UBC Theses and Dissertations

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UBC Theses and Dissertations

Wave propagation in rarefied gases Bejar Hurtado, Jose Antonio 1969

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WAVE PROPAGATION IN RAREFIED GASES BY JOSE ANTONIO BEJAR HURTADO A T H E S I S SUBMITTED IN PARTIAL FULFILMENT OF THE 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 as c o n f o r m i n g t o t h e r e q u i r e d s ^ a n d ^ r d THE UNIVERSITY OF BRITISH COLUMBIA MARCH, 1969 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and S t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my Department o r by h i s r e p r e s e n t a t i v e s . It i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t The U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8, Canada r ABSTRACT T h i s t h e s i s d e s c r i b e s an i n v e s t i g a t i o n o f t h e p r o p a g a t i o n o f a p r e s s u r e w a v e , t h r o u g h a l o n g s t r a i g h t p i p e , i n a r a r e f i e d a t m o s p h e r e o f A r g o n . T h i s phenomenon i s a p p r o x i m a t e l y a n a l o g o u s t o t h e r m a l wave p r o p a g a t i o n i n t h e b o u n d a r y s c a t t e r i n g l i m i t . C a s i m i r a n d Zirnan s t u d i e d t h i s l a t t e r phenomenon f o r t h e s t e a d y s t a t e c a s e . P . W . Matthews e x t e n d e d t h e i r t h e o r y t o t h e n o n - s t e a d y s t a t e a n d f o u n d a d i f f u s i o n c o e f f i c i e n t D d e p e n d i n g on L . S m o l u c h o w s k i s t u d i e d t h e c o r r e s p o n d i n g phenomenon i n g a s e s f o r t h e s t e a d y s t a t e a n d a g a i n P . W . Matthews e x t e n d e d t h e t h e o r y t o t h e n o n - s t e a d y s t a t e , and a g a i n f o u n d a d e p e n d e n c e o f D on L . T h e r e -f o r e a t t h e low f r e q u e n c y l i m i t a wave p r e s s u r e v e l o c i t y w o u l d d e p e n d o n L . A n e x p e r i m e n t was s e t up i n o r d e r t o p r o d u c e a wave p r e s s u r e and m e a s u r e i t s v e l o c i t y as a f u n c t i o n o f t h e f r e q u e n c y and t h e p r e s s u r e and h e n c e L . R e s u l t s have been o b t a i n e d and no d e p e n d e n c e on t h e p r e s s u r e has b e e n o b s e r v e d . TABLE OF CONTENTS A b s t r a c t i i Tab Te o f C o n t e n t s i i i T a b l e o f Tab l e s v T a b l e o f F i g u r e s v i T a b l e o f Graphs '." v i i L i s t o f Symbols v i i i A c k n o w l e d g e m e n t * i x Introduction x C h a p t e r I — T h e r m a l D i f f u s i o n 1. T h e F o u r i e r H e a t E q u a t i o n 1 2. M o d i f i e d F o u r i e r Heat E q u a t i o n 1 3. H e a t F l o w i n t h e B o u n d a r y - S c a t t e r i n g L i m i t .- . . . . 2 k. T h e Matthews M o d i f i e d E q u a t i o n A 5. Brown M o d i f i e d E q u a t i o n 6 - 6. Comments 7 C h a p t e r t8 — Gas D i f f u s i o n .1. H i g h P r e s s u r e R e g i o n 9 2. Gas D i f f u s i o n i n the B o u n d a r y S c a t t e r i n g L i m i t , M o l e c u l a r Flow 10 3. T h e Matthews E q u a t i o n f o r t h e M o l e c u l a r F l o w T h r o u g h L o n g T u b e s 12 C h a p t e r \tl — D i f f u s i o n Waves i n a R a r e f i e d Gas 1. I n t r o d u c t i o n 15 2. Gas C h o i c e 15 3. P r e s s u r e Range and P i p e D i m e n s i o n s 17 k. C o m p u t e r R e s u l t s 17 5. Comments on t h e P r e d i c t i o n s 17 i i i C h a p t e r IV — T h e A p p a r a t u s 1. G e n e r a l D e s c r i p t i o n 21 2 . A r g o n F e e d i n g S y s t e m 21 3 . F low C o n t r o l 21 k. O p e r a t i o n 23 5 . T h e Wave G e n e r a t o r Chamber 25 6 . E x t e r n a l D r i v i n g 29 7 . T h e M a i n P i p e 30 8 . Vacuum S y s t e m 30 9 . P r e s s u r e Measurement S y s t e m 34 Chapter V — E l e c t r o n i c s and Method o f Phase Measurement 1 . I n t r o d u c t i o n 38 2 . T h e D e t e c t o r 38 3 . Ion Gauge C o n t r o l Box 43 A . T h e S i g n a l C h a n n e l *»8 5 . T h e R e f e r e n c e C h a n n e l 49 6 . Method o f Phase Measurement 50 C h a p t e r VI — R e s u l t s and D i s c u s s i o n A n a l y s i s o f D a t a 53 C o n c l u s i o n 53 TABLE OF TABLES I. I o n i z a t i o n gauge s e n s i t i v i t y f o r d i f f e r e n t g a s e s 15 I I . Computer r e s u l t s f o r Matthews p r e d i c t i o n s 18 I I I . C o m p u t e r r e s u l t s f o r Matthews p r e d i c t i o n s - c o n t i n u a t i o n . . . . 19 v TABLE OF FIGURES I . F u n c t i o n a l d i a g r a m o f t h e e x p e r i m e n t . . . 22 I I . D i a g r a m o f f l o w c o n t r o l 2h I I I . I n t e r i o r o f t h e wave g e n e r a t o r chamber 26 IV. G e n e r a l v i e w o f t h e wave g e n e r a t o r a s s e m b l y 28 V . End o f t h e m a i n p i p e , b e l l o w , h i g h vacuum v a l v e and d i f f u s i o n pump 31 V I . Vacuum l i n e s . . 33 V I I . D e t a i l o f t h e V e e c o gauge and d e t e c t o r a t t a c h m e n t t o t h e m a i n p i p e 35 V I I I . M a i n p i p e 37 IX. S c h e m a t i c d i a g r a m o f t h e e l e c t r o n i c s 39 X . Home-made i o n i z a t i o n gauge scheme .- h\ X I . D e t a i l o f t h e d e t e c t o r a t t a c h m e n t t o t h e main p i p e . . . . X I . I . I o n i z a t i o n gauge c o n t r o l box k(> v i TABLE OF GRAPHS I . M a t t h e w s ' F o r u m u l a p r e d i c t i o n s . 20 I I . E x p e r i m e n t a l r e s u l t s , t h e o r e t i c a l p r e d i c t i o n s f o r D = 2v Q r/3, and Matthews t h e o r e t i c a l p r e d i c t i o n f o r L = 33.2 cm. 54 I I I . E x p e r i m e n t a l r e s u l t s , t h e o r e t i c a l p r e d i c t i o n s f o r D = 2v Q r/3, and Matthews t h e o r e t i c a l p r e d i c t i o n f o r L = 12 cm 55 IV. E x p e r i m e n t a l r e s u l t s , t h e o r e t i c a l p r e d i c t i o n s f o r D = 2v Q r/3, and Matthews t h e o r e t i c a l p r e d i c t i o n f o r L = 5.66 cm. 56 V . E x p e r i m e n t a l r e s u l t s f o r L = 33-2, L = 12, and L = 5.66 cm 57 v i i L I S T OF SYMBOLS a = v Q ( l / v Q ) C v = h e a t c a p a c i t y p e r u n i t volume d = d i a m e t e r o f t h e main p i p e D = d i f f u s i o n c o e f f i c i e n t V = s p a t i a l g r a d i e n t o p e r a t o r F = c o n d u c t a n c e o f a p i p e f = f r e q u e n c y k = wave v e c t o r K = t h e r m a l c o n d u c t i v i t y L = mean f r e e p a t h 1 = l o n g i t u d e o f t h e s a m p l e o r p i p e X = wave l e n g t h m = mass o f an A r g o n atom a) = 2irf = p r e s s u r e q = t h e r m a l h e a t c u r r e n t d e n s i t y R = gas c o n s t a n t f o r a gram r = r a d i u s o f t h e c y l i n d r i c a l s a m p l e o r p i p e p = d e n s i t y S = a r e a t = t i m e T = r e l a x a t i o n t i m e T = a b s o l u t e t e m p e r a t u r e v Q = phonon v e l o c i t y V Q = a v e r a g e phonon v e l o c i t y v = v e l o c i t y x = s p a t i a l c o o r d i n a t e v i i i ACKNOWLEDGEMENT It i s a p l e a s u r e t o e x p r e s s my g r a t i t u d e t o my s u p e r v i s o r D r . P . W. Matthews f o r h i s i n d i s p e n s a b l e a d v i s e and e n c o u r a g e m e n t . I am a l s o g r a t e f u l t o M r . A . K s h a t r i y a who made t h e p r e l i m i n a r y d e s i g n o f t h e w a v e - g e n e r a t o r and t h e m a i n - p i p e . M r . C R . Brown f o r s e v e r a l c o n v e r s a t i o n s on t h e b a s i c p h y s i c s o f t h e p r o b l e m . M r . D . L . J o h n s o n f o r many v a l u a b l e d i s c u s s i o n s and p r a c t i c a l h e l p i n t h e f i r s t s t a g e o f t h i s r e s e a r c h . M r . R. W e i s s b a c h , who made t h e w a v e - g e n e r a t o r c h a m b e r , t h e f l o w - c o n t r o l s y s t e m , and who h e l p e d me i n many p r a c t i c a l d e t a i l s . M r . R. H a i n e s and M r . G . B r o o k s f o r t h e i r h e l p w i t h the shop m a c h i n e w o r k . M r . E r n i e W i l l i a m s f o r t h e g l a s s w o r k . The P h y s i c s D e p a r t m e n t o f the U n i v e r s i t y o f B r i t i s h C o l u m b i a f o r i t s f i n a n c i a l s u p p o r t . I s p e c i a l l y want t o t h a n k my w i f e and " p r i v a t e s e c r e t a r y " F l a u r y f o r h e r v a l u a b l e h e l p i n d i s c u s s i n g and t y p i n g t h e f i r s t d r a f t o f t h i s t h e s i s . I a l s o want t o t h a n k M i s s Bev L i t t l e f o r t h e o r g a n i z a t i o n and t y p i n g o f t h e t h e s i s . i x -INTRODUCTION Steady state heat c o n d u c t i o n i n d i e l e c t r i c s f o r t h e phonon boundary - s c a t t e r i n g (low t e m p e r a t u r e ) l i m i t was s t u d i e d i n 1938 by Casimir: in 195*1 Ziman p o i n t e d o u t t h a t i f t h e C a s i m i r r e s u l t s are applied to t h e p r o p a g a t i o n o f a t e m p e r a t u r e d i s t u r b a n c e , t h i s 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 D = 2v Q r/3. 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 i s v = (2Du>)l/2, 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 i n c r e a s e s ; t h i s i s physically u n a c c e p t a b l e , •By adding a wave 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. Matthews (1966) f o u n d a s u i t a b l e way o f limiting the t h e r m a l wave v e l o c i t y a t h i g h f r e q u e n c i e s . S e e k i n g to j u s t i f y this a d d e d ' w a v e ' t e r m , P.W. Matthews (1966) e x t e n d e d the Casimir theory t o t h e n o n - s t e a d y c a s e but f o u n d i n s t e a d a correction term o f t h e f o r m 6 3 T / 6 x 26x. One o f t h e main f e a t u r e s of this treatment i s t h a t t h e t e m p e r a t u r e wave v e l o c i t y depends on the mean free path a t a l l f r e q u e n c i e s . Since thermal d i f f u s i o n i n a c r y s t a l a t v e r y low t e m p e r a t u r e s and gas diffusion in a l o n g t u b e i n t h e m o l e c u l a r f l o w r e g i o n a r e approximately a n a l o g o u s p h e n o m e n a , t h e same t h e o r y can be a p p l i e d t o a "Knud#sen gas 1, w i t h some m o d i f i c a t i o n s . A g a i n t h e gas d i f f u s i o n wave velocity a t low f r e q u e n c i e s i s found t o d e p e n d on t h e mean f r e e path. T h e p u r p o s e o f t h i s t h e s i s i s t o s t u d y the f r e q u e n c y d e p e n d e n c e of t h e d i f f u s i o n wave v e l o c i t y i n a gas a n d , i n p a r t i c u l a r , i t s d e p e n d e n c e on t h e mean f r e e p a t h . In t h e f i r s t c h a p t e r t h e T h e r m a l D i f f u s i o n e q u a t i o n i n t h e b o u n d a r y s c a t t e r i n g l i m i t i s d i s c u s s e d . C h a p t e r II d e a l s w i t h gas d i f f u s i o n c a s e . C h a p t e r III d e s c r i b e s t h e d e s i g n o f t h e e x p e r i m e n t and c h a p t e r IV and V d e a l w i t h e x p e r i m e n t a l t e c h n i q u e s . R e s u l t s a r e p r e s e n t e d i n c h a p t e r V I . 1 CHAPTER 1 THERMAL DIFFUSION l . T T h e F o u r i e r H e a t E q u a t i o n T h e d i f f u s i o n e q u a t i o n f o r t h e r m a l t r a n s p o r t r e s u l t s f r o m 1 : (a) C o n t i n u i t y e q u a t i o n f o r h e a t t r a n s p o r t i n a u n i f o r m medium, C V |I + Vq = 0 (1) (b) F o u r i e r e q u a t i o n o f t h e r m a l c o n d u c t i v i t y f o r h e a t f l o w q = • - K V T (2) C o m b i n i n g ( l ) a n d (2) we g e t t h e d i f f u s i o n e q u a t i o n £ S '- ™ v ° It. i s u s u a l t o d e f i n e a d i f f u s i o n c o n s t a n t D = K / C v and t h e n t o w r i t e t h i s e q u a t i o n i n t h e f o r m 3T _ n 3 2 T , . at " D 3 ^ ( 3 ) 1.2 M o d i f i e d F o u r i e r Heat E q u a t i o n E q u a t i o n (3) p r e d i c t s t h a t t h e v e l o c i t y f o r t h e h e a t waves i n c r e a s e s w i t h o u t l i m i t as ( f r e q u e n c y ) 1 > ^ 2 . T h i s i s p h y s i c a l l y i n a d m i s s i b l e s i n c e rao d i s t u r b a n c e c a n t r a v e l f a s t e r t h a n t h e phonon v e l o c i t y . V ' e r n o t t e 2 i n 19^ 8 p r o p o s e d a p h e n o m e n o l o g i c a l e q u a t i o n w h i c h i s a c o m b i n e d d i f f u s i o n e q u a t i o n and wave e q u a t i o n , x = " c o n s t a n t e de temps t r e s p e t i t e " ( r e l a x a t i o n t i m e ) T h e p h y s i c a l i n t e r p r e t a t i o n o f (k) i s s i m p l e . i " I t s t a t e s t h a t t h e r e i s a f i n i t e b u i l d up t ime f o r t h e o n s e t o f a t h e r m a l c u r r e n t a f t e r a t e m p e r a t u r e g r a d i e n t i s c l a m p e d o n t o a s p e c i m e n . T h e h e a t f l o w does n o t s t a r t i n s t a n t a n e o u s l y b u t r a t h e r grows g r a d u a l l y w i t h a r e l a x a t i o n t i m e T . C o n -v e r s e l y , i f a t h e r m a l g r a d i e n t i s s u d d e n l y removed t h e r e i s a l a g i n t h e d i s a p p e a r a n c e o f t h e h e a t c u r r e n t and e q u a t i o n (k) e x h i b i t s j u s t s u c h a r e l a x a t i o n , w h e r e a s e q u a t i o n (3) does n o t . T h e r e l a x a t i o n t i m e T i s a s s o c i a t e d w i t h t h e c o m m u n i c a t i o n " t i m e " between phonons ( p h o n o n - p h o n o n c o l 1 i s i o n ) f o r t h e commencement o f r e s i s t i v e f l o w . " 1 In f a c t t h e r a t e 1/T must be c o n n e c t e d w i t h t h e r m a l r e s i s t a n c e b e c a u s e x i s t h e t i m e f o r t h e e s t a b l i s h m e n t o f r e s i s t i v e f l o w , x i s p r o p o r t i o n a l t o K. A t low f r e q u e n c i e s e q u a t i o n (k) r e d u c e s t o t h e o r i g i n a l d i f f u s i o n e q u a t i o n . A t h i g h enough f r e q u e n c i e s , t h e ' w a v e ' t e r m d o m i n a t e s , g i v i n g w a v e - l i k e p r o p a g a t i o n o f t e m p e r a t u r e w a v e s , w i t h a f i n i t e u p p e r l i m i t i n g v e l o c i t y , v m = Heat F l o w i n t h e B o u n d a r y - S c a t t e r i n g L i m i t It i s known t h a t t h e t h e r m a l c o n d u c t i v i t y o f i n s u l a t i n g p e r f e c t c r y s t a l s o f i n f i n i t e e x t e n t must i n c r e a s e i n d e f i n i t e l y w i t h d e c r e a s i n g t e m p e r a t u r e , ( P e i e r l s 1929)4. However, f o r a g i v e n c r y s t a l o f f i n i t e d i m e n s i o n s t h e r e w i l l be a t e m p e r a t u r e below w h i c h t h e f r e e p a t h o f t h e e l a s t i c waves becomes c o m p a r a b l e t o t h e d i m e n s i o n s o f t h e c r y s t a l and as s o o n as t h i s i s t h e c a s e , t h e s c a t t e r i n g o f e l a s t i c waves by t h e b o u n d a r i e s o f t h e c r y s t a l w i l l g i v e r i s e t o an a d d i t i o n a l r e s i s t a n c e . T h i s has been d e m o n s t r a t e d e x p e r i m e n t a l l y 5 . C a s i m i r 6 (1938) c o n s i d e r e d t h e s t e a d y s t a t e h e a t c o n d u c t i o n i n t h e l i m i t i n g c a s e when t h e f r e e p a t h i s v e r y l o n g compared t o t h e d i m e n s i o n s o f t h e c r y s t a l . He assumed t h a t i n t h e p r o c e s s o f d i f f u s e r e f l e c t i o n t h e b o u n d a r y a c t s as a " b l a c k " s u r f a c e a b s o r b i n g a l l i n c i d e n t phohons and r a d i a t i n g phonons w i t h a " b l a c k -b o d y " d i s t r i b u t i o n c h a r a c t e r i z e d by t h e l o c a l s u r f a c e t e m p e r a t u r e . Z i m a n 7 (195*0 p o i n t e d o u t t h a t t h i s l e a d s t o a d i f f u s i o n e q u a t i o n f o r t h e p r o p a g a t i o n o f t e m p e r a t u r e d i s t u r b a n c e as e q u a t i o n (3), where t h e d i f f u s i o n c o n s t a n t i s D = 2 v Q r / 3 where r i s t h e r a d i u s o f t h e c y l i n d r i c a l s a m p l e . A g a i n as i n e q u a t i o n (3) , t h e p r o p a g a t i o n v e l o c i t y f o r a t e m p e r a -t u r e w a v e , v = (20a)) 1 / 2 , i n c r e a s e s 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 i n c r e a s e s . T h u s a g a i n i t i s n e c e s s a r y t o m o d i f y t h e C a s i m i r -Ziman e q u a t i o n i n o r d e r t o l i m i t t h e v e l o c i t y p r o p a g a t i o n o f a t h e r m a l p u l s e . J . B . B r o w n , D . Y . Chung and P.W. M a t t h e w s 8 (1966) added a wave t e r m 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 (as V e r n o t t e d i d f o r t h e h i g h -t e m p e r a t u r e r e g i o n ) , g i v i n g - h -where D = ( 2 / 3 ) v Q r . They used t h i s e q u a t i o n t o e x p l a i n t h e i r o b s e r v a t i o n s o f h e a t p u l s e p r o p a g a t i o n i n c y l i n d r i c a l samples i n t h e l o n g m e a n - f r e e - p a t h t e m p e r a t u r e r e g i o n , f o r c r y s t a l s o f A l ^ O y and f o r l i q u i d 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 0 w i t h t h e mean phonon v e l o c i t y r e q u i r e s t h a t 3t i s the mean time between boundary c o l l i s i o n s . They 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 the re s p o n s e t o a u n i t 6 - f u n c t i o n i n p u t a t x = 0 and found T ( x , t ) a t the end of t h e s a m p l e . F o r the p a r t i c u l a r c a s e o f He I I , c o m p u t a t i o n a l and e x p e r i m e n t a l r e s u l t s were i n v e r y good agreement. F o r a c y l i n d r i c a l s i n g l e c r y s t a l o f A^O^ the agreement was not so good. A few months l a t e r P.W. M a t t h e w s 9 (1366) extended the C a s i m i r t h e o r y t o a n o n - s t e a d y s t a t e , but i n s t e a d o f a d d i n g a wave term •3 2T/3t 2 a s i n e q u a t i o n (k), he found a term o f t h e form 3 3 T / 3 x 2 3 t : t h i s t erm a l s o produces t h e e x p e c t e d f o r m o f the d i s -p e r s T o n c u r v e s , w i t h 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 D i f f u s i o n E q u a t i o n P.W. M a t t h e w s 9 d e v e l o p e d a m o d i f i e d d i f f u s i o n e q u a t i o n f o r the p a r t i c u l a r c a s e o f time-dependent heat f l o w i n the boundary s c a t t e r i n g l i m i t f o r c y l i n d r i c a l samples and under t h e a s s u m p t i o n of " b l a c k " w a l l s . He f o l l o w e d the same p r o c e d u r e as C a s i m i r 6 , a p p l i e d t o t h e n o n - s t e a d y s t a t e , and o b t a i n e d the a p p r o x i m a t e e q u a t i o n where 9t 9x2 9x^9t D - f rv 0 (1 - 3T) A = r [ I n (|4 - |] E q u a t i o n (5) was d e d u c e d u n d e r t h e f o l l o w i n g r e s t r i c t i o n s : - « -Vn V X >> L L > r T h e r e f o r e X » r D i s p e r s i o n R e l a t i o n : F o r a t e m p e r a t u r e wave o f t h e f o r m T = TB + A Texp i (wt + kx) s u b s t i t u t i o n i n t o e q u a t i o n (5) g i v e t h e d i s p e r s i o n r e l a t i o n iw = - D k 2 + A k 2 i w k = M l i / 2 [(, + Aw } . ( 1 _ Aw } f ] where 6 2 = A 2 w 2 + D 2 and A = D + 6 If Aw/D i s s m a l l , e q u a t i o n (6) c a n be w r i t t e n a p p r o x i m a t e l y k = ( ^ ) 1 / 2 [(1 + $L) - (1 - %L) i ] F o r t h e c o n d i t i o n o f v e r y l o n g L (he e x p l a i n s ) , e q u a t i o n (5) does n o t t e n d t o t h e u s u a l d i f f u s i o n e q u a t i o n . As L i n c r e a s e s A i n c r e a s e s and makes t h e new t e r m more i m p o r t a n t r a t h e r t h a n l e s s i m p o r t a n t . T h e d i f f i c u l t y a p p e a r s b e c a u s e t h e c o n d i t i o n X >> L i s more i m p o r t a n t f o r terms i n A t h a n i t i s f o r D . T h e c o n d i t i o n X » L means t h a t as L t h e r a n g e o f v a l i d i t y o f t h e t h e o r y s h r i n k s t o a f r e q u e n c y r a n g e w ->• o . F o r v e r y l a r g e v a l u e s o f L / r t h e a p p r o x i m a t i o n (7) r e m a i n s v a l i d b e c a u s e Aw/D r e m a i n s s m a l l w i t h i n t h e range o f f r e q u e n c y p e r m i t t e d by X » L >> r . T h e low f r e q u e n c y l i m i t o f e q u a t i o n (7) i s t h e r e f o r e k + ( |D) 1 / 2 ( 1 - 0 as u s u a l When k = k" - i k " t h e n w / k ' i s t h e wave v e l o c i t y and W" i s t h e a t t e n u a t i o n c o n s t a n t o f t h e t e m p e r a t u r e w a v e . T h e t h e o r y i s assumed t o be v a l i d f o r f r e q u e n c i e s s u c h t h a t 2L i s l e s s t h a n o n e - h a l f w a v e l e n g t h . Brown M o d i f i e d E q u a t i o n A d i f f e r e n t m o d i f i e d h e a t e q u a t i o n f o r t h e same c a s e c o n s i d e r e d by Matthews and u n d e r t h e same " b l a c k " w a l l s a s s u m p t i o n has been d e d u c e d by Brown 1 0 ( l967) . H i s d e r i v a t i o n i s from f i r s t p r i n c i p l e s , u s i n g t h e B o l t z m a n n T r a n s p o r t E q u a t i o n f o r p h o n o n s . i n c o n t r a s t t o M a t t h e w s 9 , he f i n d s a wave t e r m s i m i l a r t o t h e t e r m assumed by Brown e t a l 8 For very steep temperature gradients, further terms wouId have to be added, but equation (8) is valid for temperature wave X > 16 r/3. tf a solution of the form T(x,t) = T exp [i(kx - wt)] Is assumed for equation (8),. then putting k = k" + i k " and solving for k' and k " k ' ? = vg T2 [1 + (1 + (wx)-2) l/2] k " 2 = [1 + 0 + (WT)-2)1/ 23 o where T = ^ r/3v 0 fa the 1-ov* frequency limit v2x O and in the high frequency limit 8c' = 2^2 H_ vo " 1 k = 2 V i v 0 T therefore at high frequencies a limiting speed of propagation and a constant attenuation coefficient k" '• = 3/21/2*»r are predicted. Comments At low frequencies equation (8) reduces to the Casimir-Ziman 8 d i f f u s i o n e q u a t i o n . In t h i s t h e s i s we a r e i n t e r e s t e d o n l y i n e f f e c t s a t v e r y low f r e q u e n c i e s , s o f o r p r e s e n t p u r p o s e s , t h e o n l y d i s c r e p a n c y between B r o w n ' s t h e o r y and M a t t h e w s ' 9 , i s t h a t t h e l a t t e r p r e d i c t s a v a l u e f o r D w h i c h depends on t h e mean f r e e p a t h , D • f r v o - ( 9 ) T h i s d e p e n d e n c e a r o s e f r o m t h e l i m i t i n g o f r a n g e s o f i n t e g r a t i o n t o tl i n s t e a d o f t°°t and i n f a c t t h e o r i g i n a l C a s i m i r t h e o r y s h o u l d be l i m i t e d i n t h e same way: t h i s l e a d s t o a C a s i m i r - Z i m a n d i f f u s i o n e q u a t i o n w i t h D g i v e n by E q u a t i o n 9. T h e p u r p o s e o f t h i s t h e s i s i s t o c h e c k t h i s one p o i n t , and i t w i l l be l e f t t o o t h e r s t o w o r k a t h i g h e r f r e q u e n c i e s where one c o u l d i n v e s t i g a t e t h e n a t u r e o f t h e h i g h e r terms s u g g e s t e d by Brown and by M a t t h e w s . - 9 -CHAPTER I I GAS DIFFUSION 2.1 High P r e s s u r e Region The p a r a b o l i c e q u a t i o n d e s c r i b e s 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 o f g a s e s , s h e a r m o t i o n , and o t h e r phenomena. The d i f f u s i o n e q u a t i o n f o r a gas can be deduced i n t h e same way as f o r heat c o n d u c t i o n 1 1 , where n i s now the d e n s i t y (or p r e s s u r e ) o f the g a s . T h i s s i m i l a r i t y i s not o n l y i n the m a t h e m a t i c a l form, but a r i s e s f rom t h e b a s i c p h y s i c s o f t h e phenomena. J u s t as heat s p r e a d s i n a c o n d u c t i n g body by the motion o f phonons, so i n d i f f u s i o n a gas s p r e a d s from one r e g i o n t o a n o t h e r by t h e motion o f m o l e c u l e s . The speed w i t h w h i c h a t e m p e r a t u r e 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 t h e thermal d i f f u s i v i t y ; s i m i l a r i l y the speed w i t h w h i c h a gas p e n e t r a t e s i n t o a n o t h e r , o r i t s e l f , i s d e t e r m i n e d by the c o e f f i c i e n t o f d i f f u s i o n . As f o r t h e F o u r i e r heat e q u a t i o n , t h e e q u a t i o n f o r d i f f u s i o n o f gases p r e d i c t s no l i m i t on t h e wave speed as one i n c r e a s e s the f r e q u e n c y o f a p r e s s u r e p e r t u r b a t i o n : c l e a r l y a m o d i f i e d e q u a t i o n i s needed as b e f o r e . As the f r e q u e n c y i n c r e a s e s , e v e n t u a l l y one 8 2n 10 r e a c h e s the normal sound r e g i o n , and the v e l o c i t y must be l i m i t e d t o t h e v e l o c i t y o f sound, c j . I t i s t h e r e f o r e r e a s o n a b l e t o w r i t e a combined d i f f u s i o n and wave e q u a t i o n w h i c h w i l l c e r t a i n l y be v a l i d i n both 1imi t s D 8 2 n 8n_ 8 2 n 9t Bx 7" T h i s e q u a t i o n has been r e c e n t l y d e r i v e d from a s i m p l e random-walk model by Weyman 1 2 (1967). The sound v e l o c i t y , c j , i s c l o s e l y r e l a t e d t o t h e a v e r a g e m o l e c u l a r speed v, i n f a c t c i = (TT / 8 ) 1 / 2 V = 0.625 " . T h i s c o r r e s p o n d s t o th e analogous s i t u a t i o n i n a therma l wave ( e q u a t i o n 4) where the wave v e l o c i t y c annot exceed the phonon v e l o c i t y . I t i s i n t e r e s t i n g t o compare t h e above gas model w i t h the t h e o r y o f h eat waves i n l i q u i d h e l i u m . In t h a t medium, t e m p e r a t u r e waves a r e p r o p a g a t e d i n a d e f i n i t e w a v e l i k e f a s h i o n c a l l e d "second sound". I t was shown by Ward and W i l k s 1 3 (1952) t h a t second sound can be re g a r d e d as a d e n s i t y f l u c t u a t i o n i n the "phonon g a s " , so t h a t t h e r e i s a c l o s e a n a l o g y w i t h o r d i n a r y sound w h i c h i s a d e n s i t y f l u c t u a t i o n i n a r e a l g a s . The second sound v e l o c i t y C 2 = ( l / 3 ) 1 ^ 2 C ] = 0 .578cj where c j i s the phonon v e l o c i t y ( f i r s t sound v e l o c i t y ) . Heat waves o f the same t y p e have s i n c e been o b s e r v e d i n a s o l i d medium - s o l i d h e l i u m by F a i r b a n k e t a l l t f (1966). 2 .2 Gas D i f f u s i o n i n the Boundary S c a t t e r i n g L i m i t , M o l e c u l a r Flow As i n the second p a r t o f th e th e r m a l a n a l o g u e , we d i s c u s s the f r e e 11 m o l e c u l e f l o w i n a l o n g c i r c u l a r tube o f r a d i u s r . The two ends a r e m a i n t a i n e d a t d i f f e r e n t p r e s s u r e s , and the t e m p e r a t u r e i s u n i f o r m t h r o u g h o u t . I f t h e p r e s s u r e i s so low t h a t L > r , a i m o l e c u l e of gas must c o l l i d e many t i m e s w i t h t h e w a l l o f the tube b e f o r e i t e n c o u n t e r s a n o t h e r mo 1 e c u l e . For t h i s r e a s o n , the f l o w of t h e gas i s d e t e r m i n e d a l m o s t e n t i r e l y by the c o l l i s i o n s w i t h t h e w a l l and i s p r a c t i c a l l y u n a f f e c t e d by i n t e r m o l e c u l a r c o l l i s i o n . A s p e c u l a r o r m i r r o r - l i k e r e f l e c t i o n o f the m o l e c u l e would r e q u i r e a s u r f a c e t h a t i s smooth on t h e s u b m i c r o s c o p i c s c a l e o f m o l e c u l a r d i m e n s i o n s . The s m o o t h e s t s u r f a c e would be on a p e r f e c t s i n g l e c r y s t a l , b u t even h e r e the r e f l e c t i o n need not t o be s p e c u l a r . In a c t u a l p r a c t i c e , t h e w a l l s u r f a c e s a r e h i g h l y i r r e g u l a r on the 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 the d i r e c t i o n o f r e f l e c t i o n f r o m t h e w a l l t o b e a r l i t t l e i f any r e l a t i o n t o the d i r e c t i o n o f i i n c i ' d e n c e 1 5 . T h i s w o u l d be the ca s e i f a gas m o l e c u l e , upon r e a c h i n g t h e s u r f a c e , were t e m p o r a r i l y adsorbed and l a t e r e v a p o r a t e d . E x p e r i m e n t s by Knudsen and o t h e r s on the r e f l e c t i o n o f mol'ecialar beams f r o m g l a s s and p o l i s h e d metal s u r f a c e s i n d i c a t e d c l o s e t o 100 p e r c e n t d i f f u s e r e f l e c t i o n . T h e r e f o r e , as i n t h e t h e r m a l c a s e , t h e " b l a c k " w a l l can be assumed. The t r a n s p o r t phenomena down a c y l i n d r i c a l p i p e i s c a l c u l a t e d by S m o T u c h o w s k i 1 s method (see P r e s e n t 1 5 ) , and the d i f f u s i v e f l o w i s c h a r a c t e r i z e d by a d i f f u s i o n c o e f f i c i e n t , D. = 27Y/3, where a v e r a g e m o l e c u l a r speed V = (8RT /TTM) 1 / 2 . In the a n a l o g o u s t h e r m a l c a s e (Ch. 1.3) c o n s i d e r e d by C a s i m i r 7 , the b l a c k body phonon d i s t r i b u t i o n p l a y s t h e p a r t o f the Maxwell d i s t r i b u t i o n f o r - 12 t h e r a r e f i e d g a s , and l e a d s t o an e x a c t l y e q u i v a l e n t v a l u e o f D. 2.3 The Matthews 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 Tubes We saw i n Ch. 1.3 how P.W. Matthews by e x t e n s i o n o f t h e C a s i m i r method t o a n o n - s t e a d y s t a t e c a s e found a c o r r e c t i o n term o f t h e f o r m 3 3 T / 3 x 2 3 t . S i m i l a r i l y f o r the gas case-he e x t e n d e d the S m o l u c h o w s k i method, f o r t h e n o n - s t e a d y s t a t e , and found a term o f t h e f o r m 3 3 n / 3 2 x 3 t as a c o r r e c t i o n t o t h e Smoluchowski f o r m u l a . S i n c e no p u b l i c a t i o n has been made f o r the gas c a s e , t h e method w i l l ' b e o u t l i n e d . C o n s i d e r m o l e c u l e s i n a r e s t r i c t e d v e l o c i t y r a n g e , v t o v + dv, o n l y , , and l e t n be t h e t o t a l m o l e c u l e s per u n i t volume I : teas « \ o JJpds - x . y . 0 \J x v../_. F o r m o l e c u l e s whose v e l o c i t y l i e s between v and v + dv the f l u x ( m o l e c u l e s / s e c . ) t h r o u g h d S Q from dS] i s 1 5 d F v = n j cos 6] ^~ v f (v) 6 V ; f ( v ) = v e l o c i t y d i s t r i but i oa-„ ' cos 01 cos 0n , c . f f \ r. = n l ^T2~ d S Q v f (v) 6 V How 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 x = 0. The f l u x t h r o u g h d S Q a t t i m e t depends on nj a t d S j . a t time ( t - p / v ) , i . e on / 8n p 3n* j _ , 2 3 2 n 2xp_ 3 2 n . p£ 3£n. ' 2 K* 3x2" " v 3x3t V2 3 t 2 ; where 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 time t . C o n s i d e r a i at -x. The net flux due to dS. and dS 1 m i r r o r e l e m e n t dS' a t - x . The net f l u x due t o dSj and dSj depends 13 on 1 _ 3n 2xp 3 2 n " l _ " l = 2 X 37 " 373t Hence the net f l u x t h r o u g h the t o t a l c r o s s s e c t i o n , S Q ( f o r m o l e c u l e s whose v e l o c i t y l i e s between v t o v + dv) i s Fv • h »'f M av [,, - | a - I , 2 |!g-a where 1^ = J c o s 0 o cos 6 j (^-) d S Q dSj 1 2 = j c o s eQ cos 8] (£•) d S Q dS] These i n t e g r a l s a r e e x a c t l y as i n the therm a l t r e a t m e n t 9 , and t h e r e f o r e t h e y a r e a l r e a d y e v a l u a t e d . The t o t a l f l u x f o r a l l m o l e c u l e s i s t h e i n t e g r a l o v e r a l l v 1 r, 3n f t, x _, . 3 2 n F " 2„ [ ' | | J } v ' ( v ) d » - l 2 J f ( . ) dv] 1 r. _ 3n . 3 2 n " 27 U 1 V 37 ' '2 a^ap The n e t f l o w i n t o r e g i o n bounded by x and x +<5x i n time St i s , <Fx + Sx " F x ) - ^ [ l l V 0 6 x - « 2 f^t 6 x ] 6 t = TT r 2 6 X 6 n T h e r e f o r e 3t" = 2Tr2r2 L ' l v a^ T " '2 8 x 2 3 t J I n t r o d u c i n g the v a l u e s o f l j and \ ^ we f i n a l l y g et 3n 1 n — 3 2 n . 3 3n •• - 14 -where a s tn the t h e r m a l a n a l o g u e . F o r t h e r m a l waves i t was assumed that at low t e m p e r a t u r e s a l l phonons .had t h e same s p e e d . F o r t h e g a s c a s e , t h e c o r r e s p o n d i n g a s s u m p t i o n o f c o n s t a n t m o l e c u l a r s p e e d i s u n s a t i s f a c t o r y . H o w e v e r , i f t h e M a x w e l l v e l o c i t y d i s t r i b u t i o n i s u s e d , t h e a n s w e r t u r n s o u t e x a c t l y t h e same, w i t h a v e r a g e m o l e c u l a r v e l o c i t y v" r e p l a c i n g p h o n o n v e l o c i t y . A l l . ' t h e d i s c u s s i o n i n C h . 1.4 i s v a l i d h e r e . T h e r e f o r e f o r t h e low f r e q u e n c y l i m i t o f a d i f f u s i o n wave the v e l o c i t y i s v = /TTO = / y to r v (1 - ^ ) (10) As p r e v i o u s l y i n d i c a t e d , t h i s d e p e n d e n c e o f v on L i n t h e low - f r e q u e n c y r a n g e i s t h e t o p i c o f t h i s t h e s i s . As n o t e d i n s e c t i o n 1.6 the d e p e n d e n c e o f v on L o f e q u a t i o n (10) does n o t depend on the v a l i d i t y o f t h e a b o v e t h e o r y , b u t o n l y on t h e q u e s t i o n o f w h e t h e r i n the s t e a d y - s t a t e t h e o r y ( C a s i m i r ' s o r S m o l u c h o w s k i 1 s ) , the r a n g e o f i n t e g r a t i o n s h o u l d be l i m i t e d t o tl i n s t e a d o f t » . 15 CHAPTER I I I DIFFUSION WAVES IN A RAREFIED GAS 3.1 I n t r o d u c t i o n From t h e e x p e r i m e n t a l p o i n t o f v i e w , t h e o b j e c t i v e w a s : (a) To g e n e r a t e a p r e s s u r e wave i n a r a r e f i e d gas i n a c y l i n d r i c a l p i p e . (b) T o measure t h e wave v e l o c i t y a t d i f f e r e n t p r e s s u r e s ( i . e . mean 3 . 2 Gas C h o i c e F i r s t l y i t was r e a s o n a b l e t o use a monatomic g a s , s o t h a t s t r a i g h t f o r w a r d K i n e t i c T h e o r y was a p p l i c a b l e and a l l t h e k i n e t i c e n e r g y was i n t h e t r a n s l a t i o n a l m o t i o n o f t h e m o l e c u l e s . A r g o n was s e l e c t e d , as i t i s r e a d i l y a v a i l a b l e , and s i n c e t h e i o n gauges u s e d i n t h e e x p e r i m e n t have a c o n s i d e r a b l y h i g h e r s e n s i t i v i t y f o r a r g o n t h a n f o r h e l i u m o r neon as shown i n t h e f o l l o w i n g t a b l e 1 ( f r o m Dushman - Vacuum T e c h n i q u e ( W i l e y ) 2 n d . E d n . p g . 323) f r e e p a t h s ) . Gas P r e s s u r e Gauge c o r r e c t i o n f a c t o r He 0 . 1 3 3 Ne 0 . 2 0 2 A 1.0 Kr 1.56 Xe 2 . 2 9 T a b l e 1 16 The a s s u m p t i o n s made a r e : ( i ) The Maxwel 1-Boltzmann d i s t r i b u t i o n can be used t o c a l c u l a t e v", 7 = ( 8 k T A r m ) l / 2 where k i s the Boltzmann C o n s t a n t . ( i i ) 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 i s L = C o n s t a n t / p where c o n s t a n t = 5 . 6 6 x 1 0 " 3 cm. t o r r . e.g. i n t h e p r e s s u r e range IO" 4 < p < 1 0 ~ 3 t o r r . , 50 cm. > L > 5 cm. A l s o i t i s assumed t h a t t h e w a l l s a r e a t u n i f o r m t e m p e r a t u r e , and g i v e p e r f e c t l y d i f f u s i v e s c a t t e r i n g . I t was n e c e s s a r y t o e n s u r e t h a t the e x p e r i m e n t a l c o n d i t i o n s were w i t h i n t h e r e s t r i c t i o n s o f t h e Matthews t h e o r y ( s e c t i o n 2 . 3 ) , as f o l l o w s : (a) A > 0 T h i s c o n d i t i o n l e a ds t o a lower l i m i t f o r L, L m j n = r (b) D > 0 T h i s c o n d i t i o n a l s o l e a d s t o a lower l i m i t f o r L, L m j n = 3 r A So t h i s c o n d i t i o n s e t s the upper p r e s s u r e l i m i t , f o r a p i p e o f g i v e n d i a m e t e r . (c) The r e s t r i c t i o n r/v << A/v X = wave l e n g t h v = v e l o c i t y o f the wave i s s a t i s f i e d so long as r << X s i n c e v < v" 17 3.3 P r e s s u r e Range and P i p e Dimensions The upper l i m i t o f the p r e s s u r e range was chosen t o be about 10"3 T o r r , as t h i s i s a r e a s o n a b l e upper l i m i t f o r the i o n gauge d e t e c t o r s . T h i s c h o i c e , c o u p l e d w i t h the c o n d i t i o n (b) ( s e c t i o n 3.2), e s t a b l i s h e s t h e p i p e d i a m e t e r , w h i c h was t h e r e f o r e t a k e n t o be two i n c h e s . Moreover f o r t h i s p i p e d i a m e t e r , the s i d e tubes f o r the i o n gauge d e t e c t o r s do not p e r t u r b the wave p r o p a g a t i o n t o o s e r i o u s l y . The lower p r e s s u r e l i m i t o f IO"1* T o r r i s s e t by t h e o r e t i c a l e x p e c t a t i o n s , as i n d i c a t e d below. 3.A Computer R e s u l t s The v e l o c i t y 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 a l o n g the two 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 e q u a t i o n . R e s u l t s f o r argon a t 300 K and v = 3.99 x 101* cm/sec a r e shown on T a b l e s II and I I I . The same r e s u l t s ( v e l o c i t y o n l y ) a r e p l o t t e d on Graph I. 3.5 Comments on t h e P r e d i c t i o n s Graph I shows t h a t t h e r e w i l l n o t be much g a i n i n i n f o r m a t i o n , r e g a r d i n g t h e s l o p e a t t h e o r i g i n , by w o r k i n g a t p r e s s u r e s such t h a t L > 30 cm. T h i s i s t h e r e a s o n why t h e p r e s s u r e range f o r t h e e x p e r i m e n t was c o n f i n e d t o v a l u e s t h a t l e a d t o 5 cm < L < 30 cm. The s l o p e a t the o r i g i n ( p r o p o r t i o n a l t o D1^2) v a r i e s by about 30% f o r 5 cm < L < 50 cm. The dependence o f D on t h e mean f r e e p a t h i s what we were l o o k i n g f o r i n our e x p e r i m e n t . 18 TABLE I I COMPUTER RESULTS FREQUENCY VELOCITY WAVELENGTH ATTENUATION X WAVELENGTH (CM/SEC) (CM) Mean F r e e Path = 5.00 cms 5. 1.619E 03 3.237E 02 6.264E 00 10. 2.286E 03 2.286E 02 6.2A6E 00 15. 2.795E 03 1.863E 02 6.227E 00 20. 3.223E 03 1.611E 02 6.208E 00 25. 3.598E 03 1.439E 03 6.189E 00 30. 3.936E 03 1.312E 02 6.171E 00 35. 4.245E 03 1.213E 02 6.152E 00 AO. 4.531E 03 1.133E 02 6.134E 00 45. 4.800E 03 1.067E 02 6.115E 00 50. 5.052E 03 1.010E 02 6.097E 00 Mean Fre e P a t h = 10.00 cms 5. 1.849E 03 3.699E 02 6.253E 00 10. 2.609E 03 2.609E 02 6.222E 00 15. 3.188E 03 2.125E 02 6.192E 00 20. 3.672E 03 1.836E 02 6.162E 00 25. 4096E 03 1.638E 02 6.132E 00 30. 4.477E 03 1.492E 02 6.102E 00 35. 4.825E 03 1.378E 02 6 .073E 00 AO. 5 .146E 03 1.287E 02 6.043E 00 45. 5.446E 03 1.210E 02 6.014E 00 50. 5.728E 03 1.146E 02 5.985E 00 Mean Fre e P a t h = 15.00 cms 5. 1.919E 03 3.839E 02 6.246E 00 10. 2.707E 03 2.707E 02 6 .209E 00 15. 3.305E 03 2.204E 02 6.173E 00 20. 3.806E 03 1.903E 02 6.136E 00 25. 4.243E 03 1.697E 02 6.100E 00 30. 4.636E 03 1.545E 02 6.064E 00 35. 4.993E 03 1.427E 02 6.029E 00 40. 5.324E 03 1.331E 02 5.993E 00 45. 5.632E 03 1.252E 02 5.958E 00 50. 5.921E 03 1.184E 02 5.923E 00 Mean Free Path = 20.00 cms 5. 1.953E 03 3.907E 02 6.242E 00 10. 2.753E 03 2.753E 02 6.200E 00 15. 3.361E 03 2.241E 02 6.159E 00 20. 3.869E 03 1.935E 02 6.118E 00 25. 4.312E 03 1.725E 02 6.078E 00 30. 4.709E 03 1.570E 02 6.037E 00 35. 5.071E 03 1.449E 02 5.997E 00 40. 5.405E 03 1 - 351E 02 5.958E 00 45. 5.716E 03 1.270E 02 5.918E 00 50. 6.008E 03 1.202E 02 5.879E 00 FREQUENCY VELOCITY WAVELENGTH ATTENUATION X WAV (CM/SEC) (CM) Fre e P a t h = = 25.00 cms 5. 1.973E 03 3.9A6E 02 6.238E 00 10. 2.781E 03 2.781E 02 6.193E 00 15. 3.39AE 03 2.263E 02 6.1*»8E 00 20. 3.905E 03 1.953E 02 6.10AE 00 25. 4.352E 03 1 .7^ +1 E 02 6.060E 00 30. 4.751E 03 1.58AE 02 6.016E 00 35. 5.H5E 03 1.A61E 02 5.973E 00 AO. 5.^51E 03 1.363E 02 5.930E 00 45. 5.763E 03 1.281E 02 5.887E 00 50. 6.056E 03 1.21 IE 02 5.8i»5E 00 TABLE I I I COMPUTER RESULTS 21 CHAPTER IV THE APPARATUS k. 1 G e n e r a l D e s c r i p t i o n The main p a r t s o f the system a r e ( F i g . 1 ) : . (1) Argon f e e d i n g s y s t e m w h i c h s u p p l i e s Argon t o the wave g e n e r a t o r (2) Wave g e n e r a t o r w h i c h produces a p r e s s u r e wave (3) Main p i p e ( t w o - i n c h ID, 12' lo n g s t r a i g h t p i p e ) (4) Vacuum s y s t e m w h i c h produces the vacuum i n the wave-guide and wave g e n e r a t o r (5) P r e s s u r e measurement s y s t e m (6) Wave d e t e c t o r s (7) E l e c t r o n i c s 4.2 A r g o n F e e d i n g System In our 12' l o n g s t r a i g h t c y l i n d r i c a l main p i p e ( 2 " ID) a p r e s s u r e g r a d i e n t , Ap , has been assumed i n o r d e r t o c a l c u l a t e t h e Argon f l o w , Q, i n grams/sec. Q = J (2 TTRT)1/2 ( R ] _ P 2 ) The maximum f l o w c o r r e s p o n d i n g t o Ap = 5 x 10 T o r r , i s o f t h e o r d e r o f 1 0 " 5 gm/sec. 4.3 Flow C o n t r o l In o r d e r t o c o n t r o l t h i s k i n d o f f l o w , a s y s t e m i n w h i c h a r e d u c t i o n i n p r e s s u r e o f 1 0 " e x t h e v a l u e o f t h a t o f the o r i g i n a l A rgon s o u r c e (a c o m m e r c i a l c y l i n d e r ) was d e s i g n e d ( F i g . I I ) . - 22 -FIGURE I Pressure measurement system A Argon supply 1 Vf av e | gene-j rator j Main pipe A Wave detectors i Va- ! cuum sys-tem FUNCTIONAL DIAGRAM OF THE EXPERIMENT 23 E s s e n t i a l l y , the f l o w i s c o n t r o l l e d by a n e e d l e v a l v e (8) w h i c h c o n n e c t s t h e low p r e s s u r e Argon s o u r c e (e) t o the wave g e n e r a t o r . a i s a s m a l l c y l i n d e r a b l e t o s u p p o r t p r e s s u r e s o f the o r d e r o f 2000 I b s / s q . i n c h . The volumes o f a, b, c, d, and e a r e a - 500 c c b = 1 c c c = 1000 c c d = 1 c c e = 1000 c c The whole f l o w c o n t r o l i s c o n n e c t e d t o t h e wave g e n e r a t o r chamber a t (8) w h i c h i s c o n n e c t e d t o t h e vacuum system t h r o u g h t h e main p i p e . 4.4 O p e r a t i o n Once a l l t h e a i r i s out o f t h e s y s t e m , c y l i n d e r a_ i s f i l l e d w i t h Argon a t h i g h p r e s s u r e , w h i l e a l l v a l v e s ( e x c e p t i n g 1 and 2) a r e c l o s e d . By o p e n i n g v a l v e 3, 1 c c o f h i g h - p r e s s u r e Argon i s t a k e n and i s o l a t e d from a_ by c l o s i n g v a l v e 3. T h i s c c o f Argon i s expanded by o p e n i n g v a l v e 4, and c i s i s o l a t e d by c l o s i n g v a l v e 4. 1 c c o f t h i s expanded Argon i s t a k e n by o p e n i n g v a l v e 5, and a g a i n t h i s c c i s i s o l a t e d by c l o s i n g v a l v e 5. I t i s expanded a g a i n 1000 t i m e s by o p e n i n g v a l v e 6. Thus, the gas i n e i s our l o w - p r e s s u r e Argon s o u r c e w h i c h i s c o n t r o l l e d t h r o u g h the wave g e n e r a t o r chamber by a n e e d l e v a l v e ( 8 ) . To wave g e n e r a t o r F i g u r e II DIAGRAM OF FLOW CONTROL 25 However, a c o n t i n u o u s s u p e r v i s i o n o f t h e p r e s s u r e i n t h e main p i p e i s n e c e s s a r y 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 e v e r y t h r e e o r f o u r m i nutes i n o r d e r t o keep t h e p r e s s u r e a t the d e s i r e d v a l u e . 4.5 The Wave G e n e r a t o r Chamber The b a s i c i d e a o f t h i s assembly i s t o chop a f l o w o f Argon gas a t a known r a t e . A aluminum d i s c , A, ( F i g . I l l ) w i t h f o u r c i r c u l a r h o l e s i s d r i v e n by B w h i c h 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. The 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 i n an aluminum c y l i n d e r , F, ( F i g . I V ) , ( 8 " I.D.), 5" l o n g w h i c h i s s e a l e d by t h e b r a s s p l a t e s D and E by means o f 0 - r i n g s ( F i g s . I l l and I V ) . The d i s c D has two h o l e s , a_ and b. a_ i s t h e h o l e w h i c h communicates w i t h t h e 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 t h e l i g h t beam fro m a r e f l e c t i n g p h o t o c e l l , P, ( F i g . IV) t o a r r i v e t o the m i r r o r , M, ( F i g . I l l ) and be r e f l e c t e d t o t h e p h o t o c e l l vacuum t u b e . The purpose o f t h i s p h o t o c e l l i s t o produce a r e f e r e n c e s i g n a l w i t h t h e same f r e q u e n c y o f t h e wave p r e s s u r e . The d e s i g n 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 the f o l l o w i n g n a t u r e : (1) a b i l i t y t o s t a n d up t o p r e s s u r e o f 15 lbs»/sq. i n . on e x t e r n a l s u r f a c e o f the g e n e r a t o r (2) s h a f t and i t s s u p p o r t d e s i g n e d t o g i v e a r i g i d and b a l a n c e d p e r f o r m a n c e o f the d i s c a t a speed up t o 600 rpm (3) c a r e f u l m a c h i n i n g o f the aluminum chopper i s d i c t a t e d by the need t o have a c l e a r a n c e between t h e chopper and p l a t e s o f •Hi f Argon Entrance V A (Chopper D i s c ) F i xed M i r r o r M 8 4' "/ i f-b e a r i ngs_ ~ S u p p o r t s •"-it / I 'HQ .'.!.'! I " i l l fli ."2 i P h o t o c e l l Main Pipe ro F i g u r e . M l SCHEMATIC DIAGRAM OF THE WAVE GENERATOR - 2? -1/8". This clearance determines the "DC" leak of Argon to the pipe (k) the 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 ) serves to 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" pipe from a d i r e c t flow of Argon from the v a l v e . This arrangement makes i t p o s s i b l e f o r Argon gas molecules to reach thermodynamic e q u i l i b r i u m w i t h the w a l l s of the wave generator by a process of c o l l i s i o n . The other s i d e of the s t a i n l e s s s t e e l p l a t e serves to support the mi r r o r which r e f l e c t s the beam from the pho t o c e l l assembly. (5) I t produces a wave pressure q u i t e c l o s e to a s i n wave. Assuming that the flow i s p r o p o r t i o n a l to the open area S between h o l e s , the r a t i o dS/dt w i l l give us the r a t i o dP/dt which i s What i s being looked f o r s s r 2(2a - s i n 2a) dS dt r2 ( l - cos 2a) d2a dt r cos a = R s i n 3 - r s i n a da Wave G e n e r a t o r I C o n t r o l V a l v e f o r Argon Flow F i g u r e IV DRIVING SYSTEM, MAGNET AND WAVE GENERATOR - 29 -dB_ d t U) = 2 I f f T h e r e f o r e dS d t r2 ( l - cos 2a) 2 R cosB 2uf - r s i n a - * » T T Rr 1 - cos 2a cosB s i n a. -8ir Rf s i n a / T - ' ( ^ c o s a ) 2 = dS d t S i n c e We can say t h a t . t h e s q u a r e r o o t i s c l o s e t o 1 t h e r e f o r e d S / d t - a s i n j w a v e . . E x t e r n a l D r i v i n g The magnet, C, ( F i g . I l l and IV) i s d r i v e n by a h i g h l y s t a b l e 'motor GKH t y p e NSH 5 4 R L . The power o f t h e motor i s 1/8 H.P. The motor s t a b i l i t y i s o b t a i n e d by means o f a p r e c i s i o n t a c h o -g e n e r a t o r f e e d b a c k c o n t r o l l e r . A z e n e r - d i o d e r e f e r e n c e c i r c u i t s p e c i a l l i n e v o l t a g e c o r r e c t i o n 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 combined w i t h a t r a n s i s t o r 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 a d j u s t m e n t o f motor speed o v e r ranges o f 100:1 w i t h s t a b i l i t y o f t \% r e g a r d l e s s o f lo.ad o r l i n e v o l t a g e c h a n g e s . The magnet i s c o n n e c t e d t o the motor by means o f a g e a r w h i c h i n c r e a s e d t h e f u l l speed o f t h e d r i v e f o u r t i m e s . Thus f r e q u e n c i e s between 0 and 25 cps can be o b t a i n e d . 30 -The Main P i p e I t c o n s i s t s o f a s t r a i g h t p i p e 2 " I.D. and 12 ' l o n g . T h i s a s s u r e s t h a t t h e p i p e does i n f a c t behave l i k e an i n f i n i t e (no r e f l e c t i o n f rom t h e end) tube f o r t h e l o n g e s t w a v e - l e n g t h . The d i a m e t e r ( 2 " ) s a t i s f i e s t h e r e s t r i c t i o n s on t h e Matthews e q u a t i o n (see S e c t i o n 3«3) O r i g i n a l l y t h e whole p i p e was made o f p y r e x g l a s s . I t was d i v i d e d i n t o t h r e e p a r t s , V e a c h . The f i r s t p a r t ( 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 p i p e s . These s i d e p i p e s c o n t a i n t h e d e t e c t o r s . The s e p a r a t i o n between d e t e c t o r s a r e g i v e n on the f i g u r e V I I I . One p r o b l e m was t h e a l i g n m e n t o f t h e t h r e e p a r t s o f the p i p e . A v e r y s m a l l e r r o r i n t h e a l i g n m e n t was enough t o br e a k t h e g l a s s p i p e . So we d e c i d e d t o s u b s t i t u t e t h e second and t h i r d segments w i t h an 8 ' ( 2 " I.D.) copper p i p e . The vacuum pump end was co n n e c t e d t o a b e l l o w s u n i t w h i c h a b s o r b e d any v i b r a t i o n s , and a l l o w e d some f l e x i b i l i t y f o r e a se o f a s s e m b l y . The c o n n e c t i o n o f the copper p i p e t o t h e b e l l o w s u n i t i s shown i n F i g . V. A l l c o n n e c t i o n s were made vacuum t i g h t by means o f a g a s k e t o r an 0 - r i n g . Vacuum System The c a l c u l a t i o n o f the r e q u i r e d pumping speed was made by c a l -c u l a t i n g t he p i p e c o n d u c t a n c e F F i g u r e V 32 -F~ (378d 3 + K F S d ^ . / . T s c c / 1 , t ' 1 fe = 365 cm. • M ar g o n = 40 T = 300 K *•''...''. - D . = 5 cm. ; and t h e n the speed S was c a l c u l a t e d • s = F (L.- P -M p where p i s t h e p r e s s u r e a t t h e end o f t h e p i p e b e s i d e the chamber, and Pp t h e p r e s s u r e b e s i d e t h e d i f f u s i o n pump. The p r e s s u r e drop 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 than 10% o v e r t h e p o r t i o n o f p i p e f o r w h i c h the measurements were a c t u a l l y t a k e n : hence t a k i n g a r a t i o o f P/Pp = 3» the speed r e q u i r e d was S = 2 2 , 4 1/sec. 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 d i f f u s i o n pump t y p e VMF-20 was i n s t a 11 ed..This pump has a speed o f 22 1/sec at \0~h T o r r . The l i m i t i n g f o r e - p r e s s u r e i s 0 . 1 0 0 T o r r and the u l t i m a t e p r e s s u r e 1 0 ~ 6 T o r r . The d i f f u s i o n pump i s c o o l e d by c o n t i n u o u s r u n n i n g w a t e r . Two m e c h a n i c a l pumps were used a and b_ ( F i g . V I ) and bo t h a r e Cenco, Hyvac 2. These pumps have a f a i r l y c o n s t a n t speed o f 20 l i t e r s / m i n u t e o v e r the range between 10~ 2 T o r r , t o a t m o s p h e r i c p r e s s u r e . Pump b i s a l s o used t o pump down t h e mercury from the McLeod gauge. V a l v e s 1 and 2 a r e S a l i s b u r y . high-vacuum v a l v e s . V a l v e 2 i s o l a t e s t h e pump b fr o m t h e system when i t Is used t o 33 FIGURE VI Vacuum Lines To main pipe CD • o O CD 05 - 34 -pump t h e McLeod gauge. V a l v e s 3 and 4 a r e a i r i n l e t s . k.S P r e s s u r e Measurement System The measurement o f the p r e s s u r e i s h i g h l y i m p o r t a n t i n the e x p e r i m e n t s i n c e t h e wave s p e e d ' i s g o i n g t o be measured as a f u n c t i o n o f t h e p r e s s u r e and f r e q u e n c y . The p r e s s u r e i s c o n t i n u o u s l y read i n a Veeco vacuum gauge c o n t r o l p a n e l ( t y p e RG). The i o n i z a t i o n gauge used was a Veeco RG-75. The p r e s s u r e meter i n t h e box was c a l i b r a t e d f o r Argon a g a i n s t a McLeod gauge w h i c h reads a b s o l u t e p r e s s u r e . The Veeco i o n i z a t i o n gauge was s i t u a t e d i n the f i r s t o f the s m a l l s i d e p i p e s , J , ( F i g . V I M ) . I t was mounted by 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 the c e n t r e where t h e gauge was s o l d e r e d • ( F i g . V I I ) . O p p o s i t e t h e Veeco gauge a s m a l l g l a s s p i p e i s d e r i v e d t o t h e McLeod gauge ( F i g . V I I I ) so t h a t b o t h (Veeco and McLeod gauges) measure the p r e s s u r e a t t h e same p o i n t o f the main p i p e . The McLeod gauge has a b u l b volume o f 408 c c . and a bore d i a m e t e r of 0.625 mm. measured w i t h a t r a v e l l i n g m i c r o s c o p e . The McLeod gauge was c a l i b r a t e d from i t s o b s e r v e d d i m e n s i o n s . T h i s was c h e c k e d f o r g r o s s e r r o r s a g a i n s t a Veeco i o n gauge u n i t . The s e n s i t i v i t y was F i g u r e VII DETAIL OF VEECO I . G . AND DETECTOR - 36 -cross section _ ir x (0.625 x iO 1) 2cm 2 bulb volume .~ 408~ cc • = 7.6 x 10'6 cm Hg cm"1 ISince the range of pressure for the experiment is between 10~3 Torr -IO"*4 Torr, the corresponding range for the McLeod gauge reading x, is, 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. If p and Pp are the pressures at each extremity of the pipe 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 G e n e r a t o r M a i n P i p e •qciru~ini ~' ., . . D e t e c t o r s v e e c o l . G . F i g u r e V I I I MAIN PIPE 38 -CHAPTER V ELECTRON ICS AND METHOD OF PHASE MEASUREMENT 5.1 I n t r o d u c t i o n T h i s c h a p t e r d e s c r i b e s the e l e c t r o n i c s used f o r the phase measure-ment and t h e method o f measurement i t s e l f . A s c h e m a t i c d i a g r a m i s g i v e n on t h e next page, F i g . IX. The s i g n a l from one o f the 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 i s a p p l i e d t o t h e Y - p l a t e s o f an o s c i l l o s c o p e . A r e f e r e n c e s i g n a l i s taken from a p h o t o c e l l t h a t d e t e c t s the r o t a t i o n o f the d i s c i n the wave-g e n e r a t i n g chamber. T h i s i s a p p l i e d t o the X - p l a t e s o f the o s c i l l o s c o p e , w h i c h i s used as a phase n u l l d e t e c t o r , the phase b e i n g measured from a phase s h i f t system i n t h e r e f e r e n c e channel d e s c r i b e d below. Thus t h e phase o f each d e t e c t o r i s measured r e l a t i v e t o a f i x e d 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 . The d e t e c t o r s used were 31 cm. a p a r t . L a r g e r s e p a r a t i o n s were not used because o f i n c r e a s i n g a t t e n u a t i o n , the s i g n a l t o n o i s e r a t i o would d e c r e a s e r a p i d l y . On t h e o t h e r hand, t h i s s e p a r a t i o n a l l o w s us t o assume t h a t b o t h d e t e c t o r s a r e a t the same mean p r e s s u r e , w i t h o u t a b i g e r r o r . The measured phase d i f f e r e n c e s between the gauges ranged from 20° t o 60° f o r a f r e q u e n c y range from 3 t o 25 Hz. 5.2 The D e t e c t o r Most i m p o r t a n t o f a l l t h e e l e c t r o n i c components i s the d e t e c t o r . Wave i G e n e r a t o r P h o t o c e l 1 Main P i p e D e t e c t o r 1 o3 D e t e c t o r 2 S i g n a l Channel D e t e c t o r i C o n t r o l Box fi I t e r i ng fompl ijpat i * n R e f e r e n c e Channel O s c i 1 l o s c o d e to i 1 t e r i n g \mpl i i c a t i o n P h ase^-Shifte F i g u r e X ELECTRONICS. FUNCTIONAL DIAGRAM 40 -The most s u i t a b l e d e t e c t o r f o r d e t e c t i o n o f a s m a l l v a r i a t i o n i n p r e s s u r e i s an i o n i z a t i o n gauge. Our p r e s s u r e wave had a p p r o x i m a t e l y a peak t o peak a m p l i t u d e o f 1/50 t h e p r e s s u r e l e v e l a t t h e main p i p e . That means t h a t the Ton c u r r e n t o f t h e c o l l e c t o r had a peak t o peak ac c u r r e n t o f I/SO o f t h e dc l e v e l . . . • . A l i m i t a t i o n f o r t h e c h o i c e o f an i o n i z a t i o n gauge v/as i t s volume s i n c e we d i d n ' t want t o i n t e r f e r e w i t h the gas f l o w . The AEI 29D15 has a 49 mm l e n g t h and 20 mm d i a m e t e r , and t h e power d i s s i p a t e d I s o n l y 10 w a t t s . N o r m a l l y o t h e r commercial i o n i z a t i o n gauges h a v e a volume 6 t o 10 t i m e s b i g g e r and the power d i s s i p a t e d i s a b o u t 40 w a t t s . T h i s d i s s i p a t i o n c o u l d l e a d t o a non z e r o tem-p e r a t u r e g r a d i e n t and r e l a t e d e f f e c t o f gas f l o w i n the v i c i n i t y o f t h e i o n gauge f i l a m e n t . A l t h o u g h t h e s e n s i t i v i t y o f t h e 29D15 i s s m a l l e r than t h a t o f o t h e r c o m m e r c i a l g a u g e s , i t was enough f o r t h e d e t e c t i o n o f the s i g n a l w i t h t h e e l e c t r o n i c s used. The s e n s i t i v i t y o f t h e 29D15 gauge i s 5.7 microamp/mi1iamp e m i s s i o n / m i c r o n p r e s s u r e a p p r o x i m a t e l y . t ^ f f r The poss i b i 1 i t y o f us i ng a Bayard" and A l p e r t gauge was c o n s i d e r e d . . - . . .-. - • -. • . • S i n c e I was i n t e r e s t e d i n a s i g n a l as good as p o s s i b l e , I made a m i n i a t u r e B a y a r d and A l p e r t i o n i z a t i o n g a u g e 1 6 ( F i g . X ) . I t was mounted on a 9 ~ p l n s o c k e t . - kl -The c a thode and c o l l e c t o r a r e .025 cm (0.010" n o m i n a l ) t u n g s t e n w i r e . The anode cage i s assembled 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 about .1 cm s p a c i n g around f o u r p a r a l l e l .16 cm s t a i n l e s s s t e e l p o s t s . The t h r e e e l e c t r o d e s a r e s u p p o r t e d a t each end by a s p o t - w e l d i n g t o the l e a d s o f a s t a n d a r d 9 - p i n vacuum s e a l e d s o c k e t . The s e n s i t i v i t y o f t h i s gauge was about t h r e e times b i g g e r than t h a t o f t h e 29D15. The reason f o r t h i s i s t h a t the g r i d w i r e s o f t h e cage a r e q u i t e s m a l l , so most o f t h e e l e c t r o n s miss the cage and shoot a c r o s s t he i n n e r p a r t o f the cage u n t i l d e f l e c t e d by the c o l l e c t o r p o t e n t i a l , perhaps passed back and f o r t h t h rough t h e g r i d s e v e r a l t i m e s , but e v e n t u a l l y a r e c o l l e c t e d by the cage. A l l t h i s t r a v e l l i n g p roduces a h i g h e r q u a n t i t y o f i o n s i n s i d e t h e cage w h i c h w i l l p r o c e e d q u i c k l y t o the c o l l e c t o r w h i c h i s n e g a t i v e w i t h r e s p e c t t o t he anode. Ions formed o u t s i d e the cage w i l l have i n s u f f i c i e n t e n e rgy t o p e n e t r a t e t h e anode g r i d ; they w i l l be a t t r a c t e d e l s e w h e r e t o more n e g a t i v e s u r f a c e s . T y p i c a l v o l t a g e s a p p l i e d were: E l e c t r o d e P o t e n t i a l c a t h o d e 200 V anode *»00 V c o l l e c t o r 0 In s p i t e o f the good r e s u l t s from t h e "home-made" i o n i z a t i o n gauge, I had t r o u b l e s p o t - w e l d i n g t he g r i d t o the p o s t s , t he p o s t s t o the A3 -p i n s o f the s t a n d a r d s o c k e t , e t c . And s i n c e t he s e n s i t i v i t y o f th e 29D15 was good enough, I d e c i d e d t o use i t . The 29D15 AEI gauge 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 : g r i d +200 V c o l l e c t o r 0 f i lament + 20 V. f i l a m e n t c u r r e n t 1 t o 1.5 A e m i s s i o n c u r r e n t 2 t o 5 niA The i n s t a l l a t i o n o f t h i s d e t e c t o r i s . c l e a r l y shown i n F i g . XI. 5.3 Ion Gauge C o n t r o l Box In o r d e r t o have t h e i o n c u r r e n t depending on t h e p r e s s u r e o n l y , we need a f i x e d g r i d c u r r e n t . ( I t i s assumed t h a t a l l the e m i s s i o n c u r r e n t i s c o l l e c t e d by t h e g r i d . ) The g r i d c u r r e n t t e n d s 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 i s h i g h l y f l u c t u a t i n g and any l i t t l e change i n the f i l a m e n t tem-p e r a t u r e w i l l p r o d u c e a g r e a t v a r i a t i o n i n the g r i d c u r r e n t . T h i s s t a b i l i t y p r o b l e m i s the b a s i c p r o b l e m o f a l l i o n gauge c i r c u i t s , and i t i s n e c e s s a r y t o d e s i g n a c i r c u i t t o m a i n t a i n the g r i d c u r r e n t w i t h i n c l o s e l i m i t s . ' A s t a n d a r d c i r c u i t f o r r e g u l a t i n g t h e e l e c t r o n e m i s s i o n c u r r e n t i n an i o n i z a t i o n gauge employs a t r a n s f o r m e r i n s e r i e s w i t h t h e f i l a m e n t . The t r a n s f o r m e r i s o p e r a t e d on i t s s a t u r a t i o n curve and i t s s e c o n d a r y w i n d i n g i s lo a d e d by a vacuum t u b e , t he b i a s o f w h i c h i s c o n t r o l l e d by the i o n gauge e m i s s i o n c u r r e n t . I t o f t e n p r o v e s d i f f i c u l t t o o b t a i n s u f f i c i e n t " l o o p g a i n " i n t h i s f e e d b a c k system because s m a l l F i g u r e XI ATTACHMENT OF DETECTOR TO MAIN PIPE kS -tubes w i l l not l o a d t h e t r a n s f o r m e r s u f f i c i e n t l y , w h i l e h i g h c u r r e n t t u b e s r e q u i r e t o o g r e a t a g r i d swing f o r c o n t r o l . As a r e s u l t , one commercial c i r c u i t uses two power tubes (6 BA's) and a dc pentode a m p l i f i e r (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 . R i c h a r d and T u t h i l l 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 they got an even b i g g e r g a i n l o o p than the commercial boxes. The c i r c u i t was t r i e d f o r on o u r gauge but t h e r e g u l a t i o n was not good enough; a v a r i a t i o n i n g r i d c u r r e n t up t o \5% was o b s e r v e d . Then we t r i e d a m o d i f i c a t i o n o f the c i r c u i t d e s i g n e d by H o l m e s 1 7 t o c o n t r o l t h e e m i s s i o n c u r r e n t i n mass s p e c t r o m e t e r s used on r o c k e t s . W i t h t h i s c i r c u i t we got a c o n s t a n t g r i d c u r r e n t (no o b s e r v e d v a r i a t i o n ) t h r o u g h hours o f a r u n . We made a two channel box, each c h a n n e l t o t a l l y independent from the o t h e r , so t h a t t h e c i r c u i t i s sy m m e t r i c w i t h r e s p e c t t o AB ( F i g . X I I ) . The t r a n s i s t o r PNP 2N1502 (power t r a n s i s t o r ) passes t h r o u g h i t s e m i t t e r - c o l l e c t o r c i r c u i t a l l t h e c u r r e n t (1 t o 1.5 A) r e q u i r e d f o r t h e 29D15 gauge f i l a m e n t . T h i s f i l a m e n t c u r r e n t i s c o n t r o l l e d by the 2N1502 base c u r r e n t w h i c h i s s u p p l i e d by th e e m i t t e r o f t h e PNP 2N1038. The 2N1038 base c u r r e n t i s p r o v i d e d by the c o l l e c t o r o f NPN 2N2192 t r a n s i s t o r , t h e base o f whi c h i s c o n n e c t e d d i r e c t l y t o t h e e m i s s i o n c u r r e n t l i n e . Thus v a r i a t i o n s i n t h e e m i s s i o n c u r r e n t a r e a m p l i f i e d by tandem a m p l i f i e r 2N2192, 2N1038 and 2N1502. The phase o f t h e a m p l i f i e d o u t p u t s i g n a l i s such t h a t a s l i g h t F i g u r e X I I IONIZATION GAUGE CONTROL kl -i n c r e a s e i n e m i s s i o n c u r r e n t d e c r e a s e s the f i l a m e n t c u r r e n t . The d e s i r e d v a l u e o f e m i s s i o n i s s e t by p o t e n t i o m e t e r R j . T h i s p o t e n t i o -meter c o n t r o l s a c o n s t a n t b a c k i n g c u r r e n t s u p p l i e d by a 3 v o l t b a t t e r y . The a l g e b r a i c d i f f e r e n c e between t h e b a c k i n g c u r r e n t and t h e e m i s s i o n c u r r e n t i s the c o n t r o l 1ing s i g n a l f o r the r e g u l a t o r . The v a l u e o f Rj has been chosen so t h a t t h e e m i s s i o n c u r r e n t may be a r b i t r a r i l y s e t t h r o u g h o u t from 2.5 t o more than 30 mA. A 100 uf c a p a c i t o r c o n n e c t e d between t h e o u t p u t o f the r e g u l a t o r and t h e base o f t h e 2N1038 t r a n s i s t o r l o w e r s the ac g a i n o f the c i r c u i t . O s c i l l a -t i o n o c c u r s i n t h e absence o f t h i s c a p a c i t o r because o f t h e time l a g between a p p l i c a t i o n o f f i l a m e n t c u r r e n t and appearance o f e m i s s i o n c u r r e n t . The 100 tt r e s i s t o r i n t h e 2N2192 e m i t t e r c i r c u i t a l s o h e l p s t o a v o i d o s c i l l a t i o n by f u r t h e r l o w e r i n g t h e o v e r - a l l g a i n o f t h e f e e d -back l o o p . The 6 . 8 K r e s i s t o r p r o v i d e s c u t - o f f b i a s f o r t h e 2N1038 t r a n s i s t o r . When t h e f i l a m e n t b a t t e r y s w i t c h i s f i r s t c l o s e d , c u r r e n t t h r o u g h t h e 6 . 8 K r e s i s t o r c h a r g e s t h e 100 u f c a p a c i t o r . However, e m i s s i o n t e m p e r a t u r e o f t h e i o n i z a t i o n gauge f i l a m e n t i s exceeded b e f o r e t h e c a p a c i t o r i s f u l l y c h a r g e d . M eanwhile, t r a n s i s t o r 2N1038 i s b i a s e d so t h a t i t s e m i t t e r c u r r e n t i s a t v e r y h i g h l e v e l w h i c h i n t u r n sends e x c e s s i v e c u r r e n t t o t h e f i l a m e n t v i a 2N1502. T h i s e x c e s s i v e c u r r e n t c o n t i n u e s t o f l o w u n t i l t h e c a p a c i t o r has c h a r g e d , w h i c h can be 1 second a f t e r the p r o p e r e m i s s i o n t e m p e r a t u r e has been matched. 48 The v a l u e o f t h e maximum f i l a m e n t c u r r e n t (1.6A) i s f i x e d by means o f a n o n - l i n e a r c i r c u i t element ( t h r e e s i l i c o n d i o d e s ) c o n n e c t e d a c r o s s t h e 2N1502 e m i t t e r r e s i s t o r . These d i o d e s t a k e o v e r a l a r g e p o r t i o n o f t h e 2N1038 base c u r r e n t , thus l i m i t i n g the g a i n o f t h e f e e d b a c k loop when c u r r e n t exceeds t h i s v a l u e . The d i o d e s a l s o e f f e c t i v e l y shunt t h e 6.8 K r e s i s t o r d u r i n g f i l a m e n t warmup so t h a t t h e c a p a c i t o r c h a r g i n g time i s c o n s i d e r a b l y r e d u c e d . 20% v a r i a t i o n s on t h e v o l t a g e s u p p l y from t h e b a t t e r y do not produce an o b s e r v a b l e v a r i a t i o n i n g r i d c u r r e n t , and v a r i a t i o n i n p r e s s u r e from 1 0 ~ 3 t o 1 0 " 5 T o r r does not produce o b s e r v a b l e g r i d c u r r e n t change. The v o l t a g e a c r o s s R2 i s t h e s i g n a l from our d e t e c t o r , i n which we a r e i n t e r e s t e d . The S i g n a l Channel The c o l l e c t o r c u r r e n t from t h e d e t e c t o r produces a v o l t a g e a c r o s s R2 ( F i g . X I I ) o f 1 t o 20 mV (depending on t h e f r e q u e n c y and p r e s s u r e ) , w h i c h i s t h e s i g n a l o f i n t e r e s t . T h i s s i g n a l was brought t o the CR-4 PARC a m p l i f i e r . T h i s a m p l i f i e r has two (low and h i g h ) band r e j e c t i o n f i 1 t e r s . T h e r e f o r e the f r e q u e n c y o f t h e s i g n a l can be a l m o s t reduced t o the f i r s t h a r monic. The a m p l i f i c a t i o n f a c t o r can be v a r i e d : i n our c a s e t h e s i g n a l was a m p l i f i e d 50 times a p p r o x i -m a t e l y . The s i g n a l from 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 t o the r e f e r e n c e kS -i n p u t o f t h e JB-6 PARC phase s e n s i t i v e d e t e c t o r w h i c h when tuned t o t h e w o r k i n g 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 the " m o n i t o r o u t p u t " a s i n wave a m p l i f i e d about 100 times (thus we a r e j u s t u s i n g t h e tuned a m p l i f i o f the r e f e r e n c e channel as a tuned a m p l i f i e r t h a t was c o n v e n i e n t l y a v a i l a b l e ) . T h i s s i g n a l i s c o n n e c t e d t o t h e v e r t i c a l i n p u t o f the o s c i 1 l o s c o p e . The R e f e r e n c e Channel The r e f e r e n c e s i g n a l i s produced by the Beckman (model 443) R e f l e c t i n g P h o t o c e l l P ( F i g . I l l ) (see S e c t i o n 4 .5). T h i s s i g n a l i s a p p r o x i m a t e l y 3 v o l t s p-p and i t has a t o o t h form w i t h s m a l l r i p p l e s a t t h e c r e s t . The r a t i o r i p p l e t o s i g n a l i s a p p r o x i m a t e l y ' 1/20. The p h o t o c e l l o u t p u t d r i v e s a Feedback O s c i 1 l a t o r ( t y p e VPO 230) w h i c h remains t i g h t l y l o c k e d i n f r e q u e n c y and phase t o t h e p h o t o c e l l s i g n a l . The a m p l i t u d e o f t h e s i n wave from the o s c i l l a t o r i s a maximum when i t i s tuned t o the same f r e q u e n c y as th e d r i v i n g s i g n a l . Once the o s c i l l a t o r i s t u n e d , t h e o u t p u t can be v a r i e d from 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 main f e a t u r e s o f t h i s o s c i l l a t o r i s t h a t i t has t h r e e o u t p u t c h a n n e l s . One o f the c h a n n e l s i s i n phase w i t h the d r i v i n g s i g n a l , t h e second channel i s +90° out o f phase, and the o t h e r has a v a r i a b l e p h a s e - s h i f t o u t p u t , c o n t r o l l e d by a l i n e a r c a l i b r a t e d p h a s e - s h i f t e r w i t h a c a l i b r a t i o n e r r o r o f l e s s than ^0.5° 1 8 . The s i n wave from the o s c i l l a t o r i s c o n n e c t e d t o the X i n p u t o f the 50 -o s c i l l o s c o p e , w h i c h t h e r e f o r e produced a L i s s a j o u s f i g u r e c h a r a c t e r i s t i c o f t h e phase d i f f e r e n c e between the two c h a n n e l s . Method o f Phase Measurement In o r d e r t o make a measurement we have t o e n s u r e , f i r s t o f a l l , t h a t t h e motor f r e q u e n c y i s s t a b i l i z e d . The c i r c u i t c o n t r o l l i n g t h e speed o f t h e motor has a S i l i c o n C o n t r o l R e c t i f i e r whose g a t e v o l t a g e i s s u p p l i e d by a u n i j u n c t i o n t r a n s i s t o r . S i n c e the t r a n -s i s t o r ' s c h a r a c t e r i s t i c depend on the t e m p e r a t u r e , i t i s n e c e s s a r y t o w a i t about 10 minutes u n t i l t h e e q u i l i b r i u m t e m p e r a t u r e i s r e a c h e d . A f t e r t h i s w a i t i n g p e r i o d , the o s c i l l a t o r i s tuned t o the r e f e r e n c e f r e q u e n c y . In t h e s i g n a l c h a n n e l a l l l e a d s were f i x e d i n p o s i t i o n , s i n c e i t was o b s e r v 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 o f the l e a d s produced n o i s e w h i c h c r e a t e d a n o n - u n i f o r m a m p l i t u d e i n the f i n a l wave. The f i l t e r s i n t h e CR -4 PARC a m p l i f i e r were s e t a t 10 Hz. f o r low band r e j e c t i o n f i l t e r , and 30 Hz. f o r the h i g h band r e j e c t i o n f i l t e r . T a k i n g t h e p r e c a u t i o n s o f s h i e l d i n g and f i x i n g a l l l e a d s , the a m p l i -tude o f t h e f i n a l s i n wave was a l m o s t c o n s t a n t ( f l u c t u a t i o n o f 5%). T h i s s m a l l f l u c t u a t i o n i n a m p l i t u d e o f t h e f i n a l s i n wave i s i n f a c t a f l u c t u a t i o n o f t h e f r e q u e n c y . T h e r e f o r e , the e l l i p s e axes o f t h e L i s s a j o u s f i g u r e a r e s l i g h t l y f l u c t u a t i n g , g i v i n g an u n c e r -t a i n ! t y o f -2° i n t h e phase measurement. In o r d e r t o measure the phase s h i f t between the two i o n i z a t i o n gauges 50 o s c i l l o s c o p e , w h i c h t h e r e f o r e produced a L i s s a j o u s f i g u r e c h a r a c t e r i s t i c o f t h e phase d i f f e r e n c e between the two c h a n n e l s . Method o f Phase Measurement In o r d e r t o make a measurement we have t o e n s u r e , f i r s t o f a l l , t h a t t h e motor f r e q u e n c y i s s t a b i l i z e d . The c i r c u i t c o n t r o l l i n g t h e speed o f t h e motor has a S i l i c o n C o n t r o l R e c t i f i e r whose g a t e v o l t a g e i s s u p p l i e d by a u n i j u n c t i o n t r a n s i s t o r . S i n c e the t r a n -s i s t o r ' s c h a r a c t e r i s t i c depend on the t e m p e r a t u r e , i t i s n e c e s s a r y t o w a i t about 10 minutes u n t i l t he e q u i l i b r i u m t e m p e r a t u r e i s r e a c h e d . A f t e r t h i s w a i t i n g p e r i o d , the o s c i l l a t o r i s tuned t o the r e f e r e n c e f r e q u e n c y . In t h e s i g n a l c h a n n e l a l l l e a d s were f i x e d i n p o s i t i o n , s i n c e i t was o b s e r v 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 o f the l e a d s produced n o i s e w h i c h c r e a t e d a n o n - u n i f o r m a m p l i t u d e i n the f i n a l wave. The f i l t e r s i n t h e CR-4 PARC a m p l i f i e r were s e t a t 10 Hz. f o r low band r e j e c t i o n f i l t e r , and 30 Hz. f o r the h i g h band r e j e c t i o n f i l t e r . T a k i n g t h e p r e c a u t i o n s o f s h i e l d i n g and f i x i n g a l l l e a d s , the a m p l i -tude o f t h e f i n a l s i n wave was a l m o s t c o n s t a n t ( f l u c t u a t i o n o f 5%). T h i s s m a l l f l u c t u a t i o n i n a m p l i t u d e o f t h e f i n a l s i n wave i s i n f a c t a f l u c t u a t i o n o f t h e f r e q u e n c y . T h e r e f o r e , the e l l i p s e axes o f the L i s s a j o u s f i g u r e a r e s l i g h t l y f l u c t u a t i n g , g i v i n g an u n c e r -t a i n ! t y o f -2° i n t h e phase measurement. In o r d e r t o measure the phase s h i f t between t h e two i o n i z a t i o n gauges 51 we pr o c e e d as f o l l o w s : (a) W i t h t h e s i g n a l from gauge I ( t h e c l o s e r t o the chamber), we s e t a z e r o phase L i s s a j o u s f i g u r e between t h e two c h a n n e l s . The c a l i b r a t e d p h a s e - s h i f t e r i n the o s c i l l a t o r was s e t t o z e r o . Then the z e r o phase was s e t i n t h e s i g n a l c h a n n e l u s i n g 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 d e t e c t o r . (b) W i t h t h e s i g n a l from gauge II ( 3 1 c m from gauge l ) we r e t u r n e d t h e L i s s a j o u s f i g u r e t o z e r o phase c o n d i t i o n a g a i n by moving th e 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 o s c i l l a t o r . Then t h e r e a d i n g o f the c a l i b r a t e d 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 r e p e a t e d f o r each f r e q u e n c y a t each p r e s s u r e . Each measurement was performed f i v e t i m e s and the ave r a g e t a k e n . The r e p r o d u c i b i l i t y o f measurements was w i t h i n ± 2% o f t h e v a l u e o b t a i n e d . T h i s method o f measurement does i n t r o d u c e s p u r i o u s phase s h i f t s , but t h e s e a r e e l i m i n a t e d by t h e n u l l method used ( e x c e p t t h a t i t i s assumed t h a t any phase s h i f t o f t h e p r e s s u r e wave, due t o the f i n i t e r e s p o n s e t i m e o f t h e gauge, i s t h e same f o r each o f the g a u g e s ) . T h i s method o f measurement i s s i m p l e and more a c c u r a t e than o t h e r methods t r i e d b e f o r e . The phase s e n s t i v e d e t e c t o r c o u l d not be used f o r phase measurement d i r e c t l y because o f t h e s l i g h t f r e q u e n c y f l u c t u a t i o n o f o u r s i g n a l . The t u n i n g i s v e r y c r i t i c a l when making 52 a measurement w i t h a phase s e n s i t i v e d e t e c t o r , and t h e f r e q u e n c y f l u c t u a t i o n e n c o u n t e r e d l e d t o measurement e r r o r s up t o t 10°. 53 CHAPTER VI RESULTS AND DISCUSSION A n a l y s i s o f Data The. c a l c u l a t i o n o f the p r e s s u r e wave v e l o c i t y was made u s i n g the s i m p l e r e l a t i o n v = Xf where X ~ 360 x d i s t a n c e between gauges/measured phase and f = f r e q u e n c y t h e r e f o r e v = 360 x 31xf/p cm/sec The r e s u l t s o f t h e s e c a l c u l a t i o n s a r e shown i n t h e next graphs - t o g e t h e r w i t h e r r o r s . T h e o r e t i c a l c u r v e s a r e a l s o shown f o r the d i f f u s i o n e q u a t i o n (3) and f o r the d i f f u s i o n e q u a t i o n w i t h l i m i t e d r a nge o f i n t e g r a t i o n ( i . e . t h e low f r e q u e n c y l i m i t o f Matthews' p r e d i c t i o n (10)). C o n c l u s i o n 1. En t h e s e e x p e r i m e n t s , no dependency o f wave p r e s s u r e v e l o c i t y on mean f r e e p a t h was found f o r the range o f p r e s s u r e and f r e q u e n c y i n v e s t i g a t e d . Matthews' f o r m u l a p r e d i c t s about 30% d i f f e r e n c e , i n t h e s l o p e o f t h e graph o f v e l o c i t y a g a i n s t f r e q u e n c y 1 ^ 2 , f o r the extreme v a l u e s o f mean f r e e p a t h s a t w h i c h I have been w o r k i n g ; t h i s d i f f e r e n c e s h o u l d be e a s i l y m s e c AO v e l o c i t y CD > -o 30 20 10 I 1 1 _ i ft > / - - - - - - - - . . . l 1 : J - • - / i- - - - — _ - -l i v. y i i l ! 11 , ' r i 1 i ' i 1 _ i :• ) -! 7 / 1,1 1 -- -I L r i i - - l / i if i -i — -t--\-q :•: - - - - - -r A / ' < ) y \ - -- - - - - -- - - - - \/ I / I T r + , , - - - -- -- - - - u 1 's - j - -- --- -• i ~r — r -! i — - - r / T- - - - - - — - - -- -- -f / i I S - - -s r i i -- -_, i 7 • • + . . . -._ - _ •-- - . . . - - > . . . - - - - - - --- -- - V + - - -1 . / I 1 1 1 ?~ 1 - 1 ~ - TO rir,!< 1 1 / 1 — 1 - - - - K - 0" r '') U § !l i - r r - "3 r r \ —i 1 i —j—, - f L I i - -L - -> 1 i i r • . i / \ -- - . t J t" i •o n 1 1 1 - V rpm -1 n -h - - t \ i " T - \ - - - - - - + 7 •U J ) 1 T - — 3 ri z -"1 'J / na 1 1 I - - -- X i --c - -! 1 1 YW 1 _ - I 1 1 l _i ' 1// -4 - - T 8 -- — p 1 - - - -- 1 J c 1 --s-| - - - -I ' M X , i - r. - U X ( f r e q u e n c y ) 4.8 13 17 23 ON 58 -o b s e r v a b l e w i t h t h e e r r o r s t h a t were p r e s e n t i n the measure-ment. 2. The e x p e r i m e n t a l s l o p e v / f 1 ^ 2 i s s i i g h t l y g r e a t e r than t h a t o f t h e one p r e d i c t e d by t h e c l a s s i c a l D i f f u s i o n e q u a t i o n , v / f 1 / 2 = ( A f D T r ) 1 / 2 • w i t h D = 2rV/3 V = 3.99 x I O 4 cm/sec t t was t h o u g h t t h a t t h e main re a s o n f o r t h i s was the s t e a d y f l o w o f t h e gas due t o the mean p r e s s u r e g r a d i e n t , w h i c h would • p r o d u c e a m o t i o n o f the medium i n w h i c h the p r e s s u r e wave was p r o p a g a t i n g . However, measurements were performed u s i n g d i f f e r e n t f l o w r a t e s a t the same mean p r e s s u r e by a d j u s t i n g t h e d i f f u s i o n pump v a l v e from f u l l open to a l m o s t c o m p l e t e l y c l o s e d ; no d i f f e r e n c e i n the measured phase s h i f t was o b s e r v e d . Comments The s t a t e m e n t number one o f the c o n c l u s i o n i s c l e a r l y shown by a s i n g l e measurement i n t h e f o l l o w i n g way: once the L i a s s a j o u s f i g u r e has been s e t t o z e r o phase, th e n w i t h o u t t o u c h i n g any o f t h e e l e c t r o n i c c o n t r o l s , we changed t h e p r e s s u r e from 2 x 10 4 T o r r t o 10 3 T o r r by o p e n i n g t h e v a l v e c o n t r o l i n g the Argon f l o w . The L i s s a j o u s f i g u r e does not change a t a l l d u r i n g the whole p r o c e s s . T h i s has been r e p e a t e d f o r each f r e q u e n c y and f o r d i f f e r e n t gauge 59 -p o s i t i o n s , and t h e r e s u l t s a r e the same - no phase change o b s e r v e d when t h e p r e s s u r e c h a n g e s . In o r d e r t o c a l c u l a t e t h e f l u x o f m o l e c u l e s t h r o u g h a s e c t i o n S q, a t x = 0, o f the p i p e ( o r phonons t h r o u g h a s e c t i o n o f t h e c r y s t a l ) S m o l o u c h o s k i ( C a s i m i r & ZIman, i n the c o r r e s p o n d i n g t h e r m a l c a s e ) i n t e g r a t e d dF (see S e c t i o n 2.3) from x = -» t o x - +°° and g o t a D = 2 v r / 3 . Matthews assumed t h a t o n l y the m o l e c u l e s between +L produced t h e f l u x t h rough S Q. T h i s a s s u m p t i o n a l o n e l e a d s t o D =• 2/3 V r ( l - 3 T / 4 L ) , and t h i s was the e x p e c t e d dependence o f wave p r e s s u r e v e l o c i t y on L a t the low f r e q u e n c y l i m i t . T h i s a s s u m p t i o n , t h a t i n p r i n c i p l e seems r e a s o n a b l e f o r a f i r s t a p p r o x i m a t i o n , has been proved e x p e r i m e n t a l l y not t o be an a d e q u a t e way o f d e s c r i b i n g the s y s t e m . Perhaps the m o l e c u l e s s c a t t e r e d by m o l e c u l a r c o l l i s i o n s h o u l d be t a k e n i n t o a c c o u n t . T h i s must p r e s u m a b l y produce a dependence o f D on L t h a t i s much s m a l l e r t h a n Matthews e q u a t i o n p r e d i c t s . W i t h r e s p e c t t o s t a t e m e n t number two i n the c o n c l u s i o n , no e x p e r i -m ental o r t h e o r e t i c a l r e a s o n has been found t o e x p l a i n t h e d i f f e r e n c e i n s l o p e o f t h e c u r v e s o b t a i n e d e x p e r i m e n t a l l y when compared v / i t h t h a t o f t h e c l a s s i c a l d i f f u s i o n e q u a t i o n . I t was checked t h a t the s i g n a l c h a n n e l was not " p i c k i n g up" t h e r e f e r e n c e s i g n a l . S i n c e the f i r s t d e t e c t o r was v e r y c l o s e t o t h e chamber (10 cm) i t c o u l d be t h a t some m o l e c u l e s a r r i v e d a t the d e t e c t o r w i t h a speed c h a r a c -t e r i s t i c not o f a wave p r o p a g a t i o n but o f a m o l e c u l a r beam. In o r d e r t o i n v e s t i g a t e t h i s p o s s i b i 1 i t y measurements were p e r f o r m e d t 60 u s i n g gauges p o s i t i o n e d a t the t h i r d and f i f t h s i d e p i p e . The v e l o c i t y so measured was t h e same as i t was f o r t h e o r i g i n a l measurement. No doubt t h e t h e o r e t i c a l i n t e r p r e t a t i o n o f t h e s e r e s u l t s needs much f u r t h e r t h ought t o f i n d a c o n v i n c i n g e x p l a n a t i o n . 61 BIBLIOGRAPHY 1. C h e s t e r H. 1963, P h y s . Rev. 131, 2013 2. V e r n o t t e P., 1958, Compt. Rend. 246, 3154 3. C a t t a n e o M.C., 1958 Compt. Rend. 246, 431 4 . P e i e r l s R. 1929 Ann P h y s i k (5) 3_» 1055 5. Haas and B i e r m a s z 1938 Phys i c s 5, 320 6. C a s i m i r H.B.G., 1938. P h y s i c a 5, 495 7. Ziman J.M., 1954 P h i l . Mag. 45, 100 8.. Brown J . B . , Chung D.Y. and Matthews P.W., I966 P h y s . L e t t . 21, 2.41 9 . Matthews P.W. 1967 Can. Jo.urn. o f Phys. 45, 323 1 0 . Brown C.R. 1967 P h y s i c a 35, 114 1 1 . Kennard E.H., K i n e t i c Theory o f Gases (McGraw H i l l 1938) 1 2 . Weymann, H.D. A7). J . P. 35_ 488 (1967) 13. Ward J.C. and 'Wi Iks J . , 1951 P h y l . Mag. 42_, 314 14. F a i r b a n k 1966 P h y s . Rev. L e t t j_6 789 1 5 . P r e s e n t . K i n e t i c Theory o f Gases (McGraw M i l l ) 16. F i s h e r S . S . 1967 R e p o r t No. 675 UCLA March 1967 17. Holmes J.C. 1957 Rev. S c i e n t . I n s t r u m . 28, 290 18. Brown C.R. P r i v a t e c o m m u n i c a t i o n . 

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