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

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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 in the Department of Physics We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1972 In present ing th i s thes is in pa r t i a l f u l f i lmen t o f the requirements for an advanced degree at the Un ive rs i t y of B r i t i s h Columbia, I agree that the L ib ra ry sha l l make it f r ee l y ava i l ab le for reference and study. I fu r ther agree that permission for extensive copying o f th i s thes i s fo r s cho l a r l y purposes may be granted by the Head of my Department or by h is representat ives . It is understood that copying or pub l i c a t i on o f th i s thes i s fo r f i nanc ia l gain sha l l not be allowed without my wr i t ten permiss ion. Department of The Un ive rs i t y of B r i t i s h Columbia Vancouver 8, Canada - i -ABSTRACT Dielectric relaxation of Fluoroform (CHF-j) i s measured carefully in the density region (0.0009-0.28 amagat) where the molecular reorientational 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 characterized by a single relaxation time, i s found to be inadequate to explain the experimental data. The Cole-Cole model, which assumes a distribution of relaxation times i s t r i e d . Agreement with experiment i s only improved at densities below 0.03 amagat. - i i -TABLE OF CONTENTS Page ABSTRACT i TABLE OF CONTENTS i i LIST OF ILLUSTRATIONS i i i LIST OF TABLES i v ACKNOWLEDGEMENTS • V Chapter I Introduction 1 Chapter II Theory of Dielectric Relaxation 2;1 Macroscopic Theory - The Debye 4 Equation and the Cole-Cole Equation 2.2 Symmetric-top molecule 8 2.3 Microscopic Theory 12 Chapter III The Experiment 3.1 The Apparatus 16 3.2 Experimental Procedures 20 Chapter IV Results and Analysis 4.1 Calibration of the Oscillator 22 4.2 Fluoroform Data Analysis 28 Chapter V Discussion and Suggestions 40 APPENDIX A 41 APPENDIX B 43 BIBLIOGRAPHY 45 - i i i -LIST OF ILLUSTRATIONS FIGURE PAGE 2.2 Component of the Dipole Moment & Classical Motions of a Symmetric-Top Molecule .10 3.1 The 30 MHz. Oscillator > 17 4.1.1 A Typical Argon Calibration Run 23 4.1.2 LCo values of the Oscillator 24 4.1.3 An Examination of the Pulse-Gas Valve 26 4.2.1 (JO/f vs.'w for CHFj 31 4.2.2 Debye F i t for CHF3 (0.0009-0.28 amagat) 33 4.2.3 Debye F i t for CHF^ (0.0009-0.045 amagat) 34 4.2.4 Cole-Cole F i t for CHF^ (0.0009-0.045 amagat) 3£ 4.2.5 Cole-Cole F i t for CHF3 (0.0009-0.28 amagat) 37 . - i v -L I S T OF TABLES TABLE PAGE 4.2.1 C o m p a r i s o n w i t h C l a r k ' s r e s u l t s 38 4.2.2 C o m p a r i s o n o f v a l u e s 39 ACKNOWLEDGEMENTS I w o u l d l i k e t o th a n k Dr. Bloom f o r s u p e r v i s i n g t h i s 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 7 w h i l e Dr. Bloom was on s a b b a t i c a l l e a v e (Aug.71-Sept.72)• I a l s o t h a n k S h i r l e y J a c k s o n and Bob A l b r e c h t i n a i d i n g my computer programming, and J.L e e s i n t h e g l a s s w o r k s . T h i s r e s e a r c h i s s u p p o r t e d , i n p a r t , by t h e 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 c o u p l i n g between an a p p l i e d r - f e l e c t r i c f i e l d and 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 has o n l y a s m a l l p r o b a b i l i t y o f i n d u c i n g 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 e n e r g y d i f f e r e n c e between d i f f e r e n t n o n - d e g e n e r a t e 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 . f r e q u e n c i e s . Hence t h e r e i s no a p p r e c i a b l e a b s o r p t i o n o f e n e r g y f r o m t h e f i e l d . However, due t o m o l e c u l a r c o l l i s i o n s , a s m a l l amount o f en e r g y i s a b s o r b e d . T h i s comes about as a r e s u l t o f c o l l i s i o n s , i n w h i c h t h e a n i s o t r o p i c , o r a n g u l a r dependent p a r t o f t h e i n t e r m o l e c u l a r f o r c e s p r o d u c e s 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 m o l e c u l e s . . I t can be shown t h a t t h e e n e r g y a b s o r b e d f r o m an o s c i l l a t i n g e l e c t r i c f i e l d i s p r o p o r t i o n a l t o t h e s p e c t r a l d e n s i t y o f t h e c o r r e l a t i o n f u n c t i o n o f t h e 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 s u c h s p e c t r a l d e n s i t i e s g i v e s i n f o r m a t i o n on 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 i n t e r m o l e c u l a r f o r c e s . A c a l c u l a t i o n b a s e d on microwave n o n r e s o n a n t a b s o r p t i o n e x p e r i m e n t a l r e s u l t s shows t h a t t h e amount o f e n e r g y a b s o r b e d i n t h e r a d i o - f r e q u e n c y r e g i o n i s 2 v e r y s m a l l . In f a c t , i t i s beyond t h e c a p a b i l i t y o f modern t e c h n o l o g y . On the o t h e r hand, t h e r e a l p a r t o f t h e d i e l e c t r i c s u s c e p t i b i l i t y can be measured v e r y e a s i l y and s h o u l d c o n t a i n t h e same amount o f i n f o r m a t i o n . No work on d i e l e c t r i c r e l a x a t i o n i n gases has 5 been done below 400 MHz. except f o r C l a r k ' s study near 30 MHz. i n t h i s l a b o r a t o r y . T h i s t h e s i s i s i n t e n d e d t o check h i s r e s u l t s and t o improve t h e a c c u r a c y o f the experiment. The change i n resonant f r e q u e n c y o f the c a p a c i t o r as a r e s u l t o f t h e change i n d e n s i t y o f f l u o r o f o r m gas i s measured c a r e f u l l y i n the i n f o r m a t i o n - r i c h r e g i o n , i n which t h e f r e q u e n c y o f m o l e c u l a r r e o r i e n t a t i o n due t o c o l l i s i o n i s o f t h e same o r d e r o f magnitude as the o s c i l l a -t i n g f i e l d . In t h e p r e c i s i o n o f t h e p r e s e n t improved equipment, a d e p a r t u r e from t h e Debye shape has been o b s e r v e d . - The Debye e q u a t i o n i s based on t h e assumption t h a t t h e approach t o e q u i l i b r i u m o f t h e m o l e c u l a r p o l a r i z a t i o n i s 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 t i m e . In an attempt t o f i t t h e r e s u l t s , the C o l e - C o l e e q u a t i o n i s adopted, which assumes a d i s t r i b u t i o n o f r e l a x a t i o n t i m e s i n s t e a d o f a s i n g l e r e l a x a t i o n t i m e . The agreement v/ith experiment i s o n l y improved below 0.03 amagat. The form o f t h e i n t e r m o l e c u l a r p o t e n t i a l i s not d i s c u s s e d i n t h i s experiment. 3 The remaining chapters of this work are as follows. Chapter II i s divided into three sections. Section 1 serves as an introduction for those who are not familiar with the phenomena of d i e l e c t r i c relaxation. Section 2 discusses the symmetric-top molecule in some d e t a i l . Section 3 gives addition insight into the relaxation process. In Chapter III, the improvement of the apparatus i s discussed in d e t a i l . The results obtained from the experiment are translated to information about the relaxation process. This i s given in Chapter IV. The p o s s i b i l i t y of doing further work in this f i e l d i s discussed in Chapter V. 4 CHAPTER II THEORY OF DIELECTRIC RELAXATION 6 2.1 Macroscopic Theory - The Debye Equation and the Cole-Cole Equation When an elect 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 distortion polarization P^ w i l l be established almost instantaneously; but the dipolar part of the polarization 1*2 takes a f i n i t e time to reach i t s equilibrium. It i s often assumed that the rate of change of ? 2 i s proportional to i t s difference from i t s equilibrium value P - P^ dP Q _ P - P, - P ' — " - ± * (2.1.1) dt ^ where To i s the di e l e c t r i c relaxation time. I f we assume that the f i e l d E i s applied suddenly at t=0, and that P2=0 at t=0, then the solution of (2.1.1) ? 2 ( P - ?±) (1 - e" t / t o) ( 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 ' 1 - 3 ) where Eo i s the amplitude of the o s c i l l a t i n g f i e l d . The static relative permittivity Cr(0) and the relative permittivity 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) - l f e (2.1.4a) 41^= 6 (Moo) - 1)E (2.1.4b) where £ i s the permittivity of vacuum; and as before the rate of change of P 2 i s proportional to i t s difference from the equilibrium value P ^ 2 - F - P l ~ P2 = J _ ( MO) - &(OD) )exp( > t ) -. 12 dt To 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 ; solving for A gives : p £ CCr(O) - 6,(00) )E (2.1.6) 2 4 ^ ( 1 + jwTo) Since P 2 i s a complex quantity, i t implies that the dipolar part of the polarization i s out of phase with the f i e l d , we can write, P l + P2 = p t " P " = Jftf ( € r ( 0 O ) " 1 ) E * g ( 6 r ( 0 ) -^(oo))E " 4 ( 1 + j w ^ ) "'' (2.1 .7) t . f t where both P and P are r e a l . When the polarization becomes complex, the permittivity must also become complex, t* = - J S r " - M B ) + V - < 0 ' - C ' W (2.1.8) 1 + j " T . 1 t f where both and 6V are real, 6 *r x « e r ( o o ) + (eAo) -€,(©)) ^ — ^2-1-9) -., 1 + « * r t r - ( 6V(0) -&(*)) ? (2.1.10) i + a) 7,8,^ The above equations are known as the Debye Equations , which can be expressed as an equation of a c i r c l e by e l i m i n a t i n g to to, and rearranging : 2 2 v 2 ' (2.1.11) Only the s e m i - c i r c l e above the £ r axis has p h y s i c a l s i g n i f i c a n c e . Each point i n the locus of t h i s s e m i - c i r c l e represents one frequency measurement* However, there i s a considerable amount of evidence that i n many f l u i d s the points do not l i e i n that s e m i - c i r c l e , but instead they l i e very c l o s e l y to a c i r c u l a r arc with i centre below the € r axis, and the diameter of the arc making an angle *TT/2 below the <Sr a x i s . Cole and Cole suggested that the above case can be represented by the empirical equation : e/ - MCO) - _ i d o L ^ M _ (2.1.12) i + ( jwr . ) 1 i.e.'the r e a l and imaginary part of €r are : 7 e: - Moo) = ( M O ) i+W1 '« s i n -or/a; ) ( 2 - 1 - 1 3 ) 1 + 2(wn) 1 -*sin*77/2 + "(wr.)^ U « t _ ( & ( 0 ) - C-r(co) ) (ar,) 1 ~ * cos*7/2 (2.1.14) 1 + 2(wr.)1 " " s i n ^ / 2 + (wr.) 2 1 1 - u ) The parameter <x can o n l y assume v a l u e s between 1 and 0. I t can be a t t r i b u t e d t o a measure o f the spread o f r e l a x a t i o n times around the most probable value To. U F r e n k e l , Kryder, and Maryott l a t e r suggested t h a t may be a measure o f whether the c o l l i s i o n s r e s p o n s i b l e f o r mo l e c u l a r r e o r i e n t a t i o n s are, i n some sense, " s t r o n g " or "weak". 8 2.2 Symmetric-Top Molecule The Fluoroform molecule CHF^ i s a symmetric-top in which I A = Ig ± I c (2.2.1) where I Q i s the moment of i n e r t i a along the symmetry axis, and Ig, I A , are the moment of in e r t i a about axes perpen-dicular to the symmetry axis. The molecule has a permanent el e c t r i c dipole moment along the symmetry axis due to the fact that the centre of the positive charge does not co-inside with the negative charge. The energy of rotation of the molecule i s found by solving the Schroedinger equation, which gives, £ J K = BJ(J + 1) •+ (A - B)K 2 (2.2.2) where B » h/S/^Ig , A = h/87T2IA , .J i s the tot a l angular momentum vector. For a given value of J, K can takes values K = J, J - l , - U - l ) , - J (2.2.3) The selection rules for transition between the various rotational .states induced by the applied f i e l d C 8 n 12 be shown to be A J = 0, +1 , and^K = 0. (2.2.4) 9 The component of the dipole moment /f- in the 13 direction of J i s (see Fig. 2.2) ( J ( J + i ) ) 1 / z When a ele c t r i c f i e l d E i s present, i t exerts a torque on this component of £ and so _J precesses about the f i e l d direction. The stationary or diagonal component of the dipole moment i n the direction of the applied f i e l d i s given by . • -A J K M = M O K M 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) If the angular frequency 00 of the applied f i e l d i s not resonant with any pair of the rotational levels, the f i e l d cannot induce rotational transition. As a result of molecular c o l l i s i o n s , transitions J-*J', M-»M' can occur, which si g n i f i e s a reorientation of the stationary component of the dipole moment, a change in i t s magnitude and direction. The 10 CONE A CONE B CONE C Fixed i n space in the absence of E with J as axis and cm. as vertex and angle 2( d - <j> ). Fixed to the molecule with the symmetry axis as axis and angle 2<Pi i f this cone r o l l s without slipping 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. Represents the precession of _J about E . FIG. 2.2 COMPONENTS OF THE DIPOLE MOMENT & CLASSICAL MOTIONS OF A SYMMETRIC-TOP MOLECULE 11 transitions that increase the dipole moment of the gas in the direction of the applied f i e l d are favoured. Eventually, the gas becomes polarized. This phenomenon i s called d i e l e c t r i c relaxation. There are, in a broad sense, three different classes 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 closely, thereby producing large changes in the internal angular momentum due to the short-range anisotropic intermolecular forces. The state of the molecule after-the c o l l i s i o n has no connection with i t s state before t t i the c o l l i s i o n . The transitions JKM-* J K M occur and large reorientation results. Also the c o l l i s i o n takes place over an interval of time which i s short compared to the period of o s c i l l a t i o n in 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 orientation effect i s not so great as in 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 in the quantum state of the molecule after the c o l l i s i o n . An individual c o l l i s i o n has l i t t l e effect in disturbing the original orientation or polarization of the molecule. 12 2.3 M i c r o s c o p i c T h e o r y o f D i e l e c t r i c R e l a x a t i o n The e x p r e s s i o n s f o r t h e d i e l e c t r i c r e l a x a t i o n phenomena can be o b t a i n e d e i t h e r f r om t h e c o r r e l a t i o n f u n c t i o n a p p r o a c h o r f r o m t h e d i s t r i b u t i o n f u n c t i o n method. The f o l l o w i n g i s t h e l a t t e r v i e w and i s e s s e n t i a l l y b a s e d •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 s e m i -c l a s s i c a l l y s u c h t h a t t h e t r a n s l a t i o n a l : m o t i o n i s c o n s i d e r e d c l a s s i c a l l y and t h e i n t e r n a l m o t i o n quantum m e c h a n i c a l l y . F o r e a ch 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 t h e i n t e r n a l s t a t e o f t h e m o l e c u l e , t h e r e c o r r e s p o n d s a d i s t r i b u t i o n • f u n c t i o n f ^ V j t ) , w h i c h s a t i s f i e s i t s own Boltzrnann e q u a t i o n . — = Z [ [ I fl' ( vl' ) f2'< V2' ) vl'2» tfl21'2«(vlt2«)dvl,dV d t i'22' - J J " f l ( v l ) f 2 ( v 2 ) v 1 2 6 ' l « 2 ' 1 2 { v 1 2 ) d v l d V 2 ) (2.3.2) where 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 and i=2, t h e p e r t u r b i n g m o l e c u l e ; v-^ 2 i s t h e i n i t i a l r e l a t i v e v e l o c i t y 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 ^ ' V g 1 t h e f i n a l r e l a t i v e v e l o c i t y a f t e r c o l l i s i o n ; 5" -p 2'12^ v12^ * s t l i e c o l l i s i o n c r o s s - s e c t i o n i n w h i c h t h e m o l e c u l e s i n i t i a l l y i n s t a t e 1 and 2 end up i n s t a t e s 1' and 2 1, r e s p e c t i v e l y . 13 E q u a t i o n (2.3.1) can be w r i t t e n i n t h e f o r m , df-i ^— dtT - 2 . ( ^ 1 . % , - f ^ i n ) (2.3.2) where A l f l r e p r e s e n t s t h e p r o b a b i l i t y p e r u n i t t i m e t h a t m o l e c u l e 1 w i l l 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 by an u n s p e c i f i c t r a n s i t i o n o f m o l e c u l e 2. : Assuming s t r o n g c o l l i s i o n s s u c h t h a t a f t e r c o l l i s i o n t h e p r o b a b i l i t y of d i s t r i b u t i o n of a d i p o l e i s d i s t r i b u t e d i n a c c o r d a n c e w i t h t h e B o l t z m a n n lav/, and i n t h e p r e s e n c e o f an a p p l i e d f i e l d E = E o e ^ c 0 t , d e t a i l e d 9 b a l a n c i n g i s p r e s e r v e d v/hen A 1 , 1 e x p ( - ( E 1 - yUjE)/kT) = A n f e x p ( - ( E l t - / ^ . E j / k T ) ; ..(2.3.3) where 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 d i p o l e m a t r i x e l e m e n t . ~ • P r o v i d e d t h a t t h e f i e l d i s n o t v e r y s t r o n g , (2.3.2) can be shown t o g i v e 1 « d t ' (2.3.4) where t h e s u p e r s c r i p t z e r o d e n o t e d t h e q u a n t i t y i n t h e absence o f t h e f i e l d . 14 t h The p o l a r i z a t i o n i n the J s t a t e i n terras of the c r o s s - s e c t i o n i s given by ( 2 . 3 . 5 ) and ( 2 . 3.4) g i v e s the r a t e equation f o r the p o l a r i z a t i o n i n the s t a t e dP = E o J - 2 GJIM» M- j M AjMJtM? " 6~JM / * J M A J I M ? ° t / 2 . 0 - 0 , , .. . 0 " k f " u JM ^JM AJMJ'M» " 1J'M« ^J'M'^JM AJMJ'M« ) ( 2 . 3 .6 ) The s o l u t i o n o f ( 2 . 3 . 6 ) can be shown t o be P - - J - Y f '^Wj where AtOj i s the h a l f - w i d t h f o r the s t a t e J . When the c o l l i s i o n s are so s t r o n g t h a t they cause c o n s i d e r a b l e change of the 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 the J s t a t e , i , e ' ' AUj = 4 t O J t = ALA = -L- ( 2 . 3.8) where "His the relaxation time characterizing the gross relaxation process. 15 i tt The complex d i e l e c t r i c c o n s t a n t c = 6 - j <= i s r e l a t e d t o t h e p o l a r i z a t i o n by = -A2L V f O ^ 2 ^ { 2 3 o) kT ^ JM ^JM. + j w When fcJ= 0, t h e s t a t i c d i e l e c t r i c c o n s t a n t i s g i v e n by kT I J M ^-JM (2.3.10) From (2.3.9) -since Aw_-_ , £y - £ (« ) = (6 r(0) -€",(co)) 5-5- (2.3.11) l + ^ 1 , A O ) 2 6, - <*(0) -e(C0)) A c j 2 + f a ) 2 t g : (2.3.12) w h i c h i s t h e f a m i l i a r Debye E q u a t i o n . 16 CHAPTER III THE EXPERIMENT 3.1 The Apparatus The equipment used in this 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 transistor 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 for this was transistors were very sensitive to temperature fluctuations and also there was a tendency towards low-frequency oscillations 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 essentially the S8me as Clark's ( see Fig. 3.1 ) except that a few modifi-cations had been incorporated in rebuilding this 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 fluctuations due to voltage variations were eliminated by using a l i n e of constant A.C. and regulated D.C. power supplies for the o s c i l l a t o r power input. In order to avoid high-frequency mechanical vibrations 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 III - 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 r u b b e r . On t h e w h o l e , t h e a p p a r a t u s was v e r y r i g i d , t o s u c h an e x t e n t t h a t even a s l i g h t t a p on any e x t e r n a l p a r t d i d n o t a f f e c t t h e f r e q u e n c y a t a l l . D i f f e r e n t o u t p u t l o a d i n g s a f f e c t t h e r e s o n a n t 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 i s o l a t e d f r om t h e o s c i l l a t o r v i a t h e c a t h o d e f o l l o w e r a m p l i f i e r . A c o n s t a n t l o a d i n g was t h e r e f o r e m a i n t a i n e d d u r i n g t h e e x p e r i m e n t . As t h e o s c i l l a t o r was e x t r e m e l y s e n s i t i v e t o t e m p e r a t u r e , e x t r a c a r e was t a k e n t o m i n i m i z e t e m p e r a t u r e v a r i a t i o n s . S i l v e r - m i c a c a p a c i t o r s and s i n g l e l a y e r r a d i o -f r e q u e n c y c h o k e s were u s e d t h r o u g h o u t t h e c i r c u i t and p l a c e d as f a r away f r o m t h e t u b e as p o s s i b l e i n o r d e r t o a v o i d t h e h e a t c u r r e n t s . The w h ole o s c i l l a t o r was e n c l o s e d i n a box f i l l e d w i t h f i b r e g l a s s wool and t e m p e r a t u r e 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 r e g u l a t o r . The m o d i f i c a t i o n i m p r o v e d t h e l o n g t e r m s t a b i l i t y c o n s i d e r a -b l y . The whole c i r c u i t was e n c l o s e d i n a m e t a l h o u s i n g t o p r e v e n t r a d i a t i o n . I n c o m i n g D.C. l e a d s were w e l l f i l t e r -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 and r . f . c h o k e s . The t u b e i t s e l f i s s e a t e d i n s i d e a p e r f o r a t e d b r a s s t u b e . There was no s i g n o f any u l t r a h i g h o r l o w f r e -quency p a r a s i t i c o s c i l l a t i o n s . 19 With these modifications, a frequency d r i f t of l e s s than 1 0 Hz. i n one hour and a h a l f has been observed a f t e r a twelve hour warm-up period The gas handling system was s l i g h t l y modified. A 'pulse-gas' valve was incorporated i n t o the system. This valve i s a modified three-way glass stop cock i t has the feature of l e t t i n g the gas in t o the system * shot-by-shot ' instead of continuously. An extra valve was placed between the d i f f u s i o n pump and the rotary pump to prevent too high pressure build-up on the exhaust side of the d i f f u s i o n pump when the gas was being pumped out of the system a f t e r the experiment. 20 3 . 2 E x p e r i m e n t s ! P r o c e d u r e s The gas 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 one m i c r o n . T h i s p r e s s u r e was m a i n t a i n e d t h r o u g h o u t t h e e x p e r i m e n t t o t h e r e f e r e n c e p o r t o f t h e p r e s s u r e m e a s u r i n g 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 measurements, t h i s r e f e r e n c e p r e s s u r e was t a k e n as z e r o . N e x t , r e g u l a t e d e l e c t r i c pox^er was a p p l i e d t o t h e o s c i l l a t o r . The f r e q u e n c y o u t p u t was o b s e r v e d v i s u a l l y on t h e H e w l e t t P a c k a r d 5 2 4 5 L E l e c t r o n i c C o u n t e r , and r e c o r d e d i n t h e H e w l e t t P a c k a r d 5 6 2 A d i g i t a l r e c o r d e r . The l a s t two d i g i t s 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 c h a r t r e c o r d e r v i a t h e d i g i t a l t o a n a l o g o u t p u t . T h i s gave a good v i s u a l i m p r e s s i o n o f t h e f r e q u e n c y d r i f t p a t t e r n . A f t e r a warm-up p e r i o d o f about t w e l v e h o u r s and when a smooth f r e q u e n c y d r i f t o f l e s s t h a n 1 0 Hz. p e r h o u r was o b s e r v e d , t h e sys t e m was t h e n r e a d y f o r t a k i n g d a t a . The t e m p e r a t u r e o f t h e gas c a p a c i t o r was measured by m e a s u r i n g t h e v o l t a g e d r o p a c r o s s a c o p p e r - c o n s t a n t a n t h e r m o c o u p l e w i r e u s i n g a K e i t h l e y 1 4 8 Nanometer. The f r e q u e n c y j u s t p r i o r t o t h e i n t r o d u c t i o n o f t h e gas was n o t e d and t a k e n t o be f. . A p u l s e o f gas was t h e n l e t i n t o t h e system v i a t h e ' p u l s e - g a s ' v a l v e . A f t e r a few se c o n d s when t h e f r e q u e n c y and p r e s s u r e were s t a b l e , t h e i r v a l u e s were r e c o r d e d . T h i s p r o c e s s was r e p e a t e d about two hundred and f i f t y t i m e s u n t i l t h e p r e s s u r e o f about o n e - t h i r d o f an atmosphere was r e a c h e d . The gas was t h e n pumped q u i c k l y o u t o f t h e sys t e m t o l e s s t h a n one 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 . The d i f f e r e n c e o f f» and f 0 f i n d i c a t e d t h e amount o f d r i f t t h a t had been o c c u r e d d u r i n g t h e e x p e r i m e n t . The sys t e m was l e f t e v a c u a t e d and t h e f r e q u e n c y d r i f t p a t t e r n was a g a i n o b s e r v e d f o r one o r two h o u r s . It--was o n l y when t h e d r i f t b e f o r e end a f t e r t h e e x p e r i m e n t was t h e same t h a t t h e d a t a was c o n s i d e r e d r e l i a b l e . The r e s u l t s were 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 i n t h e U.B.C. c o m p u t i n g l i b r a r y . The g r a p h s were p l o t t e d u s i n g t h e U.B.C. P l o t s u b r o u t i n e v i a t h e PDP-8/L computer and t h e Calcomp P l o t t e r . 22 CHAPTER IV RESULTS AND ANALYSIS 4.1 C a l i b r a t i o n o f t h e O s c i l l a t o r Argon 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 i t s 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 as a f u n c t i o n o f p r e s s u r e . 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 -p t i b i l i t y a t s t a n d a r d t e m p e r a t u r e and p r e s s u r e i s (5542+ 9 ) x l O ~ 7 . The r e l a t i o n s h i p between t h e d e n s i t y o f t h e gas 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) = Af/27r 2LC 0f?X S Tp (4.1.1) s i n c e A f f„ , f a i s t h e r e s o n a n t f r e q u e n c y o f t h e o s c i l l a t o r when no a r g o n gas was p r e s e n t . Hence a p l o t o f A f v s . p s h o u l d be a s t r a i g h t l i n e w i t h c o n s t a n t s l o p e 21T 2 LC 0 f ? X S T p and p a s s i n g t h r o u g h t h e o r i g i n . 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 were e v a l u a t e d a t each e x p e r i m e n t a l p o i n t by assuming a l l t h e q u a n t i t i e s i n e q u a t i o n 4.1.1 were e x a c t . The r e s u l t o f L C 0 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 c o n s t a n t w i t h t h e o . CM a to. —i a IE a a . 0 3 Lu ARSON A-f vs p f o - 3 0 . 7 8 MHz. T = 2 9 5 J 6 ° K a a ' 0.0 0.04 0.08 ~ I 0.12 0.16 P (RMRGATS) 0.2 0i24 0.28 FIG. 4.1.1 A T y p i c a l Argon C a l i b r a t i o n Run a " a " ID —a I *'«-•>• UJ"1. ( H a o LJ ID —a a a 0 . 0 o.oa I— 1 — 0.12 0.16 P IfitffiGflTS) 0 . 2 —I— 0.24 0.2S 0, FIG. 4.1.2 LC 0 values of the O s c i l l a t o r 25 value (1.573+0.002)xl0"x/ sec. at d e n s i t i e s above 0.06 amagats. The large deviations at the lower density regions could have been due to e i t h e r pressure or frequency measure-ments. However, as we s h a l l now show the u n c e r t a i n t i e s i n the frequency measurements are p r i m a r i l y responsible f o r the observed behavior. There was a d r i f t i n the resonant frequency at zero pressure of 8 Hz. i n 1 hour and 10 minutes i n the r e s u l t of Fig.4.1.2. A constant d r i f t c o r r e c t i o n was applied to the experimental data. The s t r a i g h t l i n e approximation was used because the d r i f t pattern before and a f t e r the experiment was s i m i l i a r . However the exact nature of the frequency d r i f t pattern during the experiment was unknown. It i s estimated that t h i s type of approximation could introduce an e r r o r of + 1 Hz. i n A f at any i n d i v i d u a l measurement. The gas was admitted i n pulses i n t o the system. The amount of gas i n each pulse was f a i r l y constant and decreased l i n e a r l y as a function of the number of pulses. A t y p i c a l r e s u l t 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 the pressure measurement were ne g l i g i b l e , with the present Texas P r e c i s i o n Gauge over the e n t i r e range. Hence the almost constant gas volume i n each pulse and the accuracy of the 3^ M O > W X CD 3 H* 3 ct H-O 3 O c+ sr CD Tl M 0) CD I o CO 0) <j &3 CD D.O a o 4V a oo I D cn ~a c: r~ cn m co C T a a • a 4^' (Vi co' o a PRESSURE (TQRRS) 4.0 8.0 12.0 1B.0 20.0 o CD * cn * a ro co LO 0.728 0.738 0.1*® The Amount of Gas i n Each Pulse 27 pressure gauge r u l e d out the deviation i n the LC 0value due to u n c e r t a i n t i e s i n the pressure measurement at lower densit-i e s . The frequency change between pulses showed a almost constant value of 4 H15. (from 0 - 2 0 t o r r s ) . Therefore an uncertainty of + 1 Hz. i n the end of the f i r s t pulse could a f f e c t the LC Dvalue by + 25%, the second pulse by + 12.5%, and so on. The e r r o r i n Af , and hence LC Q, diminished p r o g r e s s i v e l y as more gas was l e t into the system. The above explains the r e s u l t of Fig.4.1.2. 28 4.2 Fluoroform Data Analysis The Fluoroform used in this experiment had a minimun purity of 98.0% according to the manufacturer Matheson of Canada Ltd. The change in resonant frequency with respect to change in 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 this experiment, the relationship between d i e l e c t r i c susceptibility and frequency change due to change in the density of the gas i s given in (A-4) in Appendix A, which c:an be written as The LC 0 value was determined using Argon as in the previous section prior to the fluoroform measurement. The uncertainly in the pressure measurement was negligible. There was a d r i f t in the resonant frequency at zero pressure of 12 Hz. i n 1 hour and 30 minutes. The uncertainty in the frequency was estimated to not more than 1 Hz., in any individual measurement. The error i n A f was, however, much less than that in Argon becauseA£ was 47 Hz. in the f i r s t pulse of the gas and 95 Hz. in the second, f A f (4.2.1) 29 an uncertainty of + 1 Hz. in A f accounted for only 2% and 1% in /./^> respectively for the two pulses of gas. The -uncertainty of X'/^ at densities above 0.003 amagat became negligible. 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) in'section 2.3 i s written as 2 X' - X(») = ( X(0) - X(®)) » a (4.2.2) > f since A. = 6?Y- 1 , =27^, where V i s the frequency in MHz., divided by $ , * V ? _ X[<o) _ ( X ( o ) X ( o o ) , ^  . r r . p r + ^  Microwave nonresonant absorption measurement of symmetric-top molecules showed that for dilute gases in which bimolecular co l l i s i o n s were predominant, the rel a t -ionship between line-width and density i s given by Ai^ = a p , where a i s the line-width parameter in MHz/amagat, 30 Equation (4.2.3) can be rewritten as f P f f a 2 p 2 + It i s obvious that L i m JSkL- = X(0? (4.2.5) p-*ce P.. P T . X.'(y) _ 7l(oo) (4.2.6) The value X(0)/f was determined using (4.2.1). An almost constant value of (8.6767 + 0.0004) x 10"3/amagat was observed.at d e n s i t i e s above 0.25 amagat. The f o l l o w i n g steps were followed to determine the other parameters. Equation (4*2.4) was f i r s t transformed i n t o a s t r a i g h t l i n e form 2^1 = a2 W + X(OD) (4.2.7) where w = *(0) _ X-(^) x (4.2.8) V2 ? f A p l o t of xV)/p vs. w should be a s t r a i g h t l i n e of slope a 2 and in t e r c e p t X(a>)/f> . The r e s u l t i s shown i n F i g . 4.2.1. 31 '0 8'0 ZL'Q W O 9S'0 WQ ( z - o r x ) r s i u a u w u / n a / x 32 The small difference between X(0)/p andX(V )/p at higher densities implies that the f i t i s very sensitive to a small departure from the Debye shape. A rough estimate of the parameters a and X.(co)/p can only be made by performing a least square f i t for equation (4.2.7) at densities below 0.04 amagat.. The .UBC DLSQFT subroutine was used. The results were a •» 1986 + 200 MHz/amagat (4.2.9) X(co)/p= 0.00568 + 0.00002 /amagat The large error in a indicates that i t i s not a good f i t , implying that' there i s a deviation 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 likelihood 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 in this f i t t i n g . Convergence was defined to occur when the change in the value of a l l the parameters was simultaneously less than 0.1% between successive iterations. After three iterations, this convergence requirement was met; and the values of the parameters obtained were a = 1992 MHz/amagat (4.2.10) *(°°)/p = 0.00566 /amagat D E B Y E F I T a = 1992 MHz /amagat = 0.00566 / amagat -2^=0.0086767 /amagat .... - Experiment = Theory 0.12 0.16 DENSITY (RMflGflTS) 0.2 0.24 0.28 0.32 FIG. 4.2.2 Debye F i t f o r CHF^ (o.0009-0.28 amagat) 34 35. The real part of the Cole-Cole equation in density form i s A±D. = X(oo? , ( X(0) „ X (co) f • f e P . ( a p)2(l-«) - + { s p i j ) l - < x R ± n o i i r / ? fe Cap) 2 U-°' ) + Z\e?»)1-%±n«*/2 + (4.2.11) It can be shown that L i m - -2Lt^I (4.2.12) p-*» P P Lira l'(i>) = X(<o) • , p O p p V^.^'-L^/ The parameters a, X(° ° )/( ? > and were varied individually while keeping the others constant. The effects of each variation of the parameter on the f i t t i n g of experimental data were examined carefully. For positive values of c ( , i t was found that i t would give a worse f i t for density regions above 0.03 amagat; but an improved f i t at lower densities. The best combination of values obtained were a » 2030 MHz./amagat •*(oo)/p= 0.00566 /amagat (4.2.14) <* = 0.02 They were substituted into equation (4.2.11), and the results plotted in Fig.4.2.4. Fig 4.2.5 i s the same f i t but covering the whole density region measured. 36 o.o COLE-COLE FIT •a = 2030 MHz / amagat = 0.00566 /amagat 2ll°l = 0.0086767/amagat e * =0.02 ~ i 1 1 — 0.01 0.02 0.03 DENSITY IflMflGflTS) ~ i — 0.04 0.C5 FIG 4.2.4 Cole-Cole F i t for CHF.J (0.0009-0.045 amagat) 3a The e f f e c t i v e molecular r e o r i e n t a t i o n cross-23 s e c t i o n < o e : f . £ i s given b y e f f (4.2.15) v LPo where L i s the Loschmidt's number; fQ i s the density at which i*^= 1, i e . A l ) = ^  = a.(?Q .; v i s the mean r e l a t i v e v e l o c i t y of a c o l l i d i n g p a i r of molecules of reduced mass jJ-( 8 k T / i r / 0 1 / 2 (4.2.16) In Table 4.2.1, the r e s u l t s obtained from t h i s experiment are compared with Clark's. In Table 4.2.2, the e f f e c t i v e molecular r e o r i e n t a t i o n cross-section i s compared with N.M.R. and microwave experiments of the same gas. TABLE4.2.1 COMPARISON V/ITH CLARK'S RESULTS Clark 309°K 30.6 MHz Present 295.6 K 30.77 MHz * ( 0 0 ) / f 0.00563+4% per amagat % ( 0 ) / ? per amagat 0.00835+2$ M U / * 1920+5% MHz/amagat -0.00566+1% 0.0086767+0.0000004 1992 (Debye) 2030 (Cole-Cole,<X=0.02) 39 TABLE 4-2.2 COMPARISON OF 6" f f VALUES 28 2. r> r* N.M.R. Mic r o w a v e R.F. 297°K 28.2MHz 299.5*K 1193MHz 295.6 K 30.77MHz amagat 0.045+0.002 0.69 0.0153+0.00015 6-eff 3 4 , 7 9 ^ 9 5 , 5 7 ^ 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 amagat) can be o b t a i n e d by h a v i n g an e l e c t r i c f i e l d o s c i l l a t i n g a t a h i g h e r f r e q u e n c y . However, t h e p r e c i s i o n o f t h e p r e s e n t e x p e r u m e n t a l r e s u l t s o f F l u o r o f o r m gas do i n d i c a t e t h a t t h e Debye e q u a t i o n i s i n a d e q u a t e t o e x p l a i n t h e d i e l e c t r i c r e l a x a t i o n 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 r e g i o n ( l e s s t h a n 0,03 amagat). I t may be i n t e r e s t i n g t o f u r t h e r i n v e s t i g a t e t h e r e l a x a t i o n p r o c e s s . A r e c e n t e m p i r i c a l s t u d y a l o n g t h e s e l i n e s i s g i v e n by Birnbaum and Cohen (1970) who s u g g e s t e d t h a t t h e c o r r e l a t i o n f u n c t i o n s h o u l d have z e r o s l o p e a t t h e o r i g i n and become e x p o n e n t i a l a t s u f f i c i e n t l o n g t i m e s . 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 and t h e m a g n e t i c moment as a p r o b e i n i n v e s t i g a t i n g t h e m o l e c u l a r r e l a x a t i o n p r o c e s s can be compared. However, no t h e o r e t i c a l work has y e t been a t t e m p t e d e x p l i c i t l y t o r e l a t e t h e two c o r r e l a t i o n t i m e s f o r any s y s t e m . 41 APPENDIX A Relation of Resonant Frequency, D i e l e c t r i c Constant and  Gas Pressure From elementary e l e c t r i c i t y theory, the resonant frequency f© of a LCo o s c i l l a t o r i s given, i n vacuum, by fo = 1 / 2 7T(L(Co +AC) ) 1 / / 2 (A - 1) where AC i s some constant capacitance assumed to be i n p a r a l l e l with the capacitor Co, L: i s the inductance of the c i r c u i t . When a d i e l e c t r i c material i s introduced between the pl a t e s of the capacitor C 0 , the resonant frequency i s given by f £ = 1 / 27T(L(€,Co+ A C ) ) 1 / 2 (A - 2) Rearranging A-l, A-2 gives 6, - 1 = - V - ( h> - h > (A - 3) 47TLC0 f£ In the present experiment, A f <^ f 0 , A-3 can be approximated by 6r . i = * f (A - 4) where A f = f e - f 6 r 2 7r2LCofo3 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 , the r e l a t i o n -s h i p between ( 1) at S.T.P. t o a r b i t r a r y temperature and pressure i s given b y 2 5 ( e,- 1) = ( £r- l ) x x 2 7 3 * 2 (A - 5) 760 T where P i s the pressure i n mm. Hg. and T i s the temperature i n degrees K e l v i n . D e n s i t i e s are expressed i n terms of the 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 n o rmalized t o 1 at NOP f o r the gas o f i n t e r e s t . At low p r e s s u r e s , below 1 atmosphere, i d e a l gas law can be assumed and, « 1 * > P(mm.Hg. ) 273.2 / . /• \ P (amagats) = —• t a—- x — U i — \ A - o; * 760 T Combining A-4,A-5,A-6 g i v e s f ( amagat s) = — \ — K ? Ti T.n ^  (A - 7) 27rLC 6 f o " ( £ r - 1) STP 43 APPENDIX B 2.6, *7 Maximun L i k e l i h o o d Method t o E v a l u a t e P a r a m e t r i c V a l u e s I n an e x p e r i m e n t , x^, y^ a r e measured, and a t h e o r y has come up s u c h t h a t Y ± = g(3 - L,a 2» '-.a M, x ± ) • ( B - 1) where 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 t h e o r e t i c a l e q u a t i o n g. Suppose t h e d i s t r i b u t i o n o f y^ has a G a u s s i a n 2 s t a t i s t i c s , w i t h 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 y^ f t h e n t h e p r o b a b i l i t y o f f i n d i n g y^ i n t h e i * * * 1 v a l u e i s P i - [sium^rKxpi - | ( Y . " - Y i ) 2 / ^ 2 ) ( B - 2) The maximun l i k e l i h o o d f u n c t i o n i s d e f i n e d as L = TT P, " (B - 3) The b e s t v a l u e 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-^, a 2 c o n s i s t e n t w i t h t h e t h e o r y a r e t h o s e v a l u e s w h i c h y i e l d a maximun i n L. However, i t i s more c o n v e n i e n t t o maxmize l o g L, where l o g L = WQ - -2( \ (Y. - Y i ) 2 / ^ 2 + log ( ( 2 7 r } 1 / V . ) ) ( B - 4) f o r G a u s s i a n s t a t i s t i c s . 44 st WQ i s an extremun when a l l the 1 order p a r t i a l derivatives of W w.r.t. a^, j = 1 , . . , M vanish. i. e . — - 0 , j = 1 , . . . , M (B - 5) •be. Solving the -above set of equations y i e l d the best value of a. . -If the parameters are not linear then iteration method has to be used instead. The iterative scheme assumes some i n i t i a l estimates a 9 of a l l the parameters, and then a f i r s t order Taylor's series expansion of each of the par t i a l derivatives in B-4 employing these i n i t i a l estimates, i.e. G ( —-) + (a*, j = l , M ) * a j 3=1 C * a j * a k (a°,a°; j,k«l,M) vrhere A a k = a^ - a^ (B - 6) Solution of B-5 then yiel d the f i r s t order correct-ionsAa, , which, when added to the i n i t i a l estimates a9 , K j y i e l d an improved set of a. , which may again be used in equation B-5 to yie l d a better set. The iteration process i s complete when the individually parameter obtained i s ; within a certain percentage of the previous one. 45 BIBLIOGRAPHY 1. Gordon, R.G. (1968). Advances i n M a g n e t i c Resonance, 2i 1. E d i t e d by J.S.Waugh, N.Y. Academic P r e s s 2. Birnbaum, G. (1957). J.Chem.Phys., 2£, 360 3. J o n e s , F. (1963). E l e c t r o n i c Engng., 35.. 733 4. B a r r i e , T.. (1965). P r o c . I E E , 112, 408 5. Boggs, J.E., Deam, A.P..(1959). J.Phys.Chem., 64., 248 6. H i l l , N.E. (1969). D i e l e c t r i c P r o p e r t i e s and M o l e c u l a r S t r u c t u r e , Van N o s t r a n d L t d . N.Y. 7. Debye, P. (1929). P o l a r M o l e c u l e , Dover P u b l i c a t i o n 8. F r o e l i c h , H. (1958). T h e o r y o f D i e l e c t r i c s , O x f o r d U n i v e r s i t y P r e s s 9. Van V i e c k , J.H. and W e i s s k o p f , V.F. (1945). Rev. Mod. P h y s . , 12, 227 10. C o l e , K. and C o l e , R. (1941). J.Chem.Phys., £ ,341 11. F r e n k e l , L., K r y d e r , S.J.-, and M a r y o t t , A. A. (1966). J.Chem.Phys., 4^, 2610 12. Townes, C. and Schalow, A. (1955). M i c r o w a v e S p e c t r o s c o p y , M c G r a w - H i l l 13. H e r z b e r g , G. (1966). I n f r a r e d and Raman S p e c t r a , Van N o s t r a n d 14. C l a r k , R.B. (1971). M.Sc. t h e s i s , The U n i v e r s i t y o f B r i t i s h C o l u m b i a , u n p u b l i s h e d 15. H i r s h f e l d e r , J . , C u r t i s s , C , and B i r d , R. (1954). M o l e c u l a r T h e o r y o f Gases and L i q u i d s , J o h n ' W i l e y & Son 16. Gordon, R. (1966). J.Chem.Phys., 1635 46 17. "Birnbaum, G. (1966). Phys. Rev., 1^0, 101 18. 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 Electronic Circuits, McGraw H i l l 2 2 . Birnbaum, G. and Cohen, E.R. (1970). J.Chem.Phys., 21, 2885 2 3 . Armstrong, R.L. and Courtney, J.A. (1971). Can. J. Phys. £0, 1262 2 4 . 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 St a t i s t i c s for Physicists, UCRL-8417 2 7 . Orth, P.H.R., Falk, N.R., and Jones, G. (1968). Nucl. Inst, and Methods, 6£, 301 2 8 . Dong, R. (1969). Ph.D. Thesis, The University of Br i t i s h Columbia, unpublished 

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