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Coexistence curve of sulfur hexafluoride in the critical region. Ohrn, Kenneth Edward 1972

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(i)  THE COEXISTENCE CURVE OF SULFUR HEXAFLUORIDE IN THE CRITICAL REGION by KENNETH EDWARD OHRN B . A . S c , U n i v e r s i t y o f B r i t i s h Columbia, 1966  A THESIS SUBMITTED IN PARTIAL FULFIMLENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in ENGINEERING  PHYSICS  i n t h e Department of PHYSICS  We a c c e p t t h i s required  t h e s i s as conforming t o t h e  s t a n d a r d from c a n d i d a t e s f o r t h e  degree o f MASTER OF APPLIED SCIENCE  THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1972  In  presenting  this  thesis  an advanced degree at the L i b r a r y s h a l l I  f u r t h e r agree  in p a r t i a l  fulfilment of  the U n i v e r s i t y of B r i t i s h  make i t f r e e l y a v a i l a b l e  that permission  for  the requirements f o r  Columbia,  I agree  r e f e r e n c e and  f o r e x t e n s i v e copying o f  this  study. thesis  f o r s c h o l a r l y purposes may be granted by the Head of my Department by h i s of  this  written  representatives. thesis  for financial  gain s h a l l  Pf)-f5>ICS  The U n i v e r s i t y o f B r i t i s h Vancouver 8, Canada  Ift  ^AR  12  or  i s understood that copying or p u b l i c a t i o n  permission.  Department of  Date  It  that  Columbia  not be allowed without my  (ii)  ABSTRACT  T h i s t h e s i s s t u d i e s t h e shape o f the c o e x i s t e n c e curve of s u l f u r h e x a f l u o r i d e i n the c r i t i c a l  region.  The d i f f e r e n c e i n index o f r e f r a c t i o n between  the l i q u i d  and vapour phases i s shown t o be p r o p o r t i o n a l t o t h e difference i n density. " |3 " i s measured. f i t s to log-log  Thus t h e c r i t i c a l  exponent  These v a l u e s were found from  linear  data:  /3 = 0.347 +_ 0.002  - £ > 10* r  y3 * 0.339 +_ 0.003  - £ < 3 x 10  2  -3  T—Tc Here, "Tc" i s t h e c r i t i c a l  temperature and C =  T  c  The temperature range c o v e r e d i s 3 x 10" 4 - 6 6  The c r i t i c a l  <  index o f r e f r a c t i o n  with the r e s u l t  n  c  - 1.093 + 0.002  6 x 10"  2  .  ( n ) i s measured, c  (iii)  TABLE OF CONTENTS  I.  Introduction.  II.  History.  III.  A.  General  Introduction.  B.  Classical  C.  Modern  Theory.  D.  Recent  Experiments.  Theory.  Description of Optics. A.  Qualitative  B.  I n f o r m a t i o n from the F r a u n h a f e r  IV.  The L o r e n t z - L o r e n z  V.  A n a l y s i s of Optics. A. B.  VI.  VII.  Explanation.  Relationship.  C o e x i s t e n c e Curve. Isotherms.  Experimental  Details.  A.  The Sample C e l l and F l u i d .  B.  The F i l l i n g  C.  Temperature C o n t r o l .  D.  Technique  Process.  and Other Equipment.  Index o f R e f r a c t i o n Measurement. A.  General D e s c r i p t i o n .  B.  Experimental  C.  Prism Index o f R e f r a c t i o n .  D.  A n a l y s i s of Optics.  E.  E x t e r n a l Angle Measurement.  Technique.  Pattern.  (iv)  VII.  Index o f R e f r a c t i o n Measurement, (cont'd.) F.  Data  Analysis.  G.  Final  H.  S i g n i f i c a n c e of Results.  Results.  V I I I . Data A n a l y s i s . IX.  Error  Discussion.  (v)  LIST OF TABLES Page I.  D e f i n i t i o n of C r i t i c a l  Exponents.  14  II.  C l a s s i c a l and I s i n g Model V a l u e s o f C r i t i c a l  15  Exponents. III.  The C r i t i c a l  Exponent " j 3 .  18  IV.  Prism Index o f R e f r a c t i o n f o r  V.  L e a s t Square F i t o f S F  VI.  V a r i a t i o n o f Chosen Tc w i t h Assumed  n  g  A = 6328 A.  R e f r a c t i v e Index Data. Q Value.  49 55 63  1  I.  INTRODUCTION. T h i s t h e s i s i s the examination  of the coexistence  curve o f s u l f u r h e x a f l u o r i d e i n the c r i t i c a l An o p t i c a l i n t e r f e r e n c e t e c h n i q u e  region.  i s used t o measure  the d i f f e r e n c e i n index o f r e f r a c t i o n between t h e l i q u i d and vapour phases.  The c r i t i c a l  index o f r e f r a c t i o n and  the sum o f t h e l i q u i d and vapour i n d i c e s a r e a l s o measured w i t h a wedge-shaped f l u i d The  sample.  o p t i c a l method used a l l o w s c o n s i d e r a b l e  p r e c i s i o n t o be reached  i n the d e t e r m i n a t i o n o f t h e  shape o f the c o e x i s t e n c e  curve.  A more thorough i n t r o d u c t i o n t o c r i t i c a l  phen(29)  omena can be o b t a i n e d from t h e a r t i c l e s o f F i s h e r Sette  ( 3 1 )  (1969),  Smith  and H e l l e r ^ ^ ( 1 9 6 7 ) . 3 0  ( 3 2 }  ( 1 9 6 9 ) , Kadanoff e t  al  (1967), ( 5 )  (1967)  These l e n g t h y a r t i c l e s o f f e r a  d e t a i l e d p i c t u r e o f t h e o r y and experiment i n c r i t i c a l phenomena.  2 II.  HISTORY A.  General Critical  Introduction. phenomena r e s e a r c h began c i r c a  when Thomas A n d r e w s ^ r e p o r t e d he c o u l d not  liquify  t h a t , above 304°K.  carbon d i o x i d e .  S i m i l a r observ-  a t i o n s on o t h e r m a t e r i a l s showed t h i s b e h a v i o r b a s i c p r o p e r t y common t o most m a t e r i a l s . continued  on these h i g h e r - o r d e r  The critical of  1869  t o be  Research  has  phase t r a n s i t i o n s .  g e n e r a l b e h a v i o r a l s i m i l a r i t i e s known as  phenomena are shown by w i d e l y d i f f e r e n t  phase t r a n s i t i o n s :  liquid-gas;  order-disorder transformations magnetic-paramagnetic systems; paramagnetic systems; electrics;  a  binary  mixtures;  i n binary alloys;  ferro-  anti-ferromagnetic-  polarized-unpolarized ferro-  super conductors-normal c o n d u c t o r s ;  superfluid-normal  kinds  and  fluid.  T h i s t h e s i s i s concerned with the l i q u i d - g a s t r a n s i t i o n o f a simple f l u i d . therms o f a t y p i c a l f l u i d where two  The  curve  The  composition of  we  i s the l o c u s of p o i n t s g i v i n g the  of the l i q u i d  and  vapour phases.  c u r v e u s i n g van der Waals' e q u a t i o n rule.  are shown  i s the c r i t i c a l r e g i o n .  t h i s c h a p t e r d e r i v e s the shape of the  equal a r e a  iso-  find a region  Such isotherms  dotted curve's'top  itself  examine the  such as SFg,  phases c o - e x i s t .  in F i g . l .  I f we  and  Section B  coexistence the Maxwell  3  Fig. 1  The g e n e r a l b e h a v i o r o f the i s o t h e r m a l c u r v e s o f a r e a l gas, and the g e n e r a l shape of the c o e x i s t e n c e c u r v e .  4 T h i s t h e s i s seeks t o measure the shape of the c o e x i s t e n c e c u r v e o f SF^ t o a h i g h e r degree o f a c c u r a c y than p r e v i o u s  experiments.  S i m i l a r coexistence curves e x i s t  i n other  systems.  C e r t a i n p a i r s o f f l u i d s , c a l l e d b i n a r y f l u i d s , mix any p r o p o r t i o n above a c r i t i c a l Below t h i s temperature, whose c o m p o s i t i o n  temperature  in  (Tc).  they s e p a r a t e i n t o two .phases  i s g i v e n by a c o e x i s t e n c e c u r v e .  The  d i f f e r e n c e i n t h e i r c o n c e n t r a t i o n s decreases with i n c r e a s i n g temperature  and d i s a p p e a r s a t the c r i t i c a l  F i g . 2 a shows such a c o e x i s t e n c e c u r v e f o r a b i n a r y  point. fluid  system. A f e r r o m a g n e t i c m a t e r i a l (Fig.2b) a t low  temp-  e r a t u r e m a i n t a i n s a magnetic moment per u n i t volume when the magnetic f i e l d  i s removed.  d e c r e a s e s r e g u l a r i l y as temperature goes t o z e r o a t the C u r i e  behaviour  i s raised  and  temperature.  Similar observations types of c r i t i c a l  Such m a g n e t i z a t i o n  :  can be made f o r o t h e r  systems and the s i m i l a r i t y i n the  of q u a n t i t i e s d e s c r i b i n g t h e i r  equilibrium  p r o p e r t i e s can be e s t a b l i s h e d . The r e v i v a l o f i n t e r e s t  in critical  has both t h e o r e t i c a l and e x p e r i m e n t a l b a s i s .  phenomena Improve-  ments on the van der Waals e q u a t i o n have proved s i n c e a dense gas or a f l u i d ally.  difficult  i s i n t r a c t a b l e mathematic-  The major o b s t a c l e , e v a l u a t i o n o f c o n f i g u r a t i o n a l  i n t e g r a l s , has  seen some p r o g r e s s r e c e n t l y .  Also,  5  »  X£  O  Coneentration of F l u i d A (X ) C o e x i s t e n c e c u r v e of a b i n a r y f l u i d  1  A  Fig.- 2a  1 Temp  i  /  /  Ti M  i n - F i g . 2b  Spontaneous - m a g n e t i z a t i o n  \  system  6 thermodynamical occurred.  problems i n t h e c r i t i c a l  F o r example,  r e g i o n have  the T a y l o r s e r i e s expansion  of  the f r e e energy about the c r i t i c a l  p o i n t has proven  to  not be a wise t h e o r e t i c a l p r o c e d u r e . U s e f u l c r o s s - f e r t i l i z a t i o n between branches o f  p h y s i c s has o c c u r r e d .  F o r example,  Onsager's e x a c t  s o l u t i o n o f the two-dimensional I s i n g model o f a f e r r o magnet may  be t r a n s l a t e d simply and d i r e c t l y i n t o t h e  q u i v a l e n t model i n f l u i d lattice  p h y s i c s , t h e two-dimensional  gas. These important l i n k s , the new  theoretical  under-  s t a n d i n g and the r e f i n e m e n t o f e x p e r i m e n t a l t e c h n i q u e s have made c r i t i c a l  phenomena a s u b j e c t o f r e v i v e d  interest.  7  B.  Classical  Theories.  C l a s s i c a l t h e o r i e s , based on t h e e x i s t e n c e o f a t t r a c t i v e f o r c e s between p a r t i c l e s , c o n s i d e r  cooperative  e f f e c t s and a r e a b l e t o p r e d i c t t h e e x i s t e n c e o f c r i t i c a l points.  T h e o r i e s o f t h i s type a r e : van d e r Waals'  t h e o r y f o r f l u i d s i n t e g r a t e d with t h e Maxwell equal r u l e i n t h e two-phase r e g i o n ; t h e f e r r o m a g n e t i c based on the Weiss n o t i o n o f m o l e c u l a r  field  Bragg and W i l l i a m s t h e o r y f o r s u b s t i t u t i o n a l  area  theory  and the solid  solutions. To show how c l a s s i c a l t h e o r i e s can p r e d i c t critical  p o i n t s , c o n s i d e r t h e f o l l o w i n g development o f  van d e r Waals'  equation:  ( ? + a p * X p - b ) = KT We d e f i n e Pc, Tc, pc as t h e c r i t i c a l  .  (ii-D  pressure,  temper-  a t u r e and d e n s i t y and we r e w r i t e E q u a t i o n  We equate c o e f f i c i e n t s o f E q u a t i o n ing  (II-l):  ( I I - 2 ) and t h e f o l l o w -  "equation  (II-3)  8 These e q u a t i o n s r e s u l t :  ~3p  = -£  c  (II-4)  ,  3pi - %. + 1£Tc -pc r Equation  (n-5)  = ~-% <xb (II-4)  .  (II-6)  g i v e s , on i n s p e c t i o n :  . b p c ^ 7}  .  (II-7)  Combining E q u a t i o n s ( I I - 5 ) and ( I I - 6 ) t o e l i m i n a t e Pc, we  obtain:  S u b s t i t u t i n g Equation  "RT C  ~  S i n c e Equation  ( I I - 7 ) , we  8a.  (II-9)  .  (II-7) g i v e s :  we can r e w r i t e E q u a t i o n  ape -  obtain:  (II-9) to g i v e :  RTc  S u b s t i t u t i n g Equation  (II-7) i n t o Equation  (II-6),  we  9 obtain:  a zi b Using Equations  (11-10)  x  ( I I - 7 ) , ( I I - 9 ) , (11-10) i n t h e f o l l o w i n g  forms:  , and m u l t i p l y i n g  ex.~ 3?c , Equation  2jk 3p e  t  ( I I - l ) by 2 7b/a,  we  obtain:  (n-ii)  This  gives:  i- _|^)_3(p/ 8  Equation  f t  f.F(^y  (11-12)  r c  (11-12) i s a u n i v e r s a l e q u a t i o n o f  s t a t e | i m p l y i n g a "law o f c o r r e s p o n d i n g  states".  two f l u i d s w i t h the same v a l u e s o f P/Pc, T/Tc, may be s a i d t o be i n c o r r e s p o n d i n g A double about p/jP  c  power  Any  p/pc  states.  s e r i e s expansion  of Equation  (11-12)  = T/Tc = 1 g i v e s :  tn  (11-13)  Tc  Here, P'(T/Tc) has two i n t e r p r e t a t i o n s . i s the p r e s s u r e on the c r i t i c a l  isochore.  If T>Tc, i t If T<Tc,  10 it  i s t h e vapour p r e s s u r e on t h e c o e x i s t e n c e  curve:  When T < Tc, we s e t :  t [Pjtf -  6  =  (^  =  T  O  (11-14)  and we g e t :  (11-15)  Equation  (11-15) g i v e s , upon rearrangement:  p«  p [l t c  2 (-6^]  (11-16)  Tc  a t P - p isotherms  Looking can  we  V  see t h a t t h e two r o o t s o f  t h i s equation describe the  /  - p j  i f J  p o i n t s on t h e isotherms o c c u p i e d by t h e a c t u a l l i q u i d or. vapour d e n s i t y .  / i  !  Maxwell's  equal area r u l e ensures t h a t ,  pt  f o r example, AB=BC and t h e r e f o r e t h a t t h e symmetry  i m p l i e d by E q u a t i o n  i n d i c a t i v e o f t h e shape o f t h e c o e x i s t e n c e  p  (11-16) i s curve.  We may t h e r e f o r e w r i t e :  p L *  p  v  =  Pc(l pc(l-2(-£V  (11-17) / a  J  (11-18)  T h i s g i v e s us the more u s u a l f o r m u l a t i o n of the c o e x i s t e n c e curve.  or  p -pv  ~  L  4(-eV^  The e q u a t i o n  .  (ii-i9)  o f van der Waals g i v e s us another  parameter - t h e c o e f f i c i e n t o f i s o t h e r m a l c o m p r e s s i b i l i t y » i 3 it Po •  on the c r i t i c a l Pc PjO  isochore. i s dimensionless  *  6"  •  showing the d i v e r g e n c e ibility  and:  (11-20)  o f the c r i t i c a l  i s o c h o r e compress-  as T —>• T c . (2) Data c o m p i l a t i o n by E.A. Guggenheim  i n 1945  suggested t h a t a r e p r e s e n t a t i o n o f the c o e x i s t e n c e curve c o u l d be: pL~pv  -  3.5  (-€.) °  .  (11-21)  T h i s e m p i r i c a l e q u a t i o n o b t a i n s from a r e d u c e d - v a r i a b l e p l o t o f a l l a v a i l a b l e d a t a on v a r i o u s g a s e s .  Thus c l a s -  12 s i c a l theory f a i l s f o r f l u i d s . In fields  1907, P i e r r e Weiss suggested  i n ferromagnets  £  internal  and d e r i v e d :  r / ( x ) « £ * ( l - f e * % . . . )  Here JX = average jiL  large  -  (H-22)  p r o j e c t e d magnetic moment,  = m o l e c u l a r d i p o l e moment, L ( x ) = Langevin's  function,  = c o t h ( x ) - 1/x and  WT  T  I W  (n-23)  I f we expand L ( x ) i n a power s e r i e s about x=0, an analogy d e v e l o p s between Weiss' f e r r o m a g n e t i c e q u a t i o n and van der Waals' gas e q u a t i o n .  JTTC.  O  \  TC  We o b t a i n :  15  I f we' n e g l e c t h i g h e r o r d e r terms, vanishes. occur,  ;  (11-24)  the b r a c k e t e d  Then spontaneous m a g n e t i z a t i o n  factor  (H=0) can  with:  "3"  T h i s i s t o be compared t o E q u a t i o n van d e r Waals e q u a t i o n .  (n-25)  (II-7), the  A f u r t h e r analogy can be drawn. susceptibility X  e v a l u a t e the i n i t i a l by u s i n g E q u a t i o n  I f T > T c , we c  in a  may  ferromagnet  (11-12):  (11-26)  T h i s i s the C u r i e - W e i s s Equation  law and i s t o be compared t o  (11-20). The two  correspond  symmetric r o o t s which o c c u r  t o m a g n e t i z a t i o n "up"  or "down".  No e x t e n s i o n or r e f i n e m e n t o f c l a s s i c a l t h e o r y a b l e t o account  f o r the c o e x i s t e n c e or m a g n e t i z a t i o n  exponents b e i n g o t h e r than  N e i t h e r was  a c c o u n t i n g f o r the c o m p r e s s i b i l i t y or d i v e r g i n g f a s t e r than  (T -  there  was curves'  any  susceptibility  T^" . 1  Thus c l a s s i c a l t h e o r y f a i l e d and more modern developments were needed.  C.  Modern Theory. The  first  s t e p away from the c l a s s i c a l t h e o r i e s  came with Onsager's paper on the two-dimensional model i n zero f i e l d .  Each atom on a l a t t i c e s i t e  be i n a s p i n up or a s p i n down s t a t e . between n e i g h b o u r i n g  p a i r s o n l y and  p o i n t , the s p e c i f i c  heat  can  Interaction i s  i s +_ J depending  whether the s p i n s are p a r a l l e l o r a n t i p a r a l l e l . taneous m a g n e t i z a t i o n i s p r e d i c t e d .  Ising  At the  on  Spon-  critical  i s logarithmically i n f i n i t e .  No exact s o l u t i o n has been found  f o r the t h r e e  14 d i m e n s i o n a l model.  Computer approximations  have been  made whose r e s u l t s are u s u a l l y s t a t e d as l i m i t i n g ponents.  If  *-*o  we  ex-  d  -x.  IUK  y  (11-2 7)  say,  -to -  x—>  as  O  (11-28)  F o l l o w i n g c o n v e n t i o n s e s t a b l i s h e d by M.E.  Fisher  a s e t of l i m i t i n g exponents are s e t out i n T a b l e I .  TABLE 1 D e f i n i t i o n of C r i t i c a l Exponent  Magnet  Mo £  j3 T  Fluid  Ap Po  (-6^  (€>0)  ir' (6<0) A (€>o)  A>  ^ (- r* C M » (-fey*' CM  *« S_  Exponents  H  (fi-o)  Here,-£ ment o f j3  =  T  c  * .  Cv Cv  6  M  cr  8  T h i s t h e s i s i s the measure-  f o r a f l u i d , using  a p r e c i s i o n o p t i c a l method. (4)  T.D.  Lee and C H .  Yang  show t h a t the mathemat-  i c a l d i s c u s s i o n of the l a t t i c e gas model i s isomorphic, t o the magnetic problem  i n which a s p i n p o i n t s up or down.  T h i s s t r e n g t h e n s the f l u i d - m a g n e t analogy.  In f a c t ,  analogies to superconductors, s u p e r f l u i d s , f e r r o e l e c t r i c s , b i n a r y a l l o y s , and same c o n t e x t ^ \  b i n a r y f l u i d s can Thus the  be  modern e r a  discussed i n  of  the  critical  phenomena b e g i n s . In T a b l e I I are  listed  the  classical  values  (28) derived  by  potential  Landau  , by  expanding the  i n a power s e r i e s about the  A l s o r e s u l t s o f computations u s i n g the are shown.  thermodynamic  critical  point. .  l a t t i c e gas  model  TABLE I I C l a s s i c a l and  I s i n g Model V a l u e s o f C r i t i c a l  Exponent  Landau Theory  r<t S  These v a l u e s are  h  0.313  + 0.004  1  1.250  +.  1  1.31  3  5.2  i n 1965  0.001  +  0.05  ,+  0.15  i n t r o d u c e d the  theoretical discussion.  proposed between the  d+  vSince  Ising  from Kadanoff e t a l . (1967)  B. Widom t o the  3-D  Exponents  critical  r+  ap  0 , ^ , ^ ' and  -  &  Certain  "scaling  relationships  law" are  exponents.  Z  can  ,  (11-29)  i n p r i n c i p l e be  measured  16 by the method o f t h i s t h e s i s , a p r e c i s e t e s t o f the scaling  laws i s p o s s i b l e .  D.  Recent  Experiments.  As i n t e r e s t  in critical  methods d e v e l o p and new  p r e s s u r e and d e n s i t y ,  near t o the c r i t i c a l  temperature.  Good r e s u l t s have been o b t a i n e d by Weinberger Schneider  ( 7)  new  a n a l y s e s o f o l d d a t a a r e done.  C o n v e n t i o n a l methods, measuring become v e r y d i f f i c u l t  phenomena grows,  and Habgood and  (8)  Schneider  and  f o r Xenon  (9) and a l s o f o r SF^ by Atack and of  Schneider  .  The  precision  t h e s e methods, however, l e f t much t o be d e s i r e d i n  the r e g i o n near  critical.  O p t i c a l methods a r e c o n v e n i e n t f o r the study o f fluids,  and c o n s i d e r a b l e i n g e n u i t y has gone i n t o o t h e r  methods.  0. M a a s s ^ ^ 10  Weinberger and  studied C0  2  w i t h bouyant b a l l s  S c h n e i d e r ^ ^ measured the count r a t e 1 1  and as  a f u n c t i o n o f h e i g h t i n a s e a l e d sample o f Xenon mixed with a r a d i o a c t i v e t r a c e r .  G.D'Abramo, F.P.  Ricci  and  (12) F. Menzinger s t u d i e d the Ga-Hg b i n a r y f l u i d system by a method which i s e s s e n t i a l l y n e u t r o n r a d i o g r a p h y . (13) H.L.  Lorentzen  s t u d i e d CO^  near  critical  u s i n g a p r i s m a t i c v e s s e l w i t h v e r t i c a l prism  axis.  He measured a h o r i z o n t a l r e f r a c t i o n a n g l e Q (Z) proportional to by Schmidt,  |0(z).  Traube and  A s i m i l a r method has been used Straub  ( 1 4 - 1 6 )  .  H . Palmer  ( 1 7 }  used a p l a n e - p a r a l l e l - w i n d o w e d optical  v e s s e l i n a Z-Type S c h l i e r e n  system t o study ethane,  CC>2» and Xenon.  He  n e g l e c t e d the c o e x i s t e n c e c u r v e . T h i s t h e s i s uses a prism method s i m i l a r t o t h a t o f L o r e n t z e n t o measure the i n d i c e s o f r e f r a c t i o n of the l i q u i d and vapour phases o f SFg as a f u n c t i o n o f  temperature.  T a b l e I I I i s a summary of d e t e r m i n a t i o n s of the critical  exponent S  f o r v a r i o u s systems and methods.  18 TABLE I I I THE CRITICAL EXPONENT " j} "  VALUE  SYSTEM  REFERENCE  0.333  compilation  Guggenheim  (2K1945)  0.33  Xenon  Weinberger and Schneider  (7)(1952)  0.345 .+ 0.015  Xenon-analysis o f (7)  0.344 +. 0.01  co  0.346 +_ 0.008  Xenon  D.A. B a l z a r i n i  0.362 0.370  compilation f o r fluids compilation f o r ferromagnets  M. V i c e n t i n i Missoni et a l  0.354 + 0.010  He  Roach and  2  Fisher Lorentzen  4  (3)(1964) (13)(1953) (18)(19) (1968) (20)(1970)  Douglas  (21)(1966)  Ho and L i t s t e r  (22)(1969)  0.368 + 0.005  CrBr  0.349 + 0.006  N 0 ~~  0.348 +_ 0.002  co  0.354 +, 0.007  CC1F-  0.340 + 0.010  3-MethylpentaneN i t r o ethane  Wims, M c l n t y r e and Hynne  0.373 + 0.005  Ga-Hg  D'Abramo, R i c c i (12X1972) and Menzinger  3  2  2  3  J  Sengers, S t r a u b (23K1971) and ViscentiniMissoni (24X1969)  19 III.  DESCRIPTION OF OPTICS A.  Qualitative Explanation The  (  light  source used was a helium-neon  laser  X = 6328 A) whose f l u x was a t t e n u a t e d by s t a n d a r d  methods  ;  t o below 0.1 mW.  diagram  o f the a p p a r a t u s .  F i g . 3 shows a schematic The i n c i d e n t f i e l d  e r e d uniphase over the c e l l  was r e n d -  a p e r t u r e by means o f an  i n v e r t e d t e l e s c o p e and p i n h o l e f i l t e r . A glass c e l l  with p a r a l l e l windows was f i l l e d  w i t h t h e f l u i d under s t u d y .  The average d e n s i t y p  made as c l o s e as p o s s i b l e t o t h e c r i t i c a l The c e l l  density  p c  was arranged i n s i d e apparatus t h a t m a i n t a i n e d  temperature  w i t h i n 0.0002 degrees C e n t i g r a d e f o r  several hours.  A distinctive  i n t e r f e r e n c e . p a t t e r n was  observed on t h e back f o c a l p l a n e o f an o b j e c t i v e If  p = p c , then e a r t h ' s g r a v i t a t i o n a l  ensures t h a t at  was  p> pc  a t t h e c e l l ' s bottom and  the top. Therefore,  the c e l l  Z (T). o  f*'^  o  c  c  u  r  s  a  t  lens.  field p * j^c.  some h e i g h t i n  A c c u r a t e f i l l i n g puts Z near t o t h e o  c e n t r e o f volume o f t h e c e l l .  I f T >T  , the f l u i d i s c  i n t h e one-phase r e g i o n and the d e n s i t y resembles  the sketch o f F i g . 4 .  The of  distribution  i n t e r f e r e n c e p a t t e r n formed  was the s u b j e c t  this thesis.  The f i r s t r e f e r e n c e t o t h i s method o f (25) s t u d y i n g f l u i d s i s Gouy (1880). Detailed analysis and d e s c r i p t i o n o f e x p e r i m e n t a l d e t a i l s i s found i n the t h e s i s o f D.A. B a l z a r i n i  (18)  and i n W i l c o x and B a l z a r i n i  (19)  CROSSED TDCARIIERS  LENS  1  "PLANE. WAVE h= 6328 -5  (-focal len 4k -T'5 W  M ' C R O scope  3  OBTecn«/e L£M5  1  1  s—  pi ASKS  Fig.  3  O p t i c a l Apparatus  Schematic  M O  21  Fig. 4  Density D i s t r i b u t i o n ,  T  >  T.  g. 5  F o r m u l a t i o n of the Fraunhofer P a t t e r n  23 At Z o ( T ) , t h e maximum d e n s i t y g r a d i e n t  occurs  because t h e f l u i d ' s c o m p r e s s i b i l i t y i s l a r g e s t t h e r e . Fig.5  shows a plane wave i n c i d e n t on t h e c e l l .  bundle e n t e r i n g t h e c e l l  a t Zo i s r e f r a c t e d downward..  Above and below, bundles a t Z  +  and Z  are r e f r a c t e d  through a s m a l l e r a n g l e , and a r e f o c u s e d p o i n t on the F - p l a n e . shifted  A ray  t o a common  There, they i n t e r f e r e  with  phases. Assuming anti-symmetry o f t h e d e n s i t y about (o)  the f r i n g e minima should be n u l l s . imperfections,  However, s m a l l  such as s l i g h t d e p a r t u r e s  cell  from a b s o l u t e l y  p a r a l l e l windows, c a n cause l o s s o f c o n t r a s t o f f r i n g e s .  B.  Information If a f i l m  slit  from t h e F r a u n h o f e r  i s s l o w l y t r a n s p o r t e d past a v e r t i c a l  i n the F-plane while  simultaneously  i s r a i s e d through t h e c r i t i c a l o f I ( k , T) v e r s u s  Pattern.  region, a representation  T i s obtained.  taken as d a t a f o r t h i s t h e s i s .  t h e temperature  Fig.6  i s one such  Any v e r t i c a l  through the f i l m r e p r e s e n t s t h e F r a u n h o f e r  film,  section  pattern at  t h a t temperature, save f o r e f f e c t s o f l a g . If  the f l u i d  g r a d i e n t , t h e angle constant  i n t h e c e l l had a c o n s t a n t  density.,  o f r e f r a c t i o n would measure t h e  compressibility.  However, t h e d e n s i t y g r a d i e n t  changes with temperature and, t h e r e f o r e , t h e c o e f f i c i e n t of c o m p r e s s i b i l i t y , which depends upon d e n s i t y , i s not constant.  The most s t r o n g l y r e f r a c t e d f r i n g e o f t h e  pattern, nevertheless,  should be a measure o f the  Figure 6  Kymograph f o r  SF  &  25 maximum c o m p r e s s i b i l i t y .  As the f l u i d  approaches  from above, t h i s f r i n g e extends t o l a r g e r a n g l e s d i v e r g e s as  £-*• O  T  c  and  .  For n e g a t i v e £, (T< T c ) , t h e r e i s , a t Zo,  a  meniscus i n the f l u i d which i s a d i s c o n t i n u i t y o f d e n s i t y between the vapour and the l i q u i d  phases.  Thus, f o r  the p a t t e r n f a d e s away, b e g i n n i n g w i t h lowest fringes.  order  Each f r i n g e seems t o d i v e r g e to i n f i n i t e  at a p a r t i c u l a r  £<0,  angle  temperature.  The most r e f r a c t e d band i n the F r a u h o f e r p a t t e r n i s formed by r a y s which pass j u s t above and below the meniscus.  For £ > 0  f r i n g e was  formed by r a y s which t r a v e l l e d t h e c e l l  , ( s u p e r c r i t i c a l ) the h i g h e s t  and below the r e g i o n o f h i g h e s t g r a d i e n t .  For  angle above  £ < O  ( s u b c r i t i c a l ) t h e r e i s a d d i t i o n a l o p t i c a l path d i f f e r e n c e caused  by the d i s c o n t i n u i t y a t the meniscus.  vapour d e n s i t y d i f f e r e n c e may  be measured by  t h i s f a d i n g o f f r i n g e s as a f u n c t i o n o f  The  liquid-  plotting  temperature.  T h i s t h e s i s measures the c o e x i s t e n c e c u r v e o f s u l f u r h e x a f l u o r i d e by measuring the l i q u i d - v a p o u r r e f r a c t i v e index d i f f e r e n c e a t temperatures  below  Tc.  A d d i t i o n a l i n f o r m a t i o n on the a c t u a l i n d i c e s o f r e f r a c t i o n was  o b t a i n e d from the scheme d e s c r i b e d i n Chapter  VII.  26 IV.  THE LORENTZ-LORENZ  RELATIONSHIP.  R e f r a c t i v e index, "n"» and d e n s i t y "p" a r e r e l a t e d by the L o r e n t z - L o r e n z formula, w i t h "L" a's d e f i n e d by t h i s e q u a t i o n : = pL  p*-'.  The assumption close to Tc.  (iv-D  t h a t L i s a c o n s t a n t i s suspect  The L o r e n t z - L o r e n z f u n c t i o n may be regarded  as a c o n s t a n t o n l y i n a homogeneous medium i n which t h e c o r r e l a t i o n l e n g t h a s s o c i a t e d with d e n s i t y f l u c t u a t i o n s i s much s m a l l e r than the i n c i d e n t wavelength.  These  f l u c t u a t i o n s do e x i s t and g i v e r i s e t o phenomena as c r i t i c a l  such  opalescence.  Assuming L c o n s t a n t , we can expand t h e L o r e n t z Lorenz formula i n a T a y l o r s e r i e s about  j3c •  *  p'-  (IV-2)  p-pc  From the r e f r a c t i v e index measurements o f Chapter V I I , the f o l l o w i n g v a l u e s o f a.^ and a^ were c a l c u l a t e d : n  c  = 1.093  a  1  = 1.0195 +. 0.0002,  a Where  2  0.002,  = 0.0215 .+ 0.0002,  ai = (n  c  + D(n 6n  and  a  2  = (n  2 c  + 2)  2 c  ,  (IV-3)  c  - l)(3n 12n  2 c  2 c  - 2) .  (IV-4)  27 The v a l i d i t y o f the assumption is crucial.  of constant L  Some t h e o r e t i c a l a n a l y s e s , n o t a b l y L a r s e n , (27)  Mountain, is  and Zwanzig  c o n c l u d e t h a t the v a r i a t i o n  s m a l l enough t o have l i t t l e  e f f e c t upon the meaning-  f u l l n e s s of these experiments.  Comparisons on Xenon o f  independent d e t e r m i n a t i o n s of index o f r e f r a c t i o n (26) d e n s i t y by Chapman, F i n n i m o r e and Smith  (1968) show  a v e r y s l i g h t r i s e i n L as Tc i s approached They "...conclude t h a t  from  above.  (1) i t i s u n l i k e l y t h a t L f o r Xenon  v a r i e s by more than +. 1% throughout the e n t i r e range,  and  fluid,  (2) t h e r e i s no e v i d e n c e t h a t a l a r g e anomaly  e x i s t s near the c r i t i c a l p o i n t  They c a l c u l a t e  Thus et xe ar ct dependence L on their err o r s the i n de min i n g L t o beo f 0.5%. clear.  However, the a p p r o x i m a t i o n i s not a bad  one  and index of r e f r a c t i o n d a t a c o n t i n u e t o be u s e f u l i n coexistence curve  investigation.  28  V.  ANALYSIS OF  OPTICS.  In t h i s c h a p t e r , Fraunhofer  an a n a l y s i s i s made o f  p a t t e r n d e s c r i b e d i n Chapter I I I .  the  The  rel-  a t i o n s h i p o f the p a t t e r n t o the d e n s i t y d i s t r i b u t i o n ^ ( z ) and  t o the shape of the c o e x i s t e n c e  Geometrical  A.  Curve.  L i g h t from the He-Ne l a s e r into a pinhole.  f r o n t was  i s shown.  o p t i c s are used i n t h i s a n a l y s i s .  Coexistence  focussed  curve  The  (6328 A) was  r e s u l t i n g expanded wave-  passed through a l e n s t o o b t a i n a plane wave-  f r o n t t r a v e l l i n g down the o p t i c a l a x i s o f the The  i n t e n s i t y of t h i s wave f r o n t was  p o l a r i z e r and  made p e r p e n d i c u l a r The  The  attenuated  with  crossed  cell  c e l l ' s v e r t i c a l axis  was  to the o p t i c a l a x i s .  density d i s t r i b u t i o n  (Fig.4).  p(z)  varies inside  This gives r i s e to a varying  o f r e f r a c t i o n dn/dz, as d e s c r i b e d by the formula.  system.  i t ' s s i z e made much s m a l l e r than a  width with v a r i o u s masks.  the c e l l  first  A d i f f e r e n t i a t i o n o f S n e l l ' s law  of constant  index  Lorentz-Lorenz gives:  See F i g . 7 .  Planes  r e f r a c t i v e index  are h o r -  i z o n t a l and  i t i s assumed t h a t the path's c u r v a t u r e  is  s m a l l i n the d i f f e r e n t i a t i o n which g i v e s E q u a t i o n V - l . A r a y which e n t e r s h o r i z o n t a l l y (9  = 0) i s then  29  INCIDENT  PLANE WAV/6  Fig.  7  A n a l y s i s of o p t i c s f o r c o e x i s t e n c e curve determination  30 bent through the s m a l l angle 0;  -  (V-2)  dn Here, n and - j - a r e e v a l u a t e d a t t h e p o s i t i o n Z and 9 ^ i s the i n t e r i o r  angle.  At t h e c e l l boundary, S n e l l ' s law i s used t o f i n d the o u t s i d e a n g l e 0 . O  Using u n i t y as a i r ' s r e f r a c t i v e  index and assuming ©i £ s i n O i :  (V-3)  The  symmetry o f t h e d e n s i t y  d i s t r i b u t i o n i s such t h a t t h e  r e f r a c t i v e index g r a d i e n t i s e q u a l a t p o s i t i o n and Z  above and below the o p t i c a l a x i s .  that traverse  Z  +  Two r a y s  such p o s i t i o n s a r e r e f r a c t e d through t h e  same a n g l e and a r e f o c u s e d a t t h e same p o i n t  i n the f o c a l  p l a n e o f the l e n s . The o p t i c a l path d i f f e r e n c e w i l l c o n s i s t o f two parts.  There i s a d i f f e r e n c e  to the d i f f e r e n c e  i n e f f e c t i v e path l e n g t h due  i n r e f r a c t i v e index a t Z  +  and Z~ .  There i s a l s o the s t a n d a r d d i f f e r e n c e due t o a i r path inequalities.  Since the ray at Z  t r a v e l s through the  l e s s dense medium:  (V-4)  where n  +  and n"~ a r e t h e r e f r a c t i v e i n d i c e s a t Z  +  and Z .  31 U s i n g n a i r = 1 and s i n 9 £ 9  and 9 ^  j£ if (V-5)  Destructive  i n t e r f e r e n c e o c c u r s when  (V-6)  Using the Lorentz-Lorenz r e l a t i o n  _ n(j>)-n c  where  P  and  (nc-/)-a,-p*  (V-7)  *_  (V-8)  (V-9)  4>n  c  and n  c  i s the c r i t i c a l r e f r a c t i v e index, we  obtain  phase d i f f e r e n c e by m u l t i p l y i n g by  o(v-io)  .  (V-ll)  o it  F i g u r e s 8a and b i l l u s t r a t e t h e phase d i f f e r e n c e e q u a t i o n f o r sub and s u p e r c r i t i c a l d e n s i t y d i s t r i b u t i o n s . The phase d i f f e r e n c e i n each f i g u r e i s r e p r e s e n t e d by t h e c r o s s h a t c h e d a r e a .  The f i r s t  term i s the a r e a  Fig.  8  Illustration  of  phase d i f f e r e n c e E q u a t i o n  (V-ll)  under t h e c u r v e between Z_ and Z ;  t h e second terra  +  s u b t r a c t s t h e a r e a o f t h e hatched  rectangle.  minima occur when  Since  , t h e minima  are g r a p h i c a l l y constructed  i n t h e manner shown.  At s u b c r i t i c a l temperatures,  there i s a d i s -  c o n t i n u i t y i n t h e d e n s i t y d i s t r i b u t i o n a t t h e meniscus. The  angles o f t h e lower o r d e r minima d i v e r g e t o " i n f i n i t y "  depending on t h e " i n f i n i t e n e s s " o f t h e g r a d i e n t t h e r e . I t i s seen t h a t t h e l i q u i d - v a p o u r d e n s i t y d i f f e r e n c e i n f o r m a t i o n needed t o measure the c o e x i s t e n c e c u r v e i s r e p r e s e n t e d by t h e number o f f r i n g e s which have under t h e " i n f i n i t e " g r a d i e n t a t t h e meniscus. made o f t h e F r a u n h o f e r  A kymograph  p a t t e r n as temperature i s s l o w l y  r a i s e d r e v e a l s t h e t o t a l number o f f r i n g e s . i s a photographic  "disappeared"  Fig.6  p r i n t made from one such a c t u a l —5  kymograph w i t h  £ = -8.1 x 10  The number o f f r i n g e s c a n g i v e t h e c o e x i s t e n c e curve only i f the Lorentz-Lorenz  formula  i s v a l i d ' with  c o n s t a n t L and i f t h e r e c t i l i n e a r diameter ence c u r v e were c o n s t a n t . relating  n  - vap n  l  i  q  t  o  That  j^ - ~ 1  *  of the c o e x i s t -  i s , there are e r r o r s i n These a r e d i s c u s s e d  i n Chapte r IV. T  r  I t i s easy  t o see t h a t i f N i s t h e number o f  f r i n g e s p r e s e n t a t a p a r t i c u l a r temperature,  then t h e  d i f f e r e n c e i n index o f r e f r a c t i o n between t h e l i q u i d and vapour i s g i v e n by:  n -nv L  N-A  I  1  (V-ll)  where " A " i s the l a s e r  wavelength  (6328 A) and  is  the c e l l t h i c k n e s s . These d e t e r m i n a t i o n s of t h e c o e x i s t e n c e c u r v e a l l measure N, the t o t a l path d i f f e r e n c e between r a y s t r a v e r s i n g above and below the Chapter  meniscus.  IV d i s c u s s e s the L o r e n t z - L o r e n z  s h i p and e v i d e n c e of i t s a p p l i c a b i l i t y .  relation  35 VI.  EXPERIMENTAL DETAILS. A.  The Sample C e l l s  and F l u i d  Sulfur hexafluoride  (SFg) was chosen as t h e ex-  p e r i m e n t a l f l u i d because o f i t s c o n v e n i e n t p r o p e r t i e s . It  i s optically  transparent, i t s c r i t i c a l  i s a p p r o x i m a t e l y 45°C. and i t s c r i t i c a l 37 atmospheres.  It i s fairly  temperature  p r e s s u r e i s about  easy t o c o n s t r u c t  experiment-  a l v e s s e l s t o handle t h a t k i n d o f p r e s s u r e and t h e r e latively  low temperatures a r e easy t o r e a c h and m a i n t a i n . Two t y p e s o f c e l l  One  were used i n t h i s  experiment.  was c o n s t r u c t e d from pyrex g l a s s t u b i n g and t h e o t h e r ,  more e l a b o r a t e c e l l , has a metal body made o f Kovar and plane p a r a l l e l  s a p p h i r e windows brazed t o t h e body  with copper. The g l a s s t u b i n g was s e a l e d a t one end and j o i n e d to  an 8-10 cm. p i e c e o f 2-mm.  diameter c a p i l l a r y .  Two  opposing o u t e r f a c e s o f t h e tube were p o l i s h e d and made optically  flat.  No method was d e v i s e d t o p o l i s h t h e  i n n e r f a c e s , however. i s q u i t e rough.  Thus t h e i n t e r i o r o f t h e c e l l  This introduces a variable  optical  path l e n g t h t h a t reduces t h e a c c u r a c y o f t h e measurements near the c r i t i c a l  temperature.  The metal c e l l , however, has plane p a r a l l e l windows whose f l a t n e s s i s s u s p e c t o n l y t o t h e degree t h a t s m a l l warps may have been i n t r o d u c e d d u r i n g t h e b r a z i n g o p e r a t i o n s o f i t s c o n s t r u c t i o n and under t h e h i g h p r e s s u r e s encountered near  critical.  36 F i g . 9 shows t h e two e x p e r i m e n t a l v e s s e l s a f t e r filling.  B.  The F i l l i n g Cell f i l l i n g  shown i n F i g . 1 0 .  Process. was accomplished  w i t h t h e apparatus  A Welch #1400 Vacuum pump was a t t a c h e d  t o a v a l v e and c o n n e c t o r  assembly, as was a Matheson  l e c t u r e b o t t l e o f SFg. During t h e f i l l i n g cell,  o f t h e metal  a V i r t i s thermocouple vacuum guage was a l s o a t t a c h e d . The g l a s s c e l l s , a f t e r a n n e a l i n g , were t h o r o u g h l y :  c l e a n e d w i t h S p a r k l e e n , t a p water and r e s e a r c h grade acetone was  and then baked o v e r n i g h t .  then d r i l l e d t o t h e diameter  A b r a s s gas p l u g of the c a p i l l a r y  and a t t a c h e d t o t h e vacuum m a n i f o l d w i t h t e f l o n i n the t h r e a d s . was  stem  tape  T o r r S e a l low vapour p r e s s u r e epoxy r e s i n  then used t o connect  g l a s s stem t o b r a s s , and 20  hours were a l l o w e d f o r s e t t i n g and c u r i n g .  After 8  h o u r s , t h e vacuum pump was s t a r t e d and l e f t on f o r t h e remaining  12 h o u r s .  A f t e r c u r i n g t h e epoxy, t h e c e l l was p r e s s u r e t e s t e d w i t h SFg.  Leaks a t t h e g l a s s - b r a s s j o i n t c o u l d be  detected a t t h i s time. completely f i l l was  L i q u i d SFg was then a l l o w e d t o  the c e l l ,  then pumped away.  t o a c t as a f l u s h .  The l i q u i d  The p r o c e s s was r e p e a t e d .  The  c e l l was then pumped down, u n d i s t u r b e d , f o r 8 h o u r s . P r e s s u r e s as low as 0-3 microns stage w i t h t h e metal c e l l .  were a t t a i n e d d u r i n g  These p r e c a u t i o n s  ensured  this  VACUUM  PUMP  00 F i g . 10  Cell f i l l i n g ;  apparatus  39 t h a t a l e v e l o f p u r i t y o f e x p e r i m e n t a l f l u i d was reached t h a t was commensurate w i t h t h e a c c u r a c y d e s i r e d o f t h e measurement o f t h e c o e x i s t e n c e c u r v e . L i q u i d was next t r a n s f e r r e d the  c e l l ' s valve closed.  The c e l l  removed from t h e m a n i f o l d .  and i t s v a l v e were then  SFg was b l e d out o f t h e c e l l ,  using the needle valve, u n t i l ensure c r i t i c a l  i n t o t h e c e l l and  s u f f i c i e n t SFg remained t o  d e n s i t y c o u l d be r e a c h e d .  Liquid  n i t r o g e n was used t o f r e e z e the c e l l ' s c o n t e n t s and the g l a s s c a p i l l a r y was pinched o f f and s e a l e d . Rough o b s e r v a t i o n s o f meniscus made on t h e s e a l e d c e l l .  movement were  The 6-8 cm. o f r e m a i n i n g  c a p i l l a r y were then s e l e c t i v e l y pinched o f f , a c c o r d i n g to  c a l c u l a t i o n s made on t h e temperature a t which t h e  meniscus  "disappeared".  Eventually, c e l l  SFg-3A was  c o n s t r u c t e d t o such a c c u r a c y t h a t meniscus movement was absent a t temperatures  0.09°C below c r i t i c a l .  (£S10~ ) 4  A s i m i l a r procedure was used f o r t h e metal cell.  However, t h e needle v a l v e assembly  from t h i s  C.  was n o t removed  cell.  Temperature  Control  The c e l l s a r e p l a c e d i n s i d e a p p a r a t u s shown schematically i n Fig.11.  The water c i r c u l a t e d  b r a s s o u t e r can t u b i n g i s r e g u l a t e d  i n the  t o w i t h i n 0.05°C.  The r e l a x a t i o n time from t h e o u t e r can t o t h e r e a t i s about  3.5 h o u r s .  sink  The 40 second h u n t i n g c y c l e s o f t h e  40  CELL  WINDOWS  H£AT SINK - COPPER OTUNDER WOUMO VRVTTVI I4EATIS.G COIL., SMSCDOED THERMISTORS.  THERMISTORS  STYROPOAKI IMEOLATOR  water bath are reduced t o about mass.  0.0005°C a t the i n n e r  During more s e n s i t i v e p a r t s o f the  a plywood box was prevent d r i f t i n g changes.  used t o e n c l o s e the apparatus a t the i n n e r mass due t o room  Thus rough temperature P r e c i s e temperature  ically.  experiment, and temperature  c o n t r o l i s achieved.  c o n t r o l i s achieved e l e c t r o n -  Tungsten wire i s i n s e r t e d i n t o t e f l o n s p a g h e t t i  and i s wound i n a double heat s i n k .  s p i r a l i n grooves c u t i n t o the,,  T h e r m i s t o r s are e p o x i e d i n t o s t e e l or b r a s s  screws which screw i n t o tapped h o l e s i n the heat  sink.  An e r r o r s i g n a l lis d e r i v e d from one t h e r m i s t o r , which i s one arm of a d.c. b r i d g e . by a d.c. a m p l i f i e r .  The b r i d g e output i s a m p l i f i e d  The a m p l i f i e d d e t e c t o r output i s  then c o n d i t i o n e d by a l e a d - l a g network, which makes the l o o p g a i n i n f i n i t e a t zero f r e q u e n c y and p r o v i d e s the roll  o f f needed t o ensure s t a b i l i t y .  This conditioned  d.c. b r i d g e output d r i v e s the c o n t r o l i n p u t o f a s t a b ilized to  power s u p p l y , which p r o v i d e s power  the t u n g s t e n h e a t i n g w i r e .  (SlOOmw)  A g i v e n temperature  be s e l e c t e d by s e t t i n g a decade r e s i s t a n c e box on arm o f the d.c. b r i d g e .  can one  The network w i l l h o l d the therm-  i s t o r ' s r e s i s t a n c e t o w i t h i n 0.05J1 , which c o r r e s p o n d s t o 0.0015°C. (1.5 m i l l i d e g r e e ) almost i n d e f i n i t e l y . shows t h i s network i n schematic Near temperature hours.  Tc»  Fig.12  form.  i n order to reach e q u i l i b r i u m ,  c o n t r o l i s mandatory f o r p e r i o d s o f  C o r r e c t i n g f o r the i n s t a b i l i t y o f room  12-24 temperature,  REGULATED  "POWER  SUPW.Y.  F i g . 12  E l e c t r o n i c temperature c o n t r o l  network  the  drift  i n temperature over such a p e r i o d can  easily  be kept t o l e s s than 0.0002°C by e n c l o s i n g the apparatus in a V  plywood  box.  Thus p r e c i s e temperature  control  i s achieved. A second t h e r m i s t o r i s used as a m o n i t o r . circuit  The  uses another d.c. b r i d g e , d.c. d e t e c t o r , and a  chart recorder.  T h i s t h e r m i s t o r i s c a l i b r a t e d with a  p l a t i n u m r e s i s t a n c e thermometer, which i n t u r n i s c a l ibrated i n a t r i p l e are  point c e l l .  The b r i d g e components  a c c u r a t e enough to g i v e a b s o l u t e temperature t o  w i t h i n 0.02°C.  The t h e r m i s t o r ' s c h a r a c t e r i s t i c s a r e  determined t o such p r e c i s i o n t h a t the e r r o r i n temperature i s  S £ t  (o.ot V  €  £ -  .  Much o f the d a t a f o r t h e metal c e l l was  taken  u s i n g a Dymec d i g i t a l - r e a d o u t q u a r t z thermometer.  The  a c c u r a c y c l a i m e d f o r t h i s machine i s 0.0001°C. f o r s h o r t term work. w i t h changes drift  However, t h e r e i s a d r i f t i n room temperature.  i n i t associated  There i s 0.001°C.  f o r a 1.0°C. d r i f t  i n room temperature.  D. Other Equipment  and T e c h n i q u e s .  The c o n t r o l network was m a i n t a i n e d a t brium below Tc  fo** s e v e r a l hours w h i l e the sample -  came t o e q u i l i b r i u m . used, up t o 8-10 the  equili-  For fc. < 10  hours.  cell  -4 , l o n g e r p e r i o d s were  A c h a r t r e c o r d i n g was made o f  monitor t h e r m i s t o r ' s r e s i s t e n c e d u r i n g t h i s  time.  44 The F r a u n h o f e r p a t t e r n was f o c u s s e d onto a 35 mm. body used as a f i l m riate  transport.  camera  A s m a l l motor and approp-  g e a r i n g a l l o w e d f i l m t o be drawn past a narrow  slit  i n t h e F - p l a n e a t about h. r e v . per hour. When e q u i l i b r i u m was e s t a b l i s h e d , t h e l a s e r was  t u r n e d on and the camera was s t a r t e d .  was  then swept s l o w l y upwards t o above Te and t h e  r e s u l t i n g kymograph made  The temperature  (Fig.6).  The r e s i s t a n c e o f t h e monitor t h e r m i s t o r a t e q u i l i b r i u m , o r t h e q u a r t z thermometer r e a d o u t , and t h e number o f f r i n g e s counted on t h e f i l m for  the c a l c u l a t i o n of 6  form t h e raw d a t a  f o r the coexistence curve.  45 VII.  INDEX OF REFRACTION MEASUREMENT. A.  General In  Description.  o r d e r t o measure the c r i t i c a l  f r a c t i o n and t o measure the temperature n  L  .+ n  (the average  v  scheme was The cell  out.  scheme i n v o l v e d c o n s t r u c t i o n of a g l a s s  i d e n t i c a l t o those i n v o l v e d i n t h e c o e x i s t e n c e A 30°„ crown g l a s s prism was  i n s e r t e d i n the c e l l . the c e l l  d e n s i t y and  inserted  The apparatus was platform. Fig.14  c u t and  F i g . 1 3 i s a schematic diagram  and p r i s m , showing t h e i r The c e l l was  it  dependence o f  index o f r e f r a c t i o n ) , the f o l l o w i n g  d e v i s e d and c a r r i e d  c u r v e measurement.  used  index of r e -  of  orientation.  then f i l l e d with. SFg t o i n the temperature  critical  control  apparatus.  l a i d h o r i z o n t a l and mounted on a r i g i d  The He-Ne l a s e r was  used as a l i g h t  shows the arrangement between c e l l  i n t h i s determination.  and  S i n c e the c e l l  c o u l d be e a s i l y arranged t o impinge  source. l i g h t beam  i s horizontal,  the beam on  the  mutual i n t e r s e c t i o n of the meniscus and  prism edge so  t h a t t h r e e beams emerge from the c e l l .  One  through o n l y the meniscus and  thus emerged  beam passed laterally  u n d e v i a t e d , t o serve as a z e r o - a n g l e r e f e r e n c e . o t h e r two beams passed  through the l i q u i d  The  and vapour  and  p r i s m and thus were r e f r a c t e d through a n g l e s which were c h a r a c t e r i s t i c o f the d e n s i t y of t h e i r r e s p e c t i v e media. The emergence a n g l e was  measured w i t h a f r o n t -  s i l v e r e d m i r r o r mounted on an arm  attached to a spectro-  46  •A  CELL.  "POLISHeD FACES  FACES  Fig.  13  F i g . 14  C e l l and prism  Beam-prism  orientation  orientation  47 scope t u r n t a b l e .  The beams were r e f l e c t e d  scope which was f o c u s s e d a t i n f i n i t y . and t e l e s c o p e were r i g i d l y clamped same r a i l s  into a t e l e -  Both t u r n t a b l e  to o p t i c a l r a i l s , the  t o which the l a s e r , c o l l i m a t i n g d e v i c e s and  c e l l h o l d e r were a t t a c h e d .  B.  E x p e r i m e n t a l Technique.  A s e r i e s o f measurements o f t h e emergence a n g l e s of  the beams was made.  Temperature  c o n t r o l was o b t a i n e d  from t h e same e l e c t r o n i c apparatus as p r e v i o u s l y d e s c r i b e d i n Chapter VI-C.  Readings were taken from as  low a temperature as p o s s i b l e and proceeded upwards at  regular small  intervals.  A s e r i o u s drawback t o t h i s scheme i n v o l v e d t h e glass c e l l s .  As i n Fig.13,^when t h e prism was i n s e r t e d  i n t o the c e l l , a c e l l the  prism.  end had t o be c u t o f f t o accomodate  Subsequent  stem onto the c e l l  left  j o i n i n g o f a 2mm.  capillary  considerable residual  tube  stress i n  the  r e g i o n o f t h e j o i n between the pyrex c e l l body and  the  c a p i l l a r y tube.  G l a s s c e l l s used i n the c o e x i s t e n c e  c u r v e measurements were annealed o v e r n i g h t t o remove these s t r e s s e s before f i l l i n g  was a c c o m p l i s h e d .  However,  s i n c e the prism's crown g l a s s m e l t s a t a temperature below t h a t o f t h e a n n e a l i n g temperature  o f pyrex, t h e s e  index o f r e f r a c t i o n c e l l s c o u l d not be s i m i l a r i l y made stress-free.  Thus, as temperature was r a i s e d , the i n -  c r e a s i n g p r e s s u r e i n s i d e t h e c e l l became s u f f i c i e n t  48  to  cause c r a c k s t o  plosion.  form, r e s u l t i n g  Three c e l l s were d e s t r o y e d  i n a minor ex-  i n this  manner.  However, c o n s i d e r a b l e d a t a was o b t a i n e d from each before i t s d e s t r u c t i o n .  Annealed c o e x i s t e n c e  c e l l s have c o n s i d e r a b l y g r e a t e r s t r e n g t h .  cell  curve  Future  u s e r s o f t h i s scheme a r e c a u t i o n e d t o o b t a i n pyrex o r q u a r t z prisms and anneal  t h e i r c e l l s i f pressures i n  the neighborhood o f SFg (310-550 p s i ) a r e The  encountered.  raw d a t a was a n a l y z e d on UBC's IBM 360/67.  Two e x t r a p o l a t i o n s were made, one t o y i e l d  a critical  temperature Tc, t h e o t h e r t o y i e l d t h e c r i t i c a l index, n .  A least  c  squares  refractive  f i t r o u t i n e was used on t h e  l a t t e r t o get a l i n e a r f i t t o the d a t a .  C.  Prism The  Index o f R e f r a c t i o n .  index o f r e f r a c t i o n o f t h e prism was determ-  ined experimentally.  I t was a l s o checked a g a i n s t f i g u r e s  g i v e n f o r crown g l a s s by both the manufacturer and by a standard r e f e r e n c e t e x t . E x t r a p o l a t i o n s o f f i g u r e s from the Chemical Rubber Company Handbook o f Chemistry  and P h y s i c s g i v e s  a c o r r e c t i o n f a c t o r f o r t h e index o f r e f r a c t i o n a t t h e wavelength of our He-Ne l a s e r  ( X = 6328 A ) .  This  c o r r e c t i o n f a c t o r was a l s o a p p l i e d t o nn (n f o r t h e sodium "D" l i n e ) as g i v e n by t h e A f u r t h e r experimental by t h e standard elementary  manufacturer.  d e t e r m i n a t i o n was made  o p t i c a l technique  o f the angle  49 o f minimum d e v i a t i o n .  The l a s e r was used as a l i g h t  source f o r t h i s measurement. T a b l e IV shows t h e r e s u l t s .  TABLE IV Prism Index o f R e f r a c t i o n f o r A = 6328A  SOURCE  INDEX  CRC Handbook  1.515 ± 0.0005  Manufacturer  1.517 _+  Experimental  1.518 +. 0.002  The  e x p e r i m e n t a l l y determined  was used i n the d a t a  ?  v a l u e 1.518 _+ 0.2%  analysis.  D. A n a l y s i s o f O p t i c s . The  symbols used i n t h i s a n a l y s i s a r e shown i n  Fig.15. n-^ - index o f r e f r a c t i o n o f a i r n  2  = 1.0002 ,  = index o f r e f r a c t i o n o f crown g l a s s = 1.518 +_ 0.2% ,  n3 = index o f r e f r a c t i o n o f SFg ,  a  = 30° .  From S n e l l ' s law we g e t  «*n^, sin  »  n  3  s i n <j>i  .  (vii-i)  S i n c e t h e beam i s a l i g n e d t o impinge a t 90° t o t h e c e l l body, 0 j i s then a c o n s t a n t and i s e q u a l t o <X .  50  rtt  (air)  Fig.  15  Schematic diagram of o p t i c s o f index o f r e f r a c t i o n d e t e r m i n a t i o n scheme  51 Thus  <k  «h  - f t  Similarily,  II3 -  ..net  since  m  _  2  )  :  (VII-3)  t  (<p ~oL) a  Expanding s i n  d)  cos<^ = ( 1 - s i n ^ f o b t a i n the f i n a l  i  T =(f>Z~~& >•  n • Sin <~f>3 sin  we  .  2  and u s i n g  -  (vn-4)  (\~  result  (VII-5)  We of  now  have, i n E q u a t i o n  ( V I I - 5 ) , the SFg  index  r e f r a c t i o n as a f u n c t i o n o f t h e e x t e r i o r angle o f  emergence o f the l a s e r beam.  E.  E x t e r n a l Angle Measurement. F i g u r e s 16a and  16b  show the a n g l e s © and  0  and a l s o the p h y s i c a l l a y o u t o f the a p p a r a t u s used i n t h i s index o f r e f r a c t i o n measurement scheme. For a plane s u r f a c e d m i r r o r , i t can be shown t h a t i f beams one and two then © = 0/2.  are p a r a l l e l  after  reflection,  (Fig.16a).  I f the t e l e s c o p e i s f o c u s s e d a t i n f i n i t y  and  kept  MIRROR SURFACE  F i g . 16a  F i g . 16b  Angle  definitions  Apparatus  layout  r i g i d l y mounted, then a l l beams f a l l i n g on the c r o s s h a i r s are  p a r a l l e l , thus a measurement o f ©i,  and ©2  ( f o r the v a p o r ) , can e a s i l y be The  of about  accomplished.  T h e r e f o r e the a c c u r a c y o f measured  a n g l e s i n the v i c i n i t y o f 7° i s about  Data  a n g l e s ©^ and ©2 were used to c a l c u l a t e A  for  v  culated.  0.5%.  Analysis.  The raw nL and n  liquid)  spectroscope t u r n t a b l e allowed accuracy  1 minute.  F.  ( f o r the  = 6328A.  The temperature  was  also c a l -  S i n c e the L o r e n t z - L o r e n z r e l a t i o n when expanded  gives  .n(p)-n  (n -n ^ L  v  + c  ( h c - O - c L , -fp/jpc)  cc  (pc-pv)  ,  (vn-6)  ,  (vn-7)  we can o b t a i n a rough d e t e r m i n a t i o n o f a r e l a t i v e ical  temperature  ( T ) by c  (n -rwf  ™.  L  plotting  T  CO  An e x t r a p o l a t i o n o f t h i s i n f o r m a t i o n t o ( n - n ) L  gives T  c  crit-  v  = 0,  = 45.125°C. T h i s graph i s F i g . 1 7 . A second graph i s then p l o t t e d , u s i n g t h i s i n (2)  formation.  Guggenheim  found, i n h i s a n a l y s i s of  SFg c o e x i s t e n c e curve d a t a the f o l l o w i n g evidence of a s l i g h t l y skewed  distribution.  55  (VII-8)  Combining t h i s with E q u a t i o n (VII-6) we o b t a i n t h e f o l lowing .  Where  (VII-10)  From t h e s e d e t e r m i n a t i o n s , n~.= 1.093 and thus  r w  =  (l.iz MO"" ) ( T - T ) C  n  L  Thus a p l o t o f n  +- n  4  + n  (VII-11)  c  y  7;—-  v . s . (T -T) w i l l extrapolate to c  a t ( T - T ) = 0.  c  c  Such p l o t s were made and t h e UBC LQF l e a s t squares f i t r o u t i n e was used t o f i t t h e d a t a t o a l i n e a r equation. is  The f o l l o w i n g r e s u l t s were, o b t a i n e d .  the c r i t i c a l  the n  c  temperature  Fig.17  e x t r a p o l a t i o n and F i g . 1 8 i s  extrapolation. TABLE V  CELL SF -Rx 6  S  -R  F  SF -R 6  2  5  SLOPE  RMS ERROR  NO. OF POINTS  rv  RMS ERROR  1.09213  0.00004  >-4 2.68x10  0.05x10  1.09338  0.00002  3.l6xl0~  4  0.03xl0~  4  1.09238  0.00006  2.87xl0""  4  0.04xl0"  4  -4  11 36 20 67  I  ft  0  0  0"  0  O  a  <3 = u cc23= UJ oci—  0 •  idux:  —z  UJ a  00"  j  O ^  "tot-  "  ^T--r-<>-W-  0~  !—-o-io-  —r  —o  -t-<Ho-o-  0  : *-g:.jS-S4r^: —S-s-s—f-— :  .  : £  0 -0  m  0  0 0  i _, - In lo j- !. - • -b*-i*1  o'ocr  ot  O-  i  a'oor  0 00  -i D OL  1 0 03  1 O'OS  oooorxreoT-tz/fAN+iNn  0  00=)  1 a'Ut>  1 0'0£  +1  r 0 02  o'qi  The  s m a l l RMS  e r r o r s i n d i c a t e t h a t the d a t a i s  v e r y w e l l f i t t e d by a s t r a i g h t l i n e and t h a t s c a t t e r i s quite small.  G.  Final Results. A' weighted average o f the c r i t i c a l  r e f r a c t i o n was  index of  taken, u s i n g t h e number of d a t a p o i n t s  f o r each c e l l as weight f a c t o r . f o r the c r i t i c a l  The  final result  then  index of r e f r a c t i o n o f SFg i s  n  c  = 1.0929.  S i n c e the a c c u r a c y o f the prism index o f r e f r a c t i o n i s l i m i t e d t o 0.2%,  a more r e a l i s t i c  f i g u r e would  = 1.093  .  then  be  n  The ligible  H.  c  + 0.002 —  e r r o r s i n temperature measurement are neg-  i n t h i s index o f r e f r a c t i o n d e t e r m i n a t i o n  scheme.  S i g n i f i c a n c e of R e s u l t s . 25 The C.R.C. handbook g i v e s n  This determination gives N I Q L  a s i d e from the n e c e s s a r y the sodium "D"  D  =1.167 f o r  a t 25°C as 1.1655.  SFg. Thus,  wavelength c o r r e c t i o n between  l i n e and the He-Ne l a s e r ' s 6328A, t h e r e i s  e x c e l l e n t agreement t o s t a n d a r d  data.  58 VIII.  DATA ANALYSIS AND  RESULTS.  The main d a t a a n a l y s i s fits  to a log-log  was  done by l e a s t - s q u a r e s  plot.  As d i s c u s s e d i n Chapter  IV and Chapter  VI, the  number o f f r i n g e s p r e s e n t under the d i s c o n t i n u i t y  at  the meniscus and the L o r e n t z - L o r e n z r e l a t i o n s h i p  form  the b a s i s  of t h i s  analysis.  Given the d e f i n i t i o n o f j8 ,  PL-PV  =  A / T C~ -T Tj ^  7*  v  g i v e n the f i r s t  n  (viii-D  Tc  o r d e r expanded L o r e n t z - L o r e n z  (vm-2)  and g i v e n the r e l a t i o n s h i p o f E q u a t i o n  ( V - l l ) between the  c  +  ( n - i ) - a , £±  relationship,  ;  L  = n  ;  c  number o f f r i n g e s p r e s e n t and  the width o f SFg  i n the c e l l  n -n L  we  can  v  =  (Jl),  N |  (VIII—3)  say  pL-p*  =  /°c  Thus we  (N), the l a s e r wavelength(  X  "N  .  (VIII-4)  Jc-a,-(nc-t)  can w r i t e (VIII-1) a s :  N  *  (-O^  .  (VIII-5)  59 It  i s w e l l known t h a t /3= ]/Q  (see T a b l e I I I ) . We can  say r o u g h l y t h a t , s i n c e (-£ ) =  N  T  ,  c  * (T -T) .  3  (VIII-6)  C  3  An e x t r a p o l a t i o n o f t h e p l o t N  v . s . T should then  yield  T c  With t h i s T , we can proceed c  t o more  sensitive  analyses.,, Given E q u a t i o n  £g  ( V I I I - 5 ) , we can see t h a t  .  1  (VIII-7)  Thus a l o g - l o g p l o t g i v e s a s l o p e o f 3 |3 -1  - (^-'vio t- ). 3  And  £  ( V I I I  .  8 )  i f t h i s s l o p e v a l u e i s "m":  '  Fig.19  shows one such graph  IBM/360 - Calcomp f a c i l i t i e s examination  (VIII-9)  produced  a t UBC.  o f t h i s d a t a , compiled  A cursory  from  several individual  d a t a d e t e r m i n a t i o n runs on the metal c e l l the g l a s s c e l l  by t h e  and one on  SFg-3A, shows two r e g i o n s o f d i f f e r e n t  slope. The  s l o p e o f these two r e g i o n s was found by u s i n g  .[  • I !  ——  I  I-  — T  i.o  tn-  O.M _l  0.08 _J .  0.12  -  L0GIN3/EPS) -10 0.16 0.2  0.24  B  •  -!|  • • 'rx  dtz  0.32 [  0.28 _J  3  §  »  0.36  01.4 _J  61 a least was  squares method.  An a r b i t r a r y d i v i s i o n  s e l e c t e d and the d a t a above or below t h i s p o i n t  compiled.  was  The " b e s t s t r a i g h t l i n e s " through these p o i n t s  were found and the r e s u l t i n g  s l o p e s taken as a good  e s t i m a t e o f the s l o p e s o f the two r e g i o n s . was  point  f$  then used t o f i n d  Equation (VIII-9)  f o r each r e g i o n .  F u r t h e r examination of F i g . 1 9 shows t h a t d a t a run i s i d e n t i f i e d by a d i f f e r e n t  special  each  symbol.  An e s t i m a t e d e r r o r bar i s p l o t t e d above each p o i n t w i t h the square s p e c i a l symbol. t o the combined e r r o r o f  The e r r o r bar  corresponds  h~ e x t r a f r i n g e and a temp-  e r a t u r e e r r o r of 0.0005®C.  Only e r r o r b a r s g r e a t e r than  5% are p r i n t e d . The c a l c u l a t e d v a l u e s o f an e s t i m a t e o f RMS d a t a p o i n t s , and  p  error.  are accompanied  by  i f ( x i , y i ) are the  i f the c a l c u l a t e d  straight  line i s  m -xx +- b  =  (VIII-10)  then  These r e s u l t s are found  Cr  cr  The Thus, /S  larger  p  1  goes w i t h the lower  decreases c l o s e to Chapter  = 0.000001  x  = 0.000029'  temperature.  T . c  IX d i s c u s s e s e r r o r s .  However, the  low  62  v a l u e s o f Q~  i n d i c a t e t h a t t h e d a t a i s q u i t e smooth and  t h a t s t r a i g h t l i n e s a r e good c h o i c e s t o f i t t h i s  data.  63 IX.  ERROR DISCUSSION. A.  Choosing t h e C r i t i c a l  Temperature.  o  Graphs o f l o g (N /( - £ much v a r i a b i l i t y below l o g ( - £ ) due T . c  -4.0. T h i s i s  As i n Chapter V I I I (Eq. V I I I - 6 ) , T ^3=  1/3, p l o t t i n g N ^  trapolating to N ^  =0.  i t s l i m i t a t i o n s were S i n c e the ^>  and  ~  i n part to small i n a c c u r a c i e s i n the choice o f  by assuming  and  )) v . s . l o g (-£ ) show  can be found  c  v . s . T, and ex-  T h i s procedure works w e l l explored.  found i n t h i s study  s i n c e t h i s i s compatible  with o t h e r  i s not 1/3,  researchers'  f i n d i n g s , a computer study was made o f t h e v a r i a t i o n o f extrapolated T simple  c  as a f u n c t i o n o f c h o i c e o f /S .  least-squares  A  l i n e a r f i t was done t o v a r y i n g  numbers o f data p o i n t s .  The r e s u l t s a r e p r e s e n t e d i n  Table V I . TABLE VI V a r i a t i o n o f chosen T Assumed 0.333 10 p o i n t s Tc l o g (-€) A Tc < -4.7 30 p o i n t s Tc l o g (-O A Tc < -3.4 49 p o i n t s Tc l o g (-O A Tc < -3.0 69 p o i n t s Tc l o g (-€ ) ATc < -2.5  |3  c  with assumed ji v a l u e  Value 0.336  0 .339  0.342  0.345  45.433 29  45.433 30 0.000 01  45 .433 33 0 .000 03  45.433 36 0.000 03  45.433 0.000  45.432 91  45.433 09 0.000. ,18  45 .433 29 0 .000 20  45.433 49 0.000 20  45.433 0.000  45.432 60  45.433 03 0.000 43  45 .433 46 0 .000 43  45.433 88 0.000 42  45 434 0.000  45.430 28  45.431 75 0.001 47  4 5 .433 21 0 .001 46  45.434 69 0.001 48  45.436 0.001  -  -  -  64 The  e x t r a p o l a t e d v a l u e s o f Tc show extremely  small v a r i a t i o n .  The v a l u e s converge a t /S » 0.339  w i t h d i f f e r e n c e s i n Tc b e i n g o f t h e o r d e r o f 0.0002°C. A c c o r d i n g l y , t h e l e a s t squares P  f i t was re-computed and  was found f o r Tc = 45.4333 ± 0.0002°C,  d a t a p o i n t s below l o g (-£)  = -.2.5.  using a l l  Thus a l l d a t a below  the s l o p e change r e g i o n i s used. These r e s u l t s a r e found, f o r l o g ( - £ ) <  -.2.5;  Tc 45.43 31  0.336  16.8 x 10  45.43 33  0.339  2.9 x 10  45.43 35  0.342  4.3 x 10  T h e r e f o r e the b e s t v a l u e o f j3 i s 0.339 ± 0.003. 2  The  low v a l u e o f 0 - '  f o r Tc = 45.4333 i n d i c a t e s t h a t  t h i s Tc g i v e s the s t r a i g h t e s t  l i n e f o r the data involved  i n t h e computation. Until  l e s s experimental  s c a t t e r i s p o s s i b l e we  s h a l l have t o be c o n t e n t with a s t r a i g h t l i n e f i t t o the d a t a below l o g (-6 ) =-2.5. B.  E r r o r s i n R e l a t i n g D e n s i t y t o R e f r a c t i v e Index.  As d i s c u s s e d i n Chapter  IV, t h e r e l a t i o n s h i p o f  d e n s i t y t o r e f r a c t i v e index i s approximated Lorentz-Lorenz r e l a t i o n s h i p . s h i p are unknown.  by t h e  The d e t a i l s o f t h i s  relation  65 However, i n the a n a l y s i s o f the o p t i c s , o n l y the first  term o f an expanded L o r e n t z - L o r e n z f u n c t i o n i s used.  Taking the second term, E q . ( I V - 2 ) , we  obtain:  (IX-l)  I + f c  f  L  Our d a t a a n a l y s i s attempts t o f i t the f o l l o w i n g equation  (IX-2)  We o b t a i n from Eq. ( I X - l ) , t o second o r d e r :  (IX-3)  n -n L  v  =  and from Eq. (IX-2) and Eq.  (V-ll):  (IX-4)  (2 ) Taking Guggenheim's  w e l l known r e s u l t s as  approximations  fit.  I p.  = %  H)  >  (IX-5)  66 •Or,'  ~,l !  (IX-6)  S u b s t i t u t i o n of equation  ( I X - 5 , IX-6)  into  (IX-4)  gives  4  (IX-7)  F o l l o w i n g the l o g - l o g scribed  i n Chapter  a j . = O.OZ , we  *  _h£ /•-/M  talcing  |3 * V3 > A=  and  o b t a i n , upon c u b i n g E q . ( I X - 7 ) :  A (-<0 " 3  V I I I and  d a t a a n a l y s i s scheme de-  P  3.Q(-e)  * o.ic>(-£f-(o.ooiJ(-6) . 3  T Y  (IX-8J  t~ < 0.06,.  S i n c e t h i s t h e s i s d e a l s w i t h a d a t a range o f 0.000001 the l a s t two  terms a r e c l e a r l y  i n s i g n i f i c a n t , thus we  can  say: r  3  (IX-9)  Talcing l o g s o f both s i d e s and g r a p h i n g , the s l o p e o f the graph  i s taken as 3 |3 -1.  Thiei l a r g e s t the c o r r e c t i o n term  (-£.)  i s approximately  0.01.  Thus  i s c o m p l e t e l y n e g l i g i b l e , as the v a l u e  3  o f N / ( - € ) i s c o m p a r a t i v e l y huge and c o n s t a n t with T h e r e f o r e , the d a t a range o f t h i s a l l o w s us t o i g n o r e the second the L o r e n t z - L o r e n z  term  relationship.  (-€  experiment  i n t h e expansion  of  ).  67  References  (1)  C. Domb; P h y s i c s Today, ,21, 23 (1968).  (2)  E.A. Guggenheim; J . Chem. Phys.,  (3)  M.E. F i s h e r ; J . Math. Phys.,  (4)  T.D. Lee and C.N. Yang;  (5)  L.P. Kadanoff, W. Gotze, D. Hamblen, R. Hecht, E.A.S. Lewis, V.V. P a l c i a u s k a s , M. R a y l , J . S w i f t , D. Aspens, and J . Kane; Rev. Mod. Phys., 39, 395 (1967).  (6)  B. Widom; J . Chem. Phys., 43, 3898 (1965).  (7)  M.A. Weinberger and W.G. S c h n e i d e r ; Can. J . Chem., 30, 422 (1952).  (8)  H.W. Habgood and W.G. S c h n e i d e r ; Can. J . Chem., 32., 98 (1954).  (9)  D. Atack and W.G. S c h n e i d e r ; J . Phys. and C o l l . 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