UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Dielectric relaxation of fluoroform gas at 30 megahertz Tsoi, Hoi-Lum 1972

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1972_A6_7 T76.pdf [ 2.29MB ]
Metadata
JSON: 831-1.0103917.json
JSON-LD: 831-1.0103917-ld.json
RDF/XML (Pretty): 831-1.0103917-rdf.xml
RDF/JSON: 831-1.0103917-rdf.json
Turtle: 831-1.0103917-turtle.txt
N-Triples: 831-1.0103917-rdf-ntriples.txt
Original Record: 831-1.0103917-source.json
Full Text
831-1.0103917-fulltext.txt
Citation
831-1.0103917.ris

Full Text

ol  DIELECTRIC RELAXATION GF FLUOROFORM GAS -  AT 3G MEGAHERTZ by HOI-LUN TSOI  B.Sc. Notre Dame University, 1969  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n the Department of Physics  We accept t h i s t h e s i s as conforming to the required  standard  THE UNIVERSITY OF BRITISH COLUMBIA September, 1972  In p r e s e n t i n g t h i s  thesis  an advanced degree at the L i b r a r y I  in p a r t i a l  the U n i v e r s i t y  s h a l l make i t  freely  f u l f i l m e n t o f the of B r i t i s h  available  for  requirements  Columbia, I agree  for  that  r e f e r e n c e and s t u d y .  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 copying o f t h i s  thesis  f o r s c h o l a r l y purposes may be granted by the Head o f my Department o r by h i s of  this  representatives.  It  thesis for financial  i s understood that copying o r p u b l i c a t i o n gain s h a l l  written permission.  Department o f The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada  not  be allowed without my  - i ABSTRACT D i e l e c t r i c r e l a x a t i o n of Fluoroform  (CHF-j)  i s measured c a r e f u l l y i n the density region (0.0009-0.28 amagat) where the molecular r e o r i e n t a t i o n a l frequency i s of the ssme order as the o s c i l l a t i n g e l e c t r i c f i e l d of 30 MHz.  The simple Debye model, which assumes that the  dipole relaxes exponentially towards i t s equilibrium value c h a r a c t e r i z e d by a s i n g l e r e l a x a t i o n time, i s found to be inadequate to explain the experimental data. The Cole-Cole model, which assumes a d i s t r i b u t i o n of r e l a x a t i o n times i s t r i e d .  Agreement with experiment i s  only improved at d e n s i t i e s below 0.03  amagat.  - ii TABLE OF CONTENTS ABSTRACT  Page i  TABLE OF CONTENTS  i i  LIST OF ILLUSTRATIONS  i i i  LIST OF TABLES  iv  ACKNOWLEDGEMENTS  •  V  Chapter I  Introduction  1  Chapter I I  Theory of D i e l e c t r i c Relaxation 2;1 Macroscopic Theory - The Debye Equation and the Cole-Cole Equation  4  2.2 Symmetric-top  8  molecule  2.3 Microscopic Theory Chapter I I I  Chapter IV  Chapter V  12  The Experiment 3.1 The Apparatus  16  3.2 Experimental Procedures  20  Results and Analysis 4.1 C a l i b r a t i o n of the O s c i l l a t o r  22  4.2 Fluoroform Data Analysis  28  Discussion and Suggestions  40  APPENDIX A  41  APPENDIX B  43  BIBLIOGRAPHY  45  - iii LIST OF ILLUSTRATIONS FIGURE 2.2  PAGE Component of the Dipole Moment & C l a s s i c a l Motions  of a Symmetric-Top Molecule  .10  3.1  The 30 MHz. O s c i l l a t o r >  4.1.1  A Typical  4.1.2  LCo values of the O s c i l l a t o r  24  4.1.3  An Examination  26  4.2.1  17  Argon C a l i b r a t i o n  Run  of the Pulse-Gas Valve  (JO/f vs.'w f o r CHFj  23  31  4.2.2  Debye F i t f o r CHF  4.2.3  Debye F i t f o r CHF^  4.2.4  Cole-Cole F i t f o r CHF^ (0.0009-0.045 amagat)  3£  4.2.5  Cole-Cole F i t f o r CHF  37  3  (0.0009-0.28 amagat)  33  (0.0009-0.045 amagat)  34  3  (0.0009-0.28 amagat)  .  - iv L I S T OF TABLES TABLE  PAGE  4.2.1  Comparison w i t h C l a r k ' s r e s u l t s  38  4.2.2  Comparison o f  39  values  ACKNOWLEDGEMENTS I would l i k e t o thank Dr. Bloom f o r s u p e r v i s i n g this  r e s e a r c h and Dr. W a l t e r Hardy f o r h i s encouragement  and g u i d a n c e d u r i n g t h e c o u r s e o f t h e e x p e r i m e n t w h i l e 7  Dr. Bloom was on s a b b a t i c a l l e a v e I also thank S h i r l e y in  (Aug.71-Sept.72)•  Jackson and Bob A l b r e c h t  a i d i n g my c o m p u t e r p r o g r a m m i n g , a n d J . L e e s i n t h e  g l a s s works. This research i s supported, i n part, by the N a t i o n a l R e s e a r c h C o u n c i l o f Canada.  1  CHAPTER I INTRODUCTION The field  coupling  b e t w e e n an a p p l i e d r - f e l e c t r i c  a n d t h e m o l e c u l a r d i p o l e moments n o r m a l l y  a small p r o b a b i l i t y of inducing  has only  t r a n s i t i o n s between  m o l e c u l a r r o t a t i o n a l s t a t e s , s i n c e t h e energy  difference  between d i f f e r e n t non-degenerate r o t a t i o n a l s t a t e s o f most m o l e c u l e s c o r r e s p o n d s t o microwave o r i n f r a - r e d . frequencies.  Hence t h e r e  of energy from the f i e l d . collisions,  a small  i s no a p p r e c i a b l e  absorption  However, due t o m o l e c u l a r  amount o f e n e r g y i s a b s o r b e d .  This  comes a b o u t a s a r e s u l t o f c o l l i s i o n s ,  i n which the  anisotropic,  of the intermolecular  o r a n g u l a r dependent p a r t  f o r c e s produces change i n t h e r o t a t i o n a l s t a t e s o f t h e molecules..It an  oscillating  c a n b e shown t h a t t h e e n e r g y a b s o r b e d electric  spectral density  field  from  i s proportional to the  of the correlation function of the 1  m o l e c u l a r d i p o l e moment a t t h e f r e q u e n c y o f t h e f i e l d . Measurement o f such s p e c t r a l d e n s i t i e s g i v e s on  information  t h e r a t e o f m o l e c u l a r r e o r i e n t a t i o n and hence on t h e  intermolecular  forces.  A c a l c u l a t i o n b a s e d on m i c r o w a v e absorption  experimental  nonresonant  r e s u l t s s h o w s t h a t t h e amount  of energy absorbed i n t h e radio-frequency  region i s  2  very  small.  In f a c t ,  modern t e c h n o l o g y .  On  the  dielectric  and  should contain the No  t h e o t h e r hand, t h e r e a l  susceptibility  w o r k on  of  i t i s beyond t h e c a p a b i l i t y  part  c a n be m e a s u r e d v e r y  of  easily  same amount o f i n f o r m a t i o n .  dielectric  relaxation  i n gases  has  5  been  done b e l o w 400  30 MHz. check  in this  MHz.  except  laboratory.  his results  and  f o r C l a r k ' s study  This thesis  t o improve  near  i s intended  the accuracy of  to  the  experiment. The as  a result  measured  change i n r e s o n a n t  of the  frequency o f the  change i n d e n s i t y o f f l u o r o f o r m gas  carefully  i n the  information-rich  frequency  of molecular reorientation  collision  i s of the  same o r d e r o f m a g n i t u d e  field.  equipment,  In the p r e c i s i o n a d e p a r t u r e from  - The that  the  approach  is  characterized  to  f i t the  which  results,  a single  a single  is  o n l y improved  relaxation  intermolecular  the  relaxation  time.  time.  b e l o w 0.03  potential  of relaxation The  i s not  The  polarization  I n an  attempt  adopted,  times  form  observed.  assumption  agreement v/ith  amagat.  oscilla-  been  of the molecular  the Cole-Cole equation i s  assumes a d i s t r i b u t i o n  of  on  to  improved  t h e Debye s h a p e h a s  to equilibrium by  due  as t h e  of the present  Debye e q u a t i o n i s b a s e d  is  region, i n  which the  ting  capacitor  of  discussed i n this  instead experiment the experiment.  3  The remaining  chapters of t h i s work are as follows.  Chapter I I i s d i v i d e d into three s e c t i o n s . Section 1 serves as an i n t r o d u c t i o n f o r those who are not f a m i l i a r with the phenomena o f d i e l e c t r i c r e l a x a t i o n . Section 2 discusses the symmetric-top molecule i n some d e t a i l . Section 3 gives addition i n s i g h t i n t o the r e l a x a t i o n process. In Chapter I I I , the improvement of the apparatus i s discussed i n d e t a i l . The r e s u l t s obtained from the experiment are t r a n s l a t e d to information about the r e l a x a t i o n process. This i s given i n Chapter IV. The p o s s i b i l i t y of doing f u r t h e r work i n t h i s f i e l d i s discussed i n Chapter V.  4  CHAPTER I I THEORY OF DIELECTRIC RELAXATION 6  2.1 Macroscopic Theory - The Debye Equation and the ColeCole Equation When an e l e c t r i c f i e l d E i s applied to a d i e l e c t r i c , the d i s t o r t i o n p o l a r i z a t i o n P^ w i l l be established almost instantaneously;  but the d i p o l a r part o f the p o l a r i z a t i o n  1*2 takes a f i n i t e time to reach i t s e q u i l i b r i u m . often assumed that the rate o f change of ?  2  It i s  i s proportional  to i t s d i f f e r e n c e from i t s e q u i l i b r i u m value P - P^ d P _ P - P, - P ' — " -± * (2.1.1) dt ^ Q  where T o i s the d i e l e c t r i c r e l a x a t i o n time. I f we assume that the f i e l d E i s applied suddenly at t=0, and that P =0 at t=0, then the s o l u t i o n of (2.1.1) 2  ?  2  ( P  - ? ) (1 - e " ±  t/to  (2.1.2)  )  When the applied f i e l d o s c i l l a t i n g at a frequency f, angular frequency 2fff = co , E can be expressed as, E =  ( 2  where E o i s the amplitude o f the o s c i l l a t i n g  ' 1  3 )  field.  The s t a t i c r e l a t i v e p e r m i t t i v i t y C (0) and the r  r e l a t i v e p e r m i t t i v i t y at i n f i n i t i v e frequency  €r(co),  which  5 i s a constant, are defined i n terms of P and P^ by 4TP » I (<M0) 41^=  - lfe  (2.1.4a)  6 (Moo) - 1)E  (2.1.4b)  where £ i s the p e r m i t t i v i t y of vacuum; and as before the rate of change of P  i s proportional to i t s d i f f e r e n c e  2  from the e q u i l i b r i u m value P -  ^ 2  -  F  l ~ 2 = J _ ( MO)  P  P  dt  To  - &(OD) )exp(  >t  ) -. 12  4Tf'C  T  s  (2.1.5) 1«t The steady-state  of (2.1.5) i s of the form ?  2  = A e  ;  d  s o l v i n g f o r A gives : £ CCr(O) - 6,(00) )E  p Since P  +  4^(1  2  (2.1.6)  jwTo)  i s a complex quantity, i t implies that the d i p o l a r  2  part of the p o l a r i z a t i o n i s out of phase with the f i e l d , can P  we  write, l  +  P  2  =  p t  "  P  "  =  Jftf ( €  r ( 0 O )  "  1  )  *  E  g  (  6  r  (  0  "  -^(oo))E  )  4 ( 1 +  j  w  ^ ) "''  (2.1.7) t  where both P  . f t  and P  are r e a l . When the p o l a r i z a t i o n  becomes complex, the p e r m i t t i v i t y must also become complex,  t*  =  -  J S r " - MB) + -< ' V  0  1  1 where both  t f  and  6V  are r e a l ,  C  ' W  + j " T .  (2.1.8)  6  *r x  «  e r (  oo)  +  (eAo) -€,(©))  ^ — +  -., 1  - ( 6V(0) - & ( * ) )  ^-2  1  9)  «*rtr  (2.1.10)  ?  i + a) 7,8,^  The above e q u a t i o n s a r e known as the Debye E q u a t i o n s  ,  which can be e x p r e s s e d as an e q u a t i o n o f a c i r c l e by e l i m i n a t i n g to to,  and r e a r r a n g i n g :  2  2 v  2  ' (2.1.11)  Only t h e s e m i - c i r c l e above t h e significance.  £ r a x i s has p h y s i c a l  Each p o i n t i n t h e l o c u s o f t h i s  r e p r e s e n t s one f r e q u e n c y  semi-circle  measurement*  However, t h e r e i s a c o n s i d e r a b l e amount o f evidence t h a t i n many f l u i d s t h e p o i n t s do not l i e i n t h a t s e m i - c i r c l e , but i n s t e a d t h e y l i e v e r y c l o s e l y t o a c i r c u l a r a r c w i t h i  c e n t r e below t h e  € r a x i s , and t h e d i a m e t e r o f t h e a r c  making an angle *TT/2 below t h e  <Sr a x i s .  C o l e and C o l e suggested  t h a t t h e above case can  be r e p r e s e n t e d by t h e e m p i r i c a l e q u a t i o n :  e/  -  MCO)  -  _idoL^M_ i  + (jwr.)  1  i . e . ' t h e r e a l and i m a g i n a r y p a r t o f €r a r e :  (2.1.12)  7  e:  -  Moo)  =  i+W  (MO) 1 + 2(wn)  «t  _  1  -*sin*77/2 + "(wr.)^  ( & ( 0 ) - C-r(co) ) ( a r , ) 1 + 2(wr.)  1  '« s i n -or/a; )  1  1  ( 2 - 1 - 1 3 )  U  ~ * cos*7/2  " " s i n ^ / 2 + (wr.)  2 1 1  -  (2.1.14) u )  The p a r a m e t e r <x c a n o n l y assume v a l u e s between 1 and 0.  I t c a n be a t t r i b u t e d t o a measure o f t h e s p r e a d  o f r e l a x a t i o n t i m e s around t h e most p r o b a b l e v a l u e To. U  F r e n k e l , K r y d e r , and M a r y o t t  l a t e r suggested t h a t  may  be a measure o f w h e t h e r t h e c o l l i s i o n s r e s p o n s i b l e f o r m o l e c u l a r r e o r i e n t a t i o n s a r e , i n some s e n s e , " s t r o n g " o r "weak".  8  2.2 Symmetric-Top Molecule The Fluoroform molecule CHF^ i s a i n which  I  = Ig ± I  A  symmetric-top (2.2.1)  c  where I Q i s the moment of i n e r t i a along the symmetry axis, and Ig, I , are the moment of i n e r t i a about axes perpenA  d i c u l a r t o the symmetry a x i s . The molecule has a permanent e l e c t r i c d i p o l e moment along the symmetry axis due to the f a c t that the centre of the p o s i t i v e charge does not coi n s i d e with the negative charge. The energy of r o t a t i o n of the molecule i s found by s o l v i n g the Schroedinger equation, which gives, £ where  J  K  = B J ( J + 1) •+ (A - B ) K  B » h/S/^Ig  ,  A = h/87T I 2  A  .J i s the t o t a l angular momentum vector.  2  (2.2.2)  , For a given value  of J , K can takes values K = J, J - l ,  - U - l ) , - J (2.2.3)  The s e l e c t i o n r u l e s f o r t r a n s i t i o n between the various r o t a t i o n a l .states induced by the applied f i e l d C 8 n 12  be shown t o be A J = 0, +1  , and^K = 0.  (2.2.4)  9  The component of the dipole moment /f- i n the 13  d i r e c t i o n of J i s (see F i g . 2.2)  (J(J +  i))  1 / z  When a e l e c t r i c f i e l d E i s present, i t exerts a torque on t h i s component of £ the f i e l d  and so _J precesses about  direction. The stationary  or diagonal component of the  dipole moment i n the d i r e c t i o n of the applied given by  field i s  . • A  J  K  M  =  M  OKM  f cosef JKM>  where M i s the component of J along the f i e l d  direction.  M i s also quantized and takes on values M » J, J-l,  ( J - D , -J  (2.2.7)  I f the angular frequency 00 of the applied  field i s  not resonant with any p a i r o f the r o t a t i o n a l l e v e l s , the f i e l d cannot induce r o t a t i o n a l t r a n s i t i o n . As a r e s u l t of molecular c o l l i s i o n s , t r a n s i t i o n s J - * J ' , M-»M' can occur, which s i g n i f i e s a reorientation  o f the stationary  component of the dipole  moment, a change i n i t s magnitude and d i r e c t i o n . The  10  CONE A  Fixed i n space i n the absence of E with J as axis and c m . as vertex and angle 2( d - <j> ).  CONE B  Fixed to the molecule with the symmetry axis as axis and angle  2<Pi  i f t h i s cone r o l l s without  s l i p p i n g on cone A v/ith a uniform speed, i t represents the c l a s s i c s l motion of the molecule. CONE C  Represents the precession of _J about E .  FIG. 2.2 COMPONENTS OF THE DIPOLE MOMENT & CLASSICAL MOTIONS OF A SYMMETRIC-TOP MOLECULE  11  t r a n s i t i o n s that increase the dipole moment of the gas i n the d i r e c t i o n of the applied f i e l d are favoured. Eventually, the gas becomes p o l a r i z e d . This phenomenon i s c a l l e d dielectric relaxation. There are, i n a broad sense, three d i f f e r e n t c l a s s e s of c o l l i s i o n strengths : the strong, the intermediate, and the weak. In the strong c o l l i s i o n , the molecules  approach  each other very c l o s e l y , thereby producing large changes i n the i n t e r n a l angular momentum due t o the short-range a n i s o t r o p i c intermolecular f o r c e s . The s t a t e of the molecule a f t e r - t h e c o l l i s i o n has no connection with i t s state before t  t  the c o l l i s i o n . The t r a n s i t i o n s JKM-* J K M reorientation results.  i  occur and l a r g e  Also the c o l l i s i o n takes place over  an i n t e r v a l of time which i s short compared to the period of o s c i l l a t i o n  i n the applied f i e l d .  In the intermediate strength c o l l i s i o n , the J-state of the molecule i s preserved-after the c o l l i s i o n , only JM-* J M' can occur.  The o r i e n t a t i o n e f f e c t i s not so great  as i n the strong c o l l i s i o n . In the weak c o l l i s i o n , there i s only a phase-shift i n the quantum state of the molecule a f t e r the c o l l i s i o n . An i n d i v i d u a l c o l l i s i o n has l i t t l e e f f e c t i n d i s t u r b i n g the o r i g i n a l o r i e n t a t i o n or p o l a r i z a t i o n o f the molecule.  12  2.3 M i c r o s c o p i c  Theory o f D i e l e c t r i c  The e x p r e s s i o n s  f o rthe d i e l e c t r i c  phenomena c a n b e o b t a i n e d function  Relaxation relaxation  e i t h e r from t h e c o r r e l a t i o n  approach o r from t h e d i s t r i b u t i o n f u n c t i o n  The f o l l o w i n g i s t h e l a t t e r  method.  view and i s e s s e n t i a l l y based  •17  on B i r n b a u m . The k i n e t i c t h e o r y  o f gas i s t r e a t e d  classically  such that t h e t r a n s l a t i o n a l : m o t i o n  classically  and t h e i n t e r n a l m o t i o n quantum  For  function  — dt  =  f^Vjt), ( v  which s a t i s f i e s ) f  V  i t s own B o l t z r n a n n  JJ" l f  ( v  l  ) f  2  ( v  2  ) v  equation.  tfl21'2« lt2« l V  ) v  (v  i'22'  -  the internal  corresponds a d i s t r i b u t i o n •  Z[[I l' l' 2'< 2' l'2» f  i s considered  mechanically.  e a c h quantum s t a t e i , w h i c h c h a r a c t e r i s e s  state of the molecule, there  semi-  12 6'l«2'12 12 { v  ) d v  l 2 d V  )dv  ,d  (2.3.2)  )  w h e r e i = l d e s i g n a t e s t h e p o l a r m o l e c u l e a n d i=2, t h e perturbing  m o l e c u l e ; v-^ i s t h e i n i t i a l 2  relative velocity  o f m o l e c u l e s 1 and 2 b e f o r e c o l l i s i o n , v^'Vg relative velocity after collision; collision in  cross-section  the f i n a l  1  5" -p 2'12^ 12^ v  i n which the molecules  s t a t e 1 a n d 2 e n d u p i n s t a t e s 1' a n d 2 , 1  *  s  t l i e  initially  respectively.  13  Equation  (2.3.1) c a n b e w r i t t e n i n t h e f o r m ,  df-i  ^—  dtT - 2 . ( ^ 1 . % , where A  l f l  represents  molecule 1 w i l l  - f ^  i  n  (2.3.2)  )  t h e p r o b a b i l i t y p e r u n i t time  make a t r a n s i t i o n  f r o m s t a t e 1 t o 1'  accompanied b y an u n s p e c i f i c t r a n s i t i o n :  collision is  Assuming s t r o n g  collisions  t h e p r o b a b i l i t y of  and  o f m o l e c u l e 2.  such t h a t  distribution  d i s t r i b u t e d i n accordance with i n t h e p r e s e n c e o f an a p p l i e d  that  of  after  a dipole  t h e Boltzmann lav/, field  E = E e^ o  c 0 t  , detailed  9  balancing  i s p r e s e r v e d v/hen  A , exp(-(E 1  1  1  - yUjE)/kT) = A  n  f  exp(-(E  l t  -/^.Ej/kT)  ; ..(2.3.3) w h e r e E ^ i s t h e i n t e r n a l e n e r g y , JX.^ i s t h e d i a g o n a l matrix  element. Provided  dipole  ~• that the f i e l d  i s not very  strong,  (2.3.2)  c a n b e shown t o g i v e  dt  1 « 1'  (2.3.4) where t h e s u p e r s c r i p t absence o f t h e f i e l d .  zero  denoted  the quantity  i n the  14  The p o l a r i z a t i o n i n t h e J the  cross-section  (2.3.5)  s t a t e i n terras o f  i s g i v e n by  and ( 2 . 3 . 4 ) g i v e s t h e r a t e e q u a t i o n f o r t h e  p o l a r i z a t i o n i n the dP J  th  =  Eo  -2  "kf"  u  state  GJIM»  °  M- j  A M  j M J t M ? " 6~  2.  t /  JM  /*  0 - 0 , ,  A J M  ..  JIM?  .  JM ^JM JMJ'M» " J'M« ^ J ' M ' ^ J M JMJ'M« ) A  1  A  0  (2.3.6) The s o l u t i o n o f ( 2 . 3 . 6 )  P  where AtOj  - - J - Y  f  '^Wj  i s the half-width When t h e c o l l i s i o n s  cause c o n s i d e r a b l e  can be shown t o be  f o r the state J . a r e so s t r o n g t h a t  they  change o f t h e o r i e n t a t i o n f o r any  r o t a t i o n a l l e v e l , then 4 ^ i s independent o f t h e J s t a t e , i , e  ''  AUj  =4tO  J t  = ALA = -L-  (2.3.8)  where "His the relaxation time characterizing the gross relaxation process.  15  i  The c o m p l e x d i e l e c t r i c is  related to the polarization  c=  6  - j <=  by  = -A2L V kT  constant  tt  ^  When fcJ= 0, t h e s t a t i c  O ^ 2 JM ^ J M .  f  dielectric  ^  + jw  {  2  3  o)  constant i s  given by kT  From  I  JM  ^-JM  (2.3.9)  - £ ( « ) = ( 6 ( 0 ) -€",(co))  £y  5-5-  r  -since Aw_-_1  l  (2.3.11)  + ^  , ,  6, which  (2.3.10)  AO)  - <*(0)  i s the familiar  Debye  -e(C0)) Equation.  A c j  2  2  + f a ) 2 t  g:  (2.3.12)  16  CHAPTER I I I THE  EXPERIMENT  3.1 The Apparatus The equipment used i n t h i s experiment consisted mainly of two major parts, -the 30 MHz. o s c i l l a t o r and the gas handling system.  18 A Clapp type LC o s c i l l a t o r was used because of 19  i t s superior frequency s t a b i l i t y .  A few t r a n s i s t o r  20.21  c i r c u i t s had been t r i e d but the frequency was not stable enough.  The main reason f o r t h i s was t r a n s i s t o r s were  very s e n s i t i v e to temperature f l u c t u a t i o n s and also there was a tendency towards low-frequency o s c i l l a t i o n s i f the components were not c r i t i c a l l y  l a i d out.  The o s c i l l a t o r used here was e s s e n t i a l l y the S8me  as Clark's ( see F i g . 3.1 ) except that a few m o d i f i -  cations had been incorporated i n r e b u i l d i n g t h i s o s c i l l a t o r . The followings v/ere c r i t e r i a on which these  modifications  were based. Frequency f l u c t u a t i o n s due to voltage v a r i a t i o n s were eliminated by using a l i n e of constant  A.C. and  regulated D.C. power supplies f o r the o s c i l l a t o r power input. In order to avoid high-frequency  mechanical  v i b r a t i o n s the o s c i l l a t o r was seated i n a cushion of foam  +75V.  +180 V.  I.- GAS CAPACITOR II - FEED-BACK NETWORK I I I - AMPLIFIER SECTION  Co-15pf C1.C3 =100pf C2=1000pf C4=820 pf R1 =33 kn "1 R2 = 2.7kn R3=1.5kn RFC = 21(jh Z = RF. choke L = Ai r-Dux 616  f reg. count er  FIG. 3.1  THE 30 MHz. OSCILLATOR  H  18  rubber. to  On t h e w h o l e ,  s u c h an e x t e n t t h a t  part d i d not a f f e c t Different  t h e a p p a r a t u s was even a s l i g h t  isolated  t h e cathode  was  t h e r e f o r e maintained during the  follower amplifier.  As t h e o s c i l l a t o r was e x t r a c a r e was  in  a box  currents.  filled  The  experiment.  m o d i f i c a t i o n improved  temperature  the c i r c u i t  as p o s s i b l e  and  the l o n g term  radio-  and  i n order to  w h o l e o s c i l l a t o r was  w i t h f i b r e g l a s s wool  to  single layer  enclosed  temperature  r e g u l a t e d t o + 0.002°C u s i n g a t e m p e r a t u r e The  oscillator  A constant loading  S i l v e r - m i c a c a p a c i t o r s and  avoid the heat  from the  taken to minimize  a s f a r away f r o m t h e t u b e  resonant  extremely s e n s i t i v e  f r e q u e n c y chokes were used throughout placed  external  output l o a d i n g s a f f e c t the  via  variations.  t a p on a n y  the frequency at a l l .  f r e q u e n c y a l t h o u g h t h e l o a d was  temperature,  very r i g i d ,  regulator.  s t a b i l i t y considera-  bly. The to  w h o l e c i r c u i t was  prevent r a d i a t i o n .  Incoming  enclosed i n a metal D.C.  l e a d s were w e l l  housing filter-  ed f o r r . f . u s i n g f e e d - t h r o u g h c a p a c i t o r s a n d r . f . c h o k e s . The  tube  itself  i s seated inside  T h e r e was quency p a r a s i t i c  no  a perforated brass  s i g n o f any u l t r a  oscillations.  high or low  tube. fre-  19  With t h e s e m o d i f i c a t i o n s , a f r e q u e n c y l e s s than 1 0 Hz. i n one hour and a h a l f has been a f t e r a twelve The A 'pulse-gas'  observed  hour warm-up p e r i o d gas h a n d l i n g system was s l i g h t l y m o d i f i e d . v a l v e was i n c o r p o r a t e d i n t o t h e system.  T h i s v a l v e i s a m o d i f i e d three-way g l a s s has  d r i f t of  s t o p cock i t  t h e f e a t u r e o f l e t t i n g t h e gas i n t o t h e system * s h o t -  by-shot ' i n s t e a d o f c o n t i n u o u s l y . An e x t r a v a l v e was p l a c e d between t h e d i f f u s i o n pump and t h e r o t a r y pump t o p r e v e n t t o o h i g h p r e s s u r e b u i l d - u p on t h e exhaust s i d e o f t h e d i f f u s i o n pump when t h e gas was b e i n g pumped out o f t h e system a f t e r t h e experiment.  20  3.2 E x p e r i m e n t s ! The  Procedures  g a s h a n d l i n g s y s t e m was f i r s t  pumped down  t o a p r e s s u r e l e s s t h a n o n e m i c r o n . T h i s p r e s s u r e was maintained throughout port  t h e experiment  o f t h e pressure measuring  to the reference  c a p s u l e i n t h e Texas  . P r e c i s i o n Gauge.. I n a l l s u b s e q u e n t m e a s u r e m e n t s ,  this  r e f e r e n c e p r e s s u r e was t a k e n a s z e r o . Next, oscillator.  regulated electric  pox^er w a s a p p l i e d t o t h e  The f r e q u e n c y o u t p u t w a s o b s e r v e d  on t h e H e w l e t t P a c k a r d  5245L  Electronic  recorded i n the Hewlett Packard The  last  two d i g i t s  562A  Counter, and  d i g i t a l recorder.  o f t h e f r e q u e n c y were r e c o r d e d i n  a chart recorder v i athe d i g i t a l T h i s gave a good v i s u a l drift  visually  t o analog  output.  impression o f the frequency  pattern. A f t e r a warm-up p e r i o d o f a b o u t t w e l v e  a n d when a s m o o t h f r e q u e n c y d r i f t h o u r was o b s e r v e d , data.  o f l e s s t h a n 1 0 Hz. p e r  t h e s y s t e m was t h e n r e a d y f o r t a k i n g  The t e m p e r a t u r e  o f t h e g a s c a p a c i t o r was m e a s u r e d  by measuring  t h e v o l t a g e drop  thermocouple  w i r e u s i n g a K e i t h l e y 1 4 8 Nanometer.  The  hours  across a  copper-constantan  frequency just p r i o r t o the introduction  o f t h e g a s was n o t e d  and t a k e n t o be f. .  was t h e n l e t i n t o t h e s y s t e m  A p u l s e o f gas  v i a the 'pulse-gas'  valve.  After  a few seconds  stable,  t h e i r v a l u e s were r e c o r d e d .  r e p e a t e d about  two hundred  p r e s s u r e o f about The  when t h e f r e q u e n c y a n d p r e s s u r e w e r e T h i s p r o c e s s was  and f i f t y  one-third  times u n t i l the  o f a n a t m o s p h e r e was r e a c h e d .  g a s w a s t h e n pumped q u i c k l y  out o f t h e system  to less  t h a n o n e m i c r o n . The f r e q u e n c y f© was t h e n n o t e d . T h e !  difference  o f f»  and f  f 0  indicated  t h e amount o f d r i f t  t h a t had been occured d u r i n g t h e experiment. was l e f t  evacuated  again observed the d r i f t  and t h e f r e q u e n c y d r i f t  f o r one o r two h o u r s .  The  p a t t e r n was  It--was o n l y when  b e f o r e end a f t e r t h e experiment  t h a t t h e d a t a was c o n s i d e r e d  The s y s t e m  w a s t h e same  reliable.  r e s u l t s w e r e a n a l y s e d u s i n g t h e -IBM 360/67  computer w i t h t h e v a r i o u s s u b r o u t i n e a v a i l a b l e U.B.C. c o m p u t i n g t h e U.B.C. P l o t  library.  i n the  The g r a p h s w e r e p l o t t e d  using  s u b r o u t i n e v i a t h e PDP-8/L c o m p u t e r a n d  t h e Calcomp P l o t t e r .  22  CHAPTER I V RESULTS AND 4.1  Calibration  ANALYSIS  of the Oscillator  A r g o n was u s e d t o c a l i b r a t e t h e o s c i l l a t o r b e c a u s e its  d i e l e c t r i c s u s c e p t i b i l i t y i s a c c u r a t e l y known a s a  function  of pressure.  ptibility  The v a l u e o f i t s d i e l e c t r i c s u s c e -  a t s t a n d a r d t e m p e r a t u r e a n d p r e s s u r e i s (5542+  9)xlO~ . 7  The r e l a t i o n s h i p b e t w e e n t h e d e n s i t y o f t h e g a s and  f r e q u e n c y change  i s g i v e n v e r y c l o s e l y by:  f (amagats) = A f / 2 7 r L C f ? X p 2  0  since Af  f„ , f  a  Hence a p l o t  origin.  (4.1.1)  i s the resonant frequency o f the o s c i l l a t o r  when n o a r g o n g a s was  with constant  S T  present. of Af vs.p  s l o p e 21T LC f ? X 2  0  S T p  s h o u l d be a s t r a i g h t  line  and p a s s i n g t h r o u g h t h e  A t y p i c a l r e s u l t i s shown i n F i g . 4 . 1 . 1 . The L C o v a l u e s w e r e e v a l u a t e d a t e a c h  p o i n t by assuming a l l t h e q u a n t i t i e s e x a c t . The r e s u l t o f L C  0  experimental  i n e q u a t i o n 4.1.1  were  vs.j> i s shown i n F i g 4 . 1 . 2 . I t i s  seen t h a t t h e v a l u e o f L C o i s e s s e n t i a l l y constant w i t h t h e  o. CM  ARSON A-f  vs  fo-30.78  MHz.  T=295J6°K  p  a  to.  —i a  IE a a.  Lu  03  a a'  0.0  0.08  0.04  FIG.  4.1.1  ~ 0.12  P  I 0.16  0.2  (RMRGATS)  A T y p i c a l Argon C a l i b r a t i o n Run  0i24  0.28  a"  a"  ID —a I  *'«-•>•  UJ"1. (Ha  LJ  o  ID —a  a  a  0.0  o.oa  I—  1—  0.12  0.2  0.16  —I—  0.24  P IfitffiGflTS) FIG. 4.1.2  LC  0  values of the  Oscillator  0.2S  0,  25  value  (1.573+0.002)xl0"  x/  sec.  at d e n s i t i e s above  0.06  amagats. The  l a r g e d e v i a t i o n s at the lower d e n s i t y r e g i o n s  c o u l d have been due  to e i t h e r pressure or frequency  ments. However, as we  s h a l l now  measure-  show the u n c e r t a i n t i e s i n  the f r e q u e n c y measurements are p r i m a r i l y r e s p o n s i b l e f o r the observed  behavior.  There was  a drift  z e r o p r e s s u r e o f 8 Hz. result to was  i n the resonant  d a t a . The  used because the d r i f t  experiment was frequency  similiar.  drift  at  i n 1 hour and 10 minutes i n the  o f Fig.4.1.2. A c o n s t a n t d r i f t  the e x p e r i m e n t a l  frequency  c o r r e c t i o n was  straight line  applied  approximation  p a t t e r n b e f o r e and  after  the  However the e x a c t n a t u r e o f the  p a t t e r n d u r i n g the experiment was  I t i s e s t i m a t e d t h a t t h i s type o f a p p r o x i m a t i o n i n t r o d u c e an e r r o r o f + 1 Hz.  i n A f at any  unknown.  could  individual  measurement. The The  gas was  amount o f gas  decreased  admitted  i n pulses i n t o the  i n each p u l s e was  fairly  system.  constant  and  l i n e a r l y as a f u n c t i o n o f the number o f p u l s e s .  A typical result  i s shown i n Fig.4.1.3. The u n c e r t a i n t i e s  i n t h e p r e s s u r e measurement were n e g l i g i b l e , w i t h the  present  Texas P r e c i s i o n Gauge over t h e e n t i r e range. Hence the c o n s t a n t gas volume i n each p u l s e and the a c c u r a c y  almost  o f the  D.O  a o  PRESSURE (TQRRS) 4.0  8.0  12.0  1B.0  20.0 o  4V  a  ^3 M  O  CD  oo  *  > W  X CD  3 H*  3  ct HO  3 O  c+  sr  ID  cn ~a c: r~ cnC T ma co  cn *  a  CD  Tl M 0)  CD  I  a• a  o CO  0)  <j &3  4^'  CD  ro co  (Vi  co' o  LO  a  0.728  0.738  0.1*®  The Amount o f Gas i n Each P u l s e  27  p r e s s u r e gauge r u l e d out t h e d e v i a t i o n i n t h e L C v a l u e 0  t o u n c e r t a i n t i e s i n t h e p r e s s u r e measurement  due  a t lower d e n s i t -  ies. The f r e q u e n c y constant  change between p u l s e s showed a  v a l u e o f 4 H15. (from 0 - 2 0  t o r r s ) . T h e r e f o r e an  u n c e r t a i n t y o f + 1 Hz. i n t h e end o f the f i r s t a f f e c t the LC value D  and  pulse could  by + 25%, t h e second p u l s e by + 12.5%,  so on. The e r r o r i n A f , and hence L C , Q  diminished  p r o g r e s s i v e l y as more gas was l e t i n t o t h e system. e x p l a i n s the r e s u l t  almost  of  Fig.4.1.2.  The above  28  4.2 Fluoroform Data Analysis The Fluoroform used i n t h i s experiment had a minimun p u r i t y of 98.0% according to the manufacturer Matheson of Canada L t d . The change i n resonant frequency with respect to  change i n pressure was measured from 0.0009 to  0.28  amagat. The resonant frequency at zero pressure was 30.78MHz. .The temperature of the gas capacitor was  22.6+0.05°C. The  i n i t i a l pressure of the system was about 1 micron, Since 4 f <& f o i n t h i s experiment, the r e l a t i o n s h i p between d i e l e c t r i c s u s c e p t i b i l i t y and frequency change due to  change i n the density of the gas i s given i n (A-4) i n  Appendix A, which c:an be written as A  f  (4.2.1)  f The LC  0  value was determined using Argon as i n  the previous section p r i o r to the fluoroform measurement. The u n c e r t a i n l y i n the pressure measurement was negligible. at  There was a d r i f t i n the resonant frequency  zero pressure of 12 Hz. i n 1 hour and 30 minutes.  uncertainty i n the frequency was  The  estimated to not more than  1 Hz., i n any i n d i v i d u a l measurement. The e r r o r i n A f  was,  however, much l e s s than that i n Argon becauseA£ was 47 Hz. i n the f i r s t pulse of the gas and 95 Hz. i n the second,  29  an uncertainty of + 1 Hz. i n A f accounted  1%  and  f o r only  2%  i n /./^> r e s p e c t i v e l y f o r the two pulses of gas.  The -uncertainty of X'/^ at d e n s i t i e s above 0.003 amagat became n e g l i g i b l e . To f i t the experimental values two equations were attempted. These were the -Debye equation and the Cole-Cole equation. The Debye equation (2.3.12) i n ' s e c t i o n 2.3 i s w r i t t e n as 2 X'  >  - X(»)  = ( X(0) - X(®))  »  a  (4.2.2)  f  since A. = 6? - 1 , Y  =27^, where V i s the frequency i n MHz.,  d i v i d e d by $ ,  * V ?_ r  X[<o) _ (  r  .  X(o)  p  r  X(oo) , ^ +^  Microwave nonresonant absorption measurement of symmetric-top molecules  showed that f o r d i l u t e gases i n  which bimolecular c o l l i s i o n s were predominant, the r e l a t ionship between l i n e - w i d t h Ai^ = a p , where  and density i s given by  a i s the l i n e - w i d t h parameter i n MHz/amagat,  30  (4.2.3) can be r e w r i t t e n as  Equation  f  P  f  f  a p + 2  2  I t i s obvious t h a t  JSkLp-*ce P.. Lim  T  .  X.'(y)  The v a l u e An almost was  =  X(0?  (4.2.5)  _  7l(oo)  (4.2.6)  P  X(0)/f was determined u s i n g (4.2.1).  constant value o f  into a straight line  = 2 a  where  W  +  Equation  2  A p l o t o f xV)/p 2  F i g . 4.2.1.  transformed  (4.2.7)  X(OD) *(0)  V  (4*2.4) was f i r s t  form  w =  of slope a  amagat.  f o l l o w i n g s t e p s were f o l l o w e d t o determine  the o t h e r parameters.  2^1  3  d e n s i t i e s above 0.25  observed.at The  (8.6767 + 0.0004) x 10" /amagat  ?  _  X-(^) x  (4.2.8)  f  v s . w s h o u l d be a s t r a i g h t  line  and i n t e r c e p t X(a>)/f> . The r e s u l t i s shown i n  31  '0  8'0  ZL'Q  (z-orx)  W O  9S'0  rsiuauwu/na/x  WQ  32  The small d i f f e r e n c e between X(0)/p a n d X ( V ) / p at higher d e n s i t i e s implies that the f i t i s very s e n s i t i v e to  a small departure from the Debye shape. A rough estimate of the parameters a and  can only be made by performing equation DLSQFT  a l e a s t square f i t f o r  (4.2.7) at d e n s i t i e s below 0.04 subroutine was  amagat.. The  .UBC  used. The r e s u l t s were  a •» 1986 X(co)/p=  X.(co)/p  + 200 MHz/amagat  (4.2.9)  0.00568 + 0.00002 /amagat  The l a r g e e r r o r i n a i n d i c a t e s that i t i s not a good f i t , implying that' there i s a d e v i a t i o n from the Debye shape below 0.04  amagat. A more accurate f i t by taking the whole density  region (0.0009-0.28 amagat)into  account was used. This i s  the maximun l i k e l i h o o d f i t t i n g technique  (see Appendix B).  The values of the parameters obtained above was  used as  an i n i t i a l estimate i n t h i s f i t t i n g . Convergence was to was  defined  occur when the change i n the value of a l l the parameters simultaneously l e s s than 0.1% between successive  i t e r a t i o n s . A f t e r three i t e r a t i o n s , t h i s convergence requirement  was met;  and the values of the parameters  obtained were a = 1992 MHz/amagat (4.2.10) *(°°)/p = 0.00566 /amagat  DEBYE F I T a = 1992 MHz /amagat = 0.00566 / amagat -2^=0.0086767 /amagat .... - Experiment = Theory  0.12  DENSITY FIG.  0.16  0.2  0.24  (RMflGflTS)  4.2.2 Debye F i t f o r CHF^  (o.0009-0.28 amagat)  0.28  0.32  34  35.  The r e a l part of the Cole-Cole equation i n density form i s A±D. f •  =  X(oo?  ,  X(0) „  (  X (co)  e  f (ap  P  .  )2(l-«) -  Cap) -°' 2 U  )  +  { s p i j )  l-<x  R ± n o i i r  + Z\e ») -%±n«*/2 1  ?  /? fe  +  (4.2.11)  It can be shown that L  i  - -2Lt^I  m  p-*» Lira p  O  P  (4.2.12)  P  l'(i>)  =  p  X(<o)  ••  p  The parameters a, X(°°)/( > and ?  ,  V^.^'-L^/  were v a r i e d  i n d i v i d u a l l y while keeping the others constant. The  effects  of each v a r i a t i o n of the parameter on the f i t t i n g of experimental data were examined c a r e f u l l y . For p o s i t i v e values of c ( , i t was  found that i t would give a worse f i t  f o r density regions above 0.03 f i t at lower d e n s i t i e s .  amagat; but an improved  The best combination  of values  obtained were a » 2030 MHz./amagat •*(oo)/p= 0.00566 /amagat <* =  (4.2.14)  0.02  They were s u b s t i t u t e d into equation and the r e s u l t s p l o t t e d i n Fig.4.2.4. F i g 4.2.5  (4.2.11), i s the  same f i t but covering the whole density region measured.  36  COLE-COLE FIT •a == 2030 MHz / amagat = 0.00566 /amagat 2ll°l = 0.0086767/amagat  e  * =0.02  o.o  ~i  0.01  1 0.02  1 — 0.03  ~ i —  0.04  0.C5  DENSITY IflMflGflTS)  FIG 4.2.4 Cole-Cole F i t f o r CHF.J  (0.0009-0.045 amagat)  3a  The e f f e c t i v e m o l e c u l a r r e o r i e n t a t i o n c r o s s section  <o  e : f  .£  i s given  eff  b y  v L  23  (4.2.15)  Po  where L i s t h e Loschmidt's number; f Q i s t h e d e n s i t y a t which i*^= 1, i e .  A l )  = ^ = a.(?Q .; v  i s t h e mean r e l a t i v e  v e l o c i t y o f a c o l l i d i n g p a i r o f m o l e c u l e s o f reduced mass jJ(8kT/ir/0  (4.2.16)  1 / 2  In T a b l e 4.2.1, t h e r e s u l t s o b t a i n e d from  this  experiment a r e compared w i t h C l a r k ' s . In T a b l e 4.2.2, t h e e f f e c t i v e m o l e c u l a r r e o r i e n t a t i o n c r o s s - s e c t i o n i s compared w i t h N.M.R. and microwave experiments o f t h e same g a s .  TABLE4.2.1 COMPARISON V/ITH CLARK'S  Clark 309°K * per  ( 0 0 )  /f amagat  %(0)/? per  amagat  30.6 MHz  0.00563+4%  0.00835+2$  RESULTS  Present 295.6 K  30.77 MHz  0.00566+1%  0.0086767+0.0000004  1992 (Debye) M U / * MHz/amagat  1920+5% -  2030  (Cole-Cole,<X=0.02)  39  TABLE 4-2.2 COMPARISON OF 6"  f  28  N.M.R. 297°K 28.2MHz amagat  6-eff  4  ,  7  9  ^  VALUES  2.  Microwave 299.5*K 1193MHz  0.045+0.002  3  f  0.69  9  5  ,  5  7  ^  r*  r>  R.F. 295.6 K 30.77MHz 0.0153+0.00015  1  1  1  ,  9  ^  40  CHAPTER V DISCUSSION  AND  SUGGESTIONS  H i g h e r a c c u r a c y i n t h e l o w e r d e n s i t y measurements ( l e s s t h a n 0.003 a m a g a t ) c a n b e o b t a i n e d b y h a v i n g electric  field  the p r e c i s i o n  oscillating  at a higher frequency.  o f the present experumental  F l u o r o f o r m g a s do i n d i c a t e inadequate  to explain  t h a t t h e Debye  the d i e l e c t r i c  an However,  results  of  equation i s  relaxation  phenomena.  The C o l e - C o l e e u q a t i o n i m p r o v e s o n l y i n t h e l o w e r d e n s i t y region(  l e s s t h a n 0,03  further  investigate  empirical and  Cohen  a m a g a t ) . I t may b e i n t e r e s t i n g t o  the relaxation  process.  A recent  study a l o n g these l i n e s i s g i v e n by Birnbaum (1970) who  suggested  that the correlation  function  s h o u l d h a v e z e r o s l o p e a t t h e o r i g i n a n d become e x p o n e n t i a l at  s u f f i c i e n t long times. The s i m i l a r i t y  l 6  >  2 3  » ^ 2  between t h e method o f  u s i n g t h e e l e c t r i c moment a n d t h e m a g n e t i c moment a s a probe i n i n v e s t i g a t i n g  the molecular relaxation  process  can be compared. However, no t h e o r e t i c a l work h a s y e t been attempted t i m e s f o r any  e x p l i c i t l y t o r e l a t e t h e two system.  correlation  41  APPENDIX A R e l a t i o n o f Resonant Frequency, D i e l e c t r i c Constant and Gas P r e s s u r e From elementary frequency  e l e c t r i c i t y theory, the resonant  f© o f a LCo o s c i l l a t o r i s g i v e n , i n vacuum, by  fo =  1 / 2 7T(L(Co +AC) )  (A - 1)  1 / / 2  where A C i s some c o n s t a n t c a p a c i t a n c e assumed t o be i n p a r a l l e l w i t h t h e c a p a c i t o r Co, L: i s t h e i n d u c t a n c e o f the c i r c u i t . When a d i e l e c t r i c m a t e r i a l i s i n t r o d u c e d between the p l a t e s o f the c a p a c i t o r C  0  , the resonant  frequency  i s g i v e n by f = £  Rearranging  1 / 27T(L(€,Co+ A C ) )  (A - 2)  1 / 2  A - l , A-2 g i v e s  6, - 1 =  V ( h> 47TLC f£  -  (A - 3  >  h  0  In t h e p r e s e n t experiment, A f <^ f  0  , A-3 can be  approximated by  6  . i =  r  where  Af = f  e  -f  6  r  * f  2 7r2LCofo3  (A - 4)  42  For r a r e gases, the d i s p e r s i o n i s n e g l i g i b l e between  r a d i o - f r e q u e n c i e s and v i s i b l e l i g h t , t h e r e l a t i o n -  s h i p between  (  1) a t S.T.P. t o a r b i t r a r y t e m p e r a t u r e  and p r e s s u r e i s g i v e n ( e,-  by  2 5  1) = ( £ r - l )  x  x  2 7 3  760 where P i s t h e p r e s s u r e i n mm.  *  2  (A - 5)  T  Hg. and T i s t h e t e m p e r a t u r e  i n degrees K e l v i n . D e n s i t i e s a r e e x p r e s s e d i n t e r m s o f t h e amagat u n i t , which i s p r o p o r t i o n a l t o d e n s i t y normalized t o 1 a t NOP f o r t h e g a s o f i n t e r e s t . At l o w p r e s s u r e s ,  below  1 atmosphere, i d e a l g a s l a w c a n be assumed a n d , « 1 * > P (amagats) = *  P(mm.Hg. ) 273.2 —• —- x — — 760 T ta  U  i  /. /• \ \ A - o;  C o m b i n i n g A-4,A-5,A-6 g i v e s  f  ( amagat s ) = — \ ? Ti T.n ^ 27rLC 6  — K f o " ( £ r - 1)STP  (A - 7)  43  APPENDIX B  *7  2.6,  Maximun L i k e l i h o o d Method t o E v a l u a t e In theory  ±  Values  an e x p e r i m e n t , x ^ , y ^ a r e m e a s u r e d , a n d a  h a s come u p s u c h  Y  Parametric  that  = g(3-L,a »  '-.a , x ) • M  2  ( B  ±  1)  -  w h e r e a ^ , j = 1,..,M a r e M p a r a m e t e r s c h a r a c t e r i z i n g t h e theoretical  equation  g.  Suppose t h e d i s t r i b u t i o n  o f y^ h a s a  Gaussian  2 statistics, then  with  as t h e v a r i a n c e a s s o c i a t e d w i t h  the probability Pi -  o f f i n d i n g y^ i n t h e i * * *  [sium^rKxpi  Y i )  - |(Y."-  The m a x i m u n l i k e l i h o o d  2  / ^  a  value  which y i e l d  ( B  where L = W  Gaussian  2)  as  (B - 3)  o f t h e u n d e t e r m i n e d p a r a m e t e r s a-^, are those  values  Q  -  i t i s more c o n v e n i e n t  -2(  \ (Y. -  Y i )  2  / ^  2  t o m a x m i z e l o g L,  +  log((27r} V.)) 1/  ( B  for  -  a m a x i m u n i n L.  However,  log  )  "  consistent with the theory  2  f  value i s  1  function i s defined  L = TT P, The b e s t  2  y^  statistics.  -  4)  44  st WQ i s an extremun when a l l the 1 d e r i v a t i v e s of W w.r.t. a^, j = i.e. —  -  0  ,  j =  •be.  1,..,M  order p a r t i a l  vanish.  1,...,M  (B  -  5)  Solving the -above set of equations y i e l d the best value o f a. .  -  I f the parameters are not l i n e a r then method has to be used instead. The i t e r a t i v e a9  some i n i t i a l estimates  iteration  scheme assumes  o f a l l the parameters,  and then  a f i r s t order Taylor's s e r i e s expansion o f each of the p a r t i a l d e r i v a t i v e s i n B-4 employing  these i n i t i a l estimates,  i.e. G  ( —-) j=l,M)  (a*, vrhere  A a  k  *  a  j  +  3=1 C * j * k a  a  (a°,a°; j,k«l,M)  = a^ - a^  (B - 6)  Solution o f B-5 then y i e l d the f i r s t order c o r r e c t ionsAa,  , which, when added t o the i n i t i a l estimates a9 ,  K  j  y i e l d an improved set o f a. , which may again be used i n equation B-5 to y i e l d a better s e t . The i t e r a t i o n i s complete when the i n d i v i d u a l l y  process  parameter obtained i s  within a c e r t a i n percentage o f the previous one.  ;  45  BIBLIOGRAPHY 1.  G o r d o n , R.G. (1968). A d v a n c e s i n M a g n e t i c 2i  1. E d i t e d b y J.S.Waugh, N.Y. A c a d e m i c  Resonance, Press  2.  B i r n b a u m , G. (1957). J . C h e m . P h y s . , 2£,  3.  J o n e s , F. (1963). E l e c t r o n i c  4.  Barrie,  5.  B o g g s , J . E . , Deam, A.P..(1959). J . P h y s . C h e m . , 64., 248  6.  Hill,  E n g n g . , 35.. 733  T.. (1965). P r o c . I E E , 112,  N.E. (1969). D i e l e c t r i c  S t r u c t u r e , Van N o s t r a n d  408  P r o p e r t i e s and M o l e c u l a r  L t d . N.Y.  7.  D e b y e , P. (1929). P o l a r M o l e c u l e , D o v e r  8.  F r o e l i c h , H. (1958). T h e o r y University Press  9.  Van V i e c k , J.H. and W e i s s k o p f , Phys.,  12,  360  Publication  of Dielectrics,  Oxford  V . F . (1945). R e v . Mod.  227  10.  C o l e , K. a n d C o l e , R. (1941). J . C h e m . P h y s . , £  11.  Frenkel,  L . , K r y d e r , S.J.-, a n d M a r y o t t , A. A.  J . C h e m . P h y s . , 4^, 12.  ,341 (1966).  2610  T o w n e s , C. a n d S c h a l o w ,  A. (1955). M i c r o w a v e  Spectroscopy,  McGraw-Hill 13.  Herzberg, Van  14.  Nostrand  C l a r k , R.B. (1971). M.Sc. t h e s i s , British  15.  G. (1966). I n f r a r e d a n d Raman S p e c t r a ,  Columbia,  Hirshfelder,  16.  unpublished  J . ,Curtiss,  M o l e c u l a r Theory  The U n i v e r s i t y o f  C,  and B i r d ,  o f Gases and L i q u i d s ,  G o r d o n , R. (1966). J . C h e m . P h y s . ,  R.  (1954).  John'Wiley  1635  & Son  46  17. 18.  "Birnbaum, G. (1966). Phys. Rev., 1^0, 101 Clapp, J . (1948). Proc.I.R.E., ^6, 356 (1954). Proc.I.R.E., ^2, 1295  19.  Roberts, W.A. (1948). Proc.I.R.E., 136, 1261  20.  Demaw, D. (1971). Q.S.T., Jan., 11  21.  Markus, J . (1968). Source-Book of E l e c t r o n i c McGraw H i l l  22.  Birnbaum, G. and Cohen, E.R. (1970). J.Chem.Phys., 21,  Circuits,  2885  23.  Armstrong, R.L. and Courtney, J.A. (1971). Can. J . Phys. £0, 1262  24.  Bloom, M. (1972). MTP Int.Rev.of Science, Magnetic Resonance, Ed. C.A.McDonell  25.  J e l a t i s , J . (1948). J.Appl.Phys., 1£, 419  2$.  Orear, J . (1958). Notes on S t a t i s t i c s f o r P h y s i c i s t s , UCRL-8417  27.  Orth, P.H.R., Falk, N.R., and Jones, G. (1968). Nucl. Inst, and Methods, 6£, 301  28.  Dong, R. (1969). Ph.D. Thesis, The U n i v e r s i t y o f B r i t i s h Columbia, unpublished  

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0103917/manifest

Comment

Related Items