d i s p e r s i o n r e l a t i o n f o r s o l u t i o n s  (63)  z  to equation (63)  of the  f orm exp i ( kz - cot) i s 2  1 -  v is  O) /OJ 2  (64)  Z  p  the phase v e l o c i t y of the wave, and ojp is  the plasma  frequency d e f i n e d by 4TT N  e m  e  2  1 + Z  m — i  (65)  m  The r e f r a c t i v e index  (66)  now f o l l o w s  from equation  (64):  43  n  W  n  For an argon  = 1 -  2  plasma with N  g  = TO  1 7  2  (67)  cm , w  ~ 1.5  - 3  x 10  s" .  1 3  1  o  For the 6328 A l a s e r circumstances,  ojp/w  shock waves of t h i s  r a d i a t i o n , to ~ 3 x 1 0 ~ 5 x 10  - 3  .  1 5  s  - 1  .  In  Electron densities  experiment did not exceed 1 0  1 7  f o r the c m , and - 3  so the approximation oo  << t o was always v a l i d .  This  the c a s e , and t a k i n g ~  as  (65)  2  (67)  2  negligible,  n-1  and  the same r e s u l t as  the d i s p e r s i o n  2  N  -A  = m  is  being  i  render  2 TT e  This  equations  these  e  OJ  (68)  2  that which would f o l l o w  equation of the n e u t r a l gas  from  taking  and e v a l u a t i n g  it  o  for  the case of no resonances.  the constants N in cm - 3 e  of equation  (68)  Again  f o r 6328 A r a d i a t i o n ,  may be e v a l u a t e d .  n - 1 = -1 .796 x l O " e  4.3  Numerical  2 2  N  Then,  with  e  (69)  Calculations  The r e s u l t s  of t h i s  chapter were used to e v a l u a t e  the r e f r a c t i v e index v a r i a t i o n with temperature and pressure of a-general plasma .  argon  plasma and of a shock wave produced argon  \  44  In  performing the thermodynamic c a l c u l a t i o n s , g  and T were obtained as data but from f i r s t  functions  principles  f  H = 1 P  Equation simple  (70)  kinetic  follows  of P and H not from e x t e r n a l  using  P  N.  +  (70)  i  from the d e f i n i t i o n of enthalpy and  theory.  The p a r t i t i o n f u n c t i o n s the method of D r e l l i s h a k tures  E  were c a l c u l a t e d  et al. (Reference 18).  considered here were not high enough f o r  further  ionizations  to be of  t i o n made in equation  (49)  n e u t r a l i t y of the o r i g i n a l  significance,  refractive 14(a).  The temperasecond and  and so the assump-  remained a v a l i d one, given test  Computer programmes I.B.M. 370 computer.  following  charge  gas.  were w r i t t e n in F o r t r a n f o r an  P vs T curves  index were determined.  f o r constant  values  These are drawn in  of  Figure  A l s o , ( n - 1) was determined f o r a shock wave plasma  a f u n c t i o n of the Mach number Mi of the shock wave and the pressure 14(b).  Pi of the t e s t The c a l c u l a t i o n s  and f o r l i g h t  gas.  The curves are drawn in  were performed throughout  of wavelength  o  6328 A.  for  as initial  Figure argon  \  45  15-0 * T E M P E R A T U R E  Figure  14(a).  (DEG REE S K x lO  V a r i a t i o n of P with T f o r f i x e d values of the r e f r a c t i v e index of an argon plasma .  3  )  46  Figure  14(b).  R e f r a c t i v e index as a f u n c t i o n o f  o f an a r g o n P and Mi. x  shock  wave  plasma  \  Chapter 5  EXPERIMENT AND RESULTS  5.1  V e l o c i t y Measurements The  shock wave v e l o c i t y , necessary f o r the c a l c u l a -  t i o n of the shock wave parameters, was measured with a smear camera.  The c o n s t r u c t i o n of the smear camera i s  d e s c r i b e d in  Reference 13. The of  shock wave was f i r s t  photographed in the part  the pyrex shock tube through which the shock wave  just  before e n t e r i n g the t e s t s e c t i o n .  passed  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 s p a t i a l writing  progress  of the shock wave.  speed of the camera was 46.96 cm/s,  this  figure  The follow-  ing from a r o t a t i o n frequency measurement on the drum bearing the  mirror.  to  make t h i s  A Monsanto  measurement.  photograph of an argon M  Programmable Counter-Timer was used Figure 15 shows a c o l o u r smear  shock wave f o r which P i = 4 t o r r and  = 14.3.  47  CM  Figure  15.  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 t e s t in  s e c t i o n by the smear camera f e l l  the e a r l i e r s e c t i o n of the shock  tube.  from i t s  A typical  value  smear  camera photograph f o r the measurement at the t e s t  section  shown in Figure 16.  is  The t h i n v e r t i c a l  of a marker on the shock tube; i t  black  line  the  shows the end of the  is image  test  s e c t i o n at which the shock wave e n t e r e d .  D I S T A N C E <~  R=4  11  TORR  TEST S E C T ION  THIN  PYREX  MARKER  SHOCK  u  TUBE  Figure 16.  T y p i c a l smear photograph f o r at the t e s t s e c t i o n .  v e l o c i t y measurement  \  50  The shock wave v e l o c i t y was measured from the on the time a x i s  positions  of the l e a d i n g edge of the shock wave as  j u s t entered and j u s t  l e f t the t e s t  section.  were made f o r each pressure P i that was numbers were c a l c u l a t e d using sound in argon at 20°C i s  it  Five measurements  investigated.  Mach  the f a c t that the v e l o c i t y of  0.318 km/s.  The r e s u l t s  are shown  i n Figure 17.  SHOCK WAVE MACH NUMBER  40  BO >p,  Figure 17.  Mi vs Pi f o r shock waves  (IN  in argon.  !60 TORR)  \  51  5.2  Shock Wave Parameters Using  for the  the r e s u l t s  of S e c t i o n s  4.1  and 4.2, and  shock wave v e l o c i t i e s , the thermodynamic  5.1  parameters  and the r e f r a c t i v e indexes of the shock wave plasmas were calculated  for various  the i n i t i a l Table was  I  test  gas  is  values  of P .  important, and t h i s  was a l s o c a l c u l a t e d .  shows the shock wave parameters f o r Pi = 16 t o r r ;  the f i r s t  shock wave to be i n v e s t i g a t e d .  temperature of 20°C was assumed. is  The r e f r a c t i v e index of  x  this  An i n i t i a l  The atomic weight  of argon  39.944 f o r the i s o t o p i c mixture found in the environment.  5.3  Density It  Measurements remains  at Pi = 16 T o r r  to t r e a t of the a p p l i c a t i o n of the den-  s i t o m e t e r to check that the measured d e n s i t y jump across shock  f r o n t was in accordance with the value expected from  shock wave theory and the v e l o c i t y measurements. carry  the  this  out,  the r e f r a c t i v e index jump across  f r o n t was measured f o r a Pi = 16 t o r r  was observed during  order to  the shock  shock wave.  was compared with the expected value of 1.51 It  In  The r e s u l t  x 10" . 5  the course of the experiment  that the quartz windows of the t e s t  s e c t i o n darkened with  successive  This darkening was  shock wave d e t o n a t i o n s .  caused  by the d e p o s i t of t h i n l a y e r s  of carbon by the d r i v e r gas.  consequence, i t was necessary  to c a l i b r a t e the densitometer  before every measurement.  In  52  INITIAL TEST  SHOCK  WAVE  M A C H  VELOCITY  S  9-90  (DYNE/crYv ) v  DENSITY  (3/ ^ c  COMPRESSION  E F F E C T I V E  )  (DEGREES  ADIABATIC  K)  N,  2-64 x l O  3-50x10"*  1-27 x l O " "  293  8'96 I0  1-6  EXPONENT  b  3  X  1-63  7  5-2 8 xlO  (ch," ) 3  A  2-13x10^  3^62  RATIO  TEMPERATURE  N  3-15  (K^/ )  NUMBER  P R E S S U R E  SHOCK WAVE PLASMA  GAS  2-1 2 x l d '  n  8-33 x l d  ( c h ^ )  8-33 x  Ne ( c m - ) 3  5-49  x IO"  1 Cf  2-20xlO"  k  5-8  K  S  5  lO*  - l-50xlO" l-5| IO"  b  5  x  Table  I.  The test  Shock  length  W a v e P a r a m e t e r s f o r P i = 16 T o r r .  6 o f t h e p a t h o f t h e l a s e r beam  s e c t i o n was measured  a s 2.45 cm.  A typical oscillogram lower trace integrated  i s shown i n F i g u r e  i n the  ' 18. T h e  i s t h e s c h l i e r e n s i g n a l ; t h e upper t r a c e s i g n a l , which  i s the  i sa l i n e a r analogue f o r the density  \  53  profile.  The v o l t a g e  also  for density.  also  for distance  scale  of the upper t r a c e  Similarly, along  the time a x i s  the shock  is  is  calibrated  calibrated  tube.  SIGNAL V O L T A G E(-VE)  2<OOmv(2-36x  t  DENSITY ANALOGUE SIGNAL S C H L I E REN SIGNAL  10'emf 100 m V ^ T  l/JS ( 0 . 2 9 c r * )  ^ T I M E (SHOCK  WAVE  DISPLACEMENT) (J~\  C A L I BRAT I O N  Figure  18.  =  1 - 0 8 x 1 0 m V / RADIAN b  S c h l i e r e n s i g n a l and d e n s i t y analooue s i g n a l f o r a Mach9-9shock wave in a r g o n , Pi = 16 torr  Four o s c i l l o g r a m s average  value of An was,  This d i f f e r s about  were taken f o r  from equation  Pj = 16 t o r r .  (28),  An = 1.45  from the expected v a l u e , An = 1.51  - 5  x 10" . 5  ,  by  4%. There were three major c o n t r i b u t i o n s  c r e p a n c y , each one d i f f i c u l t to estimate were f i r s t l y In  x 10  The  the i n a c c u r a c i e s  particular,  the high  to  this  dis-  quantitatively.  There  i n t r o d u c e d by the e l e c t r o n i c s .  pass f i l t e r may have had an e f f e c t on  the lower frequency F o u r i e r components  of  the s c h l i e r e n  signal,  54  thereby s h i f t i n g integrator  the e f f e c t i v e c a l i b r a t i o n constant  f o r that  signal.  Secondly,  the  the flow f i e l d of  shock wave may not have been uniform across diameter of the shock  of  the  the whole of  tube t r a v e r s e d by the l a s e r  the  beam.  Laminar flow e f f e c t s would serve to e f f e c t i v e l y decrease 6. Lastly,  the measured value of Pi may have been in e r r o r ,  causing  an e r r o r in the c a l c u l a t i o n of the expected value  An.  In view of these p o s s i b i l i t i e s ,  t i o n and measurement  5.4  Qualitative  greater  predic-  limits.  d e c r e a s e d , the shock wave v e l o c i t y  ratio,  temperature and s p e c i f i c  i n c r e a s e , and i o n i z a t i o n the negative  to be w i t h i n a c c e p t a b l e  of  I n v e s t i g a t i o n at Pi = 2 Torr  As Pi is The compression  seems  the agreement  of  becomes a p p r e c i a b l e .  c o n t r i b u t i o n to  than the p o s i t i v e  increases.  enthalpy For Pi = 2 t o r r ,  (n - 1) from the e l e c t r o n s  is  c o n t r i b u t i o n from the n e u t r a l  atoms.  The approximate e l e c t r o n r e f r a c t i v e index is  (n - 1)  ~ -  For the n e u t r a l  For the  ions,  atoms,  (n - 1 ) ,  ~ 5 x 10" . 6  10  - 5  .  a  (n - 1 ) .  ~ 0.5 x 1 0 " . 6  0.7 x 1 0 ~ . 6  of n e u t r a l gas  is  The change particles  For the undisturbed  in the shock wave and the o r i g i n a l 4 x 10" . 6  in r e f r a c t i v e index i s  e f f e c t of the ions Figure  (n - 1) ~  in r e f r a c t i v e index between the  then approximately  the change  t e s t gas,  For the e l e c t r o n  approximately  f o r a Pi = 2 t o r r shock wave.  test assembly,  - 10" .  on the r e f r a c t i v e index change  19 shows an o s c i l l o g r a m  assembly  5  is  small.  of the s c h l i e r e n  No attempt was made to  The  signal  integrate  55  the  signal  electronically.  was 49 mV/us  (7704 C.R.O.  from S e c t i o n  5.1, was 5.01  The c a l i b r a t i o n curve trace).  gradient  The shock wave  mm/us, Mach  velocity,  15.7.  SIGNAL V O L T A G E  (-VE)  I•  1  T -X  k>TIME (SHOCK  C A L  I B R A T  F i g u r e 19.  I O N  \  : ( 4 j - \ = 0 - 9 9 5 mV/  Schlieren signal Pi = 2 t o r r .  A positive  voltage  WAVE  DISPLACEMENT)  RADIAN  f o r a Mach 16 shock wave i n  deflection indicates  argon,  a schlieren  d e f l e c t i o n due to the passage in the d i r e c t i o n of  the  wave of an i n c r e a s e  to say,  neutral  atom c o n t r i b u t i o n d e f l e c t s  while the change assembly In  in r e f r a c t i v e index.  That is  the s i g n a l  in r e f r a c t i v e index due to the e l e c t r o n  of the shock wave plasma  the o s c i l l o g r a m  of Figure  through which the t r a c e runs. is  tion.  almost Making  refractive  the  positively,  deflects  the s i g n a l  19, the e a r l i e s t  signal  above the noise can be seen in the second g r a t i c u l e  It  shock  This  is  a positive  immediately followed by a n e g a t i v e an estimate of the area of  index jumps corresponding  negatively. deflection  square  deflection.  signal  deflec-  these s i g n a l s ,  to the f i r s t  and  the  second  \  56  deflections  r e s p e c t i v e l y are about + 2 x 10~  These are roughly  half  the values  probably accounted f o r p a r t l y c a n c e l l a t i o n of the opposing the  neutral  causes  to the n e u t r a l length tive  p r e d i c t e d above, which  by the l i k e l i h o o d of  .  is  some  e f f e c t s of the e l e c t r o n s and  t r a c e of Figure  19 a l s o  a schlieren signal atoms  by about  shows that  which f o l l o w s  1 ys .  This  of 5 mm on the shock wave a x i s .  r e f r a c t i v e index g r a d i e n t s  almost  p a r t i c l e density  SIGNAL i V O L T A G E (-ve)  /  profiles  \  is  the e l e c t r o n the s i g n a l  corresponds  The p o s i t i v e  due  to a and  c e r t a i n l y overlap  c e r t a i n e x t e n t , and so a d e t a i l e d d i s t i n c t i o n neutral  - 6  atoms.  The density  and -6 x 1 0  6  negato a  of e l e c t r o n and  not p o s s i b l e .  The form  A  *TIME  P A R T I C L E DENSITIES  1- i  •  • + I ~J~ ! T | % t N 4 ' 3 K I O + "j"  J \1  4-  1=  N  e  = Nj-1  t  Figure 20.  5 rntn  k  io'  •NEUTRAL  -tH  x  M  -^DISTANCE  ATOMS  -ELECTRONS +IONS  P a r t i c l e d i s t r i b u t i o n s i m p l i e d by the s c h l i e r e n s i g n a l f o r the Mach 16 shock wave in argon, Pi = 2 t o r r .  of the p a r t i c l e d i s t r i b u t i o n s indicated  in Figure  the e x p e c t a t i o n approximately  20.  (Section  2 mm.  i m p l i e d by the  oscillogram  The r e s o l u t i o n obtained 2.2) of a s p a t i a l  supports  resolution  of  Chapter 6  CONCLUSIONS  6.1  General  Conclusions  The attempt to c o n s t r u c t a s c h l i e r e n utilizing  the wedge technique was  been demonstrated w i t h i n  the l i m i t s  d e n s i t y measured and the d e n s i t y tive  index changes  The a b i l i t y negative  p r e d i c t e d by t h e o r y .  Refrac-  were measurable  5  A spatial  of the densitometer to t r a c k  r e f r a c t i v e index changes  the conventional  advantage  was  resolution  by  of  system  it  accuracy due to geometric c o m p l e x i t i e s equation  (28)  of  both p o s i t i v e  also  and  demonstrated.  of the wedge technique over  k n i f e edge techniques  and f i d e l i t y of the o p t i c a l  assumptions.  having  2 mm was e x p e c t e d , and found to be a c h i e v e d .  The p r i n c i p a l  operating  agreement  of accuracy between the  of the order of 10~  the device without d i f f i c u l t y . approximately  successful,  densitometer  lies  in the  employs. is  the d e v i c e is  simplicity  Loss of  negligible,  and the  based on very few  The wedge need not even be l i n e a r over the  whole region of  i n t e r s e c t i o n of the l a s e r  58  beam; i t  needs  to  \  59  be l i n e a r only over the s c h l i e r e n d e f l e c t i o n of rays.  Diffraction  with  k n i f e edge  6.2  Future Work  problems are not encountered as  the beam is  ways in which the device  A double wedge  used by P o t t e r  they are  techniques.  There are several be improved.  individual  (References  divided into  technique s i m i l a r  11 & 12)  might  to the one  could be i n t r o d u c e d .  two channels which are i d e n t i c a l  for opposite  senses  two channels  are e x a c t l y b a l a n c e d , when t h e i r s i g n a l s  subtracted ing  of the wedge g r a d i e n t .  the plasma  s c h l i e r e n signal  light  measures  with twice the s e n s i t i v i t y single  contributions  As long as  cancel.  Here but  the are  The remain-  the r e f r a c t i v e index  that would be obtained  gradient using a  wedge. The s e n s i t i v i t y  the d i s t a n c e  from the t e s t  advantage  gained  expansion  of the beam's  divergence.  might a l s o section  by going t h i s  Distances  cross  is  be improved by  to the wedge.  However, the  severely o f f s e t  s e c t i o n due to i t s  by the natural  of between 1 and 2 metres are a p p r o p r i a t e .  The a p p r o p r i a t e wedge g r a d i e n t s  are then not great  create d i f f r a c t i o n e f f e c t s .  should  arrangement techniques  offers  increasing  It  enough  be noted that  a more compact system than most  to  this  conventional  al1ow. The s p a t i a l  r e s o l u t i o n could be improved by reduc-  t i o n of the diameter of the beam by o p t i c a l  means,  such as  \  \  60  is  d e s c r i b e d in Reference 6.  However, an attempt to reduce  the beam diameter was made and proved u n s u c c e s s f u l . i n d i c a t e d that a great make a s i g n i f i c a n t  deal  This  of e f f o r t would be necessary  to  improvement.  The instrument could be adapted to measure the change i n r e f r a c t i v e index across  stationary  beam through the o b j e c t in q u e s t i o n . arrangement Reference  1.  to perform t h i s  o b j e c t s by sweeping A possible  function is  suggested  optical in  the  \  BIBLIOGRAPHICAL REFERENCES  1.  Boye Ahlborn and C h r i s t o p h e r A.M. Humphries, , I n s t r . , Vol . 4J7_, No. 5 ( 1 976 ) .  Rev.  Sci.  2.  Michiru  3.  E'.L.  4.  Jerome Daen and P.C.T. ( 1 962 ) .  5.  P.C.T.  6.  John H. K i e f e r and Robert W. L u t z , Phys. ( 1 965) .  7.  John H. K i e f e r and Robert W. L u t z , J . Chem. Phys., 658 (1966).  8.  Robert W. Lutz and John H. K i e f e r , Phys. ( 1 966).  9.  Laurence S.  Yasuhara, Kenji Yoneda and Susumu Sato, J . Soc . Japan , 3_6 , 555 ( 1 974) .  R e s l e r , J r . and M. S c h e i b e , J . A c o u s t , 27, 932 (1955).  de Boer, Rev.  10.  G. Dodel  11. *  Michael  12.  M.U.  H a l l , Rev.  Soc. Am.,  de Boer, J . Chem. Phys.,  Sci.  Instr.,  Sci.  Phys.  3_6, 1222  3_6 , 1 1 35 ( 1 965 ).  Instr.,  F l u i d s , 8,  Fluids,  37, 1735  and W. Kunz, A p p l i e d O p t i c s ,  1393  44,  9,  1638  (1966).  1_4, 2537  ( 1 975).  U. P o t t e r , M. Sc. T h e s i s , Department of P h y s i c s , The U n i v e r s i t y of B r i t i s h Columbia, 1967.  P o t t e r and B. A h l b o r n , AIAA J o u r n a l , 6, 2227  61  (1968).  \  62  13.  Jean-Paul R. Hum', Ph.D. T h e s i s , Department of P h y s i c s , The U n i v e r s i t y of B r i t i s h Columbia, 1970.  14.  P. Redfern and B. A h l b o r n , Can. J . Phys.,  15.  Ya. B. Z e l ' d o v i c h and Yu. P. R a i z e r , Elements of Gasdynamics and the C l a s s i c a l Theory of Shock Waves, Academic Press I n c . , 1968.  16.  A.G. Gaydon and I.R. H u r l e , The Shock Tube in HighTemperature Chemical Physics, Chapman and Hall Ltd. London, 1963.  17.  Boye A h l b o r n , Can. J . Phys.,  18.  K.S.  Drellishak, Phys.  CF.  1280  (1 972 )  5_3, 976 (1 975).  Knopp and A l i  F l u i d s , 6,  5_0_, 1 771  Bulnet Campbell,  (1963).  19.  V. Hermoch, Czech. J . Phys.,  20.  Joseph 0. H i r s c h f e l d e r , C h a r l e s F. C u r t i s s and R. Byron B i r d , Molecular Theory of Gases and Liquids, John Wiley and Sons I n c . , 1954. Ralph A. Alpher and Donald R. White, Phys. F l u i d s , 2, 153 (1959).  21..  1_2 , 939 (1 970).  22.  Ralph A. Alpher and Donald R. 1 62 ( 1 959 ) .  White, Phys.  23.  Lyman S p i t z e r , J r . , Physics of F u l l y Wiley and Sons Inc., 1962.  Ionized  F l u i d s , 2,  Gases,  John  \  APPENDIX A  Reference 1: Boye AhlOTrn and C h r i s t o p h e r A.M. Humphries, "A Q u a n t i t a t i v e S c h l i e r e n Densitometer Employing a Neutral Wedge," Rev. S c i . I n s t r . , V o l . 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^AA  Quantitative schlieren densitometer employing a neutral density wedge  hectic  Of^'^ •  Boye Ahlborn and Christopher A. M. Humphries Department of Physics, University of British Columbia, Vancouver, British Columbia V6T 1W5, Canada  (Received 29 September 1975; infinalform, 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. T i m e integration of the attenuated probe beam intensity yields the absolute refractive index variation across moving objects. T h e device is tested by measuring the density j u m p across a M a c h 9 shock wave.  INTRODUCTION  ported p r e v i o u s l y . T h e lens L focuses the deflected beam to roughly the same area o n the p h o t o m u l t i p l i e r ( P M ) independently o f the deflection a, so that the signal varies w i t h deflection o n l y o n a c c o u n t o f the neutral wedge. L i m i t e d use was made o f the d y n o d e c h a i n , the seventh d y n o d e being used as the anode in o r d e r to reduce the signal-to-noise ratio. A n a r r o w - b a n d interference filter I is interposed to e l i m i n a t e p l a s m a light emitted during the measuring process. 0  S c h l i e r e n techniques have been used in s h o c k w a v e studies f o r a long time to detect density steps a n d to measure density v a r i a t i o n s . " " ' l n these diagnostic systems one uses the fact lhat an o p t i c a l density gradient (schliere) deflects a probing light beam and the deflection angle a is d i r e c t l y p r o p o r t i o n a l to the gradient o f the refractive index. In the standard arrangement the probing beam is initially b l o c k e d b y a k n i f e edge or slit. W h e n the density gradient is i n t r o d u c e d , the beam is deflected to pass the knife edge and reach a detector. If the beam deflection is small c o m p a r e d t o the beam w i d t h at the knife edge, the deflection angle a will be p r o p o r t i o n a l to the amount o f flux w h i c h passes the edge. T o obtain a signal well a b o v e the noise, o n e w o u l d like to have as large an angle a as possible, so that the measuring beam w o u l d have to be quite w i d e in turn. A w i d e signal b e a m , h o w e v e r , i m p l i e s a l o w spatial resolution. In c o n v e n tional s c h l i e r e n systems one c a n find an o p t i m u m tradeo f f b e t w e e n spatial r e s o l u t i o n a n d angular deflection a . T h e uncertainty o f the angular measurement is essentially due to d i f f r a c t i o n o f the slit o r knife edge c m p l o y e d in these s c h l i e r e n systems. T h e r e f o r e , to m i n i m i z e the d i f f r a c t i o n effects, o n e ought to eliminate the slit o r knife edge. 1  DENSITY STEPS ACROSS MOVING OBJECTS T h e s c h l i e r e n deflection a o f a ray o f light passing through a length 5 o f a material w i t h transverse r e f r a c tive index gradient dn/dx is given b y  _ S dn n dx  W e have d e v e l o p e d a quantitative schlieren s y s t e m in w h i c h the knife edge is r e p l a c e d b y a neutral density wedge w h i c h essentially e l i m i n a t e s the diffraction effects. T h e flux behind the wedge is p r o p o r t i o n a l to the location o f the center o f g r a v i t y o f the p r o b i n g beam, and varies therefore w i t h the d e f l e c t i o n angle a. T h e b e a m shape is u n i m p o r t a n t , as was d i s c u s s e d in a p r e v i o u s a p p l i c a t i o n o f neutral density wedges to the measurement o f s m a l l shifts o f w i d e s p e c t r a l l i n e s . 8,7  (1)  w h i c h f c r laboratory plasmas (n — 1) m a y be t a k e n as  dn a~8dx  5  (2)  If the deflection is caused b y material m o v i n g at u n i f o r m v e l o c i t y v — dx/dt, one has  8 dn v dt  (3)  H e n c e , one finds the change o f r e f r a c t i v e index A/i between t = -so a n d t s i m p l y b y integrating E q . (3): l  a(t) dt = —Anitt).  (4)  T h i s time integration m a y be c a r r i e d out e l e c t r o n i c a l l y so that the v a r i a t i o n o f the index o f r e f r a c t i o n a n d , hence, the density c a n be m e a s u r e d i m m e d i a t e l y .  APPARATUS CALIBRATION  A s k e t c h o f the apparatus is given in F i g . 1. T h e probing beam is p r o v i d e d b y a S p e c t r a P h y s i c s 115 H e O u r apparatus is designed to measure density steps N e laser of 0.5-mW beam p o w e r and 6328-A w a v e l e n g t h . ' a c r o s s s h o c k waves. T h e o b s e r v a t i o n region P ( F i g . 1) T h e neutral density wedge N w i t h transmission v a r y i n g is a section o f a 2.5-cm-diam s h o c k tube w i t h p a r a l l e l in the X d i r e c t i o n is p r o d u c e d p h o t o g r a p h i c a l l y , as re- w i n d o w s . S h o c k waves propagate in the ^ d i r e c t i o n . T h e 570  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  6k  Sfr  570  .  M  I  2  HeNe  shock 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 M , and the stationary mirror M : The angle between M, and M (95°), the distance between rotation axis 0, signal beam axis, and the position of M are carefully chosen so that the beam appears to be rotating 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 frequency <VQ. V, corresponds to the voltage signal for the light path shown in Fig. 1 (solid line). The narrow 3  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 v 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 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 V> indicating 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,, M , and M . 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,JV . With this precaution the variation of intensity of the measuring' beam V - V = A V can be transformed into a schlieren deflection angle, 0  -3  2  2  2  0  2  3  t  0  (a)  a - BAV, fi = ^ = dV  1.2 x 10-  (5) mV so that the change of refractive index is found with Eqs. (4) and (5):  u—  10msec  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. 2p  EXPERIMENTAL TEST OF THE DEVICE  sec  FIG. 2. (a) Photomultiplier signal obtained when the signal beam is chopped by rotating prism (V,) and deflected by mirrors M „ M . and M (V ). (b) Calibration signal obtained with rotating prism. Time scale ( is also calibrated in deflection angle a for / = 7.17 cm. 2  3  571  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  1976  To test the schlieren wedge system the density variation across shock waves was measured. The shocks were generated with a gaseous detonation driver deSchlieren  65  densitometer  571  scribed previously. T h e initial pressure of the driver gas, an equimolar mixture of C H plus 0 , was 350Torr. The shock wave velocity was measured for various pressures of the argon test gas by using a smear camera. The shock tube reaches Mach 17.9 in 1 T o r r of argon test gas, Mach 14.3 in 4 Torr argon, and Mach 9.2 in 16 T o r r argon. T h e dependence of the refractive index upon the plasma composition is well understood. The subject is treated by Alpher and White. For argon gas the relationships at the wavelength of 6328 A are 8  2  2  2  9  N=  FIG. 4. Arrangement to sweep the probing beam through a stationary object ftP. The glass blocks G, and C rotate synchronously in opposite directions, thus producing a constant sweep speed for small angles y. 2  1 + 1.04 x 10- «„-, 23  a  N, = 1 + 0.70 x 1 0 - « , 23  N  e  = 1 -  = 2.2 x 10 c m " and N = N = 9.7 x 10 cm" . 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 n = 1 + 2.25 x 10 . Subtracting the refractive index for the gas in the initial stage,/i, = 1 + 5.5 x 10 , one finds that the theoretical increase across the shock is  (  18  18.0 x 1 0 - « , 23  e  where «„, n , and n are the refractive indexes, and N„, Nf, and N are the number densities in c m 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 t  e  - 3  e  n -  1 = £  («, -  1).  2  2  2  a2  14  i 2  1  -5  _G  (7)  sll0c:k  a  t  An = 1.7 x 10" ,  (8)  5  j In the test experiment, shocks of Mach 9.2 in argon of Po = 16 T o r r 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 Af. With standard shock wave theory one then finds for the shock-compreSsed gas a pressure p - 2.0 x 10" dyn/cm , an equilibrium temperature of T = 7500 K , and the composition N  3  which causes the schlieren deflection and is measured in the experiment. Entering this result into E q . (3), written in the form aAt = (S/v)An, and using S = 2.5 cm and v = 2.94 x 10 cm/sec, we have 5  aAt = 1.4 x i d " ' rad sec. 0  (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 rad sec ( ± 1 0 % ) . The agreement is reasonably good considering that the calculations depend critically on the initial filling pressure. Since no absolute calibration of the pressure manometer was available, no error margin for the expected value in E q . (8) could be quoted. -10  MODIFICATION FOR STATIONARY OBJECTS  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.  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 v is known, one can again replace dX by v dt in E q . (3) and find the refractive index through the integration of E q . (4). A possible arrangement for sweeping the observation beam through the stationary object (such as an arc) is shown in F i g . 4. In analyzing cylindrical (or nonplanar) objects, one must account " 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.  572  Schlieren densitometer  s  s  2  R e v . S c i . I n s t r u m . , V o l . 47, N o . 5 , M a y  3  1976  t  1  66  572  ACKNOWLEDGMENTS We thank B. Armstrong for valuable comments on the manuscript. This work was supported by a grant from the A E C B of Canada. 1 ! 3  7  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). A. G. Gaydon and I. R. Hurie, The Sliock Tube in High  Temperature ChemicalI'hysics (Chapman and Hail, London, 1961).  573  R e v . S c i . Instrum., V o l . 47, N o . 5, M a y 1976  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). 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). 4  Schlieren  67  densitometer  573  \  APPENDIX B  THE NEUTRAL DENSITY WEDGE  The emulsion  is  important  its  H & D curve.  D of the negative exposure. the  c h a r a c t e r i s t i c of a This  is  photographic  a p l o t of the  as a f u n c t i o n of the l o g a r i t h m  The exposure  E is  density  of  the  r e l a t e d to the i n t e n s i t y  I and  time t of exposure by  E = It  The  transmission  (B.l )  T of the negative  is  r e l a t e d to i t s  density  by D = - log  A typical the  H & D curve is  1 0  T  (B.2)  shown in Figure  B.l.  The shape  of  curve depends upon the type of developer used and the  time of  development. The  to as  range  of l i n e a r i t y of  the gamma of the f i l m .  exposures  for ordinary f i l m s .  It In  68  the H & D curve i s  covers this  the useful region  referred  range  of  69  D = k + ylog ! o E  Substituting  equation (B.2)  i n t o equation  T = k'  E"  (B. 3)  (B.3)  gives:  (B.4)  Y  where  log  1 0  k  1  - -k  The wedge was produced on a Kodak type 649, developed 5 minutes  (B.5)  photographic  in Kodak D 19 d e v e l o p e r .  plate, A  \  70  one-dimensional  l i n e a r g r a d i e n t of l i g h t  j e c t e d on to the p l a t e by a h a l f be seen from equation v a r i a t i o n of E. of t r a n s m i s s i o n  (B.l),  Equation  shadow t e c h n i q u e .  this  (B.4)  However y i s  mission-distance  a  with d i s t a n c e  small  variation  should not be  r a i s e d to the  negative  number and so the  trans-  curve should be approximately l i n e a r .  was demonstrated by the c a l i b r a t i o n curves which i n d i c a t e d that between t r a n s m i s s i o n s 64%/mm.  As can  shows, t h e n , t h a t the  T of the p l a t e  pro-  brought about a l i n e a r  l i n e a r but should vary as d i s t a n c e power of y•  i n t e n s i t y was  This  (e.g.  Figure  the wedge was approximately  linear  of 25%'and 75%, with a g r a d i e n t  of  12),  APPENDIX C  OPTICS OF THE CALIBRATION SYSTEM  in  The  geometry of the m i r r o r system  is  illustrated  Figure C l .  The r o t a t i n g m i r r o r assembly  is  drawn in two  positions:  in the p o s i t i o n drawn with f u l l  pass the beam in such a way that its  original  the  m i r r o r assembly  the  first  that The is  is  r o t a t e d by a small  Rays i n d i c a t e d by f u l l  lines.  angle  lines  Real  and v i r t u a l  rays  Here i t  is  6 from the  show the  reflecposition  i n d i c a t e d by  hypo t h e t i c a 1 1 y  the second m i r r o r of the prism assembly path of rays  assumed  has not r o t a t e d .  r e f l e c t e d by both m i r r o r s when r o t a t e d by 6  shown by dotted  reflection  lines,  show the e f f e c t on the beam of the r o t a t i o n of  m i r r o r by 6.  The  the  u n d e f l e c t e d from  by the system when the m i r r o r s are in the  drawn with f u l l dashed l i n e s  emerges  the m i r r o r s  p a t h ; i n the p o s i t i o n drawn with dotted  above p o s i t i o n . t i o n of l i g h t  it  lines,  lines.  dashed ray i s  at the f i r s t  d e v i a t e d through an angle  mirror.  It  appears  to diverge  26 by  its  from  u n d e f l e c t e d ray from a p o i n t X a f t e r f u r t h e r r e f l e c t i o n s  at the second and t h i r d m i r r o r s .  71  The r e a l  point of divergence  Figure C.l  Optics  of  the m i r r o r  system of  the c a l i b r a t i o n  device.  73  is  Y.  This  rotates  is  slightly  displaced  about 0 and not B.  It  from B because  is  easy to see  the m i r r o r  that  C'Y = C ' Z  It  follows  from equation  (Cl)  D'Z  Applying  that  = D ' C '. + C'Y  it  follows  that  D'X  Hence, s u b s t i t u t i n g  equation  D'X  However, the a c t u a l  = D'Z  (C.2)  = D'C'  by 8 causes  (C.3)  in equation  (C.3),  + C'Y'  (C.4)  r e f l e c t i o n of the r o t a t e d beam is  i n d i c a t e d by the dotted r a y .  surface  (C.2)  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 a t D',  as  (Cl )  the dotted and dashed rays  to  diverge  26.  Hence, the dotted and  rays  6 is  a small  between D'  Now i f  and D" may be n e g l e c t e d .  D"A  D",  The r o t a t i o n of the m i r r o r  from each other by an angle d i v e r g e by 46.  at  = iD'X  angle,  the  full  distance  Then, again f o r small  8,  (C5)  74  From equation  (C.4),  this  becomes  D" A = i D ' C  Finally, (C,6)  if  DD  1  + C'Y  (C6)  and BY are both taken as  small,  equation  becomes  DA = i(DC  + CB)  (C.7)  = DC  This  is  the r e s u l t  have been taken as 6.  (C.8)  quoted in S e c t i o n 2.3. small  are a l l  of f i r s t  The d e f l e c t i o n s encountered in t h i s  order of 10~  4  The q u a n t i t i e s  that  and higher order  in  experiment are of the  r a d i a n , 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 i s shown in Figure D . l .  SPARK  GAP D E  - 0 O - 4 O T O R R  N ^0OI6OTORR  S  V A C U U M  I  PUMP  T  - ® — - A R G O N  O M E  -0O-2OOO/;  T  (PI  E R -®  0 0-2400 -0  AIR  TORR  AIR VACUUM (  )<—PUMP  ACETYLENE] OXYGEN  Figure D . l .  General layout of shock tube a p p a r a t u s .  75  RANYl)  \  76  The detonation chamber and the mixing about three l i t r e s  in volume.  gauge monitored the pressure filling 0-40  tank were both  The 0 - 2400 t o r r in the mixing  pressure  tank and the  pressure when the d e t o n a t i o n chamber was f i l l e d ; t o r r pressure  gauge was  used to monitor  i n the detonation chamber when i t Similarly  the 0 -  160 t o r r and the 0 - 4 0  gauges measured the pressure Piranyi  was being  the  pressure  pumped down.  torr  of the argon t e s t  pressure gas,  while  gauge monitored the e v a c u a t i o n of the shock  tube.  In order to minimize the movement of the t e s t due to the d e t o n a t i o n , the d e t o n a t i o n chamber and the tube were mounted on separate f i r m l y f i x e d to i t s support  extended).  friction  table,  tables.  to s l i d e  of  50 lbs  s t r u c t u r e of the d e t o n a t i o n and to reduce the r e c o i l  produced at  capacitor. vide a spark  and sandbags f o r  the spark  tube was  that would set o f f system  chamber in order to  velocity.  were  vibrations.  ignited  by an  electric  of a 1.72  sufficient  the d e t o n a t i o n . given  increase  The t a b l e s  r e d u c t i o n of  of 18 kV was  is  its  would be o v e r -  gap by the d i s c h a r g e  A charging v o l t a g e  diagram of the i g n i t i o n  shock  wt was appended to the  The oxygen-acetylene mixture was spark  section  on the s u r f a c e f o r a  (beyond which the bellows  A lead weight  laden with b r i c k s  The shock  the  but the d e t o n a t i o n chamber and  s t r u c t u r e were allowed  c e r t a i n distance  support  the  yF  to p r o -  The c i r c u i t  in Reference 13.  \  \  APPENDIX E  APPLICATION OF THE R.C.A. 931A PHOTOMULTIPLIER  In  normal  use,  the ten dynodes  tube serve as a m p l i f i c a t i o n which o r i g i n a t e s is  at the photocathode.  picked up by the anode and d r i v e s  producing the  stages f o r  a voltage  output.  of the p h o t o m u l t i p l i e r  the e l e c t r o n c u r r e n t , The a m p l i f i e d  signal  a load r e s i s t o r ,  thus  Each p h o t o e l e c t r o n produced at  photocathode e v e n t u a l l y produces a pulse of output  and the output v o l t a g e pulses  of a l l  waveform i s  the photons  Consider  the sum of the  of the i n c i d e n t l i g h t  a light  beam of c o n s t a n t  length.  Electrons  constant  r a t e , and with the same e n e r g i e s .  voltage  beam.  intensity  and wave-  are l i b e r a t e d from the photocathode at a  e x p e c t a t i o n of a constant statistical  voltage,  signal.  leads  to the  But there is a  in the number of e l e c t r o n s  in the  cascade  produced f o r each p h o t o e l e c t r o n by a m p l i f i c a t i o n  at the  dynodes.  Because  variation  voltage  This  of t h i s  f l u c t u a t i o n , the output voltage  component superimposed on the c o n s t a n t to noise  ratio  becomes l e s s  voltage.  f a v o u r a b l e with each  stage of dynode a m p l i f i c a t i o n .  77  has a noise And the  signal  successive  78  In intensity level  the s c h l i e r e n d e n s i t o m e t e r , the l a s e r  had to be reduced by a n e u t r a l  which was  of a v a i l a b l e  suitable  light  filter  f o r the p h o t o m u l t i p l i e r .  made p o s s i b l e  use of the p h o t o m u l t i p l i e r . were employed, l e a d i n g  density  beam  The excess  a m o d i f i c a t i o n of the normal  A lesser  number of dynode  to an improved s i g n a l  to noise  The l o s s of a m p l i f i c a t i o n was compensated by changing neutral  density  filter  of i n c i d e n t l i g h t dynode s t a g e s , 4.7 to 95. circuit  to allow  falling  the s i g n a l  an i n c r e a s e  is  in the  on the photocathode. to noise  The seventh dynode was  diagram  to a  shown in Figure  r a t i o was used as  stages ratio. the  fraction  Using 6  improved from  the anode.  The  E.l.  O  -700 V IOO  K  IOOK  IOO  Figure  E.l  K  C i r c u i t diagram f o r o p e r a t i o n of the R.C.A. 931A p h o t o m u l t i p l i e r with l i m i t e d dynode a m p l i f i c a t i o n .  \  79  The dynode r e s i s t o r 1 ma.  The p h o t o m u l t i p l i e r  voltage  vs l i g h t  chain  current is  characteristic  intensity)  should  the output voltage characteristic  curve  be l i n e a r  c u r r e n t never exceeds one tenth of t h i s .  In  must not exceed 100 mV.  curve was measured.  It  is  approximately  if  (output the tube  these The  circumstances,  photomultiplier  shown in Figure  E.2.  . 600-  OUT P U T ' V O L T A G E  (my)  500H  4 0 0 -  300-  200-  IOO-  O  Figure  o  0-2  O  E. 2  0-3  0-4  > LIGHT  INTENSITY  0-5  07  06  (ARBITRARY  UNITS)  C h a r a c t e r i s t i c cut-ve of an R.C.A. 931A photomult i p l i e r with 6 dynode a m p l i f i c a t i o n stages in use.  During  the experiment,  ated on a n o n - l i n e a r  part of  its  the p h o t o m u l t i p l i e r characteristic.  But  wa~s operit  was  80  the v a r i a t i o n  in s i g n a l  being measured, and t h i s teristic  due to the s c h l i e r e n e f f e c t that was was small  enough that the c h a r a c -  was e f f e c t i v e l y l i n e a r over i t s  range.  was i n c r e a s e d by the b u f f e r a m p l i f i e r c i r c u i t . the voltage .signal,  The noise Considering  v a r i a t i o n due to the s c h l i e r e n e f f e c t as  experimental  signal  to noise r a t i o s  the  were about 20.  

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