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Wave propagation in rarefied gases Bejar Hurtado, Jose Antonio 1969

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WAVE  PROPAGATION  IN RAREFIED  GASES  BY  J O S E ANTONIO  A THESIS THE  SUBMITTED  BEJAR HURTADO  IN PARTIAL  FULFILMENT OF  REQUIREMENTS FOR THE DEGREE OF MASTER OF  SCIENCE  IN THE DEPARTMENT OF PHYSICS  We a c c e p t t h i s t h e s i s required s^and^rd  THE UNIVERSITY  as  conforming  OF B R I T I S H  MARCH,  1969  to  COLUMBIA  the  In p r e s e n t i n g an the  advanced  degree  Library shall  I further for  this  agree  scholarly  by  his  of  this  written  thesis  in p a r t i a l  fulfilment  of  at  University  of  Columbia,  the  make  that  it  permission  purposes  may be  representatives. thesis  for  freely  It  financial  available for  permission.  Department The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, Canada  by  the  understood  gain  Columbia  for  extensive  granted  is  British  shall  Head o f  be  requirements  reference copying  that  not  the  of  I agree and this  or  allowed  without  that  Study. thesis  my D e p a r t m e n t  copying  for  or  publication my  r  This  thesis  pressure  ABSTRACT  describes  wave,  through  atmosphere  of  to  wave  thermal  Casimir  Argon.  case.  state  and  P.W.  found  and  at  depend  on  low  and  its  hence  pressure  the  straight  phenomenon in  this  the  is  latter  coefficient  corresponding  frequency  found  limit  in a  to  D depending  phenomenon  extended  the  a dependence  a wave  for  theory  pressure  of  of  a  rarefied analogous  scattering  phenomenon  their  P.W. Matthews  propagation  approximately  boundary  extended  and a g a i n  the  limit.  the the on  steady non-steady  L.  in gases theory  for  to  the  D on L .  velocity  the  Therewould  L.  An e x p e r i m e n t measure  This  again  state,  the  pipe,  a diffusion  steady  fore  long  Matthews  studied  non-steady  of  studied  Smoluchowski state  a  investigation  propagation  a n d Zirnan  state  an  L. has  was s e t  velocity Results been  up as  in o r d e r a function  have  observed.  been  to of  produce the  obtained  a wave  frequency  pressure and  the  and no d e p e n d e n c e  and pressure  on  the  TABLE  OF CONTENTS  Abstract  i i  Tab Te o f C o n t e n t s  iii  T a b l e o f Tab l e s  v  Table o f Figures  vi  T a b l e o f Graphs  '."  vii  L i s t o f Symbols  viii  Acknowledgement  *  ix  Introduction Chapter  I  x —  Thermal  Diffusion  1.  The F o u r i e r Heat  2.  Modified  3.  Heat  F o u r i e r Heat  Flow  in the  1  Equation  Boundary-Scattering  Limit  -  Chapter  t8  —  .-  k.  The Matthews  5.  Brown M o d i f i e d  6.  Comments  Gas  Diffusion  .1. 2.  High Gas  Chapter  \tl  —  Modified  Equation  2 A  Equation  6  Diffusion  Region  9  i n the Boundary  Molecular  10  Equation  Flow Through  Long Tubes  Waves  in  a  Scattering  Flow  T h e Matthews  Diffusion  ....  7  Pressure  Limit, 3.  1  Equation  for the  Rarefied  Molecular 12  Gas  1.  Introduction  15  2.  Gas  15  3.  Pressure  Range a n d P i p e  k.  Computer  Results  17  5.  Comments  on t h e P r e d i c t i o n s  17  Choice  i ii  Dimensions  17  Chapter  Chapter  Chapter  IV  V  VI  —  —  —  The  Apparatus  1.  General  Description  2.  Argon  3.  Flow  k.  Operation  5.  T h e Wave G e n e r a t o r  6.  External  Driving  29  7.  The Main  Pipe  30  8.  Vacuum S y s t e m  30  9.  Pressure  Feeding  21  System  21  Control  Electronics  21 23 Chamber  Measurement  and Method o f  25  System Phase  34 Measurement  1.  Introduction  38  2.  The D e t e c t o r  38  3.  Ion Gauge C o n t r o l  A.  The  Signal  5.  The  Reference  6.  Method o f  Results  and  Box  Channel Channel  Phase  Measurement  43 *»8 49 50  Discussion  Analysis  of  Conclusion  Data  53 53  TABLE OF TABLES  I.  Ionization  gauge s e n s i t i v i t y  for  different  II.  Computer  results  for  Matthews  predictions  III.  Computer  results  for  Matthews  predictions  v  gases  15 18  -  continuation  ....  19  TABLE  I.  Functional  diagram of  II.  Diagram o f  flow  III. IV. V.  Interior General End o f  of  diffusion VI. VII.  experiment  of  control  t h e wave  main p i p e ,  Detail  Main  of  chamber  generator  bellow,  high  assembly vacuum v a l v e  28 and 31  .. Veeco  gauge and d e t e c t o r  attachment  33 to  main p i p e  35  pipe  37  Schematic  diagram of  the  X.  Home-made  ionization  g a u g e scheme  Detail  the  XI.I.  26  pump  the  of  Ionization  22 2h  IX.  XI.  ...  Vacuum l i n e s  the VIII.  the  t h e wave g e n e r a t o r  view the  OF FIGURES  detector  gauge  electronics  attachment  control  box  vi  39 .-  to  the  main p i p e  h\ .... k(>  TABLE  I. II.  Matthews'  Forumula  Experimental  OF GRAPHS  predictions  results,  theoretical  D = 2v r/3, and Matthews Q  L = III.  33.2  D = 2v r/3, Q  IV.  12  theoretical  and Matthews  predictions  theoretical  for  prediction for  55 results,  Q  5.66  theoretical  5.66  predictions  theoretical  for  prediction for  56  cm.  Experimental L =  prediction for  cm  D = 2v r/3, and Matthews  V.  theoretical  for  54 results,  Experimental  L =  predictions  20  cm.  Experimental  L =  .  results  f o r L = 33-2,  L = 12, a n d  57  cm  vi i  LIST  a  =  v  v  =  heat  d  =  diameter  D  =  diffusion  V  =  spatial  F  =  conductance  f  =  k  =  wave  K  =  thermal  L  =  mean f r e e  path  1  =  longitude  of  X  =  wave  m  =  mass  a)  =  2irf  =  pressure  q  =  thermal  R  =  gas  r  =  radius  p  =  density  S  =  area  t  =  time  T  =  relaxation  T  =  absolute  C  Q  (l/v ) Q  capacity of  per  the  of  pipe  pipe  frequency vector conductivity  the  sample  or  pipe  length of  an A r g o n  heat  constant of  atom  current for  the  density  a gram  cylindrical  sample  time  temperature  phonon  velocity  V  Q  =  average  phonon  =  volume  operator  a  =  x  main  gradient  Q  =  unit  coefficient  v  v  OF SYMBOLS  velocity  velocity spatial  coordinate vi i i  or  pipe  ACKNOWLEDGEMENT  It Dr.  is  a pleasure  P.  to  W. M a t t h e w s  I am a l s o Mr.  for  grateful A.  physics  Mr.  Brown f o r of  the  in  R.  the  R.  of  The  Physics  for  its  supervisor  advise  and  encouragement.  preliminary design  of  the  main-pipe.  conversations  many v a l u a b l e of  this  who made  on  the  basic  discussions  and  practical  research.  the w a v e - g e n e r a t o r  and who h e l p e d  and M r . G .  Williams  want  for  Department  financial  her v a l u a b l e  Brooks  for  me  chamber,  the  i n many p r a c t i c a l  their  help with  the  to  the  of  glass  of  details.  shop  the  University  of  British  Columbia  support.  t h a n k my w i f e  help  work.  and " p r i v a t e  in d i s c u s s i n g  and  typing  secretary" the  first  Flaury draft  thesis.  I a l s o want typing  my  work.  Ernie  this  the  several  system,  Haines  Mr.  I specially for  the  stage  Weissbach,  machine  to  problem.  first  flow-control  Mr.  and  D . L . Johnson for  help  indispensable  K s h a t r i y a who made  CR.  Mr.  his  my g r a t i t u d e  to  wave-generator  Mr.  express  to  the  thank Miss  Bev  Little  thesis.  ix  -  for  the  organization  and  INTRODUCTION  Steady s t a t e heat c o n d u c t i o n i n d i e l e c t r i c s boundary-scattering  by C a s i m i r : are  (low temperature)  f o r t h e phonon  l i m i t was s t u d i e d  in 195*1 Z i m a n p o i n t e d o u t t h a t  a p p l i e d to t h e p r o p a g a t i o n  i n 1938  i f the Casimir  o f a temperature  disturbance,  leads to a d i f f u s i o n e q u a t i o n w i t h d i f f u s i o n c o n s t a n t  this  D = 2v r/3. Q  i t follows that t h e t e m p e r a t u r e wave v e l o c i t y  is  which increases i n d e f i n i t e l y as t h e f r e q u e n c y  increases;  physically  results  v = (2Du>)l/2, this  is  unacceptable,  •By adding a w a v e t e r m t o t h e Cas imi r - Z i m a n e q u a t i o n , J . B . B r o w n , O.Y. Chang, and P.W. M a t t h e w s limiting  the t h e r m a l  (1966)  wave v e l o c i t y  to j u s t i f y t h i s a d d e d ' w a v e '  at high  term,  o f t h i s treatment  is that  case  6 T/6x 6x. 2  3  way o f  frequencies.  P.W. Matthews  the Casimir theory t o t h e n o n - s t e a d y c o r r e c t i o n term o f t h e f o r m  found a s u i t a b l e  (1966)  but found  Seeking extended  instead a  One o f t h e m a i n  t h e t e m p e r a t u r e wave v e l o c i t y  features d e p e n d s on  the mean f r e e path a t a l l f r e q u e n c i e s .  Since thermal  diffusion  in a crystal  gas  diffusion  in a l o n g  tube  approximately  analogous  phenomena,  a "Knud#sen g a s , 1  with  low t e m p e r a t u r e s and  in the molecular flow t h e same  theory  some m o d i f i c a t i o n s .  Again  wave v e l o c i t y a t l o w f r e q u e n c i e s path.  at very  i s found  region are c a n be a p p l i e d t h e gas  to  diffusion  t o d e p e n d on t h e mean  free  The of  purpose the  the  chapter  on  first  scattering case.  this  diffusion  dependence  In  of  thesis  mean  chapter  limit III  is  free  describes  in chapter  VI.  the  frequency  and,  dependence  in p a r t i c u l a r ,  its  path.  discussed.  with  study  i n a gas  the Thermal  IV a n d V d e a l  presented  to  wave v e l o c i t y  the  Chapter  is  the  Diffusion  equation  Chapter  II  deals  design  of  the  experimental  in the  with  gas  experiment  techniques.  boundary diffusion and  Results  are  1  CHAPTER  1  THERMAL D I F F U S I O N  l.T  The  F o u r i e r Heat  The  diffusion  (a)  equation  Continuity  C  Equation  equation  |I  V  +  Vq  Fourier equation  (b)  q Combining  =  (l)  for  for  of  is  write  usual this  1.2  to  we  (2)  define  equation  _ "  thermal  combined  predicts as  get  the  the  flow  diffusion  equation  constant  D = K/C  and  v  then  to  form  , ( 3 )  Equation that  the  can  1 >  velocity  ^ . 2  travel  proposed a  19^8  =  heat  v°  (frequency)  diffusion  x  for  3 ^  s i n c e rao d i s t u r b a n c e  in  conductivity  2  (3)  2  i n a u n i f o r m medium,  3 T  n  D  Equation  V'ernotte  1  (1)  a diffusion  in  F o u r i e r Heat  limit  from :  (2)  Modified  without  transport  results  •-KVT and  3T at  heat  transport  0  =  £ S '- ™ It.  thermal  equation  "constante  This faster  for is  the  physically  than  the  phenomenological a n d wave  de  temps  heat  waves  increases  inadmissible  phonon  velocity.  equation which  is  a  equation,  tres  petite"  (relaxation  time)  .  The  physical  i n t e r p r e t a t i o n of  is  (k)  simple.  i "It of  states a  thermal  onto but  rather  lag  if  exhibits  a  just  the  because  x  the  v  Heat It  is  Flow  with  a  a  gradient of  of  the  is  T  the  must  time to  be  start  whereas  onset  clamped  instantaneously T.  removed  Con-  there  is  and e q u a t i o n  equation  with  is  time  current  for K.  the At  diffusion  waves, with  connected  the  a (k)  (3) does  not.  communication  for  with  the  "time"  commencement  thermal  establishment  low  giving  a finite  of  frequencies  equation.  At  resistive  equation  high  wave-like upper  resistance  enough  reduces  (k)  frequencies,  propagation  limiting  flow,  of  velocity,  =  m  that  Boundary-Scattering the  infinite  thermal extent  must  finite free  dimensions  path  of  the  Limit  conductivity  crystal  the  not  the  1  temperature,(Peierls  which  heat  for  gradient  suddenly  associated  decreasing of  time  relaxation  is  relaxation,  1/T  in the  known  crystals  gradually  term d o m i n a t e s ,  temperature  temperature does  flow."  original  'wave'  a  up  (phonon-phonon c o l 1 i s i o n )  proportional  the  after  build  flow  time  rate  is  finite  The heat  such  phonons  In f a c t  to  a  disappearance  resistive  is  is  thermal  relaxation  between  x  current  grows  in the  The  there  a specimen.  versely,  of  that  of  increase 1929) .  there  elastic  4  will waves  insulating  indefinitely However,  be  a  for  perfect with  a  temperature  becomes  given below  comparable  to  the  dimensions  o f the crystal  the  scattering  of elastic  crystal been  will  give  rise  demonstrated  Casimir  (1938)  6  dimensions  body"  the free  reflection  t h e boundary  a l l incident  the diffusion  radius  phohons  as i n e q u a t i o n  constant  (3),  t u r e wave,  v = (20a)) / ,  increases.  Thus a g a i n  thermal  T h i s has  1  i n order  2  state  heat conduction i n  is very  long compared t o  acts  that  is  i n the process  as a " b l a c k "  and r a d i a t i n g  leads  surface  phonons w i t h  by t h e l o c a l  o f temperature  o f the c y l i n d r i c a l  Ziman e q u a t i o n  o f the  resistance.  He a s s u m e d  (195*0 p o i n t e d o u t t h a t t h i s  7  where  J.B.  path  of the c r y s t a l .  the propagation  Again  the steady  distribution characterized  Ziman for  t o an a d d i t i o n a l  considered  the  absorbing  by t h e b o u n d a r i e s  5  l i m i t i n g c a s e when  diffuse  waves  is the case,  experimentally .  the  of  and as soon as t h i s  surface  D = 2v r/3 Q  equation  as e q u a t i o n  where  "black-  temperature.  to a diffusion  disturbance  a  (3),  r is the  sample.  the propagation increases  indefinitely  i t is necessary to limit  velocity  t o modify  the velocity  f o r a tempera-  as t h e frequency the Casimir-  propagation  of a  pulse.  B r o w n , D . Y . Chung a n d P . W . M a t t h e w s  term t o t h e C a s i m i r - Z i m a n e q u a t i o n temperature  region),  giving  8  (1966)  (as V e r n o t t e  a d d e d a wave did f o r the high-  -  where  D =(2/3)v r.  h -  They used  Q  this  equation to explain  o b s e r v a t i o n s o f heat p u l s e propagation i n c y l i n d r i c a l in the long mean-free-path Al^Oy  and f o r l i q u i d  samples  region, f o r crystals of  h e l i u m I I . The i d e n t i f i c a t i o n o f v  t h e mean p h o n o n v e l o c i t y boundary  temperature  their  requires  0  with  t h a t 3 t i s t h e mean t i m e b e t w e e n  collisions.  T h e y o b t a i n e d t h e s o l u t i o n o f e q u a t i o n (h) f o r t h e r e s p o n s e t o a unit  6 - function  of t h e s a m p l e .  input a t x = 0 and found  For the p a r t i c u l a r  and e x p e r i m e n t a l r e s u l t s were cylindrical  single  A few months theory •3 T/3t 2  P.W.  t o a non-steady 2  2  :  this  c a s e o f He I I , c o m p u t a t i o n a l  i n v e r y good a g r e e m e n t .  o f A^O^  Matthews  t h e a g r e e m e n t was n o t s o g o o d .  (1366) e x t e n d e d  9  the Casimir term  he f o u n d a t e r m o f t h e f o r m  term a l s o produces  persTon c u r v e s , w i t h  For a  s t a t e , b u t i n s t e a d o f a d d i n g a wave  a s i n e q u a t i o n (k),  3 T/3x 3t 3  later  crystal  T ( x , t ) a t t h e end  t h e expected form o f the d i s -  t h e t e m p e r a t u r e wave v e l o c i t y  reduced  below  t h a t f o r a p u r e d i f f u s i o n wave f o r h i g h e r f r e q u e n c i e s .  ,k  The Matthews M o d i f i e d P.W.  Matthews  9  Diffusion  Equation  developed a modified d i f f u s i o n equation f o r the  p a r t i c u l a r c a s e o f time-dependent scattering  limit  of " b l a c k " w a l l s . to the non-steady  forcylindrical He f o l l o w e d  heat flow samples  i n the boundary  and under  the assumption  t h e same p r o c e d u r e a s C a s i m i r , 6  s t a t e , and o b t a i n e d t h e a p p r o x i m a t e e q u a t i o n  applied  9t  9x2  9x^9t  where  Equation  D  -  A  =  f  «  L  L  >  r  Therefore  X  »  r  Dispersion  Relation:  temperature =  substitution  the  form  wave o f  + A Texp  TB  into  iw  =  equation  -  Dk  +  2  = Mli/2  k  the  following  restrictions:  -  >>  T  under  V  X  a  (|4 - |]  [In  deduced  Vn  For  (1 - 3T)  0  r  (5) was  -  rv  i  (wt  (5)  give  +  kx)  the  dispersion  relation  Ak iw 2  [(,  Aw  +  }  .  _ Aw  (1  }  f]  where 6  If  Aw/D  For  the  not  tend  2  = A w 2  is  small,  k  =  ( ^ )  condition  increases  +  2  to  the  and  2  equation  1  of  /  [(1  2  very  usual  and makes  D  (6)  diffusion  the  new  can  $L)  +  long  A  L  (he  D  be w r i t t e n  -  (1  As  +  6  approximately  %L)  -  explains),  equation.  t e r m more  =  i]  equation  L increases  important  rather  (5)  does  A  than  less  important. X  >>  The  The d i f f i c u l t y i s more  L  condition  important  »  X  L  validity  o f the theory  For  large  very  valid  permitted  The  The is  k  +  =  k"  -  theory less  than  A different Matthews deduced using  constant  and under  heat  t h e range  ->• o .  remains of  frequency  therefore  as u s u a l  i s t h e wave v e l o c i t y  o f the temperature  a n d W"  wave.  f o r frequencies  t h e Boltzmann  equation  t h e same  10  8  (7)  w  such  t h a t 2L  wavelength.  by B r o w n ( l 9 6 7 ) .  Brown e t a l  range  Equation  modified  9  is f o r D.  t h e range o f  is  (7)  w/k'  t o be v a l i d  one-half  Matthews ,  then  it  .  ( 1 - 0  i k "  i s assumed  as L  within  of equation  1/2  i n A than  the approximation  small r  the condition  t o a frequency  L/r  >>  limit  (|D)  that  shrinks  L  the attenuation  Brown M o d i f i e d  to  »  X  low f r e q u e n c y  When  means  of  because  f o r terms  Aw/D r e m a i n s  by  k  is  values  because  appears  "black"  walls  His derivation  Transport  he f i n d s  f o r t h e same  Equation  case  assumption  has been  i s from f i r s t  f o r phonons.  a wave t e r m s i m i l a r  c o n s i d e r e d by  in  principles, contrast  t o t h e t e r m assumed by  For very steep temperature gradients, further terms wouId have to be added, but equation X  >  (8) is v a l i d f o r temperature wave  16 r / 3 .  t f a s o l u t i o n of the form T(x,t)  =  T exp [i(kx  Is assumed f o r equation  - wt)]  (8),. then putting  k  =  k" +  i k " and  s o l v i n g f o r k' and k "  k  '  ?  k"  where  T  =  vg 2  =  T  =  2  [1  o  ^ /3v  [1  +  (1 +  +  0  (wx)-2)l/2]  + (WT)-2)1/ 2 3  r  0  f a the 1-ov* frequency  limit  v2x O  and in the high frequency 8c'  =  limit  2 ^ 2 H_ o v  " k  =  1 2Viv T 0  therefore a t high frequencies a limiting speed of propagation and  a constant attenuation c o e f f i c i e n t  k" '• = 3/2 *»r 1/2  are p r e d i c t e d .  Comments At low frequencies equation  (8) reduces to the Casimir-Ziman  8  diffusion effects  equation.  at  very  discrepancy latter  This  predicts  tl  • f  be  limited  instead  diffusion  The be  v  t°°  in the  t  to others  Matthews.  the  for  interested  present  only  purposes,  and M a t t h e w s ' , 9  D which depends  from  is  that  in the  only  the  on t h e mean f r e e  path,  and  ( 9 )  the  limiting of  in fact  same w a y :  this  so  theory  for  we a r e  o -  equation with  investigate by  r  arose  of  purpose of left  Brown's  a value  dependence  to  thesis  low f r e q u e n c i e s ,  between  D  In t h i s  this  D given  thesis  is  to work a t  nature of  the  the  to  original  leads  by  to  Equation  check  higher higher  ranges  this  integration  Casimir theory  should  a Casimir-Ziman 9.  one p o i n t ,  frequencies terms  of  where  suggested  and one  by  it  will  could  Brown and  -  9  -  CHAPTER I I  GAS  2.1  DIFFUSION  High P r e s s u r e Region The  parabolic equation 8n 2  describes  not o n l y t h e heat c o n d u c t i o n but a l s o the d i f f u s i o n  g a s e s , s h e a r m o t i o n , and  The  diffusion  of  o t h e r phenomena.  e q u a t i o n f o r a gas  as f o r h e a t c o n d u c t i o n  1 1  c a n be d e d u c e d  , w h e r e n i s now  the d e n s i t y  similarity  i s not o n l y  i n the mathematical  f r o m t h e b a s i c p h y s i c s o f t h e phenomena.  s p r e a d s f r o m one  w h i c h a gas  form, but  arises  in diffusion  a  gas  r e g i o n t o a n o t h e r by t h e m o t i o n o f m o l e c u l e s .  the thermal d i f f u s i v i t y ;  similarily  penetrates into another, or  itself,  coefficient  of  the speed  with  i s determined  p r e d i c t s no  by  diffusion.  f o r the F o u r i e r heat e q u a t i o n , the e q u a t i o n f o r d i f f u s i o n  gases  limit  on  t h e wave s p e e d  frequency of a pressure perturbation: is  (or pressure)  speed w i t h which a temperature d i s t u r b a n c e i s t r a n s m i t t e d , i s  d e t e r m i n e d by  As  way  J u s t as h e a t s p r e a d s i n  a c o n d u c t i n g body by t h e m o t i o n o f p h o n o n s , s o  the  i n t h e same  the gas.  This  The  of  needed as b e f o r e .  As  as one  clearly  increases  of  the  a modified equation  the frequency i n c r e a s e s , e v e n t u a l l y  one  10  r e a c h e s the normal to  the v e l o c i t y  a combined valid  sound  r e g i o n , a n d t h e v e l o c i t y m u s t be l i m i t e d  o f sound,  diffusion  c j .  It is therefore  a n d wave e q u a t i o n w h i c h w i l l  8 n  D  8 n Bx "  8n_  2  2  9t  7  T h i s e q u a t i o n h a s been r e c e n t l y model by W e y m a n  ci  situation  /  The sound  V  2  0.625 " .  t h e phonon  It  i sinteresting  of  h e a t waves  sound  (l/3) ^ C] 1  helium.  velocity).  As  the theory  I n t h a t medium, t e m p e r a t u r e waves  fluctuation  1 3  (1952)  gas.  "second  t h a t second  sound  i n t h e "phonon g a s " , s o sound w h i c h  The second  sound  is a  velocity  0 . 5 7 8 c j w h e r e c j i s t h e phonon v e l o c i t y  medium - s o l i d  i n the Boundary  i n the second  velocity  H e a t w a v e s o f t h e same t y p e h a v e s i n c e  in a solid  Gas D i f f u s i o n  in a real =  2  analogous  velocity.  in a definite wavelike fashion called  fluctuation  observed  2.2  This corresponds t o the  there i s a c l o s e analogy with ordinary  =  c j , is closely  v, in fact  I t was shown by Ward a n d W i l k s  density C2  velocity,  ( e q u a t i o n 4) w h e r e t h e wave  can be r e g a r d e d as a d e n s i t y that  from a s i m p l e random-walk  t o c o m p a r e t h e a b o v e g a s model w i t h  inliquid  propagated  sound".  =  i n a t h e r m a l wave  cannot exceed  are  (1967).  1 2  derived  t o t h e average m o l e c u l a r speed ( T T / 8 )1  =  c e r t a i n l y be  1imi t s  i n both  related  reasonable to w r i t e  been  h e l i u m by F a i r b a n k e t a l  Scattering  Limit,  (first  l t f  (1966).  M o l e c u l a r Flow  p a r t o f t h e t h e r m a l a n a l o g u e , we d i s c u s s  the  free  11  molecule  flow  i n a long  are maintained uniform  before of  at different  throughout.  i m o l e c u l e of  the w a l l  gas must  collide  that  dimensions.  actual  mirror-like  many t i m e s  unaffected  reflection  with  For t h i s  from the w a l l  to bear  iinci'dence .  This would  surfaces  little  thermal  The  i f any  relation  per cent  1  temporarily  phonon  case  (see Present  speed V  ( C h . 1.3)  distribution  =  5  In  reflection of  upon  later  on t h e r e f l e c t i o n surfaces  Therefore,  of  indicated as  i n the  assumed.  pipe  i s c a l c u l a t e d by  ) , and t h e d i f f u s i v e  coefficient, (8RT/TTM) / .  considered  plays  1  of  a d s o r b e d and  diffuse reflection.  by a d i f f u s i o n  average molecular  require  molecular  i f a gas m o l e c u l e ,  t h e " b l a c k " w a l l c a n be  S m o T u c h o w s k i s method characterized  collision.  t o the d i r e c t i o n  E x p e r i m e n t s by K n u d s e n a n d o t h e r s  case,  with  i r r e g u l a r on t h e  the d i r e c t i o n  t r a n s p o r t p h e n o m e n a down a c y l i n d r i c a l  thermal  scale of  flow  need n o t t o be s p e c u l a r .  are highly  be t h e c a s e  t h e s u r f a c e , were  t o 100  the  by t h e c o l l i s i o n s  m o l ' e c i a l a r beams f r o m g l a s s a n d p o l i s h e d m e t a l close  reason,  of the molecule would  m b l e c i a l a r s c a l e , and one w o u l d e x p e c t  evaporated.  the w a l l o f the tube  by i n t e r m o l e c u l a r  the r e f l e c t i o n  p r a c t i c e , the w a l l  reaching  two e n d s  s m o o t h e s t s u r f a c e w o u l d be on a p e r f e c t s i n g l e  but even here  1 5  The  i s s o low t h a t L > r , a  i s smooth on t h e s u b m i c r o s c o p i c  The  r.  and t h e t e m p e r a t u r e i s  a n o t h e r mo 1 e c u l e .  is practically  A specular or  crystal,  pressures,  i s determined almost e n t i r e l y  and  a surface  tube o f radius  If the pressure  i t encounters  t h e gas  circular  1  2  D. =  flow i s  27Y/3, w h e r e  In t h e a n a l o g o u s  by C a s i m i r , 7  the b l a c k  t h e p a r t o f the Maxwell  body  distribution for  -  2.3  12  the  r a r e f i e d g a s , and  The  M a t t h e w s E q u a t i o n f o r t h e M o l e c u l a r Flow Through Long  We  saw  i n Ch.  1.3  l e a d s t o an  how  P.W.  method t o a non-steady form  3 T/3x 3t 3  .  2  exactly  e q u i v a l e n t value of  state  case found a c o r r e c t i o n  S i m i l a r i l y f o r t h e gas  case-he  s t a t e , and  of  3 n/3 x3t  to the  formula. the  as a c o r r e c t i o n  2  S i n c e no  publication  method w i l l ' b e  has  only,, and  the t o t a l  l e t n be  :  teas  -x.y  v  term  Smoluchowski case,  =  nj cos  n  l  cos  t i m e t d e p e n d s on  lies  between v and  f r o m dS] ^~  vf  is  , dS  '  2  K  8n  3 n 3x2"  p  2xp_  2  "  v  c  .  Q  vf  f  \  r.  (v)  6  f  The  (t-p/v),  1  a t --x. x. at  the  flux  V  f l u x through i.e  dS  Q  on  3n*  3 n 3x3t 2  .  w h e r e d i f f e r e n t i a l s a r e e v a l u a t e d a t x = 0 and dS'  v + dv  ; f ( v ) = v e l o c i t y d i s t r i b u t i oa-„  V  x = 0.  nj a t d S j . a t time  j_ , 2 *  volume  1 5  (v) 6  01 c o s 0n ^T2~  /  dv,  x v../_.  Q  6]  range, v to v +  per u n i t  o  c o n s i d e r a T a y l o r e x p a n s i o n about  m i r r o r element  the  I JJpds  0 \J  through dS  =  at  « \  .  '  How  molecules  m o l e c u l e s whose v e l o c i t y  dF  the  outlined.  in a restricted velocity  (molecules/sec.)  found a  b e e n made f o r t h e gas  Consider molecules  For  term of  extended  method, f o r the non-steady 3  Tubes  M a t t h e w s by e x t e n s i o n o f t h e C a s i m i r  Smoluchowski the form  D.  p£ V2  3£n. 3t2  time t .  ;  Consider a i  T h e net n e t fflux l u x due o ddS. S j aand n d ddS Sj depends The due tto  13  on _  1  "l  " l  _  =  2  3n 37 "  X  Hence t h e n e t f l u x t h r o u g h m o l e c u l e s whose v e l o c i t y  where  lies  •  h »' M  1^  =  Jcos 0  1  =  j c o s eQ  2  These i n t e g r a l s  2  cross section,  av [,, - | a - I  cos 6j  (^-) d S  Q  c o s 8]  (£•)  Q  dS  i s the integral  F  "  1 2„  "  1 . U 27 1  The  net flow into <x  +  2  dS]  t r e a t m e n t , and 9  flux for a l l  over a l l v  r,  3n f  x _,  t,  _ 3n 37  . '2  '  V  .  3 n 2  J f ( . )  2  " x)  3 n 2  a^ap  ^  -  F  =  [llV06x r  TT  2  6  X  6  - « f^t 2  n  Therefore 3n  1  3t"  Introducing  =  dv]  r e g i o n b o u n d e d by x a n d x +<5x i n t i m e St i s ,  Sx  F  , |!g-a  The t o t a l  [ ' | | J } v ' ( v ) d » - l  r  Q  dSj  a r e e x a c t l y as i n t h e thermal  t h e r e f o r e they a r e a l r e a d y e v a l u a t e d . molecules  S (for  between v t o v + dv) i s  f  o  3 n 373t  the total  v  F  2xp  2Tr2r2  n L  ' l  — v  3 n  .  2  a^T  "  '2  3 n  ••  3  8x23t  J  t h e v a l u e s o f l j a n d \ ^ we f i n a l l y g e t  6  x  ]  6  t  -  14  -  For  thermal  where  as  tn  that a t gas  thermal  the  low  case,  analogue.  temperatures  the  all  corresponding  is  unsatisfactory.  However,  is  used,  turns  the  velocity All.' t h e low  v"  answer  replacing  discussion  frequency  v As  dependence  the  the  validity  whether  is  in  range  of  the of  the  the of  assumption if  out  the  is  = / y  this  integration  same,  assumed  speed.  For  molecular  velocity  here.  wave  the speed  distribution  with  average  molecular  of  this  theory  should  be  velocity  for  the  is  of  (10) v on L  thesis.  equation but  the  Therefore  (1 - ^ )  dependence  theory,  steady-state  same  constant  the  valid  to r v  v on L o f  above  of  i t was  velocity.  1.4  topic  the  Maxwell  exactly  a diffusion  indicated,  range  1.6  of  /TTO  =  previously  -frequency  the  limit  phonons .had  phonon  in Ch.  waves  (10)  only  on  (Casimir's limited  As  noted  does the or  t o tl  in  not  the in  low section  depend  question  on  of  Smoluchowski s), 1  instead  of  t».  15  CHAPTER  DIFFUSION WAVES  3.1  II I  IN A RAREFIED GAS  Introduction From t h e (a)  experimental  To g e n e r a t e  point  of  view,  a p r e s s u r e wave  in a  t h e wave v e l o c i t y  at  the  objective  rarefied  gas  was: in a  cylindrical  pipe. (b)  To measure free  3.2  Gas  pressures  (i.e.  mean  paths).  Choice  Firstly  it  forward  K i n e t i c T h e o r y was  was  different  was  in the  selected, used  reasonable  to  use  applicable  translationalmotion as  in the  it  is  readily  experiment  argon  than  for  (from  Dushman -  of  and a l l  the  available,  have  helium or  a monatomic g a s ,  and s i n c e  as  shown  Vacuum T e c h n i q u e  i n the  (Wiley)  2nd.  Pressure  Gas  Argon  the  higher  that  kinetic  molecules.  a considerably  neon  the  so  ion  was gauges  following  table  Edn. pg.  323)  Gauge  He  0.133  Ne  0.202  A  1.0  Kr  1.56  Xe  2.29 1  energy  sensitivity  correction  factor  Table  straight  for 1  16  The a s s u m p t i o n s (i)  made a r e :  The M a x w e l 1 - B o l t z m a n n d i s t r i b u t i o n 7  =  (8kTArm)  t o c a l c u l a t e v",  l / 2  where k i s t h e Boltzmann (ii)  c a n be u s e d  Constant.  The mean f r e e p a t h L ( u n r e s t r i c t e d  by b o u n d a r i e s ) a t p r e s s u r e  p is L  =  Constant/p  where c o n s t a n t = 5 . 6 6 x 10"3 range  Also  IO"  <  4  < 10~ 3  p  torr.,  i t i s assumed t h a t t h e w a l l s  give perfectly  diffusive  I t was n e c e s s a r y within  e.g. i n t h e pressure  50 cm. >  L > 5 cm.  a r e a t u n i f o r m t e m p e r a t u r e , and  scattering.  t o ensure  the restrictions  cm. t o r r .  that  the experimental conditions  o f t h e Matthews theory  were  ( s e c t i o n 2 . 3 ) , as  follows: (a)  A  >  This (b)  D  >  This  0 condition  (c)  limit  f o r L, L j m  n  = r  0 condition also  So t h i s given  leads t o a lower  condition  leads t o a lower  sets  t h e upper  limit  f o r L, L j m  pressure l i m i t ,  diameter.  The r e s t r i c t i o n  r/v  X  =  wave l e n g t h  v  =  velocity  is s a t i s f i e d  <<  A/v  o f t h e wave  s o l o n g as r  <<  X  since v  <  v"  n  = 3rA  f o r a pipe of  17  3.3  P r e s s u r e Range and The 10"  upper  limit  T o r r , as  3  detectors. (section taken  o f the p r e s s u r e  this  3.2),  t o be  i s a reasonable  inches.  f o r the too  limit  about  f o r the  i o n gauge  Moreover f o r t h i s  was  therefore  pipe diameter,  i o n g a u g e d e t e c t o r s do  the  n o t p e r t u r b t h e wave  pressure  limit  of  indicated  IO" * T o r r 1  i s s e t by  theoretical  below.  Computer R e s u l t s The  velocity  t h e two  and  a t t e n u a t i o n o f t h e wave p r e s s u r e p r o p a g a t e d  R e s u l t s f o r argon  a r e shown on T a b l e s are plotted  on  Comments on  the  Graph  II and  Graph  regarding >  30  was  This  not  same r e s u l t s  for  5 cm  <  be much g a i n by w o r k i n g  i s t h e r e a s o n why  confined to values  s l o p e at the o r i g i n  i s w h a t we  3.99  x  10*  cm/sec  1  (velocity  only)  Predictions  cm.  L  =  I.  The  <  K and v  I I I . The  the s l o p e at the o r i g i n ,  the experiment  path  a t 300  I shows t h a t t h e r e w i l l  that L  along  i n c h e s p i p e were computed a c c o r d i n g t o the Matthews  equation.  3.5  upper  c h o s e n t o be  seriously.  e x p e c t a t i o n s , as  3.A  r a n g e was  e s t a b l i s h e s the pipe diameter, which  two  propagation  lower  Dimensions  T h i s c h o i c e , c o u p l e d w i t h t h e c o n d i t i o n (b)  s i d e tubes  The  Pipe  50  were  cm.  (proportional The  in information, at pressures  the pressure  that  1  2  varies  d e p e n d e n c e o f D on  l o o k i n g f o r i n our  range f o r  l e a d t o 5 cm  to D^)  <  L  by a b o u t  t h e mean  experiment.  such  free  < 30  30%  cm.  18 TABLE FREQUENCY  Mean F r e e P a t h 5. 10. 15. 20. 25. 30. 35. AO. 45. 50. Mean F r e e P a t h  VELOCITY (CM/SEC) =  5.00 cms  =  1.619E 0 3 2.286E 03 2.795E 0 3 3.223E 0 3 3.598E 0 3 3.936E 0 3 4.245E 0 3 4.531E 0 3 4.800E 0 3 5.052E 0 3 10.00 cms 1.849E 03 2.609E 0 3 3.188E 03 3.672E 0 3 4096E 0 3 4.477E 03 4.825E 0 3 5.146E 03 5.446E 03 5.728E 03  5. 10. 15. 20. 25. 30. 35. AO. 45. 50. Mean F r e e P a t h  =  Mean F r e e P a t h  =  COMPUTER  WAVELENGTH (CM)  RESULTS ATTENUATION X WAVELENGTH  02 02 02 02 03 02 02 02 02 02  6.264E 00 6.2A6E 00 6.227E 00 6.208E 00 6.189E 00 6.171E 00 6.152E 00 6.134E 00 6.115E 00 6.097E 00  3.699E 02 2.609E 02 2.125E 02 1.836E 02 1.638E 02 1.492E 02 1.378E 02 1.287E 02 1.210E 02 1.146E 02  6.253E 00  3.237E 2.286E 1.863E 1.611E 1.439E 1.312E 1.213E 1.133E 1.067E 1.010E  6.222E 00 6.192E 00 6.162E 00 6.132E 00 6.102E 00 6.073E 00 6.043E 00 6.014E 00 5.985E 00  15.00 cms 1.919E 2.707E 3.305E 3.806E 4.243E 4.636E 4.993E 5.324E 5.632E 5.921E  5. 10. 15. 20. 25. 30. 35. 40. 45. 50.  II  03 03 03 03 03 03 03 03 03 03  02 02 02 02 02 02 02 02 02 02  6.246E 6.209E 6.173E 6.136E 6.100E 6.064E 6.029E 5.993E 5.958E 5.923E  00 00 00 00 00 00 00 00 00 00  3.907E 02  6.242E 6.200E 6.159E 6.118E 6.078E 6.037E 5.997E 5.958E 5.918E 5.879E  00 00 00 00 00 00 00 00 00 00  3.839E 2.707E 2.204E 1.903E 1.697E 1.545E 1.427E 1.331E 1.252E 1.184E  20.00 cms  5. 10. 15. 20. 25.  1.953E 2.753E 3.361E 3.869E 4.312E  03 03 03 03 03  30. 35. 40. 45. 50.  4.709E 5.071E 5.405E 5.716E 6.008E  03 03 03 03 03  2.753E 02 2.241E 02 1.935E 02 1.725E 02 1.570E 02 1.449E 02 1 - 351E 02 1.270E 02 1.202E 02  FREQUENCY  Free Path 5. 10. 15. 20. 25. 30. 35. AO. 45. 50.  VELOCITY (CM/SEC) ==  WAVELENGTH (CM)  ATTENUATION X WAV  2 5 . 0 0 cms 1.973E 0 3  3.9A6E 0 2  2.781E 03  2.781E 02  3.39AE 3.905E 4.352E 4.751E 5.H5E  03 03 03 03 03  5.^51E 0 3 5.763E 0 3 6.056E 03  2.263E 1.953E 1 .7^+1 E 1.58AE 1.A61E 1.363E 1.281E 1.21 IE  TABLE I I I  02 02 02 02 02 02 02 02  COMPUTER RESULTS  6.238E 6.193E 6.1*»8E 6.10AE 6.060E 6.016E 5.973E 5.930E 5.887E 5.8i»5E  00 00 00 00 00 00 00 00 00 00  21  CHAPTER  THE  k. 1  IV  APPARATUS  General D e s c r i p t i o n The .  main  (1)  p a r t s of the system  Argon  are  feeding system which  ( F i g . 1): s u p p l i e s Argon  t o t h e wave  generator (2)  Wave g e n e r a t o r w h i c h  (3)  Main  (4)  Vacuum s y s t e m w h i c h  pipe  and w a v e  4.2  (6)  Wave d e t e c t o r s  ID, 12'  long s t r a i g h t  produces  t h e vacuum  pipe)  i n the  wave-guide  system  Electronics  Argon  Feeding 12'  f l o w , Q,  System  long s t r a i g h t  gradient,  Ap  , has  cylindrical  been assumed  main  in order  pipe  ( 2 " ID) a  to calculate  pressure  the  Argon  Torr,  i s of  i n grams/sec. Q  The  a p r e s s u r e wave  generator  P r e s s u r e measurement  In o u r  =  J  (2 TTRT)1/2  (  maximum f l o w c o r r e s p o n d i n g  the order of  4.3  (two-inch  (5)  (7)  produces  10"  5  to  R]  Ap  _  =  P 2  5  )  x  10  gm/sec.  Flow C o n t r o l In o r d e r reduction Argon  to control  this  kind o f flow, a system  i n pressure of 10"  source  (a c o m m e r c i a l  e  x  i n which  the value of that of the  c y l i n d e r ) was  designed  (Fig.  a original II).  -  22  -  FIGURE I  Pressure measurement system  Argon supply  A  A  1 Vf av e | gene-j rator j  i  Main pipe  Wave d e t e c t o r s  FUNCTIONAL DIAGRAM OF THE EXPERIMENT  Va- ! cuum system  23  Essentially,  the  flow  connects  low  pressure  a  the  i s a small  2000 I b s / s q .  The  at  cylinder  by  a needle  Argon source  able  to support  valve  (8) w h i c h  (e) t o t h e wave  pressures  of  generator.  the o r d e r  of  inch.  v o l u m e s o f a , b,  The  is controlled  a  -  500  b  =  1 cc  c  =  1000  d  =  1 cc  e  =  1000  c , d, and  are  cc  cc  cc  whole flow c o n t r o l (8) w h i c h  e  i s connected  i s connected  t o t h e wave g e n e r a t o r  t o t h e vacuum s y s t e m t h r o u g h  chamber  the  main  pipe.  4.4  Operation Once a l l t h e a i r i s o u t Argon at high closed. and  By o p e n i n g v a l v e 3,  e x p a n d e d by 1 cc of t h i s  times  cc by  expanded Argon  a needle  which valve  v a l v e 3.  and  i s taken  (excepting  1 and Argon  2)  are  is  closing  by o p e n i n g v a l v e 5,  through  with  taken  T h i s cc o f Argon i s  i s i s o l a t e d by  T h u s , t h e gas  is controlled (8).  c  c l o s i n g v a l v e 5.  o p e n i n g v a l v e 6.  a_ i s f i l l e d  1 cc of high-pressure  closing  o p e n i n g v a l v e 4,  i s i s o l a t e d by  Argon source by  the system, c y l i n d e r  pressure, while a l l valves  i s o l a t e d f r o m a_ by  this  of  valve  and  again  I t i s expanded a g a i n in e  i s our  4.  1000  low-pressure  t h e wave g e n e r a t o r  chamber  T o wave  Figure  II  DIAGRAM OF FLOW CONTROL  generator  25  However, a c o n t i n u o u s pipe  i s necessary  4.5  i n t h e main  and t h e n e e d l e v a l v e has t o be a d j u s t e d  three o r four minutes desired  supervision of the pressure  i n order  t o keep t h e p r e s s u r e  every  at the  value.  T h e Wave G e n e r a t o r Chamber The a  basic  idea o f t h i s  assembly  i s t o chop a f l o w o f Argon gas a t  known r a t e .  A a l u m i n u m d i s c , A, ( F i g . I l l ) w i t h by  The F,  B which  four c i r c u l a r  ( F i g . I V ) , ( 8 " I . D . ) , 5" l o n g w h i c h D a n d E by means o f 0 - r i n g s  i n an aluminum c y l i n d e r , i s sealed  (Figs.  by t h e b r a s s  I l l and I V ) . The d i s c D  has  two h o l e s , a_ a n d b .  12'  main p i p e , and b i s a g l a s s window w h i c h a l l o w s  from a r e f l e c t i n g M,  a_ i s t h e h o l e w h i c h c o m m u n i c a t e s w i t h t h e  p h o t o c e l l , P, ( F i g . IV) t o a r r i v e  ( F i g . I l l ) a n d be r e f l e c t e d  purpose o f t h i s  photocell  the  same f r e q u e n c y  The  design  (1)  is driven  i s i n t u r n d r i v e n by t h e e x t e r n a l magnet C.  d i s c A and i r o n d i s c B a r e e n c l o s e d  plates  holes  signal  t o stand  The  with  pressure.  was d i c t a t e d by c o n s i d e r a t i o n o f t h e f o l l o w i n g  ability  beam  to the mirror,  t o t h e p h o t o c e l l vacuum t u b e .  i s t o produce a reference  o f t h e wave  the l i g h t  up t o p r e s s u r e  nature:  o f 15 lbs»/sq. i n . o n e x t e r n a l  surface of the generator (2)  s h a f t and i t s s u p p o r t  designed  to give a rigid  and  balanced  p e r f o r m a n c e o f t h e d i s c a t a s p e e d up t o 600 rpm (3)  c a r e f u l machining o f t h e aluminum chopper need t o have a c l e a r a n c e  i s d i c t a t e d by t h e  between t h e c h o p p e r and p l a t e s o f  A  •Hi  V F i xed  f8  Mirror  M  Photocell  I ro  •"-it  'HQ .'.!.'! I  "/if-  /  Entrance ~ Supports  Figure.Ml  Disc)  b e a r i ngs_  4'  Argon  (Chopper  "ill  fli."2  Main Pipe  i  SCHEMATIC DIAGRAM OF THE WAVE GENERATOR  -  1/8".  2?  -  T h i s c l e a r a n c e determines  t h e "DC" l e a k o f Argon t o  the p i p e (k)  t h e s t a i n l e s s s t e e l p l a t e , S, ( F i g . I l l ) s e r v e s t o d i f f u s e l y r e f l e c t the Argon gas molecules  i n t o the chamber and screens  o f f the 2" p i p e from a d i r e c t f l o w o f Argon from t h e v a l v e . T h i s arrangement makes i t p o s s i b l e f o r Argon gas molecules t o reach thermodynamic e q u i l i b r i u m w i t h t h e w a l l s o f the wave g e n e r a t o r by a p r o c e s s o f c o l l i s i o n .  The o t h e r s i d e o f t h e  s t a i n l e s s s t e e l p l a t e s e r v e s t o support t h e m i r r o r which r e f l e c t s t h e beam from t h e p h o t o c e l l assembly. (5)  I t produces a wave p r e s s u r e q u i t e c l o s e t o a s i n wave.  Assuming t h a t t h e f l o w i s p r o p o r t i o n a l t o t h e open a r e a S between h o l e s , t h e r a t i o dS/dt w i l l g i v e us the r a t i o dP/dt which i s What i s b e i n g looked f o r  s  s  r (2a  dS dt  r (l  2  r cos a -r sin a  2  = da  -  s i n 2a)  - cos 2a) R sin 3  d2a dt  Wave  I Control Figure  IV  Valve f o r Argon  Flow  DRIVING SYSTEM, MAGNET AND WAVE GENERATOR  Generator  -  dB_ dt  U)  dS dt  r (l  2  =  29  -  Iff  Therefore - c o s 2a)  2  -*»TT  Rr  -8ir  Rf  2 R cosB 2uf - r si n a  1 - c o s 2a cosB s i n a. sina / T - '(^ cosa)  2  =  dS dt  Since  We c a n s a y t h a t . t h e dS/dt  -  square root  a sinjwave.  External  .  Driving  The m a g n e t , C,  ( F i g . I l l a n d I V ) i s d r i v e n by a h i g h l y  'motor GKH t y p e NSH 5 4 R L . The  i s close to 1 therefore  motor  stability  generator  feedback  The power o f t h e motor  i s obtained  by means o f a p r e c i s i o n  circuit  of  The  lo.ad o r l i n e  increased between  combined w i t h  a  a m p l i f i e r and l e a d network a l l o w p r e c i s i o n adjustment  s p e e d o v e r r a n g e s o f 100:1  magnet  tacho-  special line voltage correction  n e t w o r k and t e m p e r a t u r e c o m p e n s a t i o n c i r c u i t r y  o f motor  H.P.  controller.  A zener-diode reference  transistor  i s 1/8  stable  voltage  with  regardless  changes.  i s c o n n e c t e d t o t h e motor the f u l l  s t a b i l i t y o f t \%  by means o f a g e a r w h i c h  speed o f t h e d r i v e f o u r times.  0 a n d 25 c p s c a n be  obtained.  Thus  frequencies  30  The  Main  Pipe pipe 2"  It c o n s i s t s o f a s t r a i g h t assures  t h a t t h e p i p e does  reflection  The  i n f a c t behave  (2")  satisfies  This  l i k e a n i n f i n i t e (no  the restrictions  t h e w h o l e p i p e was made o f p y r e x  into three parts, V  wave-length.  on t h e Matthews  first  part  glass.  ( F i g . V I M ) has f i v e s m a l l  ( 1 " I.D.) s h o r t s i d e  These s i d e pipes c o n t a i n t h e d e t e c t o r s .  p r o b l e m was t h e a l i g n m e n t  small e r r o r i n the alignment  pipe.  So we d e c i d e d  connected  (2"  VIII.  was e n o u g h t o b r e a k t h e g l a s s  t o s u b s t i t u t e t h e second and t h i r d  I.D.) c o p p e r p i p e .  t o a bellows  any v i b r a t i o n s , and  f o r ease o f assembly.  the copper pipe t o t h e bellows  unit  w e r e made vacuum t i g h t  segments  T h e vacuum pump e n d was  u n i t which absorbed  a l l o w e d some f l e x i b i l i t y  connections  The s e p a r a t i o n  of the three parts of the pipe.  A very  w i t h an 8 '  I t was d i v i d e d  each.  between d e t e c t o r s a r e g i v e n on t h e f i g u r e  One  long.  ( s e e S e c t i o n 3«3)  Originally  pipes.  I.D. a n d 1 2 '  from t h e end) tube f o r t h e l o n g e s t  diameter  equation  The  -  The c o n n e c t i o n o f  i s shown i n F i g . V. A l l by means o f a g a s k e t  o r an  0-ring.  Vacuum The  System  c a l c u l a t i o n o f t h e r e q u i r e d p u m p i n g s p e e d was made b y c a l -  culating  the pipe conductance F  Figure V  32  F~  1fe= •  KFSd^./.T  +  s  c  c  /  1  ,  t  '  40  =  300 K  =  *•''...''.  5 cm.  - D. = then  378d 3  365 cm.  M argon  T  and  (  -  ;  t h e speed S was  •s  =  calculated  (L.- -M P  F  p where p i s the pressure  a t t h e end o f t h e p i p e b e s i d e t h e  c h a m b e r , a n d Pp t h e p r e s s u r e p r e s s u r e drop  beside the diffusion  i n t h e p i p e was r e q u i r e d t o be l e s s  the p o r t i o n o f pipe f o r which taken: was  at  \0~  h  Torr.  The l i m i t i n g  continuous  6  running  Two m e c h a n i c a l  = 3» t h e s p e e d r e q u i r e d  pump h a s a s p e e d o f 22 1 / s e c  The d i f f u s i o n  pumps w e r e u s e d a a n d b_ ( F i g . V I ) a n d b o t h a r e T h e s e pumps h a v e a f a i r l y  20  over  liters/minute  Valve  t h e range between  Pump b i s a l s o u s e d  McLeod gauge.  pump i s c o o l e d by  water.  C e n c o , Hyvac 2.  pressure.  diffusion  f o r e - p r e s s u r e i s 0.100 T o r r and t h e  10~ T o r r .  ultimate pressure  actually  A C o n s o l i d a t e d Vacuum C o r p o r a t i o n  pump t y p e VMF-20 was i n s t a 11 ed..This  The 10% o v e r  than  t h e measurements were  h e n c e t a k i n g a r a t i o o f P/Pp  S = 22,4 1/sec.  pump.  Valves  2 isolates  constant  10~ T o r r , t o 2  speed o f atmospheric  t o pump down t h e m e r c u r y  from t h e  1 and 2 a r e S a l i s b u r y . high-vacuum v a l v e s .  t h e pump b f r o m  t h e system  when  i t Is used t o  33  FIGURE V I Vacuum L i n e s  To main pipe  CD  • CDo  O  05  -  pump t h e M c L e o d g a u g e .  k.S  34 -  Valves  3 and 4 a r e a i r i n l e t s .  P r e s s u r e Measurement System The m e a s u r e m e n t experiment function  is highly  s i n c e t h e wave s p e e d ' i s  going  important  i n the  t o be m e a s u r e d a s a  o f t h e pressure and frequency.  The p r e s s u r e panel  o f the pressure  i s c o n t i n u o u s l y read  ( t y p e RG).  The i o n i z a t i o n  i n a V e e c o vacuum gauge  g a u g e u s e d was a V e e c o RG-75.  The p r e s s u r e m e t e r i n t h e box was c a l i b r a t e d McLeod gauge w h i c h  reads  The Veeco i o n i z a t i o n  absolute  f o r Argon a g a i n s t a  pressure.  gauge was s i t u a t e d  side pipes, J , (Fig. V I M ) .  control  i n the f i r s t  o f the small  I t w a s m o u n t e d b y means o f a b r a s s  p l a t e w i t h a h o l e a t t h e c e n t r e w h e r e t h e g a u g e was s o l d e r e d •(Fig. V I I ) .  Opposite  t h e V e e c o gauge a s m a l l g l a s s p i p e  ( F i g . V I I I ) so that both  i s d e r i v e d t o t h e McLeod  (Veeco and McLeod gauges) measure t h e  p r e s s u r e a t t h e same p o i n t o f t h e m a i n p i p e .  The McLeod gauge has a b u l b  v o l u m e o f 408 c c . a n d a b o r e  of 0.625 mm. m e a s u r e d w i t h a t r a v e l l i n g  T h e M c L e o d g a u g e was c a l i b r a t e d was c h e c k e d  f o r gross  T h e s e n s i t i v i t y was  from  diameter  microscope.  i t sobserved  dimensions.  e r r o r s a g a i n s t a V e e c o i o n gauge  unit.  This  gauge  Figure  VII  DETAIL  OF VEECO  I.G.  AND  DETECTOR  -  36 -  cross section _ ir x (0.625 x iO ) c m bulb volume .~ 408~ cc 1  • =  2  2  7.6 x 10'6 cm Hg cm"  1  ISince the range of pressure for the experiment is between 10~ Torr 3  -IO"* Torr, the corresponding range for the McLeod gauge reading 4  x , i s , 1.14 cm  < x < 3.64 cm  The gauge was connected to the two extremes of the main pipe (Fig. VII l ) .  So we were able to measure the gradient of pressure  i n the pipe and therefore to make the corresponding correction for the pressure at the detectors. each extremity of the pipe  If p and Pp are the pressures at  p - p / x (detector distance from the  Veeco I.G.) is the correction to be added to the pressure read at the Veeco gauge.  1n general the pressure at the end of the pipe beside the chamber was measured to be three times the pressure at the end beside the diffusion pump, so that the pressure varied by only -5% over the length of pipe used to take measurements.  McLeod Gauge  1  Wave Generator  Main  Pipe  •qciru~ini  ., veeco  . . l.G.  Figure  VIII  Detectors  MAIN  PIPE  ~'  38  -  CHAPTER V  ELECTRON ICS AND METHOD OF PHASE MEASUREMENT  5.1  Introduct ion This  chapter  describes  t h e e l e c t r o n i c s used f o r t h e phase measure-  ment a n d t h e m e t h o d o f m e a s u r e m e n t i s g i v e n on t h e next  itself.  page, F i g . IX.  The s i g n a l  diagram  f r o m one o f t h e  i o n gauges d e t e c t o r s , a f t e r a m p l i f i c a t i o n and f i l t e r i n g  is applied  to the Y-plates  i s taken  o f an o s c i l l o s c o p e .  from a p h o t o c e l l that detects generating  chamber.  o s c i l l o s c o p e , which being  This  A reference signal  t h e r o t a t i o n o f t h e d i s c i n t h e wave-  i s applied to the X-plates  i s used as a phase n u l l  measured from a phase s h i f t  described  of the  d e t e c t o r , t h e phase  system i n the reference  i s measured  r e f e r e n c e , and t h e phase d i f f e r e n c e f o u n d . w e r e 31 cm. a p a r t .  Larger  relative  to a  The d e t e c t o r s  hand, t h i s  to noise  r a t i o would  On t h e o t h e r  that both  d e t e c t o r s a r e a t t h e same mean p r e s s u r e , w i t h o u t  s e p a r a t i o n a l l o w s us t o assume  The measured phase d i f f e r e n c e s between t h e gauges  f r o m 20° t o 60° f o r a f r e q u e n c y  Most  used  decrease  rapidly.  The  fixed  s e p a r a t i o n s were n o t used because o f  increasing attenuation, the signal  error.  channel  below.  Thus t h e phase o f each d e t e c t o r  5.2  A schematic  a big ranged  r a n g e f r o m 3 t o 25 H z .  Detector important  of a l l the electronic  components  i s the detector.  Wave iGenerator  Main  o3  Pipe  Detector  1  Detector  Signal Detector iControl Box  2  Channel  fi  I t e r i ng  fompl ijpat i * n Osci1loscode Reference Channel  i1tering  Photocel 1  \mpl i i c a t i o n  Figure  X  ELECTRONICS.  FUNCTIONAL  Phase^-Shifte  DIAGRAM  to  40 -  The most  suitable  pressure  i s an  Our  detector  ionization  f o r d e t e c t i o n of a small gauge.  p r e s s u r e wave had a p p r o x i m a t e l y  1/50  the pressure  level  a peak t o peak a m p l i t u d e  a t t h e main p i p e .  Ton c u r r e n t o f t h e c o l l e c t o r I/SO  s i n c e we  had a p e a k t o p e a k a c c u r r e n t o f .. • .  f o r the choice  d i d n ' t want  Is only  o f an  ionization  to interfere with  T h e A E I 2 9 D 1 5 h a s a 49 mm dissipated  10 w a t t s .  Normally  a b o u t 40 w a t t s .  perature the  g r a d i e n t and  related  diameter,  other  bigger  dissipation  flow.  a n d t h e power  commercial  ionization  a n d t h e power d i s s i p a t e d  could  l e a d t o a non z e r o  e f f e c t o f gas f l o w  the sensitivity  commercial with  o f t h e 29D15 i s s m a l l e r  g a u g e s , i t was  i n the v i c i n i t y  of  than  that of other  enough f o r t h e d e t e c t i o n o f t h e s i g n a l  the e l e c t r o n i c s used.  sensitivity  micron pressure  o f t h e 29D15 g a u g e approximately.  t  i s 5.7  ^ff  microamp/mi1iamp  .  Since  -  .  I was  miniature  .  .-.  -  interested in a signal  Bayard  and A l p e r t  m o u n t e d on a 9 ~ p l n  ••  -.  •  .  considered.  •  a s good a s p o s s i b l e , I made a  ionization  socket.  emission/  r  T h e p o s s i b i 1 i t y o f us i ng a Bayard" a n d A l p e r t g a u g e was  was  tem-  i o n gauge f i l a m e n t .  Although  The  This  g a u g e v/as i t s v o l u m e  t h e gas  l e n g t h a n d 20 mm  g a u g e s h a v e a v o l u m e 6 t o 10 t i m e s is  of  T h a t means t h a t t h e  o f t h e dc l e v e l .  A limitation  variation in  gauge  1 6  (Fig.  X  ).  It  -  The  kl -  cathode and c o l l e c t o r  wire.  The anode cage  a r e .025 cm (0.010" n o m i n a l )  i s a s s e m b l e d by w i n d i n g  and s p o t - w e l d i n g  0.008 cm (0.003" n o m i n a l ) t u n g s t e n w i r e a t a b o u t around four p a r a l l e l electrodes  a r e supported  leads o f a standard  The  .16 cm s t a i n l e s s  steel  .1 cm s p a c i n g  posts.  The t h r e e  a t each e n d by a s p o t - w e l d i n g  9 pin -  vacuum s e a l e d  The reason  f o rthis  t o the  socket.  s e n s i t i v i t y o f t h i s g a u g e was a b o u t t h r e e t i m e s  o f t h e 29D15.  tungsten  b i g g e r than  that  i s that the grid wires o f the  cage a r e q u i t e s m a l l , s o most o f t h e e l e c t r o n s m i s s t h e cage and shoot  across  collector several  potential,  to  quickly  t h e anode.  Ions  energy t o penetrate t o more n e g a t i v e  inside  the grid A l l this  t h e cage which  i s negative with  formed o u t s i d e t h e cage w i l l t h e anode g r i d ;  through  by t h e c a g e .  t o the c o l l e c t o r which  they w i l l  have  respect  insufficient  be a t t r a c t e d  elsewhere  surfaces.  v o l t a g e s a p p l i e d were: Electrode  Potent i a l  cathode  200 V  anode  *»00 V 0  col lector  In s p i t e  d e f l e c t e d by t h e  back and f o r t h  produces a higher q u a n t i t y o f ions  proceed  Typical  perhaps passed  times, but eventually are collected  travelling will  t h e i n n e r p a r t o f t h e cage u n t i l  o f t h e good r e s u l t s  I had t r o u b l e s p o t - w e l d i n g  f r o m t h e "home-made" i o n i z a t i o n  the grid  t o the posts, the posts  gauge, t o the  A3  pins o f the standard  -  socket, etc.  And s i n c e t h e s e n s i t i v i t y o f  the  29D15 was g o o d e n o u g h , I d e c i d e d t o u s e i t .  The  29D15 AEI g a u g e  i s run under t h e f o l l o w i n g c o n d i t i o n s :  grid  The  5.3  +200 V  collector  0  f i lament  + 20 V.  filament  current  1 t o 1.5 A  emission  current  2 t o 5 niA  installation  of this  detector  is. c l e a r l y  shown  i n F i g . XI.  I o n Gauge C o n t r o l B o x In o r d e r  t o h a v e t h e i o n c u r r e n t d e p e n d i n g on t h e p r e s s u r e  we n e e d a f i x e d current The  grid  c u r r e n t . ( I t i s assumed  iscollected  grid  is highly fluctuating  stability and  The  by  circuit  ionization  t o design  a circuit  to maintain  sufficient  current  in series with  current in thefilament.  on i t s s a t u r a t i o n curve and i t s  b y a vacuum t u b e ,  "loop  This  circuits,  the grid  for regulating the e l e c t r o n emission  t h e i o n gauge e m i s s i o n  obtain  current.  '  i s operated  i s loaded  i n t h e f i l a m e n t tem-  i n the g r i d  gauge employs a t r a n s f o r m e r  transformer  winding  change  close limits.  A standard an  and any l i t t l e  i s t h e b a s i c p r o b l e m o f a l l i o n gauge  i t i s necessary  within  t o v a r y w i t h t h e p r e s s u r e , and moreover i t  produce a great v a r i a t i o n  problem  that a l l the emission  by t h e g r i d . )  c u r r e n t tends  perature w i l l  only,  current.  the bias o f which I t o f t e n proves  gain" i n this  feedback  secondary  i s controlled  difficult to  system because small  Figure  XI  ATTACHMENT OF DETECTOR TO MAIN  PIPE  kS  tubes w i l l tubes  amplifier  Richard they  The  not load t h e transformer  require too great  commercial  circuit  a grid  gain  As a r e s u l t , o n e and a dc pentode  T h e n we t r i e d  loop  than t h e commercial  ingrid  current  up t o \5% was o b s e r v e d .  a modification of thec i r c u i t  theemission  current  through hours o f a run.  grid  current  by H o l m e s  to  1 7  used on r o c k e t s . (no o b s e r v e d  We made a t w o c h a n n e l  box,  each  independent from t h e o t h e r , so that t h e c i r c u i t i s  symmetric w i t h  respect  The  PNP  transistor  designed  i n mass s p e c t r o m e t e r s  c i r c u i t we g o t a c o n s t a n t  totally  boxes.  f o r o n o u r g a u g e b u t t h e r e g u l a t i o n was n o t  good enough; a v a r i a t i o n  channel  (6 BA's)  current  used a t h y r a t r o n f o r l o a d i n g t h e t r a n s f o r m e r and  c i r c u i t was t r i e d  variation)  swing f o r c o n t r o l .  high  (6SH7) t o p r o v i d e t h e r e q u i r e d f e e d b a c k .  and T u t h i l l  With t h i s  sufficiently, while  u s e s two power t u b e s  g o t an even b i g g e r  control  -  emitter-collector  t o AB ( F i g . X I I ) .  2N1502 ( p o w e r t r a n s i s t o r ) p a s s e s t h r o u g h i t s  circuit  a l l thecurrent  (1 t o 1.5 A ) r e q u i r e d  for  the  29D15 g a u g e f i l a m e n t .  the  2N1502 b a s e c u r r e n t w h i c h i s s u p p l i e d by t h e e m i t t e r o f t h e  PNP  2N1038. T h e 2N1038 b a s e c u r r e n t i s p r o v i d e d by t h e c o l l e c t o r  o f NPN  The  filament current  i s c o n t r o l l e d by  2N2192 t r a n s i s t o r , t h e b a s e o f w h i c h i s c o n n e c t e d d i r e c t l y  to theemission current  This  current  line.  Thus v a r i a t i o n s i n t h e e m i s s i o n  a r e a m p l i f i e d by tandem a m p l i f i e r  phase o f t h e a m p l i f i e d output s i g n a l  2N2192, 2N1038 a n d 2N1502. i s such t h a t a s l i g h t  Figure XII  IONIZATION GAUGE CONTROL  -  kl  increase  in emission  desired value  current  of emission  i s s e t by  meter c o n t r o l s a constant battery.  The  the emission The  value  backing  the  filament  potentiometer  current  current  i s the  c o n t r o l 1ing s i g n a l  been chosen so t h a t  s e t t h r o u g h o u t f r o m 2.5  the  2N1038 t r a n s i s t o r  t i o n occurs  l a g between a p p l i c a t i o n o f emission  The  back  oscillation loop.  2N1038  6.8  filament  be  by  f u r t h e r lowering K resistor  battery  the  f o r the  regulator.  current A  the  and  100  be uf  the  circuit.  base  Oscilla-  the  appearance  and  may  r e g u l a t o r and  time of  a l s o helps  the o v e r - a l l  provides  gain of  cut-off bias  switch  charged.  i t s emitter  current  current  continues  1 second a f t e r  the  is f i r s t  for  to the  feed-  the  100  closed, current  uf c a p a c i t o r .  i o n i z a t i o n gauge f i l a m e n t  is fully  sends e x c e s s i v e current  current  2N2192 e m i t t e r c i r c u i t  K r e s i s t o r charges  capacitor that  in the  6.8  The  temperature of  so  volt  transistor.  When t h e the  current  potentio-  backing  c a p a c i t o r because of  filament  The  current.  100 tt r e s i s t o r  avoid  this  a 3  t o more t h a n 30 mA.  lowers the ac g a i n of  in the absence of  This  by  the emission  c a p a c i t o r c o n n e c t e d between the o u t p u t of the of  current.  Rj.  supplied  a l g e b r a i c d i f f e r e n c e between the  o f Rj h a s  arbitrarily  decreases  the  is at very  until  high  filament the  2N1038 i s  level which  v i a 2N1502.  c a p a c i t o r has  proper emission  emission  is exceeded before  Meanwhile, t r a n s i s t o r  to the  to flow  However,  through  biased  in turn  This  excessive  charged, which  t e m p e r a t u r e has  the  can  been m a t c h e d .  48  The  value  o f t h e maximum f i l a m e n t c u r r e n t  means o f a n o n - l i n e a r c i r c u i t connected across  element  t h e 2N1502 e m i t t e r  (1.6A) i s f i x e d by  (three s i l i c o n  resistor.  diodes)  These diodes  o v e r a l a r g e p o r t i o n o f t h e 2N1038 b a s e c u r r e n t , t h u s gain o f the feedback diodes  l o o p when c u r r e n t e x c e e d s t h i s  also effectively  s h u n t t h e 6.8  20%  v a r i a t i o n s on t h e v o l t a g e  p r o d u c e an o b s e r v a b l e pressure current  from  10~  3  5  Torr  are  interested.  The  Signal  The  collector current  R2  CR-4  The  filament  i s considerably  reduced.  f r o m t h e b a t t e r y do n o t c u r r e n t , and v a r i a t i o n i n  does not produce o b s e r v a b l e  grid  PARC a m p l i f i e r .  r e j e c t ion f i 1 t e r s . reduced  varied:  from our d e t e c t o r ,  i n which  we  Channel  is the signal  almost  time  in grid  i s the signal  from t h e d e t e c t o r produces a v o l t a g e  ( F i g . X I I ) o f 1 t o 20 mV  which  value.  change.  voltage across  be  variation  t o 10"  The  R2  supply  l i m i t i n g the  K r e s i s t o r during  warmup s o t h a t t h e c a p a c i t o r c h a r g i n g  take  ( d e p e n d i n g on t h e f r e q u e n c y  of interest. This  and p r e s s u r e ) ,  s i g n a l was b r o u g h t t o t h e  a m p l i f i e r h a s two ( l o w a n d h i g h ) band  Therefore  t o the f i r s t  This  across  the frequency  harmonic.  i n our case the signal  of the signal  c a n be  The a m p l i f i c a t i o n f a c t o r c a n  was a m p l i f i e d 50 t i m e s  approxi-  mately.  The  signal  f r o m t h e CR-4  PARC a m p l i f i e r i s c o n n e c t e d  to the reference  kS  -  i n p u t o f t h e J B - 6 PARC p h a s e s e n s i t i v e d e t e c t o r w h i c h when to the working  tuned  f r e q u e n c y , and w i t h t h e r e f e r e n c e channel i n  " s e l e c t i v e e x t e r n a l " , g i v e s a t t h e " m o n i t o r o u t p u t " a s i n wave amplified of  about  100 t i m e s  the r e f e r e n c e channel  available).  ( t h u s we a r e j u s t as a tuned  This signal  u s i n g t h e tuned  amplifier  i s connected  amplifi  t h a t was c o n v e n i e n t l y  to the v e r t i c a l  input of the  osci1loscope.  The  Reference  The  reference signal  Reflecting  Channel  Photocell  i s produced  1/20.  a t the crest. The p h o t o c e l l  VPO 2 3 0 ) w h i c h the photocell oscillator  be v a r i e d  ripple  to signal  signal.  tightly  (type  l o c k e d i n f r e q u e n c y and phase t o  T h e a m p l i t u d e o f t h e s i n wave f r o m t h e  Once t h e o s c i l l a t o r  t o t h e same f r e q u e n c y as i s tuned, t h e output can  f r o m 0 t o a p p r o x i m a t e l y 10 v o l t s .  One o f t h e m a i n f e a t u r e s o f t h i s o s c i l l a t o r output channels. signal,  i s approximately'  o u t p u t d r i v e s a Feedback O s c i 1 l a t o r  i s a maximum when i t i s t u n e d  the d r i v i n g  signal  p-p a n d i t h a s a t o o t h f o r m w i t h s m a l l  The r a t i o  remains signal.  443)  P ( F i g . I l l ) (see S e c t i o n 4.5). This  is approximately 3 v o l t s ripples  by t h e Beckman (model  t h e second  One o f t h e c h a n n e l s channel  i s that  i s i n phase w i t h the d r i v i n g  i s +90° o u t o f p h a s e ,  a variable phase-shift output, controlled  error of less  The  i s connected  the o s c i l l a t o r  and t h e o t h e r has  by a l i n e a r  phase-shifter with a calibration  s i n wave f r o m  i t has t h r e e  calibrated  t h a n ^0.5°  1 8  .  t o the X input o f the  50  oscilloscope,  which  characteristic  t h e r e f o r e produced a L i s s a j o u s  o f t h e phase d i f f e r e n c e  Method o f Phase In o r d e r  b e t w e e n t h e two  t o make a m e a s u r e m e n t  we  is stabilized.  i s s u p p l i e d by a u n i j u n c t i o n  sistor's  characteristic  reached.  After  reference  frequency.  observed  this waiting  channel  transistor.  a l l leads were f i x e d  t h a t o s c i l l a t i o n s and v i b r a t i o n s  i n t h e CR-4  rejection Taking  s i n w a v e was  This  small  fluctuation  fact  a fluctuation  the Lissajous  of the leads  t a i n ! t y o f -2°  i t was  produced  rejection  and f i x i n g a l l l e a d s ,  wave.  The  filter. the ampli-  (fluctuation  in amplitude of the f i n a l  of  5%).  s i n wave i s i n  T h e r e f o r e , the e l l i p s e axes  are s l i g h t l y fluctuating,  i n t h e phase  since  i n the f i n a l  almost constant  of the frequency.  figure  temperature i s  w e r e s e t a t 10 H z . f o r low band  the precautions of s h i e l d i n g  tude of the f i n a l  of  PARC a m p l i f i e r  tran-  i t i s necessary  in position,  and 30 Hz. f o r t h e h i g h band  filter,  Since the  the o s c i l l a t o r i s tuned t o the  noise which created a non-uniform amplitude filters  R e c t i f i e r whose g a t e  the equilibrium  period,  of a l l ,  The c i r c u i t c o n t r o l l i n g  d e p e n d on t h e t e m p e r a t u r e ,  t o w a i t a b o u t 10 m i n u t e s u n t i l  In t h e s i g n a l  channels.  have t o e n s u r e , f i r s t  s p e e d o f t h e m o t o r has a S i l i c o n C o n t r o l  voltage  figure  Measurement  t h a t t h e motor frequency the  -  giving  an  uncer-  measurement.  In o r d e r t o m e a s u r e t h e p h a s e s h i f t b e t w e e n t h e two  ionization  gauges  50  oscilloscope,  which  characteristic  o f t h e phase d i f f e r e n c e  Method o f Phase In o r d e r that the  t h e r e f o r e produced a L i s s a j o u s  t o make a m e a s u r e m e n t  we  is stabilized.  i s s u p p l i e d by a u n i j u n c t i o n characteristic  reached.  After  reference  frequency.  observed  this waiting  channel  transistor.  a l l leads were f i x e d  t h a t o s c i l l a t i o n s and v i b r a t i o n s  i n t h e CR-4  rejection Taking  filter,  s i n w a v e was  small  fluctuation  fact  a fluctuation  the L i s s a j o u s  of the leads  t a i n ! t y o f -2°  i t was  produced  rejection  and f i x i n g a l l l e a d s ,  wave.  The  filter. the ampli-  (fluctuation  i n amplitude of the f i n a l  of  5%).  s i n wave i s i n  T h e r e f o r e , the e l l i p s e axes  are s l i g h t l y fluctuating,  i n the phase  since  i n the f i n a l  almost constant  of the frequency.  figure  temperature i s  w e r e s e t a t 10 Hz. f o r l o w band  the precautions of s h i e l d i n g  This  of  PARC a m p l i f i e r  tran-  i t i s necessary  in position,  and 30 Hz. f o r t h e h i g h band  tude o f the f i n a l  Since the  the o s c i l l a t o r i s tuned t o the  noise which created a non-uniform amplitude filters  R e c t i f i e r whose g a t e  the e q u i l i b r i u m  period,  of a l l ,  The c i r c u i t c o n t r o l l i n g  d e p e n d on t h e t e m p e r a t u r e ,  t o w a i t a b o u t 10 m i n u t e s u n t i l  the signal  channels.  have t o e n s u r e , f i r s t  s p e e d o f t h e m o t o r has a S i l i c o n C o n t r o l  sistor's  In  b e t w e e n t h e two  Measurement  t h e motor frequency  voltage  figure  giving  an  uncer-  measurement.  In o r d e r t o m e a s u r e t h e p h a s e s h i f t b e t w e e n t h e two  ionization  gauges  51  we p r o c e e d a s f o l l o w s : (a)  With  the signal  f r o m gauge I ( t h e c l o s e r t o t h e c h a m b e r ) ,  we s e t a z e r o p h a s e L i s s a j o u s f i g u r e b e t w e e n t h e two The  calibrated  zero.  (b)  phase-shifter i n the o s c i l l a t o r  channels.  was s e t t o  T h e n t h e z e r o p h a s e was s e t i n t h e s i g n a l  channel  using  the u n c a l i b r a t e d p h a s e - s h i f t e r i n t h e phase s e n s i t i v e  detector.  With  returned  the signal  from gauge  the L i s s a j o u s f i g u r e the c a l i b r a t e d  II ( 3 1 c m f r o m g a u g e  l ) we  t o z e r o p h a s e c o n d i t i o n a g a i n by m o v i n g  phase-shifter i n the o s c i l l a t o r .  reading of the c a l i b r a t e d  Then t h e  p h a s e - s h i f t e r was t h e r e q u i r e d  phase  measurement.  T h i s o p e r a t i o n was  repeated  f o r each frequency  E a c h m e a s u r e m e n t was p e r f o r m e d The  reproducibility  five  times  a t each  pressure.  and t h e a v e r a g e  taken.  o f m e a s u r e m e n t s was w i t h i n ± 2% o f t h e v a l u e  obtained.  T h i s method o f measurement does but  these  introduce spurious  a r e e l i m i n a t e d by t h e n u l l  method used  phase  (except  shifts, that  it  is  a s s u m e d t h a t a n y p h a s e s h i f t o f t h e p r e s s u r e w a v e , due t o t h e f i n i t e response time o f t h e gauge,  T h i s method o f measurement methods t r i e d for  before.  i s t h e same f o r e a c h o f t h e g a u g e s ) .  i s simple  a n d more a c c u r a t e  than  other  T h e p h a s e s e n s t i v e d e t e c t o r c o u l d n o t be u s e d  phase measurement d i r e c t l y  f l u c t u a t i o n of our s i g n a l .  because o f t h e s l i g h t  The t u n i n g  i s very  frequency  critical  when m a k i n g  52  a measurement w i t h a phase s e n s i t i v e fluctuation  encountered  d e t e c t o r , and t h e f r e q u e n c y  l e d t o measurement e r r o r s  up t o t  10°.  53  CHAPTER VI  RESULTS AND DISCUSSION  Analysis  o f Data  The. c a l c u l a t i o n  o f t h e p r e s s u r e wave v e l o c i t y was made u s i n g t h e s i m p l e  relation  v  =  Xf  X  ~  360 x d i s t a n c e b e t w e e n g a u g e s / m e a s u r e d  f  =  frequency  v  =  where  phase  and  therefore  The  360 x 31xf/p  r e s u l t s o f these  calculations  -together w i t h e r r o r s . diffusion  equation  prediction  a r e shown  Theoretical  (3)  range o f i n t e g r a t i o n  cm/sec  i n t h e next  graphs  c u r v e s a r e a l s o shown f o r t h e  and f o r t h e d i f f u s i o n e q u a t i o n w i t h  ( i . e . t h e low f r e q u e n c y  limit  limited  o f Matthews'  (10)).  Conclus ion 1.  En t h e s e e x p e r i m e n t s ,  no d e p e n d e n c y o f wave p r e s s u r e  o n m e a n f r e e p a t h was f o u n d frequency  investigated.  difference, 1  which  f o r t h e range o f p r e s s u r e and  Matthews' formula p r e d i c t s  i n t h e s l o p e o f t h e graph  frequency ^ , 2  velocity  of velocity  a b o u t 30%  against  f o r t h e e x t r e m e v a l u e s o f mean f r e e p a t h s a t  I have been w o r k i n g ;  this  d i f f e r e n c e s h o u l d be e a s i l y  I  m sec  1 _  l  :-  1 _  AO  i ' i  i  /  -  •  !  :•  !  i i  30  > -o  -t- q -\-  +,,  I -  -  :•:  -  -  ...  -  1  I  1  1  -  -  -  - -- -  -  1  --  "1 1 1  i  -  -  -  i )  1  ...  -  --  —  -  y  r  ••  >  1  I  L  i  -  I'M (frequency)  -  -- -  -  +  ...  -._  -  i i i  1  -  I  -  4.8  --  -  -  --  '')  -r  -L  •  . V  —  r.  13  17  -  _  i  - -  -  -  -  -  --  TO  §  -  - -•  -  —  rir,!<  -  -  !l r  ->  i  i  t"  --  -  •- -- -  -  \  'J  -1 •on  1n  •U J 1T - 3rina  /  —  c)  -  l  -  -  z-  -  | - U  23  _  - -  -  1~  -  J  +7 X 1  -s-  p  t  -  --  U  rpm  1  J c  -  +  - 0"r  1  T 8  , i-  ...  K  1 --  X  -  -  I  i  -  -  1  -  i  -  I  4 -  T-  - --  1  - -  --  -j-  —  —i  -  T r  1  \  \  i  -  t  -  if  -  I  I  r  ?~  /'  i  s  ./  --  's  f/  7  "3 \  1  i  "T  ! 1 1 YW  _i ' 1//  -  /  -  f  -  /  r  1/ 1  1  / -h - -  r  A  /  u  -  V  1  r  .  10  11,'  r  /i  < ) \ \/  -  -  i  i -  -rr  -  --  L  l  —  -  1  —j—, 1  - --  i  !  20  -  -  --  - -  S  --  -  --  -  i ~r —r-  _,  --  - -  1  i  —  -  -  i-  y  v.  1,1 7  v e l o c i ty  CD  > /  J  l i l _ i  1  1  i  ft  -  --  -X  -  -  ON  58  observable  with  -  the e r r o r s that were present  i n the  measure-  ment.  2.  The  experimental  that of  t h e one  v/f / 1  v/f ^ 1  p r e d i c t e d by  =  2  slope  (AfDTr) / 1  is siightly  2  the c l a s s i c a l  greater  Diffusion  than equation,  2  • w i t h  D  =  2rV/3  V  =  3.99  t t was  x  thought  flow of  the  IO  t h a t t h e main  gas  due  different  t h e medium  rates at  in which  the  from f u l l  steady  g r a d i e n t , which  would  t h e p r e s s u r e wave  was  t h e same mean p r e s s u r e  t h e d i f f u s i o n pump v a l v e no  f o r t h i s was  However, measurements were performed  flow  closed;  reason  t o t h e mean p r e s s u r e  • produce a motion of propagating.  cm/sec  4  using  by a d j u s t i n g  open to a l m o s t  completely  d i f f e r e n c e i n the measured phase s h i f t  was  observed.  Comments The  statement  single  n u m b e r one  measurement  f i g u r e has  of  in the  10  3  T h i s has  once the  been set t o z e r o phase, then w i t h o u t  Torr  Lissajous  conclusion is clearly  f o l l o w i n g way:  t h e e l e c t r o n i c c o n t r o l s , we to  the  by  opening  figure been  does not  repeated  f o r each  frequency  and  any  f r o m 2 x 10  the Argon f l o w .  change at a l l d u r i n g  a  Liassajous  touching  changed the p r e s s u r e  the v a l v e c o n t r o l i n g  shown by  the whole  of 4  Torr The  process.  for different  gauge  59  p o s i t i o n s , and t h e r e s u l t s when t h e p r e s s u r e  In  order  S ,  thermal  the f l u x of molecules  -  Smolouchoski case)  +°°  molecules alone  ( o r phonons  first  =  2vr/3.  b e t w e e n +L p r o d u c e d to D  =•  a section of the  i n the corresponding  M a t t h e w s assumed  the flux  through  2/3 V r ( l - 3 T / 4 L ) ,  This assumption, approximation,  that  by m o l e c u l a r  collision  With  S .  This  Q  in  or theoretical  seems r e a s o n a b l e  signal first  channel  e x p e r i m e n t a l l y n o t t o be Perhaps the molecules  s h o u l d be t a k e n  into account.  order  diffusion  equation.  t  to explain the difference  I t was c h e c k e d  c l o s e t o t h e chamber  arrived  this  (10 cm)  Since the  i t c o u l d be  a t the d e t e c t o r w i t h a speed  charac-  b u t o f a m o l e c u l a r beam.  possibi1ity  measurements  v/ith  that the  up" t h e r e f e r e n c e s i g n a l .  n o t o f a wave p r o p a g a t i o n  to investigate  This  i s much s m a l l e r  o b t a i n e d e x p e r i m e n t a l l y when c o m p a r e d  d e t e c t o r was v e r y  teristic  h a s been f o u n d  was n o t " p i c k i n g  t h a t some m o l e c u l e s  fora  number two i n t h e c o n c l u s i o n , no e x p e r i -  reason  slope o f the curves  that o f the c l a s s i c a l  assumption  predicts.  respect t o statement  mental  -» t o  that only the  must p r e s u m a b l y p r o d u c e a dependence o f D on L t h a t than Matthews e q u a t i o n  =  on L a t t h e low f r e q u e n c y  in principle  has been p r o v e d  x  a n d t h i s was t h e e x p e c t e d  a n a d e q u a t e way o f d e s c r i b i n g t h e s y s t e m . scattered  a section  i n t e g r a t e d dF ( s e e S e c t i o n 2.3) f r o m  dependence o f wave p r e s s u r e v e l o c i t y limit.  through  through  ( C a s i m i r & ZIman,  and g o t a D  leads  observed  changes.  to calculate  crystal)  x  a r e t h e same - no p h a s e c h a n g e  a t x = 0, o f t h e p i p e  q  -  were  In  performed  60  using  gauges p o s i t i o n e d a t t h e t h i r d and f i f t h  velocity  side pipe.  The  s o m e a s u r e d was t h e same a s i t was f o r t h e o r i g i n a l  measurement.  No d o u b t t h e t h e o r e t i c a l much f u r t h e r  interpretation  of these results  thought t o f i n d a convincing  explanation.  needs  61 BIBLIOGRAPHY  1963,  1.  C h e s t e r H.  2.  V e r n o t t e P.,  3.  C a t t a n e o M.C.,  4.  Peierls  R.  5.  Haas and  Biermasz  6.  Casimir  7.  Z i m a n J.M.,  8..  B r o w n J . B . , Chung D.Y.  1958,  9.  1958  1938  1954  Physica  5, 45,  Mag.  1055 320  495 100 I966  a n d M a t t h e w s P.W.,  1967  C a n . Jo.urn. o f P h y s . 35,  B r o w n C.R.  11.  Kennard  12.  Weymann, H.D.  13.  Ward J . C . and 'Wi I k s J . ,  14.  Fairbank  1966  15.  Present.  K i n e t i c Theory  16.  FisherS.S.  1967  R e p o r t No. 675  17.  Holmes J.C.  1957  Rev. S c i e n t .  18.  B r o w n C.R.  Physica  K i n e t i c Theory A7). J . P.  35_  323  o f G a s e s (McGraw H i l l 488  1951  (1967)  P h y l . Mag.  P h y s . Rev. L e t t  Private  45,  114  10.  E.H.,  431  2.41  1967  M a t t h e w s P.W.  3_» 5,  Phys i c s  Phil.  21,  (5)  Ann P h y s i k  3154  246,  Compt. R e n d .  1938.  H.B.G.,  2013 246,  Compt. R e n d .  1929  Phys. L e t t .  131,  P h y s . Rev.  j_6  o f Gases  314  789 (McGraw  UCLA Instrum.  communication.  42_,  Mill)  March 28,  1967 290  1938)  

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