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Measurements of the elastic constants of a liquid crystal Morris, Stephen William 1985

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MEASUREMENTS OF THE ELASTIC CONSTANTS OF A LIQUID CRYSTAL / by STEPHEN WILLIAM MORRIS B . S c . , U n i v e r s i t y Of B r i t i s h Columbia,1981 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n THE FACULTY OF GRADUATE STUDIES Department Of P h y s i c s 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 the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA A p r i l 1985 © Stephen W i l l i a m M o r r i s , 1985 2> 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 of the r e q u i r e m e n t s f o r an advanced degree a t the U n i v e r s i t y of B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree 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 of t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by the Head of my Department or by h i s or her r e p r e s e n t a t i v e s . I t i s unde r s t o o d t h a t c o p y i n g or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department of P h y s i c s The U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 Date: 25 A p r i l 1985 i i A b s t r a c t The bend and s p l a y e l a s t i c c o n s t a n t s of the nematic l i q u i d c r y s t a l o c t y l c y a n o b i p h e n y l (8CB) a r e measured as a f u n c t i o n of temperature u s i n g an e l e c t r i c - f i e l d - i n d u c e d d e f o r m a t i o n . The c a p a c i t a n c e and b i r e f r i n g e n c e of a sample c e l l , t e mperature c o n t r o l l e d t o 0.1mK, were s i m u l t a n e o u s l y measured as a f u n c t i o n of a p p l i e d v o l t a g e . The s p l a y c o n s t a n t i s d e t e r m i n e d from the c r i t i c a l v o l t a g e a t the onset of the d e f o r m a t i o n ( t h e F r e e d e r i c k s z t r a n s i t i o n ) . The bend c o n s t a n t i s found by f i t t i n g the d a t a above the c r i t i c a l v o l t a g e t o the t h e o r y of D e u l i n g , which i s d e r i v e d i n f u l l . The data i s a l s o a n a l y s e d i n the h i g h - and l o w - f i e l d l i m i t s of the t h e o r y . The bend e l a s t i c c o n s t a n t d i s p l a y s a p r e t r a n s i t i o n a l d i v e r g e n c e near the n e m a t i c - s m e c t i c A phase t r a n s i t i o n due t o s m e c t i c f l u c t u a t i o n e f f e c t s . The d i v e r g e n t p a r t i s f i t by a power law w i t h a c r i t i c a l exponent of 1.0dt.0.1. The D e u l i n g t h e o r y , which i s based on the assumption of l i n e a r e l a s t i c i t y , shows s y s t e m a t i c d e v i a t i o n from the d a t a a t h i g h v o l t a g e s a t a l l t e m p e r a t u r e s , w i t h the disagreement i n c r e a s i n g r a p i d l y as the s m e c t i c phase i s approached. T h i s r e s u l t s u g g e s t s t h a t t h e l i n e a r e l a s t i c t h e o r y of D u e l i n g / f a i l s f o r l a r g e d e f o r m a t i o n s and where s m e c t i c f l u c t u a t i o n s c o n t r i b u t e s i g n i f i g a n t l y t o the bend e l a s t i c i t y . The f a i l u r e near the s m e c t i c t r a n s i t i o n may be e x p l a i n a b l e by the quenching of s m e c t i c f l u c t u a t i o n s by the d e f o r m a t i o n . T a b l e of C o n t e n t s A b s t r a c t i i L i s t of T a b l e s v L i s t of F i g u r e s v i Acknowledgement i x Chapter I INTRODUCTION 1 1.1 The L i q u i d C r y s t a l l i n e S t a t e 1 1.2 The Nematic Phase 5 1.3 The Smectic A Phase 12 1.4 The Nematic - Smectic A Phase T r a n s i t i o n 18 Chapter I I THEORY OF THE FREEDERICKSZ TRANSITION 22 2 . 1 The D i r e c t o r •. 22 2.2 The C a p a c i t a n c e 34 2.3 The O p t i c a l Phase D i f f e r e n c e 42 2.4 Problems With The Theory 48 Chapter I I I THE EXPERIMENT 56 3.1 The C e l l 56 3.2 The Temperature C o n t r o l 61 3.3 The E l e c t r o n i c And O p t i c a l Systems 67 Chapter IV DATA ANALYSIS 73 4.1 F i t S t r a t e g y And D e c o n v o l u t i o n Of F r i n g e s 73 4.2 D i e l e c t r i c And R e f r a c t i v e Index R e s u l t s 81 4.3 E l a s t i c C onstant R e s u l t s 94 4.4 R e s u l t s In The C r i t i c a l Region 110 Chapter V CONCLUSION 118 APPENDIX A - COMPUTER PROGRAMS 122 A.1 Main R o u t i n e For C a p a c i t a n c e A n a l y s i s CF 122 A.2 S u b r o u t i n e REDCAP 131 A.3 S u b r o u t i n e CHISQC 131 A.4 S u b r o u t i n e EXTRAP 133 A.5 S u b r o u t i n e HFC And F u n c t i o n s HFZ And HFARG 134 A. 6 Main R o u t i n e For Phase Data A n a l y s i s PF 135 A.7 S u b r o u t i n e REDPHS 145 A.8 S u b r o u t i n e CHISQP 145 A.9 S u b r o u t i n e HFLIN 148 A.10 S u b r o u t i n e s HFO And FEW And R e l a t e d F u n c t i o n s ...148 A.11 S u b r o u t i n e PLTFIT 151 A.12 F u n c t i o n TEMP(R) 152 A.13 S u b r o u t i n e VFIND And F u n c t i o n VZ 153 A.14 S u b r o u t i n e FAINT, FBINT And FCINT And R e l a t e d F u n c t i o n s 153 i v A.15 F u n c t i o n GAM(T) 155 A.16 F u n c t i o n ENBAR(T) 155 A.17 F u n c t i o n DZFIT(T) 156 A.18 F u n c t i o n CZFIT(T) 156 APPENDIX B - DATA TABLES 158 REFERENCES 161 V L i s t of T a b l e s I . - A Survey of the p u b l i s h e d v a l u e s of the c r i t i c a l exponent V;/ 121 I I . R e s u l t s of C a p a c i t a n c e Data F i t s 158 I I I . R e s u l t s of Phase Data F i t s 159 IV. E l a s t i c C o n s t ant R e s u l t s 160 v i L i s t of F i g u r e s 1. The s t r u c t u r e of 4 , 4 — n - o c t y l c y a n o b i p h e n y 1 1 2. Schematic diagram of the phases of 8CB 2 3. The t h r e e independent d e f o r m a t i o n s of a nematic and t h e i r a s s o c i a t e d e l a s t i c c o n s t a n t s 9 4. Schematic diagram of the F r e e d e r i c k s z t r a n s i t i o n 23 5. Graph of d i r e c t o r a n g l e <^  v e r s u s p o s i t i o n i n the c e l l , f o r 3 = 1.4 X = 1. and v a r i o u s reduced v o l t a g e s 33 6. Graph of reduced c a p a c i t a n c e v e r s u s reduced v o l t a g e , f o r V =1.4 and v a r i o u s "K v a l u e s 36 7 i Graph showing the weak dependence of h i g h f i e l d s l o p e on the h i g h f i e l d reduced e l a s t i c c o n s t a n t 41 8. Graph of reduced phase v e r s u s reduced v o l t a g e , f o r * K = 1 , V=0.2, and v a r i o u s v a l u e s of 45 9. Graph of ^ X ft v e r s u s p o s i t i o n i n the c e l l , f o r the same c o n d i t i o n s as i n F i g . 5 ...54 10. The c o n s t r u c t i o n of the sample c e l l , 57 11. O p t i c a l system used t o measure t h i c k n e s s of f i l l e d c e l l 59 12. The c o n s t r u c t i o n of the t h e r m o s t a t 62 13. The temperature c o n t r o l c i r c u i t ....63 14. The p u l s e - w i d t h m o d u l a t i o n c i r c u i t used t o c o n t r o l the b a t h temperature 66 15. B l o c k diagram of the e l e c t r o n i c , o p t i c a l and d a t a a c q u i s i t i o n system 68 16. An example of raw c a p a c i t a n c e - v o l t a g e d a t a 74 17. F l o w c h a r t of c a p a c i t a n c e d a t a a n a l y s i s 75 18. F l o w c h a r t of o p t i c a l d a t a a n a l y s i s 77 19. Examples of raw f r i n g e d a t a 78 20. An example of raw o p t i c a l phase d a t a 80 v i i 21. Graph of the c r i t i c a l v o l t a g e v e r s u s temperature 83 22. Graph of the z e r o v o l t a g e c a p a c i t a n c e v e r s u s temperature 84 23. Graph of h i g h f i e l d c a p a c i t a n c e d a t a p l o t t e d a g a i n s t i n v e r s e v o l t a g e 86 24. Graph of the p r i n c i p a l d i e l e c t r i c c o n s t a n t s v e r s u s temperature 87 25. Graph of the reduced d i e l e c t r i c c o n s t a n t % v e r s u s temperature ....88 26. Graph of the z e r o v o l t a g e phase d i f f e r e n c e d ^ v e r s u s temperature 90 27. Graph of h i g h f i e l d o p t i c a l phase d a t a 91 28. Graph of the p r i n c i p a l i n d i c e s of r e f r a c t i o n v e r s u s temperature 92 29. Graph of reduced index of r e f r a c t i o n V v e r s u s temperature 93 30. Graph of the s p l a y e l a s t i c c o n s t a n t v e r s u s temperature 95 31. Graph of the reduced e l a s t i c c o n s t a n t ~K v e r s u s temperature 99 32. Graph of t h e o r e t i c a l f i t t o reduced phase d a t a 101 33. Graph of p e r p e n d i c u l a r r e s i d u a l s of phase d a t a f i t ..102 34. Graph of t h e o r e t i c a l f i t t o reduced c a p a c i t a n c e d a t a 103 35. Graph of p e r p e n d i c u l a r r e s i d u a l s of c a p a c i t a n c e d a t a f i t 1 04 36. The temperature v a r i a t i o n of the s t a n d a r d d e v i a t i o n of the f i t s . 106 37. The temperature v a r i a t i o n of the i n s e n s i t i v i t y of the f i t t o *X 107 38. Graph of the bend e l a s t i c c o n s t a n t v e r s u s temperature 109 39. Graph of the s p l a y e l a s t i c c o n s t a n t v e r s u s the square of the nematic o r d e r parameter 112 40. Graph of the bend e l a s t i c c o n s t a n t v e r s u s the square of the nematic o r d e r parameter 113 v i i i 41. Graph of the best f i t of the d i v e r g e n t p a r t of the bend e l a s t i c c o n s t a n t t o a power law i n the reduced t e m p e r a t u r e 115 42. Graph of the best f i t of the reduced e l a s t i c c o n s t a n t t o . a power law i n the reduced temperature 117 Acknowledgement T h i s experiment was proposed by Dr. P e t e r P a l f f y - M u h o r a y , w i t h o u t whose c o n t i n u e d i n s p i r a t i o n i t would never have been completed. Thanks a r e a l s o due t o Dr. A. J . B e r l i n s k y , who i n t r o d u c e d me t o the t h e o r y of c r i t i c a l phenomena, and t o Dr. D a v i d B a l z a r i n i who g e n e r o u s l y p r o v i d e d space, equipment and i d e a s . I am a l s o i n d e b t e d t o Dr. D a v i d Dunmur f o r s u p p l y i n g l i q u i d c r y s t a l samples, and f o r h i s h o s p i t a l i t y on my v i s i t t o E n g l a n d . I am g r a t e f u l t o John de Bruyn f o r many u s e f u l d i s c u s s i o n s , p a r t i c u l a r l y on the a f t e r n o o n of 9 May 1984, the day of "The G r e a t D i s a s t e r " . Many pe o p l e k i n d l y l e n t equipment f o r t h i s e x p e r i m e n t . Dr. M. C r o o k s , Dr. J . C a r o l a n and Dr. W. Hardy s u p p l i e d the main e l e c t r o n i c s . The computer was borrowed from Matt C h o p t u i k and S i n g Chow l e n t v o l t m e t e r s and some o t h e r bus d e v i c e s . I would l i k e t o thank D^r. B. Bergersen and Dan Zimmerman f o r d i s c u s s i o n s of the d a t a a n a l y s i s and t h e o r y . The b a t h temperature c o n t r o l l e r was d e s i g n e d and b u i l t by Kent Lundgren. A l e c Sandy and Don Mathewson, who a l l o w e d themselves t o be i n t e r f a c e d t o the computer f o r days a t a t i m e , were r e s p o n s i b l e f o r most of the c a p a c i t a n c e d a t a c o l l e c t i o n . I would l i k e t o thank N i c o l e Salmon, who t y p e d t h i s tome. And then t h e r e ' s Mary, who was v e r y p a t i e n t . 1 I . INTRODUCTION . . . t h i n g s t h a t seem t o us hard and s t i f f must be composed of d e e p l y i n d e n t e d and hooked atoms and h e l d f i r m by t h e i r i n t e r t a n g l i n g b r a n c h e s . . . L i q u i d s , on the o t h e r hand, must owe t h e i r f l u i d c o n s i s t e n c y t o component atoms t h a t a re smooth and round. For poppy seed can be poured as e a s i l y as i f i t were water... 1.1 The L i q u i d C r y s t a l l i n e S t a t e L i q u i d c r y s t a l s a r e indeed l i q u i d i n the sense t h a t they f l o w and do not support shear s t r e s s e s . However, u n l i k e f a m i l i a r l i q u i d s they have a n i s o t r o p i c o p t i c a l and m e c h a n i c a l p r o p e r t i e s r e m i n i s c e n t of some c r y s t a l l i n e s o l i d s . L i q u i d c r y s t a l l i n e phases occur a t te m p e r a t u r e s i n t e r m e d i a t e between a h i g h temperature normal l i q u i d phase and a low temperature s o l i d phase. The source of the symmetry-breaking o r d e r p r e s e n t i n l i q u i d c r y s t a l s i s the asymmetry of the m o l e c u l e s . The m a t e r i a l used i n t h i s s tudy was 4, 4'«-n-octylcyanobiphenyl which has the s t r u c t u r e shown i n F i g . 1. L u c r e t i u s rv >C8H17 F i g u r e 1 - The s t r u c t u r e of 4 , 4 - n - o c t y l c y a n o b i p h e n y l 2 T h i s m a t e r i a l , which we w i l l h e r e a f t e r r e f e r t o as 8CB, i s t y p i c a l of a l a r g e c l a s s of m a t e r i a l s which show l i q u i d c r y s t a l l i n e phases. The e s s e n t i a l c h a r a c t e r i s t i c s of these m a t e r i a l s a r e t h a t they are composed of l a r g e o r g a n i c m o l e c u l e s which a r e l o n g compared t o t h e i r w i d t h and r e l a t i v e l y i n f l e x i b l e . We w i l l c o n c e n t r a t e on the m a c r o s c o p i c p r o p e r t i e s of the l i q u i d c r y s t a l , and not attempt t o s o l v e the d i f f i c u l t problem of how the m o l e c u l a r s t r u c t u r e c o n s p i r e s t o produce them. For our p u r p o s e s , one may p i c t u r e the m o l e c u l e as a hard r o d . The o b s e r v e d phases of 8CB are shown s c h e m a t i c a l l y i n F i g . 2. The s t r u c t u r e of these phases has been d e t e r m i n e d by X-ray and l i g h t s c a t t e r i n g e x p e r i m e n t s , by t h e i r o p t i c a l p r o p e r t i e s and by the n a t u r e of the d e f e c t s v i s i b l e under a m i c r o s c o p e . Nematic A F i g u r e 2 Schematic diagram of the phases of 8CB 3 At t e m p e r a t u r e s g r e a t e r than 40.5 C 8CB i s a normal l i q u i d . T h i s i s the " i s o t r o p i c " phase. The i s o t r o p i c l i q u i d i s c l e a r and s l i g h t l y v i s c o u s , l i k e a v e r y f i n e machine o i l . At 40.5°C t h e r e o c c u r s a l i q u i d - l i q u i d phase t r a n s i t i o n t o the "nematic" phase. T h i s l i q u i d i s n o t i c e a b l y more v i s c o u s than t h e ~ i s o t r o p i c phase and o f t e n has a c l o u d y , t u r b i d appearance. In t h i s phase the l o n g m o l e c u l e s tend t o p o i n t i n one d i r e c t i o n on the avera g e . The c e n t r e s of mass of the m o l e c u l e s a re s t i l l p o s i t i o n a l l y d i s o r d e r e d as i n a normal l i q u i d . T h i s , the s i m p l e s t l i q u i d c r y s t a l phase, i s the main s u b j e c t of t h i s study and i s d e s c r i b e d a t g r e a t e r l e n g t h i n a l a t e r s e c t i o n . Of p a r t i c u l a r i n t e r e s t i s the phase t r a n s i t i o n between t h i s and the " s m e c t i c A" phase. The s m e c t i c A phase i s the s i m p l e s t of the " l a y e r e d " phases which a r e c a l l e d , not s u r p r i s i n g l y , s m e c t i c A, B, C e t c . Many of t h e s e phases are known, a l t h o u g h not enough t o exhaust the a l p h a b e t . Some m a t e r i a l s go d i r e c t l y i n t o a s m e c t i c phase from the i s o t r o p i c w h i l e o t h e r s show s e v e r a l d i f f e r e n t s m e c t i c phases as the temperature i s l o w e r e d . The s m e c t i c A phase has a l a y e r s t r u c t u r e w i t h the normal to the " l a y e r s p a r a l l e l t o the l o n g m o l e c u l a r axes on the ave r a g e . The t h i c k n e s s of t h e s e l a y e r s i n 8CB i s 49 ? which i s about t w i c e the l e n g t h of the 8CB m o l e c u l e . 8CB i s t h e r e f o r e c a l l e d a " b i l a y e r s m e c t i c " . In appearance s m e c t i c s A a r e l e s s t u r b i d - l o o k i n g than n e m a t i c s but a r e more v i s c o u s . In the s m e c t i c A phase t h e r e i s s t i l l no p o s i t i o n a l o r d e r w i t h i n the l a y e r s . The " h i g h e r " s m e c t i c s , B, C, e t c . a r e 4 p r o g r e s s i v e l y more o r d e r e d w i t h i n the l a y e r s . These phases w i l l not c o n c e r n us any f u r t h e r . In any event t h e r e a r e no h i g h e r s m e c t i c phases i n 8CB, as i t f r e e z e s t o a waxy s o l i d from the s m e c t i c A phase at 21.5° C. Modern work i n l i q u i d c r y s t a l s i s m o t i v a t e d m a i n l y by t h e o r e t i c a l i n t e r e s t i n t h e i r many phase t r a n s i t i o n s and by t h e i r w idespread a p p l i c a t i o n as low-power e l e c t r o n i c d i s p l a y d e v i c e s . L i q u i d c r y s t a l s have, however, been known and s y s t e m a t i c a l l y s t u d i e d s i n c e the middle of the 19th c e n t u r y . A h i s t o r i c a l review of the work b e f o r e 1945 has been g i v e n by K e l k e r * . The modern f i e l d of l i q u i d c r y s t a l s has been the s u b j e c t of s e v e r a l books > n o t a b l y those of de Gennes and C h a n d r a s e k h a r ^ . The emphasis of the de Gennes book makes i t p a r t i c u l a r l y u s e f u l f o r u n d e r s t a n d i n g the nematic phase. B l i n o v ^ d i s c u s s e s the v a r i o u s responses of l i q u i d c r y s t a l s t o e l e c t r i c and magnetic f i e l d s and the d e v i c e a p p l i c a t i o n s of n e m a t i c s . An e x c e l l e n t r e v i e w of e x p e r i m e n t a l r e s u l t s has been er g i v e n i n the book by de J e u . We now t u r n t o the s p e c i f i c s of the nematic and s m e c t i c A phases and the phase t r a n s i t i o n between them. 5 1.2 The Nematic Phase In t h i s s e c t i o n we g i v e a b r i e f r e v i e w of the nematic phase. The a x i s a l o n g which the l o n g m o l e c u l e s t e n d t o p o i n t i n the nematic phase i s a s p e c i a l d i r e c t i o n i n the l i q u i d . P r o p e r t i e s of the b u l k l i q u i d such as d i e l e c t r i c c o n s t a n t or index o f r e f r a c t i o n a r e d i f f e r e n t a l o n g t h i s a x i s than i n the p l a n e p e r p e n d i c u l a r t o i t . The l i q u i d i s r o t a t i o n a l l y symmetric o n l y about t h i s s p e c i a l d i r e c t i o n . I t i s c o n v e n i e n t t o d e f i n e a u n i t v e c t o r , c a l l e d the " d i r e c t o r " which p o i n t s a l o n g t h i s symmetry a x i s of the c r y s t a l . The d i r e c t o r , which we denote by n, i s i n g e n e r a l a f u n c t i o n of p o s i t i o n . Of c o u r s e , the concept of a s p a t i a l l y v a r y i n g d i r e c t o r i s o n l y u s e f u l on s c a l e s which a r e l a r g e compared t o the m o l e c u l a r s i z e . The c r y s t a l d e s c r i b e d by n i s i d e n t i c a l t o t h a t d e s c r i b e d by -n. That i s , we r e q u i r e t h a t a l l the p h y s i c a l q u a n t i t i e s be i n v a r i a n t under the t r a n s f o r m a t i o n n-»-n. T h i s e x p r e s s e s the f a c t t h a t i n a nematic the l o n g range m o l e c u l a r o r i e n t a t i o n a l o r d e r does not d i s t i n g u i s h between one end of the m o l e c u l e and the o t h e r . The l i q u i d has t h i s symmetry even i f the m o l e c u l e s do n o t . Thus any c o r r e l a t i o n of the m o l e c u l e s t h a t depends on t h e i r e n d - f o r - e n d o r i e n t a t i o n s e x t e n d s o n l y t o d i s t a n c e s on the o r d e r of a m o l e c u l a r l e n g t h . In the s m e c t i c A phase, where the concept of the d i r e c t o r i s s t i l l u s e f u l , i t i s p r o b a b l y t h e s e s h o r t range c o r r e l a t i o n s t h a t l e a d t o the " b i l a y e r " s t r u c t u r e of the s m e c t i c l a y e r s i n 8CB. In the nematic phase the m o l e c u l e s a r e not a l l p e r f e c t l y 6 a l i g n e d with the d i r e c t o r . There i s i n general some d i s t r i b u t i o n of o r i e n t a t i o n s . As a f i r s t approximation, we may suppose that t h i s d i s t r i b u t i o n only depends on the p o l a r angle Q. between the long a x i s of the molecule and the d i r e c t o r and not on the azimuthal angle. T h i s i s e q u i v a l e n t to assuming that the molecules are u n c o r r e l a t e d with r e s p e c t to r o t a t i o n s about t h e i r long axes. In t h i s approximation a measure of the degree of nematic order , a s o - c a l l e d "order parameter", i s given by where < > denotes an average over the d i s t r i b u t i o n of o r i e n t a t i o n s . Note that the e q u i v a l e n c e of n and -n makes <cosft > z e r o . P e r f e c t alignment, a l l $ =o or Tt , g i v e s S=1, while i f $ i s randomly d i s t r i b u t e d S=0. A simple mean-field theory, the Maier-Saupe theory ^ , of the nematic phase can be c o n s t r u c t e d by assuming that a p a r t i c u l a r molecule, l a b e l l e d by i , has an o r i e n t a t i o n a l energy p r o p o r t i o n a l to (2) — ±(ZCOSZ&i - | ) g In the s i m p l e s t v e r s i o n of t h i s theory the p r o p o r t i o n a l i t y constant i s temperature independent. T h i s theory c o r r e c t l y p r e d i c t s t h a t the i s o t r o p i c - n e m a t i c t r a n s i t i o n i s f i r s t order, that i s , that S jumps d i s c o n t i n u o u s l y to a nonzero value at the t r a n s i t i o n temperature T W j . I t a l s o g i v e s a q u a l i t a t i v e l y c o r r e c t temperature dependence of S below T^j . More on t h i s 7 t h e o r y can be found i n the g e n e r a l r e f e r e n c e s The e f f e c t of w a l l s or f r e e s u r f a c e s on the nematic d i r e c t o r i s of some importance i n . l i q u i d c r y s t a l e x p e r i m e n t s and a p p l i c a t i o n s . The i n t e r a c t i o n s between a l i q u i d c r y s t a l and the s u r f a c e of a m a t e r i a l a r e v e r y s e n s i t i v e t o the c h e m i c a l and p h y s i c a l s t a t e of the s u r f a c e . T h i s l o n g s t u d i e d problem i s the s u b j e c t of a r e c e n t monograph^ . The most d e s i r a b l e s o r t of s u r f a c e i s one which c o n s t r a i n s the d i r e c t o r t o one g i v e n d i r e c t i o n ("strong a n c h o r i n g " ) which i s e x p e r i m e n t a l l y known and i s c o n t r o l l a b l e by s u r f a c e p r e p a r a t i o n . S u i t a b l y p r e p a r e d s u r f a c e s produce w e l l known boundary c o n d i t i o n s on the d i r e c t o r f i e l d and make the p r e p a r a t i o n of o r i e n t e d s i n g l e c r y s t a l s p o s s i b l e . The g e n e r a l continuum e q u a t i o n s f o r the f l o w and d e f o r m a t i o n of n ematics ("nematodynamics") a r e q u i t e complex but the s t a t i c case i s remarkably s i m p l e . " D e f o r m a t i o n " of the c r y s t a l s i m p l y means a s p a t i a l l y v a r y i n g d i r e c t o r f i e l d . As l o n g as the s c a l e of t h e s e s t a t i c d e f o r m a t i o n s i s l a r g e compared t o a m o l e c u l a r l e n g t h we can c o n s i d e r the o r d e r parameter as i f i t was u n d i s t u r b e d by them. The "continuum l i m i t " i s where "a" i s a m o l e c u l a r l e n g t h . The c r y s t a l r e s i s t s such d e f o r m a t i o n s because t o produce them one must do work a g a i n s t the i n t e r m o l e c u l a r f o r c e s t h a t t e n d t o a l i g n the m o l e c u l e s . One can c o n s t r u c t a l i n e a r e l a s t i c (3) 1 8 t h e o r y of the d i r e c t o r f i e l d by c o n s i d e r i n g o n l y l o w e s t o r d e r g r a d i e n t s of n. A remarkable f a c t i s t h a t the symmetry of the d i r e c t o r f i e l d a l l o w s o n l y t h r e e independent d e f o r m a t i o n s and c o n s e q u e n t l y o n l y t h r e e e l a s t i c c o n s t a n t s . These d e f o r m a t i o n s are i l l u s t r a t e d i n F i g . 3. The e l a s t i c f r e e energy per u n i t volume a s s o c i a t e d w i t h these d e f o r m a t i o n s i s g i v e n by (4) £ N =^ Ku(v.A)+iKai(h.V*n) +iK,3(nXVxA) The t h r e e e l a s t i c c o n s t a n t s are g i v e n the p i c t u r e s q u e names " s p l a y " , " t w i s t " and "bend" and are known as the "Frank-Oseen c o n s t a n t s " a f t e r t h e i r o r i g i n a t o r s 0 . I t i s of i n t e r e s t t o e s t i m a t e the magnitude of the e l a s t i c c o n s t a n t s . S i n c e n i s d i m e n s i o n l e s s and f ^ has u n i t s energy/volume the e l a s t i c c o n s t a n t s must have u n i t s of f o r c e . The e l a s t i c c o n s t a n t s s h o u l d be of o r d e r U/a where U i s a c h a r a c t e r i s t i c energy of the nematic o r d e r i n g i n t e r a c t i o n and a i s the m o l e c u l a r l e n g t h . For U we may take k g T N l , where Ty^ i s the n e m a t i c - i s o t r o p i c t r a n s i t i o n t e mperature and kg i s Boltzmann's c o n s t a n t , s i n c e t h i s t r a n s i t i o n o c c u r s when the temperature i s such t h a t the t h e r m a l energy k^T^ j per m o l e c u l e i s comparable t o the i n t e r a c t i o n energy which d r i v e s the nematic o r d e r . In t h e s e m a t e r i a l s T N, ^ 300K and a=* 50A*, k f t =10~ JK -12. so R " kgT N, /a ^ 1 0 Newton. As we w i l l see l a t e r the e l a s t i c c o n s t a n t s a r e indeed of t h i s o r d e r except a t t e m p e r a t u r e s c l o s e t o the n e m a t i c - s m e c t i c A t r a n s i t i o n . 9 F i g u r e 3 - The t h r e e independent d e f o r m a t i o n s of a nematic and t h e i r a s s o c i a t e d e l a s t i c c o n s t a n t s Splay K 1 1 Twist K 22 Bend K 33 10 The t a s k o f t h e e x p e r i m e n t t o be d e s c r i b e d i s t o improve on t h i s e s t i m a t e . E l e c t r i c and m a g n e t i c f i e l d s have i n t e r e s t i n g and u s e f u l e f f e c t s on n e m a t i c s and a l m o s t a l l e x p e r i m e n t s and a p p l i c a t i o n s i n v o l v e them. We w i l l c o n s i d e r t h e e l e c t r i c f i e l d c a s e s i n c e i t i s r e l e v a n t t o our e x p e r i m e n t . The m a g n e t i c c a s e i s g e n e r a l l y s i m p l e r however, s i n c e l i q u i d c r y s t a l s have v e r y s m a l l m a g n e t i c s u s c e p t i b i l i t i e s . The e l e c t r i c c o n t r i b u t i o n t o t h e f r e e e n e r g y d e n s i t y i s g i v e n by (5) - P E = - i e o E - | ( ^ - E where E i s t h e e l e c t r i c f i e l d , i s t h e p e r m i t t i v i t y of f r e e s p a c e and £ ( n ) i s t h e d i e l e c t r i c c o n s t a n t t e n s o r . S i n c e the-n e m a t i c i s a n i s o t r o p i c t h e d i e l e c t r i c p r o p e r t i e s must be e x p r e s s e d as s e c o n d rank t e n s o r s and t h e e l e c t r i c d i s p l a c e m e n t v e c t o r D = £ » E i s n o t , i n g e n e r a l , p a r a l l e l t o E . We have w r i t t e n t h i s t e n s o r as a f u n c t i o n o f n s i n c e c l e a r l y i t s p r i n c i p a l a x e s f o l l o w t h e c r y s t a l a x e s w h i c h a r e r e p r e s e n t e d by A n. In t h e p r i n c i p a l a x i s frame we have / € X o o \ ( 6 ) £ = ( O 6 j L O j V O O € . , 1 / where €-n i s t h e d i e l e c t r i c c o n s t a n t f o r f i e l d s p a r a l l e l t o n and £ ^ t h e c o n s t a n t f o r f i e l d s p e r p e n d i c u l a r t o n. B o t h €n and £ L . d e p e n d on t h e d e g r e e o f n e m a t i c o r d e r and hence on t e m p e r a t u r e . 11 In 8CB the d i e l e c t r i c a n i s o t r o p y /\£. = £ - £ ± i s p o s i t i v e . The e l e c t r i c d i s p l a c e m e n t i n terms of the d i r e c t o r f i e l d i s (7) ]) = e0e±£ + e0be (E*h)n so t h a t e q u a t i o n (5) reduces t o (8) -f e = e^E 1 +A6(l«r\) / Only the second term depends on n. In m a t e r i a l s w i t h p o s i t i v e e q u a t i o n (8) i m p l i e s t h a t the f r e e energy i s m i n i m i z e d f o r n p a r a l l e l t o E . I f n i s not a l i g n e d w i t h E the c r y s t a l w i l l f e e l a t o r q u e 22 per u n i t volume g i v e n by r = DXE = e o A e ( E . n ) ( A x E ) T h i s t o r q u e w i l l r e o r i e n t the c r y s t a l s u b j e c t t o a g e n e r a l i z e d form of v i s c o u s d r a g , the e l a s t i c s t r e s s e s , g i v e n by e q u a t i o n ( 4 ) , and the boundary c o n d i t i o n s on n. We w i l l d e s c r i b e i n the next c h a p t e r how we w i l l measure the e l a s t i c c o n s t a n t s by measuring the d e f o r m a t i o n of the d i r e c t o r f i e l d when the e l a s t i c f o r c e s a r e b a l a n c e d a g a i n s t known e l e c t r i c f i e l d s and s u r f a c e boundary c o n d i t i o n s . 1 2 1.3 The Smectic A Phase In t h i s s e c t i o n we g i v e a more q u a n t i t a t i v e d e s c r i p t i o n of the l a y e r e d s t r u c t u r e of s m e c t i c s and i n t r o d u c e the de Gennes model of the s m e c t i c A phase. q I t has been found by X-ray s c a t t e r i n g t h a t t o a good a p p r o x i m a t i o n the " l a y e r s " of a s m e c t i c may be d e s c r i b e d as a o n e - f o u r i e r - c o m p o r i e n t mass d e n s i t y wave of the form (10) s\f — s\ I \ J _ 1IJ / u/ ^lTl>*~ Y\ pit) =pj\ e**)) where i s a background d e n s i t y and C^=2TT/d where d i s the l a y e r s p a c i n g , means "take the r e a l p a r t " . The c o o r d i n a t e s are such t h a t the l a y e r s l i e i n the xy p l a n e and n i s p a r a l l e l t o the z a x i s . The complex a m p l i t u d e X i s g i v e n by (11) where the r e a l v a l u e d f u n c t i o n u ( r ) g i v e s the d i s p l a c e m e n t of the l a y e r s and ty, a r e a l number, g i v e s t h e i r a m p l i t u d e . $ i s the o r d e r parameter f o r the s m e c t i c A phase i n the de Gennes model In o r d e r t o get some i d e a of how the e l a s t i c p r o p e r t i e s of the s m e c t i c A phase d i f f e r from the nematic phase we c o n s i d e r a l i n e i n t e g r a l 13 (12) - L 1 A • For a d e f e c t - f r e e s m e c t i c t h i s i n t e g r a l , no ma t t e r what i t s p a t h , s i m p l y c o u n t s the number of l a y e r s between p o i n t s A and B. Hence f o r a c l o s e d c o n t o u r <13> ~ <P h * c U = O from which we deduce t h a t ^ X n =0 by Stokes theorem. We co n c l u d e t h a t i n a d e f e c t - f r e e s m e c t i c " t w i s t " or "bend" d e f o r m a t i o n s , which i n v o l v e S7x n by e q u a t i o n ( 4 ) , a r e not a l l o w e d . T h i s i s e q u i v a l e n t t o s a y i n g t h a t the e l a s t i c c o n s t a n t s and a r e i n f i n i t e i n a s m e c t i c . The de Gennes m o d e l ^ was i n s p i r e d by an an a l o g y between s m e c t i c s A and s u p e r c o n d u c t o r s . T h i s model has s t i m u l a t e d most of ,the r e c e n t work on s m e c t i c s and the n e m a t i c - s m e c t i c A phase t r a n s i t i o n . I t i s not a c o i n c i d e n c e t h a t de Gennes has a l s o w r i t t e n a book on s u p e r c o n d u c t o r s " . The c o n n e c t i o n between s m e c t i c s A and s u p e r c o n d u c t o r s can be made because both have an o r d e r parameter which i s a complex number . In the s u p e r c o n d u c t o r 5f i s the w a v e f u n c t i o n of the Cooper p a i r s . In the s m e c t i c $ d e s c r i b e s the magnitude and phase of the mass d e n s i t y wave. The f r e e energy d e n s i t y i n the de Gennes model i s a g e n e r a l i z a t i o n of the phen o m e n o l o g i c a l Landau - G i n z b u r g e x p a n s i o n f o r a s u p e r c o n d u c t o r i n powers of $ 1 4 and i t s g r a d i e n t s (14) m z + _L . T i (v T - i^ ) ) f i where 0/ changes s i g n a t T ^ f l ( 1 5 ) # =• ^ ' ( T — T N A ) as u s u a l f o r a Landau t h e o r y of a second o r d e r phase t r a n s i t i o n , "'^r i s the g r a d i e n t o p e r a t o r i n the xy p l a n e , t r a n s v e r s e t o the d i r e c t o r . The g r a d i e n t terms c o n t a i n phenomenological " a n i s o t r o p i c masses" My and M -^ , which a r e equ a l f o r a s u p e r c o n d u c t o r , and the t r a n s v e r s e g r a d i e n t term c o n t a i n s a c o u p l i n g t o s m a l l v a r i a t i o n s i n the d i r e c t o r r e s u l t i n g from v a r i a t i o n s of u ( r ) i n the pl a n e of the l a y e r s . To the lo w e s t o r d e r they a r e ( 1 6 ) f|g i s the nematic c o n t r i b u t i o n t o the f r e e energy, e q u a t i o n ( 4 ) , which now o n l y c o n t a i n s the s p l a y term 1 5 (17) E q u a t i o n (14).has the same form as the Landau-Ginzburg f r e e energy of a s u p e r c o n d u c t o r i f we i d e n t i f y n w i t h the v e c t o r p o t e n t i a l A , and a l l o w a n i s o t r o p i c Cooper p a i r masses. The q u a n t i t y CJ, T F r e p l a c e s 2e/c f o r the s u p e r c o n d u c t o r , where 2e i s the charge of a Cooper p a i r and c is.' the v e l o c i t y of l i g h t . We now r e c o g n i z e our e a r l i e r r e s u l t , ^ X n =0, as the a n a l o g of the e x p u l s i o n of a magnetic f i e l d B = 7 x A by a su p e r c o n d u c t o r (the w e l l known M e i s s n e r e f f e c t ) . L e t us c o n s i d e r a s i m p l e case of e q u a t i o n ( 1 4 ) , when j ^ J i s s p a t i a l l y u n i f o r m . Then (18) and the t r a n s v e r s e g r a d i e n t terms drop o u t . T h i s l e a v e s where the new e l a s t i c c o n s t a n t B i s g i v e n by B = ^ o ^ / M y * T h e B e l a s t i c c o n s t a n t i s a s s o c i a t e d w i t h d i s p l a c e m e n t s of the l a y e r s p a r a l l e l t o n w h i l e the nematic s p l a y c o n s t a n t Kjj i s a s s o c i a t e d w i t h the bending of the l a y e r s . B can be measured by 12 \% X - r a y ' and l i g h t s c a t t e r i n g . We can form a c h a r a c t e r i s t i c l e n g t h 16 (20) A = / which i s the a n a l o g of the magnetic p e n e t r a t i o n depth i n s u p e r c o n d u c t o r s . In the s m e c t i c , ?1 measures the p e n e t r a t i o n depth of bend and t w i s t d e f o r m a t i o n s imposed as boundary c o n d i t i o n s . Note t h a t 7i d i v e r g e s as we approach T N f l from below. A s i m i l a r e f f e c t o c c u r s i n the s u p e r c o n d u c t i n g c a s e . There i s a second l e n g t h s c a l e i n the problem, namely the coherence l e n g t h (21 ) where we have i g n o r e d the a n i s o t r o p y of M. The coherence l e n g t h measures t h e range over which a l o c a l p e r t u r b a t i o n of e x t e n d s " . The r a t i o of A and ^ i s the s o - c a l l e d "Landau G i n z b u r g parameter" (22) I f a s u p e r c o n d u c t o r i s s a i d t o be "type I " and the a p p l i c a t i o n of a magnetic f i e l d w i l l lower t h e normal-s u p e r c o n d u c t i n g t r a n s i t i o n t e m p e r a t u r e and make i t f i r s t o r d e r ^ . I f k > / £ the m a t e r i a l i s a "Type I I " s u p e r c o n d u c t o r and the magnetic f i e l d f i r s t produces an i n t e r m e d i a t e "Shubnikov" phase where normal and s u p e r c o n d u c t i n g a r e a s c o e x i s t 17 b e f o r e the c o n d u c t i v i t y i s d r i v e n c o m p l e t e l y normal at h i g h e r f i e l d s . Smectic l i q u i d c r y s t a l s w i t h second o r d e r n e m a t i c - s m e c t i c A t r a n s i t i o n s appear t o be analogous t o type I s u p e r c o n d u c t o r s ^ . The d e p r e s s i o n of T N f l w i t h the a p p l i c a t i o n of t w i s t p r e d i c t e d by de Gennes'^ was r e c e n t l y o b s e r v e d e x p e r i m e n t a l l y ^ . With any a n a l o g y , t h e r e i s a p o i n t where the s i m i l a r i t y ends. S e v e r a l f e a t u r e s of the s m e c t i c a r e q u i t e u n l i k e the s u p e r c o n d u c t o r and i t i s t h e s e f e a t u r e s which make the s m e c t i c phase i n t e r e s t i n g . Because of the p r e s e n c e of the s p l a y term i n e q u a t i o n (19) the t h e r m a l f l u c t u a t i o n s of u(r_) d e s t r o y the l o n g range o r d e r of the s m e c t i c l a y e r s . One f i n d s t h a t the d i s p l a c e m e n t u ( j r ) d i v e r g e s l o g a r i t h m i c a l l y (23) where L i s the sample d i m e n s i o n > . S c a t t e r i n g X-rays from t h i s " q u a s i - l o n g range o r d e r " produces s h a r p power-law s i n g u l a r i t i e s i n the i n t e n s i t y r a t h e r than the ^ - f u n c t i o n l i k e Bragg s p o t s of t r u e l o n g range o r d e r . These have been observed 9 by s e n s i t i v e X-ray e x p e r i m e n t s T h i s d i v e r g e n c e i s a consequence of the l i n e a r e l a s t i c t h e o r y . To u n d e r s t a n d q u a s i -l o n g range o r d e r one must t u r n t o an anharmonic t h e o r y . T h i s 16 has been c a r r i e d out by G r i n s t e i n and P e l c o v i t s Another p o i n t of d i s s i m i l a r i t y , a g a i n due t o the s p l a y term, i s t h a t n does not have the gauge s y m m e t r y ^ of A . T h i s makes the t h e o r i e s of the nematic t o s m e c t i c A phase t r a n s i t i o n 18 more c o m p l i c a t e d than t h o s e of the s u p e r c o n d u c t i n g t r a n s i t i o n . 1.4 The Nematic - Smectic A Phase T r a n s i t i o n In t h i s s e c t i o n we c o n s i d e r the e f f e c t of p r e t r a n s i t i o n a l s m e c t i c f l u c t u a t i o n s on the n e m a t i c . In the nematic phase the e x p e c t a t i o n v a l u e of the s m e c t i c A o r d e r parameter i s z e r o but thermodynamic f l u c t u a t i o n s of s m e c t i c - l i k e order can have a l a r g e e f f e c t on the b u l k nematic. Very near a second o r d e r nematic s m e c t i c A t r a n s i t i o n the nematic has " i s l a n d s " of l o c a l s m e c t i c A - l i k e o r d e r w i t h d i m e n s i o n s g i v e n by a c o r r e l a t i o n l e n g t h . Roughly s p e a k i n g t h e s e i s l a n d s s t r o n g l y r e s i s t t w i s t and bend d e f o r m a t i o n s l i k e a b ulk s m e c t i c . As a r e s u l t the o v e r a l l t w i s t and bend e l a s t i c c o n s t a n t s of the nematic are i n c r e a s e d . The de Gennes model p r o v i d e s us w i t h an analogous p r o c e s s i n a s u p e r c o n d u c t o r , the p r e t r a n s i t i o n a l i n c r e a s e i n d i a m a g n e t i c i Q s u s c e p t i b i l i t y . The r e s u l t of a c a l c u l a t i o n by Schmid Q t r a n s l a t e d i n t o l i q u i d c r y s t a l l i n e terms g i v e s (24) K2/2.— "t" 2 M T T ll 19 2 where ^ and a r e the c o r r e l a t i o n l e n g t h s p a r a l l e l and A O p e r p e n d i c u l a r t o n and denotes a background nematic \°[ c o n t r i b u t i o n . From the g e n e r a l t h e o r y of c r i t i c a l phenomena we expect the c o r r e l a t i o n l e n g t h s t o d i v e r g e a c c o r d i n g t o a power law ( 2 6 ) where the reduced temperature t i s g i v e n by ( 2 7 ) ± = Jl | F u r t h e r m o r e , a c c o r d i n g t o the s t a n d a r d dogma of r e n o r m a l i z a t i o n group t h e o r i e s ^ ) ^ we expect t h i s phase t r a n s i t i o n t o be i n the same " u n i v e r s a l i t y c l a s s " as the three, d i m e n s i o n a l XY s p i n model s i n c e t h i s i s a t h r e e d i m e n s i o n a l c r y s t a l whose o r d e r parameter i s the complex number $ . Thus the p r e d i c t e d c o r r e l a t i o n l e n g t h exponents a r e 20 C 2 8 ) > ) ) = O.(o7 A number of e x p e r i m e n t s have measured t h e s e exponents ^^~3^by a v a r i e t y of t e c h n i q u e s and the concensus i s now t h a t the nematic s m e c t i c A t r a n s i t i o n f a i l s t o f i t the s t a n d a r d p i c t u r e o u t l i n e d above. T h i s f a i l u r e has s t i m u l a t e d a l a r g e t h e o r e t i c a l and e x p e r i m e n t a l e f f o r t which i s s t i l l o n going. We d e f e r u n t i l t h e l a s t c h a p t e r a r e v i e w of the r e l e v a n t e x p e r i m e n t a l r e s u l t s of o t h e r a u t h o r s . The g e n e r a l d i s a g r e e m e n t s a r e two. F i r s t , the p a r a l l e l and p e r p e n d i c u l a r c o r r e l a t i o n l e n g t h exponents a r e not found t o be e q u a l . T h i s i s i n c o m p r e h e n s i b l e i n any model which obeys " i s o t r o p i c s c a l i n g " . A r e n o r m a l i z a t i o n group t h e o r y w i t h " a n i s o t r o p i c s c a l i n g " was d e v e l o p e d by Lubensky '"^  t o account f o r t h i s r e s u l t . I t p r e d i c t s (29) , „ - A. y X 2.0-m The observed exponent r a t i o i s s m a l l e r than t h i s . The second major disagreement i s t h a t the exponents *^)// and a r e not found t o be u n i v e r s a l f o r d i f f e r e n t m a t e r i a l s ' ^ . T h i s has l e d t o the s u g g e s t i o n t h a t these m a t e r i a l s , which have second o r d e r • n e m a t i c - s m e c t i c A t r a n s i t i o n s , l i e c l o s e t o a t r i c r i t i c a l p o i n t where the t r a n s i t i o n becomes f i r s t o r d e r . Near such a p o i n t the c r i t i c a l exponents would be expected t o e x h i b i t " c r o s s o v e r " t o t r i c r i t i c a l v a l u e s . 21 I t i s the o b j e c t i v e of t h i s experiment t o measure the nematic e l a s t i c c o n s t a n t s K J J and throughout the nematic range and p a r t i c u l a r l y near the nematic t o s m e c t i c A t r a n s i t i o n w i t h s u f f i c i e n t a c c u r a c y t o d e t e r m i n e the c r i t i c a l exponent " ^ j . 22 I I . THEORY OF THE FREEDERICKSZ TRANSITION I t i s a c a p i t a l m i s t a k e t o t h e o r i z e b e f o r e one has d a t a . S i r A r t h u r Conan Doyle S c a n d a l i n Bohemia 2.1 The D i r e c t o r In t h i s c h a p t e r we w i l l d e s c r i b e a p a r t i c u l a r e l e c t r i c f i e l d i n d u c e d d e f o r m a t i o n of the nematic and o u t l i n e how i t can be used t o determine two of the e l a s t i c c o n s t a n t s . T h i s d e f o r m a t i o n i s one of s e v e r a l s i m i l a r e l e c t r i c and magnetic e f f e c t s known c o l l e c t i v e l y as " F r e e d e r i c k s z t r a n s i t i o n s " a f t e r t h e i r d i s c o v e r e r " ^ who f i r s t o b s e r v e d them i n 1927. They a r e c a l l e d " t r a n s i t i o n s " because, as we s h a l l see, the onset of the d e f o r m a t i o n i s f o r m a l l y a f i e l d - i n d u c e d second o r d e r phase t r a n s i t i o n . D i s c u s s i o n s of the F r e e d e r i c k s z 2.-5" t r a n s i t i o n can be found i n the g e n e r a l r e f e r e n c e s and thorough r e v i e w papers have been w r i t t e n by D e u l i n g ^ and C h i g r i n o v C o n s i d e r the e x p e r i m e n t a l s e t up shown i n F i g . 4. A nematic sample i s e n c l o s e d between the p l a t e s of a p a r a l l e l p l a t e c a p a c i t o r . We assume t h a t the two p l a t e s have been s u r f a c e t r e a t e d so t h a t the d i r e c t o r i s r i g i d l y anchored i n the p l a n e of the p l a t e p a r a l l e l t o the x a x i s . The c a p a c i t o r i s formed by the p l a n e s z=0 and z=L. I f the a p p l i e d v o l t a g e i s z e r o the s u r f a c e t r e a t m e n t w i l l s u f f i c e t o produce a d e f e c t - f r e e 23 Figure 4 - Schematic diagram tr a n s i t i o n of the Freedericksz a b c L z v<v c v ~ V c v » v c nematic c r y s t a l with n p a r a l l e l to x everywhere. Now consider what happens when the voltage across the capacitor i s slowly increased, s t a r t i n g from zero. I n i t i a l l y nothing happens. The equilibrium director configuration for small voltages is uniform (Fig. 4a). Then at a " c r i t i c a l voltage" Vc the director begins to deform (Fig. 4b). The deformation increases from zero u n t i l at voltages much larger than Vc the director i s nearly aligned with E except in thin regions near the walls (Fig. 4c). The. existence of a c r i t i c a l voltage below which there i s no deformation can be explained q u a l i t a t i v e l y as follows. A deformed region with length, scale j£ has an e l a s t i c energy density of order K/£ where K i s the magnitude of the e l a s t i c constants ( r e c a l l that K has units of force). If t h i s deformation i s just balanced by the e l e c t r i c energy of the same A "2. region, the e l e c t r i c energy density i s of order €o£»6 E and 24 we f i n d a c h a r a c t e r i s t i c l e n g t h S e t t i n g j£=L, the c e l l t h i c k n e s s and u s i n g V =E L, we f i n d a c h a r a c t e r i s t i c v o l t a g e f o r the b a l a n c e of e l e c t r i c and e l a s t i c energy (31 ) In due c o u r s e we w i l l see t h a t the a c t u a l c r i t i c a l v o l t a g e i s g i v e n by (32) For v o l t a g e s l e s s t h a n V c the e l a s t i c energy dominates, above V c the e l e c t r i c energy i s dominant. We now c o n s i d e r i n d e t a i l the f u l l s o l u t i o n f o r the e q u i l i b r i u m d e f o r m a t i o n f o r v o l t a g e s above Ve . The s o l u t i o n of t h i s problem i s due t o D e u l i n g * ^ . The r e l e v a n t e q u a t i o n s a r e the f r e e energy per u n i t volume, e q u a t i o n s (4) and (5) (33) - f = i k u ( 7 . f i f * ^ K i . ( A ^ « n ) * + l K B ( A « V l f f i ) * - i € o B ' E w i t h M a x w e l l ' s e q u a t i o n s i n the medium 25 ( 3 4 ) VD-O , VXE = 0 ( we assume the c r y s t a l i s n o n c o n d u c t i n g ) and the c o n s t i t u e n t e q u a t i o n (35) , , A X A s u b j e c t t o the r i g i d boundary c o n d i t i o n s on the w a l l s . I t i s c l e a r from symmetry t h a t none of the v e c t o r q u a n t i t i e s w i l l have y components. The d i r e c t o r f i e l d w i l l be g i v e n by (36) where i s the a n g l e between n* and the x a x i s . Symmetry d i c t a t e s t h a t <J> and a l l o t h e r s c a l a r q u a n t i t i e s can be f u n c t i o n s of z o n l y . The f r e e energy per u n i t a r e a i s then (37) F Note t h a t t h i s d e f o r m a t i o n i s independent of K.2Z • C o n s i d e r the e l e c t r i c term f i r s t . From ^ X E =0 we deduce t h a t E i s the g r a d i e n t of a p o t e n t i a l w h ich, by symmetry can o n l y depend on z. T h e r e f o r e E =(0, 0, E^ ). S i m i l a r l y from ^ • D =0 we deduce t h a t the z component of D i s independent of 26 z. From e q u a t i o n (35) i t i s (38) = € 0 E 2 ( € n W<t> + € J L C O S 1 4 0 We now i n t r o d u c e the v o l t a g e V ( 3 9 ) V=(MZ = ^ f ^ s i i ^ + e i c p s * * ) " ' ^ thus we f i n d t h a t j (40) Note t h a t , w h i l e i s independent of z e x p l i c i t l y i t i s s t i l l a " f u n c t i o n a l " of <^(z). The e l e c t r i c term of e q u a t i o n (37) can now be i n t e g r a t e d as i r (41 ) 2. Jo The f r e e energy as a f u n c t i o n a l of <|>(z) u s i n g e q u a t i o n s (37) and (40) i s i F = J. I (KuCos^+KaSin^cj)) dz H e r e a f t e r we use primes t o denote d /dz. 27 Our t a s k i s t o f i n d <|>(z) which m i n i m i z e s F. To do t h i s we make $ F (the f i r s t v a r i a t i o n of F) z e r o s u b j e c t t o the c o n d i t i o n s ( 4 3 ) q S ( o ) = o <\>lLfr) = 4\v> where denotes the maximum a n g l e a t the c e n t r e of the c e l l . We f i n d f o r SF (44) £ o f e . c o s 1 * + £ v S I M 1 * ) * J We then i n t e g r a t e the $tf term by p a r t s and a r e l e f t w i t h an i n t e g r a n d where a f a c t o r . S i n c e was a r b i t r a r y ( the s t a n d a r d argument goes) the o t h e r f a c t o r i n the i n t e g r a n d must be z e r o . T h i s g i v e s the n o n l i n e a r d i f f e r e n t i a l e q u a t i o n : (KwCos^ + Ki j S i M 1 * } 4>x/ i / V 2 . (45) ( - K ' I , ) s i 4 cos4> (4>)' M u l t i p l i c a t i o n of t h i s e q u a t i o n by 2 and u s i n g 2 <p<p = ( ( < p ) ) r e s u l t s i n an e q u a t i o n w h i c h , when i n t e g r a t e d over z i s 28 (46) -+ 2 . (K 3 3 • K ' n ^ sine}) cos The f i r s t and t h i r d terms can be i n t e g r a t e d by p a r t s . The second term then c a n c e l s out l e a v i n g = C + €<.(£« s m l 4 + 6 x c o s 2 * ) where C i s the c o n s t a n t of i n t e g r a t i o n . We use the boundary c o n d i t i o n s from e q u a t i o n (43) w i t h the a d d i t i o n a l c o n d i t i o n t h a t <|>(L/2)=0 ( which can be deduced by symmetry ) t o deter m i n e C. (48) C — The f i n a l s o l u t i o n w i l l be i n p a r a m e t r i c form w i t h CD. as the parameter. P u t t i n g C i n t o e q u a t i o n (47) and i n t r o d u c i n g reduced q u a n t i t i e s U9» * - T - I K~ — I we f o r g e t , a f t e r some rearrangement, a s i m p l e d i f f e r e n t i a l e q u a t i o n 4> 29 (50) 1 ' = D p y " 1 ( s i v f J r - s ' w H Note that t h i s equation admits the solution (|)(z)=0 for a l l voltages. We can now relate the voltage to the parameter. Rewriting equation (39) s l i g h t l y One can now use equation (50) for d^>/dz to change the variable of integration to<^ . drops out giving Using equation (52) we can determine 4^  for any given values of the voltage, the d i e l e c t r i c constants and e l a s t i c constants. Returning to equation (50) we find that the variables are separated. Integration of the inverse of equation (50) gives (53, 2 = I f i g ra^fc^^'^XH^'^)? A special case of t h i s equation i s z=L/2 and <|)=(|> . This 30 d e t e r m i n e s • U s i n g t h i s e x p r e s s i o n f o r D-^  i n e q u a t i o n (53). g i v e s The parameter (h appears i n both the i n t e g r a n d and the l i m i t s of t h e s e two i n t e g r a l s . I t i s a u s e f u l s i m p l i f i c a t i o n t o d e f i n e a new parameter 1^  and change v a r i a b l e s t o ty where ( 5 6 ) . i h Then e q u a t i o n (52) becomes • • ' • V - <J§ ^ C v ^ t ^ j ^ from which we can see t h a t i n the l i m i t 1^=0 ( the l i m i t of no d e f o r m a t i o n ) we have 31 (58) V Q i s the c r i t i c a l v o l t a g e a t which the d e f o r m a t i o n b e g i n s , as we foreshadowed i n e q u a t i o n ( 3 2 ) . Below V C the z e r o s o l u t i o n <^>(z)30 i s the g l o b a l minimum of F and t h e r e i s no d e f o r m a t i o n . I t w i l l be c o n v i e n i e n t t o d e f i n e a reduced v o l t a g e v by These i n t e g r a l s a r e r e l a t e d t o complete e l l i p t i c i n t e g r a l s of the f i r s t k i n d , a f a c t we make use of l a t e r . We may d e f i n e an " i n c o m p l e t e " v e r s i o n of ^  as $ t ( *f) which i s the same a s l B e x c e p t t h a t the upper l i m i t of i n t e g r a t i o n i s V i n s t e a d of TT/2. In t h i s n o t a t i o n the s o l u t i o n f o r t h e e q u i l i b r i u m d e f o r m a t i o n i s 32 v = ZsT^/A -I (62) TT (63) The p r o c e d u r e t o d e s c r i b e a p a r t i c u l a r d e f o r m a t i o n i s the f o l l o w i n g . Given the m a t e r i a l c o n s t a n t s *K and 2^, s o l v e e q u a t i o n (62) f o r the parameter \Tj f o r a g i v e n reduced v o l t a g e v. Then e q u a t i o n (63) d e f i n e s the f u n c t i o n (|)(z) i m p l i c i t l y . In t h i s n o t a t i o n dz/d<^ i s g i v e n by <64) 4i- L gi+W^X^^mV)?'^ We w i l l make use of t h i s i n l a t e r s e c t i o n s . F i g . 5 shows how v a r i e s w i t h z f o r the c h o i c e of parameters L=2, $ = 1 . 4 , 7 ( = 1 and v a r i o u s reduced v o l t a g e s . Note t h a t a t h i g h e r v o l t a g e s approaches 7 T / 2 except i n t h i n "boundary l a y e r s " near the w a l l s . The t h i c k n e s s of these l a y e r s i s the or d e r of the l e n g t h s c a l e g i v e n i n e q u a t i o n (30) and i s i n v e r s e l y p r o p o r t i o n a l t o the v o l t a g e a p p l i e d . 33 F i g u r e 5 - Graph of d i r e c t o r a n g l e v e r s u s p o s i t i o n i n the c e l l , f o r 5 = 1 . 4 K = 1 . and v a r i o u s reduced v o l t a g e s (px>u) d> 9|6UD JOpSJIQ 34 2.2 The C a p a c i t a n c e In the l a s t s e c t i o n we developed a t h e o r y f o r the d e f o r m a t i o n of a d i r e c t o r f i e l d s u b j e c t t o c e r t a i n boundary c o n d i t i o n s by an e l e c t r i c f i e l d . U n f o r t u n a t e l y we have no. e x p e r i m e n t a l probe which can measure the o r i e n t a t i o n of the d i r e c t o r a t a p o i n t i n s i d e the sample. We must c o n s i d e r the i n t e g r a t e d e f f e c t of the d e f o r m a t i o n on the e x t e r n a l l y measurable p r o p e r t i e s of the c e l l . The two p r o p e r t i e s we measured were the c a p a c i t a n c e and the b i r e f r i n g e n c e ( o p t i c a l phase d i f f e r e n c e ) . These two i n t e g r a t e d p r o p e r t i e s , a l o n g w i t h t h e r m a l and e l e c t r i c a l c o n d u c t i v i t y , a r e d i s c u s s e d by the D e u l i n g review-* • . The c a p a c i t a n c e of the c e l l i s the s i m p l e s t of the two so we t r e a t i t f i r s t . R e c o n s i d e r e q u a t i o n (38) c e l l was composed of i n f i n i t e s i m a l s e r i e s c a p a c i t o r s of c a p a c i t a n c e dC, t h i c k n e s s dz and a r e a A, then 4> + £ j C O S 2 ( j ) ) (65) E v i d e n t l y £ ( z ) i s an " e f f e c t i v e " d i e l e c t r i c c o n s t a n t . I f the (66) The t o t a l c a p a c i t a n c e i s then C where 35 C d C J A € o € C z > A e 0 e J . \ ( » - + ^ m 7 - 4 ) ) u s i n g e q u a t i o n (64) f o r dz/d<|> t o change the v a r i a b l e of i n t e g r a t i o n t o CD ./ 72 < 6 8 > C " Jf((i«MX\ + « * ^ X % Note t h a t a t z e r o v o l t a g e the c a p a c i t a n c e i s (69) C D = so i n the n o t a t i o n of the l a s t s e c t i o n B «,o, C = A where we have i n t r o d u c e d the reduced c a p a c i t a n c e c by F i g . 6 shows a p l o t of. reduced c a p a c i t a n c e vs reduced v o l t a g e f o r 7 f = 1 . 4 and v a r i o u s v a l u e s of * K . 36 F i g u r e 6 - Graph of reduced c a p a c i t a n c e v e r s u s reduced v o l t a g e , f o r #=1.4 and v a r i o u s K. v a l u e s 90UD|JODCIDO paonpay 3 7 Note t h a t at l a r g e v a l u e s an i n f l e c t i o n p o i n t appears near the o r i g i n . T h i s was f i r s t n o t i c e d by Schad e_t a _ l ^ . To use t h i s t h e o r y t o e x t r a c t and from C-V data the procedure i s as f o l l o w s . Suppose , which c o n t a i n s €|| and i s known, perhaps from a s e p a r a t e e x p e r i m e n t . From measured c a p a c i t a n c e s and v o l t a g e s one must e x t r a c t V C and C Q from d a t a near the ' k i n k ' i n the C-V c u r v e which marks the F r e e d e r i c k s z t r a n s i t i o n . Then one produces reduced d a t a a c c o r d i n g t o e q u a t i o n s (59) and ( 7 1 ) . V C , CJL and % combine t o g i v e K|| v i a e q u a t i o n ( 5 8 ) . Note t h a t £|_ can be o b t a i n e d from C 0 and the a r e a t o t h i c k n e s s r a t i o A/L. Then a n o n l i n e a r f i t of reduced d a t a t o e q u a t i o n s ( 6 2 ) and ( 7 0 ) w i t h 7C as a f i t t e d p a r a m e t e r ^ f i x e d , g i v e s one the b e s t - f i t v a l u e of "X . The bend c o n s t a n t i s then ( 7 2 ) KM (I +*) We r e f e r t o t h i s p r o c e d u r e as the f u l l n o n l i n e a r f i t O b v i o u s l y the i n f l e c t i o n p o i n t which appears at h i g h v a l u e s of *X makes d e t e r m i n a t i o n of ^ d i f f i c u l t , s i n c e i t i s h a r d t o t e l l when the d e f o r m a t i o n has s t a r t e d . An e r r o r i n V C a f f e c t s t h e r e s t of the d a t a v i a the r e d u c t i o n f o r m u l a . A l s o note t h a t the shape of the C-V c u r v e depends as much on ^ as on*X . Thus a c c u r a t e v a l u e s of ^ a r e needed and ^ i s a p o t e n t i a l source of s y s t e m a t i c e r r o r . M u l t i p a r a m e t e r n o n l i n e a r f i t t i n g t o b o t h and % i s d i f f i c u l t because the i n t e g r a l s i n q u e s t i o n a r e p o o r l y 38 c o n d i t i o n e d . F a l s e and/or s h a l l o w minima abound. These p i t f a l l s have been e x p l o r e d by Maze . Because of the c o m p l e x i t y of n o n l i n e a r f i t t i n g and the i n t e g r a l s i n v o l v e d , many a u t h o r s have r e s o r t e d t o the l i m i t i n g 31 MO c a s e s of h i g h and low v o l t a g e ' . The low v o l t a g e l i m i t , v a l i d ' j u s t above V c i s the case 0. I t i s s t r a i g h t f o r w a r d t o expand the i n t e g r a n d s f o r s m a l l and i n t e g r a t e . For t h e v o l t a g e , e q u a t i o n ( 6 2 ) , we get t o lowest o r d e r i n Y)i (73) v ^ 1 ( K + y + i ) y S i m i l a r l y f o r e q u a t i o n (70) ( 7 4 ) C whence we f i n d t h a t the reduced C-V da t a c u r v e l i n e a r i z e d near v=0 i s ( 7 6 ) c = v = S L P C v W i t h t h i s we can e x t r a c t a low f i e l d kappa T h i s seems l i k e a s i m p l e way t o get However the range of v a l i d i t y of t h e l i n e a r i z a t i o n i s not easy t o d e c i d e . T h i s 39 approach may g i v e erroneous r e s u l t s i f t h e r e i s an i n f l e c t i o n p o i n t i n the cu r v e near V c . The h i g h f i e l d l i m i t i s more d i f f i c u l t t o a r r i v e a t because the i n t e g r a l s d i v e r g e i n the 1^=1 l i m i t . T h i s problem can be c i r c u m v e n t e d i n the f o l l o w i n g way. We can r e a r r a n g e the i n t e g r a n d of J B t o get T D _ / x / \ /A v l !> U - f K l p S W H r ; £ (77) Now u s i n g (62) A = (78) T V 2-\£ >/!+*!?' We can e x p r e s s (70) as (79) Co T"V" J jfl-9*«t» i n which we can s a f e l y put ^ = 1. So t h a t (80) C = C o o + S HFC('/V) ( 8 D <: 'HFC Our n o t a t i o n d i f f e r s somewhat from t h a t of U c h i d a . One can use a l i n e a r f i t t o h i g h f i e l d d a t a t o get S^.^ and C ^ . From 40 CQ and CgQ we can e x t r a c t 0 • S o l v i n g (81) n u m e r i c a l l y w i t h t h i s ^ , g i v e s ' K and hence . We c a l l the reduced e l a s t i c c o n s t a n t a r r i v e d a t i n t h i s w a y * K y p £ . To see the l i m i t a t i o n s of t h i s method c o n s i d e r how weakly the s l o p e S ^ p t depends o n " K . T h i s i s made c l e a r i n a p l o t of S R F C / \ C Q f o r v a r i o u s £f shown i n F i g . 7. Thus a s m a l l e r r o r i n S H p t t r a n s l a t e s t o a l a r g e e r r o r i n XHFC* A l s o , note t h a t V c and C 0 a r e s t i l l needed t o use t h i s method so an i n f l e c t i o n p o i n t may s t i l l be a problem. However t h i s e x t r a p o l a t i o n i s v e r y u s e f u l t o get ^  . We have a n a l y z e d our c a p a c i t a n c e d a t a a l l t h r e e ways, low f i e l d , h i g h f i e l d and f u l l n o n l i n e a r f i t u s i n g $ from the h i g h f i e l d i n t e r c e p t . 41 F i g u r e 7 - Graph showing the weak dependence of h i g h f i e l d s l o p e on the h i g h f i e l d reduced e l a s t i c c o n s t a n t 3A V s *do|S 42 2.3 The O p t i c a l Phase D i f f e r e n c e L i g h t p o l a r i z e d p a r a l l e l t o the d i r e c t o r samples m a t e r i a l w i t h index of r e f r a c t i o n njj . P e r p e n d i c u l a r t o t h i s the index i s n ^ . We d e f i n e the b i r e f i n g e n c e A n by A n = n (| -nj_ . One would c a l l n ^ the " o r d i n a r y i n d e x " n e , and njj the " e x t r a o r d i n a r y i n d e x " n g i n more customary n o t a t i o n . In 8CB An>0 . C o n s i d e r a c a p a c i t o r w i t h t r a n s p a r e n t e l e c t r o d e s , l i k e the one shown i n F i g . 4. L i g h t p o l a r i z e d a l o n g the y d i r e c t i o n e n c o u n t e r s index n ^ whatever the v o l t a g e because ri i s always p e r p e n d i c u l a r t o the y a x i s . L i g h t p o l a r i z e d a l o n g x samples a v a r i e t y of i n d i c e s t h r o u g h the c e l l . I f the two beams e n t e r the c e l l i n phase they emerge w i t h an o p t i c a l phase d i f f e r e n c e g i v e n by (82) where ^ i s the wavelength and n ( z ) i s the e f f e c t i v e index sampled by the beam p o l a r i z e d a l o n g x. Note t h a t our n o t a t i o n i s d i f f e r e n t than t h a t of D e u l i n g . At z e r o v o l t a g e the phase d i f f e r e n c e i s (83) d o = ' Z i r L A n and t h e c a p a c i t o r a c t s l i k e a wavepl a t e w i t h i t s " f a s t a x i s " a l o n g y. C l e a r l y a t i n f i n i t e v o l t a g e b oth beams sample n ^ and d =0. In p r a c t i c e we use one beam p l a n e p o l a r i z e d a t 45° t o 43 the x a x i s which has e q u a l in-phase components a l o n g x and y. From el e m e n t a r y c r y s t a l o p t i c s , the e f f e c t i v e index n ( z ) i s g i v e n by (84) As b e f o r e we change the v a r i a b l e of i n t e r g r a t i o n i n (82) u s i n g e q u a t i o n 64 f o r dz/d<j> . A f t e r some a l g e b r a we a r r i v e a t ( 8 5 ) l _ < J — n i l / i _ ( L I _ A - r l " / | — L.\ where the new i n t e g r a l i s < 8 6 ) " ~ J o ( ( l + ^ ^ ' r X l - ^ ' M l i r T J w i t h the reduced index of r e f r a c t i o n as a new parameter , 8 7 ) - 0 nL-.\ We d e f i n e a q u a n t i t y £ as the reduced phase (88) £ = I d o where we have chosen the s i g n s t o make & p o s i t i v e . Note t h a t n \\ / A n can be r e l a t e d t o "9 by 44 A h r f ^ v P -I and t h a t a t i n f i n i t e v o l t a g e & approaches 1 because d->0. Graphs of S v e r s u s v a r e shown i n F i g . 8. They have the same g e n e r a l shape as the c-v graphs shown i n F i g . 6 but they have a somewhat d i f f e r e n t dependence on li . For l a r g e *K the problem caused by i n f l e c t i o n p o i n t problem i s a l s o p r e s e n t i n the o p t i c a l c a s e ^ . A l s o , a n o n l i n e a r f i t t o S vs v i n v o l v e s knowing an a d d i t i o n a l m a t e r i a l parameter not needed i n the c a p a c i t a n c e c a s e . Three i n t e g r a l s ( A , B and now (C ) must be c a l c u l a t e d i n s t e a d of two. When the two beams emerge from the c r y s t a l the beam p o l a r i z e d a l o n g the x - a x i s has an a d d i t i o n a l phase d over the phase of the beam p o l a r i z e d a l o n g the y - a x i s . Assuming no l o s s e s , the s u p e r p o s i t i o n of the two beams i s an e l l i p t i c a l l y p o l a r i z e d beam w i t h the same i n t e n s i t y as the i n g o i n g l i n e a r l y p o l a r i z e d beam. I t would be p o s s i b l e t o measure d d i r e c t l y u s i n g , f o r example, a v a r i a b l e compensator. However, t o s i m p l i f y the da t a a q u i s i t i o n i t was thought best t o measure an i n t e n s i t y . Thus we p l a c e d the l i q u i d c r y s t a l between c r o s s e d p o l a r i z e r s and measured the t r a n s m i t t e d i n t e n s i t y . The e l l i p t i c a l l y p o l a r i z e d l i g h t emerging from the l i q u i d c r y s t a l passed t hrough a second p o l a r i z e r ( the " a n a l y s e r " ) which i s a l i g n e d a t 90^ t o t h e f i r s t . In t h i s arrangement the measured i n t e n s i t y i s 45 F i g u r e 8 - Graph of reduced phase v e r s u s reduced f o r "X = 1, >p=0.2, and v a r i o u s v a l u e s of ^ 9SDi|d paonpay 4 6 (90) where I Q i s the i n g o i n g i n t e n s i t y . As d changes from d ^ t o z e r o w i t h i n c r e a s i n g v o l t a g e the t r a n s m i t t e d i n t e n s i t y o s c i l l a t e s f o r m i n g a s e r i e s of i n t e r f e r e n c e f r i n g e s . The number of t h e s e f r i n g e s N i s the number of wavelengths "A i n the z e r o v o l t a g e o p t i c a l p a th l e n g t h d i f f e r e n c e < " » N = ^ = k * H to e x t r a c t d we must "d e c o n v o l v e " the f r i n g e s by (92) Some c a r e must be taken w i t h the p r i n c i p a l v a l u e s of the i n v e r s e c o s i n e t o get the c o r r e c t v a r i a t i o n of d. Once the f r i n g e s have been de c o n v o l v e d t o get the phase d i f f e r e n c e d one can e x t r a c t d ^ and . Then the reduced phase d a t a can be f i t t o the f u l l n o n l i n e a r t h e o r y g i v e n e a r l i e r r Ml f o r £ and v . C o n s i d e r a t i o n s s i m i l a r t o the c a p a c i t a n c e case a p p l y t o t h i s f u l l f i t . Once a g a i n we can c o n s i d e r t h e h i g h and low f i e l d l i m i t s . E x panding e q u a t i o n (86) f o r s m a l l we f i n d 47 0 3 ) $ ^ ^ n u An i\ < U s i n g ( 7 3 ) the low f i e l d l i m i t i s <94» $ = 4^ ^ s V = S ^ V Ah (K + t f + 0 The low f i e l d s l o p e i s Sypo > s o t h a t the low f i e l d reduced e l a s t i c c o n s t a n t i n t h i s case i s - *~-(*4J-<M,>. The h i g h f i e l d e x p r e s s i o n i s ha r d e r t o a r r i v e a t than i n the c a p a c i t a n c e case . One f i n d s t h a t f o r * ? - > 1 d x / l ± 2 2 L * ' d x U F O T h i s i s q u i t e d i f f e r e n t from the h i g h f i e l d c a p a c i t a n c e r e s u l t which gave C l i n e a r i n 1/V. In p r i n c i p l e one can get *K from b o t h the s l o p e and the i n t e r c e p t . However the i n t e r c e p t l\\^o i s a c t u a l l y the i n f i n i t e phase d i f f e r e n c e l i m i t of e q u a t i o n (96) which o c c u r s a t s m a l l v o l t a g e s f a r o u t s i d e the range of v a l i d i t y 48 of Ip^ 1 . To determine the s m a l l v a l u e I ^ F O r e q u i r e s a r a t h e r l o n g e x t r a p o l a t i o n from the d a t a , and so i t i s s u b j e c t t o c o n s i d e r a b l e e r r o r . IHR ) a c t u a l l y c o n t a i n s the same i n t e g r a l as the c a p a c i t a n c e h i g h f i e l d s l o p e , e q u a t i o n ( 8 1 ) , and so s u f f e r s from the same weak dependence. To use S^po t o get r e q u i r e s V c , d G ,V and Y . Once a g a i n the dependence on " K i s r a t h e r weak. An i m p o r t a n t d i f f e r e n c e between the o p t i c a l case and the c a p a c i t a n c e case i s t h a t t h e r e i s not enough i n f o r m a t i o n i n the o p t i c a l d a t a t o determine a l l the pa r a m e t e r s . There i s no i n f o r m a t i o n on £JL or . However we have 2fT L &n/"\ which a l l o w s us t o deduce g i v e n L / 7 i . The f i t s t r a t e g y w i l l be f u l l y e x p l a i n e d i n c h a p t e r 4. We have a n a l y z e d our o p t i c a l phase d a t a i n both the h i g h and low f i e l d r e g i o n s , and by the f u l l n o n l i n e a r f i t . 2.4 Problems With The Theory In t h i s s e c t i o n we c r i t i c a l l y examine the assumptions and l i m i t a t i o n s of the t h e o r y . We w i l l d e a l w i t h a p u r e l y c o m p u t a t i o n a l problem f i r s t . The t h r e e i n t e g r a l s we have d e r i v e d , A B and £ , must a l l be computed a c c u r a t e l y and e f f i c i e n t l y many tim e s t o c a r r y out the n o n l i n e a r f i t s . They a r e a l l of the form 49 (98) 6 * r w i t h v a r i o u s w e l l - b e h a v e d f u n c t i o n s f . For v o l t a g e s much g r e a t e r than V c one f i n d s t h a t Yj becomes v e r y n e a r l y e q u a l t o 1 because 4»n i s c l o s e t o If/2 a n d ^ =s "^2,4\l%* S i n c e f ( 1 Y / 2 ) i s n o n z e r o , t h i s means t h a t the l a r g e s t c o n t r i b u t i o n t o the i n t e g r a l comes from v e r y near ty = TT / 2 . T h i s makes 3E d i f f i c u l t t o compute w i t h s t a n d a r d n u m e r i c a l i n t e g r a t i o n r o u t i n e s . F u r t h e r m o r e , ^ e v e n t u a l l y becomes c o m p u t a t i o n a l l y i n d i s t i n g u i s h a b l e from 1, even u s i n g d o u b l e p r e c i s i o n a r i t h m e t i c . These n u m e r i c a l problems can be d e a l t w i t h by n o t i n g t h a t where ( ¥1) i s the < complete e l l i p t i c i n t e g r a l of the f i r s t k i n d . I t t u r n s out t h a t ^ ^ f o r Irj - > 1 d o o ) K(f7^ ^ lo^ O/d-*?)) and e f f i c i e n t a l g o r i t h i m s e x i s t f o r c a l c u l a t i n g K c 1-x) t o h i g h p r e c i s i o n . E q u a t i o n ( 1 0 1 ) s u g g e s t s a more n a t u r a l parameter i s 0( , where 50 (101) ^ =. / — For h i g h v o l t a g e s 0( r i s e s a p p r o x i m a t e l y l i n e a r l y w i t h v o l t a g e . (102) The i n t e g r a l i s now w e l l behaved and can be done by a s t a n d a r d r o u t i n e w i t h h i g h p r e c i s i o n . The p l a c e s which were n u m e r i c a l l y d i f f i c u l t w i t h V) have been r e - e x p r e s s e d w i t h OL . In p r a c t i c e f o r V = 20V^ and a c t u a l v a l u e s of X, 3f and we found C(. c o u l d be as h i g h as f i f t y . We now t u r n our a t t e n t i o n t o the l i m i t a t i o n s of the t h e o r y . The t h e o r y d e s c r i b e d i s o n l y v a l i d f o r the e q u i l i b r i u m s t a t e of the d i r e c t o r . I f we sweep the v o l t a g e a n y t h i n g but q u a s i -s t a t i c a l l y t h e r e w i l l be dynamic e f f e c t s i n v o l v i n g v i s c o s i t i e s . In p a r t i c u l a r the time c o n s t a n t f o r the d i r e c t o r f i e l d t o a c h i e v e e q u i l i b r i u m a f t e r a s t e p change i n v o l t a g e i s i n f i n i t e a t V"c . For p a r t i c u l a r l y l a r g e s t e p s one may a c t u a l l y produce f l u i d f l o w as w e l l as r e o r i e n t a t i o n ^ . In p r a c t i c e i t s u f f i c e s t o sweep v e r y s l o w l y t h r o u g h and somewhat f a s t e r a t h i g h e r v o l t a g e s . We assumed t h a t we c o u l d produce s u r f a c e s where n was c o m p l e t e l y f i x e d p a r a l l e l t o the s u r f a c e . On a r e a l s u r f a c e t h i s assumption may f a i l i n two ways, n may not be p a r a l l e l and i t may not be r i g i d . Motooka and Fukuhara * have c o n s i d e r e d 51 the t h e o r y w i t h t h e s e assumptions r e l a x e d . In the case of n o n r i g i d a l i g n m e n t , so c a l l e d "weak a n c h o r i n g " , one s t i l l f i n d s a sharp c r i t i c a l v o l t a g e below which t h e r e i s no d e f o r m a t i o n but i t s v a l u e i s s m a l l e r than i n the s t r o n g l y anchored case^"^ . In the case when n i s not p r e c i s e l y p e r p e n d i c u l a r t o E a t the w a l l s , which may be caused by a " t i l t e d " a l i g n m e n t by the •7 s u r f a c e t r e a t m e n t one f i n d s some d e f o r m a t i o n o c c u r s a t a l l v o l t a g e s w i t h a l a r g e i n c r e a s e near V c f o r the u n t i l t e d c a s e . The onset of the d e f o r m a t i o n i s no l o n g e r a b r u p t . One c o u l d m i s t a k e the i n f l e c t i o n p o i n t s ' ^ d i s c u s s e d e a r l i e r f o r t h i s ef f e c t . The d i f f i c u l t y w i t h t r y i n g t o q u a n t i t a t i v e l y account f o r th e s e e f f e c t s i s t h a t new and undetermined parameters must be i n t r o d u c e d t o d e s c r i b e the s u r f a c e . We t r i e d t o a v o i d t h i s problem by s e l e c t i n g the best s u r f a c e treatment known and assuming s t r o n g a n c h o r i n g . L e t us t u r n now t o problems a r i s i n g from e f f e c t s i n the b u l k l i q u i d . The l i q u i d has a s m a l l but f i n i t e e l e c t r i c a l c o n d u c t i v i t y due t o i m p u r i t i e s . G e n e r a l l y s p e a k i n g c o n d u c t i v i t y i s o n l y i m p o r t a n t i n DC and low f r e q u e n c y AC e x p e r i m e n t s . At s u f f i c i e n t l y h i g h f r e q u e n c y ( a kHz or so ) the l i q u i d c r y s t a l r e a c t s as i f the f i e l d had i t s RMS v a l u e ^ and c o n d u c t i v i t y e f f e c t s a r e m i n i m i z e d . An i n s i d i o u s e f f e c t f o r e x p e r i m e n t s w i t h a c c u r a t e temperature c o n t r o l i s ohmic h e a t i n g . The c o n d u c t i v i t y of our m a t e r i a l was s u f f i c i e n t l y low t h a t t h i s was i n s i g n i f i c a n t ( R c t L 1 > 20Mfl} . A more i n t e r e s t i n g space-charge e f f e c t i s due t o 52 " f l e x o e l e c t r i c i t y " . T h i s i s the l i q u i d a n a l o g of p i e z o e l e c t r i c i t y whereby a deformed nematic a q u i r e s an e l e c t r i c a l p o l a r i z a t i o n . The e f f e c t of t h i s on the shape of the d e f o r m a t i o n above the F r e e d e r i c k s z t r a n s i t i o n has been i n c l u d e d i n the t h e o r y by D e u l i n g n . However t h i s i s r e a l l y a dynamic e f f e c t i n d i s g u i s e because any space-charge w i l l , w i t h enough t i m e , decay due to c o n d u c t i o n . Furthermore we expect t h i s t o be a s m a l l e f f e c t i n 8CB s i n c e i t i s most pronounced i n pear or banana shaped m o l e c u l e s (8CB i s more l i k e a cucumber ). Next we c o n s i d e r the p o s s i b l e e f f e c t s of the f i e l d and the d e f o r m a t i o n on the nematic i t s e l f . R e c a l l t h a t our t r e a t m e n t of the d e f o r m a t i o n was p r e d i c a t e d on the assumption t h a t a | ^ 7 n | < < 1 where av*50 I t i s a s i m p l e m a t t e r t o show t h a t t h i s a s s u m p t i o n i s not v i o l a t e d u n t i l v ery l a r g e f i e l d s f o r r e a s o n a b l e v a l u e s of *K and V . (103) CL By e q u a t i o n (101) IB ^ 0( f o r l a r g e v o l t a g e s . S i m i l a r l y V\/»^v>et# Hence we must re a c h reduced v o l t a g e s of o r d e r L/a f o r a | ^ n | v * i . For t h i s experiment Lv».50yu. so L/a 10*' . The h i g h e s t reduced v o l t a g e a c h i e v e d was about 25. By t h i s argument the d e f o r m a t i o n i s not g r e a t enough t o a f f e c t the m o l e c u l a r o r d e r parameter S d i r e c t l y . In p r i n c i p l e the e l e c t r i c f i e l d can i n c r e a s e S d i r e c t l y but t h i s i s a l s o a s m a l l e f f e c t a t the f i e l d s we used. A l t h o u g h the d e f o r m a t i o n and the f i e l d a re not g r e a t enough 53 t o a f f e c t the m o l e c u l a r o r d e r parameter, they can have an i m p o r t a n t e f f e c t on the o r i e n t a t i o n a l f l u c t u a t i o n s of the d i r e c t o r f i e l d . T h i s e f f e c t i s somewhat c o m p l i c a t e d because of the presence of s m e c t i c f l u c t u a t i o n s near T / ^ f l ) which cause an i n c r e a s e d K33. T h i s a l s o has a d i r e c t o r f l u c t u a t i o n - q u e n c h i n g e f f e c t . Inasmuch as b u l k m a t e r i a l p r o p e r t i e s are averages over the f l u c t u a t i o n s of n they may be a l t e r e d i f the spectrum of f l u c t u a t i o n s i s changed. A c t u a l l y , the d i s t i n c t i o n between d i r e c t o r f l u c t u a t i o n s and nematic o r d e r parameter f l u c t u a t i o n s i s an a r t i f a c t of our continuum model. The most i n t e r e s t i n g and c o m p l i c a t e d e f f e c t not c o n t a i n e d i n our continuum t h e o r y i s the e f f e c t of on the s m e c t i c o r d e r , which i n t u r n e f f e c t s K 3 $ r which i n t u r n e f f e c t s the d i r e c t o r , down our c h a i n of models. We f i n d (,04) |vy^= z U v - ^ t s m b . T h i s i s p l o t t e d i n F i g . 9 f o r ^ = 1 V= 1 .4 and a sequence of reduced v o l t a g e s . 54 55 For l a r g e v o l t a g e s the e f f e c t i s c o n c e n t r a t e d near the w a l l s . Thus the c e n t r e of the c e l l has a h i g h e r e f f e c t i v e T ^ than r e g i o n s near the w a l l s . Madhusudana and S r i k a n t a ^  have c a r r i e d out exp e r i m e n t s on t h i s e f f e c t . I t must be p o i n t e d out t h a t t h e r e a r e i m p o r t a n t d i f f e r e n c e s between an experiment where the l i q u i d i s deformed a t t e m p e r a t u r e s f a r from T ^ , the undeformed t r a n s i t i o n t e m p e r a t u r e , and then c o o l e d through T N^ and one i n which th e s e o p e r a t i o n s a re r e v e r s e d . Having d e s c r i b e d the t h e o r e t i c a l f o u n d a t i o n of the experiment we next take up i t s r e a l i z a t i o n . 56 I I I . THE EXPERIMENT ...our p r o g r e s s i n n a t u r a l p h i l o s o p h y i s c h i e f l y r e t a r d e d by the want of p r o p e r e x p e r i m e n t s . . . Hume 3.1 The C e l l Our experiment c e n t e r e d around a sample c e l l which h e l d about s i x c u b i c m i l l i m e t e r s of 8CB l i q u i d c r y s t a l . I t was a c i r c u l a r p a r a l l e l p l a t e c a p a c i t o r w i t h t r a n s p a r e n t e l e c t r o d e s whose i n s i d e s u r f a c e s were t r e a t e d t o produce the o r i e n t a t i o n e f f e c t s d i s c u s s e d i n t h e l a s t c h a p t e r . T h i s s u r f a c e t r e a t m e n t i s d e s c r i b e d below. The sample of 8CB we used was k i n d l y p r o v i d e d by Dr. D a v i d Dunmur. I t was manufactured by BDH c h e m i c a l s and was used w i t h o u t f u r t h e r p u r i f i c a t i o n . The c e l l and i t s p a r t s a r e shown i n F i g . 10. The p l a t e s of the c a p a c i t o r were formed by t h i n c o a t i n g s of indium t i n - o x i d e ( ITO ), a t r a n s p a r e n t c o n d u c t o r , on g l a s s b l o c k s . The g l a s s b l o c k s , c o a t e d and c u t t o s i z e , were manufactured commercialy . The ITO c o a t i n g was a s t a n d a r d one used i n d i s p l a y d e v i c e s . We masked the e l e c t r o d e p a t t e r n w i t h S c o t c h tape and e t c h e d o f f the s u r r o u n d i n g ITO w i t h a s t r o n g HC1 s o l u t i o n and z i n c powder. The r e s u l t i n g p l a t e s were 0.5 i n c h a c r o s s w i t h a t a b .125 i n c h wide f o r making e l e c t r i c a l c o n t a c t , which extended t o 57 F i g u r e 10 - The c o n s t r u c t i o n of the sample c e l l 58 the edge of the g l a s s . The g l a s s b l o c k s had dime n s i o n s 1 i n c h by 1.25 i n c h and were .250 i n c h e s t h i c k . A f t e r masking and e t c h i n g , the p l a t e s were s u r f a c e t r e a t e d to produce d i r e c t o r a l i g n m e n t . T h i s was done by e v a p o r a t i n g s i l i c o n monoxide o b l i q u e l y onto the s u r f a c e " ^ . We used a f i l m t h i c k n e s s of about 400 A*. The a n g l e between the s u r f a c e and the beam of SiO was 30° . T h i s p r o c e d u r e produces a l i g n m e n t w i t h the d i r e c t o r i n the p l a n e of the s u r f a c e w i t h a d i r e c t i o n p e r p e n d i c u l a r t o the beam d i r e c t i o n . T h i s t r e a t m e n t i s thought to produce the s t r o n g e s t a n c h o r i n g w i t h the l e a s t t i l t a n g l e of any t r e a t m e n t known"^ . The two g l a s s b l o c k s were s e p a r a t e d by a t h i n mylar g a s k e t . T h i s gasket was c u t from 0.001 i n c h t h i c k M y l a r of the s o r t used w i t h overhead p r o j e c t o r s . I t had a c i r c u l a r h o l e c u t i n i t which was s l i g h t y l a r g e r than the c a p a c i t o r p l a t e s , so t h a t the p l a t e s d i d not e n c l o s e any gasket m a t e r i a l . The r e s u l t i n g t h i n c y l i n d r i c a l volume was the sample space. A narrow gap i n the gasket l e d from the i n t e r i o r of the c e l l t o the o u t s i d e . T h i s was i n c l u d e d t o a l l o w f o r the s l i g h t t h e r m a l e x p a n s i o n of the l i q u i d . The b l o c k s of g l a s s were squeezed t o g e t h e r by two h a l v e s of a copper c e l l body as shown i n F i g . 10. Ho l e s i n the c e l l body a l l o w e d a l a s e r beam t o pass through the t r a n s p a r e n t c a p a c i t o r . The g l a s s was o v e r l a p p e d so t h a t the two ITO c o n t a c t t a b s were a c c e s s i b l e . We put s m a l l b l o b s of indium metal onto th e s e t a b s to make s o f t s e a t s f o r two machine screws which came through n y l o n i n s u l a t e d h o l e s i n the c e l l body. The two h a l v e s of the 59 c e l l body were t i g h t l y screwed t o g e t h e r by s i x cap screws. The f i n i s h e d c e l l f i t i n s i d e the temperature c o n t r o l system which i s d e s c r i b e d i n the next s e c t i o n . The t h i c k n e s s of the f i l l e d c e l l was measured u s i n g an i n t e r f e r o m e t r i c a p p a r a t u s ^ which i s shown i n F i g . 1 1 . The c e l l was wrapped w i t h p l a s t i c p i p e and e n c l o s e d i n a foam b l o c k . Temperature c o n t r o l l e d water was c i r c u l a t e d t h r o u g h the p i p e so t h a t the c e l l c o u l d be h e l d a t 4 5 ° C . At t h i s temperature the 8CB i s i s o t r o p i c w i t h a n e a r l y t emperature independent index of r e f r a c t i o n of 1 . 5 6 7 0 . 0 0 0 2 . We measured the index w i t h a r e f T a c t o m e t e r which was temperature c o n t r o l l e d u s i n g the same c i r c u l a t i n g water. laser F i g u r e 11 - O p t i c a l system used t o measure t h i c k n e s s of f i l l e d c e l l 60 A l a s e r beam ( 7^  = 6328 $ ) , expanded and s p a t i a l l y f i l t e r e d , was passed t h r o u g h a l o n g f o c a l l e n g t h l e n s and r e f l e c t e d o f f the c e l l a t an a n g l e & b e f o r e i t came t o f o c u s . At the f o c u s a l i n e of t h r e e c l o s e l y spaced s p o t s were produced. The o u t s i d e two, which c o n t a i n e d most of the i n t e n s i t y , were due to r e f l e c t i o n s from the o u t s i d e a i r - g l a s s s u r f a c e s of the c e l l . -The c e n t r a l dim spot was the s u p e r p o s i t i o n of the two beams r e f l e c t i n g from the f r o n t and back s u r f a c e s of the sample volume. I t was dim because the index of the l i q u i d i s not ve r y d i f f e r e n t from t h a t of the g l a s s . The o u t s i d e s p o t s were masked by a s l i t w h ich a l l o w e d the c e n t r a l spot t o be p r o j e c t e d onto a s c r e e n . On the s c r e e n one c o u l d c l e a r l y o b s e r v e the a n g l e s © at which the f r o n t and back r e f l e c t i o n s i n t e r f e r e d t o produce a dark f r i n g e . T h i s o c c u r s a t a n g l e s g i v e n by £ ) - s/ - s«tfe " where m i s an i n t e g e r , iTl^ c, i s the l i q u i d i n d e x and L i s the sample t h i c k n e s s . Our t u r n t a b l e had an a n g u l a r r e s o l u t i o n of 0.01 . We observed f i v e f r i n g e s which were r e p r o d u c i b l e t o .03° . A f i t t o the d a t a gave a t h i c k n e s s L of 39.5^M.±.2 a t the c e n t r e of the c e l l . The t h i c k n e s s u n i f o r m i t y of t h e c e l l was checked by o b s e r v i n g the f r i n g e p a t t e r n w i t h a beam expanded t o the a r e a of the c e l l . We found t h a t the c e l l was s l i g h t l y l e n s shaped w i t h the v a r i a t i o n of t h i c k n e s s about two f r i n g e s , or about 1.5^u. t o the o u t e r edge, o u t s i d e the p l a t e s . A l l our measurements were c a r r i e d out on the same c e l l . (105) YY\ 61 3.2 The Temperature C o n t r o l The "nematic range", - T N ^ , of 8CB i s 7 ° C . The p r e t r a n s i t i o n a l e f f e c t s we wish t o observe o c c u r w i t h i n about 1 0 C of Tflfl. Thus good temperature measurement and c o n t r o l a r e e s s e n t i a l t o the e x p e r i m e n t . Our t e m p e r a t u r e c o n t r o l system had two s t a g e s . The innermost was a feedback c o n t r o l l e d copper c y l i n d e r (the " b l o c k " ) which e n c l o s e d the c e l l d e s c r i b e d e a r l i e r . The b l o c k was surrounded by foam i n s u l a t i o n and e n c l o s e d i n a c y l i n d r i c a l "can" wound w i t h p i p e s through which temperature c o n t r o l l e d water was pumped. The can, p i p e s and c i r c u l a t i n g water system were the m s e l v e s i n s u l a t e d from the room. The complete t h e r m o s t a t i s shown i n F i g . 12. The i n n e r b l o c k was a copper c y l i n d e r f o u r i n c h e s l o n g and f o u r i n c h e s i n d i a m e t e r . I t had a t h e r m a l c a p a c i t y of about 2.7 k J / K. A t h r e a d e d a x i a l h o l e w i t h two keyways a c c e p t s the c e l l which was h e l d i n p l a c e by two p l u g s which were screwed i n from each end. The keyways se r v e d t o f i x the c e l l o r i e n t a t i o n and p r o v i d e e l e c t r i c a l a c c e s s . O p t i c a l a c c e s s was p r o v i d e d by a s m a l l e r a x i a l h o l e through t h e p l u g s . The o u t s i d e s u r f a c e of the c y l i n d e r b l o c k was t h r e a d e d and wound n o n i n d u c t i v e l y w i t h a h e a t e r w i r e which was v a r n i s h e d down f o r good t h e r m a l c o n t a c t . I n t o the end f a c e s of the c y l i n d e r were i n s t a l l e d t h r e e Fenwal t h e r m i s t o r s e p o x i e d i n s i d e copper b o l t s , as w e l l as the sensor probe of a H e w l e t t P a c k a r d model 2804A q u a r t z thermometer. The i n n e r t emperature c o n t r o l e l e c t r o n i c s a r e shown i n F i g . 13. The c o n t r o l t h e r m i s t o r was b a l a n c e d a g a i n s t a G e n e r a l R a d i o decade r e s i s t o r i n a Wheatstone b r i d g e . 62 F i g u r e 12 - The c o n s t r u c t i o n of the t h e r m o s t a t -C L A S E R . B E A M O U T C R B O X 63 F i g u r e 13 - The temperature c o n t r o l c i r c u i t s > t£ uS 6 4 The decade r e s i s t o r had a minimum s t e p of O . O I i l . The b r i d g e was d r i v e n by a 1.35 v o l t mercury b a t t e r y . The e r r o r s i g n a l was a m p l i f i e d by a H e w l e t t Packard model 419A DC n u l l m e t e r which was used s i m p l y as a h i g h g a i n a m p l i f i e r w i t h g a i n about 300 000. The DC n u l l m e t e r was "chopper s t a b i l i z e d " and had e x c e l l e n t n o i s e r e j e c t i o n . The output of the n u l l m e t e r was g i v e n t o a Kepco OPS-7-2 programable power s u p p l y w i t h the feedback shown. T h i s feedback produces a c o m b i n a t i o n p r o p o r t i o n a l and i n t e g r a l c o n t r o l which d r i v e s the b l o c k h e a t e r . T h i s feedback network has been used by B a l z a r i n i ^ and some t h e o r e t i c a l j u s t i f i c a t i o n 52. f o r i t has been g i v e n by Forgan A second t h e r m i s t o r , Wheatstone b r i d g e and DC n u l l m e t e r were used t o m o n i t o r s h i f t s of the t e m p e r a t u r e on a c h a r t r e c o r d e r . T h i s monitor b r i d g e c o u l d be a c c u r a t e l y n u l l e d w i t h a second G e n e r a l Radio decade r e s i s t o r a l s o w i t h 0.01 TL minimum s t e p . U s i n g t h i s system and a g a i n of 100 000 one c o u l d e a s i l y o b serve changes as s m a l l as 50yu,K i n the t e m p e r a t u r e . The q u a r t z thermometer was used t o measure the a b s o l u t e t e m p e r a t u r e of the b l o c k . I t was c a l i b r a t e d t o a J a r r e t t water t r i p l e - p o i n t c e l l and i s b e l i e v e d t o be a b s o l u t e l y a c c u r a t e t o O.lmK over the c o u r s e of the e x p e r i m e n t . Far from T N B i n runs when the q u a r t z thermometer was on another experiment we used the r e s i s t a n c e of the r e m a i n i n g b l o c k t h e r m i s t o r measured by a K e i t h l e y 177 d i g i t a l m u l t i m e t e r t o get the a b s o l u t e t e m p e r a t u r e . T h i s t h e r m i s t o r was c a l i b r a t e d a g a i n s t the q u a r t z thermometer p r e v i o u s l y . The foam t h a t s u r r o u n d s the b l o c k was about two i n c h e s 65 t h i c k . A temperature d i f f e r e n c e between the b l o c k and the o u t e r can r e l a x e d e x p o n e n t i a l l y w i t h a time c o n s t a n t of 17 hours. The o u t e r can was b r a s s and was c o m p l e t e l y wrapped w i t h copper p i p e which was s o l d e r e d i n p l a c e . The c i r c u l a t i n g water was p r o v i d e d by a l a r g e r e f r i g e r a t e d bath which c i r c u l a t e d i t s 35 l i t r e s of water about once an hour. The water bath i s temperature c o n t r o l l e d by s w i t c h i n g between a r e f r i g e r a t o r and a h e a t e r . We r e p l a c e d the crude temperature c o n t r o l p r o v i d e d by the m a n u f a c t u r e r w i t h a c i r c u i t shown i n F i g . 14. T h i s c i r c u i t m o n i t o r s the can temperature w i t h a t h e r m i s t o r which i s compared to a decade r e s i s t o r i n a b r i d g e . The e r r o r s i g n a l from the b r i d g e i s used t o d e t e r m i n e the duty c y c l e of a square wave which s w i t c h e s between the h e a t e r and the f r i d g e . T h i s c o n t r o l l e r can h o l d the water temperature s t a b l e t o b e t t e r than 1mK. ' We m a i n t a i n e d a c o n s t a n t temperature d i f f e r e n c e of a few degrees between the can and the b l o c k . F i n a l l y , t o p r e v e n t room temp e r a t u r e s h i f t s from a f f e c t i n g the e l e c t r o n i c s we e n c l o s e d them i n a l a r g e i n s u l a t e d c r a t e . The c o n t r o l system j u s t d e s c r i b e d c o u l d h o l d the t e m p e r a t u r e of the sample s t a b l e t o 0.1 mK or b e t t e ^ w i t h no d r i f t over a p e r i o d of days. We c a l i b r a t e d the m o n i t o r b r i d g e decade r e s i s t o r v a l u e a t n u l l a g a i n s t the a b s o l u t e temperature measured by the q u a r t z thermometer and the c a l i b r a t e d t h e r m i s t o r . The p o l y n o m i a l f i t t o t h i s d a t a which was used t o get the temperature i s i n c o r p o r a t e d i n t o the d a t a a n a l y s i s programs reproduced i n appendix A, as the f u n c t i o n TEMP (R ) . 66 Figure 14 - The pulse-width modulation c i r c u i t used to control the bath temperature 67 3.3 The E l e c t r o n i c And O p t i c a l Systems The experiment c o n s i s t e d of a p p l y i n g an AC v o l t a g e t o the c a p a c i t o r f i l l e d w i t h nematic l i q u i d c r y s t a l and measuring the c a p a c i t a n c e and t h e i n t e n s i t y of l i g h t t r a n s m i t t e d between c r o s s e d p o l a r i z e r s . The a p p l i e d v o l t a g e must be changed v e r y s l o w l y i n o r d e r t h a t the nematic o r i e n t a t i o n be i n e q u i l i b r i u m . i n the f i e l d . D u r i n g each of t h e s e sweeps the temperature must be h e l d c o n s t a n t . The o p t i c a l and e l e c t r o n i c system t o do t h i s i s shown i n F i g . 15. The experiment i s c o n t r o l l e d and the da t a c o l l e c t e d by a Commodore PET minicomputer v i a an IEEE bus system. The v o l t a g e a p p l i e d t o the c e l l o r i g i n a t e s as a d i g i t a l number w r i t t e n t o a s p e c i f i c a d d r e s s i n t h e PET memory. T h i s number was c o n v e r t e d t o a DC l e v e l by an e x t e r n a l 12 b i t d i g i t a l t o a n a l o g c o n v e r t e r which was c o n n e c t e d t o the memory e x p a n s i o n p o r t p r o v i d e d i n s i d e the PET. The D/A p r o v i d e d a DC v o l t a g e v a r i a b l e between z e r o and t e n v o l t s i n 4096 s t e p s . T h i s DC s i g n a l was d e l i v e r e d t o the AM m o d u l a t i o n i n p u t of a H e w l e t t - P a c k a r d model 3312 A s i g n a l g e n e r a t o r v i a an op-amp c i r c u i t which c o n v e r t e d the DC l e v e l from 0-10 v o l t s t o -2.7 - +2.7 v o l t s . T h i s v o l t a g e range a l l o w e d one t o c o n t r o l the AC o u t p u t l e v e l of the s i g n a l g e n e r a t o r from z e r o t o about 8 V R ( (^. The op-amp c i r c u i t had b u i l t i n t o i t an RC network w i t h a time c o n s t a n t of 30 seconds. T h i s smooths the D/A s t e p s so t h a t no abr u p t 68 F i g u r e 15 - B l o c k diagram of the e l e c t r o n i c , o p t i c a l and da t a a c q u i s i t i o n system . C A P A C I T A N C E ^ B R I D G E ^ PHOTOMETER P H O T O M E T E R • S I G T N A L TO D/A 5ERIAL FAST LINE HP Q U A R T Z T H E R M . 1 f TAPE. 69 jumps i n v o l t a g e were p r e s e n t e d t o the c e l l . The output of the s i g n a l g e n e r a t o r was a m p l i f i e d so t h a t the f i n a l maximum v o l t a g e was about 20 V R M S . The PET computer's i n t e r n a l c l o c k was used f o r t i m i n g the D/A s t e p r a t e . The ou t p u t v o l t a g e c o u l d be ramped as s l o w l y as 0.05 mV/sec. The output v o l t a g e of the a m p l i f i e r was measured by a K e i t h l e y 175 a u t o r a n g i n g v o l t m e t e r c o n t r o l l e d by the PET v i a the IEEE bus. Most runs were done w i t h a freq u e n c y of 1500 Hz and some e a r l i e r ones a t 10kHz. The c e l l response was found t o be independent of frequency i n t h i s range. The d i e l e c t r i c c o n s t a n t data of Dunmur^ , r e f e r e d t o l a t e r , was o b t a i n e d a t 1500 Hz. The output of the a m p l i f i e r was a p p l i e d t o the c e l l v i a a r a t i o t r a n s f o r m e r which forms p a r t of a G e n e r a l Radio model 1615-A c a p a c i t a n c e b r i d g e . The b r i d g e was used i n a mode such t h a t , when n u l l e d , the t r a n s f o r m e r r a t i o was 1. The c a p a c i t a n c e b r i d g e was conn e c t e d t o the c e l l i n the " s h i e l d e d unknown" arrangement so t h a t a l l the c a p a c i t a n c e of the l e a d s was e x c l u d e d . We used c o a x i a l l e a d s a l l the way i n t o the temperature c o n t r o l l e r so t h a t a l l s t r a y c a p a c i t a n c e s were e x c l u d e d except those from the t a b s i n s i d e the c e l l . The copper c e l l body completed the s h i e l d i n g . The o t h e r arm of the c a p a c i t a n c e b r i d g e c o n t a i n s s t a n d a r d c a p a c i t o r s and t a p s onto the r a t i o t r a n s f o r m e r s e l e c t a b l e by s w i t c h e s . The e r r o r s i g n a l of the b r i d g e was f e d t o a P r i n c e t o n A p p l i e d R e s e a r c h model 5204 l o c k - i n a m p l i f i e r which was used i n v e c t o r mode. The r e f e r e n c e s i g n a l f o r the l o c k - i n was taken 70 from the s i g n a l g e n e r a t o r . N u l l i n g the b r i d g e gave the c e l l c a p a c i t a n c e from the p o s i t i o n of the s w i t c h e s . The procedure f o r c o l l e c t i n g c a p a c i t a n c e and v o l t a g e d a t a was as f o l l o w s . As the computer s l o w l y i n c r e a s e d the v o l t a g e , changes i n c e l l c a p a c i t a n c e were f o l l o w e d by manually n u l l i n g the b r i d g e . Whenever a c a p a c i t a n c e d a t a p o i n t was d e s i r e d , the D/A t i m i n g l o o p was manually i n t e r u p t e d and, a f t e r w a i t i n g f o r the v o l t a g e t o s t a b i l i z e , the measured c a p a c i t a n c e was typed i n t o the computer. The computer then read the v o l t a g e v i a the IEEE bus and r e c o r d e d i t and the c a p a c i t a n c e on t a p e . The D/A s t e p p i n g was then r e s t a r t e d . There were s e v e r a l problems w i t h t h i s . F i r s t , the b r i d g e must be kept near n u l l a t a l l times or the v o l t a g e read by t h e the K e i t h l e y 175 does not c o r r e s p o n d t o the a c t u a l c e l l v o l t a g e . The l o c k - i n a m p l i f i e r i s so s e n s i t i v e , however, t h a t v e r y s m a l l lOOyuV) e r r o r s i g n a l s c o u l d be m a i n t a i n e d . A second problem a r o s e because the c a p a c i t a n c e b r i d g e c o n t r o l s which n u l l e d the r e s i s t i v e p a r t of the unknown impedance ( the " d i s s i p a t i o n f a c t o r " ) were damaged by a p r e v i o u s u s e r . I t was sometimes d i f f i c u l t t o c o m p l e t e l y n u l l t h e r e s i s t i v e p a r t . L u c k i l y , i t was not a s t r o n g f u n c t i o n of the v o l t a g e . We used th e magnitude-phase o p t i o n of the l o c k - i n i n o r d e r t o observe when we had s u c c e s s f u l l y n u l l e d the c a p a c i t i v e p a r t of the e r r o r s i g n a l , which o c c u r s when the phase of the e r r o r s i g n a l i s z e r o . Of c o u r s e , we s t i l l had t o n u l l t he r e s i s t i v e p a r t w e l l enough t h a t the o v e r a l l e r r o r s i g n a l was s m a l l . 71 Near the c r i t i c a l v o l t a g e we used ramp r a t e s of about 0.05 mV/s. At l a r g e r v o l t a g e s we c o u l d use l a r g e r ramp r a t e s but the r e q u i r e m e n t of f o l l o w i n g the c a p a c i t a n c e changes on the b r i d g e meant t h a t we had t o use a slow ramp r a t e even f a r from V c . I t o f t e n took 12 hours t o complete a c a p a c i t a n c e run w i t h s e v e r a l p e o p l e n u l l i n g the b r i d g e i n s h i f t s . Each s e t of c a p a c i t a n c e d a t a u s u a l l y c o n t a i n e d about 300 c a p a c i t a n c e - v o l t a g e measurements at v o l t a g e s from 0 t o 2 0 V R M J . The c a p a c i t a n c e was i n the range 200pf t o 450pf and was measured t o O.Olpf. On many runs the c a p a c i t a n c e was not measured and the s i g n a l was a p p l i e d t o the c e l l d i r e c t l y from the a m p l i f i e r and the b r i d g e and l o c k -i n not used. The o p t i c a l measurements were t a k e n a u t o m a t i c a l l y s i m u l t a n e o u s l y w i t h the c a p a c i t a n c e measurements. The l i g h t from a 10 mW helium-neon l a s e r was a t t e n u a t e d and passed t h r o u g h the p o l a r i z e r s and the c e l l t o f a l l on a p h o t o t r a n s i s t o r i n the op-amp c i r c u i t shown i n F i g . 15. The l a s e r beam was not expanded so t h a t i t sampled o n l y a s m a l l spot a t the c e n t r e of the c e l l , where the t h i c k n e s s was measured p r e v i o u s l y . We a t t e n u a t e d the beam t o about 0.5 yu.W. At t h e s e i n t e n s i t i e s the o u t p u t of the p h o t o t r a n s i s t o r c i r c u i t shown i s l i n e a r and the o u t p u t v o l t a g e i s about 1 v o l t . S i n c e we o n l y needed r e l a t i v e i n t e n s i t i e s the p h o t o t r a n s i s t o r c i r c u i t was u n c a l i b r a t e d . The o u t p u t of t h i s photometer was measured by a K e i t h l e y 177 v o l t m e t e r i n t e r f a c e d t o the PET v i a the bus. The same program l o o p which t i m e d the D/A s t e p s p e r i o d i c a l l y measured the i n t e n s i t y and the v o l t a g e and r e c o r d e d them on the t a p e . 72 Because of the complex shape of the f r i n g e s i t was n e c e s s a r y t o r e c o r d over a thousand d a t a p o i n t s a t each t e m p e r a t u r e . P r o g r e s s of the experiment c o u l d be m o n i t o r e d by w a t c h i n g the f r i n g e s on an XY r e c o r d e r which used the a n a l o g o u t p u t s of the v o l t a g e and photometer output v o l t m e t e r s . On runs when c a p a c i t a n c e was not measured the experiment c o u l d be run unattended under the c o n t r o l of the computer. O f t e n t h i s was done o v e r n i g h t . The computer a l s o p e r i o d i c a l l y r e c o r d e d the temperature on the tape e i t h e r by r e a d i n g the q u a r t z thermometer or the c a l i b r a t e d t h e r m i s t o r v i a the K e i t h l e y ohmmeter. At the end of a run the d a t a on the c a s s e t t e tape was t r a n s f e r r e d t o the main UBC computer v i a .the bus and a s e r i a l f a s t l i n e . A l l l a t e r a n a l y s i s was done u s i n g the mainframe. The next c h a p t e r c o n c e r n s the a n a l y s i s of the d a t a . 73 IV. DATA ANALYSIS Go, c l e a r t h y c r y s t a l s ! Shakespeare, Henry V 2 , i i i , 5 4 4.1 F i t S t r a t e g y And D e c o n v o l u t i o n Of F r i n g e s In t h i s c h a p t e r we f i t the e x p e r i m e n t a l d a t a t o the t h e o r y i n the v a r i o u s ways d e s c r i b e d i n c h a p t e r 2. We w i l l e x t r a c t s e v e r a l m a t e r i a l parameters from the d a t a , namely €.\\, njj , and njL , i n a d d i t i o n t o the s p l a y and bend e l a s t i c c o n s t a n t s , K|| and K 3 3 . F i n a l l y , we w i l l examine the b e h a v i o u r of the e l a s t i c c o n s t a n t s and the o t h e r parameters near the nematic s m e c t i c A phase t r a n s i t i o n . S i n c e the a n a l y s i s i s somewhat complex, we w i l l o u t l i n e i t i n t h i s s e c t i o n . F i g . 16 shows some t y p i c a l raw c a p a c i t a n c e -v o l t a g e d a t a . The i n s e t shows the r e g i o n near the c r i t i c a l v o l t a g e V c . We g e n e r a l l y found t h a t the c a p a c i t a n c e , f o r v o l t a g e s between z e r o and the c r i t i c a l v o l t a g e , showed a s l i g h t p o s i t i v e s l o p e . T h i s may be due t o edge e f f e c t s o r , more l i k e l y , t o s m a l l r e g i o n s of poor a l i g n m e n t . The F r e e d e r i c k s z t r a n s i t i o n i t s e l f showed o n l y a v e r y s l i g h t r o u n d i n g , v i s i b l e i n the i n s e t . A f l o w c h a r t f o r the c a p a c i t a n c e d a t a a n a l y s i s i s shown i n F i g . 17. As was mentioned i n c h a p t e r 2, one can e x t r a c t from the c a p a c i t a n c e data i t s e l f a l l the parameters needed f o r the 74 An example of raw c a p a c i t a n c e - v o l t a g e data 75 F i g u r e 17 - Flowchart of c a p a c i t a n c e data a n a l y s i s Raw C V data t F i t to data near VC , e x t r a c t V c and S ~ t Use f i t to C vs 1/V to get Coo and S H F t Use C© and A/L to get €x F i n d *6 parameter 3= (Q./Q»)-l Use and £j L t o 9 e t ^» Use $ and S H K,to g e t ^ p c . Reduce the c a p a c i t a n c e s c = (C/C© )-1 Use $ and Sup Cto get X L F C Reduce the v o l t a g e s f o r 5 t r i a l value of \^ v = (V/\fc )~1 F u l l n o n l i n e a r f i t to c,v g i v e s *K , with % f i x e d . t ~  Use best and to get K|| *  76 f u l l n o n l i n e a r f i t . S i n c e our a r e a t o t h i c k n e s s r a t i o A/L was not v e r y w e l l known, we chose i t so t h a t the v a l u e s of €JL produced a r e i n agreement w i t h those of Dunmur f o r 8CB. F i g . 17 i s e s s e n t i a l l y the f l o w c h a r t of the program CF which c a r r i e d out the c a p a c i t a n c e data a n a l y s i s . I t i s reproduced i n appendix A. The outcome of t h i s a n a l y s i s a r e v a l u e s f o r the d i e l e c t r i c c o n s t a n t s £ w, £j_, and $, and the s p l a y e l a s t i c c o n s t a n t Kj| as w e l l as t h r e e v a l u e s f o r the reduced e l a s t i c c o n s t a n t kappa ; HLFC » ^ H P t a n d ^ from the f u l l f i t , a l l as a f u n c t i o n of t e m p e r a t u r e . The next s e c t i o n c o n c e r n s the r e s u l t s o t h e r than the e l a s t i c c o n s t a n t s . The the e l a s t i c c o n s t a n t r e s u l t s a re p r e s e n t e d i n the s e c t i o n f o l l o w i n g t h a t . The a n a l y s i s of the o p t i c a l d a t a i s somewhat more c o m p l i c a t e d than t h a t of the c a p a c i t a n c e d a t a . A f l o w c h a r t f o r t h i s a n a l y s i s i s shown i n F i g . 18. The d e c o n v o l u t i o n of the f r i n g e s t o get the phase t u r n e d out t o be more d i f f i c u l t than we a n t i c i p a t e d . F i g . 19 shows some raw f r i n g e d a t a . E q u a t i o n (91) which r e l a t e s the t r a n s m i t t e d l i g h t i n t e n s i t y t o the o p t i c a l phase d i f f e r e n c e assumes t h a t t h e r e i s no e x t i n c t i o n or d e p o l a r i z a t i o n of the l i g h t . I t i s e v i d e n t from the shape of the f r i n g e s i n F i g . 19 t h a t some l o s s o c u r r e d because the e n v e l o p e s of the f r i n g e s a r e not c o n s t a n t w i t h v o l t a g e . The e f f e c t i s s i g n i f i c a n t near and becomes se v e r e near T^a . We used the f o l l o w i n g scheme t o de c o n v o l v e the f r i n g e s . A q u a d r a t i c f u n c t i o n was f i t t o d a t a near each of the maxima and minima of the f r i n g e s and the extrema were found. 77 Figure 18 - Flowchart of o p t i c a l data analysis Raw fringe data I Deduce envelope from f i t s to peak positions and li n e a r interpolation and extrapolation I Deconvolve fringes to get raw phase d I F i t near V c extract d© Vc and S u P O Use d 0 to get An I Use n from Dunmur data and An to get nW f hjL, \> Use Su^and f i t to #(T) V to get X^PC, High f i e l d f i t to V vs 1/d use ty?and S H P O t o get K.HPO Reduce the phases &= 1-(d/do) Reduce the voltages for 5 t r i a l values of VC v = <V/\fe)-l F u l l nonlinear f i t to 5,v to get * K with tf.V fixed Use best V c and #(T),£ X(T) to get K u K 33 = KI\ ( X + 1) 78 F i g u r e 19 - Examples of raw f r i n g e d a t a 79 An approximate upper envelope f o r the f r i n g e s was c o n s t r u c t e d by l i n e a r l y i n t e r p o l a t i n g between the p o s i t i o n s of the maxima. For d a t a a t v o l t a g e s below the f i r s t f r i n g e we took the h e i g h t of the f i r s t f r i n g e t o be the v a l u e of the upper e n v e l o p e . S i m i l a r l y we e x t r a p o l a t e d the upper envelope t o v o l t a g e s above the l a s t f r i n g e . The same procedure was used t o c o n s t r u c t the lower envelope of the f r i n g e s . We c o u l d then n o r m a l i z e the d a t a so t h a t the f r i n g e s c o u l d be d e c o n v o l v e d t o get the raw phases d. T h i s procedure worked w e l l f o r f r i n g e s whose h e i g h t s were n e a r l y c o n s t a n t . Some t y p i c a l raw phase d a t a produced by t h i s d e c o n v o l u t i o n i s shown i n F i g . 20. The e x t r a p o l a t i o n of the envelope below the f i r s t f r i n g e a l l o w e d us t o d e c o n v o l v e the d a t a i n the r e g i o n of the F r e e d e r i c k s z t r a n s i t i o n . A n a l y s i s proceeded a c c o r d i n g t o the f l o w - c h a r t i n F i g . 18. U n f o r t u n a t e l y , some runs near T N ^ and near T N j c o u l d not be d e c o n v o l v e d because the envelope was not w e l l d e s c r i b e d by the l i n e a r i n t e r p o l a t i o n used above. On t h e s e runs we f i t the f r i n g e maxima and minima t o p a r a b o l a e as b e f o r e , then o n l y d e c o n v o l v e d these extrema. T h i s i s the u s u a l method used by o t h e r a u t h o r s * . To get V c f o r t h e s e c a s e s we used f i t s t o i n t e n s i t y d a t a near the F r e e d e r i c k s z t r a n s i t i o n . In t h i s method one cannot get d D , e x c e p t a p p r o x i m a t e l y by c o u n t i n g f r i n g e s and u s i n g e q u a t i o n ( 9 2 ) . We used a p o l y n o m i a l f i t t o d e v a l u e s from d a t a a t o t h e r t e m p e r a t u r e s t o i n t e r p o l a t e t o f i n d the v a l u e &Q needed t o produce reduced phases t o complete the f i t . The a n a l y s i s then proceeded as b e f o r e . (PDJ) 90U0J9JJ1P QSDLJd 81 The phase d a t a a n a l y s i s as in F i g . 18 was c a r r i e d out by the program PF and i t s s u b r o u t i n e s reproduced in appendix A. The d e c o n v o l u t i o n was done by a s e p a r a t e program. Note t h a t in the o p t i c a l case we make use of f i t s t o the d i e l e c t r i c c o n s t a n t £j_and t o $ from the c a p a c i t a n c e d a t a . The parameter V was found by co m b i n i n g v a l u e s of the b i r e f r i n g e n c e An o b t a i n e d from do and t h e t h i c k n e s s , and v a l u e s of the average index "n" = ( n)|+2nj( )/3 from the measurements of Dunmur . n i s n e a r l y temperature independent. As b e f o r e we get t h r e e v a l u e s of kappa, KUFO, *KHPO and the f u l l f i t K as w e l l as njj , nj_ and V , a l l as a f u n c t i o n of tem p e r a t u r e . 4.2 D i e l e c t r i c And R e f r a c t i v e Index R e s u l t s T h i s s e c t i o n concerns the e x t r a c t i o n of m a t e r i a l parameters o t h e r than t h e e l a s t i c c o n s t a n t s , which a re needed f o r the r e s t of the a n a l y s i s . Both o p t i c a l and c a p a c i t a n c e d a t a near the F r e e d e r i c k s z t r a n s i t i o n were handl e d s i m i l a r l y . Two p o i n t s were s e l e c t e d which s t r a d d l e d the c r i t i c a l v o l t a g e so t h a t the s m a l l i n t e r m e d i a t e r e g i o n of r o u n d i n g was e x c l u d e d . Then the da t a below the lower of these v o l t a g e s were f i t t o a s t r a i g h t l i n e . The z e r o v o l t a g e i n t e r c e p t of t h i s l i n e was taken t o be the z e r o v o l t a g e c a p a c i t a n c e ( C 0 ) or o p t i c a l phase d i f f e r e n c e ( d 0 ) , as the case may be. The d a t a above the upper v o l t a g e p o i n t were f i t t o a q u a d r a t i c , up t o a c u t - o f f v o l t a g e of about 2 V C . The 82 i n t e r s e c t i o n of t h i s q u a d r a t i c and the l i n e a r f i t below V c was taken t o be the c r i t i c a l v o l t a g e . L a t e r i n the f u l l f i t t h i s v a l u e was a l l o w e d t o v a r y s l i g h t l y t o improve the f i t . The low f i e l d s l o p e , S t F C or , was found from the d e r i v a t i v e of the q u a d r a t i c f i t a t V c . T h i s procedure i s c a r r i e d out i n t e r a c t i v e l y i n the r o u t i n e s PF and CF g i v e n i n appendix A. A graph of the c r i t i c a l v o l t a g e vs temperature i s shown i n F i g . 21. Away from T N A , the c r i t i c a l v o l t a g e shows the g e n t l e t e mperature dependence which i s due t o the v a r i a t i o n i n the deg ree of nematic o r d e r p r e s e n t . However, near T^^ i t shows an unexpected sharp i n c r e a s e . Our method of f i n d i n g V c i s s u s c e p t i b l e t o s y s t e m a t i c e r r o r i f the i n c r e a s e i n near T ^ i s s u f f i c i e n t l y l a r g e t h a t the i n f l e c t i o n p o i n t d e s c r i b e d i n c h a p t e r 2 i s p r e s e n t . T h i s c o u l d l e a d t o an o v e r e s t i m a t e of V c near T N ^ . The i n c r e a s e shown i n F i g . 21 i s however much too l a r g e t o be e n t i r e l y due t o t h i s e r r o r . T h i s i n c r e a s e i n V c i s due t o a c o m b i n a t i o n of the p r e t r a n s i t i o n a l b e h a v i o u r of KJJ , €Lj. and $ . We w i l l d i s c u s s the e r r o r i n V c more f u l l y i n the next s e c t i o n when we c o n s i d e r the e l a s t i c c o n s t a n t F i g . 22 shows a graph of the z e r o v o l t a g e c a p a c i t a n c e , CQ v e r s u s t e m p e r a t u r e . N o t i c e t h a t C 0 , which i s p r o p o r t i o n a l t o Cj., a l s o shows a s m a l l decrease, near T N f t . We f i t the C0 d a t a t o a p h e n o m e n o l o g i c a l t e m p e r a t u r e dependence, shown by the s o l i d l i n e i n F i g . 22, i n o r d e r t o i n t e r p o l a t e t o get v a l u e s of £± needed f o r the phase d a t a a n a l y s i s . T h i s p h e n o m e n o l o g i c a l f i t i s the f u n c t i o n CZFIT(T) i n appendix A. 1.4 1.3 H w 1.2 H o > 1.1 U) O 1 ~o > D 0.9 O 0.8 H 0.7 H 0.6 33 • Capacitance A Phase a A A A A 8 g • A A A • ^ A ' A • 34 35 ~~l 1 1 — l i 36 37 38 39 40 41 Temperature C T l C ' CD K> I o 0) O r t i-h CD 3 rr T) tr CD CD •1 0> O rt- n C I--n r t CD t - " n 0> <o l O CD < CD cn c in 9 84 F i g u r e 22 - Graph of the z e r o v o l t a g e c a p a c i t a n c e v e r s u s temperature 85 The h i g h f i e l d a n a l y s i s i n the c a p a c i t a n c e case i s handled by the s u b r o u t i n e EXTRAP. A s t r a i g h t l i n e was f i t t e d t o c a p a c i t a n c e v e r s u s i n v e r s e v o l t a g e f o r v o l t a g e s ' l a r g e r than a minimum v o l t a g e which was u s u a l l y s e l e c t e d t o be 17 v o l t s . F i g . 23 shows some t y p i c a l c a p a c i t a n c e d a t a p l o t t e d v e r s u s i n v e r s e v o l t a g e . The graph i s q u i t e s t r a i g h t even down t o v o l t a g e s o n l y a few t i m e s V c . T h i s i s because the a p p r o x i m a t i o n ^7=1 i s a good one even a t modest v o l t a g e s . The s l o p e Sjjp£ was used t o get the h i g h f i e l d reduced e l a s t i c c o n s t a n t T^HFC which i s d i s c u s s e d i n the next s e c t i o n . The i n t e r c e p t C ^ i s p r o p o r t i o n a l t o t h e d i e l e c t r i c c o n s t a n t 6|| . We used C© and Ceo t o deduce €\\ and €j. by a d j u s t i n g the are a t o t h i c k n e s s r a t i o A/L so t h a t our r e s u l t s agreed w i t h those of Dunmur f o r £ .L . The d i e l e c t r i c c o n s t a n t s we o b t a i n e d a r e shown on the graph i n F i g . 24. Note t h a t the v a l u e s we o b t a i n from the e x t r a p o l a t i o n t o i n f i n i t e v o l t a g e a re s y s t e m a t i c a l l y l a r g e r than Dunmur's measurements. We r e t u r n t o t h i s r e s u l t i n the next s e c t i o n when we d i s c u s s the d e t e r m i n a t i o n of the reduced e l a s t i c c o n s t a n t 7 C H F t f r o m S y F t . The d i m e n s i o n l e s s parameter $ , which i s independent of A/L, i s shown i n F i g . 25. Once a g a i n we f i t a ph e n o m e n o l o g i c a l temperature dependence t o t h i s d a t a , the f i t may be found i n the f u n c t i o n GAM(T) g i v e n i n appendix A. 86 F i g u r e 23 - Graph of h i g h f i e l d c a p a c i t a n c e data p l o t t e d a g a i n s t i n v e r s e v o l t a g e oo To o o o o o o o CN| O C O t O T+ C N ro ro ro - K ) jd 9DUD|jODdD3 8 7 F i g u r e 24 - Graph of the p r i n c i p a l d i e l e c t r i c c o n s t a n t s v e r s u s temperature • • • o CO 13 Q • r ^ ro CN ;z o o o o o O) oo o ro oo CJ ro o <D D ro O CD Q_ ro p: m ro ro ro ro JUDJSUOQ 0|jp9|9ig 89 T u r n i n g t o the o p t i c a l c a s e , we now c o n s i d e r the r e s u l t s f o r the z e r o v o l t a g e phase d i f f e r e n c e C\Q , which i s p r o p o r t i o n a l t o A n . T h i s d a t a i s shown i n F i g . 26 w i t h a p h e n o m e n o l o g i c a l f i t g i v e n as the f u n c t i o n DZFIT(T) i n appendix A. ' A t y p i c a l p l o t of v o l t a g e vs 1/d i s shown i n F i g . 27. Once a g a i n we f i n d a s t r a i g h t l i n e as e x p e c t e d from the h i g h f i e l d t h e o r y . The s l o p e and i n t e r c e p t , which g i v e i n f o r m a t i o n about the reduced e l a s t i c c o n s t a n t , w i l l be c o n s i d e r e d i n the next s e c t i o n . T h i s a n a l y s i s was done by the s u b r o u t i n e HFLIN. F i t s S3 t o the index d a t a of Dunmur were used t o get the average index "n, g i v e n i n the f u n c t i o n ENBAR(T). U s i n g the t h i c k n e s s measured b e f o r e , we f i n d the two i n d i c e s of r e f r a c t i o n and n j _ . These are shown i n F i g . 28. F i n a l l y , the two i n d i c e s were used t o f i n d the reduced q u a n t i t y V needed i n the o p t i c a l f i t s . T h i s parameter i s shown i n F i g . 29. The temperature dependences of d 0 ,V and a r e v e r y s i m i l a r . A l l show p r e t r a n s i t i o n a l i n c r e a s e s near T N f t . A l l the d a t a d i s c u s s e d i n t h i s s e c t i o n a r e t a b u l a t e d i n appendix B. 90 F i g u r e 26 - Graph of the z e r o v o l t a g e phase d i f f e r e n c e d 0 v e r s u s temperature > 2 0 - , 1 9 H 1 8 H <D 1 7 H o 16 H 1 5 - ^ • dP • • • • dp d? • 1 4 XL 0 . 3 0 0 . 3 5 0 . 4 0 0 . 4 5 0 . 5 0 0 . 5 5 Inverse phase difference d~1 rad 0 . 6 0 iQ C fl) I o n tr I — 1 a o 0) •a cr w tn n> a r t OJ 92 F i g u r e 28 - Graph of the p r i n c i p a l i n d i c e s of r e f r a c t i o n v e r s u s temperature • • X • X • o o O • • • X • X • • 00 co co N -co O CO • C N C O E c Q o C O 00 m C O O O -P O o o t> o o o <b o + m o cn ro oo O ro o CD D ro CD Q_ to 0 m ro ro ro ro CM i n 93 94 4.3 E l a s t i c C o n s tant R e s u l t s The s p l a y e l a s t i c c o n s t a n t Kjj i s de t e r m i n e d by the c r i t i c a l v o l t a g e V c and the d i e l e c t r i c c o n s t a n t s Gj. and % . The c a p a c i t a n c e and o p t i c a l phase d i f f e r e n c e a t v o l t a g e s above the F r e e d e r i c k s z t r a n s i t i o n depend on the d i m e n s i o n l e s s r a t i o \+*K of the bend and s p l a y c o n s t a n t s and on the m a t e r i a l c o n s t a n t s $ and . doe) K „ = " V £ 2 e 0 Ae fax (107, K 3 3 = K u ( » - • • * ) From the d a t a shown i n the l a s t s e c t i o n we f i n d the s p l a y e l a s t i c c o n s t a n t shown i n F i g . 30. As the temperature i s lowered the e l a s t i c c o n s t a n t i n c r e a s e s . T h i s i s e x p e c t e d from mean f i e l d t h e o r y , which p r e d i c t s t h a t the e l a s t i c c o n s t a n t s s h o u l d be p r o p o r t i o n a l t o S , where S i s the nematic o r d e r parameter . Near we f i n d a sudden i n c r e a s e i n K|| not p r e d i c t e d by mean f i e l d t h e o r y . R e c a l l t h a t the de Gennes model p r e d i c t s t h a t K J J i s f i n i t e i n the s m e c t i c phase. T h i s does not p r e c l u d e an i n c r e a s e i n K|| , o n l y t h a t i t s h o u l d not d i v e r g e a t T^p . In the next s e c t i o n we examine t h i s and o t h e r p r e t r a n s i t i o n a l e f f e c t s o b s e r v e d . » 95 F i g u r e 30 - Graph of the s p l a y e l a s t i c c o n s t a n t v e r s u s temperature suojMaN Zi_Q[ % •jsuoo XD|ds 96 The l a r g e s t s o u r c e of u n c e r t a i n t y i n Kj| i s due t o the u n c e r t a i n t y of the c r i t i c a l v o l t a g e V c . I t can be seen from F i g s . 20,21 and 24 t h a t i s more s c a t t e r e d than €j_ or *]$ . A l s o , K|| depends on the square of V c and o n l y l i n e a r l y on € j _ and K . The random e r r o r i n V f c was e s t i m a t e d from the b e h a v i o u r of the f u l l n o n l i n e a r f i t , t o be d e s c r i b e d l a t e r i n t h i s s e c t i o n . A l t h o u g h the p a r a m e t e r s €JL and If have a s m a l l random e r r o r compared t o V c , they may c o n t a i n s i g n i f i c a n t s y s t e m a t i c e r r o r as seen i n the l a s t s e c t i o n . The e r r o r b a r s shown i n F i g . 30 were a r r i v e d a t by n e g l e c t i n g the e r r o r s i n € x and V compared t o the e r r o r i n V c . I t must be s t r e s s e d t h a t t h e s e a r e o n l y e s t i m a t e s of the s t a t i s t i c a l e r r o r and do not r e f l e c t the p o s s i b l e s y s t e m a t i c e r r o r s . We now t u r n t o the a n a l y s i s r e q u i r e d t o o b t a i n the reduced e l a s t i c c o n s t a n t kappa. The low f i e l d v a l u e s of kappa, TKUFC and Xtro > were deduced from t h e s l o p e of the d a t a j u s t above V c as d e s c r i b e d i n the l a s t s e c t i o n . In the c a p a c i t a n c e c a s e , the h i g h f i e l d v a l u e of kappa X H F C w a s found from the s l o p e S«pc by the s u b r o u t i n e HFC and t h e f u n c t i o n s HFZ and HFARG g i v e n i n appendix A. In the o p t i c a l case we a t t e m p t e d t o get *KHFO from both the s l o p e and the i n t e r c e p t . They were found by the s u b r o u t i n e HFO and t h e f u n c t i o n s HFSZ, HFSARG, HFIZ and HFIARG. The r e s t of the f u n c t i o n s and s u b r o u t i n e s i n appendix A a r e c o n c erned w i t h the f u l l n o n l i n e a r f i t . The c a p a c i t a n c e a n a l y s i s program CF c o n t a i n s a l o o p which c a r r i e s out the n o n l i n e a r f i t f o r f i v e t r i a l v a l u e s of V. . On the f i r s t pass through the 97 l o o p the s u b r o u t i n e VFIND, which c a l l s FAINT and uses the f u n c t i o n VZ, f i l l s a r r a y s w i t h v a l u e s of , and the e l l i p t i c i n t e g r a l "KT( Y]) . FAINT and FBINT and f u n c t i o n s FA and FB c a r r y out the e v a l u a t i o n of / \ andB . S u b r o u t i n e REDCAP reduces the c a p a c i t a n c e d a t a . The a c t u a l f i t r o u t i n e i s a s i m p l e one-55 d i m e n s i o n a l v e r s i o n of p a r a b o l i c e x t r a p o l a t i o n a r r a n g e d t o m i n i m i z e c a l l s t o CHISQC, which e v a l u a t e s the " g o o d n e s s - o f - f i t " of the t h e o r y t o the d a t a . To reduce computation we re-use the v a l u e s of 0(, , and the e l l i p t i c i n t e g r a l and l i n e a r l y i n t e r p o l a t e the t h e o r e t i c a l f u n c t i o n . CHISQC c a l c u l a t e s the sum of the squares of the p e r p e n d i c u l a r d i s t a n c e s between a d a t a p o i n t and the t h e o r y . T h i s approach was m o t i v a t e d by the d e s i r e t o ensure t h a t the d a t a was not s y s t e m a t i c a l l y w e i g h t e d by the use of o n l y the c a p a c i t a n c e r e s i d u a l s , as i s c u s t o m a r i l y done i n *X f i t s . The c u r v e t o be f i t t o has a s t e e p r i s e and a l o n g f l a t t a i l . I f o n l y " v e r t i c a l " ( c a p a c i t a n c e ) r e s i d u a l s were a l l o w e d t o c o n t r i b u t e t o the g o o d n e s s - o f - f i t c r i t e r i o n , the i n i t a l s t e e p r i s e would be s t r o n g l y weighted over the f l a t t a i l p o r t i o n . U s i n g " p e r p e n d i c u l a r " r e s i d u a l s removed t h i s b i a s . The o p t i c a l phase program PF i s s i m i l a r t o CF except i n a few d e t a i l s . To reduce c o m p u t a t i o n , the s u b r o u t i n e FEW r e p l a c e s the l a r g e number of d a t a p o i n t s w i t h about 250 averaged p o i n t s . The d a t a a r e reduced by REDPHS. The g o o d n e s s - o f - f i t s u b r o u t i n e i s CHISQP which uses the s u b r o u t i n e FCINT and FC t o c a l c u l a t e the a d d i t i o n a l i n t e g r a l (C . Upon e x i t i n g the main l o o p both PF and CF p r o v i d e the b e s t - f i t v a l u e s of and the c u r v a t u r e of 98 the g o o d n e s s - o f - f i t f u n c t i o n about the minimum. T h i s p r o v i d e s a 4T5" way of e s t i m a t i n g the s t a t i s t i c a l e r r o r s i n and 7< . The s u b r o u t i n e PLTFIT can be used to produce p l o t s 'of the t h e o r e t i c a l f i t s and the r e s i d u a l s . To t e s t the programs we produced " s i m u l a t e d d a t a " which obeyed the t h e o r y w i t h known v a l u e s of the p a r a m e t e r s . Running our f i t programs on the s i m u l a t e d d a t a reproduced the c o r r e c t i n p u t parameters i n a l l c a s e s , i n c l u d i n g the h i g h and low f i e l d s l o p e and i n t e r c e p t s . Thus we were c o n f i d e n t t h a t , i n the happy event t h a t the r e a l d a t a obeyed the t h e o r y , we would be a b l e t o e x t r a c t the c o r r e c t e l a s t i c c o n s t a n t s . The v a r i o u s r e s u l t s f o r the parameter kappa a r e c o l l e c t e d t o g e t h e r i n F i g . 31. The g e n e r a l r e s u l t i s t h a t kappa i s near z e r o except w i t h i n about 1 ° C of T where the d i v e r g e n c e of K^j causes i t t o g r e a t l y i n c r e a s e . The low f i e l d and f u l l f i t r e s u l t s a r e g e n e r a l l y i n agreement w i t h one a n o t h e r f o r both the c a p a c i t a n c e and o p t i c a l c a s e s , a l t h o u g h the low f i e l d v a l u e s have more s c a t t e r . The h i g h f i e l d r e s u l t s show a l a r g e s y s t e m a t i c d i s a g r e e m e n t , w i t h the c a p a c i t a n c e v a l u e s l a r g e r than the f u l l f i t v a l u e s and the o p t i c a l " v a l u e s s m a l l e r . In the o p t i c a l case we have o n l y p l o t t e d the h i g h f i e l d r e s u l t s d e r i v e d from the s l o p e . The r e s u l t s f o r kappa from the i n t e r c e p t were s c a t t e r e d randomly. T h i s was not unexpected g i v e n the l o n g e x t r a p o l a t i o n t o get the i n t e r c e p t . 99 F i g u r e 31 - Graph of the reduced e l a s t i c c o n s t a n t 7K. v e r s u s t e m p e r a t u r e CD O C _o *o D Q. • O O D O o O x X + o CM CO O O CD O O X CM CO ~r c o X } O O O O O XGJ> + O X<3> o+ + xo + o+ & X O + o + o + XD X2> + >8f X ^ X ^ H ^ N " CM O o 00 O ro o CD D ro O CD Q_ c o CD m ro ro ro ro CM I Dddo» 100 The s y s t e m a t i c d i f f e r e n c e i n kappa between the h i g h f i e l d f i t and the f u l l f i t s can be a t t r i b u t e d t o b i a s i n t r o d u c e d d u r i n g the d e c o n v o l u t i o n of the f r i n g e s i n the case of the o p t i c a l d a t a . R e c a l l t h a t we assumed t h a t the envelope of the f r i n g e s was independent of v o l t a g e a f t e r the h i g h e s t v o l t a g e f r i n g e . The a c t u a l envelope p r o b a b l y i n c r e a s e s w i t h v o l t a g e . T h i s a p p r o x i m a t i o n i s p a r t i c u l a r l y g r o s s near T ^ where the f r i n g e s a r e most a f f e c t e d by s c a t t e r i n g and d e p o l a r i z a t i o n . The i n e x a c t d e c o n v o l u t i o n combined w i t h the s e n s i t i v i t y of X^FO T O s m a l l changes i n the h i g h f i e l d s l o p e makes the e l a s t i c c o n s t a n t s o b t a i n e d from t h i s d a t a e x t r e m e l y u n r e l i a b l e . The r e s u l t s from the h i g h f i e l d a n a l y s i s of the c a p a c i t a n c e d a t a a r e more s i g n i f i c a n t . Here one cannot blame the l a r g e r kappa v a l u e s on d e c o n v o l u t i o n . The l i n e a r f i t s t o h i g h f i e l d d a t a unambiguously g i v e s y s t e m a t i c a l l y l a r g e r kappa v a l u e s than low f i e l d f i t . In the l a s t s e c t i o n we saw t h a t the v a l u e s of €|| o b t a i n e d from the h i g h f i e l d e x t r a p o l a t i o n were s y s t e m a t i c a l l y l a r g e r than the measurements of D u n m u r ^ t T h i s s u g g e s t s t h a t the h i g h f i e l d a p p r o x i m a t i o n of U c h i d a and T a k a h a s h i i s obeyed, but w i t h " e f f e c t i v e " parameters ^ and " K which a r e l a r g e r than the low f i e l d v a l u e s . The f u l l n o n l i n e a r f i t s t o the D e u l i n g t h e o r y were g e n e r a l l y " e y e b a l l " p e r f e c t . O f t e n t h e r e was no v i s i b l e d e v i a t i o n of t h e o r e t i c a l f i t from the d a t a . Near T ^ however the q u a l i t y of ' the f i t got p r o g r e s s i v e l y worse. Examples of d i r e c t c omparisons between t h e o r e t i c a l f i t s and reduced d a t a are shown i n F i g . 32 and F i g . 34. S y s t e m a t i c d e v i a t i o n between the Reduced voltage a o u D j j o D d D Q paonpay tn I o 7.0-> 6.0-§ 5.0 '(/) CD 4.0 -D D O TJ 3.0 c CD CL L_ CD « « a. 2.0 1.0 Residuals for data shown in Fig. 34 0.0 8 6 10 reduced voltage 12 14 16 TJ M • i£> C •-1 ft> Co cn I a CU Xi n fD <~\ a. a> n> 3 r t CL CD o r t 0> fO 10 t-t* a c Q) I—1 W o o 0) TJ 0) r> rt CU D O ro 105 d a t a and the be s t f i t can be seen on p l o t s of the " p e r p e n d i c u l a r " r e s i d u a l s , which a r e the minimum d i s t a n c e s between each reduced d a t a p o i n t and the t h e o r e t i c a l c u r v e . These r e s i d u a l s would be randomly d i s t r i b u t e d i f the t h e o r y f i t the d a t a t o w i t h i n e r r o r s . A p l o t of the r e s i d u a l s of the o p t i c a l phase d a t a shown i n F i g . 32 i s shown i n F i g . 33. Note t h a t f o r low v o l t a g e s the r e s i d u a l s appear t o be random, but a t h i g h e r v o l t a g e s they show a d e v i a t i o n from the t h e o r y which i n c r e a s e s w i t h v o l t a g e . In the case of the o p t i c a l phase f i t s some of t h i s d e v i a t i o n c o u l d be e x p l a i n e d by the d e c o n v o l u t i o n e r r o r mentioned p r e v i o u s l y . We f i n d , however, a s i m i l a r t r e n d i n t he r e s i d u a l s of the c a p a c i t a n c e d a t a which are shown i n F i g . 35. In the c a p a c i t a n c e case the r e s i d u a l s t y p i c a l l y show s y s t e m a t i c d e v i a t i o n a t lower v o l t a g e s than i n the o p t i c a l c a s e . The same d e v i a t i o n s from the t h e o r y have been p r e v i o u s l y •a g o b s e r v e d by Maze The t r e n d i n the q u a l i t y of the f i t s w i t h temperature " i s i n t e r e s t i n g . We have c a l c u l a t e d two q u a n t i t i e s which measure the q u a l i t y of the f i t s . The f i r s t i s the s t a n d a r d d e v i a t i o n , which i s the the average magnitude of the p e r p e n d i c u l a r r e s i d u a l s such as those shown i n F i g . 33. The s t a n d a r d d e v i a t i o n i s o b t a i n e d from the minimum v a l u e of the goodness-of-f i t c r i t e r i o n , i t i s c a l c u l a t e d by the s u b r o u t i n e PLTFIT i n the ap p e n d i x . 1 06 F i g u r e 36 - The temperature v a r i a t i o n of the s t a n d a r d d e v i a t i o n of the f i t s o c _o O «o D _C CJ O L < • • • • in o o o O o • < • • • • • • • • • <• a • • < • • • • • <D <D 3i • o ro oo (J ro o U £ O <D CL ro m ro ro CN O ro ro O o o o U0l|DIA9p pjDpUDJS 1 07 F i g u r e 37 - The t e m p e r a t u r e v a r i a t i o n t h e f i t t o X of t h e i n s e n s i t i v i t y of CD O c 8 ® o co S- ° o SZ O 0_ • • o o I D O o K) O CM O • • i <2 Loo O K> O • • m • o o • ro ro lO ro 3 "o CD CL E .CD ro ro ro 9Jn|DAJnQ 9SJ8AU| 108 The second q u a n t i t y i s the c u r v a t u r e of the g o o d n e s s - o f - f i t about the minimum, as a f u n c t i o n of the parameter *K . T h i s i s a measure of the s e n s i t i v i t y of the f i t t o the parameter "7^ , the 5*5" s h a r p e r the minimum the l e s s the s t a t i s t i c a l e r r o r i n *K. S t r i c t l y s p e a k i n g , t h i s e s t i m a t e of s t a t i s t i c a l e r r o r i s o n l y v a l i d i f the r e s i d u a l s a re randomly d i s t r i b u t e d . The v a r i a t i o n of the s t a n d a r d d e v i a t i o n and i n v e r s e c u r v a t u r e w i t h temperature a r e shown i n F i g s . 3 6 and 3 7 . We found t h a t the d a t a showed a s y s t e m a t i c d e v i a t i o n from the D e u l i n g t h e o r y a t h i g h v o l t a g e s . The f a i l u r e of the t h e o r y was enhanced i n the c r i t i c a l r e g i o n c l o s e t o T N p . U s i n g the be s t f i t v a l u e s of kappa and the e r r o r e s t i m a t e d from the c u r v a t u r e we a r r i v e a t the r e s u l t s f o r K 3 5 shown i n F i g . 3 8 . As w i t h KJJ , away from T ^ q ^ K ^ j shows a temperature dependence a t t r i b u t a b l e t o the c l a s s i c a l i n c r e a s e of the nematic o r d e r parameter. Near T ^ j i n c r e a s e s d r a m a t i c a l l y due t o the s m e c t i c f l u c t u a t i o n s . T h i s i s the s u b j e c t of the next s e c t i o n . 109 F i g u r e 38 - Graph of the bend e l a s t i c c o n s t a n t v e r s u s temperature CD O C _o o D Q . O C J CD CO O JC CL • • E • s • • B B e B B o CD ro _ 00 (J ro o CD l . < D ro D CD Q_ to 0 ro if) ro -e—B- - B O -ro o ro ro O O CN I o co i o ID O CN O O O 00 o CD I O o CN 110 4.4 R e s u l t s In The C r i t i c a l Region T h i s s e c t i o n i s concerned w i t h phenomena observed a t t e m p e r a t u r e s j u s t above the nematic t o s m e c t i c A phase t r a n s i t i o n . S e v e r a l e f f e c t s made i t d i f f i c u l t t o get u s a b l e data c l o s e t o T N R . As p r e v i o u s l y mentioned, the f r i n g e s o b t a i n e d from the o p t i c a l measurements became d i s t o r t e d due t o l o s s of i n t e n s i t y . A l s o , we observed t h a t the time taken f o r the c r y s t a l t o r e o r i e n t i n the f i e l d became so l o n g t h a t even v e r y slow ramping of the v o l t a g e ( < 0.05»»V/sec) produced h y s t e r e s i s . T h i s slowness i s presumably a consequence of the i n c r e a s e i n the v i s c o s i t y which opposes r e - o r l e n t a t i o n . We were unable t o c o l l e c t u s e f u l d a t a w i t h i n about 20 mK of T N n . We c o u l d not d i r e c t l y d e t e r m i n e T^ n . Indeed, the e x a c t meaning of T N^ i s somewhat ambiguous i n t h i s experiment because of the d e p r e s s i o n of T N f t by j V X n| ^ . Our r e s u l t s a r e d e r i v e d from d a t a t a k e n a t a range of f i e l d s , and hence d e f o r m a t i o n s , so t h a t the a p p r o p r i a t e T ^ i s some average over the e f f e c t i v e t r a n s i t i o n t e m p e r a t u r e s i n v a r i o u s p a r t s of the c e l l . An i n d i r e c t d e t e r m i n a t i o n of T N ^ w i l l be d e s c r i b e d below. Near T ^ we o b s e r v e d s m a l l i n c r e a s e s i n An, , and K J J These may be due t o changes i n the s h o r t - r a n g e c o r r e l a t i o n s of the m o l e c u l e s brought about by the s m e c t i c - l i k e f l u c t u a t i o n s . In mean f i e l d t h e o r y An i s p r o p o r t i o n a l t o the o r d e r parameter S, g i v e n i n e q u a t i o n ( 1 ) . An i t s e l f can be taken as the o r d e r parameter . We may view the p r e t r a n s i t i o n a l i n c r e a s e of An as d i r e c t e v i d e n c e t h a t the nematic o r d e r 111 parameter i s i n c r e a s e d near the n e m a t i c - s m e c t i c A t r a n s i t i o n by non m e a n - f i e l d , t h a t i s , f l u c t u a t i o n e f f e c t s . The e l a s t i c c o n s t a n t s a re p r o p o r t i o n a l t o S i n mean f i e l d t h e o r y ^ . Thus, t o d e t e c t the e f f e c t s of f l u c t u a t i o n s , i t i s 2. i n t e r e s t i n g t o p l o t KJJ and K<j^ v e r s u s d G , which i s p r o p o r t i o n a l t o (Anf and hence S^" . T h i s i s shown i n F i g . 3 9 and 4 0 . We now f i t the d i v e r g e n t p a r t of K^-j t o a power law and e x t r a c t the c r i t i c a l exponent. R e c a l l t h a t the de Gennes model p r e d i c t s ( 1 0 8 ) =" C "t-where Ky^ i s the " n o n d i v e r g e n t " p a r t , t i s the reduced t e m p e r a t u r e , S>/f i s a c r i t i c a l exponent and C i s a c o n s t a n t . We assume t h a t the temp e r a t u r e dependence of K 3 3 i s the c o n t i n u a t i o n of the m e a n - f i e l d K - j ^ i n t o the r e g i o n near T ^ . The tem p e r a t u r e dependence of d© , away from the s m a l l i n c r e a s e near Tjgfl, i s w e l l d e s c r i b e d by (109) - T - = a - + b d 0 - f c d o U s i n g t h i s , and a f i t t o the l i n e a r p a r t of the graph i n F i g . 4 0 , g i v e s an e x p r e s s i o n f o r K9,(T). 1 12 Figure 39 - Graph of the-splay e l a s t i c constant versus the square of the nematic order parameter O O m % o o o ro •D O • • D • • cm no • CD • •D • ° n m ^ C M * " "D CD o O C r o co 8 o o o i n • C D CM 00 C D LTD T C M O O O 1 13 F i g u r e 40 - Graph of the bend e l a s t i c constant versus the square of the nematic order parameter OD O o C N O 0 0 o to o o C N O O O 0 0 o to o o C N • o o D o o • D o D o o o o •c _ CM o o m o o o ro m C N CD D o O" o c/) 8 <= O o in o o o SU0JM9N oi » P u s 8 1 1 4 A graph of l o g ( - K ^ ( T ) ) v e r s u s l o g ( t ) , f o r the c o r r e c t v a l u e of T Nu , s h o u l d be a s t r a i g h t l i n e w i t h s l o p e -V/j and i n t e r c e p t l o g ( C ). We t h e r e f o r e v a r i e d t o produce the best s t r a i g h t l i n e f i t , w e i g h t i n g the d a t a p o i n t s by t h e i r e s t i m a t e d e r r o r . We o m i t t e d the data above 36° C which was so c l o s e t o the background t h a t i t c o n t r i b u t e d no i n f o r m a t i o n . The r e s u l t s of the f i t were not s t r o n g l y dependent on t h i s upper c u t o f f . From the f i t shown i n F i g . 41 we f i n d a c r i t i c a l exponent of 1.0 and a c r i t i c a l t e m p e r a t u r e of 33.507°C. The "~ W <v/Z a m p l i t u d e C was 1.3 X 10 newtons. From the c u r v a t u r e of A w i t h T^ft near the minimum we e s t i m a t e t h e s t a t i s t i c a l e r r o r i n T N R t o be 5 mK. The s t a t i s t i c a l e r r o r e s t i m a t e f o r the exponent i s o n l y 0.02 . A l l o w i n g f o r the s e n s i t i v i t y of the f i t t o c h o i c e of (T) and upper c u t o f f , a more r e a s o n a b l e e r r o r e s t i m a t e i s 0.1. I t i s d i f f i c u l t t o see how our data c o u l d be c o n s i s t e n t w i t h an exponent lower than t h i s . In p a r t i c u l a r our r e s u l t i s c l e a r l y i n c o n s i s t e n t w i t h the de Gennes p r e d i c t i o n ^ of 0.67 and w i t h the l i g h t s c a t t e r i n g and X-ray measurements by the MIT group , which g i v e v a l u e s near 0.7 . 1 1 5 F i g u r e 41 - Graph of the best f i t of the d i v e r g e n t p a r t of the bend e l a s t i c c o n s t a n t t o a power law i n the reduced temperature 116 I t i s i n t e r e s t i n g t o f i t the r a t i o of the e l a s t i c c o n s t a n t s K ^ / K j | ='X+1 t o a power law. The e r r o r i n kappa i s s m a l l e r than t h a t of e i t h e r e l a s t i c c o n s t a n t and, s i n c e K | j and have the same temperature dependence i n mean f i e l d , the n o n d i v e r g e n t p a r t of "K+ 1 i s e x p e c t e d t o be temperature independent. F u r t h e r m o r e , the v a l u e of kappa f a r from TNn i s e s s e n t i a l l y z e r o . Thus we may t a k e the n o n d i v e r g e n t p a r t t o be 1. P r o c e e d i n g as b e f o r e we f i n d the f i t shown i n F i g . 42. We i n c l u d e a l l the data up t o T N| . T h i s g i v e s an exponent of 1.0 dt .1 and a c r i t i c a l t e m perature of 3 3 . 5 0 3 0 C ±.005, which i s i n agreement w i t h the p r e v i o u s r e s u l t . 1 1 / 118 V. CONCLUSION In r e a l i t y we know n o t h i n g f o r t r u t h l i e s i n the aby s s . Democritus We have measured the s p l a y and bend e l a s t i c c o n s t a n t s of 8CB u s i n g ah e l e c t r i c f i e l d induced F r e e d e r i c k s z t r a n s i t i o n . The s p l a y c o n s t a n t was d e t e r m i n e d from the c r i t i c a l v o l t a g e at the F r e e d e r i c k s z t r a n s i t i o n and the bend c o n s t a n t from the subsequent d e f o r m a t i o n of the d i r e c t o r up t o v o l t a g e s about twenty t i m e s the c r i t i c a l v o l t a g e . We measured the d e f o r m a t i o n s i m u l t a n e o u s l y from i t s e f f e c t s on both the b i r e f r i n g e n c e and c a p a c i t a n c e of the sample. The p r i n c i p a l d i e l e c t r i c c o n s t a n t s and i n d i c e s of r e f r a c t i o n were deduced from the d a t a , making use of the c e l l d i mensions and p u b l i s h e d v a l u e s of the average i n d e x ^ . As w e l l as d e t e r m i n i n g the v a r i o u s c o n s t a n t s as a f u n c t i o n of t e m p e r a t u r e , t h i s method i s a l s o a s t r o n g t e s t of the l i n e a r e l a s t i c theory, of the d e f o r m a t i o n , p a r t i c u l a r l y near th e s m e c t i c A phase where the bend e l a s t i c i t y i s dominated by s m e c t i c f l u c t u a t i o n e f f e c t s . We a l s o t e s t e d the i n t e r n a l c o n s i s t e n c y of the t h e o r y by examining the d a t a i n the h i g h and low f i e l d l i m i t s . I t was found t h a t the o p t i c a l and c a p a c i t a n c e r e s u l t s a g r e e d w i t h one another w i t h i n e r r o r s , w i t h the o p t i c a l d a t a h a v i n g l e s s u n c e r t a i n t y . In f i t s over the whole range of 119 v o l t a g e s we found some d e v i a t i o n from the D e u l i n g t h e o r y a t a l l t e m p e r a t u r e s . A s i m i l a r t r e n d i n the r e s i d u a l s was p r e v i o u s l y o b s e r v e d by M a z e J O . F u r t h e r m o r e , e l a s t i c c o n s t a n t v a l u e s d e t e r m i n e d from the h i g h f i e l d c a p c i t a n c e d a t a were s y s t e m a t i c a l l y l a r g e r than the f u l l and low f i e l d f i t s . The e x t r a p o l a t i o n of the c a p a c i t a n c e t o i n f i n i t e v o l t a g e produced £|| v a l u e s which were l a r g e r than those measured by Dunmur^ . Near the nematic - s m e c t i c A phase t r a n s i t i o n we observed a d e t e r i o r a t i o n i n the q u a l i t y of the f i t t o the D e u l i n g t h e o r y . A l l t h e s e r e s u l t s suggest a the breakdown of l i n e a r e l a s t i c i t y a t l a r g e d e f o r m a t i o n s and c l o s e t o T N p . T h i s breakdown c o u l d be due t o the s u p p r e s s i o n of the s m e c t i c f l u c t u a t i o n s , which a r e r e s p o n s i b l e f o r the l a r g e v a l u e s of K-gj , by d e f o r m a t i o n s w i t h nonzero 7 x n, By t h i s f l u c t u a t i o n - q u e n c h i n g mechanism, K 3 3 becomes s t r a i n dependent near T ^ . A mean f i e l d t h e o r y of t h i s n o n l i n e a r e l a s t i c i t y have been g i v e n by Chu and M c M i l l a n The v a l i d i t y of the l i n e a r e l a s t i c t h e o r y has been t e s t e d c l o s e t o T N f j by Ma j o r o s e t a_l ^  u s i n g m a g n e t i c a l l y produced d e f o r m a t i o n s . They found no disagreement w i t h the l i n e a r t h e o r y down t o tempe r a t u r e s v e r y c l o s e t o T w f l, where t h e y observed the onset of a " s t r i p e i n s t a b i l i t y " . T h i s i n s t a b i l i t y i s obser v e d when one c o o l s a deformed nematic t o j u s t above the s m e c t i c t r a n s i t i o n " ^ w . I t i s h y p o t h e s i z e d t h a t the i n s t a b i l i t y s i g n a l s the o n s et of n o n l i n e a r b e h a v i o u r . Our experiment has a somewhat d i f f e r e n t geometry than t h o s e d i s c u s s e d above and we used an e l e c t r i c , r a t h e r than m a g n e t i c , f i e l d . D i r e c t comparisons w i t h magnetic d e f o r m a t i o n e x p e r i m e n t s 1 20 are made d i f f i c u l t by the f a i l u r e of the t h e o r y a t h i g h f i e l d s i n the e l e c t r i c c a s e , even f a r from T ^ . We d i d not observe a s t r i p e i n s t a b i l i t y above the s m a l l e s t reduced temperature we reached, which was about 8 X 10^". N e v e r t h e l e s s , i n c o n t r a s t w i t h Majoros e_t a_l, we d i d observe an i n c r e a s e d d e v i a t i o n from the l i n e a r t h e o r y c l o s e t o T^p. The c r i t i c a l exponent S)^  f o r the d i v e r g e n c e of was found t o be 1.0 ± 0.1. The c r i t i c a l t e m p e r a t u r e T N B was o b t a i n e d from the f i t and was found t o be 33.507 °C dt .004. T h i s e r r o r bar on T^R i s p r o b a b l y too s m a l l , s i n c e we d i d not attempt t o account f o r the d r i f t due t o sample d e g r a d a t i o n . The c r i t i c a l exponent has been measured by many workers u s i n g a l a r g e v a r i e t y of m a t e r i a l s and t e c h n i q u e s . T h e i r p u b l i s h e d r e s u l t s a r e summarized i n t a b l e 1. Our r e s u l t i s i n agreement w i t h some e l e c t r i c a l F r e e d e r i c k s z t r a n s i t i o n e x p e r i m e n t s ^ ^ , ? a l t h o u g h the exp e r i m e n t s and a n a l y s i s a re d i f f e r e n t i n many d e t a i l s . I t i s apparent from the t a b l e t h a t t h e r e i s wide disagreement about the v a l u e of t h i s exponent. Some r e s u l t s a r e c l o s e t o the de Gennes p r e d i c t i o n of 0.67 but most a r e l a r g e r . I t seems u n l i k e l y t h a t f u r t h e r e x p e r i m e n t s l i k e the one d e s c r i b e d here w i l l shed much l i g h t on the c r i t i c a l exponents of the nematic - s m e c t i c A t r a n s i t i o n u n t i l the d e f o r m a t i o n i t s e l f i s b e t t e r u n d e r s t o o d . 121 Table I - A Survey of the p u b l i s h e d values of the c r i t i c a l exponent "^ // . REFERENCE (no.) Ma t e r i a 1 Method Sprunt et a l (22) Ocko et a l (23) Garland et a l (24) Janossy et a_l (25) B i r e k i et a l (26) Pindak et a l (27) Leger (28) Cheung et a l (30) C l a d i s (32) Cheung et a_l (31) 80CB 8CB 8S5 8S5 9S5 "5CB 10S5 "5CB 4 0 . 7 8CB CBOOA CN BBOA CBOOA CBOOA BBMBA 0.75*0.04 0.72*0.05 0.89*0.05 0.83*0.01 0.71*0.03 0.67 0.61*0.03 0.57 0.78*0.02 1 . 0.75*0.04 0.675*0.025 0.65*0.05 0.65*0.05 0.52*0.03 1.0±0.1 l i g h t s c a t t , x ray s c a t t x ray s c a t t . e l e c t r i c a l F r e e d e r i c k s z l i g h t s c a t t . c h o l e s t e r i c p i t c h magnet i c F r e e d e r i c k s z magnetic F r e e d e r i c k s z magnetic F r e e d e r i c k s z e l e c t r i c a l F r e e d e r i c k s z 122 APPENDIX A - COMPUTER PROGRAMS A.1 Main R o u t i n e For C a p a c i t a n c e A n a l y s i s CF C C Program CF f o r f i t t i n g C a p a c i t a n c e d a t a C C d e v i c e 4 d a t a f i l e C d e v i c e 11 output f i l e f o r f i t r e s u l t s C C IMPLICIT REAL*8(A - H,0 - Z) LOGICAL FIRST C C A r r a y s needed f o r p l o t s C REAL*4 VPLOT(100), BPLOT(100), APLOTOOO), VAPLOT(100) REAL*4 V( 1 0 0 0 ) , C(1000) C C A r r a y s t o h o l d d a t a and reduced d a t a and r e s u l t s C DIMENSION VDATA(500), CDATA(500), VRED(500), CRED(500) DIMENSION VCARAY(5), HFKRAY( 5) , FTKRAY(5), CHIFIN(5) DIMENSION NRAY(5) , CURV(5) , VRAY(500,5), CRAY(500,5) DIMENSION VTHRAY(500,5), CTHRAY(500,5), RCRAY(500,5) DIMENSION RVRAY(500,5), RESV(500), RESC(500), 1 VTH(500), CTHC500) C C A r r a y s needed f o r BEFORE f i t C DIMENSION CBFOR(200), VBFOR(200), YFBFOR(200), WT(200) DIMENSION YDBFOR(200), SBFOR(5), SIGBFR(5), ABFOR(5), 1 BBFOR(5) DIMENSION PBFOR(2) C C A r r a y s f o r AFTER f i t C DIMENSION CAFT(500), VAFT(500), YFAFT(500) DIMENSION YDAFT(500), SAFT(5), SIGAFT(5), AAFT(5), 1 BAFT(5) DIMENSION PAFT(3) C DIMENSION ALFRAY(500), ETARAY(500), ELARAY(500) COMMON AK, ANU, GAMMA, PIBY2, ALPHA, ETA COMMON /PARAM/ ALFRAY, ETARAY, ELARAY COMMON /SAVEK/ E L I P C PIBY2 = DARCOS(0.0D0) PI = 2.DO * PIBY2 C C Read code t o t a g o u t p u t . FREAD i s a f r e e format READ, C 1 23 WRITE (5,10) 10 FORMAT (1X, ' C a p a c i t a n c e f i t f o r : ' ) WRITE (6,20) 20 .FORMAT (1X, ' RUN NAME DATA 1 f o r JULY, 2 f o r AUG' 1 ) CALL FREAD(6, ' I : ' , MONTH) WRITE (6,30) 30 FORMAT (1X, .'Input number p a r t of run name') CALL FREAD(6, 'R*8:', RNUM) IF (MONTH .EQ. 1) WRITE (5,40) RNUM IF (MONTH .GE. 2) WRITE (5,50) RNUM 40 FORMAT (1X, ' JULY', F10.5) 50 FORMAT (1X, ' AUG', F10.5) C C Read the Temperature C WRITE (6,60) 60 FORMAT (1X, 'What i s the MBOX v a l u e ( i n ohms)?*) CALL FREAD(6, 'R*8:', EMBOX) C C Read i n c a p a c i t a n c e f i l e C K = 1 70 CONTINUE READ (4,90,END=80) VDATA(K), CDATA(K) K = K + 1 GO TO 70 80 CONTINUE NDATA = K - 1 90 FORMAT (2F30.15) C C Read i n l o c a t i o n s of t r a n s i t i o n C 100 CONTINUE WRITE (6,110) 110 FORMAT (1X, ' L i n e # j u s t b e f o r e t r a n s i t i o n ? ' 1 ) CALL FREAD(6, ' I : ' , NBFOR) WRITE (6,120) 120 FORMAT (1X, 'What l i n e # j u s t a f t e r ? ' ) CALL FREAD(6, ' I : ' , NAFTER) C C F i l l a r r a y f o r BEFORE f i t C DO 130 J = 1, NBFOR CBFOR(J) = CDATA(J) VBFOR(J) = VDATA(J) 130 CONTINUE NB = NBFOR C C Deduce l a r g e s t c a p a c i t a n c e C CMAX = O.OD0 DO 140 K = 1, NDATA 1 24 IF (CDATA(K) .GE. CMAX) CMAX = CDATA(K) 140 CONTINUE C C F i l l AFTER a r r a y up t o f i x e d f r a c t i o n of CMAX C WRITE (6,150) 150 FORMAT (1X, ' i n p u t a FRACT (used .625)') CALL FREAD(6, 'R*8:', FRACT) CF = FRACT * CMAX J = NAFTER K = 1 160 CONTINUE CAFT(K) = CDATA(J) VAFT(K) = VDATA(J) J = J + 1 K = K + 1 IF (CDATA(J) .GE. CF) GO TO 170 GO TO 160 170 CONTINUE NA = K - 1 NM = J - 1 C C Do t h e two f i t s b e f o r e and a f t e r VCRIT. C DOLSF i s a p o l y n o m i a l f i t l i b r a r y s u b r o u t i n e . C CALL DOLSF(1, NB, VBFOR, CBFOR, YFBFOR, YDBFOR, WT, 0, 1 SBFOR, SIGBFR, ABFOR, BBFOR, SSBFOR, .TRUE., 2 PBFOR) C C CALL DOLSF(2, NA, CAFT, VAFT, YFAFT, YDAFT, WT, 0, 1 SAFT, SIGAFT, AAFT, BAFT, SSAFT, .TRUE., PAFT) C C ALGRAF i s a l i b r a r y p l o t t i n g r o u t i n e . C DO 180 M = 1, NM V(M) = VDATA(M) C(M) = CDATA(M) 180 CONTINUE CALL ALGRAF(V, C, NM, -1) C C DO 190 J = 1, 50 VPD = DFLOAT(J) * VDATA(NM) / 5.D1 VPLOT(J) = VPD BPLOT(J) = PBFOR(1) + PBFOR(2) * VPD 190 CONTINUE DO 200 J = 1, 100 CPD = (DFLOAT(J)*(CMAX - PBFOR( 1 ) ) / l .D2) + 9.D-1 * 1 PBFOR(1) APLOT(J) = CPD VAPLOT(J) = PAFT(1) + PAFT(2) * CPD + PAFT(3) * CPD 1 * CPD 1 25 200 CONTINUE C c CALL ALGRAF(VPLOT, BPLOT, -100, 0) CALL ALGRAF(VAPLOT, APLOT, -100, 0) C ALDONE i s a l i b r a r y r o u t i n e which i n i t i a t e s p l o t s CALL ALDONE C C WRITE (6,210) 210 FORMAT ( I X , 'You l i k e t h i s f i t ? 0.=YES...') CALL FREAD(6, 'R*8:', YESNO) IF ( .NOT. (YESNO .EQ. 0.D0)) GO TO 100 C C C a l c u l a t e c r i t i c a l v o l t a g e VCRIT C AAA = PAFT(3) * PBFOR(2) * PBFOR(2) BBB = 2.DO * PBFOR(1) * PBFOR(2) * PAFT(3) + PAFT(2) * 1 PBFOR(2) - 1.D0 CCC = PAFT(3) * PBFOR( 1 ) * PBFOR(l) + PAFT(2) * PBFOR( 1 1 ) + PAFT(1) IF (PBFOR(2) .EQ. 0.0D0) GO TO 220 C C DISC = DSQRT(BBB*BBB - 4.D0*AAA*CCC) VCRIT = (-1.D0/(2.D0*AAA)) * (DISC + BBB) GO TO 230 C 220 CONTINUE VCRIT = CCC C C F i n d low f i e l d s l o p e SLFC C 230 CONTINUE CEE = PBFOR(1) + PBFOR(2) * VCRIT SLFC = 1.DO / (2.D0*PAFT(3)*CEE + PAFT(2)) C C deduce GAMMA w i t h e x t r a p o l a t i o n t o i n f i n i t e f i e l d C VMIN = 17.DO C CALL EXTRAP(NDATA, VDATA, CDATA, VMIN, NBFOR, VCRIT, 1 CZERO, CINF, SHFC, SQR, GAMMA) T = TEMP(EMBOX) EPZERO = 8.85418782D0 CEMPTY = 3.524D0 * EPZERO EPSPRP = CZERO / CEMPTY C C use i n i t i a l s l o p e as f i r s t guess of AK = kappa C AK = (GAMMA*(((2.0D0*CZERO)/(SLFC*VCRIT)) - 1.0D0)) -1 1.0D0 C C w r i t e out t h i n g s so f a r 1 26 240 250 C C c WRITE ( WRITE ( FORMAT 260 270 280 290 300 310 320 330 340 350 C c c c c c c FORMAT WRITE WRITE WRITE WRITE WRITE WRITE WRITE WRITE WRITE WRITE WRITE FORMAT FORMAT FORMAT FORMAT FORMAT FORMAT FORMAT FORMAT FORMAT FORMAT 1 5,240 5,250 (IX, , F10 ( 1X, 5,260 5,270 5,290 5,280 5,290 5,300 5,310 5,320 5,330 5,340 5,350 ( IX, ' F r a (IX, (1X, ( i x , (1X, (IX, (1X, (IX, (IX, (IX, EMBOX, T EPSPRP Mbox= ', F10.5, 6) E perp = ' ) NBFOR, NAFTER, TRACT SSBFOR SSAFT CZERO CINF GAMMA SLFC VCRIT AK Nbfor= ' , 15, ' Na f t e r = t= ' , F i t t o Temp= \ 5 , F10.5) capac i tance 1 5 , F i t a f t e r V c r i t Sum of squares= CZERO= CINF = GAMMA= i n i t i a l SLFC= ' VCRIT= e s t i m a t e d KAPPA= b e f o r e V c r i t : ' ) ( was q u a d r a t i c i n V)') F30.15) F30.15) F30.15) F30.15) F30.15) F30.15) F30.15) (+/-) around VCRIT..') Try a 5 v a l u e s of VCRIT WRITE (6,360) 360 FORMAT (1X, 'Give me a range CALL FREAD(6, 'R*8:', VCRAN) VCORIG = VCRIT FIRST = .TRUE. DO 620 NVC = 1 , 5 VCRIT = (VCORIG - VCRAN) + DFLOAT(NVC - 1) * (VCRAN/ 1 2.DO) VCARAY(NVC) = VCRIT c a l l h i g h f i e l d s u b r o u t i n e 370 CONTINUE AKHF = AK CALL HFC(SHF, GAMMA, VCRIT, CZERO, AKHF) HFKRAY(NVC) = AKHF WRITE (5,380) VCRIT, CZERO 380 FORMAT (1X, " H i g h f i e l d f i t s f o r VCRIT=', F16.8, 1 ' CZERO=', F16.8) 127 WRITE (5,390) AKHF, SHF, SQR 390 FORMAT (1X, 'Kappa h i g h f i e l d = ' , F16.8, ' slope=', 1 F16.8, ' Sqr=*, F16.8) C C reduce the d a t a C CALL REDCAP(NDATA, VDATA, CDATA, CZERO, VCRIT, NRED, 1 VRED, CRED) NRAY(NVC) = NRED C C f i l l ALFRAY,ETARAY and ELARAY which s t o r e a l p h a , e t a and K( C IF ( .NOT. FIRST) GO TO 410 ETA = 0.0D0 ALPHA = 0.0D0 DO 400 1 = 1 , NRED C VFIND f i n d s the c o r r e c t a l p h a f o r the g i v e n v o l t a g e CALL VFIND(VRED(D) ALFRAY(I) = ALPHA C DELI KM i s a l i b r a r y f u n c t i o n which c a l c u l a t e s C the complete e l l i p t i c i n t e g r a l K ( l - e t a ) ELARAY(I) = DELIKM(DEXP(-1.DO*ALPHA),IND) ETARAY(I) = 1.0D0-DEXP(-1.0D0*ALPHA) 400 CONTINUE FIRST = .FALSE. 410 CONTINUE C C Do the f i t t i n g C ITMAX = 30 ITER = 0 DELK = 0.5D0 AK2 = AK AK1 = AK2 - DELK AK3 = AK2 + DELK CALL CHISQC(NRED, CRED, VRED, AK1, F1, CTH, VTH, 1 RESC, RESV) CALL CHISQC(NRED, CRED, VRED, AK2, F2, CTH, VTH, 1 RESC, RESV) CALL CHISQC(NRED, CRED, VRED, AK3, F3, CTH, VTH, 1 RESC, RESV) 420 CONTINUE WRITE (5,430) ITER, DELK 430 FORMAT (1X, '####### i t e r = \ 15, ' ###### delK=' 1 , F20.15) WRITE (5,440) 440 FORMAT (14X, 'kappa', 20X, ' c h i s q r ' ) WRITE (5,460) AK1, F1 WRITE (5,450)-WRITE (5,460) AK2, F2 WRITE (5,450) WRITE (5,460) AK3, F3 450 FORMAT ( 1X, ' ' ) 460 FORMAT (1X, F30.15, F30.15) 128 C C C ODO*DELK*DELK) 2.0D0*AK2*DELK)) / FNEW, CTH, VTH, c r i t e r i o n C C C C A1 = (F1 + F3 - 2.0D0*F2) / (2. IF (A1 .LE. 0.0D0) GO TO 500 B1 = (F2 - F1 + A1 *(DELK*DELK -1 DELK AKNEW = -1.0D0 * B l / (2.0D0*Al) CALL CHISQC(NRED, CRED, VRED, AKNEW, 1 RESC, RESV) DELK = DABS(AK2 - AKNEW) t h i s i s the convergence CONV = 1.OD-4 IF (DELK .LE. CONV) GO TO 540 I t e r a t i o n ITER = ITER + 1 IF (ITER .GE. ITMAX) GO TO 520 IF (AKNEW .GT. AK3) GO TO 480 IF (AKNEW .LT. AK1) GO TO 490 IF (AKNEW .GT. AK2) GO TO 470 case t h a t AK1<AKNEW<AK2 F3 = F2 AK3 = AK2 F2 = FNEW AK2 = AKNEW AK1 = AK2 - DELK CALL CHISQC(NRED, 1 RESC, RESV) GO TO 420 470 CONTINUE case t h a t F1 = F2 AK1 = AK2 F2 = FNEW AK2 = AKNEW AK3 = AK2 + DELK CALL CHISQC(NRED, 1 RESC, RESV) GO TO 420 480 CONTINUE case t h a t DELK=DABS(AK2-AK3) F1 = F3 AK1 = AK3 F2 = FNEW • AK2 = AKNEW AK3 = AK2 + DELK CALL CHISQC(NRED, CRED, VRED, AK3, F3, CTH, VTH, 1 RESC, RESV) GO TO 420 490 CONTINUE case t h a t DELK=DABS(AK2-AK1) CRED, VRED, AK1, F1, CTH, VTH, AK2<AKNEW<AK3 CRED, VRED, AK3, F3, CTH, VTH, AK3<AKNEW AKNEW<AK1 129 F3 = F1 ' AK3 = AK1 F2 = FNEW AK2 = AKNEW AK1 = AK2 - DELK CALL CHISQC(NRED, CRED, VRED, AK1, F1, CTH, VTH, 1 RESC, RESV) GO TO 420 C C 500 CONTINUE WRITE (5,510) 510 FORMAT (1X, ' P a r a b o l i c f i t c u r v e s wrong way!') GO TO 620 C 520 CONTINUE WRITE (5,530) 530 FORMAT ( I X , 'Too many i t e r a t i o n s ! ' ) GO TO 620 C 540 CONTINUE WRITE (5,550) CONV 550 FORMAT (1X, ' F i t has converged. CONV=', F20.10) C C . w r i t e out r e s u l t s of f i t C AK = AKNEW FTKRAY(NVC) = AKNEW CHIFIN(NVC) = FNEW CURV(NVC) = A1 C DO 560 J = 1, NRED VTHRAY(J,NVC) = VTH(J) CTHRAY(J,NVC) = CTH(J) VRAY(J,NVC) = VRED(J) CRAY(J,NVC) = CRED(J) RVRAY(J,NVC) = RESV(J) RCRAY(J,NVC) = RESC(J) 560 CONTINUE C WRITE (5,570) AK WRITE (5,580) A1 WRITE (5,590) CHIFIN(NVC) WRITE (5,600) VCRIT 57 0 FORMAT ( I X , 'KAPPA= ', F20.15) 580 FORMAT (1X, 'CURVATURE= ', F20.15) 590 FORMAT (1X, 'CHISQC= ', F20.15) 600 FORMAT (1X, 'VCRIT= ', F20.15) C C p l o t C WRITE (6,610) 610 FORMAT (1X, 'Do you wanna p l o t ? (0=NO)') CALL FREAD(6, 'R*8:', YESNO) 1 30 IF (YESNO .EQ. 0.D0) GO TO 620 C CALL PLTFIT(NRED, VRED, CRED, VTH, CTH, RESV, RESC, 1 STD) C C 620 CONTINUE C C WRITE (5,630) 630 FORMAT (1X, 'VCRIT,.HIGH FIELD K,.FIT K,.CHISQ' C JMIN = 1 CHIMIN = CHIFIN(1) DO 650 J = 1, 5 WRITE (5,660) VCARAY(J), HFKRAY(J), FTKRAY(J), 1 CHIFIN(J) IF (CHIFIN(J) .LT. CHIMIN) GO TO 640 GO TO 650 640 CONTINUE JMIN = J CHIMIN = CHIFIN(J) 650 CONTINUE 660 FORMAT (1X, 4F16.8) C C bes t one now l a b e l l e d by JMIN C WRITE (6,670) JMIN 670 FORMAT (1X, 'Best c h i s q r v a l u e i s number', 16) WRITE (6,680) 680 FORMAT (1X, 'Wanna l o o k a t plots?..0=NO') CALL FREAD(6, 'R*8:', YESNO) IF (YESNO .EQ. 0.0D0) GO TO 700 C NRED = NRAY(JMIN) DO 690 J = 1, NRED VTH(J) = VTHRAY(J,JMIN) CTH(J) = CTHRAY(J,JMIN) VRED(J) = VRAY(J,JMIN) CRED(J) = CRAY(J,JMIN) RESV(J) = RVRAY(J,JMIN) RESC(J) = RCRAY(J,JMIN) 690 CONTINUE C CALL PLTFIT(NRED, VRED, CRED, VTH, CTH, RESV, RESC, 1 STD) C 700 CONTINUE C C W r i t e out r e l e v a n t s t u f f on d e v i c e 11 C WRITE (11,710) MONTH, RNUM, T, FTKRAY(JMIN), 1CHIFIN(JMIN), CURV(JMIN), STD, VCARAY(JMIN), 2HFKRAY(JMIN), CZERO, CINF 131 710 FORMAT (16, F10.5, 9E20.10) C C STOP END A.2 S u b r o u t i n e REDCAP C C s u b r o u t i n e t o produce reduced c a p a c i t a n c e data C SUBROUTINE REDCAP(NDATA, VDATA, CDATA, CZERO, VCRIT, 1 NRED, VRED, CRED) IMPLICIT REAL*8(A - H,0 - Z) DIMENSION VDATA(500), CDATA(500), VRED(500), CRED(500) J = 1 DO 10 K = 1, NDATA VR = (VDATA(K)/VCRIT) - 1.0D0 IF (VR .LE. 0.0D0) GO TO 10 VRED(J) = VR CRED(J) = (CDATA(K)/CZERO) - 1.0D0 j = a + 1 10 CONTINUE NRED = J - 1 RETURN END A.3 S u b r o u t i n e CHISQC C C Goodness of f i t f o r c a p a c i t a n c e d a t a C SUBROUTINE CHISQC(NRED, CRED, VRED, TESTK, F, CTH, 1 VTH, RESC, RESV) IMPLICIT REAL*8(A - H,0 - Z) DIMENSION CRED(500), VRED(500), VTH(500), CTH(500), 1 RESC(500) DIMENSION ALFRAY(500), ETARAY(500), ELARAY(500), 1 RESV(500) COMMON AK, ANU, GAMMA, PIBY2, ALPHA, ETA COMMON /PARAM/ ALFRAY, ETARAY, ELARAY COMMON /SAVEK/ ELIP C AK = TESTK DO 10 J = 1, NRED ALPHA = ALFRAY(J) ETA = ETARAY(J) ELIP = ELARAY(J) 1 32 CALL FAINT(A) CALL FBINT(B) VTH(J) = ((1.DO/PIBY2)*DSQRT(1.DO+GAMMA*ETA)*A) - 1. 1 ODO CTH(J) = (B/A) - 1.ODO 10 CONTINUE C c LV = 1 LC = 1 SUM = 0.0D0 C DO 120 M = 1, NRED C C deduce LC LV s . t . C VTH(LV)>VRED(M)>VTH(LV-1) C CTH(LC)>CRED(M)>CTH(LC-1) C NV = LV IF (VRED(M) .GT. VTH(NRED)) GO TO 40 IF (VRED(M) .LE. VTH(1)) GO TO 50 20 CONTINUE NV = NV + 1 IF (NV .EQ. NRED + 1) NV = 2 IF ((VRED(M) .LE. VTH(NV)) .AND. (VRED(M) .GT. VTH( 1 NV - 1))) GO TO 30 GO TO 20 30 CONTINUE C LV = NV SL = (CTH(LV) - CTH(LV - 1)) / (VTH(LV) - VTH(LV -1 D ) CTHINT = CTH(LV - 1) + SL * (VRED(M) - VTH(LV - 1)) C GO TO 60 C 40 CONTINUE C case VRED(M)>VTH(NRED) SL = (CTH(NRED) - CTH(NRED - 1)) / (VTH(NRED) - VTH( 1 NRED - 1)) CTHINT = CTH(NRED) + SL * (VRED(M) - VTH(NRED)) GO TO 60 C 50 CONTINUE C case VRED(M)<VTH(1) SL = (CTH(2) - CTH(1)) / (VTH(2) - V T H ( D ) CTHINT = CTH(1) - SL * (VTH(1) - VRED(M)) C 60 CONTINUE C RESC(M) = CTHINT - CRED(M) CRESQR = RESC(M) * RESC(M) C C 1 33 NC = LC IF (CRED(M) .GT. CTH(NRED)) GO TO 90 IF (CRED(M) .LE. C T H ( l ) ) GO TO 100 C 70 CONTINUE NC = NC + 1 IF (NC .EQ. NRED + 1) NC = 2 IF ( (CRED(M) .LE. CTH(NC)). .AND. (CRED (M) .GT. CTH ( 1 NC - 1))) GO TO 80 GO TO 70 C 80 CONTINUE LC = NC SL = (VTH(LC) - VTH(LC - 1)) / (CTH(LC) - CTH(LC -1 1 ) ) VTHINT = VTH(LC - 1) + SL * (CRED(M) - CTH(LC - 1)) C GO TO 110 C 90 CONTINUE C case CRED(M)>CTH(NRED) SL = (VTH(NRED) - VTH(NRED - 1)) / (CTH(NRED) - CTH( 1 NRED - 1)) VTHINT = VTH(NRED) + SL * (CRED(M) - CTH(NRED)) GO TO 110 C 100 CONTINUE C case CRED(M)<CTH(1) SL = (VTH(2) - VTH(1)) / (CTH(2) - CTH(1)) VTHINT = VTH(1) - SL * (CTH(1) - CRED(M)) C C 110 CONTINUE RESV(M) = VTHINT - VRED(M) IF (RESV(M) .GT. 1 . D I0) RESV(M) = 1 . D I0 VRESQR = RESV(M) * RESV(M) RESQR = 1.0D0 / ((1.0D0/VRESQR) + (1.D0/CRESQR)) SUM = SUM + RESQR 120 CONTINUE F = SUM RETURN END A.4 S u b r o u t i n e EXTRAP C C s u b r o u t i n e t o do the h i g h f i e l d a n a l y s i s of cap d a t a C SUBROUTINE EXTRAP(NDATA, VDATA, CDATA, VMIN, NBFOR, 1 VCRIT, CZERO, CINF, SLOPE, SS, GAMMA) 1 34 IMPLICIT REAL*8(A - H,0 - Z) DIMENSION VDATA(500), CDATA(500), V F I T ( 5 0 0 ) , 1 V I F I T ( 5 0 0 ) DIMENSION CFIT 1 ( 3 0 0 ) , C F I T 2 ( 3 0 0 ) , YF(500), YD(500), 1 WT(500) DIMENSION S ( 3 ) , S I G ( 3 ) , A ( 2 ) , B ( 2 ) , P(2) C C Get CZERO from f i t C DO 10 J = 1, NBFOR V F I T ( J ) = VDATA(J) C F I T 1 ( J ) = CDATA(J) 10 CONTINUE C C DOLSF i s a l i b r a r y p o l y n o m i a l f i t r o u t i n e CALL DOLSF(1, NBFOR, VFIT, CFIT1, YF, YD, WT, 0, S, 1 SIG, A, B, SS, .TRUE., P) C CZERO = P(1) C C FIT INVERSE VOLTAGES V>VMIN C K = 1 DO 20 J = NBFOR, NDATA IF (VDATA(J) .LT. VMIN) GO TO 20 V I F I T ( K ) = 1.DO / VDATA(J) CFIT2CK) = CDATA(J) K = K + 1 20 CONTINUE NFIT = K - 1 C C DOLSF i s a l i b r a r y p o l y n o m i a l f i t r o u t i n e CALL DOLSF(1, NFIT, V I F I T , CFIT2, YF, YD, WT, 0, S, 1 SIG, A, B, SS, .TRUE., P) C CINF = P(1) SLOPE = P(2) GAMMA = (CINF/CZERO) - 1.DO RETURN END A.5 S u b r o u t i n e HFC And F u n c t i o n s HFZ And HFARG C S u b r o u t i n e t o c a l c u l a t e h i g h f i e l d kappa C SUBROUTINE HFC(SHF, GAMMA, VCRIT, CZERO, AKHF) IMPLICIT REAL*8(A - H,0 - Z) LOGICAL LZ EXTERNAL HFZ COMMON /HFCOM/ TK, G, ST TK = AKHF 1 35 G = GAMMA PI = 2.DO * DARCOS(O.DO) ST = (PI*SHF/(2.D0*G*CZERO*VCRIT*DSQRT(1.DO+G))) HFERR = 1.D-8 AKMAX = 1.D3 AKZ = -1.DO C ZER02 i s a l i b r a r y r o o t f i n d i n g r o u t i n e CALL ZER02(AKZ, AKMAX, HFZ, HFERR, LZ) AKHF = AKZ RETURN END C C F u n c t i o n HFZ=0 when c o r r e c t kappa i s found C DOUBLE PRECISION FUNCTION HFZ(AKHF) IMPLICIT REAL*8(A - H,0 - Z) COMMON /HFCOM/ TK, G, ST EXTERNAL HFARG TK = AKHF HFZ = (DGAU16(0.0D0,1.0D0,HFARG)) + ST RETURN END C C HFARG i s the i n t g r a l argument C DOUBLE PRECISION FUNCTION HFARG(X) IMPLICIT REAL*8(A - H,0 - Z) COMMON /HFCOM/ TK, G, ST TOP = 1.DO+TK * X * X BOT = 1.DO+G * X * X HFARG = DSQRT(TOP/BOT) RETURN END A.6 Main R o u t i n e For Phase Data A n a l y s i s PF C Program PF f o r f i t t i n g phase d a t a C C d e v i c e 4 i n p u t phase d a t a C d e v i c e 11 output f i t r e s u l t s C C IMPLICIT REAL*8(A - H,0 - Z) LOGICAL FIRST C C A r r a y s needed f o r p l o t s C REAL*4 VPLOT(lOO), BPLOT(lOO), APLOT(lOO), VAPLOTO00) REAL*4 V(5000), PH(5000) 1 36 A r r a y s t o h o l d d a t a and reduced d a t a and r e s u l t s HFKIRA(5), FTKRAY(5) DIMENSION VDATA(5000), PHDAT(5000), VRED(500) 1 PHRED(500) DIMENSION VCARAY(5), HFKSRA(5) 1 CHIFIN(5) DIMENSION NRAY(5), CURV(5), VRAY(300,5), PHRAY(300,5) DIMENSION VTHRAY(300,5), PHTHRA(300,5), RPHRAY(300,5) DIMENSION RVRAY(300,5), RESV(500), RESPH(500) DIMENSION VTH(500), PHTH(500) A r r a y s needed f o r BEFORE f i t DIMENSION PHBFOR(500), VBFOR(500), YFBFOR(500), 1 WT(500) DIMENSION YDBFOR(500), SBFOR(5), SIGBFR(5), ABFOR(5), 1 BBFOR(5) DIMENSION PBFOR(2) A r r a y s . f o r AFTER f i t DIMENSION PHAFT(IOOO), VAFT(1000), YFAFT(1000) DIMENSION YDAFT(1000), SAFT(5), SIGAFT(5), AAFT(5), 1 BAFT(5) DIMENSION PAFT(3) DIMENSION ALFRAY(500), ETARAY(500), ELARAY(500) COMMON AK, ANU, GAMMA, PIBY2, ALPHA, ETA COMMON /PARAM/ ALFRAY, ETARAY, ELARAY COMMON /SAVEK/ ELIP PIBY2 = DARCOS(O.ODO) PI = 2.DO * PIBY2 Read i n a code t o t a g o u t p u t w i t h WRITE (5,10) FORMAT (1X, 'Phase f i t f o r : ' ) WRITE (6,20) FORMAT (1X, ' run name 1 = JULY 2 = AUG' 1 ) CALL FREAD(6, ' I : ' , MONTH) WRITE (6,30) FORMAT (1X, 'Input number p a r t of run name') CALL FREAD(6, *R*8:*, RNUM) IF (MONTH .EQ. 1) WRITE (5,40) RNUM IF (MONTH .GE. 2) WRITE (5,50) RNUM FORMAT ( I X , ' JULY', F10.5) FORMAT (1X, ' AUG', F10.5) Read t h e Temperature 1 37 WRITE (6,60) 60 FORMAT (1X, 'What i s the MBOX v a l u e ( i n ohms)?') CALL'FREAD(6, 'R*8:', EMBOX) C C Read i n phase f i l e C K = 1 7 0 CONTINUE READ (4,100,END=80) VDATA(K), PHDAT(K) K = K + 1 GO TO 70 80 CONTINUE NDATA = K - 1 WRITE (5,90) NDATA 90 FORMAT (1X, 'There are ', 110, ' data p o i n t s . . . ' ) 100 FORMAT (2F30.15) C C Read i n l o c a t i o n s of t r a n s i t i o n C 110 CONTINUE WRITE (.6,120) 120 FORMAT (1X, ' L i n e # j u s t b e f o r e t r a n s i t i o n ? ' 1 ) CALL FREAD(6, ' I : ' , NBFOR) WRITE (6,130) 130 FORMAT (1X, 'What l i n e # j u s t a f t e r ? ' ) CALL FREAD(6, ' I : ' , NAFTER) C C F i l l a r r a y f o r BEFORE f i t C DO 140 J = 1, NBFOR PHBFOR(J) = PHDAT(J) VBFOR(J) = VDATA(J) 140 CONTINUE NB = NBFOR C C F i l l AFTER a r r a y down t o f i x e d f r a c t i o n of PHDAT(1) C WRITE (6,150) 150 FORMAT (1X, ' i n p u t a FRACT (used .625)') CALL FREAD(6, 'R*8:', FRACT) PHFIN = FRACT * PHDAT(1) J = NAFTER K = 1 160 CONTINUE PHAFT(K) = PHDAT(J) VAFT(K) = VDATA(J) J = J + 1 K = K + 1 IF (PHDAT(J) .LE. PHFIN) GO TO 170 GO TO 160 170 CONTINUE NA = K - 1 NFIN = J - 1 1 38 C C Do the f i t s t o b e f o r e and a f t e r VCRIT C C DOLSF i s a l i b r a r y p o l y n o m i a l f i t r o u t i n e CALL DOLSF(1, NB, VBFOR, PHBFOR, YFBFOR, YDBFOR, WT, 1 0, SBFOR, SIGBFR, ABFOR, BBFOR, SSBFOR, .TRUE., 2 PBFOR) C C CALL DOLSF(2, NA, PHAFT, VAFT, YFAFT, YDAFT, WT, 0, 1 SAFT, SI'GAFT, AAFT, BAFT, SSAFT, .TRUE., PAFT) C C Do the p l o t s C DO 180 M = 1, NFIN V(M) = VDATA(M) PH(M) = PHDAT(M) 180 CONTINUE CALL ALGRAF(V, PH, NFIN, -1) C C C C C C DO 190 J = 1, 5 VPD = DFLOAT(J) * VDATA(NFIN) / 5.DO VPLOT(J) = VPD BPLOT(J) = PBFOR(1) + PBFOR(2) * VPD 190 CONTINUE DO 200 J = 1, 100 PHD = (DFL0AT(J)*(1.2D0*PBFOR(1) - PHFIN)/1.D2) + 1 PHFIN APLOT(J) = PHD VAPLOT(J) = PAFT(1) + PAFT(2) * PHD + PAFT(3) * PHD 1 * PHD 200 CONTINUE CALL ALGRAF(VPLOT, BPLOT, -100, 0) CALL ALGRAF(VAPLOT, APLOT, -100, 0) CALL ALDONE WRITE (6,210) 210 FORMAT (1X., 'You l i k e t h i s f i t ? 0.=YES. CALL FREAD(6, 'R*8:', YESNO) IF ( .NOT. (YESNO .EQ. 0.D0)) GO TO 110 C C C a l c u l a t e c r i t i c a l v o l t a g e C AAA = PAFT(3) *-PBFOR(2) * PBFOR(2) BBB = 2.DO * PBFOR(1) * PBFOR(2) * PAFT(3) + PAFT(2) * 1 PBFOR(2) - 1.DO CCC = PAFT(3) * PBFOR(1) * PBFOR(l) + PAFT(2) * PBFOR( 1 1 ) + PAFT(1) IF (PBFOR(2) .EQ. 0.0D0) GO TO 220 1 39 C C DISC = DSQRT(BBB*BBB - 4.DO*AAA*CCC) VCRIT = (-1,D0/(2.D0*AAA)) * (DISC + BBB) GO TO 230 C 220 CONTINUE VCRIT = CCC C C C a l c u l a t e low f i e l d s l o p e C 230 CONTINUE CEE = PBFOR(1) + PBFOR(2) * VCRIT SLFO = 1.D0 / (2.D0*PAFT(3)*CEE + PAFT(2)) C C deduce GAMMA from f i t t o c a p a c i t a n c e gammas C T = TEMP(EMBOX) GAMMA = GAM(T) C C Get d e l t a N and ANU from ENBAR and DZERO C DZERO = PBFOR(1) THICK = 37.2D0 / 0.6328D0 DELN = DZERO / (2.D0*PI*THICK) ENB = ENBAR(T) C ENPAR = ENB + (2.0D0/3.D0) * DELN ENPERP = ENB - (1.DO/3.DO) * DELN ANU = ((ENPAR*ENPAR)/(ENPERP*ENPERP)) - 1.DO C C F i t h i g h v o l t a g e vs 1/phase t o a l i n e C WRITE (5,240) VDATA(NDATA) 240 FORMAT (1X, 'Give me VMIN l a s t v o l t a g e i s ' , F10.5) CALL FREAD(6, 'R*8:', VMIN) C C CALL HFLIN(NDATA, VDATA, PHDAT, VMIN, SHF, YHF, SQR) C C use i n i t i a l s l o p e as f i r s t guess of KAPPA C AK = ((-1.D0*ENPAR*ANU*DZERO)/(DELN*VCRIT*SLFO)) - ( 11.D0+GAMMA) C C w r i t e out t h i n g s so f a r C WRITE (5,250) EMBOX, T WRITE (5,260) ENPAR, ENPERP WRITE (5,270) GAMMA 250 FORMAT (1X, 'Mbox= F10.5, ' Temp= 1 , F10.6) 260 FORMAT ( I X , ' N p a r a l l e l = F10.5, ' N p e r p e n d i c u l a r = 1 , F10.6) 140 C C C C c c c c c c c c c 270 FORMAT (IX, Gamma = ' ) WRITE (5,280 > NBFOR, NAFTER, FRACT WRITE(5,502) WRITE (5,310 SSBFOR WRITE (5,300 WRITE (5,310 SSAFT WRITE (5,320 ) DZERO WRITE (5,330 ) DELN WRITE (5,340 ) ANU WRITE (5,350 ) SLFO WRITE (5,360 ) VCRIT WRITE (5,370 ) AK 280 FORMAT (IX , Nbfor= ' , 15, ' N a f t e r = ', 15, 1 •' Frac :t= ' , F10.5) 290 FORMAT O x , F i t t o c a p a c i t a n c e b e f o r e V c r i t : 300 FORMAT O x , F i t a f t e r V c r i t ( was q u a d r a t i c 310 FORMAT O x , Sum of squares= ', F30.15) 320 FORMAT O x , DZERO= ', F30.15) 330 FORMAT O x , DELN= ', F30.15) 340 FORMAT O x , NU= ', F30.15) 350 FORMAT O x , ' i n i t i a l SLFO= ', F30.15) 360 FORMAT O x , 'VCRIT= *, F30.15) 370 FORMAT O x , ' e s t i m a t e d KAPPA = ' , F30.15) ') i n V ) ' ) Reduce the number of da t a p o i n t s a f t e r V c r i t t o 250 NFEW = 250 CALL FEW(VDATA, PHDAT, NDATA, VCRIT, NFEW) Try a range of VCRIT WRITE (6,380) 380 FORMAT (1X, 'Give me a range (*/-) around VCRIT..') CALL FREAD(6, 'R*8:', VCRAN) VCORIG = VCRIT FIRST = .TRUE. DO 650 NVC = 1 , 5 VCRIT = (VCORIG - VCRAN) + DFLOAT(NVC - 1) * (VCRAN/ 1 2.DO) VCARAY(NVC) = VCRIT c a l l h i g h f i e l d s u b r o u t i n e 390 CONTINUE AKHFS = AK AKHFI = AK CALL HFO(SHF, YHF, VCRIT, GAMMA, ANU, DZERO, AKHFS, 1 AKHFI) HFKSRA(NVC) = AKHFS HFKIRA(NVC) = AKHFI 141 • _ i WRITE (5,400) VCRIT WRITE (5,410) SHF, YHF WRITE (5,420) AKHFS, AKHFI, SQR 400 FORMAT (1X, ' V c r i t = ', F16.8) 410 FORMAT (1X, 'HFslope= ', F16.8, ' H F I n t e r c e p t = 1 F16.8) 420 FORMAT (1X, 'Slope HFk=', F16.8, ' I n t e r c e p t HFk=', 1 F16.8, ' SQR= ', F16.8) C C reduce the d a t a C CALL REDPHS(NDATA, VDATA, PHDAT, DZERO, VCRIT, NRED, 1 VRED, PHRED) NRAY(NVC) = NRED C C f i l l ALFRAY,ETARAY and ELARAY C IF ( .NOT. FIRST) GO TO 440 ETA = 0.0D0 ALPHA = 0.0D0 DO 430 1 = 1 , NRED CALL VFIND(VRED(I)) ALFRAY(I) = ALPHA C DELIKM i s a l i b r a r y e l l i p t i c i n t e g r a l K ( l - x ) ELARAY(I) = DELIKM(DEXP(-1.DO*ALPHA),IND) ETARAY(I) = 1.0D0-DEXP(-1.0D0*ALPHA) 430 CONTINUE FIRST = .FALSE. 440 CONTINUE C C Do the f i t t i n g C ITMAX = 30 ITER = 0 DELK = 0.5D0 AK2 = AK AK1 = AK2 - DELK AK3 = AK2 + DELK CALL CHISQP(NRED, PHRED, VRED, AK1, F1, PHTH, VTH, 1 RESPH, RESV) CALL CHISQP(NRED, PHRED, VRED, AK2, F2, PHTH, VTH, 1 RESPH, RESV) CALL CHISQP(NRED, PHRED, VRED, AK3, F3, PHTH, VTH, 1 RESPH, RESV) 450 CONTINUE WRITE (5,460) ITER,- DELK 460 FORMAT ( I X , '####### i t e r = \ 15, ' ###### delK=' 1 , F20.15) WRITE (5,470) -470 FORMAT (14X, 'kappa', 20X, ' c h i s q r ' ) WRITE (5,490) AK1, F1 WRITE (5,480) WRITE (5,490) AK2, F2 WRITE (5,480) 142 C WRITE (5,490) AK3, F3 480 FORMAT ( 1X, ' ' ) 490 FORMAT (1X, F30.15, F30.15) A1 = ( F l + F3 - 2.0D0*F2) / (2.0D0*DELK*DELK) IF (A1 .LE. 0.0D0) GO TO 530 B1 = (F2 - F1 + A1 *(DELK*DELK - 2.0D0*AK2*DELK)) / 1 DELK AKNEW = -1.0D0 * B1 / (2.0D0*A1) CALL CHISQP(NRED, PHRED, VRED, AKNEW, FNEW, PHTH, 1 VTH, RESPH, RESV) DELK = DABS(AK2 - AKNEW) C CONV = 2.0D-3 C C IF (DELK .LE. CONV) GO TO 570 C C Next i t e r a t i o n C ITER = ITER + 1 IF (ITER .GE. ITMAX) GO TO 550 IF (AKNEW .GT. AK3) GO TO 510 IF (AKNEW .LT. AK1) GO TO 520 IF (AKNEW .GT. AK2) GO TO 500 C case t h a t AK1<AKNEW<AK2 F3 = F2 AK3 = AK2 F2 = FNEW AK2 = AKNEW AK1 = AK2 - DELK CALL CHISQP(NRED, PHRED, VRED, AK1, F1, PHTH, VTH, 1 RESPH, RESV) GO TO 450 500 CONTINUE C case t h a t AK2<AKNEW<AK3 F1 = F2 AK1 = AK2 F2 = FNEW AK2 = AKNEW AK3 = AK2 + DELK CALL CHISQP(NRED, PHRED, VRED, AK3, F3, PHTH, VTH, 1 RESPH, RESV) GO TO 450 510 CONTINUE C case t h a t AK3<AKNEW C DELK=DABS(AK2-AK3) F1 = F3 AK1 = AK3 F2 = FNEW AK2 = AKNEW AK3 = AK2 + DELK . CALL CHISQP(NRED, PHRED, VRED, AK3, F3, PHTH, VTH, 1 RESPH, RESV) 143 GO TO 450 520 CONTINUE C case t h a t AKNEW<AK1 C DELK=DABS(AK2-AK1) F3 = F1 AK3 = AK1 F2 = FNEW AK2 = AKNEW AK1 = AK2 " DELK CALL CHISQP(NRED, PHRED, VRED, AK1, F 1 , PHTH, VTH, 1 RESPH, RESV) GO TO 450 C c 530 CONTINUE WRITE (5,540) 540 FORMAT (1X, ' P a r a b o l i c GO TO 650 f i t c u r v e s wrong way!') 550 CONTINUE WRITE (5,560) 560 FORMAT (1X, 'Too GO TO 650 many i t e r a t i o n s ! ' ) C C C 570 CONTINUE WRITE (5,580) CONV s 580 FORMAT (1X, ' F i t has converged. w r i t e out r e s u l t s of f i t AK = AKNEW FTKRAY(NVC) = AKNEW CHIFIN(NVC) = FNEW CURV(NVC) = A1 CONV=' , -F20.10) 590 DO 590 J = 1, NRED VTHRAY(J,NVC) = VTH(J) PHTHRA(J,NVC) = PHTH(J) VRAY(J,NVC) = VRED(J) PHRAY(J,NVC) = PHRED(J) RVRAY(J,NVC) = RESV(J) RPHRAY(J,NVC) = RESPH(J) CONTINUE C C WRITE (5,600) WRITE (5,610) . WRITE (5,620) WRITE (5,630) 600 FORMAT (1X, 610 FORMAT (1X, 620 FORMAT (1X, 'CHISQP= 630 FORMAT (1X, 'VCRIT= p l o t AK A1 CHIFIN(NVC) VCRIT 'KAPPA= 'CURVATURE= F20.15) F20.15) F20.15) F20.15) 144 WRITE (6,640) 640 FORMAT (1X, 'Do you wanna p l o t ? (0=NO)') CALL FREAD(6, 'R*8:', YESNO) IF (YESNO .EQ. 0.D0) GO TO 650 ' C CALL PLTFIT(NRED, VRED, PHRED, VTH, PHTH, RESV, 1 RESPH, STD) C C 650 CONTINUE C C WRITE (5,660) 660 FORMAT (1X, 'VCRIT... S l o p e H F k . . . I n t c p t H F k . . . F i t K . . . 1 . . . c h i s q r ' ) C JMIN = 1 CHIMIN = CHI FIN(1) DO 680 J = 1, 5 WRITE (5,690) VCARAY(J), HFKSRA(J), H F K I R A ( J ) , 1 FTKRAY(J), CHIFIN(J) IF (CHIFIN(J) .LT. CHIMIN) GO TO 670 GO TO 680 67 0 CONTINUE JMIN = J CHIMIN = CHIFIN(J) 680 CONTINUE 690 FORMAT (1X, 5F14.7) C C b e s t one now l a b e l l e d by JMIN C WRITE (6,700) JMIN 700 FORMAT (1X, 'Best c h i s q r v a l u e i s number', 16) WRITE (6,710) 710 FORMAT (1X, 'Type any number t o go on t o p l o t s ' ) CALL FREAD(6, 'R*8:', YESNO) C NRED = NRAY(JMIN) DO 720 J = 1, NRED VTH(J) = VTHRAY(J,JMIN) PHTH(J) = PHTHRA(J,JMIN) VRED(J) = VRAY(J,JMIN) PHRED(J) = PHRAY(J,JMIN) RESV(J) = RVRAY(J,JMIN) RESPH(J) = RPHRAY(J,JMIN) 720 CONTINUE C ' CALL PLTFIT(NRED, VRED, PHRED, VTH, PHTH, RESV, RESPH, 1 STD) C C C W r i t e out r e l e v a n t s t u f f on d e v i c e 11 C 1 45 WRITE (11,730) MONTH, RNUM, T, FTKRAY(JMIN), ICHIFIN(JMIN), CURV(JMIN), STD, VCARAY(JMIN), 2HFKSRA(JMIN), HFKIRA(JMIN), DZERO 730 FORMAT (16, F10.5, 9E20.10) C C STOP END A.7 S u b r o u t i n e REDPHS C C s u b r o u t i n e t o produce reduced data C SUBROUTINE REDPHS(NDATA, VDATA, PHDAT, DZERO, VCRIT, 1 NRED, VRED, PHRED) IMPLICIT REAL*8 (A - H,0 - Z)' DIMENSION VDATA(5000), PHDAT(5000), VRED(500), 1 PHRED(500) J = 1 DO 10 K = 1, NDATA VR = (VDATA(K)/VCRIT) - 1.0D0 IF (VR .LE. 0.0D0) GO TO 10 VRED(J) = VR PHRED(J) = (1.0D0 -(PHDAT(K)/DZERO)) J = J + 1 10 CONTINUE NRED = J - 1 RETURN END A.8 S u b r o u t i n e CHISQP C C Goodness of f i t , Phase case C SUBROUTINE CHISQP(NRED, PHRED, VRED, TESTK, F, PHTH, 1 VTH, RESPH, RESV) IMPLICIT REAL*8(A - H,0 - Z) DIMENSION PHRED(500), VRED(500), VTH(500), PHTH(500), 1 RESPH(500) DIMENSION ALFRAY(500), ETARAY(500), ELARAY(500), 1 RESV(500) COMMON AK, ANU, GAMMA, PIBY2, ALPHA, ETA COMMON /PARAM/ ALFRAY, ETARAY, ELARAY COMMON /SAVEK/ ELIP C AK = TESTK FNU = (DSQRT(1.D0+ANU)) / (DSQRT(1.D0+ANU) - 1.D0) DO 10 J = 1, NRED 146 C C ALPHA = ALFRAY(J) ETA = ETARAY(J) ELIP = ELARAY(J) CALL FAINT(A) CALL FBINT(B) CALL FCINT(C) VTH(J) = ((1.DO/PIBY2)*DSQRT(1.DO+GAMMA*ETA)*A) - 1. 1 ODO PHTH(J) = FNU * (1.D0-(C/B)) 10 CONTINUE LV = 1 LP = 1 SUM = 0.0D0 C DO 120 M = 1, NRED C • C deduce LP LV s . t . C VTH(LV)>VRED(M)>VTH(LV-1) C PHTH(LP)>PHRED(M)>PHTH(LP-1) C NV = LV IF (VRED(M) .GT. VTH(NRED)) GO TO 40 IF (VRED(M) .LE. VTH(1)) GO TO 50 20 CONTINUE NV = NV + 1 IF (NV .EQ. NRED + 1) NV = 2 IF ((VRED(M) .LE. VTH(NV)) .AND. (VRED(M) .GT. VTH( 1 NV - 1))) GO TO 30 GO TO 20 30 CONTINUE C LV = NV SL = (PHTH(LV) - PHTH(LV - l ) ) / (VTH(LV) - VTH(LV -1 1 ) ) PTHINT = PHTH(LV - 1) + SL * (VRED(M) - VTH(LV - 1)) C GO TO 60 C 40 CONTINUE C case VRED(M)>VTH(NRED) SL = (PHTH(NRED) - PHTH(NRED - l ) ) / (VTH(NRED) -1 VTH(NRED - 1)) PTHINT = PHTH(NRED) + SL * (VRED(M) - VTH(NRED)) GO TO 60 C 50 CONTINUE C case VRED(M)<VTH(1) SL = (PHTH(2) - PHTH ( 1 ) ) / (VTH(2) - VTH(O) PTHINT = PHTH(1) - SL * (VTH(1) - VRED(M)) C C 60 CONTINUE 1 47 C C c c RESPH(M) = PTHINT - PHRED(M) PHRESQ = RESPH(M) * RESPH(M) NC = LP IF (PHRED(M) .GT. PHTH(NRED)) GO TO 90 IF (PHRED (M) .LE. PHTH (D) GO TO 100 7 0 CONTINUE NC = NC + 1 IF (NC .EQ. NRED + 1) NC = 2 IF ((PHRED(M) .LE. PHTH(NC)) .AND. (PHRED(M) .GT. 1 PHTH(NC - 1 ) ) ) GO TO 80 GO TO 70 80 CONTINUE LP = NC SL = (VTH(LP) - VTH(LP - 1)) / (PHTH(LP) - PHTH(LP -1 1 ) ) VTHINT = VTH(LP - 1) + SL * (PHRED(M) - PHTH(LP - 1) 1 ) GO TO 110 C C 90 CONTINUE C case PHRED(M)>PHTH(NRED) SL = (VTH(NRED) - VTH(NRED - 1)) / (PHTH(NRED) -1 PHTH(NRED - 1)) VTHINT = VTH(NRED) + SL * (PHRED(M) - PHTH(NRED)) GO TO 110 C 100 CONTINUE C case PHRED(M)<PHTH(1) SL = (VTH(2) - VTH(1)) / (PHTH(2) - PHTH(1)) VTHINT = VTH(1) - SL * (PHTH(1) - PHRED(M)) 110 CONTINUE RESV(M) = VTHINT - VRED(M) I F (RESV(M) .GT. 1.D10) RESV(M) = 1.D10 VRESQR = RESV(M) * RESV(M) RESQR = 1.0D0 / ((1.0D0/VRESQR) + (1.D0/PHRESQ)) SUM = SUM + RESQR 120 CONTINUE F = SUM RETURN END 0 148 A.9 S u b r o u t i n e HFLIN C C C SUBROUTINE HFLIN(NDATA, VDATA, PHDAT, VMIN, SHF, YHF, 1 SQR) IMPLICIT REAL*8(A - H,0 - Z) DIMENSION VDATA(5000), PHDAT(5000), VFIT(2000) DIMENSION P I F I T ( 2 0 0 0 ) , YF(2000), YD(2000), WT(2000) DIMENSION S ( 3 ) , S I G ( 3 ) , A ( 2 ) , B ( 2 ) , P(2) C C FIT INVERSE VOLTAGES V>VMIN C K = 1 DO 10 J = 1, NDATA IF (VDATA(J) .LT. VMIN) GO TO 10 VFIT(K) = VDATA(J) P I F I T ( K ) = 1.D0 / PHDAT(J) K = K + 1 10 CONTINUE NFIT = K - 1 C C DOLSF i s a l i b r a r y p o l y n o m i a l f i t r o u t i n e CALL DOLSF(1, NFIT, P I F I T , VFIT, YF, YD, WT, 0, S, 1 SIG, A, B, SS, .TRUE., P) C SHF = P(2) YHF = P(1) SQR = SS RETURN END A.10 S u b r o u t i n e s HFO And FEW And R e l a t e d F u n c t i o n s C C High f i e l d s u b r o u t i n e t o get AKHFS and AKHFI C SUBROUTINE HFO(SHF, YHF, VCRIT, GAMMA, ANU, DZERO, 1 AKHFS, AKHFI) IMPLICIT REAL*8(A - H,0 - Z) LOGICAL LZ1, LZ2 EXTERNAL HFSZ, HFIZ COMMON /HFCOM/ TK, G, U, ST TK = AKHFS G = GAMMA U = ANU PI = 2.DO * DARCOS(O.ODO) C DEN = 2.DO * VCRIT * DZERO ST = (-1.D0*SHF*PI*(DSQRT(1.D0+U) - 1.DO)*DSQRT(1. 1D0+G)) / DEN 149 C HFERR1 = 1.D-9 AKMAX1 = 1.D2 AKZ1 = -1.DO C C ZER02 i s a l i b r a r y r o o t f i n d i n g r o u t i n e CALL ZER02(AKZ1, AKMAX1, HFSZ, HFERR1, LZ1) IF ( .NOT. LZ1) WRITE (5,10) 10 FORMAT (1X, * NO ROOT FOUND i n s l o p e High F i e l d f i t . . . ' 1 ) AKHFS = AKZ 1 C ST = (PI*YHF*DSQRT(1.DO+G)) / (2.DO*G*VCRIT) C HFERR2 = 1.D-9 AKMAX2 = 1.D2 AKZ2 = -1.DO C C ZER02 i s a l i b r a r y r o o t f i n d i n g r o u t i n e CALL ZER02(AKZ2, AKMAX2, HFIZ, HFERR2, LZ2) IF ( .NOT. LZ2) WRITE (5,20) 20 FORMAT (1X, ' NO ROOT FOUND i n i n t e r c e p t High F i e l d f i t . . . ' 1 ) AKHFI = AKZ2 RETURN END C C HFSZ i s z e r o when s l o p e HFK i s found C DOUBLE PRECISION FUNCTION HFSZ(AKHFS) IMPLICIT REAL*8(A - H,0 - Z) COMMON /HFCOM/ TK, G, U, ST EXTERNAL HFSARG TK = AKHFS HFSZ = ST - DGAU16(0.ODO,1.ODO,HFSARG) RETURN END C C DOUBLE PRECISION FUNCTION HFSARG(X) IMPLICIT REAL*8(A"- H,0 - Z) COMMON /HFCOM/ TK, G, U, ST X2 = X * X T = 1.D0-(DSQRT((1.D0+U)/(1.D0+X2*U))) HFSARG = T * DSQRT((1.D0+TK*X2)*(1.D0+G*X2)) / ( 1 . 1D0-X2) RETURN END C C C DOUBLE PRECISION FUNCTION HFIZ(AKHFI) IMPLICIT REAL*8(A - H,0 - Z) COMMON /HFCOM/ TK, G, U, ST 150 EXTERNAL HFIARG TK = AKHFI HFIZ = ST - DGAU16(0.0D0,1.0D0,HFIARG) RETURN END C C DOUBLE PRECISION FUNCTION HFIARG(X) IMPLICIT REAL*8(A - H,0 - Z) COMMON /HFCOM/ TK, G, U, ST X2 = X * X HFIARG = DSQRT((1.D0+TK*X2)/(1.D0+G*X2)) RETURN END C C S u b r o u t i n e t o t h i n d a t a a f t e r VCRIT t o save $$ C SUBROUTINE FEW(VDATA, PHDAT, NDATA, VCRIT, NFEW) IMPLICIT REAL*8(A - H,0 - Z) DIMENSION VDATA(5000), PHDAT(5000), VFEW(500), 1 PHEW(500) DIMENSION VBIN(50), PHBIN(50), Y F ( 5 0 ) , YD(50), WT(50), 1 S(50) -DIMENSION S I G ( 5 0 ) , A ( 5 0 ) , B ( 5 0 ) , P(3) C J = 1 10 CONTINUE IF (VDATA(J) .GE. VCRIT) GO TO 20 J = J + 1 GO TO 10 20 CONTINUE NV = J C C Determine the b i n s i z e and p r i n t i t o u t . . C NBIN = (NDATA - J ) / NFEW IF (NBIN .LE. 3) NBIN = 4 WRITE (6,30) NV, NFEW, NBIN 30 FORMAT (1X, 'Number b f o r Vc=', 16, ' chop t o ' , 16, 1 ' b i n s = ' , 16) C DO 50 L = 1, NFEW C VSUM = 0.0D0 C DO 40 M = 1, NBIN VBIN(M) = VDATA(J) PHBIN(M) = PHDAT(J) VSUM = VSUM + VDATA(J) J = J + 1 40 CONTINUE C VFEW(L) = VSUM / DFLOAT(NBIN) C 151 C DOLSF i s a l i b r a r y p o l y n o m i a l f i t r o u t i n e CALL DOLSF(2, NBIN, VBIN, PHBIN, YF, YD, WT, 0, S, 1 SIG, A, B, SQR, .TRUE., P) C PHEW(L) = P(1) + P(2) * VFEW(L) + P(3) * VFEW(L) * 1 VFEW(L) C 50 CONTINUE C L = 1 NDATA = NV + NFEW - 1 C DO 60 K = NV, NDATA VDATA(K) = VFEW(L) PHDAT(K) = PHEW(L) L = L + 1 60 CONTINUE C RETURN END A.1'1 S u b r o u t i n e PLTFIT C s u b r o u t i n e t o p l o t r e s u l t s of f i t C SUBROUTINE PLTFIT(NRED, VRED, CRED, VTH, CTH, RESV, 1 RESC, STD) IMPLICIT REAL*8(A - H,0 - Z) REAL*4 V ( 5 0 0 ) , C ( 5 0 0 ) , V F I T ( 5 0 0 ) , C F I T ( 5 0 0 ) , RV(500), 1 RC(500) REAL*4 PRES(500) DIMENSION VRED(500), CRED(500), ALFRAY(500), 1 ETARAY(500) DIMENSION ELARAY(500), VTH(500), CTH(500), RESV(500), 1 RESC(500) C SUMPR2 = 0.0D0 C DO 10 J = 1, NRED V F I T ( J ) = VTH(J) C F I T ( J ) = CTH(J) RV(J) = RESV(J) RC(J) = RESC(J) PR2 = 1.D0 / ((1.D0/RESV(J))**2 + (1.D0/RESC(J))**2) SUMPR2 = SUMPR2 + PR2 PRES(J) = DSQRT(PR2) V ( J ) = VRED(J) C ( J ) = CRED(J) 10 CONTINUE STD = DSQRT(SUMPR2/NRED) 1 52 20 CONTINUE WRITE (5,30) STD 30 FORMAT (1X, 'Standard d e v i a t i o n = ', F30.15) WRITE (6,40) 40 FORMAT (1X, '0=Quit 1 = f i t graph 2 = V r e s i d 3=Cresid 4=Pre 1 ) CALL FREAD(6, 'R*8:', GRAPH) IF (GRAPH .EQ. 0.0D0) RETURN IF (GRAPH .EQ. 1.D0) GO TO 70 IF (GRAPH .EQ. 2.DO) GO TO 50 IF (GRAPH .EQ. 3.DO) GO TO 60 C CALL ALGRAF(V, PRES, NRED, 0) CALL ALDONE GO TO 20 C 50 CONTINUE C CALL ALGRAF(V, RV, NRED, 0) CALL ALDONE GO TO 20 C 60 CONTINUE C CALL ALGRAF(V, RC, NRED, 0) CALL ALDONE GO TO 20 C 70 CONTINUE CALL ALGRAF(V, C, NRED, -1) CALL ALGRAF(VFIT, CFIT, -1*NRED, 0) CALL ALDONE GO TO 20 END A.12 F u n c t i o n TEMP(R) C F u n c t i o n t o e v a l u a t e Temperature C DOUBLE PRECISION FUNCTION TEMP(R) IMPLICIT REAL*8(A-H,0-Z) A=91.356938270024003D0 B=-0.068776762624791 DO C=0.000027138365424D0 D=-0.000000004534137D0 TEMP=A+B*R+C*R*R+D*R*R*R RETURN END 153 A.13 S u b r o u t i n e VFIND And F u n c t i o n VZ C C f i n d s best a l p h a f o r a g i v e n v o l t a g e C SUBROUTINE VFIND(V) IMPLICIT REAL*8(A - H,0 - Z) LOGICAL LZ EXTERNAL VZ COMMON AK, ANU, GAMMA, PIBY2, ALPHA, ETA COMMON /VOLT/ VD VD = V VERROR = 1.OD-9 ALFMAX = 100.DO ALFMIN = 0.0D0 C ZER02 i s a l i b r a r y r o o t f i n d i n g r o u t i n e CALL ZER02(ALFMIN, ALFMAX, VZ, VERROR, LZ) ALPHA = ALFMIN RETURN END C C VZ=0 when c o r r e c t ALPHA i s found C DOUBLE PRECISION FUNCTION VZ(ALPHA) IMPLICIT REAL*8(A - H,0 - Z) COMMON AK, ANU, GAMMA, PIBY2, ALFDUM, ETA COMMON /SAVEK/ ELIP COMMON /VOLT/ V ALFDUM = ALPHA. ETA = 1.0D0-DEXP(-1.0D0*ALPHA) C DELIKM i s a l i b r a r y e l l i p t i c i n t e g r a l K(1-x) ELIP = DELIKM(DEXP(-1.ODO*ALPHA),IND) CALL FAINT(A) VZ = (V + 1.0D0) - (1.0D0/PIBY2) * DSQRT(1.OD0+GAMMA* 1 ETA) * A RETURN END A.14 S u b r o u t i n e FAINT, FBINT And FCINT And R e l a t e d F u n c t i o n s C SUBROUTINE FAINT(A) C IMPLICIT REAL*8(A - H,0 - Z) COMMON AK, ANU, GAMMA, PIBY2, ALPHA, ETA COMMON /SAVEK/ EL I P EXTERNAL FA T = DSQRT((1.0D0+AK*ETA)/(1.0D0+GAMMA*ETA)) EIP = T * ELIP A = DEXP(ALPHA/2.ODO) * DGAU16(0.ODO,PIBY2,FA) + EIP 154 RETURN END C DOUBLE PRECISION FUNCTION FA(PHI) IMPLICIT REAL*8(A - H,0 - Z) COMMON AK, ANU, GAMMA, PIBY2, ALPHA, ETA ES2 = ETA * (DSIN(PHI)) ** 2 T1 = DSQRT((1.0D0+AK*ES2)/(1.0D0+GAMMA*ES2)) T2 = DSQRT((1.0D0+AK*ETA)/(1.0D0+GAMMA*ETA)) FA = (T1 - T2) / DSQRT(1.0D0+(DEXP(ALPHA) - 1.D0)*( 1DCOS(PHI))**2) RETURN END C SUBROUTINE FBINT(B) C IMPLICIT REAL*8(A - H,0 - Z) COMMON AK, ANU, GAMMA, PIBY2, ALPHA, ETA COMMON /SAVEK/ ELIP EXTERNAL FB T = DSQRT((1.0D0+AK*ETA)*(1.0D0+GAMMA*ETA)) EIP = T * ELIP B = DEXP(ALPHA/2.DO) * DGAU16(0.0D0,PIBY2,FB) + EIP RETURN END C DOUBLE PRECISION FUNCTION FB(PHI) IMPLICIT REAL*8(A - H,0 - Z) COMMON AK, ANU, GAMMA, PIBY2, ALPHA, ETA ' ES2 = ETA * (DSIN(PHI)) ** 2 T1 = DSQRT((1.0D0+AK*ES2)*(1.0D0+GAMMA*ES2)) T2 = DSQRT((1.0D0+AK*ETA)*(1.0D0+GAMMA*ETA)) FB = (T1 - T2) / DSQRT(1.0D0+(DEXP(ALPHA) - 1.D0)*( 1DCOS(PHI))**2) RETURN END C SUBROUTINE FCINT(C) C IMPLICIT REAL*8(A - H,0 - Z) COMMON AK, ANU, GAMMA, PIBY2, ALPHA, ETA COMMON /SAVEK/ ELIP EXTERNAL FC T = DSQRT((1.D0+AK*ETA)*(1.DO+GAMMA*ETA)/(1.DO+ANU* 1 ETA)) EIP = T * ELIP C = DEXP(ALPHA/2.DO) * DGAU16(0.0D0,PIBY2,FC) + EIP RETURN END C DOUBLE PRECISION FUNCTION FC(PHI) IMPLICIT REAL*8(A - H,0 - Z) COMMON AK, ANU, GAMMA, PIBY2, ALPHA, ETA ES2 = ETA * (DSIN(PHI)) ** 2 155 T1 = DSQRT((1.D0+AK*ES2)*(1.D0+GAMMA*ES2)/(1.DO+ANU* 1ES2)) T2 = DSQRT((1.DO+AK*ETA)*(1.DO+GAMMA*ETA)/(1.DO+ANU* 1 ETA)) FC = (T1 - T2) / DSQRT(1.DO+(DEXP(ALPHA) - 1.DO)*( 1DCOS(PHI))**2) RETURN END A.15 F u n c t i o n GAM(T) C C F u n c t i o n f i t t o GAMMA from c a p a c i t a n c e d a t a C DOUBLE PRECISION FUNCTION GAM(T) IMPLICIT REAL*8(A - H,0 - Z) X = DLOG(2.D0-DLOG(T - 33.53D0)) C A = -0.658253255683873D0 B = 0.597035483376152D0 C = 0.037285118112837D0 D = -0.001866105990918D0 E = -0.003583587750158D0 F = -0.000485871203777D0 G = 0.000526397253193D0 C P = A+B*X + C*X*X + D*X*X*X + E*X*X 1*X*X + F*X*X*X*X*X + G*X*X*X*X* 2X*X C GAM = 1.D0 / (P - 0.7D0*X + 1.36D0) RETURN END A.16 F u n c t i o n ENBAR(T) C C F u n c t i o n f o r f i t t o Dunmur d a t a t o g i v e average index C DOUBLE PRECISION FUNCTION ENBAR(T) IMPLICIT REAL*8(A - H,0 - Z) C C A = 0.742523521312106D0 B = 0.056009345076744D0 C = -0.000840433621095D0 C D = -1.939980547157384D0 156 E = 0.294319650826645D0 F = -0.008355771448850D0 G = 0.000079379753787D0 C ENPAR = A + B*T + C*T*T ENPERP = D + E*T + F*T*T + G*T*T*T C ENBAR = (2.0D0*ENPERP + ENPAR) / 3.ODO RETURN END A.17 F u n c t i o n DZFIT(T) C C F u n c t i o n t o e v a l u a t e DZERO from f i t t o o p t i c a l d a t a C DOUBLE PRECISION FUNCTION DZFIT(T) IMPLICIT REAL*8(A - H,0 - Z) C X = DLOG(2.D0-DLOG(T - 33.53D0)) A = 50.476158877086664D0 B = 5.395037153990744D0 C = -1 .025087215412876D0 D = -0.2410467372979D0 E = 0.051954473728999D0 F = 0.02330878762354D0 c DZFIT = A + B*X + C*X*X + D*X*X*X + E*X 1*X*X*X + F*X*X*X*X*X RETURN END A. 18 F u n c t i o n CZFIT(T) C C F u n c t i o n t o deduce CZERO from f i t t o c a p a c i t a n c e d a t a C DOUBLE PRECISION FUNCTION CZFIT(T) IMPLICIT REAL*8(A - H,0 - Z) X = DLOG(2.D0-DLOG(T - 33.53D0)) C A = 0.005439002O97344D0 B = 0.00036605922011ODO C = -0.000035032443337D0 D = -0.000021504535889D0 E = -0.000001795711260D0 F = 0.000001446343032D0 157 FX = A + B*X + C*X*X + D*X*X*X + E*X* 1X*X*X + F*X*X*X*X*X CZFIT = 1.DO / FX RETURN END 158 APPENDIX B ~ DATA TABLES Table I I - R e s u l t s of C a p a c i t a n c e Data F i t s Temp S K ^HFC co °C v o l t s v o l t s P F pF 33.5370 16.05 0.95 1.0794 0.0372 8.56 170.18 457.41 33.5560 9.88 0.66 1 .0024 0.0747 8.51 170.63 455.07 33.6691 3.32 0.30 0.9780 0.0510 3.77 171.85 451.22 33.8264 1 .79 0.25 0.9692 0.0431 2.10 172.88 450.44 34.2147 0.86 0.20 0.9452 0.0289 1.15 174.61 449.60 35.1230 0.49 0.23 0.8966 0.0342 0.79 179.22 448.73 35.8381 0.27 0.20 0.8798 0.0157 0.60 182.46 446.95 36.7519 0.13 0.16 0.8514 0.0285 .0.44 186.01 443.80 37.6426 0.11 0.13 0.8182 0.0311 0.46 190.78 439.84 37.9336 0.09 0.12 0.8092 0.0312 0.46 192.61 438.19 38.3761 0.23 0.15 0.7643 0.0259 0.76 195.94 435.19 38.8270 0.23 0.15 0.7459 0.0309 0.75 199.68' 431.44 39.3084 0.27 0.28 0.7256 0.0468 0.69 203.79 426.22 40.3936 1 .21 0.43 0.6269 0.0438 4.09 228.81' 402.83 159 T a b l e I I I - R e s u l t s of Phase Data F i t s Temp K d O °C v o l t s v o l t s r a d 33 .5334 11. 96 0. 78 1 .3000 0 .0461 56. 889 33 .5370 18. 68 1 . 15 1 .0540 0 .0285 56. 703 33 .5437 16. 84 1 . 06 1 .0200 0 .0295 56. 517 33 .5560 9. 12 0. 70 1 .0612 0 .0289 56. 318 33 .5819 6. 68 0. 68 1 .0070 0 . 1 175 56. 065 33 .6691 3. 19 0. 38 0 .9846 0 .0577 55. 642 33 .6858 3. 18 0. 55 0 .9836 0 .1082 55. 710 33 .7124 2. 48 0. 41 0 .9767 0 .0646 55. 480 33 .7561 2. 10 0. 33 0 .9773 0 .0669 55. 335 33 .7914 1 . 90 0. 31 0 .9709 0 .0650 55. 279 33 .8089 1 . 75 0. 42 0 .9755 0 .0591 55. 086 33 .8264 1 . 65 0. 26 0 .9750 0 .0674 55. 096 34 .1255 0. 91 0. 15 0 .9510 0 .0420 54. 459 34 .2147 0. 85 0. 23 0 .9409 0 .0392 54. 281 35 .1230 0. 26 0. 20 0 .9.103 0 .0320 52. 670 36 .1640 0. 14 0. 18 0 .8697 0 .0669 50. 529 36 .4562 0. 1 1 0. 21 0 .8601 0 .0763 49. 998 36 .4562 0. 1 1 0. 26 0 .8616 0 .0840 50. 014 36 .7519 0. 06 0. 14 0 .8540 0 .0355 49. 400 37 .0319 0. 10 0. 23 0 .8363 0 .0763 48. 809 37 .3356 0. 01 0. 17 0 .8394 0 .0559 47. 906 37 .6426 0. 13 0. 14 0 .8055 0 .0231 47. 697 37 .9336 0. 20 0. 14 0 .7909 0 .0253 46. 560 37 .9758 0. 04 0. 15 0 .8093 0 .0474 46. 324 38 .3761 0. 05 0. 16 0 .7905 0 .0391 45. 193 38 .6113 0. 02 0. 15 0 .7857 0 .0465 44. 275 38 .8270 0. 04 0. 14 0 .7718 0 .0392 43. 826 39 .0665 0. 06 0. 15 0 .7571 0 .0465 42. 749 39 .3084 0. 06 0. 23 0 .7459 0 .0415 41 . 471 39 .5529 - 0 . 02 0. 15 0 .7400 0 .0246 40. 387 39 .7778 - 0 . 01 0. 14 0 .7241 0 .0445 39. 093 40 .0034 - o . 06 0. 13 0 .7100 0 .0319 37. 445 40 .2325 - o . 01 0. 16 0 .6801 0 .0437 35. 373 40 .3936 - 0 . 12 0. 18 0 .6619 0 .0386 33. 287 160 T a b l e IV - E l a s t i c t y p e Temp Kj| ±. °C 10 N F 33. 5334 14. 014 F 33. 5370 9. 172 C 33. 5370 9. 619 F 33. 5437 8. 548 F 33. 5560 9. 202 C 33. 5560 8. 21 1 F 33. 5819 8. 234 F 33. 6691 7. 795 C 33. 6691 7. 691 F 33. 6858 7. 771 F 33. 7124 7. 650 F 33. 7561 7. 642 F 33. 7914 7. 530 F 33. 8089 7. 596 F 33. 8264 7. 583 C 33. 8264 7. 493 F 34. 1255 7. 151 F 34. 2147 6. 986 C 34. 2147 7. 050 F 35. 1230 6. 417 C 35. 1230 6. 225 C 35. 8381 5. 896 F 36. 1640 5. 710 F 36. 4562 5. 556 F 36. 4562 5. 537 F 36. 7519 5. 405 C 36. 7519 5. 373 F 37. 0319 5. 132 F 37. 3356 5. 108 F 37. 6426 4. 641 C 37. 6426 4. 788 F 37. 9336 4. 410 C 37. 9336 4. 617 F 37. 9758 4. 609 F 38. 3761 4. 298 C 38. 3761 4. 017 F 38. 61 13 4. 181 F 38. 8270 3. 972 C 38. 8270 3. 710 F 39. 0665 3. 749 F 39. 3084 3. 556 C 39. 3084 3. 366 F 39.5529 3. 405 F 39. 7778 3. 158 F 40. 0034 2. 908 F 40. 2325 2. 497 F 40. 3936 2. 191 C 40. 3936 1 . 966 Constant R e s u l t s -11 -12 - i t 10 N 10 N 10 N 0.99 181.66 16.89 0.50 180.48 14.38 0.66 164.00 14.55 0.49 152.51 1 2.68 0.50 93. 17 8.23 1 .22 89.35 14.38 1 .92 63.25 15.79 0.91 32.67 4.83 0.80 33.26 4.16 1.71 32.52 8.32 1 .01 26.60 4.73 1 .05 23.70 4.08 1.01 21 .84 3.74 0.92 20.92 4.10 1 .05 20.07 3.42 0.67 20.89 2.65 0.63 13.63 1 .59 0.58 12.93 1 .94 0.43 13.14 1 .60 0.45 8.11 1 .42 0.48 9.27 1 .58 0.21 7.48 1 .23 0.88 6.52 1 .42 0.98 6.15 1 .60 1 .08 6.17 1 .86 0.45 5.71 0.87 0.36 6.08 0.93 0.94 5.65 1 .59 0.68 5.18 1 .09 0.27 5.26 0.70 0.36 5.31 0.73 0.28 5.28 0.69 0.36 5.03 0.67 0.54 4.80 0.88 0.43 4.51 0.82 0.27 4.95 0.68 0.50 4.25 0.79 0.40 4.14 0.71 0.31 4.58 0.67 0.46 3.96 0.74 0.40 3.77 0.93 0.43 4.28 1.08 0.23 3.34 0.57 0.39 3.12 0.58 0.26 2.75 0.46 0.32 2.47 0.51 0.26 1 .93 0.46 0.27 4.35 1.04 161 REFERENCES 1. H. K e l k e r , M o l . C r y s t . L i q . C r y s t . , 2J_, 1 ( 1973). 2. P. G. de Gennes, The P h y s i c s Of L i q u i d C r y s t a l s , C l a r e n d o n P r e s s , O x f o r d (1974). 3. S. Chandrasekhar, L i q u i d C r y s t a l s ,Cambridge U. P r e s s , Cambridge (1977). 4. L. M. B l i n o v , E l e c t r o - o p t i c a l And M a g n e t o - o p t i c a l  P r o p e r t i e s of L i q u i d C r y s t a l s , W i l e y , New York (1983) . 5. W. H. de J e u , P h y s i c a l P r o p e r t i e s Of L i q u i d C r y s t a l l i n e  M a t e r i a l s , Gordon and Breach, New York (1979). 6. W. M a i e r , A. Saupe, Z. N a t u r f o r s c h , 14a, 882 (1959). Z. N a t u r f o r s c h , 15a, 287 ( i 9 6 0 ) . 7. J . Cognard, Al i g n m e n t Of Nematic L i q u i d C r y s t a l s And T h e i r  M i x t u r e s (MoT] C r y s t . L i q . C r y s . sup. 1), Gordon and B r e a c h , New York (1982). 8. C. W. Oseen, T r a n s . Faraday S o c , 29, 883 (1933). F. C. Frank, D i s c . Faraday S o c , 25, 19 (1958). 9. J . D. L i t s t e r , R. J . B i r g e n e a u , M. K a p l a n , C. R. S a f i n y a , J . A l s - N i e l s e n , Order In S t r o n g l y F l u c t u a t i n g Condensed  M a t t e r Systems , T. R i s t e ed., Plenum, New York (1980). 10. P. G. de Gennes, S o l i d S t . Comm., J_0, 753 ( 1972). 11. P. G. de Gennes, S u p e r c o n d u c t i v i t y In M e t a l s And A l l o y s , W. A. Benjamin I n c . , New York (1966). 12. M. R. F i s c h , P. S. P e r s h a n , L. B. Sorensen, Phys. Rev. A, 29, 2741 (1984). 13. H. B i r e k i , R. S c h a e t z i n g , F. R o n d e l e z , J . D. L i t s t e r , Phys. Rev. L e t t . , 36, 1376 (1976). 14. N. V. Madhusudana, B. S. S r i k a n t a , M o l . C r y s t . L i q . C r y s t . , 99, 375 (1973). 15. A. C a i l l e , Comptes rendus Ac. Sc. P a r i s , 274B, 891 (1972). 16. G. G r i n s t e i n , R. A. P e l c o v i t s , Phys. Rev. A, 26, 915 (1982). 17. T. C. Lubensky, J . de Chim. Phys., 80, 31 (1983). 18. A. Schmid, Phys. Rev., _1_80, 527 (1968). 162 19. Shang-Ken Ma, Modern Theory Of C r i t i c a l Phenomena , W. A. Benjamin I n c . , New York (1976). 20. C. Gooden, R. Mahmood, D. B r i s b i n , A. B a l d w i n , D. L. Johnson, M. E. Neubert, Phys. Rev. L e t t . , 5_4, 1035 (1985). 21. R. Mahmood, D. B r i s b i n , I . Khan, C. Gooden, A. B a l d w i n , D. L. Johnson, M. E. Neubert, Phys. Rev. L e t t . , 54, 1031 (1985). 22. S. S p r u n t , L. Solomon, J . D. L i t s t e r , Phys. Rev. L e t t . , 53, 1923 (1984). 23. B. M. Ocko, R. J . B i r g e n e a u , J.D. L i t s t e r , Phys. Rev. L e t t . , 52, 208 (1984). 24. C. W. G a r l a n d , M. M e i c h l e , B. M. Ocko, A. R. K o r t a n , C. R. S a f i n y a , L. J . Yu, J . D. L i t s t e r , R. J . B i r g e n e a u , Phys. Rev. A, 27, 3234 (1983). 25. I . J a n o s s y , L. B a t a , A c t . Phys. P o l . , A54, 643 (1978). 26. H. B i r e k i , J . D. L i t s t e r , M o l . C r y s t . L i q . C r y s t . , 42, 1043 (1977). 27. R. S. P i n d a k , C. Huang, J . T. Ho, Phys. Rev. L e t t . , 32, 43 (1974). 28. L. L e g e r , Phys. L e t t . , 44A, 535 (1973). 29. M. D e l a y e , R. R i b o t t a , G. Durand, Phys. Rev. L e t t . , 31, 443 (1973). 30. L. Cheung, R. B. Meyer, H. G r u l e r , Phys. Rev. L e t t . , 3J_, 349 (1973). 31. L. Cheung, R. B. Meyer, Phys. L e t t . , 43A, 261 (1973). 32. P. E. C l a d i s , Phys. Rev. L e t t . , 3J_, 1200 (1973). 33. V. F r e e d e r i c k s z , V. Z o l i n a , T r a n s . Faraday S o c , 23, 919 (1933). 34. H. J . D e u l i n g , L i q u i d C r y s t a l s , S o l i d S t a t e P h y s i c s Supplement 14, L. L i e b e r t ( e d . ) , Academic P r e s s , New York (1978). 35. V. G. C h i g r i n o v , Sv. Phys. C r y s t a l l o g r . , 27, 245 (1982). 36. H. J . D e u l i n g , M o l . C r y s t . L i q . C r y s t . , J J 3 , 123 (1972). 163 37. Hp. Schad, B. S c h e u b l e , J . N e h r i n g , J . Chem. Phys., 7J_, 5140 (1979). 38. C. Maze, Mol . C r y s t . L i q . C r y s t . , 48, 273 (1978). 39. T. U c h i d a , Y. T a k a h a s h i , M o l . C r y s t . L i q . C r y s t . L e t t . , 72, 133 (1981). 40. M. G. C l a r k , E. P. Raynes, R. A. Smith, R. J . A. Tough, J . Phys. D, j_3, 2151 ( 1 9 8 0 ) . 41. R. J . A. Tough, E. P. Raynes, Mol . C r y s t . L i q . C r y s t . L e t t . , 56, 19 (1979). 42. C. Maze, D. Johnson, M o l . C r y s t . L i q . C r y s t . , 3_3, 213 (1976). 43. D. A. B a l z a r i n i , D. A. Dunmur, P. P a l f f y - M u h o r a y , M o l . C r y s t . L i q . C r y s t . L e t t . , J_02, 35 ( 1 984). 44. M. Abramowitz, I . Stegun, Handbook of M a t h e m a t i c a l  F u n c t i o n s , Dover, New York (1970), p.591 45. F. B r o c h a r d , M o l . C r y s t . L i q . C r y s t . , 23, 51 (1973). 46. T. Motooka, A. Fukuhara, J . A p p l . Phys., 50, 6907 (1979). J . A p p l . Phys. L e t t . , 34, 305 (1979). 47. H. Mada, M o l . C r y s t . L i q . C r y s t . L e t t . , 82, 53 (1982). 48. R.B. Meyer, Phys. Rev. L e t t . , 22, 918 (1969). 49. H. J . D e u l i n g , M o l . C r y s t . L i q . C r y s t . , 26, 281 (1974). S o l . S t . Comm., J_4, 1073 ( 1974). 50. S. A. Casalnuovo, R. C. M o c k l e r , W. J . O ' S u l l i v a n , Phys. Rev. A, 29, 257 ( 1 9 8 4 ) . 51. D. B a l z a r i n i , A. Rosenberg, P. P a l f f y - M u h o r a y , Can. J . Phys., 61_, 1060 ( 1983). 52. E. M. Forgan, C r y o g e n i c s , J_4, 207 (1974). 53. D. A. Dunmur, M. R. M a n t e r f i e l d , W. H. M i l l e r , J . K. Dunleavy, M o l . C r y s t . L i q . C r y s t . , 46, 127 (1978). 54. R. G. P r i e s t , M o l . C r y s t . L i q . C r y s t . , 129 (1972). 1 64 55. P. R. B e v i n g t o n , Data R e d u c t i o n and E r r o r A n a l y s i s f o r the  P h y s i c a l S c i e n c e s , M c G r a w - H i l l , New York (1969). 56. K. C. Chu, W. L. M c M i l l a n , Phys. Rev. A, J_5, 337 (1977). 57. S. J . M a j o r o s , D. J . B r i s b i n , D. L. Johnson, M. E. Neubert, Phys. Rev A, 20, 1619 (1979). 58. P. E. C l a d i s , S. T o r z a , J . A p p l . Phys., 46, 584 (1975). 

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