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Electron spin relaxation of Cu(II) dithiocarbamates in solution: a study of reorientational correlation… Phillips, Paul Stewart 1978

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ELECTRON SPIN RELAXATION OF Cu(II) DITHIOCARBAMATES  IN SOLUTION: A STUDY OF REORIENTATIONAL CORRELATION  TIMES IN TOLUENE. by PAUL STEWART PHILLIPS B.Sc, University of Sussex, 1974 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES (Dept. of Chemistry) We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA Aug. 1978 © P a u l Stewart P h i l l i p s , 1978 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h C olumbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . n • , f CHEMISTRY Department o f _ _ _ _ _ _ _ _ _ _ _ The U n i v e r s i t y o f B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 28th September .1978 - i -ABSTRACT A s e r i e s of Cu(II)-63 dithiocarbamates w i t h a l k y l and c y c l o a l k y l s u b s t i t u e n t s of v a r i o u s shape, s i z e and r i g i d i t y has been prepared. Their r o t a t i o n a l c o r r e l a t i o n times ( x ^ ) were measured ( i n the f a s t motional regime) i n toluene using the l i n e - w i d t h s of t h e i r ESR spectra. The ESR spec t r a were recorded on paper tape w i t h a d i g i t a l a c q u i s i t i o n system and analysed on an Amdahl 470 computer u t i l i s i n g s p e c i a l l y w r i t t e n programs. This procedure r e s u l t s i n a s u b s t a n t i a l improvement i n p r e c i s i o n over e a r l i e r manual methods. The observed c o r r e l a t i o n times were compared w i t h those c a l c u l a t e d by the QLRF model, the c o n d i t i o n a l f r e e r o t a t i o n model, the Stokes-E i n s t e i n and P e r r i n ' s t i c k ' hydrodynamic models and the ' s l i p ' hydro- -dynamic model. The r e s u l t s i n r e l a t i o n to Pecora's extension of the hydrodynamic model are a l s o discussed. The s t a t i s t i c a l mechanical and ' s t i c k ' hydrodynamic models proved u n s a t i s f a c t o r y . The r e s u l t s are adequately described by the ' s l i p ' hydrodynamic model w i t h an a d d i t i o n a l term a s c r i b a b l e to a f r e e - r o t a t i o n c o r r e l a t i o n time ( T _ ) ; i . e . where n i s the v i s c o s i t y of the solvent and C i s a f u n c t i o n of the s i z e and shape of the molecule c a l c u l a t e d from the ' s l i p ' theory. - i i -TABLE OF CONTENTS Page 1 INTRODUCTION 1 2 THEORETICAL 9 2.1 Phenomenological Introduction to Relaxation 9 2.1.1 The Bloch Equations 9 2.1.2 The Magnetic Resonance Experiment 10 2.1.3 The ESR Experiment 13 2.2 The Isotropic ESR Spectrum 14 2.3 The mj Dependence of Line Widths i n Copper Dithiocarbamate 18 2.3.1 A Simple Two S i t e Model 18 2.3.2 Introduction to Redfield Theory 22 2.4 The Relationship Between D i f f u s i o n and Spin Rotational C o r r e l a t i o n Times 27 2.5 Spin Rotational Contribution to Relaxation 28 3 DIGITAL ACQUISITION OF ESR SPECTRA 31 3.1.1 Spectrometer D i s t o r t i o n 31 3.1.2 The A c q u i s i t i o n Process 31 3.1.3 Data Reduction 33 3.1.4 P r e c i s i o n 33 4 EXPERIMENTAL 36 4.1 Preparation of Potassium Dithiocarbamates 36 4.2 Preparation of 6 3 C u / 6 5 C u Complexes 38 4.2.1 Copper(II)-63 Chloride 38 4.2.2 Copper(II)-63 Dithiocarbamates 38 4.3 Preparation of Copper-Free Nickel Complexes for Matrix Experiments 38 4.4.1 Choice of Solvent 40 4.4.2 Preparation of the ESR Samples 41 4.4.3 Sample Tubes 41 4.5 Recording the Spectra 41 4.6 Temperature Measurement 45 4.7 F i e l d C a l i b r a t i o n 45 4.8 Analysis of Spectra 46 4.8.1 Kivelson's Method - Line Height Analysis 46 4.8.2 Direct Measurement of Line-Width 46 - i i i -4.8.3 Computer Assisted Analysis of Spectra 46 4.10 Analysis of Data 49 4.11 P o l y c r y s t a l l i n e Spectra 50 5 ERROR DISCUSSION 53 5.1 Overall Experimental Error 52 5.2 Temperature Errors 53 5.3 C a l i b r a t i o n 54 5.3.1 Modulation F i e l d from the Magnetometer Probe 54 5.4 Errors i n Analysis: SCAN - Program Errors 54 5.5 Noise, Magnet I n s t a b i l i t y , etc. 56 5.5.1 Spectral Noise 56 5.5.2 Magnet I n s t a b i l i t y 56 5.6 Errors i n Analysis 60 5.6.1 Kivelson's Method 60 5.6.2 Line Width Analysis by Hand 60 5.7 Breakdown of the Theoretical Framework 63 5.7.1 The Slow Motional Regime 63 5.7.2 Other Contributions to the Line-Width 63 5.7.2.1 Unresolved Hyperfine 63 5.7.2.2 Overlap 66 5.7.3 V a r i a t i o n of the g and A Tensor Anisotropies 66 5.7.4 Testing Kivelson's Theory 67 6 RESULTS 68 7 DISCUSSION 86 7.1 Introduction 86 7.2 The Simple Hydrodynamic Model 86 7.3 Non-Hydrodynamic Models 90 7.3.1 Conditional Free Rotation 90 7.3.2 The Quasi-Lattice-Random-Flight (QRLF) Model 92 7.4 Extensions to the Simple Hydrodynamic Model 93 7.5 Temperature Dependence of the E f f e c t i v e Volume 95 7.6 Perrin's Modification of the Simple Hydrodynamic Model 96 7.7 The S l i p Hydrodynamic Model 99 7.8 The Pecora Extension of the Simple Hydrodynamic Model 102 7.9 Some Defects of the Hydrodynamic Models 106 7.10 Conclusions and Future Work 108 BIBLIOGRAPHY 111 - i v -APPENDIX I - Flow Chart f o r the Program SCAN. L i s t i n g of the Program SCAN 117 APPENDIX I I - Experimental Data 128 APPENDIX I I I - C a l c u l a t i o n of Molecular Volumes 175 -v-LIST OF TABLES Table Page 3.1 Methods of Increasing Signal Intensity from 32 ESR Spectrometers 4.1 R e c r y s t a l l i s a t i o n Solvents for Copper Dithiocarbamates 39 4.2 Microanalyses for Copper Dithiocarbamates 39 4.3 T y p i c a l Spectrometer Operating Conditions 42 5.1 Sources of Errors i n Line-Widths and Positions 52 5.2 Temperature Gradients i n Sample Tubes 53 6.1a-d Example of Data Output from the Program KIVEL 69-71a 6.2-6.13 Data Output from KIVEL for a l l the Probes Appendix II 6.14 Apparent A c t i v a t i o n Energy of for a l l the Probes 81 6.15 Data for the T 2 . v s . n / T Plots for a l l the Probes 82 6.16 Data for the a".vs.T /n Plots for a l l the Probes 83 6.17 Anisotropy of Motion vs. Temperature for a l l the 84 Probes 63 6.18 Spin Hamiltonian Parameters for Cu(Et 2DTC) 85 7.1 Observed and Calculated Molecular Volumes for 87 a l l the Probes 7.2 Analysis of the A c t i v a t i o n Energy for a l l the Probes 90 7.3 Comparison of the Sti c k Models for Selected Probes 98 7.4 Comparison of the S l i p and St i c k Models for Selected 101 Probes - v i -LIST OF FIGURES Figure Page 1.1 Spectral Densities for Nuclear/Electron 5 Dipole Fluctuations 2.1 T y p i c a l Copper Dithiocarbamate Spectrum 14a 2.2 Schematic of Two-Site Molecular Jumping 20 2.3 Diagramatic D e f i n i t i o n of D i f f u s i o n Axes 29 4.1 Schematic of Spectrometer 44 4.2 Schematic of D i g i t a l A c q u i s i t i o n System 45 4.3 A Ty p i c a l Calibrated Spectral Line 47a 4.4 Ty p i c a l P o l y c r y s t a l l i n e Spectrum 51a 5.1 Cubic F i t to Top of Fremy's Salt Line 57a 5.2 Cubic F i t to Top of Dithiocarbamate Line 58 5.3 Cubic F i t to Top of Noisy Dithiocarbamate Line 59 5.4 Direct-Line-Width vs. Line-Height Analysis 61 63 5.6 Spectrum of Cu(0c 2DTC) i n the Slow-Motional Regime 64 5.7 Line-Width of Perdeutrated Cu(Me2DTC) vs. 65 Ordinary 6 3Cu(Me 2DTC) 6.1 Ty p i c a l Line-Width vs. Temperature Plot 72 6.2a-g Typ i c a l Plot Output from KIVEL 73-79 6.3a-j T2.vs.n/T plots for a l l the Probes Appendix II 7.1 Theoretical T 0.vs.n/T Plot 104 - v i i -ACKNOWLEDGEMENT I wish to s i n c e r e l y thank Dr. F. G. Herring f o r h i s help and guidance throughout my research. I would a l s o l i k e to thank Dr. J . M. Park f o r many h e l p f u l and s t i m u l a t i n g d i s c u s s i o n s r e l a t i n g to t h i s work; Dr. A. S t o r r f o r as s i s t a n c e i n preparing the perdeuterated s a l t s and f o r the loan of h i s s p a c e - f i l l i n g models; John Mayo and Mike Hatton f o r t h e i r work on the microprocessor; K. Sukul f o r p a t i e n t l y e f f e c t i n g many r e p a i r s to the spectrometer; and f i n a l l y , Anna Wong of the chemistry o f f i c e f o r typing t h i s t h e s i s . I a l s o wish to acknowledge the Chemistry Department of the U n i v e r s i t y of B r i t i s h Columbia f o r p r o v i d i n g f i n a n c i a l support during the course of t h i s work. - v i i i -Once upon a correlation time.... -1-CHAPTER 1  INTRODUCTION The dynamic p r o p e r t i e s of l i q u i d s i n v o l v e v i b r a t i o n a l , t r a n s l a t i o n a l and r e o r i e n t a t i o n a l modes of molecular motion. Many important p h y s i c a l p r o p e r t i e s of l i q u i d s r e l a t e to the l a t t e r two modes of motion and i n recent years there has been considerable i n t e r e s t i n studying them using a wide range of techniques (1-3). In p a r a l l e l w i t h these experimental s t u d i e s a sound t h e o r e t i c a l framework, based on time c o r r e l a t i o n f u n c t i o n s , has emerged. These a l l o w the motions of l a r g e assemblies of molecules to be described f o r m a l l y i n p h y s i c a l l y meaningful terms. The study of molecular r e o r i e n t a t i o n ( o f t e n r e f e r r e d to as r o t a t i o n ) i n f l u i d s g e n e r a l l y i n v o l v e s the a p p l i c a t i o n of a r a d i a t i o n f i e l d to a f l u i d and measuring the response of that f l u i d to the a p p l i e d f i e l d . For such measurements to r e f l e c t the r o t a t i o n a l motions of the molecules i n the f l u i d , the i n t e r a c t i o n of the molecule w i t h the f i e l d must depend on the o r i e n t a t i o n of that molecule i n the f i e l d ; i . e . the i n t e r a c t i o n must be a n i s o t r o p i c . Generally the response of the system, I ( W q ) , at angular frequency to i s the F o u r i e r transform, o I(u) ) = f° G(t)cosw t dt 1.1 ° o ° of the ensemble average time c o r r e l a t i o n f u n c t i o n G ( t ) , -2-G(t) = <[T(0).F(0)].[T(t).F(t)]> 1.2 f o r the appropriate i n t e r a c t i o n between the molecular tensor property, T_, and the tensor f i e l d , F_, ( u s u a l l y defined i n terms of the l a b o r a t o r y frame and hence s t a t i o n a r y ) . T_ and F determine the experimental technique used. I f T_ i s the d i p o l e moment, then F_ can be electromagnetic r a d i a t i o n or an e l e c t r i c f i e l d ; IR spectroscopy and d i e l e c t r i c r e l a x a t i o n r e s p e c t i v e l y . I t i s more convenient to express F_ i n the molecular frame. I f T_ and F_ are s p h e r i c a l tensors of rank j ( j ^ 2 f o r most p h y s i c a l p r o p e r t i e s ) , then _F can be transformed i n t o the molecular frame; molec, , , . 4 ^ = Z ^ D m k [ r ( t ) l X ' 3 m where D^,{r(t)} i s the transformation tensor and represents a set of Euler mk angles i n t e r e l a t i n g the l a b o r a t o r y and molecular frame. From the p r o p e r t i e s of s p h e r i c a l tensors (4a,5); T.F = E ( - l ) k T m ° l e % m ° l e C ( t ) k = J ( . l )V° l e W.[ r ( t ) ] 1.4 , -k -k mk k m as viewed from the molecular frame. The time dependence i s now c a r r i e d e n t i r e l y by the transformation tensor, hence the c o r r e l a t i o n f u n c t i o n can be expressed as a l i n e a r combination of the normalised elements of D~*.{r(t)}; -3-r J , k _ < D i k [ r ( ° ) ] - D m i c [ r ( t ) ] > |<Dmktr(0)]>r Hence the response, 1 ( 0 ) ^ ) , can be expressed as a l i n e a r combination of normalised s p e c t r a l d e n s i t i e s ; j j . k = r°° G J ' k ( t ) c o s a ) t d t 1.6 (u> o ) o ° I t i s o f t e n convenient to use j u s t the zero frequency s p e c t r a l i n t e n s i t y ; the c o r r e l a t i o n time x . This can be TJ»k = f» „j,k c o /°° G J' (t) dt 1.7 r e a d i l y r e l a t e d to hydrodynamic parameters ( f o r a x i a l d i f f u s i o n ) v i a the Hubbard (6) and other equations (2). ( r j ' V 1 = R j j j ( j + l ) - k 2 ] +R,r.k2 1.8 c Rj^ and R u are the perpendicular and p a r a l l e l components of the d i f f u s i o n tensor. Also x ^ ' k = V n/kT 1.9 c m where i s the e f f e c t i v e hydrodynamic volume of the probe, TI i s the v i s c o s i t y of the solvent and T i s the temperature. The s p e c t r a l d e n s i t i e s and c o r r e l a t i o n times observed depend on the experimental technique; t h i s should be borne i n mind when comparing r e s u l t s -4-from d i f f e r e n t techniques. For example, IR r a d i a t i o n probes changes i n d i p o l e moment; a v e c t o r i a l property ( i . e . a f i r s t rank tensor j = l ) . I t i s a property of s p h e r i c a l tensors that -j-$k$+j so only the s p e c t r a l d e n s i t i e s I"^'^(w ) and ( to ) are observable. On the other hand, o o Raman spectroscopy probes changes i n p o l a r i s a b i l i t y ; a 2nd rank tensor, 2 0 so f i v e ( i n p r i n c i p l e ) s p e c t r a l d e n s i t i e s are observable; I ' ( W Q ) , 2 + l 2+1 I (w^) and I H ( W q ) ; none of which are accessable by IR spectroscopy. The form of the c o r r e l a t i o n f u n c t i o n observed a l s o depends on the technique used (7). The motion of neighbouring molecules i s o f t e n s t r o n g l y c o r r e l a t e d due to d i p o l e i n t e r a c t i o n s . For d i e l e c t r i c r e l a x a t i o n techniques, which probe the o r i e n t a t i o n of the d i p o l e s , the c o r r e l a t i o n f u n c t i o n i s of the form; E <D (0) D (t)> G(t) = l , n n 1.10 I < D , (0) D , (0)> l ' , n ' 1 i e . one sums over s e v e r a l p a r t i c l e s and obtains a m u l t i p l e p a r t i c l e c or-r e l a t i o n time. With techniques such as IR, Raman and magnetic resonance, where the observable parameter i s only weakly coupled to adjacent p a r t i c l e s , one obtains the s i n g l e p a r t i c l e c o r r e l a t i o n f u n c t i o n (1.5). For ins t a n c e , w i t h IR one observes changes i n d i p o l e moment. Although the d i p o l e s are s t r o n g l y coupled,changes i n them are not, hence our 'observable' i s i n -dependent of adjacent molecules, i e . s i n g l e p a r t i c l e . However i t should be noted that that i n the case of IR. high probe concentrations are r e q u i r e d , hence the c o r r e l a t i o n time observed i s f o r the motion of the probe i n a -5-concentrated s o l u t i o n or a s o l u t i o n of i t s e l f . With other techniques (eg. ESR), which use low probe co n c e n t r a t i o n s , one i s measuring a c o r r e l a t i o n time f o r the probe i n some so l v e n t . Thus even when the s p e c t r a l d e n s i t i e s are d i r e c t l y comparable they may r e f l e c t motions i n d i f f e r e n t environments. S i n g l e p a r t i c l e techniques are p r e f e r a b l e to m u l t i p l e techniques as the theory i s b e t t e r developed and the c o r r e l a t i o n times are much more e a s i l y r e l a t e d to the a v a i l a b l e hydrodynamic models f o r l i q u i d s . Magnetic resonance presents an unusual case i n that the probe f i e l d i i s a f i x e d frequency and as such the c o r r e l a t i o n f u n c t i o n cannot be found. ( I t i s u s u a l l y assumed to be a simple exponential f u n c t i o n (8a)). However the s p e c t r a l d e n s i t y at the probe frequency ( W q ) i s approximately the same 'as i t i s at zero frequency (see f i g . 1.1) S p e c t r a l d e n s i t y of n u c l e a r / e l e c t r o n d i p o l e f l u c t u a t i o n s 1(0))' F i g . 1.1 In t h i s case we can de f i n e an average c o r r e l a t i o n time T (cf 1.7) (0) 1.11 -6-The x J measured i s x c , u s u a l l y denoted x . The assumed form of c the c o r r e l a t i o n f u n c t i o n i s thus; (t) ace - ( t ) 1.12 The s i t u a t i o n f o r ESR i s complicated by the f a c t that the probe frequency i s higher and (1.11) i s only v a l i d to a f i r s t approximation. However K i v e l s o n (9-11) Freed (12,13) and more r e c e n t l y Park and H e r r i n g (14) have developed t h e o r i e s which s u c c e s s f u l l y d e s c r i b e ESR l i n e widths i n terms of r e o r i e n t a t i o n a l c o r r e l a t i o n time ( T ^ ) • P a r a l l e l to t h i s development of a t h e o r e t i c a l framework there has been a considerable r i s e i n i n t e r e s t i n the use of ESR to probe molecular motions i n b i o l o g i c a l systems; the s p i n probe technique. Even though one cannot e x t r a c t an e x p l i c i t c o r r e l a t i o n f u n c t i o n from ESR data i t i s a s i n g l e p a r t i c l e technique and o f f e r s a concentration s e n s i t i v i t y of at l e a s t three orders of magnitude greater than any other technique (with the p o s s i b l e exception of resonance Raman spectroscopy). Obviously the lower the concentration of the probe the l e s s i t w i l l perturb the system being s t u d i e d . This i s o f t e n a more important c r i t e r i o n than f i n d i n g the , c o r r e l a t i o n f u n c t i o n . There have been very few ESR s t u d i e s of.simple l i q u i d systems (15-22), but there have been numerous st u d i e s i n b i o l o g i c a l systems (4b,23). These 1 s t u d i e s have shown that s p i n probes can r e f l e c t v a r i o u s p h y s i c a l changes i n biomembranes f o r instance. However, owing to the complex nature of such systems, the p h y s i c a l and p h y s i o l o g i c a l i m p l i c a t i o n s of these changes -7-are not yet f u l l y appreciated. I t i s the purpose of t h i s work to i n v e s t i g a t e the e f f e c t of the shape of the s p i n probe on i t s r o t a t i o n a l c o r r e l a t i o n time i n a simple l i q u i d , and to i n t e r p r e t the r e s u l t s i n terms of simple hydrodynamic models. This complements the work of Park and Herring on the e f f e c t of the solvent on the motion of a s p i n probe. I t i s hoped that t h i s w i l l improve our understanding of the more complex systems. Toluene was chosen as our simple l i q u i d as i t ' h a s a wide working temperature range, i s a good s o l v e n t , has w e l l defined p h y s i c a l pro-p e r t i e s and has been the subject of previous NMR motional s t u d i e s (24). 9 The probes chosen were a s e r i e s of d copper(II) complexes of the general formula -I where R, R' = a l k y l or a c y c l i c group. The a l k y l d e r i v a t i v e s (R = R') are u s u a l l y r e f e r r e d to as b i s - d i a l k y l d i t h i o c a r b a m a t e s and the c y c l i c d e r i v a t i v e s r e f e r r e d to as b i s - ( p a r e n t amine)-N-carbodithioates. (The IUPAC nomenclature i s unclear f o r these compounds and they are r e f e r r e d to simply as dithiocarbamates (DTC) throughout t h i s t h e s i s ) . The s p i n Hamiltonian parameters f o r these complexes are t y p i c a l l y (25) -8-gn - g z z = 2.087 A „ = A l g Z Z = 2.018 J A Z Z ^ * { g y y = 2.022 * l A y y v X X X X where the axes are denoted i n I. Dlthiocarbamates are stable, e a s i l y synthesised and have ESR parameters that are v i r t u a l l y independent of the substituents. The g tensor anisotropy (g|| - gj_) B q l i s f a i r l y small, but the A tensor anisotropy (A|j -AjJ i s large. This means that the probes' l i n e widths are f a i r l y s e n s i t i v e to changes i n c o r r e l a t i o n time. Also the i s o t r o p i c A value i s large, but the l i n e s are narrow*, t h i s minimises the e f f e c t of overlap. This study required the a c q u i s i t i o n of large amounts of accurate ESR data; to f a c i l i t a t e t h i s use was made of a recently developed d i g i t a l a c q u i s i t i o n system (26). This system required a number of changes i n experimental technique to avoid d i s t o r t i o n of the data. Also a number of FORTRAN programs were written to handle and interpret the d i g i t a l data. = -490I = -120 > MHz = -130 J -9-CHAPTER 2 THEORY 2.1 Phenomenological I n t r o d u c t i o n to R e l a x a t i o n 2.1.1 The Bloch Equations Spin r e l a x a t i o n i s t r e a t e d thoroughly i n a number of t e x t s (4,8,27-30) so only the s a l i e n t f eatures w i l l be d e a l t w i t h here. The c l a s s i c a l equation of motion f o r a group of spins i n a magnetic f i e l d (B) w i t h i n d i v i d u a l magnetic moments (y) and a t o t a l magnetisation (M) i s dM/dt = yMXB 2.1 I f represents a steady magnetic f i e l d B q along the z-axis 2.1 reduces to dM /dt = oi M 2.2.1 x o y dM /dy = oi M 2.2.2 y o x dM /dt = 0 2.2.3 z f o r the i n d i v i d u a l components of M where O) o=YB q, the Larmor precession frequency. I f the magnetic f i e l d i s now turned o f f the magnetization components w i l l ( i n the simplest case) decay w i t h a f i r s t order process to zero. The -10-M^ component (note that dM z/dt i s zero, not M z) decays v i a energy t r a n s f e r to the l a t t i c e . The time constant f o r the decay i s c a l l e d T , the s p i n - l a t t i c e r e l a x a t i o n time. Thus dM /At = -(M -M )/Tn 2.3.1 z z o 1 where M = YB o o M q i s the bulk magnetisation i n the f i e l d B q and x i s the s t a t i c magnetic s u s c e p t i b i l i t y . The M^ and components decay by a d d i t i o n a l mechanisms r e s u l t i n g i n dephasing of the s p i n s . This occurs w i t h a time constant T 2, the s p i n - s p i n or transverse r e l a x a t i o n time. dM I At = -M /T„ 2.3.2 x x 2 dM /At = -M /T„ 2.3.3 y x 2 These r e l a x a t i o n processes of course occur a l l the time, not j u s t when the f i e l d i s removed. To a l l o w f o r t h i s we have to combine eqns. 2.2 and 2.3 dM I At = co M - M /T„ 2.4.1 x o y x 2 dM /At = to M - M /T„ 2.4.2 y o x y 2 dM I At = -(M -M )/T, 2.4.3 z z o 1 -11-These are the Bloch equations. 2.1.2 The Magnetic Resonance Experiment In the usual magnetic resonant experiment an o s c i l l a t i n g magnetic f i e l d (B 1) i s applied to the sample at r i g h t angles to the main magnetic where ^ , j , k are the unit vectors. B^ i s usually l i n e a r l y p o l a r i s e d , but behaves l i k e two counter-ro t a t i n g f i e l d s and can be rewritten (27a,29a) where U J i s the frequency of o s c i l l a t i o n of B^. To solve 2.5 i t i s necessary to view the motion from a r o t a t i n g coordinate frame. We define two new axes u, v which rotate at 03 i n the x, y plane. The stationary and r o t a t i n g 'z' axis are taken as coincident. Using primes to denote the r o t a t i n g frame we have f i e l d . The complete equation of motion for the spins i s thus dM/dt = Y(MXB) + YCMXB^) - (iMx+j_M )/T 2 - k ^ - M ^ / ^ 2.5 JJ.. = B.. ( i c o s o j t - j s i n o ) t ) 2.6 M' •= i.'u + j 'v + k'Mz 2.7.1 2.7.2 03* = -03k' 2.7.3 One can r e a d i l y show that dM/dt = dM'/dt + 03'xM 2.8 This equivalence i s true, because only the component o s c i l l a t i n g i n phase with the precession can cause resonance. The expression 2.6 i s j u s t a mathematical convenience. -12-thus from 2.5 and 2.8 dM/dt = yM'xCB'+w'/y) + Y(M'+B') - (I'M +j'M )/T„ - k'(M -M)/T. 2.9 x x y z z _L This can be separated i n t o the three components of M' and solved f o r the s t a t i o n a r y s t a t e ; i e . when, du/dt = dv/dt = dM z/dt = 0 2.10 From our d e f i n i t i o n of our r o t a t i n g a x i s system u and v are 90° out of phase. I t i s convenient to introduce a complex transverse s u s c e p t i b i l i t y X where X = x ' - i x " 2.11.1 where x ' = u / B ; L 2.11.2 X" = W B i 2.11.3 The l o n g t i t u d i n a l s u s c e p t i b i l i t y i s simply X = M / B 2 11 4 o z o i , ± ± , H X 1 , x" a r e c a l l e d the r e a l and imaginary parts of the s u s c e p t i b i l i t y (which simply means they are 90° out of phase.) The f i n a l s o l u t i o n to 2.9 i s M = X B 2.12.1 z o o -13-(u o-o))T 2 X'(«) = x D % l+(u> - O J ) T 2 + Y 2 B 2 T 1 T 9 o / 1 ± I 2.12.2 T 2 X"(w) = x 0 " 0 2.12.3 2.1.3 The ESR Experiment In ESR the magnetic s u s c e p t i b i l i t y a r i s e s from unpaired e l e c t r o n spins and B-^  i s the magnetic ve c t o r of the i n c i d e n t microwave r a d i a t i o n . When a paramagnetic sample i s placed i n the c a v i t y of an ESR spectro-, meter the whole arrangement can be considered a simple LCR c i r c u i t (28a), where R i s the r e s i s t a n c e and represents t r a n s m i s s i o n l o s s e s e t c . , C i s the capacitance of the c a v i t y , determined by i t s s t r u c t u r e and L depends 1 p a r t l y on the magnetic s u s c e p t i b i l i t y of the sample. The impedance (Z) of the c a v i t y i s given by I t can be r e a d i l y shown from 2.13 that at resonance the impedance i s complex and of the form Z = R + i(u>L-l/u>c) 2.13 Z = a ( l + b x " ) + i c x ' 2.14 where a, b, c are simple constants. -14-An ESR spectrometer e s s e n t i a l l y measures impedance changes as the sample i s swept through resonance. At resonance the s i g n a l w i l l c o n s i s t of a r e a l and an imaginary component, the absorption and d i s p e r s i o n s i g n a l s r e s p e c t i v e l y . For t e c h n i c a l reasons i t i s e a s i e r to measure absorption than d i s p e r s i o n ; the l i n e shape measured i n ESR i s , AT ^ 1 = — — 5 „ x-7j 2.15 1 + TZ2(.-.o)Z + y 2 B 1 T 1 T 2 where A i s a c o l l e c t i o n of constants and instrumental parameters. In p r a c t i c e (determined by the microwave power) i s kept small 2 2 and y ^^^2 t^ i e s a t u r a t i ° n term can be neglected compared w i t h the other terms. The observed l i n e shape i s thus AT T , , 2 1 2.16 L(w) = — — * 2 2 l + T 2 ( a J - a 3 o ) Z This i s the l o r e n t z i a n line-shape w i t h h a l f - w i d t h at h a l f - h e i g h t of 1/^2' In p r a c t i c e the d e r i v a t i v e of 2.16 i s measured. In t h i s case 1^ i s given by 1/T„ = /3AH 12 2 PP where AH i s the peak-to-peak l i n e width. PP A t y p i c a l ESR spectrum i s shown i n f i g u r e 2.1. I t can be seen i t c o n s i s t s of four l i n e s ( i e four d i f f e r e n t to 's) each of d i f f e r e n t width o ( d i f f e r e n t T 2 ' s ) . The above d i s c u s s i o n i s purely phenomenologicaljit describes the l i n e shape, but cannot e x p l a i n the o r i g i n s of r e l a x a t i o n SPECTRUM NO. 1 CU63ME2DTC) IN CS2 AMBIENT TEMP. 1800 PT5. CALIBRATION 50.0 GAUSS/INTERVAL - 1 5 -(T^) or the values f o r a) Q. The c a l c u l a t i o n and o r i g i n of these values i s discussed below. 2.2 The I s o t r o p i c ESR Spectrum The s p i n Hamiltonian f o r an unpaired e l e c t r o n i n t e r a c t i n g w i t h a s i n g l e nucleus i s K = eB.g.jS + _S.A.^ 2.17 where B i s a s t a t i c magnetic f i e l d , g and A are the g and by p e r f i n e tensors r e s p e c t i v e l y , 8 i s the Bohr magneton and ^  and _I are the e l e c t r o n and nuclear s p i n operators r e s p e c t i v e l y . I f the g and A tensors are i s o t r o p i c and we de f i n e the main f i e l d as being along the z a x i s 2.17 reduces to K = g B B S + a I . S + f ° | S + i " + S " l + v 2.18 ° o o z o z z 2 L J where g Q and a Q are the i s o t r o p i c g and bype r f i n e constants r e s p e c t i v e l y . /\ -|- s\ s\ — /v y\. /s —J— /\ S = S + iS and S = S - i S , I , I are defined analogously, x y x y & J 2.18 can be solved e x a c t l y or more conveniently by p e r t u r b a t i o n theory. I f we l e t X = K0+*H' 2 .19 .1 and H' = S + . I + S7l +] 2.19 H' can be t r e a t e d as a p e r t u r b a t i o n ; i t i s a l s o h e r m i t i a n , so from -16-perturbation theory we have to a second order E = E° + <n|H'|n> E <n H' m> 2.20 m m " 1 , i — i ' m^ n E -E n m where E° = <m|M|m> m 1 1 n> = s,m ;I,m > = m ,m > 1 1 s i ' s i m> = S,m ';I,mT'>= m ' ,m '> s I s I m^  and m^. are the electron and nuclear spin quantum numbers. The zero order term i s simply E° = g gB m + a mT 2.21 m o o s o I The f i r s t order term i s zero, as the matrix of M has no diagonal elements. The second term i s , a2 ^ l < m s m i l s + r + S - I + l m ^ > l 2 ~ —T -T.O „o 4 mrm E E , , I s m m_ m m' s i s 3 which reduces to; -17-or I(I+l)-mI(mI-l) mI+h mI-2.22.3 for m' = -1/2 s The zero order energy terms for the denominator or 2.22 are given by; 3 - E - = ±(g BB + a mT) m+j mT+i o o o I I h Ik It is usual to neglect the smaller a,m^  term, thus llI±h E = ±g BB m,.-, °o o The selection rules for ESR are Am=±l and Am=0, thus for absorption (|l/2> -y |+l/2>) the transition energy is the difference between 2.22.2 and 2.22.3, AE ( 2 > = + a 2 —o 2gBB j~i(i + l ) - m") 2.23 The transition frequency (o>Q) to second order is thus, a) = AE/h = g BB + a mT + a o 6o o o I —o 2gBB 1(1 + 1) - mz) 2.24 For copper dithiocarbamates 1=3/2; M ' = ±3/2, ±1/2, ie. four lines will be observed. For many organic radicals a Q (in field units)<<BQ and the second -18-order term can be neglected, however f o r copper dithiocarbamates t h i s i s not a good approximation. 2.3 The nij. Dependance of Line Widths i n Copper Dithiocarbamates 2.3.1 A Simple Two-Site Model (28b) Consider an assembly of spins perturbed i n such a manner that some spins precess at w-dto and some at co+dco ( i e . there are two s i t e s ) , where co i s the reference frequency. Under these c o n d i t i o n s the vec t o r sum of magnetisation of the precessing component w i l l decay i n a time T^^l/dto. Now i f a s p i n jumps from one s i t e to another i t w i l l be out of phase by 2dto. I f n spins jump at mean time i n t e r v a l s of x, then, from random walk — 2 theory, the mean square phase d i f f e r e n c e accumulated d0 ( i n radians) i s . From the d e f i n i t i o n of the vector sum of the magnetic moments i n the x-y plane must decrease to 1/e of the o r i g i n a l value (when a l l spins were i n phase) i n a time T^. This corresponds to an rms dephasing of ^ 1 rad. Hence i n the time T„, d0 = n(xdto) 2.25 2' n(xdco) = 1 2.26 However during t h i s time there must be n = T 2/x jumps 2.27.1 •'. l/T^xdco ( x < l / d o j ) 2.27.2 -19-for two-site jumping. This is valid only i f T<l/da) ie. the environment has fluctuated before the transverse component decays as a consequence of the T2~l/dti) decay. This is known as the fast motional limit. If x<<l/do) then T^ type mechanisms also contribute to T2- Transitions between the Zeeman levels result in life-time broadening of these states; this is equivalent to dephasing (uncertainty in co) and hence contributes to T^. We now have a physical picture for relaxation which allows us to make some simple predictions. We are interested in an expression for f° r copper dithiocarbamates. These compounds have approximately axially symmetric g and A tensors whose symmetry axes are coincident with the molecular frame; e >>g zz &yy °xx and A » A = A 1 zz 1 yy 1 xx1 The motional processes that contribute to relaxation can therefore be approximated by molecular jumps between two sites, one where the laboratory and molecular frames are coincident (denoted 111) and one where the molecular x axis (or equivalently is the y axis) is parallel to the lab z axis (denoted x). ie. rotation about the z axis. -20-S C H E M A T 1 C OF TWO SITE JUMPING Molecular z Frame < > LABORATORY FRAME F i g . 2.2 Jumps between s i t e s i n the molecular x-y plane have no e f f e c t on as the x-y components are equal, i e dco=0. However these jumps do a f f e c t the s p i n - r o t a t i o n a l term. This i s discussed i n s e c t i o n 2.4. The complete Hamiltonian ( i n the high f i e l d l i m i t ) f o r a molecule i n s o l u t i o n i s H = B.T(0,0).g.T(e,0).S_+ S_.T(e,0).A.T(e,0).I. 2.28 where T(0,0) i s the transformation tensor between the molecular and lab o r a t o r y frames. (0 and 0 are the po l a r angles between the two frames and c a r r y the time dependence f o r a molecule i n s o l u t i o n ) . For the case above the transformation tensors are p a r t i c u l a r l y simple, being the u n i t tensor and the i n v e r s i o n tensor f o r the r e s p e c t i v e o r i e n t a t i o n s . To make a transformation between frames i t i s convenient to d i v i d e -21-the Hamiltonian into an isotropic (time independent) and an anisotropic (time dependent) thus in the molecular frame we have H = H 0 + H ' 2.29.1 Ho = g-B.S^  + a j s . j : 2.29.2 H 1 = B.g'.J5 + S.A.j. 2.29.3 where g Q and a Q are the isotropic g and A values;((l/3)Tr{g} and (l/3)Tr{A} respectively) and g' and A' are the traceless g and A tensors (g' + g 0I = g etc.) The trace of a tensor is invariant under any unitary transformation, hence we need only be concerned with H ' when making the transformation. Thus transforming 2.29.3 to the lab frame and expanding we get K' (Bl lz) = $Bg... S + S . A.,.1 + S.AM + S.A.I ^ v • ' & i i z z " z z xx x y yy y VT (Biz) = BBgiS + S.A.I + S . A' . I + S - A' .1 v ' b ± z z ± z y yy y x zz x where g,. = g1 , g. = g' = g' and A,, = A' , A, = A' = A' 6U zz X xx 6yy \l zz ± xx yy S and S only contribute to the off diagonal elements, so to first order; x y H'(BJlz) = BBdgu.Sz + Sx.Au.Iz 2.30.1 H'(Blz) = 3B 6g x.S z + SZ.AJ.Z 2.30.2 -22-2.30 give the d i f f e r e n c e i n precession frequencies d w = YL = ( Ag.gB Q + AAm^/Ti 2.31 where Ag = g\\ -gx a n d AA = Aj, -Aj_ s u b s t i t u t i n g i n 2.27 2.32.1 a + gm_ + ym' 2.32.2 where a, g, y a r e constants. This i s the form of equation f o r T„ that K i v e l s o n found. I t c l e a r l y shows the m^. dependence of the l i n e widths and that x can be c a l c u l a t e d from t h i s dependency. The approximate upper l i m i t of x f o r the f a s t motional approximation can a l s o be c a l c u l a t e d from 2.31. For copper dithiocarbamates Ag*0.0675 and AA=-2.24 (Grads/sec). Converting to c o n s i s t e n t u n i t s and s u b s t i t u t i n g mr=-3/2 (the broadest l i n e ) we f i n d that dco~5.3 Grads. Hence from (2.26.2) we would expect slow motional e f f e c t s f o r x>190pS. In p r a c t i c e the value i s x=150pS. 2.3.2 I n t r o d u c t i o n to R e d f i e l d Theory (4c) The d e r i v a t i o n above gives a good p h y s i c a l p i c t u r e of the cause of r e l a x a t i o n , however i t cannot be used to o b t a i n accurate c o r r e l a t i o n times; the non-secular terms are not n e g l i g i b l e and the molecules do not jump between two s i t e s , they move over an i n f i n i t e number of o r i e n t a t i o n s . -23-In r e a l i t y we have a large ensemble of molecules each with a time dependent wave function describing t h e i r spin. An exact s o l u t i o n to the problem for each molecule i s obviously not possible. However a s t a t i s t i c a l s o l u t i o n i s possible, by making use of density matrix theory. A l l the molecules are the same so the wavefunction ijj(t) of a given molecule must be expandable i n terms of a basis set u; the eigenfunctions of the s t a t i c Hamiltonian. lji. (t) = E c k ( t ) U . 2.33 k n k n k where the c o e f f i c i e n t s c now carry the time dependency. The expectation value (observable) for a given physical property A (magnetisation i n our case) i s given by = <i|)k(t)|A|t|>k(t)> = E C k Ck<n|A|m> 2.34 n m 1 1 n,m where m> denotes u > etc. 1 ' m We usually have a large number of spins ( t y p i c a l l y 10"'"') and thus require the ensemble average of A, <A>; <A> = E p A. = E E p,C kC k <n|A|m> 2.35 K. K. i i£ n m k n,m where i s a weighting factor for each wave function. It i s convenient to define a density operator p such that -24-i i k* k <m p n> = E p. C C 2.36 1 1 , k n m k then <A> = E <m| p| n>:<n| A|m> n,m I t can be e a s i l y shown that p i s Hermitian (27b) and thus; <A> = Tr{pA} 2.37 We are i n t e r e s t e d i n the time dependence of the magnetisation; d i f f e r e n t i a t i n g 2.35 and remembering that the time e v o l u t i o n of the o r i g i n a l wavefunctions, ^ ( t ) , i s described by the Hamiltonian, ^ = \ity 2.38 i dt ^ v we f i n d that f o r the ensemble average (27c); I f K i s time independent we f i n d the formal s o l u t i o n , p(t) = e-(i/h)M(t) p ( 0 ) e(i/h)K(t) 2 > 4 Q However as noted i n s e c t i o n 2.3.1 H contains a time dependent term, a l s o i t contains the e f f e c t of the microwave f i e l d e ( t ) : thus (2.39) becomes (8b) -25-= ^ [ p ( t ) , K o + H ^ t ) + e(t)] 2.41 Eqn. 2.40 suggests transforming to the i n t e r a c t i o n r e p r e s e n t a t i o n to a i d the s o l u t i o n of 2.41 (equivalent to changing to the r o t a t i n g frame i n the Bloch equations). I f we combine e(t ) w i t h f o r convenience 2.40 becomes d / dt where p + i s given by = | [ p + , Vi t t ) ] 2.42 P ( t ) = e - ( 1 / h ) H o ( t ) p H ( t ) e ( i / h ) K o ( t ) 2 ' 4 3 and "H+ i s defined analogously Eqn. 2.42 cannot be solved e x a c t l y . However R e d f i e l d proposed that an approximate s o l u t i o n can be obtained by i t e r a t i v e ..integration w i t h successive approximations, followed by s t a t i s t i c a l averaging over the ensemble. Thus i f we neglect the e f f e c t of e(t) (equivalent to i g n o r i n g s a t u r a t i o n e f f e c t s i n the Bloch equations), (2.42), to second order, becomes (8,b,27d) ^dt~ = k [ P + ( ° > > H + , ( t ) ] + (| ) 2 / t [ [ P + ( 0 ) , H + ( t ) ] , H + ( t ) ] d t ' 2.44 R e d f i e l d showed that t h i s can be expressed as (8b,27e) -26-where R = — i — f 2J 0 , a, (oi a.) - 6 0 E ., , (U>3'Y) aa'33 2^2 [_ a3a 3 a3 a3 3 Y a Y - 6a'3' I J Q (a> fl)l 2.46 a y 3 Y v Y3 J where k(o>) i s the s p e c t r a l d e n s i t y , /oo G(x)exp(iu)t)dt 2.48 G(x) i s the c o r r e l a t i o n f u n c t i o n , G(x) = <a|H(t) |a'xa|H*(t+x) |a'> 2.50 which reduces to (8c) 2.51 G(x) = |«a|H (t)|a*> 2|e ' T l / x . f o r a random s t a t i o n a r y process ( i e G(x) depends only on x and not t ) . x^ i s c a l l e d the c o r r e l a t i o n time and i s s i m i l a r to the mean jump time defined i n s e c t i o n 2.3.1 and can be i d e n t i f i e d w i t h x^ i n eqn (1.10). <aj e t c . are the b a s i s wavefunctions and are found b y ; s o l u t i o n of the s t a t i c Hamiltonian (see s e c t i o n 2.2). R e d f i e l d made the i d e n t i f i c a t i o n (8c,27e) R , , = -1/T_ , aa aa 2aa -27-i f only the r e a l part of the s p e c t r a l d e n s i t y i s taken. The f i n a l s o l u t i o n f o r T^ i s of the form (8e) 1/T2 = a + 3mx + yrn^. + Sm^ 2.52 c f . 2.32 The forms of a, 3 e t c . used i n t h i s work ( i n the program KIVEL) are due to Bruno (32) and are -28-where F = | < g n - g j e f o D = ( A j _ - A u ) / ( / 6 R ) 2 -1 ' U T2 0) X ( 0 ) T 2 ) 2 + 1 U = [1 + ( U ) T 2 ) 2 ] 1 (ax ) 2 _ P = [1 + 7— 1 2.4 The R e l a t i o n s h i p between D i f f u s i o n and Spin R o t a t i o n a l C o r r e l a t i o n  Times The r e d u c t i o n of (2.50) to (2.51) i m p l i c i t l y assumes an i s o t r o p i c d i f f u s i o n tensor (R) i e . r o t a t i o n a l d i f f u s i o n i s e q u a l l y probable about a l l axes. I f the d i f f u s i o n i s a n i s o t r o p i c then s t r i c t l y speaking f i v e 2 0 2+1 2+2 c o r r e l a t i o n times (T ' , T *~ , x *" ) are needed to describe the motion (2). However f o r dithiocarbamates i t i s reasonable to assume the molecular (and A and g) axes are c o i n c i d e n t w i t h the d i f f u s i o n axes, i e . the d i f f u s i o n tensor i s diagonal i n the molecular frame. This means 2 0 2 2 that only two x's, x ' and x ' , are needed to d e f i n e the motion. The others i n v o l v e the (zero) o f f diagonal tensor elements. -29-Rotation in the X-Y Plane, tl T Diffusion About Z-Axis i /^{•^v Rotation About Y-Axisy <Jjy Diffusion of Z-Axis Into X-Axis 2,2 x i s a s s o c i a t e d w i t h the interchange of the R and R tensor xx yy elements, i e . r o t a t i o n s i n the X-Y plane; d i f f u s i o n about the Z a x i s . As mentioned i n s e c t i o n 2.3.1 t h i s motion makes no c o n t r i b u t i o n to r e -l a x a t i o n , (but see s e c t i o n 2.5), i f the molecule has a x i a l g and A tensors, x i s associated w i t h the interchange of the diagonal tensor elements ( i e . r o t a t i o n of the z a x i s i n t o the X-Y plane). As the mole-20 cule has a x i a l ESR parameters x 1 represents the average c o r r e l a t i o n time f o r d i f f u s i o n about the X and Y axes. One can now see that x (2.51) i s equivalent to x f o r dithiocarbamates. c I t should be noted that i f the d i f f u s i o n i s a n i s o t r o p i c only the average c o r r e l a t i o n time f o r r o t a t i o n s about the X and Y a x i s are access-i b l e . However f o r dithiocarbamates the volume swept out by r o t a t i o n about these axes i s approximately the same, hence i t i s not unreasonable to assume that the d i f f u s i o n tensor i s a x i a l , w i t h the p r i n c i p a l a x i s p a r a l l e l to the molecular Z a x i s i e . R = R i R xx yy zz 2 0 2 2 The motion of the molecule i s thus a c c u r a t e l y described by x ' and x ' . -30-(The l a t t e r i s a c c e s s i b l e v i a the a" term, t h i s i s discussed i n s e c t i o n 2.5). The p h y s i c a l r e l a t i o n between the r o t a t i o n a l c o r r e l a t i o n time and r o t a t i o n a l d i f f u s i o n i s c l e a r , the mathematical r e l a t i o n s h i p i s l e s s obvious and f o r a x i a l d i f f u s i o n i t i s derived from the Hubbard r e l a t i o n (2,6) ( T 2 ' k ) _ 1 = 6R X + k2(R,|-Rj_) 2.55 where R^ = ^~zz and R l = R = R xx yy 2.5 Spin R o t a t i o n a l C o n t r i b u t i o n s to R e l a x a t i o n I f one uses equation (2.53) d i r e c t l y i t i s found that there i s a r e s i d u a l l i n e width c o n t r i b u t i o n which cannot be accounted f o r ; i e . the l i n e s appear to f i t the form 2 3 1/T2 = a' + a" + Bm1 + ym 2 + <5m2 2.56 where a' i s given by the previous equation f o r a and a" i s the r e s i d u a l l i n e width. The a" term has been a t t r i b u t e d to s p i n - r o t a t i o n i n t e r -a c t i o n s (18); that i s a coupling of the e l e c t r o n s p i n and nuclear s p i n caused by the i n e r t i a l time l a g between the r o t a t i o n of the e l e c t r o n cloud and the n u c l e i . (see (8e) f o r d e t a i l s ) The t h e o r e t i c a l expressions f o r a" f o r a x i a l d i f f u s i o n (with the unique -31-a x l s p a r a l l e l to the molecular Z a x i s ) i s given by (33) where 6 = s - g e t c . g i s the f r e e e l e c t r o n g value, gx °xx fae & e 2 2 Note the presence of the x ' term; r o t a t i o n i n the X-Y plane does i n f a c t c o n t r i b u t e to the l i n e - w i d t h , but only v i a a"; a" thus gives us a handhold on the anisotropy of the motion (N). I t i s u s e f u l to r e w r i t e (2.57) i n terms of the p r i n c i p a l a x i s of the d i f f u s i o n tensor. I f we define N = 'R^ /Rj. 2.58 and remember that (1.9) x^ = Vry/kT then a" = ^ _ ^ J E L _ ) ( 6 g ) 2 + ( S g ) 2 + (2N-1)(6g) 2 To a f i r s t approximation 2.59 a" cc Ag/x 2.60 and a" cc T/Vn 2.61 where T i s the temperature, n the v i s c o s i t y and V the hydrodynamic volume of the probe molecule. Although a" can be determined from and an assumption about motion, i t was obtained e m p i r i c a l l y from the l i n e width, and eqn. 2.53, along w i t h x„, using the program KIVEL. -32-CHAPTER 3  DIGITAL ACQUISITION OF ESR SPECTRA There are three b a s i c c r i t e r i a that have to be met f o r the r e l i a b l e d i g i t a l a c q u i s i t i o n of spectra: a) The instrument must produce an undis-t o r t e d s i g n a l ; b) the a c q u i s i t i o n process i t s e l f must not d i s t o r t the s i g n a l (34); and c) the data r e d u c t i o n methods should a c c u r a t e l y t r a n s -form the data. These c o n d i t i o n s are discussed below. At t h i s point i t should be noted that our goal here i s to q u i c k l y e x t r a c t accurate and p r e c i s e i n f o r m a t i o n from a d i g i t i s e d spectrum and not to enhance s e n s i t i v i t y . The l a t t e r may be achieved, u s u a l l y at the expense of the former, by use of a number of d i g i t a l enhancement methods (35-40) none of which w i l l be discussed here. 3.1.1 Spectrometer D i s t o r t i o n D i s t o r t i o n a r i s e s from s u b t r a c t i o n of or a d d i t i o n to the harmonic components the L o r e n t z i a n l i n e (which can be found by expanding the L o r e n t z i a n as a F o u r i e r s e r i e s ) . This u s u a l l y occurs when the s i g n a l i s perturbed by noise or the s i g n a l i s f i l t e r e d . High frequency noise (>10x the fundamental frequency of the l o r e n t z i a n ) can be f i l t e r e d o f f without d i s t o r t i n g the l i n e . Low frequency noise (< fundamental frequency) or b a s e l i n e d r i f t can be removed by simple numerical methods (41), or i n our case circumvented by the use of a good AFC u n i t and a ' s t a t e - o f - t h e -a r t ' l o c k - i n a m p l i f i e r . I t i s the mid-range noise and the attempts to remove i t that cause the d i s t o r t i o n of spectra. In general the midrange noise i s a simple f u n c t i o n of a m p l i f i e r -33-g a i n , the weaker the s i g n a l the higher the gain required and the lower the observed s i g n a l to noise r a t i o (SNR). The obvious s o l u t i o n i s to increase the s i g n a l i n t e n s i t y . There are many ways of doing t h i s (see t a b l e 3.1), but each method can lead to d i s t o r t i o n (see chapter 5 f o r f u r t h e r d e t a i l s ) . TABLE 3.1 Parameter D e l e t e r i o u s E f f e c t Upper l i m i t s Concentration D i p o l a r and s p i n exchange broadening <10 _ 3M Sample Si z e Cavity l o a d i n g , reduced Q Depends on d i -e l e c t r i c constant of the solvent Modulation Broadening of l i n e s Depends on l i n e width Microwave Power S a t u r a t i o n broadening and heating V a r i e s w i t h sample u s u a l l y <10mW Varying the time constant (RC) i s the most commonly used method to improve SNR, however i t does so by decreasing the noise and not by i n -creasing the s i g n a l . Since mid-range noise and the s i g n a l are i n d i s -t i n g u i s h a b l e f i l t e r i n g r e s u l t s i n d i s t o r t i o n . ( I t i s u s e f u l to note that the other methods do not remove in f o r m a t i o n , they j u s t transform ( d i s t o r t ) the l i n e . The c o r r e c t s i g n a l can be recovered, i n p r i n c i p l e , by the reverse transform.) U s u a l l y i t i s easy to meet the c r i t e r i a o u t l i n e d above. However -34-i n d i s c r i m i n a t e use of f i l t e r s can cause severe (and not always apparent) d i s t o r t i o n . The usual rule-of-thumb f o r the choice of time constant i s that i t should be l / 1 0 t h the time to t r a v e r s e the narrowest l i n e i n the spectrum (42). For our work t h i s was found to be inadequate and a l/100th rule-of-thumb was adopted; i e . Where LW i s the l i n e width, ST i s the scan time i n seconds and SW i s the sweep width. (LW and SW being i n the same u n i t s ) D i g i t a l a c q u i s i t i o n i s e s s e n t i a l l y a transformation process. (36) Analogue data i s transformed i n t o d i g i t a l data and then transformed back i n t o the analogue data, not n e c e s s a r i l y by the inverse transform. For the simple i n v e r s e transformation to be v a l i d the Nyquist c r i t e r i o n has to be met. (36,43) This s t a t e s that the sampling r a t e (R) must be 1 at l e a s t twice the highest frequency component of the s i g n a l . The l a t t e r i s d i c t a t e d by the a m p l i f i e r bandwidth (1/TTRC) (36,44), thus RC i s u s u a l l y chosen (see s e c t i o n 3.1.1) so as not to d i s t o r t a Lorentzian l i n e , thus RC < (LW.ST)/(SW.100) 3.1 3.1.2 The A c q u i s i t i o n Process R > 1/TTRC 3.2 R > SW/(2.LW.ST) 3.3 -35-where ST i s now the scan time i n minutes. It i s customary to have a 'safety margin' of a factor of two, but there i s nothing gained by gross oversampling; no extra information i s obtained (43). 3.1.3 Data Reduction There are three aspects of.data reduction; these are transformation, smoothing and analysis. S t r i c t l y speaking smoothing i s a transformation procedure using convolution functions. (36,42). D i g i t a l smoothing and transformation procedures were not used for t h i s work; however i t i s worthwhile pointing out that using a time constant i s equivalent to smoothing with a one-sided exponential convolution function (37). Methods of analysis are discussed i n d e t a i l i n chapter 4. The p r e c i s i o n of analysis i s a function of a c q u i s i t i o n rate, time constants etc. and i s discussed below. 3.1.4 P r e c i s i o n The f i r s t important point to note i s that smoothing improves sensi-t i v i t y and aesthetics, but does not a f f e c t the p r e c i s i o n of the analysis (45). To improve the p r e c i s i o n we must increase the information density of the s i g n a l . (Some methods of doing t h i s were discussed i n section 3.3.1). The p r e c i s i o n (P; the inverse f r a c t i o n a l error) of a given para-meter of a spectrum perturbed by white-gaussian-band-limited noise i s given by Posener (45) as - 3 6 -1/2 P = c.SNR.n ' 3.4 where c i s a constant depending on l i n e shape and the parameter to be measured and n i s the number of data p o i n t s . At f i r s t s i g h t i t appears that the p r e c i s i o n can be increased i n d e f i n i t e l y , but (46,47) 1/2 SNRcc(RC) ' and n cc R cc 1/RC 3.5 thus from eqn. 3.4; f o r a given scan r a t e the p r e c i s i o n i s f i x e d . I f the sampling r a t e i s at or above the Nyquist r a t e , the only way to improve the p r e c i s i o n i s to increase the SNR by decreasing the scan r a t e and then i n c r e a s i n g the time constant i n accord w i t h eqn. 3.1. I t can now be r e a d i l y seen that the maximum p r e c i s i o n a v a i l a b l e i s purely a f u n c t i o n of scan r a t e (SR), 1/2 Pce(SR) ' 3.6 This trade o f f between time and p r e c i s i o n i s aggravated by 1/f noise ( b a s e l i n e d r i f t ) , which i s not accounted f o r i n Posener's theory. Normally t h i s i s not a problem, but f o r very long scan times (>20 min) i t can become s e r i o u s . I d e a l l y a complete set of spectra should be c o l l e c t e d i n one day and t h i s tends to d i c t a t e the scan time used and thus the f i n a l pre-c i s i o n obtained. -37-CHAPTER 4  EXPERIMENTAL 4.1 P r e p a r a t i o n of Potassium Dithiocarbamates (48,49) A l l compounds were prepared by the standard procedure of mixing s t o i c h i o m e t r i c q u a n t i t i e s of carbon d i s u l p h i d e , potassium hydroxide and the appropriate secondary amine i n a l c o h o l or water; CS2+KOH+R2NH -y R 2NCS~ K ++H 20 The d i t h i o d e r i v a t i v e i s formed p r e f e r e n t i a l l y to the xanthate. D e t a i l s are given below. The secondary amine (0.1 moi) was d i s s o l v e d i n a l c o h o l i c potassium hydroxide (100 ml of 0.1M). Carbon d i s u l p h i d e (0.1 moi) i n ethanol (ca. 50 ml) was then added dropwise over a period of 30 min w h i l e the mixture was s t i r r e d . (N.B. The order of a d d i t i o n i s c r i t i c a l as both KOH and the amines react w i t h carbon d i s u l p h i d e s e p a r a t e l y ) . The r e s u l t a n t mixture was reduced i n volume w i t h a r o t a r y evaporater u n t i l a s l u r r y formed. Ether or heptane was added to p r e c i p i t a t e out the potassium s a l t which was then f i l t e r e d o f f . The s o l i d was r e d i s s o l v e d i n a minimum amount of hot water ( f o r the c y c l i c and octade c y l d e r i v a t i v e s ) or hot ethanol ( f o r the a l k y l d e r i v a t i v e s ) and cooled. The s o l u t i o n s were then f i l t e r e d again ( a f t e r adding ether to the e t h a n o l i c s o l u t i o n s ) and the process repeated u n t i l the c h a r a c t e r i s t i c y ellow c o l o u r a t i o n of im-pure potassium dithiocarbamates disappeared. -38-Dimethyl-dg amine (Merck, Sharpe & Dohme 99 atom %) i s a gas and the pr e p a r a t i o n procedures r e q u i r e s l i g h t m o d i f i c a t i o n . The amine (ca. 1 gm) was t r a n s f e r r e d on a vacuum l i n e i n t o a standard f l a s k , the pressure measured and the exact mass c a l c u l a t e d . The gas was then t r a n s f e r r e d i n t o a f l a s k c o n t a i n i n g a s o l u t i o n of KOH i n D^ O of the appropriate st r e n g t h . The mixture was kept i n an i c e bath to minimise gas l o s s . Carbon d i s u l p h i d e was then added slo w l y . The s o l u t i o n was then reduced as before and the f i n a l product r e c r y s t a l l i s e d from Ti^O and checked by NMR f o r r e s i d u a l hydrogen. (None was detected) Y i e l d s were high i n a l l cases and the f i n a l products were stored i n the dark under n i t r o g e n at -15°C to minimise decomposition. The f o l l o w i n g potassium dithiocarbamates were s u c c e s s f u l l y made: L i s t i n g by parent amine/imine dimethyl-dg amine diethylamine di-n-propylamine di-n-butylamine d i-n-hexylamine di-n-octylamine p i p e r i d i n e (pentamethylenimine) hexamethyleneimine octamethyleneimine dodecamethyleneimine N-methyl-octadecylamine morpholine Sodium d i e t h y l d i t h i o c a r b a m a t e , sodium dimethyldithiocarbamate and ammonium p y r o l l i d i n e - N - c a r b o d i t h i o a t e are a v a i l a b l e from A l d r i c h as were most of the s t a r t i n g m a t e r i a l s . I t should be noted that the p y r o l l e d e r i v a t i v e cannot be made by the above methods. -39-4.2 Preparation of O JCu/"~>Cu Complexes 4.2.1 Copper(II)-63 Chloride 63 I s o t o p i c a l l y enriched (99.99% AERE Harwell) Cu metal was dissolved i n cone, n i t r i c acid and the s o l u t i o n evaporated. The residue was repeatedly c r y s t a l l i s e d from cone, hydrochloric acid to form copper(II)-63 ch l o r i d e . The chloride was used i n preference to the n i t r a t e as i t i s less deliquescent. It i s also thermally stable so excess acid can be r e a d i l y removed by heating. 4.2.2 Copper(II) Dithiocarbamate Complexes A s l i g h t excess of the appropriate potassium dithiocarbamate s a l t i n aqueous sol u t i o n was added to copper-63 chloride (ca. 20 mg, 0.15 mMol) i n s o l u t i o n . The complex was extracted with chloroform. In some cases i t was necessary to add ethanol to break up the water/ chloroform emulsions that form i n the presence of dithiocarbamates. After separation the chloroform s o l u t i o n was washed several times with water and then f i l t e r e d . The s o l u t i o n was allowed to evaporate and the residue dried at 80°C. The complex was r e c r y s t a l l i s e d by d i s s o l v i n g i n a minimum amount of b o i l i n g solvent and r a p i d l y cooled i n an i c e bath. (See table 4.1 for d e t a i l s ) . Mixed isotope s a l t s were also prepared for e s t a b l i s h i n g recry-s t a l l i s a t i o n procedures and for micro-analysis, (see table 4.2) -40-.3 P r e p a r a t i o n of Copper-free N i c k e l Complexes f o r M a t r i x Experiments TABLE 4.1 * Copper Dithiocarbamate R e c r y s t a l l i s a t i o n Solvent Me2NH Chloroform Et 2NH Chloroform n-Pr 2NH Pentane n-Bu2NH Ethanol n-Hx2NH Pentane/-78°C n-0c 2NH -N-Me-Octadecylamine Toluene P y r o l l i d i n e Chloroform P i p e r i d i n e Chloroform Hexamethyleneimine Chloroform Octamethyleneimine Ethanol/Chloroform 50:50 Dodecamethylerieimine Ethanol/Chloroform 50:50 Morpholine Chloroform TABLE 4.2 Copper Dithiocarbamate Analysis calculated (Observed) Carbon Hydrogen Nitrogen Me2NH 23.71(23.65) 3.98(4.00) 9 . 2 2 ( 9 . 0 1 ) Et 2NH 33.36(33.17) 5.60(5.60) 7.80(7.79) n-Pr 2NH 40.40(40.27) 6.78(6.89) 6.73(6.62) n-Bu2NH 45.77(45.73) 7.68(7.77) 5.93(5.90) n-Hx2NH 53.43(54;23) 8.97(8.60) 4.79(4.86) n-0c2NH - - -N-Me Octadecylamine 61.52(61.59) 10.32(11.05) 3.59(3.73) P y r o l l i d i n e 33.78(33.98) 8.97(8.60) 7.88(7.87) Hexamethyleneimine 40.80(40.78) 5.88(5.96) 6.80(6.79) Octamethyleneimine 46.18(46.12) 6.89(6.77) 5.98(5.88) Dodecamethyleneimine 53.80(53.65) 8.34(8.26) 4.83(4.77) L i s t i n g by parent amine/imine. -41-In matrix experiments the copper-63 complex i s doped into i t s n i c k e l analogue at a l e v e l of 0.1% w/w. Consequently the host n i c k e l complex must contain <0.005% (50ppm) of mixed isotope copper impurity. However, (for as yet unknown reasons) a l l the n i c k e l s a l t s t r i e d gave n i c k e l complexes containing ^0.1% copper. To circumvent t h i s problem the n i c k e l s a l t s had to be treated as follows: An excess of the aqueous n i c k e l s a l t was added to a s o l u t i o n of the appropriate potassium dithiocarbamate s a l t , the s o l u t i o n f i l t e r e d and the residue discarded. The f i l t r a t e i s now copper free (<0.001% by ESR). The pure n i c k e l complex was then prepared by adding aqueous potassium d i t h i o -carbamate to the f i l t r a t e , f i l t e r i n g and r e c r y s t a l l i s i n g the residue. (See section 4.4.2 for d e t a i l s ) 4.4.1 Choice of Solvent Toluene was chosen as the solvent because i t ; a) has a low d i e l e c t r i c constant - i t does not load the cavity excessively, b) forms good glasses for p o l y c r y s t a l l i n e studies, c) i s r e l a t i v e l y non-polar; i t shows no evidence of coordination with the complex, d) has a large l i q u i d range -95°C to +110°C, e) and because a l l of i t s phy s i c a l properties are well documented. (50-54). There are several tabulations for the v i s c o s i t y of toluene, (50-54) however i t was found more convenient to use an a n a l y t i c a l expression for the v i s c o s i t y . A set of data was f i t t e d to the sum of -42-two exponentials to within 0.5%. The equation i s given below ( v a l i d from -60°C to +100°C). Vis=0.2139284xl0 _ 1 exp(0.9256913xl0 3/T)+0.3760345xl0~ 4exp(0.2221321^10 4/T) where T i s the absolute temperature 4.4.2 Preparation of the ESR Samples Spectrograde toluene was dried over Type 4A molecular sieves and then further p u r i f i e d j u s t p r i o r to use by passage through an alumina -4 column. A 10 ml s o l u t i o n of ca. 10 M, of the complex was prepared, ca. 0.5 ml of the s o l u t i o n transferred to a 'flamed out' ESR tube; degassed by several freeze-pump-thaw cycles to a pressure of <10 Torr on a greaseless vacuum l i n e and then sealed o f f . A l l s o l u t i o n manipulations were done i n a glove bag under an atmos-phere of dry nitrogen. Samples were stored i n the dark at -15°C to prevent decomposition (55,56) 4.4.3 Sample Tubes The sample tubes were constructed from s p e c i a l t h i n walled pyrex tubing (5 mm OD, 4 mm ID) to maximise the sample volume. Each tube was f i l l e d to a depth of 3-4 cm (t h i s minimises temperature gradients due to convection: see error discussion) and sealed o f f at 4-5 cm (t h i s prevents the solvent from d i s t i l l i n g out of the cavity area). Care was taken to leave a gap of ca. 1 cm between the top of the solvent and the seal to allow for l i q u i d expansion. 4.5 Recording Spectra A l l spectra were recorded on a ESR spectrometer which consisted of; -43-a Varian 12" magnet with Mk II f i e l d - d i a l c o n t r o l , an HP k l y s t r o n power supply and sweep uni t , a home-built AFC and 100 kHz modulation unit and an Ithaco 391A phase-lock ampl i f i e r . The microwave bridge was r e f l e c t i v e homodyne using a TE 102 cavity, three port c i r c u l a t o r , Schottky detector diode and a microwave bucking arm. The ca v i t y was f i t t e d with a dewar system and the con t r o l l e d by a Varian temperature control u n i t . The magnetic f i e l d was c a l i b r a t e d with a home-built proton magnetometer. The spectrometer was interfaced to a 16 b i t microprocesser-controlled d i g i t a l a c q u i s i t i o n system (26). (See f i g s . 4.1 & 4.2) The spectra, along with c a l i b r a t i o n data, were either recorded on a f l a t bed HP X-Y recorder or on paper tape v i a the microprocessor. The same recording conditions were maintained f o r a l l samples. These are summarised i n table 4.3. TABLE 4.3 Control Setting Microwave power 10-15 mW Modulation 0.8G Scan time & sweep width 5 min. for 25G, 50G or 100G for i n d i v i d u a l l i n e s depending on l i n e width or 25 min. for 250G for the 3 low f i e l d l i n e s and 25G for 5 min. for the high f i e l d l i n e . (The l a t t e r method settings were used for the ' l i n e heights method') Time constant Chosen according to the c r i t e r i a outlined i n section 3.1.1, usually 400 mS or 1.25 s depending on the noise l e v e l . Temperature N 2 flow rate ca. 15SCFH (see error discussion) also see section 5.2 Sampling rate Hex 20 skips, ca. 4 pts./sec. 819 or 882 pts. per scan. ESR SPECTROMETER FIGURE 4.1 (Fleldlal) Control from logic control + Timing unit 1> High speed paper tape punch DIGITAL ACQUISITION SYSTEM Multl plexor I Buffer | + Driver MICROCOMPUTER (F-8) 3 CPU 3850 PORT<£ PORT 1 MEMORY 3853 I M 8 RAM Clock Logic control + Tlmlng unit PSU 3851 (FAIR-BUG) Interrupt PORT Ext. Interrupt I ASR 331 TTY Start /Stop switch .Push button for calibration CD a JO m -£> -46-Th e whole spectrometer was p e r i o d i c a l l y tested with standard samples to check for s e n s i t i v i t y , amplifier phase, magnet s t a b i l i t y , etc. One complete spectrum was always recorded at room temperature to check for paramagnetic impurities. 4.6 Temperature Measurement The temperature was measured by wrapping a copper/constantan thermocouple around the sample, taking care not to decouple the ca v i t y . The EMF was measured with a high impedance 4% d i g i t microvoltmeter giving a r e s o l u t i o n of ca. 0.01°C. 4.7 F i e l d C a l i b r a t i o n Each spectrum was c a l i b r a t e d absolutely using a manually tracked proton magnetometer. The c a l i b r a t i o n procedure was as follows. The magnetometer f r e -quency was continually changed manually so that the g l y c e r o l protons i n the probe were maintained at resonance (observed on an oscilloscope) as the main magnetic f i e l d was scanned. At su i t a b l e i n t e r v a l s of time the c a l i b r a t i o n button was pressed, the spectrum was ticked and the corresponding proton frequency noted, or the proton frequency and the corresponding f i e l d d i a l voltage were stored i n the microprocesser's memory for punching out on paper tape as a c a l i b r a t i o n table. P l o t t i n g proton resonance frequency against the c a l i b r a t i o n marks (for chart recordings) or f i e l d - d i a l voltage (for d i g i t i s e d spectra) allows the l i n e positions to be read o f f and converted to absolute f i e l d values. This was done automatically by computer i n the l a t t e r case (see -47-section 4.9.3). A t y p i c a l spectrum complete with a l l the c a l i b r a t i o n data i s shown i n f i g . 4.3. 4.8 Analysis of Spectra The spectra can be analysed i n two ways: by measuring r e l a t i v e l i n e heights and the width of one l i n e , thus allowing a l l the l i n e widths to be calculated, a l t e r n a t i v e l y the l i n e widths can be measured d i r e c t l y . The former method i s a good manual method, but e f f i c i e n t computer programs based on t h i s method have proved d i f f i c u l t to imple-ment. The converse i s true of the l a t t e r method. 4.8.1 Kivelson's Method - Line Heights Analysis (18) The heights (h) of each l i n e were measured r e l a t i v e to the high f i e l d (sharpest) l i n e , i t s width ( L W q ) being determined by a separate 25G scan. The l i n e widths (LW) for each i n d i v i d u a l l i n e are calculated from the formula LW = LW Ihk o This method has a number of disadvantages (see section 5.6.1) and was used only for the p i l o t studies. 4.8.2 Direct Measurement of Line Widths Each l i n e of the spectrum was recorded and c a l i b r a t e d i n d i v i d u a l l y ( i n 5 min 25,50 or 100G scans). The gain, scan width and time constant were adjusted to s u i t each peak using the c r i t e r i a given i n section 3.1.1. The l i n e widths and positions were then simply read off the c a l i b r a t e d spectrum. "•258.06 GHU5S-30U0G 259 76 261 46 263.16 264.86 266.56 26B.26 269.96 271.65 _ 1 _ 1 I 1 1 1 — ' 1 273.35 _1 275.05 _ J 276.75 _ l 278.45 _ l 280.15 281.85 283.J55 _ l 1 SPECTRUM NO. 1 CUUD-63 IBU2DT02 IN TOLUENE HIGH FIELD LINE SWEEP WIDTH 25.4G x CALIBRATION PTS. FIGURE 4.3 305.63 0.0 T T 21.645 43.691 —1 1 65.536 87.381 -1 109.23 -1 131.07 1 1 1 152.92 174.76 196.61 FIELD (X10= ) —1 1 218.45 240.3 I 1— 262.14 283. 327.66 -48-4.8.3 Computer Assisted Analysis of Spectra Each l i n e with i t s associated c a l i b r a t i o n table, (see section 4.8) spectrum number and date were recorded on paper tape as 16 b i t X-Y data point p a i r s . The paper tape was read at the computing centre and transferred to a l i n e f i l e . The program SCAN was then invoked to read the f i l e . This program reads i n one spectral l i n e at a time and f i r s t checks for a c q u i s i t i o n errors; the data i s recorded as blocks of 63 X - Y points (one record) i f the microprocesser i s working c o r r e c t l y . Incorrect records ( i d e n t i f i e d by an incorrect length) are flagged for l a t e r c o r r e c t i o n and then skipped. The X-Y data point pairs are then converted to integers and stored i n an array. The c a l i b r a t i o n table i s also checked (the microprocesser can under some circumstances misread the frequency counter) and unreasonable points rejected. The 'edited' c a l i b r a t i o n data i s f i t t e d to a st r a i g h t l i n e and any further erroneous or poor points rejected. The f i t provides the parameters for converting f i e l d - d i a l voltages to absolute f i e l d values. The approximate baseline i s located and the data scanned i n in c r e -ments of l/20th of the scan width (eg, every 40th data point) u n t i l the peaks are located (as a turning point i n the data) and the approximate crossover found (where the l i n e i n t e r s e c t s the baseline). The extrema of the peak are f i t t e d to cubics and the crossover region i s f i t t e d to a str a i g h t l i n e . The exact peak positions are found a n a l y t i c a l l y by d i f f e r e n t i a t i n g the cubic and solving f o r the maximum or minimum, taking care to ensure there i s no ambiguity as to which s o l u t i o n i s the correct one. The baseline i s now taken to be at the average of the heights of -49-the extrema. The crossover i s taken to be h a l f way between the peak extrema and may be checked against the value obtained from the i n t e r s e c t i o n of the s t r a i g h t l i n e f i t with the baseline. This provides a test for peak asymmetry. The l i s t i n g for SCAN and an abridged flow chart are given i n Appendix I. 4.10 Analysis of Data The l i n e width data were smoothed (as a function of temperature) with a sextic polynomial and written into a computer f i l e along with the temperatures and corresponding l i n e positions and microwave frequencies. These data and the appropriate A and g tensor anisotropics are used by the program KIVEL (see below) to c a l c u l a t e the r o t a t i o n a l c o r r e l a t i o n times, spin r o t a t i o n a l terms, v i s c o s i t y data (see section 4.4.1), l e a s t squares f i t s for l n ^ . v s . 1/T, x 2 • v s . n/T, a".vs. T/n and various other pertinent data. The f i n a l r e s u l t s were tabulated and/or plotted by the computer. Relevant d e t a i l s of the equations used by KIVEL can be found i n Section 2.3.2. The o r i g i n a l version of KIVEL i s due to F. G. Herring and J . M. Park and b r i e f l y , works as follows (57): F i r s t l y the four l i n e positions and the corresponding microwave frequencies are read i n . Then by using a l e a s t squares procedure and making second order corrections to the l i n e positions the i s o t r o p i c g and A values are determined. The four line-widths, the AA and Ag values are read i n and sub-s t i t u t e d i n eqns 4.8 along with t h e i r corresponding m^'s and guesses for x 9 and a", (one could solve d i r e c t l y for a, B, y and 6, extract the -50-value f o r T c from these and then b a c k - c a l c u l a t e to get a". However t h i s can produce misleading r e s u l t s (14)). 6 = LW (obs) - LW ( c a l c ) 4.8.1 mI m-j. m-j, A standard l e a s t squares procedure i s followed (58) to minimise (4.8.1) •, 2 i e . a " and x„ are adjusted u n t i l E <5 i s a minimum. 2 J m m-. I T 2 o LW(calc) = a* + a" + 3mj + ymT + S™{, 4.8.2 However one m o d i f i c a t i o n i s r e q u i r e d , the W T ^ terms i n the c o e f f i c i e n t s of (4.8.2)(see s e c t i o n 2.3.2) are assumed to be slo w l y v a r y i n g w.r.t. and thus can be considered constant when d i f f e r e n t i a t i n g w.r.t. to x . This i s equivalent to saying that a, 3 e t c . are l i n e a r i n x ^ , thus much s i m p l i f y i n g the d i f f e r e n t i a l s f o r the l e a s t squares a n a l y s i s . However one e x t r a c y c l e of the l e a s t squares procedure i s re q u i r e d as no c o r r e c t i o n i s made to the x^ i n the ust^ terms and thus i s 'one ap p r o x i -mation poorer' than x 2 i n the l i n e a r terms. Only two l i n e s are i n f a c t needed to get a" and v , however a l l four l i n e s were used f o r t h i s work as t h i s improves our p r e v i s i o n . 'Mean f i e l d ' values of a, 3, y> 5 are a l s o c a l c u l a t e d i r e c t l y from the l i n e -widths using K i v e l s o n ' s method (18), as w e l l as being b a c k - c a l c u l a t e d from the x 2 and a" found by the l e a s t squares procedure. These were p r i n t e d out f o r d i a g n o s t i c purposes. 4.11 P o l y c r y s t a l l i n e Spectra The p o l y c r y s t a l l i n e spectra of the copper complexes were recorded i n -51-a toluene glass. The Ag and AA parameters were obtained by simulation procedures due to Tait (59). A few powder spectra were obtained in a matrix of the nickel analogue. However the g and A values obtained this way did not agree well with the toluene glass or isotropic (solution) parameters. This was attributed in part to the fact that the nickel salts are not generally isomorphous with the copper salts and also specific inter-molecular interactions may exist (60). A typical polycrystalline spectrum is shown in fig. 4.4. -52-CHAPTER 5  ERROR DISCUSSION 5.1 Overall Experimental Errors A number of the error sources discussed here are not normally apparent. D i g i t i s a t i o n of the spectra r e s u l t s i n at least a f i v e -f o l d improvement i n p r e c i s i o n and removes the damping e f f e c t of the X-Y recorder. The r e s u l t i s that many noise sources, notably the high frequency ones, can cause s i g n i f i c a n t errors. Errors due to the time constant and d i g i t i s a t i o n procedures are discussed i n chapter 3. Errors from modulation, saturation and dipolar broadening are discussed elsewhere (42). The estimated errors f o r various experimental variables are given i n table 5.1. As a general rule-of-thumb errors were taken as 0.1G i n a l l l i n e p ositions and the l i n e widths of the two lo w - f i e l d l i n e s , and as 50 mG i n the l i n e widths of the two high f i e l d l i n e s . This allows error estimates i n other para-meters to e a s i l y made. 5.2 Temperature errors The temperature of the gas flow was monitered i n the immediate v i c i n i t y of the sample and was always stable to ±0.01°C over the temperature range used (-60°C to 100°C). However as t h i s might not r e f l e c t the temperature of the sample i t s e l f a s p e c i a l dual thermocouple was constructed and inserted into a dummy sample (ca 0.5 ml of toluene sealed i n 5 cm of pyrex tubing). The r e s u l t s are given i n table 5.2. -53-TABLE 5.1 Control E f f e c t Magnitude Modulation Broadening, e f f e c t s narrowest l i n e s most <1% on sharpest l i n e MW Power Saturation broadening N e g l i g i b l e ; onset of saturation 200 mW at RT Time Constant Displacement of l i n e and broadening <5 mG D i g i t i s a t i o n Various see chapter 3 Ne g l i g i b l e Program See section 5.4 <1% of l i n e width Magnet i n s t a b i l i t y Random s h i f t i n g of l i n e p ositions and l i n e widths > <5 mG Temperature Systematic s h i f t s i n l i n e widths and positions see section 5.2 Neg l i g i b l e e f f e c t <0.6°C MW frequency Baseline d r i f t and s h i f t i n l i n e positions N e g l i g i b l e <2 mG over 5 min. scan Magnetometer Absolute p r e c i s i o n Absolute accuracy 1:107 <0.1G Personal error Modulation error (see section 5.3.1) <0.1% N e g l i g i b l e TABLE 5.2 N 2 Flow Rate Temperature Time to F i n a l Temp • Temp Gradient j ump ° C equilibrium 10SCFH -25 - -35 -35 +0.6 -35 - -25 -25 +0.6 +30 - +20 +20 +0.2 +20 - +30 +30 0.0 +60 - +50 +50 -0.6 +50 - +60 <3 min i n a l l +60 -0.9 15SFCH -25 - -35 cases -35 +0.2 -35 - -25 -25 +0.2 +30 - +20 +20 0.0 +20 - +30 +30 0.0 +60 - +50 +50 0.0 +50 - +60 +60 -0.2 -54-The presence of the thermocouple lowers the Q of the c a v i t y (and hence increases the noise l e v e l s ) . I t a l s o perturbs the magnetic and e l e c t r i c v e c t o r s of the microwaves. The consequences of t h i s are unknown. To t e s t f o r any e f f e c t s the l i n e width and p o s i t i o n of the 63 high f i e l d l i n e of Cu(Et2DTC) i n toluene were repeatedly measured w i t h and without the thermocouple present. Apart from a s h i f t i n l i n e p o s i t i o n , e x a c t l y accounted f o r by a corresponding s h i f t i n microwave frequency (caused by the presence of the thermocouple), the two sets of r e s u l t s were i d e n t i c a l . The magnetometer probe i s l o c a t e d on the outside of the c a v i t y thus the f i e l d at the sample may w e l l d i f f e r from the measured f i e l d . This was checked by c a l i b r a t i n g the against Fremy's s a l t , whose absolute g-value i s known (15). The e r r o r was n e g l i g i b l e <0.1 g (0.003% of the l i n e p o s i t i o n ) . 5.3 C a l i b r a t i o n (See s e c t i o n 4.8) For a f i x e d scan time the time to sweep through the proton's resonance i s i n v e r s e l y p r o p o r t i o n a l to the sweep width. Hence the l a r g e r the sweep width the greater the personal e r r o r (the % e r r o r of the scan width remains constant however.) The personal e r r o r can be estimated from the o s c i l l o s c o p e or from the l e a s t squares f i t of the c a l i b r a t i o n data. C a l i b r a t i o n e r r o r s were test e d by doing s e v e r a l c a l i b r a t e d scans under the same c o n d i t i o n s (25G scan i n 5 mins). The r e s u l t s are given below. -55-Slope 7.803 ±0.021 u n i t s = <0.3% of l i n e w i d t h Intercept 3259.630' ± 0.052G = <0.002% of l i n e p o s i t i o n The e r r o r i n the i n t e r c e p t a r i s e s mainly from the e x t r a p o l a t i o n performed to get t h i s v a l u e , i n p r a c t i c e the e r r o r i n the l i n e p o s i t i o n i s i n f a c t smaller. See s e c t i o n 5.4.1 f o r f u r t h e r d i s c u s s i o n of these e r r o r s . 5.3.1 Modulation F i e l d from the Magnetometer Probe The probe generates an a l t e r n a t i n g magnetic f i e l d of ca. 200 mG. This has a n e g l i g i b l e e f f e c t on the l i n e width of copper dithiocarbamate complexes, but can severely d i s t o r t (by over modulation) the l i n e s of Fremy's s a l t . I t i s a l s o necessary to keep the a m p l i f i e r ' s time constant > 40 mS or t h i s modulation shows as a high frequency o s c i l l a t i o n on top of the peaks. This problem can be overcome by mounting the probe j u s t above the c a v i t y . 5.4 E r r o r s i n A n a l y s i s SCAN - Program E r r o r s Given that the b a s i c a l g o r i t h m of SCAN was c o r r e c t the program was tested by using i t to f i n d the l i n e widths and p o s i t i o n s of s e v e r a l of the same s p e c t r a l l i n e obtained under i d e n t i c a l c o n d i t i o n s . The r e s u l t s are given below. Line widths 3.422 ± 0.013G Line p o s i t i o n s 3271.715 ± 0.021G -56-Note that the line position error is smaller than the cali-bration error as the crossover lies within the calibration points, not at an extrapolated point. Half the error in the line width is attributable to the calibration ( ± 0.009G in this case). The remainder probably arises from spectrum noise or the algorithm. The algorithm was tested in a number of ways. Approximation of the top of the peak to a cubic is good even for the narrowest peaks and the fitting procedure is very tolerant of noise. (see figs. 5.1, 5.2 and 5.3; fits and points as selected by the program). Spectral parameters such as starting point and data point density have a minor effect on the cubic f i t and this is reflected in part by the 0.009G error mentioned above. It is possible to extract the error in the line width from the fitting subroutine, however this has not been done yet. Further checks of the algorithm were made by plotting some of the spectra, measuring them by hand and comparing the results with those obtained by SCAN. In a l l cases the results of SCAN were within the experimental error of the manual measurements. Line width vs. temperature plots are normally smooth, any point that deviates considerably from a sextic polynomial f i t can be con-sidered erroneous. Lines giving such points are plotted out, measured by hand and checked against the result from SCAN. Of 1000 lines run to date <2% f a l l into this category and in no case was the error attri-butable to SCAN or the calibration. -57-5.5 Noise, Magnet I n s t a b i l i t y e t c. 5.5.1 S p e c t r a l noise Noise on the spectrum l i m i t s the p r e c i s i o n obtainable (see s e c t i o n 3.1.3) I f the noise l e v e l i s high (SNR<15:1) the program becomes u n r e l i a b l e and cannot be used as i t stands. At lower noise l e v e l s the p r e c i s i o n i s a f u n c t i o n of the noise. An SNR of 50:1 -4 was t y p i c a l f o r a 10 M sample. This l i m i t s the p r e c i s i o n to ca. 1% of the l i n e width. 5.5.2 Magnet I n s t a b i l i t y Short term magnet i n s t a b i l i t y ( i e . magnet power supply noise) manifests i t s e l f as o s c i l l a t i o n s on the peak extrema and can be severe at low time constants. Regular maintenance of the magnet power supply minimises t h i s problem. Spectra were checked f o r day-to-day consistency by running standard samples. Ne i t h e r the mean nor the standard d e v i a t i o n of the l i n e width (LW) and p o s i t i o n (LP) v a r i e d s i g n i f i c a n t l y . Day 1,20 spectra LW 3.421 ± 0.015 LP 3271.715 ± 0.021 Day 2,20 spectra LW 3.422 ± 0.013 LP 3271.910 ± 0.027* * correct e d f o r a change i n microwave frequency 5.6 E r r o r s i n A n a l y s i s 5.6.1 K i v e l s o n ' s Method This i s described i n chapter 4. The l i n e width (LW) of a given -60-l i n e i s LW = LW / h 2 o where h i s the l i n e height r e l a t i v e to the reference peak, l i n e width L W q . This formula presupposes that the l i n e s have the same shape. The broader l i n e s are e s s e n t i a l l y E o r e n t z i a n but the high f i e l d l i n e i s only 75% so. This accounts f o r the d i f f e r e n c e s between d i r e c t l i n e width a n a l y s i s and l i n e height a n a l y s i s of the same run. (see f i g 5.4). Other disadvantages are that the e r r o r i n the sharpest l i n e i s propagated to a l l the l i n e s , a l s o the l i n e p o s i t i o n s cannot be determined very a c c u r a t e l y from a s i n g l e 250 G scan (>0.5 G e r r o r ) . However one advantage i s that the square root term reduces e r r o r s from the height measurements, r e s u l t i n g i n a f a i r l y smooth set of data. 5.6.2 Line Width A n a l y s i s by Hand Some runs were analysed by measuring l i n e widths manually. Line p o s i t i o n s could be obtained f a i r l y a c c u r a t e l y by t h i s method (y ± 0.1G), but l i n e width measurements showed a l o t of s c a t t e r owing to the d i f f i -c u l t i e s of judging where the extrema were; e s p e c i a l l y i n the presence of noise. E r r o r s were t y p i c a l l y 5% of the l i n e width. The i n d i v i d u a l l i n e width e r r o r s tended to 'average out', but the f i n a l r e s u l t s were much more s c a t t e r e d than those obtained by K i v e l s o n ' s method. The poor q u a l i t y of the r e s u l t s could be p a r t i a l l y compensated f o r by t a k i n g measurements at a l a r g e r number of temperatures, but t h i s was not always p r a c t i c a l . BIS-CUUI)-63 DIETHYL DITHIOCflRBRMRTE IN CHLOROFORM -62-The program SCAN improves p r e c i s i o n by an order of magnitude making d i r e c t l i n e width a n a l y s i s more p r a c t i c a l and i n f a c t s u p e r i o r to K i v e l s o n ' s method. 5.7 Breakdown of the T h e o r e t i c a l Framework 5.7.1 The Slow M o t i o n a l Regime Ki v e l s o n ' s theory i s a f i r s t order theory (9) and r e q u i r e s that d i o . T « l where x i s the c o r r e l a t i o n time and dco i s the anisotropy of the g and A tensors. (see S e c t i o n 2.3.1). For copper dithiocarbamates the theory breaks down when the r o t a t i o n a l c o r r e l a t i o n time (x) i s ca. 150 pS (see s e c t i o n 3.2). When x i s greater than t h i s the complex i s s a i d to have entered the slow-motional regime; as x gets longer the spectrum undergoes a number of changes, these are i l l u s t r a t e d i n f i g . 5.6. A u s e f u l r u l e of thumb i s that the slow motional regime s t a r t s when the low f i e l d l i n e becomes broader than 30 G. A negative s p i n r o t a t i o n a l term i s a l s o taken as evidence of the breakdown of the theory. A l l data p o i n t s g i v i n g such r e s u l t s were r e j e c t e d . 5.7.2 Other C o n t r i b u t i o n s to Line Width The v a r i a t i o n of l i n e width w i t h temperature was assumed to be e n t i r e l y due to r e l a x a t i o n . Other p o s s i b i l i t i e s are changes i n unresolved hyperfine and peak overlap. -63-5.7.2.1 Unresolved Hyperfine 33 The S s a t e l l i t e s l i e i n the wings of the l i n e s and make no c o n t r i b u t i o n s to the l i n e width. Proton hyp e r f i n e was checked by comparing Cu(MeDTC) w i t h Cu(CT^DTOjfig. 5.7. These d i f f e r e n c e s c o n t r i b u t e s 10% to the f i n a l x , but they were temperature independent and were considered not to e f f e c t the f i n a l c onclusions. The e f f e c t 14 of the N hyper f i n e i s s t i l l u n c e r t a i n . There i s some evidence that i t c o n t r i b u t e s to the l i n e width; the hydrogen hype r f i n e only accounts f o r 10% of the 25% d e v i a t i o n from the L o r e n t z i a n shape found i n the high f i e l d l i n e (61). Experiments are being devised to i n v e s t i g a t e i t s e f f e c t s . 5.7.2.2. Overlap The e f f e c t of overlap was i n v e s t i g a t e d by s y n t h e s i s i n g s p e c t r a and comparing the width of the generated l i n e s w i t h the l i n e widths supplied to the program. The low f i e l d l i n e was the main source of e r r o r . I f the low f i e l d l i n e was <20G wide the c o r r e c t i o n s were i n s i g n i f i c a n t . For a low f i e l d l i n e 20-30G wide the c o r r e c t i o n s were always l e s s than the estimated e r r o r s f o r the i n d i v i d u a l l i n e s . Above 30G the c o r r e c t i o n s were s m a l l , but s i g n i f i c a n t ; however these sp e c t r a were r e j e c t e d f o r the reasons o u t l i n e d i n s e c t i o n 5.7.1. A numerical c o r r e c t i o n program was w r i t t e n to t e s t the e f f e c t s of overlap on x and a". The changes were small and the program was pro-h i b i t i v e l y expensive to run on a r o u t i n e b a s i s ; consequently no c o r r e c t i o n s f o r overlap were made. -66-5.7.3 V a r i a t i o n s i n the g & A tensors a n i s o t r o p i c s K i v e l s o n ' s theory a l s o assumes that the g & A tensor a n i s o t r o p i e s (Ag and AA r e s p e c t i v e l y ) are independent of temperature. Such changes are u s u a l l y assumed s m a l l , however i n our case there i s a p o s s i b i l i t y of r o t a t i o n about the carbon-nitrogen bond. Al s o the longer chains may f o l d over and 'touch' the copper atom. Both of these phenomena could i n v a l i d a t e the assumption. IR s t u d i e s of dithiocarbamates show there i s extensive double bonding character to the carbon-nitrogen bond (62). Proton NMR stud i e s i n d i c a t e that the b a r r i e r to r o t a t i o n i s very h i g h as no r o t a t i o n was observable (62-64). Thus bond r o t a t i o n was not considered to be a source of e r r o r . Chain f o l d i n g probably occurs to some extent, but space f i l l i n g models i n d i c a t e that f o l d i n g over and 'touching' of the copper i s impossible f o r p e n t y l and smaller chains (they are too short) and s t e r i c a l l y very unfavourable f o r the longer chains (or l a r g e r r i n g s ) . Any e f f e c t s on AA and Ag are probably very s m a l l . The g and A tensors were assumed a x i a l l y symmetric. A n a l y s i s of the p o l y c r y s t a l l i n e s p e c t r a showed t h i s to be a good approximation f o r a l l the copper complexes used. 5.7.4 Testing K i v e l s o n ' s Theory A u s e f u l t e s t f o r the v a l i d i t y of the theory i s to p l o t gamma vs. beta (15), c a l c u l a t e d from the l i n e widths (see s e c t i o n 4.10) and to compare t h i s w i t h a p l o t of gamma vs. beta c a l c u l a t e d from the c o r r e l a t i o n times and Ag and AA. I f AA and Ag are c o r r e c t and temperature independent -67-th en the two pl o t s w i l l be superimposable. Any deviation indicates changes i n AA and/or Ag or a complete breakdown of the theory. The theory has been thoroughly tested by Herring and Park (14) for 63 Cu(MeDTC) and i t only breaks down i n the slow motional regime as expected. -68-CHAPTER 6  RESULTS For convenience, i n t h i s chapter and chapter 7 the copper d i t h i o - -carbamates are r e f e r r e d to by t h e i r parent amine. The (four) l i n e widths of eleven compounds at ( t y p i c a l l y ) 15 temperatures were measured. This represents a l a r g e body of data and have been tabulated by computer and c o l l e c t e d i n Appendix I I along w i t h graphs of the e d i t e d and smoothed data (a t y p i c a l example i s shown i n f i g u r e 6.1). A l l data was processed using the computer program KIVEL, which uses the four l i n e - w i d t h s at a given temperature and s o l v e s f o r a" and x^ using a l e a s t squares approach (see s e c t i o n 4.10). I t a l s o provides t a b l e s of a l l the p e r t i n e n t data (see t a b l e s 6.1,a,r-,d f o r example and a l s o t a b l e s 6.2-6*13 i n appendix I I ) , l e a s t squares f i t s and p l o t s of l n ( x 2 ) . v s . 1/T, T2.vs.n/T and ct".vs.T/n. I t a l s o p l o t s a, 8, y, vs. n/T and y.vs.B (see f i g 6.2,a-g f o r example). The l a t t e r p l o t s are u s e f u l d i a g n o s t i c t e s t s . Freed (15) showed that the y-vs.B p l o t s are e s p e c i a l l y u s e f u l . These'plots w i l l be l i n e a r i f the theory i s v a l i d and b a c k - c a l c u l a t e d values of y and 8 (see s e c t i o n 4.10) w i l l l i e on the same l i n e as the experimental values i f the AA and Ag values are c o r r e c t (see below). No s i g n i f i c a n t d e v i a t i o n s from l i n e a r i t y were observed f o r any of the probes. As expected from theory (eqn. 2.32.1) the l i n e width i s greatest f o r the low f i e l d l i n e s (see f i g 6.1). Eqn. 2.32.1 a l s o shows that the l i n e width decreases w i t h x^, however the a" c o n t r i b u t i o n to the l i n e - w i d t h (eqn. 2.60) increases w i t h x 0 . As x 9ccl/T (eqn. 1.9) one then expects * * * * * C U I l I t - 6 3 BIS DIETHYL DITHIOCARBAMATE IN TOLUENE TEMP - 44 . 9 -40 .9 - 36 .4 -31 .2 -20 .2 - 10 .4 - 0 . 2 9 .0 19.7 29.1 39.2 49.2 57 .6 MW-FREO: 9.03761 9.03772 9.03755 9.03752 9.03741 9. 03733 9.03727 9.03715 9.03704 9.03710 9.03678 9.03666 9.03655 GHZ/RAOS 56.7850 56.7856 56.7846 56.7844 56.783 7 56.7832 56.7828 56.7821 56.7814 56.7818 56.7798 56.7790 56.7783 23.70 3036.32 20. 88 3036.25 18.39 3036.74 16.16 3036.32 12.91 3036.63 10.94 3036.93 9.45 3037.75 8. 52 3038.26 7.84 3038.85 7.50 3039.24 7.25 3039.93 7.07 3040.44 7.12 3040.90 LINE WIDTHS 13.00 31 13. 11.51 3113. 10.16 3113. 8.96 31 13. 7.30 31 13. 6.48 31 13. 5.97 3114. 5.69 3114. 5.49 3114. 5.42 31 14. 5.46 3114. 5.57 3114. 5.65 31 15. K POSITIONS (GAUSS) 86 6.3.1 3193. RI 47 77 63 78 92 05 27 46 60 85 99 05 5.79 3193. 77 5.32 3193.50 4 .89 3.193. 33 4.33 3193.05 4 .10 3192.78 4.01 3192.57 4 .00 3192.32 4.06 3192. 08 4 .17 3191.83 4.36 3191.68 4.61 3191.38 4.81 3191.17 -*«****«.**»** 4.45 3276. 27 4.14 3276.00 3.87 3275. 53 3.65 3275. 24 3.36 3274.42 3.26 3273.77 3.26 3273.07 3.32 3272.45 3.44 3271. 72 3.58 3271.11 3.76 3270.39 3.97 3269.75 4.23 3269.12 A-VALUE1MHZ) -228.84 W - 0 . 6 3 1 -228.789+/-0.777 -227.863+/-0.745 -227.946+/-0.420 -226.854+/-0.171 -225 . 916*/-O. 096 -224.537+/-0.226 -223 .44347 -0 . 194 -2Z2.178*/ -0 .152 - 220 .990 */ -0.096 -219 . 8 89*/-O. 03 7 -218.774+/-0.093 -217.769+/-0. 102 A(GAUSS I -79.954 - 79 .929 -79.608 - 79 . 630 - 79 .245 -78 .914 - 78 .433 - 7 8 . 050 - 77 .607 - 7 7 . 189 -76 .807 -76 .416 - 7 6 . 064 G-VALUE 2 .04498 2.04513 2.04509 2.04526 2.04535 2.04542 2.04542 2.04543 2 .04546 2.04554 2.04547 2.04551 2.04555 1 o TABLE.. 6.1 a * * * * * C U t11) -63 BIS DIETHYL DITHIOCARBAMATE IN TOLUENE ***** TEMP ALPHA ALPHAC ALPHA" BETA EETAC GAMMA GAMMAC DELTA DELTAC LW ERR - 4 4 . 9 0.9102E+01 0.8315E+01 0 .4437E*00 -0.6724E+01 - 0 . 6251E * 0 1 0.2210E + 01 0 .2440E-01 0 .1367F* •00 - 0 . 5 9 6 0 E - 01 0.2769 - 4 0 . 9 0.8167E+01 0.7948E+01 0. 6 127E+00 -0.5737E+01 -0.5435E+01 0.1930E+01 0.2106E+01 0 . 7000E -•01 - 0 . 5 1 5 0 E - 01 0.2028 - 3 6.4 0.7316E-01 0 .7173E*01 0. 7491E+00 -0.4840E->-Ol - 0 . 4 7 1 6 E * 0 l 0 .1695E*01 0. 1809E*01 - 0 . I U 0 F - 15 - 0 . 4 3 79E - 01 0.1241 - 3 1 . 2 0.6552E+01 0.6488E+01 0 .8761E*00 -0.4057E+01 -0.4069E+01 0 . U 9 0 E + 0 1 0. 1541E + 01 - 0 . 5 0 0 0 E - 01 - 0 . 3 6 88E- 01 0.0545 - 2 0 . 2 0.5525E+01 0.5551E*01 0. 1120E4-01 -0 .2943E*01 -0.3114E+01 0 . 1 1 6 0 E » 0 1 0.1139E+01 - 0 . 1067E+00 - 0 . 2 6 6 3 E - 01 0.05 81 - 1 0 . 4 0. 5064E+01 0.5 0 9 8 E » 0 1 0.1391E+01 -0.2357E+01 -0 .2509E*01 0.9050E+00 0.8778E<-00 - 0 . 9 0 0 0 E - 01 - 0 . 1 9 9 6 E - 01 0.0550 -0.2 0.4819E+01 0.4843E+01 0.1690E+01 - 0 . 1 9 47E + 01 - 0 . 2028E+01 0 .6825E*00 0.6642E+00 - 0 . 5 1 6 7 E -•01 - 0 . 1 4 6 1 E - 01 0.0313 9 . 0 0 . 4 7 U E + 01 0 .473IE + 01 0. 1 9 3 5 E * 0 l - 0 . 1 6 8 5 E * 0 1 -0.1711E+01 0.5375E+00 0 .5204E*00 - 0 . 2 I 6 7 E - 01 - 0 . 1090E- 01 0.0191 1 9 . 7 0.4667E+01 0.4701E+0I 0.2198E+01 -0.1425E+01 -0.1453E+01 0 .4325E*00 0. 4044E+00 - 0 . 1 8 3 3 E -•01 - 0 . 7 9 4 0 E - 02 0 .0297 29 .1 0.4702E+01 0.4745E + 01 0 .2430E*01 - 0 . 1 2 4 3 E * 0 1 -0.1295E+01 0.3725E+00 0 .3362E*00 - 0 . 2 8 3 3 E - 01 - 0 . 6 2 0 6 E - 02 0. 0400 39.2 0. 4836E*0 l 0.4860E+01 0.2730E+01 -0.1092E+01 - 0 . 1 1 5 1 E - 0 1 0 .2975E*00 0. 2778E + 00 -0 .3167E -01 - 0 . 4 7 8 8 E - 02 0.0270 4 9 . 2 0 .5036E*01 0.5019E+01 0.3080E + 01 - 0 . 9 5 C 8 E + 00 - 0 . 1016E+01 0 .2150E*00 0.2279E+00 - 0 . 3 6 6 7 E - 01 - 0 . 3 6 2 3 E - 02 0.0259 57 .6 0.5174E + 01 0 .5196E*01 0.3361E + 01 -0 .8246E+00 -0.9476E+00 0.2225E+00 0.2049E«-00 - 0 . 6 1 6 7 E - 01 - 0 . 3 1 1 7 E - 02 0. 0432 " C " DENOTES SACK-CALCULATED VALUE, SEE SECTION 4 . 9 . 3 "LW ERR" IS RMS ERROR OF BACK-CALCULATED LINE-WIOTHS W.R.T. OBSERVED VALUES TARLE 6.1b * * * « « CLMI I I -63 BIS DIETHYL DITHIOCARBAMATE IN TOLUENE ***** T(C I TIKI V I S /T v i s e ALPHA" TAUCtPSI AV0L*10**22 - 4 4 . 9 22 8.3 0. 8173E-02 1. 8659 0 . 4 4 4 + / - 0 . 7 U 127 .7+ / - 6 .6 2.158 - 4 0 . 9 232 .3 0 .7255E -02 1.6852 0 . 6 1 3 + / - 0 . 5 2 3 11 0. 6* / - 4.8 2.104 - 3 6 . 4 2 3 6 . 8 0. 6387E-02 1.5125 0 .749+/ -0 .321 9 5 . 3 + / - 2 .9 2.060 - 3 1 . 2 242. 0 0 .5558E -02 1.3451 0 . 8 7 6 + / - 0 . 1 4 2 8 1 . 5 + / - 1.3 2.025 - 2 0 . 2 2 5 3 . 0 0. 42 49 E- 02 1.0749 1.120+/ -0. 153 6 0 . 9 + / - 1.3 1.978 - 1 0 . 4 262 .8 0. 3428E-02 C. 9008 I .391+/ -0 .147 4 7 . 5 + / - 1 .2 1.915 - 0 . 2 2 T 3 . 0 0 .2797E -02 0.7637 1 .690+/ -0.086 3 6 . 7 * / - 0 . 7 1.810 9.0 282.2 0 .2365E -02 0. 6673 1.935 + / - 0 . 0 5 4 2 9 . 4 + / - 0.4 1.717 19.7 2 9 2 . 9 0. 19 75 E-02 0.5784 2 .198+/ -0 .086 2 3 . 5 + / - 0 .6 1.645 29 . 1 302 .3 0 . 1706E-02 0.5157 2 .430+/ -0 .118 20.0+ / - 0. 8 1.623 39.2 3 1 2 . 4 0. 1473E-02 0. 4602 2 .730+/ -0 .081 1 7 . 0 + / - 0 .5 1.594 49.2 322 .4 0 . 1286E-02 0.4147 3 .080+ / -0 .079 14 .3+ / - 0.5 1.540 57.6 3 3 0 . 8 0 . 1 1 5 5 E - 02 0. 3822 3.361 + / - 0 . 1 3 2 1 3 . 1 + / - 0 .7 1.563 "AVOL " IS THE APPARENT OR HYDRODYNAMIC VOLUME OF THE MOLECULE (ASSUMING THE STOKES-EINSTEIN EQUATION IS APPLICABLE I T A B L E ; 6 . i c ***** C U ( I I ) - 6 3 BIS DIETHYL DITHIOCARBAMATE IN TOLUENE ***** FITTING PARAMETERS FOR LNTAU . VS • 1/T , AL PHA".VS. T/VISC t TAUC .V S. V1 SC/T, LNTAU-LNT. VS. 1/T . LNV I SC.V S. 1/T. LNT.VS.l/T LNTAU.VS.l/T SLOPE=14.10+/-0.11 KJ INTERCEPT= 0.07W-0.001 PS CORRELATION COEFF = 0.9997 ALPHA".VS.T/VIS S L0PE=3. 79 5E-03+1-\. 24 E-04 INTERCEPT* 0 . 187*/-0. 202 CORRELATION C0EFF=0.9942 TAUC.VS.VISC/T SLOPE=1.630E*04*/-1.6E+02 INTERC E PT= -7.81+/-2.51 CORRELATION COEFF=0.9995 LNTAU-LNT.VS. 1/T SLOPE" 11. 84*/-0. 13 KJ I NTERCEPT= 55.4+/- 0.6 PS CORRELATION C0EFF=0.9993 LNVIS.VS.l/T SLOPE" 9.69»/-0.11 KJ I NTERC EPT = 0.0109+/-0.0001 PS CORRELATION C0EFF = 0.9993 LNT.VS.l/T SL0PE=-2.26+/-0.02 KJ INTERCEPT=744.5*/- 1.4 PS CORRELATION C0EFF=-.9995 TABLE 6.1d CUlID-63 BIS DIETHYL DITHIDCRRBRMRTE IN TOLUENE X X FIGURE 6.2a X X x x x x -I 1 1 1 1 \ 1 1 1 1 1 1 \— J " 0.07 o.pB ' " " " VIS/T 1X10~r) 0.01 0.02 0.03 O.M 0.05 0.06 Q _ 0 .ps 0.09 0.1 0.11 0.12 0.13 0.14 *X CUUI)-63 BIS DIETHYL DITHIOCRRBRMRTE IN TOLUENE x x X X X X X X X FIGURE 6.2 b ° - ° 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.D8 0.09 0.1 0.11 0.12 0.13 0.14 0. VIS/T lX10-r) X X x x X X * X CUlID-63 BIS DIETHYL DITHIOCRRBRMflTE IN TOLUENE X X FIGURE 6,2c X 1 1 1 1 1 1 1 1 1 1 1 1 1 1 " l 0.0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0. P8 VIS/T lX10" r) CUC1D-63 BIS DIETHYL DITHIOCRRBflMflTE IN TOLUENE X X X X X X X X X FIGURE 6-2 g X X X X o I , , , , , ! , , , 1 , i .24 0.26 0.2B 0.3 0.32 0.34 0.36 „ 0.38 0.4 0.42 0.44 0.46 0. 1/T IX10" 2 ) CUIID-63 BIS DIETHYL DITHIOCRRBRMflTE IN TOLUENE X X X X X x FIGURE 6.2f . -, , , , , , , , , ( , , 0.0 0.01 0.02 0.03 0.04 0.05 0.06 , 0.07 0.08 0.09 0.1 0.11 0. VIS/T (X10-1 ) CUUI)-63 BIS DIETHYL DJTHIOCflRBRMRTE IN TOLUENE X X X X X X X x FIGURE 6,2g X X X X 1 1 1 1 1 1 1 1 1 1 1 1 0.0 10.0 20.0 30.0 40.0 50.0 60.0 , 70.0 80.0 90.0 100.0 110.D 120 T/VIS (X101 J -80-the l i n e - w i d t h , vs. temperature p l o t s to show a minimum, as i n f a c t observed. The simple hydrodynamic t h e o r i e s p r e d i c t a l i n e a r dependence of ITICT^) on 1/T, w i t h a slope dependent only on the solvent and not the probe. This was observed i n a l l cases. The r e s u l t s are given i n t a b l e 6.14. The i n t e r c e p t f o r such p l o t s i s not p h y s i c a l l y meaningful as even the simplest model p r e d i c t s n on-linear behaviour at high tempera-tur e s . They have been included as they and the corresponding slopes define x f o r the experimental temperature range. The r e g r e s s i o n co-e f f i c i e n t (r) f o r t h i s data was r > 0.999 and there were no v i s i b l e d e v i a t i o n s from l i n e a r i t y i n the p l o t s . At a f i x e d temperature the c o r r e l a t i o n time i s expected to be a f u n c t i o n of the molecular volume (eqn. 1.9). An a l t e r n a t i v e way of l o o k i n g at t h i s i s to consider the temperature r e q u i r e d to o b t a i n a given c o r r e l a t i o n time. A value of 150pS was found to be u s e f u l as t h i s corresponds approximately to the f a s t motional l i m i t (see s e c t i o n 2.3.1) f o r copper dithiocarbamates. The temperature (T^) corresponding to t h i s c o r r e l a t i o n time f o r each compound i s given i n t a b l e 6.14. I t i s c l e a r from t a b l e 6.14 that the r i n g compounds r o t a t e s l i g h t l y f a s t e r than the l i n e a r compound of corresponding molecular weight. More i n t e r e s t i n g though i s the f a c t that the N-Me, Octadecylamine d e r i v a t i v e r o t a t e s f a s t e r than the hexyl d e r i v a t i v e . -81-TABLE 6.14 Copper Dithiocarbamate T°C m E , kJ/mol obs D i f f e r e n c e Me2NH <-60 13 55 + 0. 24 0. 063 + 0. 005 Et 2NH -50 14 10 + 0. 11 0. 074 + 0. 001 n-Pr 2NH -35 13 80 + 0. 11 0. 138 + 0. 002 n-Bu2NH -30 13 01 + 0. 14 0. 257 + 0. 003 n-Hx2NH 0 13 54 + 0. 05 0. 376 + 0. 001 n-0c 2NH 25 14. 05 + 0. 18 0. 453 + 0. 002 N-Me,Octadecylamine -5 15 16 + 0. 16 0. 171 + 0. 002 P y r o l l i d i n e -50 13 77 + 0. 23 0. 098 + 0. 002 Hexamethyleneimine -45 13. 93 + 0. 29 0. 081 + 0. 002 Octamethyleneimine -40 14. 22 + 0. 18 0. 098 + 0. 002 Dodecamethyleneimine -20 13. 51 + 0. 18 0. 255 + 0. 002 Mean - 12. 73 + 0. 36 -The T 2.VS.TI/T plots are i n general expected to be l i n e a r . The least squares f i t s were excellent (r > 0.999), but gave negative intercepts i n a l l cases. P h y s i c a l l y t h i s i s a completely unacceptable r e s u l t . However de t a i l e d examination of the plots shows the onset of non-linear behaviour at high temperatures (see f i g s 6.2f for example and f i g s 6.3, a-j i n appendix I I ) ) . The s i g n i f i c a n c e of t h i s n o n - l i n e a r i t y was tested by assuming that the plot was composed of two i n t e r s e c t i n g s t r a i g h t l i n e s ; the low temperature portion (< 50°C) which contains most of the data, and the high temperature portion where the points s t a r t to deviate to a shallower straight l i n e . The r e s u l t s are given i n table 6.15. -82-TABLE 6.15 Copper Dithiocarbamate Slope (low temp por t i o n ) x 10^ Slope (high temp por t i o n ) x 10^ S i g % Me^NH 1 08 + 0 02 0. 54 + 1 20 99.95 Et^NH 1 67 + 0. 02 1. 29 + 0 03 99.95 n-Pr 2NH 2 54 + 0 05 2. 09 + 0 04 99.5 n-Bu2NH 3 04 + 0 07 2. 76 + 0. 27 <90 n-Hx2NH 5 76 + 0. 05 5. 38 + 0 26 <90 n-Oc2NH 8 98 + 0. 07 6. 82 + 0 15 99.75 N-Me,Oc tadecylamine 5 88 + 0 10 3. 74 + 0. 50 97.5 P y r o l l i d i n e 1 86 + 0. 05 1. 43 + 0. 10 99 Hexamethyleneimine 1 76 + 0. 03 1. 21 + 0 20 95 Octamethyleneimine 2 42 + 0. 03 1. 69 + 0. 06 99.95 Dodecamethyleneimine 4 03 + 0. 08 3. 35 + 0. 02 95 ' K i v e l s o n showed (18) that i f the assignment of the r e s i d u a l l i n e width a" to the s p i n - r o t a t i o n a l i n t e r a c t i o n i s c o r r e c t then p l o t s of a".vs.T/n w i l l be l i n e a r . This was found except at temperatures c o r r e s -ponding to l a r g e c o r r e l a t i o n times ( x 2 > lOOpS). In these cases con-s i d e r a b l e d e v i a t i o n from l i n e a r i t y was observed; negative a" were found i n some cases. However, at high T 2 ' s the s p i n - r o t a t i o n a l c o n t r i b u t i o n s to l i n e w i d t h are very s m a l l . Also the l i n e - w i d t h e r r o r s are r e l a t i v e l y l a r g e . Consequently the e r r o r s i n a" are very l a r g e (see t a b l e s 6.2-6.13 i n appendix I I ) . The s i g n i f i c a n c e ( i f any) of t h i s n o n - l i n e a r i t y i s s t i l l u n c e r t a i n . The slopes and i n t e r c e p t s f o r the l i n e a r p o r t i o n of the data are given i n t a b l e 6.16. -83-TABLE 6.16 Copper Dithiocarbamate Slope x l O 3 Intercept Me^NH 6 .54 + 0. 19 0 .35 + 0. 18 Et 2NH 3 .35 + 0. 07 0 .48 + 0. 03 n-Pr 2NH 2 12 + 0. 02 0 .46 + 0. 01 n-Bu2NH 1 69 + 0. 08 0 .61 + 0. 06 n-Hx2NH 1 43 + 0. 05 0 .07 + 0. 05 n-Oc2NH 1 57 + 0. 12 -0 .46 + 0. 10 " *' * (1 53 + 0. 20) (-0 .21 + 0. 04) N-Me,Octadecylamine 1 55 + 0. 05 0 .25 + 0. 04 P y r o l l i d i n e 3 51 + 0. 06 0 .52 + 0. 03 Hexamethyleneimine 2. 93 + 0. 10 0 .67 + 0. 05 Octamethyleneimine 2. 28 + 0. 06 O .49 + 0. 02 Dodecamethyleneimine 1. 42 + 0. 07 0 .55 + 0. 00 * Selected points The a" term i s important as i t gives a hand hold on the anisotropy of motion N (see section 2.11), which i s given by eqn. 2.59 N = c t " T 21.400xl0" 4 - (dg) 2 - (dg)y + \ (dg) 2 where x^ i s i n pS The i n d i v i d u a l g tensor components were not determined f o r a l l the probes. The accuracy of the parameters obtained from glasses does not warrant i n d i v i d u a l measurements i f the changes are known to be small. Thus for copper dithiocarbamates -84-N = a " x 2 0 . 7 x 10 4 - (dg)* + | ( d S > l l (dg)£ = 4 x l 0 _ 4 v (dg)* = 7.2 x 10" 3 (25) •'. N = a " x 2 0 . 0 1 + 0.5 The values of N f o r a s e l e c t i o n of temperatures f o r a l l the probes i s given i n t a b l e 6.17. TABLE 6.17 Copper Dithiocarbamate Anisotropy N -40°C 0°C +40°C 1 +80°C Me2NH 1.3 1.1 1.0 -Et 2NH 1.2 1.1 1.0 i n-Pr 2NH 1.2 1.1 1.0 0.9 n-Bu2NH - 1.5(1.3) 1.2 1.0 n-Hx2NH - 1.0(1.2)- 1.2 1.1 n-0c 2NH - - 1.0(1.2) 1.2 N-Me,Octadecylamine - 1.1(1.8) 1.4 1.1 P y r o l l i d i n e -ve(1.3) • 1.1 0.9 -Hexamethyleneimine 0.9(1.5) 1.1 1.0 0.9 Octamethyleneimine - 1.2 1.0 • 0.9 Dodecamethyleneimine - 1.3(1.4) 1.2 1.0 ( ), extrapolated p o i n t s ; observed p o i n t s i n non-linear r e g i o n . The p o l y c r y s t a l l i n e s p e c t r a ( i n toluene) of a l l the probes were recorded, however the d i f f e r e n c e s were small and the AA and Ag parameters -85-63 for CuCEtjDTC) used throughout this work (see table 6.18) TABLE 6.18 63 Spin Hamiltonian parameters for Cu(Et DTC) A = -126 xx A = -108 yy A = -477 zz AA = -374 A0 = -226 > MHz g = 2.025 xx g = 2.020 yy g =2.084 "zz Ag = 0.0677 g Q = 2.045 -86-CHAPTER 7 DISCUSSION 7.1 In t r o d u c t i o n There are s e v e r a l models f o r motion i n l i q u i d s (2,4d), however we w i l l be concerned only w i t h the simpler models and t h e i r extensions. These are the Debye-Einstein-Stokes (simple or s t i c k ) hydrodynamic model (65,66); Pecora's f r e e r o t a t i o n a l extension of the simple hydro-dynamic model (67); the s l i p models (68,69); the c o n d i t i o n a l f r e e r o t a t i o n model due to A t k i n s (70,71); and the quasi-lattice-random-f l i g h t model due to O ' R e i l l y (24,72-86). These t h e o r i e s apply only to i s o t r o p i c d i f f u s i o n ; they can be extended, but there are r a r e l y enough a c c e s s i b l e experimental v a r i a b l e s to warrant the i n t r o d u c t i o n of e x t r a parameters. 7.2 The Simple Hydroynamic Model This model a p p l i e s to small-step i s o t r o p i c r o t a t i o n a l d i f f u s i o n of a hard sphere i n a continuous medium w i t h s t i c k boundary c o n d i t i o n s ( i e . the r e l a t i v e v e l o c i t y of the sphere and the medium i s zero at t h e i r i n t e r f a c e ) . I t has proved to be s u c c e s s f u l f o r a l a r g e number of l i q u i d s (65) and a cursory examination of the r e s u l t s of t h i s t h e s i s i n d i c a t e that t h i s theory may provide a s a t i s f a c t o r y model i n t h i s case a l s o . The c o r r e l a t i o n time T „ i s given by x 2 = Vn/kT 7.1.1 or x 2 = AT exp(-En/RT) 7.1.2 -87-where V i s the e f f e c t i v e or hydrodynamic volume of the probe. n i s the v i s c o s i t y of the solvent, T i s the absolute temperature, En i s the energy b a r r i e r to viscous flow and A i s a constant. From (7.1.1) a l i n e a r r e l a t i o n , passing through the o r i g i n , between T 2 and n/T i s expected. As mentioned i n chapter 6 such pl o t s do not i n fact pass through the o r i g i n . Also they show non-linear behaviour near the o r i g i n . A possible explanation for t h i s i s given i n s ection 7.8. One also expects a l i n e a r r e l a t i o n s h i p between V (which i s pro-por t i o n a l to the slope of (the l i n e a r portion of) the T2.vs.n/T plots) and molecular volume; the r e s u l t s are shown i n table 7.1. Deta i l s of the methods for c a l c u l a t i n g molecular volume can be found i n appendix I I I . TABLE 7.1 Copper Dithiocarbamate o Molecular Volumes A3 Calculated Observed Me2NH 220 149 ± 2 : Et 2NH 288 230 ± 2 n-Pr 3NH 356 350 ± 7 n-Bu2NH 424 420 ± 9 n-Hx2NH 570(760)* 794 ± 6 n-Oc2NH 696(1100)* 1239 ± 10 N-Me,Octadecylamine 798 811 ± 13 P y r o l l i d i n e 261 240 ± 7 Hexamethyleneimine 328 236 ± 4 Octamethyleneimine 394 334 ± 4 Dodecamethyleneimine 528 556 ± 10 see appendix I II -88-For the l a r g e molecules, the agreement between theory and experiment i s e x c e l l e n t , although c o r r e c t i o n s have to be made f o r dead volume (see appendix I I I ) i n the case of the dihexylamine and dioctylamine d e r i v a t i v e s . The dimethylamine, diethylamine and p y r o l l i d i n e d e r i v a t i v e s have apparent volumes smaller than the c a l c u l a t e d volumes; t h i s may represent a breakdown of the ' s t i c k ' boundary c o n d i t i o n (see s e c t i o n 7.7), anisotropy of the motion (see s e c t i o n 7.9) or i t may simply be a geometric e f f e c t ; the smaller molecules are non - s p h e r i c a l . The anomalous behaviour of the hexamethyleneimine d e r i v a t i v e may be r a t i o n a l i s e d as f o l l o w s (however see s e c t i o n 7.7); space f i l l i n g models i n d i c a t e that the r i n g i s h i g h l y s t r a i n e d when l y i n g i n the same plan as the four sulphur atoms. The p r e f e r r e d conformation i s probably a c h a i r form w i t h the r i n g s perpendicular to the sulphurs. (This a l s o a p p l i e s to some extent to the octamethylene d e r i v a t i v e , but the r i n g i s a l o t l e s s s t r a i n e d i n t h i s case). A consequence of t h i s i s that the d i f f u s i o n axes may no longer be co i n c i d e n t w i t h the molecular axes and may co n t a i n c o n t r i b u t i o n s from o f f diagonal elements of the d i f f u s i o n tensor (see s e c t i o n 2.10). The r e s u l t , i n the case of a c h a i r conformation, i s that the observed c o r r e l a t i o n time becomes shorter and hence the observed volume i s smaller than expected. The r e l a t i o n s h i p between l n ( j ^ ) and 1/T should be l i n e a r : t h i s can be r e a d i l y shown from 7.1 l n ( x 2 ) = ln(Vn/kT) 7.2.1 -89-= l n ( n ) - ln(T) + B 7.2.2 where B' = ln(V/k) For a l a r g e number of l i q u i d s n can be represented as n = n exp(-En/RT) 7.3 f o r toluene TIQ Z 0.015P and Eri/R ~ 9.0kJ/mol. These constants depend s l i g h t l y on the temperature range used and were c a l c u l a t e d i n d i v i d u a l l y f o r each set of data, hence from 7.2 and 7.3 where B = ln(Vn/k) For our experimental temperature range ln(T) i s e s s e n t i a l l y l i n e a r w.r.t. 1/T and i t s c o n t r i b u t i o n to the slope of ln(x2).vs.1/T p l o t s can be c a l c u l a t e d s e p a r a t e l y . I t can a l s o be accounted f o r more d i r e c t l y by p l o t t i n g l n ( i 2 ) - l n ( T ) . v s . l / T . The r e s u l t s are tabulated i n t a b l e 7.2 (see a l s o t a b l e 6.14). The d i f f e r e n c e between the observed and expected a c t i v a t i o n energy i s s m a l l , but outside of experimental e r r o r . This and the n o n - l i n e a r i t y of the x 2.vs.n/T p l o t s i m p l i e s that a temperature dependent mechanism a f f e c t i n g x^, not accounted f o r by t h i s simple model, i s operating. l n ( x 2 ) = En/RT - ln(T) + B 7.4 -90-TABLE 7.2 Copper Dithiocarbamate EnkJ/mol E 0 B SkJ/mol Difference Me^NH 9 .88 + 0. 08 11 .37 + 0, 26 1 49 + 0. 36 Et 2NH 9 .69 + 0. 11 11 .84 + 0. 13 2 15 + 0. 17 n-Pr2NH 9 97 + 0. 10 11 .46 + 0. 11 1 49 + 0. 15 n-Bu2NH 9 .27 + 0. 09 10 .61 + 0. 14 1 34 + 0. 17 n-Hx2NH 8 .81 + 0. 05 10 93 + 0. 05 2 12 + 0. 07 n-0c2NH 8 57 + 0. 03 11 28 + 0. 20 2 71 + 0. 20 N-Me,Octadecylamine 8 90 + 0. 06 12 59 + 0. 16 3 69 + 0. 17 Py r o l l i d i n e 9. 69 + 0. 01 11 44 + 0. 25 1. 75 + 0. 27 Hexamethyleneimine 9 59 + 0. 10 11 65 + 0. 31 2. 06 + 0. 33 Octamethyleneimine 9 37 + 0. 09 11 85 + 0. 22 2. 48 + 0. 22 Dodecamethyleneimine 9. 07 + 0. 08 11. 04 + 0. 18 1. 97 + 0. 20 Mean* 9. 35 + 0. 45 11. 35 + 0. 40 1. 96 + 0. 44 * excluding 7.3 'Non-Hydrodynamic' Models The simple hydrodynamic model contains a number of assumptions which l i m i t s i t s general a p p l i c a b i l i t y to cases where the molecules are large. Several alternative theories have been developed to overcome these deficiencies (2,70,71,87,88), but most are suitable only for describing very simple l i q u i d s such as l i q u i d argon. Two models of more general use have been developed by Atkins and O'Reilly respectively. These are discussed below. 7.3.1 Conditional Free Rotation Model Atkins proposed that the probe partitions between 'cavities' (or 'expanded l a t t i c e sites') and 'contracted l a t t i c e s i t e s ' within the l i q u i d . He supposed that the probe was motionless i n the contracted s i t e s , but underwent free rotation i n the ca v i t i e s . -91-Assuming the c a v i t y c o n c e n t r a t i o n , P, i s given by P = P Qexp(-Ep/RT) 7.5.1 then -h x 2 = AT 2exp(Ep/RT) 7.5.2 where Ep i s the energy to form a c a v i t y s i t e to accomodate f r e e r o t a t i o n and A i s p r o p o r t i o n a l to the moment of i n e r t i a of the probe. A t k i n s t e s t e d t h i s model w i t h reasonable success w i t h l i q u i d ammonia. Unfortunately h i s success seems to be due to the f a c t that f o r l i q u i d ammonia the b a r r i e r to motion between expanded and con-t r a c t e d s i t e s i s s m all so that the energy of c a v i t y formation i s the dominant f a c t o r . For more complex l i q u i d s t h i s energy b a r r i e r ( e s s e n t i a l l y the a c t i v a t i o n energy f o r v iscous flow) i s much l a r g e r and becomes the dominant f a c t o r . This i s evidenced by the f a c t that the a c t i v a t i o n energy f o r the c o r r e l a t i o n time f o r our probes i s i n -dependent of the s i z e and shape of the probe and depends only on the solvent (14). (One would expect the energy to form a c a v i t y to increase w i t h i t s s i z e ) . Since A t k i n s ' theory neglects t h i s e x t r a energy b a r r i e r i t g r o s s l y underestimates the observed a c t i v a t i o n energy (4kJ/mol against an observed value of l l k J / m o l ) and i s not s u i t a b l e f o r i n t e r p r e t i n g our data. The theory remains of i n t e r e s t however, as i t provides a f i r m e r t h e o r e t i c a l foundation f o r Pecora's a r b i t a r y assignment of T Q to a f r e e r o t a t i o n a l c o r r e l a t i o n time (see s e c t i o n 7.8). 7.3.2 The Quasi-Lattice-Random-Flight (QLRF) Model This model, based on the Ivanov (89) jump d i f f u s i o n theory, -92-assumes that the probe molecules occupy c a v i t i e s , as i n A t k i n s ' s theory, but undergo r o t a t i o n only when jumping from one c a v i t y to another. The c o r r e l a t i o n time as given by t h i s model i s x 2 = (/ 2 V ) 2 / 3 / 6 ( l - X ) D exp(-w/RT) 7.6.1 or T 2 = Aexp(E/RT) 7.6.2 where D i s the d i f f u s i o n c o e f f i c i e n t f o r the probe D = D exp(E n/RT) 7.7 w i s the energy to form a c a v i t y and X=0 f o r l a r g e angle r o t a t i o n a l jumps (>TT/3) . This model has been s t r o n g l y c r i t i c i s e d by McClung (2) as i t contains more a d j u s t a b l e parameters than experimental v a r i a b l e s , thus one can f i t any set of T 2'S to t h i s theory. O ' R e i l l y claims to be able to c a l c u l a t e a l l h i s parameters, but the accuracy of the c a l c u l a t i o n s i s f a r from s a t i s f a c t o r y . Also f o r l a r g e molecules such as the dithiocarbamates i t i s u n l i k e l y that they w i l l undergo l a r g e angle jumps, an assumption which i s c e n t r a l to the QLRF model. This model i s of i n t e r e s t though as i t i s not an extension of the simple hydrodynamic model, but i t can be r e l a t e d to i t v i a the S t o k e s - E i n s t e i n r e l a t i o n s h i p f o r n and D n = kT/VD 7.8 hence -93-T 2 = /2 (Vn/kT)/(l -A)exp(-w/RT) 7.9 This theory introduces an e x t r a temperature term, exp(w/RT), but u n f o r t u n a t e l y w = 4kJ/mol and thus i s too l a r g e to account f o r the observed discrepancy i n the a c t i v a t i o n energy. Also c a v i t y formation does not appear to play a s i g n i f i c a n t r o l e i n determining the a c t i v a t i o n energy as one would then expect a strong dependence of the a c t i v a t i o n energy on the s i z e of the probe. The jump angle may be temperature dependent, but t h i s cannot be t e s t e d experimentally and there i s no s a t i s f a c t o r y theory to p r e d i c t i t s behaviour. In c o n c l u s i o n the QLRF model can e x p l a i n some of the r e s u l t s , but i s no more s u c c e s s f u l than the simple hydrodynamic model. In view of the number of poorly defined parameters i n the QLRF theory i t i s u n l i k e l y to f i n d general a p p l i c a b i l i t y . 7.4 Extensions to the Simple Hydrodynamic Model The o v e r a l l success of the simple hydrodynamic model has l e d to many m o d i f i c a t i o n s of t h i s theory, u s u a l l y by removing the inherent assumptions. Some of these m o d i f i c a t i o n s are discussed below. The assumptions of the theory are: a) small step d i f f u s i o n , b) i s o t r o p i c d i f f u s i o n , c) uncoupled r o t a t i o n a l d i f f u s i o n , d) 'hard' spheres, e) s p h e r i c a l geometry, f ) continuous medium, g) ' s t i c k ' boundary c o n d i t i o n s . a) This r e s t r i c t i o n has been removed by E g e l s t a f f and Gordon (90,2) to give the 'jump d i f f u s i o n ' model. Large angle jumps were a l s o used i n O ' R e i l l y ' s model (see s e c t i o n 7.3.2). I t i s u n l i k e l y that molecules as l a r g e as the dithiocarbamates w i l l undergo l a r g e angle jumps and we can assume that the small jump c r i t e r i o n i s met by these -94-molecules. (Also see section 7.9). b) Anisotropic d i f f u s i o n i s discussed i n general i n section 7.9 and more s p e c i f i c a l l y i n r e l a t i o n to molecular geometry i n section 7.6. c) It has been suggested that under some circumstances the r o t a t i o n a l and t r a n s l a t i o n a l d i f f u s i o n tensors become coupled (90) i n which case the eqn. 2.55 i s no longer v a l i d . This i s d i f f i c u l t to test with ESR, but i s usually s i g n i f i c a n t only with small molecules. d) The assumption that the molecules are 'hard' spheres, or a l t e r n a t i v e l y that 'only the repulsive terms of the intermolecular p o t e n t i a l are s i g n i f i c a n t ' (91-93), has been well tested (see section 7.5) and appears to be v a l i d for a l l non-coordinating solvents. e) Eqn. 7.1.1 has been extended to the more general case of the e l l i p s o i d by P e r r i n (94) and l a t e r by Favro (95). This i s d i s -cussed i n section 7.6. f) Obviously on a microscopic l e v e l the medium i s not continuous; t h i s has led to the development of a number of s t a t i s t i c a l mechanical theories such as Atkins model. On the whole these theories are not successful due to our poor understanding of the intermolecular forces acting i n l i q u i d s . If the probe molecule i s much larger than the solvent molecules the medium appears continuous and d e t a i l s of the intermolecular potentials are unimportant; however some e f f e c t s may be apparent with the smaller dithiocarbamates (see section 7.9). g) In recent years there has been growing i n t e r e s t i n developing the hydrodynamic model with ' s l i p ' boundary conditions ( i e . the r o t a t i o n a l -95-v e l o c i t i e s of the solvent molecules and the probe molecules are not d i r e c t l y r e l a t e d ) . This theory has been moderately s u c c e s s f u l f o r d e s c r i b i n g d e v i a t i o n s from the ' s t i c k ' hydrodynamic model. This i s discussed f u r t h e r i n s e c t i o n 7.7. 7.5 Temperature Dependence of the E f f e c t i v e Volume The e f f e c t i v e volume, V, may be temperature dependent f o r a number of reasons a) f o l d i n g and u n f o l d i n g of the hydrocarbon chains i n the probes (caused by changes i n the solvent d i e l e c t r i c constant f o r instance) b) breakdown of the hard sphere approximation of the simple hydrodynamic theory; and c) the anisotropy of the motion may change thus changing the volume swept out by the probe. (The l a t t e r i s discussed i n s e c t i o n 7.9). I f chain f o l d i n g were a strong f u n c t i o n of temperature one would expect the a c t i v a t i o n energy to vary considerably from probe to probe, except f o r the r i n g d e r i v a t i v e s which are e f f e c t i v e l y r i g i d . This was not observed, thus, w i t h the p o s s i b l e exception of the Me-N-Octadecyl d e r i v a t i v e (E i s ~ 2kJ higher than the o t h e r s ) , v a r i a t i o n s i n the UlSo chain geometry, i f they occur at a l l , do not appear to change the e f f e c t i v e volume i n a temperature dependent manner. The hard sphere approximation obviously has l i m i t a t i o n s . However, Wilhelm (96) and Jonas (97) have studied the e f f e c t i v e hard sphere diameter of a number of solvents as a f u n c t i o n of temperature. I t would appear that the e f f e c t i s small (<2% f o r 150°C temperature change) and much l e s s than our experimental e r r o r . -96-7.6 Perrin's M o d i f i c a t i o n of the Simple Hydrodynamic Model Very few molecules are approximated by a sphere but f o r t u n a t e l y many can be adequatelydescribed by an e l l i p s o i d . P e r r i n extended the simple hydrodynamic model to in c l u d e the case of the general e l l i p s o i d (94). Three c o r r e l a t i o n times are required to describe the r o t a t i o n a l motion of an e l l i p s o i d , one f o r each symmetry a x i s . The c o r r e l a t i o n time, T^, f o r r o t a t i o n about the i ' t h a x i s i s given by, T. = C./6kT 7.10 x 1 where i s the f r i c t i o n a l c o e f f i c i e n t f o r r o t a t i o n about the i ' t h a x i s . For an e l l i p s o i d w i t h semi-axes lengths of a,b,c and each a x i s denoted by s u b s c r i p t s a,b,c r e s p e c t i v e l y , the f r i c t i o n c o e f f i c i e n t s are given by P e r r i n as; C 0 ^ ^ 7.11.1 a C 16rrri b +c 3 , 2 2 b Q+c R 1 6 T r r i 2^ 2 a +c 3 16irn 2 2 a P+c R 2^, 2 a +b Cfe = — 5 - 1 — — 7.11.2 7.11.3 a P+b Q where P,Q,R are given by D, below, when d=a,b,c r e s p e c t i v e l y ds 'o ( d Z + s Z ) / ( a 2 + s 3 ) ( b Z + s Z ) ( c Z + s Z ) CO ——: :  D = J -,2, 2 N 7 / 2, 3 W , 2 , 2 W 2. 2, 7 ' 1 1 A -97-t h i s cannot be solved a n a l y t i c a l l y except f o r the case of oblate (a<b=c) and p r o l a t e (a>b=c) spheroids. However i t can be r e a d i l y solved n u m e r i c a l l y (using the program, QINF, f o r i n t e g r a t i o n over a semi-i n f i n i t e range, from the UBC computing centre) and shown that i f the axes are s c a l e d , w i t h a=l, then x . = C.a 3/6kT 7.12 1 x where C^ i s now obtained from 7.11 using the scaled semi-axes and a i s the unsealed value of 'a'. Dithiocarbamates are b e t t e r approximated by a p a r a l l e l i'piped than an e l l i p s o i d and i f the molecular dimensions are used d i r e c t l y then 'a' has to be corrected by a f a c t o r of 3/4TT (see appendix I I I ) . A lso the C^'s are most conveniently c a l c u l a t e d without the 16Trn/3 term; hence 7.12 has to be modified to 2 C i a 3 n T i = f - n s r 7 - 1 3 For dithiocarbamates the c o r r e l a t i o n time observed by ESR i s the average f o r r o t a t i o n about the two longest a x i s (a and b) thus hence from 7.13 and 7.14 the t h e o r e t i c a l value of the observed cor-r e l a t i o n (T ) time i s given by 4_ a 3n ^a^b 7.15 T e ~ 3 kT C +C, a b -98-The values of T , T . , (the observed T„) and T (the values e obs 2 S p r e d i c t e d by the simple hydrodynamic model), f o r some of the probes are given below. Eqn. 7.15 h i g h l i g h t s a very important concept; the c o r r e l a t i o n time i s not r e l a t e d to the volume of the molecule, V, but only to a c o l l e c t i o n of f r i c t i o n c o e f f i c i e n t s and the l a r g e s t l i n e a r dimension of the molecule. TABLE 7.3 Copper Dithiocarbamate* Axes a,b,c(A) + T e + obs + T S Me2NH 12,6,3 42 16 21 Et 2NH 14,6,3.5 51 23 41 P y r o l l i d i n e 14,6,3 51 26 37 Hexamethyleneimine 16,6,3.4 71 24 47 Octamethyleneimine 16,6,4.5 81 33 56 Dodecamethyleneimine 20,7,4.5 99 64 75 N-Me,Octadecylamine# 57,3.8,3.8 78 86 . 114 + 20°C; u n i t s are pS # Assuming chains l i e along the l i n e of the C-N(CS 2)bond. * Only s e l e c t e d compounds have been examined owing to the d i f f i c u l t y of approximating some of the molecules by e l l i p s o i d s ; a l s o the geometry of the long chain molecules i s u n c e r t a i n i n s o l u t i o n . The c o r r e l a t i o n times p r e d i c t e d by the simple hydrodynamic models are i n reasonable accord w i t h the observed values ( t h i s i s hardly sur-p r i s i n g i n view of the good agreement between the observed and c a l c u l a t e d molecular volumes, t a b l e 7.1), but they f a i l to p r e d i c t the general trend of values, the dimethylamine and hexamethylimine d e r i v a t i v e s being very n o t i c e a b l e anomalies i n t h i s respect. The P e r r i n model not only f a i l s to p r e d i c t the general trend, but the absolute values of T O B G are i n c o r r e c t by upto a f a c t o r of three. C l e a r l y -99-the s t i c k hydrodynamic models are not applicable to the dithiocarbamates studied here. The apparent success of the P e r r i n model i n pre d i c t i n g the c o r r e l a t i o n time of the N-Me,Octylamine d e r i v a t i v e may be at t r i b u t e d to the fact that t h i s molecule i s very large and i t i s generally r e -cognised that the s t i c k models are applicable to large molecules. However a somewhat more s a t i s f a c t o r y explanation of t h i s i s given by the ' s l i p theory'. (See section 7.7). 7.7 The S l i p Hydrodynamic Model In general the s t i c k hydrodynamic models predict c o r r e l a t i o n times for small molecules which are 2-3 times too slow (see table 7.3). Hu and Zwanzig (68) proposed that the ' s l i p ' boundary condition (see 'g' i n section 7.4) was more r e a l i s t i c f o r these cases and produced a set of correction factors for the hydrodynamic equations for the case of oblate and prolate spheroids. These r e s u l t s were used su c c e s s f u l l y by Pecora (67) to predict the c o r r e l a t i o n times for a number of l i q u i d s . Youngren and Acrivos (69) extended Hu and Zwanzig's ideas to the general case of an e l l i p s o i d . Their r e s u l t s were used by Fury and Jonas (97) and also Pederson (99) to test the theory with a wider range of l i q u i d s . They only had l i m i t e d success. Fury and Jonas f e l t that t h i s was due to geometrical e f f e c t s and t r i e d an a r b i t a r y weighting procedure to allow for the discrepancies, but with only minor success. Pederson ascribed these deviations to intermolecular forces ('stick') acting along c e r t a i n molecular axes. In an e a r l i e r paper Kivelson (21) had suggested introducing a combination of s l i p and s t i c k to account for -100-such d i s c r e p a n c i e s , but f a i l e d to produce a s e l f - c o n s i s t e n t model. The i n t r o d u c t i o n of a combination of s l i p and s t i c k may p a r t l y e x p l a i n the r e s u l t s , but the hydrodynamic t h e o r i e s do not account f o r i n e r t i a l e f f e c t s (see s e c t i o n 7.8 and 7.9) and these may be important when molecules w i t h a wide range of molecular geometries are being studi e d . C o r r e l a t i o n times f o r the dithiocarbamates were c a l c u l a t e d using Youngren and Ac r i v o s ' s r e s u l t s . The method f o r doing t h i s i s o u t l i n e d below as i t i s not very c l e a r from t h e i r paper. The c o r r e l a t i o n time x. f o r r o t a t i o n about the i ' t h a x i s i s l given by, x. = L i n 7.16 6kT where i s the torque about the i ' t h a x i s and i s given by L. = X.a 3Tr 7.17 X X where X. i s the f r i c t i o n c o e f f i c i e n t obtained from the tab l e s i n Youngren x and A c r i v o s ' s paper; a TT i s a 'pseudo-volume', where a i s the semi-axis length of the longest a x i s of the e l l i p s o i d , hence x . = V^11 7.18 1 6kT Co r r e c t i n g the e l l i p s o i d a l shape and ta k i n g the average c o r r e l a t i o n time as before (see s e c t i o n 7.6) we get -101-3 X X u na a b ' s l i p 4kT X +X^ a b 7.19 The value of T ,. f o r s e l e c t e d dithiocarbamates are given i n t a b l e s l i p 7.4 along w i t h the experimental v a l u e s , x , and the r e s u l t f o r obs P e r r i n ' s t h e o r i e s , x . TABLE 7.4 * Copper Dithiocarbamate o Axes a,b,c(A) + s l i p + obs + X e Me2NH 12,6,3 14 16 42 Et 2NH 14,6,3.5 17 23 51 P y r o l l i d i n e 14,6,3 18 26 51 Hexamethyleneimine 16,6,3.4 19 24 71 Octamethyleneimine 16,6,4.5 25 33 81 Dodecamethyleneimine 20,7,4.5 31 64 99 N-Me,Octadecylamine 57,3.8,3.8 78* 86 78 + 20°C; u n i t s are pS * see t a b l e 7.3 # c a l c u l a t e d from Hu and Zwanzig's t a b l e s The s l i p model works very w e l l , both i n p r e d i c t i n g the general trends of the c o r r e l a t i o n times and t h e i r absolute values. Note e s p e c i a l l y that the hexamethyleneimine d e r i v a t i v e i s no longer anomalous (see t a b l e 7.1). The poor r e s u l t f o r the dodecamethyleneimine d e r i v a t i v e i s probably due to i n c o r r e c t assignment of the axes s i z e s , as the conformation of the r i n g s i s u n c e r t a i n . Hu and Zwanzig observed that as the a x i a l r a t i o s of a spheroid tend to zero or i n f i n i t y (a d i s k and a needle r e s p e c t i v e l y ) then x / x . , 1, which exp l a i n s the one-to-one correspondence of x , . and x s t i c k s l i p e -102-for the N-Me,0ctamethylamine d e r i v a t i v e . The apparent success success of the P e r r i n model i n predicting the c o r r e l a t i o n time for t h i s compound i s undoubtedly a consequence of t h i s and not that t h i s p a r t i c u l a r molecule ' s t i c k s ' . The s l i g h t discrepancy between the observed and calculated r e s u l t s may be due to a combination of experimental error and the method used to approximate the molecule to an e l l i p s o i d . It could also be due to the neglect of i n e r t i a l terms (see sections 7.8 and 7.8 The Pecora Extension of the Simple Hydrodynamic Model Closer examination of the s l i p model reveals two shortcomings. The most obvious i s that for oblate and prolate e l l i p s o i d s the cor-r e l a t i o n times about the polar axes are zero. This i s p h y s i c a l l y un-reasonable as the c o r r e l a t i o n time cannot be le s s than the free r o t a t i o n a l c o r r e l a t i o n time for that axis. The second short-coming i s that the s l i p theory f a i l s to predict the n o n - l i n e a r i t y of the x^.vs.n/T plo t s (see table 6.5). In the l i g h t of t h i s , Pecora suggested that the dynamics of a molecule i n s o l u t i o n might be described by a combination small step r o t a t i o n a l d i f f u s i o n (the simple hydrodynamic model) and simple free r o t a t i o n ; i e . * where t i s re l a t e d to the c l a s s i c a l free-rotor r e o r i e n t a t i o n a l c o r r e l a t i o n 7.9). x 2 = Vn/kT + T 0 7.20.1 t ime, T FR 7.20.2 -103-where I i s the moment of i n e r t i a of the probe and, A , i s the jump angle i n radians. ( S t r i c t l y speaking there are three x 's, one r R f o r each a x i s . Pecora i s presumably r e f e r r i n g to an average v a l u e ) . For dithiocarbamates x i s ^5pS f o r the dimethylamine d e r i -v a t i v e i n c r e a s i n g to ^ 20pS f o r the dioctylamine d e r i v a t i v e . A p l o t of X2.vs.n/T should give a l i n e a r p l o t at low temperature (when T 0 > > T o b s ) a n ^ becomes non - l i n e a r at higher temperatures (when X Q ^ X ^ ) . I n t e r p r e t a t i o n of the i n t e r c e p t i s d i f f i c u l t : E x t r a p o l a t i o n of the l i n e a r r e gion gives negative i n t e r c e p t s which are * This d e s c r i p t i o n i s somewhat ad-hoc as one cannot add c o r r e l a t i o n times, but only the corresponding 'rate constants'. One can draw the analogy of a r e a c t i o n w i t h two r a t e processes c h a r a c t e r i s e d by r a t e constants and t ^ l , thus the observed c o r r e l a t i o n time T ^ i s given by - 1 - 1 , - 1 T = p x • + p X 7.. obs h h o o where x =Vn/kT and p, and p are analogous to mole f r a c t i o n s , i . e . h r h <"> N h P h N,+N h o Where N i s the number of molecules undergoing f r e e r o t a t i o n and N i s the number of molecules obeying the simple hydrodynamic model. From 7.5.3 we get N.+N h o 7.22 T h T o T ° b s = v v v v 7 , 2 3 (cont.) -104-p h y s l c a l l y unreasonable. Extrapolation from the non-linear region i s hampered by lack of good data. In fact obtaining data for n/T<0.0003 i s probably impossible (present l i m i t i s n/T'vO.0008). It i s i n t h i s i n -accessable region that one might expect the points to tend assymptotically to x-p, i f n->0 fa s t e r than 1/T. The t h e o r e t i c a l plot i s shown i n f i g . 7.1. . 1 1 1 f 1 2 3 7//TX10* FIGURE 7.1 The experimental pl o t s (see f i g s 6.3.a-j) do i n fa c t appear to follow the t h e o r e t i c a l form. However, as other workers(99) have found p i s probably an exponential function of temperature (see section 7.3) tRus, since p =l-p h o l i m T„->-T, and lim p, -K),T„->T 2 h ^h ' 2 o j^ -o T-*» i e . the l i m i t i n g conditions are the same as Pecora's equation, (see over page). It i s i n t e r e s t i n g to note that the r e l a t i v e magnitudes of T q and are i r r e l e v a n t , we only have to assume that p Q i s small, a far more tenable assumption than x 0<<T n used to get the l i m i t i n g conditions from Pecora's equations. -105-the evidence i s not c o n c l u s i v e . The most d i r e c t way to t e s t Pecora's suggestion i s to p l o t n.vs.x at a f i x e d temperature. This i s r e a d i l y done by changing the s o l v e n t * and has been done f o r dimethyl and d i e t h y l dithiocarbamates by Herring and co-workers (14). The p l o t s are l i n e a r w i t h i n t e r c e p t s very c l o s e to the expected x values f o r these r K probes. The e f f e c t of x on the a c t i v a t i o n energy p l o t s may be analysed as f o l l o w s x - x = Vn/kT 7.24.1 obs o from 7.2 l n [ x , (1-x /x , 1 = En/T-ln(T) + B 7.24.2 obs o obs at low temperatures x /x , « 1 o obs 7.23.2 becomes l n ( x , ) - x /x ^ = Eri/T - ln(T) + B 7.24.3 obs o obs Over our experimental temperature range l n ( x , ) ^10 and x /x , <1, hence obs o obs the e f f e c t of x i s n e g l i g i b l e and 7.6.3 becomes (cf eqn. 7.1.2) l n ( x ^ _ ) = En/T - ln(T) + B 7.24.4 * This of course assumes that x 2 i s independent of the solvent molecular volume and there are no i n t e r m o l e c u l a r i n t e r a c t i o n s . However i t i s a b a s i c premise of the simple hydrodynamic theory that the solvent can be t r e a t e d as an i n e r t continuous medium; i f one accepts t h i s then changing the solvent i s a p e r f e c t l y v a l i d way to t e s t Pecora's theory. -106-Obviously cannot account f o r the p r e v i o u s l y mentioned discrepancy FR i n the a c t i v a t i o n energies. (See t a b l e 7.2) As w i t h the simple hydrodynamic model there should be a d i r e c t r e l a t i o n between V and the slope of the l i n e a r p o r t i o n of the T 2.vs.n/T p l o t s i f i s s m a l l . This has already been discussed i n s e c t i o n 7.2. h K Ov e r a l l , Pecora's suggestion works w e l l . Although i s too small to account f o r the 2kJ/mol discrepancy i n the a c t i v a t i o n energy* i t does account f o r the non-linear behaviour of the T 2 . v s . n / T p l o t s . 7.9 Some Defects of the Hydrodynamic Model The hydrodynamic models are u s e f u l , but somewhat o v e r s i m p l i f i e d . F i r s t l y they assume that the energy i s eq u a l l y p a r t i t i o n e d among the three r o t a t i o n axes and secondly they ignore i n e r t i a l e f f e c t s . The p a r t i t i o n i n g of energy may be unequal f o r two reasons: a) i f the probe and solvent have d i p o l e moments there may be some degree of alignment between the molecules r e s u l t i n g i n cooperative r o t a t i o n (90); i e . there i s a p r e f e r e n t i a l t r a n s f e r of energy from the solvent molecules to probe molecules a l i g n e d i n the appropriate manner. In general terms the in t e r m o l e c u l a r p o t e n t i a l i s a n i s o t r o p i c . Some the o r i e s have been developed to describe t h i s (100,101), but they have proved d i f f i c u l t to implement (13). These e f f e c t s are probably temperature dependent, l i q u i d s tend to 'open up' as the temperature i s * This i s true w i t h i n the context of Pecora's ad-hoc i n t r o d u c t i o n of t h i s term. However t h i s might not be true i f the r e s u l t s were analysed i n terms of eqns. 7.22. -107-r a i s e d (24). Such phenomena may w e l l account f o r the discrepancy of 2kJ i n the a c t i v a t i o n energy and the change i n the anisotropy of motion w i t h temperature. I t i s d i f f i c u l t to p r e d i c t the i n f l u e n c e of these a n i s o t r o p i c p o t e n t i a l s even q u a l i t a t i v e l y ; f o r example cooperative motion has been shown to be important i n cyclohexane even though the d i p o l e moment i s small (90). b) The e f f e c t i v e c o l l i s i o n diameter of most molecules i s a n i s o t r o p i c . For instance one expects more c o l l i s i o n s to occur along the long a x i s of a dithiocarbamate than perpendicular to i t . For a l a r g e molecule t h i s e f f e c t averages out as the surrounding medium i s u s u a l l y i s o t r o p i c a l l y disposed about the probe ( i n the absence of d i p o l e - d i p o l e i n t e r a c t i o n s , s o l v a t i o n e t c . ) . However f o r very small probes (smaller than or of s i m i l a r dimensions to, the solvent molecules) i t i s not c l e a r that these e f f e c t s w i l l average out and the motion may w e l l be more a n i s t r o p i c than p r e d i c t e d by the s l i p or s t i c k models. These e f f e c t s are e s p e c i a l l y n o t i c e a b l e when the medium i t s e l f i s a n i s o t r o p i c such as i n l i q u i d c r y s t a l s or b i o l o g i c a l membranes (4b,e,102,103). However i n such cases the ' s t r u c t u r e ' of the medium i s oft e n w e l l defined and as a consequence the an i s t r o p y of motion i s b e t t e r understood than f o r the case of an i s o t r o p i c medium. I n d i r e c t c o n t r i b u t i o n s of i n e r t i a l e f f e c t s have already been discussed i n s e c t i o n 7.8. One might expect there to be d i r e c t i n e r t i a l c o n t r i b u t i o n s to the r o t a t i o n a l c o r r e l a t i o n times on mechanical grounds. For example f o r a p r o l a t e spheroid one expects r o t a t i o n about the long a x i s to be -108-f a s t e r than r o t a t i o n about the short axes simply because the moment of i n e r t i a i s smaller about that a x i s . In p r a c t i c e t h i s i s im-portant only i f the jump angle i s l a r g e and the angular momentum c o r r e l a t i o n time, T , i s of s i m i l a r magnitude to the r o t a t i o n a l c o r r e l a t i o n time, T Q , (T_ i n our case: small step d i f f u s i o n c r i t e r i o n 0 Z simply means that T <<T q). AS pointed out p r e v i o u s l y there are a number of t h e o r i e s developed to account f o r t h i s , but i t i s u n l i k e l y that they apply to dithiocarbamates. I f the probes can undergo f r e e r o t a t i o n i n the solvent ' c a v i t i e s ' then obviously the s m a l l step c r i t e r i o n i s i n v a l i d ; the jump angles may exceed 360° depending on the l i f e t i m e of the probe i n the c a v i t y . F o r t u n a t e l y t h i s case can be simply d e a l t w i t h u s i n g Pecora's extension of the hydrodynamic model ( s e c t i o n 7.8). 7.10 Conclusions and Future Work In c o n c l u s i o n , the s l i p model w i t h Pecora's extension provides an e x c e l l e n t q u a l i t a t i v e d e s c r i p t i o n of the r e s u l t s of t h i s t h e s i s and i t does not c o n f l i c t w i t h the r e s u l t s of other workers; i e . the c o r r e l a t i o n times are w e l l described by the r e l a t i o n (see eqns. 7.19 and 7. 20) 3 A A . n a a b -r nc T2 = ZkT Y~+\7 T o 7 - 2 5 a b Q u a n t i t a t i v e l y the r e s u l t s are i n good agreement w i t h theory although minor d i s c r e p a n c i e s remain. D i f f e r e n c e s between the observed and p r e d i c t e d t2's probably r e f l e c t the d i f f i c u l t y i n a s s i g n i n g an - 1 0 9 -e l l i p s i o d a l volume to the probes, but may be due to neglect of the free-rotational terms, or errors i n the AA and Ag terms. The 2kJ/mol discrepancy i n the observed activation energy, although small i s probably r e a l (a temperature dependent systematic error of this magnitude i s extremely u n l i k e l y ) . I t s o r i g i n i s unclear, but may be due to the 1/kT term or the temperature dependence of the p term i n the free rotation contribution to the correlation time o (7.23). The reason for the change i n anisotropy of the motion i s also unclear. I t may be caused by changes i n the intermolecular potential i n the l i q u i d , or possibly be an a r t i f a c t of the analysis (we measure the average of two correlation times; i t i s possible that one of these i s i n the slow motional regime at low temperatures. This would explain the non-linear behaviour of the a",vs.T/ri plots as w e l l ) . The above points are r e l a t i v e l y minor and should be cleared up by further work. The scope for extending this work i s very large. The acquisition and data processing systems can be improved considerably. Automated analysis of the p o l y c r y s t a l l i n e spectra should speed up the simulation procedures enormously and allow better determinations of the AA and Ag parameters. The most important step would be to extend the work to other solvents so that x ^ . v s . r i plots can be done. Such plots give direct access to the x term and provide a rigorous test for Pecora's suggestion and the generality of eqn. 7.24. For copper dithiocarbamates, ESR i s unfortunately limited to the -110-measurement of the average of the c o r r e l a t i o n times f o r r o t a t i o n about two of the three molecular axes. One can overcome t h i s e i t h e r by changing the probe s l i g h t l y (using vanadium instead of copper (104) or monothiocarbamates (105) and other (106,107) analogs instead of dithiocarbamates: t h i s w i l l change the g and A tensors and consequently the modes of r o t a t i o n obser-13 v a b l e ) , or by changing the techniques used; C NMR and Resonance Raman spectroscopy are p o s s i b i l i t i e s . The former i s w e l l e s t a b l i s h e d f o r the measurement of r e l a x a t i o n times (108), but the l a t t e r needs considerable development. Both techniques are s i n g l e p a r t i c l e techniques g i v i n g cor-r e l a t i o n times described by second rank tensors and are hence d i r e c t l y comparable w i t h ESR data. Resonance Raman i s of e s p e c i a l i n t e r e s t as i t can be used w i t h very d i l u t e s o l u t i o n s l i k e ESR. I t i s not known at present i f the dithiocarbamates d i s p l a y the Resonance Raman e f f e c t , but i t should be p o s s i b l e to synthesise dithiocarbamates that do and which are a l s o s u i t a b l e f o r ESR s t u d i e s . I n v e s t i g a t i o n of other probes i s a l s o p o s s i b l e , the n i t r o x i d e s or acetylacetonate complexes f o r instance. However, u n l i k e the other probes the chemistry of dithiocarbamates i s r e l a t i v e l y s t r a i g h t - f o r w a r d . The synthesis of d e t e r g e n t - l i k e dithiocarbamates ( f o r membrane s t u d i e s (109)), water s o l u b l e d e r i v a t i v e s and a wide range of molecular geometries present no major problems. This makes the dithiocarbamates a much more v e r s a t i l e f a m i l y of probes than the other two. 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C r y s t . , B-30, 2928 (1974). 112. G. Pegronal and A. P i g n e d o l i . Acta. C r y s t . , 23, 398 (1967). 113. P. Newman, C. Roston and A. White. J.C.S. Dalton, 1332 (1973). -117-APPENDIX I Flow-Chart f o r KIVEL L i s t i n g of KIVEL -118-A B R 1 D G E D F L O W C H A R T  F O R S C A N Start G e t C o n v e r s i o n F a c t o r s f r o m L i n e a r L S F S t o r e R e s u l t s R e a d i n B i n a r y D a t a C o n v e r t t o D e c i m a l a n d E x t r a c t C a l i b . T a b l e . X - Y C o ordinates| o f | S p e c t r a - * R i n d B a s e l i n e B, (No i se L e v e l fix. N s t e p f rorrv L S F i 4 __, S c a n D a t a a t I n t e r v a l s o f N s t e p C o l l e c t 3pts.. ( Sub rou t i ne ) R e a d a t ' P r o g r a m ;'. P L O T ( O p t i o n a l ) ' | Subrout ine| B a s e l i n e N o I S u b r o u t i n e l P e a k Y e s F i t Y,» Y 3 t o a C u b i c A -119-V w w A Store Soln, No Locate Xove.r Yes; Store Results Ca l cu l a te LW's| in Gauss and Fi nd New Xoverl No" Scan Data 2.p.fs. at Intervals o f Nstep Yes Scan Data for 2nd Peakl Print Results Yes Expand Data" Set No ISubroutine Xover Main Reset Flags to Read New Spect rum Vfurnj - 1 2 0 -O — X • L U m o **• • m —< CD •Z » N - « C < —i sO <-» \ O L U O II — t- • —' C C W Z uj < # ll Z * < I - Z # L U II — O Of « LO h-u. w a r n <r a *- * 3 u. . cc — CO U J X m Of LO 2 *r *•* ftM •a CO • < LU < z CL U J • • w z X _ ao LU (NI L L c • —• w • < 2 O > X LU o < o Of • 2 •H a c «•-* U J — O •-^ _ J UJ > o • LU 0. n c — * m »— c c > x of • < fr- - J - o O o to — • or t U _ o Ct x a : (NI T U J CO L U Q. X —1 f- •ft- O • ft . a <o o ft— ft— o. o m • tNJ * ft • o l/l X *ft —• * m L L • — (M Z 1 cn ft* « > • LU LU o — o • w W fNI o Of Z 2 o >• * * . 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X < ft- O Z U O Of NO * - • II O. - I- I I II LU < U J N h l O Z oc — Ct — LO Of Of O "X < UJ 2 I i * CO —• — a II z X X II of at < <. X > > z z o. o n a cv a x x c o w j - w Z i - N . m - . t - * L U < U J Of < ^ L U < « ( _ J h - Z K > Z i O t - Z # L U f t . a « o c c 4 - t - - a * o f o f O n r x c u - n o f D Q. I l l 3 • L L » C J I L • O J (NI C* %0 •* *v O 3 ro w • m z U J < « — o m « O H O . *- tX # K Z O O O ftf c n O h- 2 a z a. . O L J L O L U o 3 -J o Of LU CO C O cO O z C O K — 0C Z5 L O L U 2 O Z O O Of L U U J X CO X K Z 3 — 2 O I / I O H U h-o z z LL Z C O UJ C O a <. z CO • ft a CL LU UJ a? • f - Z LL co » > Z z CO \~ U J • a ft-< ft— C O UJ 2 z < < ft-—1 C O LL "N. -s. •v. z 2 Z o O o s: z z z I z o n o o o u < t -< C O I— Q-X LL z O LU 2 o C O O z o — ft» CO z o z o c LU < rsi C O C O L O —J w ft- • ~2 - J X u <T a t - <r LU a —i U Of C O Of co <r a «• 3 *x z o • —i a . t- LU o C L iSl <r C i Q of N . 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V . WO < 13 + h- M r g — < o < — * < U J U 2 w ft *«» U-i o ^ 2 O LU o — «M O o • • h- m K — V . of w a w ft- o o • — X 00 UJ • P-4 UJ < < 2 •— u ft < LU LU < * 0D CO L U X L L 1- »* L U o Jt C O X z w L U u _J •— ft— ft* C L a LU L U co Q ft* >s UJ <— cO _ l w • ft* ft-1 o LL a C O L L - J LL z ft LL CO CO < • • z i - ts ft x. * J J Z I L Z K < L U to X • • — 2 d ft-> \ -v. -"v V . 3 O W O -1 LU o. t - L U * — • < < to o a. K - > < to o w O L O z C O z z z z z Of II w 2 II II L L o ac Z 2 Of z z CO — t - * u u II Of z < UJ CO o w » L U z a O o a o t- O LU < < 2 o o ii LU L U H II L U 2 L U <r * L U X t- II O _j < Of UJ < * 1- < a. L U z X z z z U L U h- z U LU O -J * * w O Of It 1 - * * I J h U l j Z -J X h- Z * o a z w z z z z z U J a ft-4 Of CO O a o _ J - - UJ L U ac h- ft- Of * D J < O LU "— Z OC - J w w •* QC * LU • <i ft-a a o o o o a co OC o LU CO L0 2 UJ < L L LL C L co 3 o, ac u a: -a - J < a a. ft* LU < IL IL CL a C O L O Q » u u u u CO _ i Dt L L ac • - J L U « u ft— ft— CO 2 U ft— X L L • O U te-* * J K L L • X. L O O u o O O LU Of ft— t— t - t - L- LU _ J 1 - < <— LU o I LU LU UJ L U L U C L < UJ LU Of CO CO 0" CO CO CO to m CO o to Ct K -* U) o o o a m U) U) l_> u * * o o O O O 2 — < O O (NJ C O * C • ¥-X O a o < 2 ac L U » •*> L U o. X o f- < • 2 1— X o =) C O C L • O of — O U ft-~ft L L < • •s. 0* < L U — » P O z Of C L ^> * UJ ft* o w * O Of II L L _ j a * 1- t - _> — O X C L I < L L * O Ul 2 U QC 1- < L U U J C O C O m u o U Of X o • C O L U W • O ft- > Of O UJ U O UJ UJ • LU X > • O OC o o z a of x z LU II O LU ft- — LL _J — w ac _ j — LL Of X < LL LU O O •— N T O T = 3 2 6 7 8 . 0 * 2 0 . 0 / 8 INC I F M N T O T . L T . 2 0 1 . O R . ( N T O T . G T . 2 0 0 0 H IERR=1 I F l I E R R . E 0 . 1 ) W R I T E I 6 . 1 3 9 ) I F ( I E R R . E O . l ) NT0T=900 139 F O R M A T ! / 1 0 X . • * * * WARNING B A S E L I N E DATA UNUSABLE * * * ' ) C C A L C U L A T E NO. OF INPUT P T S . ( 1 S T . INCH OF SPECTRUM1 NBASE = N T 0 T / 1 5 NSTEP=NTOT/20 C L I M I T KO. OF POINTS I F I N B A S E - N B E G . G T . 4 0 I INC = 1 - ( N B A S E - N B E G I / 4 0 IF (NBAS E - N B E G . L T . 5 I NBEG=1 437 KBASE=0 DO 27 I =NBEG, NBA S E , I N C KBAS E=KBAS E + 1 B{ KBASE I= IDATAX( I I A IKBASE 1= IDATAYI I I 27 CONTINUE C C A L C U L A T E AVERAGE DATUM INCREMENT ERROR CHECK DX = I B IKBASE I -B< I I ) / F L O ATI KBASEI W R I T E ( 6 , l l l l l D X . N S T E P l ' l l l FORMAT ( / • SCAN L I N E A R I T Y CHECK • . 3 X , 'DX = ' ,E 1 5 . 7. 5X, C 1 N S T E P = ' . 1 3 / 1 C F I T TO STRAIGHT L I N E TO F I N D B A S E L I N E AND TR IGGER L E V E L 436 I F I K B A S E - 2 . L E . 2 I GOTO 501 C A L L L S F ( K B AS EI 444 WRITE<6,22221 P 1 ( I t . S D A , P I ( 2 I .SOB 2222 FORMAT (/• BASEL INE F IT INTERC EPT= ' , IPE 16. 9 , •+/- ' , C 1 P E 1 3 . 7,3 X,» S L O P E * ' . 1 PE 16 .9 , • • / - • , 1 PE 13 . 7 / I C TEST FOR B A S E L I N E SLOPE B A S E = P 1 I H I F M A B S J P H 2 11 . L T . O . II .OR. (Bt II . L T . 2 5 0 0 . 0 1 1 GOTO 26 W R I T E ( 6 , 2 1 I 21 F O R M A T I / I O X , ' * * * WARNING SLOP ING BASEL INE * * * • ! B A S E = 1 6 3 8 4 . 0 2 6 TR IG=10.0*SDA I F ( T R I G . G T . 7 . 5 E + 0 3 I TR IG=7 .5E+03 33 WR ITE I6 .25 I TR IG ,NPTS .MAGSW.BASE 2 5 FORMAT (//•****• , 2 X , • TR I G = ' . F 6 . 0 , 2X. »NPTS= • , ci 5 . 2 x , « SWEEP WIDTH*• , 1 4 , ' G ' , 2 X , • B A S E L I N E * * , F 6 . O , C« * * * * • / ) RETURN C SET ERROR F L A G IF SPECTRUM NOT COMPLETE 501 MAGSW=-999 RETURN END C C C TH IS SUBROUTINE ED ITS THE C A L I B R A T I O N TABLE AND PRODUCES THE C CONVERSION FACTORS FOR CHANGING F I E L D DIAL VOLTAGE TO GAUSS SUBROUTINE CAL IB ICONV) DIMENSION FREOI I 0 0 ) . X I l O O ) . Y F ( l O O ) C . P I ( 2 I . C C N V I 2 I , Y D ( 1 0 0 1 , X F ( 5 0 1 . F 0 ( 5 0 1 COMMON/CALDAT/XF .FQ .NP COMMCN/L INF IT / P I , S D A , S O B C O M M C N / L I N E / F R E Q . X . Y F , YD C SET RE SCAN COUNTER KOUNT = 0 C SET READ NO. COUNTER K= 1 C SET DATA NO. COUNTER M = 0 NSAVE=NP 33 DO 47 1=1,NP C TEST FOR UNREASONABLE CAL IB . PTS. I F ( X F ( I ) . G E . 3 2 7 6 7 . 0 1 GOTO 69 IF IFO IK I .GE.14.ZE+06) GOTO 97 IF! FOI K I .LE . 12.4E+06I GOTO 97 M = M+1 FREO(MI=FOIKI X(M)=XFIK> 97 K=K+l 47 CONTINUE C TEST FOR ADEOUATE NO. OF CALIBRATIOM PTS. 57 I F IM -2 .LE .2 I GOTO 69 KERR=NP-M NP=M IFI KERR.NE.OI WRITEI6.35I KERR 35 FORMATI/IOX. • * * * WARNING ' . 1 2 . C INCORRECT CAL IB . POINTS * * *• / ) 36 CALL LSFINPI C GENERATE FITTED POINTS DO 16 1=1.NP YFI I » = P 1 I 2 I * X ( I l + P K II YD( I l=ABS( YFt I l - F R E O M 11 16 C3NTINUE J_ C CONVERT FROM MHZ • TO GAUSS CONVI1 I=P1111*234.868E-06 -CONV12I = P 1 I 2 I * 2 3 4 . 8 6 8 E - 0 6 I SDAG=S0A*234.868E-06 SDBG=S0B*2 34.868E-06 C TEST GOODNESS OF FIT EDIT IF WORSE THAN 0.41 ERROR I F I S D B / P H 2 I . L T . 4 . 0 E - 0 3 I GOTO 56 IF IKOUNT.EO. 31 GOTO 96 K0UNT=K0UNT+1 TEST=-1.0 C LOCATE PT. OF MAX ERROR 00 61 1=1.NP I F (YD (11 .GT .TEST I IMAX=I TEST=YDIIMAXI 61 CONTINUE WRITEI6.64I FREO ( I MAX I 64 FORMAT I / IOX . ' * * * ERRONEOUS CAL IB . PT. ' . E 1 5 . 7 . ' * * * • / ) C REMOVE ERRONEOUS CAL IB . PT• K=0 DO 62 1=1.NP IF 1I.EO.IMAXI GOTO 63 K=K*1 FREOIK) = FREOI I I X( K)=X(I I 63 CONTINUE 62 CONTINUE NP=NP-1 C REFIT DATA GOTO 36 C SET DEFAULT VALUES IN THE EVENT OF AN ERROR 69 CONVII I=3000.0 C0NV(2 l=7.0E-04 96 WRITEI6.70I 70 FORMATI / IOX. ' * * * ERROR CALIBRATION DATA INCDRRECT * * * • / ! WRITEI6.76I 76 FORMAT! /8X, "F IELD* , 13X , 'FREO( OBS) • /I DO 74 I=1,NSAVE 74 WRITEI6.75I X F I I I . F O I I I 75 FORMATI2 I4X.E15.7 I I 56 WRITEI6.73) CONV I 1 I , SD AG, CONV I 2 I. SDBG 73 FORMAT!/ 'CALIBRATION INTERCEPT=•,1PE16.9,•+/-•, C 1 P E 1 3 . 7 , 3 X , ' S L 0 P E = ' , 1 P E 1 6 . 9 , ' + / - ' , I P E 1 3 . 7 / ) RETURN END C C C THIS SUBROUTINE LOCATES THE EXTREMA, FITS THE TOP TO A CUBIC C AND THEN FINDS THE EXACT POSITION OF THE EXTREMA 3Y SOLVING C THE CUBIC SUBROUTINE PEAK!PMAX.NPK,KSCANI DIMENSION FP I IOOI .PK I IOOI .YF I IOOI .YD I 1001,WT! 100) . SI G( 41 C ,B1 ( 4 I , P ( 4 I . PMAX ! 21 ,YMAX I 2 I . YMX I 2 ) . A l ( 4 1 . S I 41 INT EGER*2 I DATAY(2000 I,IOATAXI 2000) COMMCN/DIGDAT/IDATAX,IDATAY COMMON/LINE/PK.FP,YF.YD COMMON/FLAGSII END,NBEG.NPTS.NFIN.NTOT COMMON / SCAN/1 PT.NSTEP, BASE COMMON/PEEK/TRIG,PR EC IS C0MMON/FIT/A1.Bl.P.S.YMAX C SET ARRAY OVER-READ FLAG IFLAG=0 C NSTEP LIMITS NARROWEST LINE TO 0 .75 IN 02NSTEP I NSTEP=NT0T/20 I F I K S C A N . E O . i l NSTEP=NPTS/10 C SET MIN USEABLE PK-PK HT. TO TRIGGER LEVEL DELX=TRIG IF(DELX.GT.4.0E+03I DELX=4.0E+03 C TEST FOR END OF SPECTRUM 202 IF( IPT.LT.NF IN) GOTO 209 IF INPK.E0.2 I RETURN IEND=1 RETURN 209 IF INPK.E0.2 I GOTO 208 MREC=IPT PK ( l »= ICATAY(MREC) IPT=IPT+NSTEP C START PEAK SEARCH ONLY IF SIGNAL > • / - DELX OF BASELINE IF ( ( PK ( l ) .GT .BASE -DELX I.AND.( PK( 1 1 . LT .BASE + DEL X) I C GOTO 202 C RESET NSTEP TO PREVENT OVERSHOOT IPT=IPT-NSTEP IEND=0 C SET DO LOOP PARAMETERS INC=1 208 NPT*2 MREC=IPT PK(1 I = IDATAY(MRECI IPT = IPT+NSTEP MREC=IPT C START PEAK SEARCH PK(2I=IDATAY(MRECI IPT=IPT*NSTEP DO 203 1 = 2 » NPT S,NSTEP NPT=NPT+1 MREC=IPT*I PK (NPT 1= IDATAYIMREC) C COMPARE RELATIVE HTS. OF DATA PTS • TO LOCATE PEAK EXT* E^ A I F ( I (PK I N P T - I I . G E . P M NPT-21 I.AMD.IPK(NPT-1 I .GE.PK(NPT))I C O R . I (PKINPT- I I.LE.PK (NPT-2) I • AM D.IPKI NPT- II .LE.PKIN1PTI I C l l GOTO 207 203 CONTINUE WRITEI6.301I 301 FORMATI/ IOX. ' * * * ERROR FAILURE TO LOCATE PEAK C (ROUTINE PEAK) * * * • ) RETURN C SET SLOPE FLASS 207 IGRAD=l I F ( P K ( 2 I . G T . P K I l ) l IGRAD=-l C SET RESCAN FLAGS I FOR NEGATIVE SORT) NTEST=0 I STEP=2*NSTEP 322 INC=1+ISTEP/100 K=0 C CHECK IF ARRAY LOCATIONS ARE VALID IF! (MREC-2*NSTEP.LT.NBEGI.0R.(MREC + ISTEI>-2*NSTEP.GT.NFtN C l l IFLAG=1 C READ IN DATA AROUND EXTREMA AND FIT TO CUBIC (1ST APPROXI DO 211 1=1.ISTEP,INC NREC=MREC+(l-2*NSTEPI K=K + 1 FP(KI=IDATAX(NRECi PK(K)=IDATAY(NRECI 211 CONTINUE NPT = K CALL PL0F I3 .NPT.S IG.SS . .TRUE. I I F ( P ( 4 I . E O . 0 . 0 l GOTO 37 IREP=0 C SOLV FOR APPROX MAXIMUM PMAX(NPK) = XMAX(P.YMX. IREP.IGRAD) C IF SORT NEGATIVE REFIT USING MORE POINTS IF (YMXd I .NE.0.0 I GOTO 30 ISTEP=2*ISTEP I F INTEST .EO. i l GOTO 30 NTEST=l GOTO 322 C CALCULATE RECORO NO. FOR APPROX MAXIMUM 30 IMAXPT = MREC-2*NSTEP* IF IX(FL0AT(NPTI * (PMAX(NPKI -FP( 1)I C/1FP INPT I - FP I 1M I IREP=IREP + 1 C READ IN AND FIT POINTS SYMETRICALLY C SET NPTS FOR 0.5X ERROR LEVEL: POSENER NPT=NSTEP*2 PPT=FLOAT(NPTI IF I PPT.LT. I TRIG/500.0 1**2) NPT = I F I X (TR IG/500 .01 * *2 425 INC = NPT/100*1 IF INPT.LT.20 I NPT = 20 C CHECK IF ARRAY LOCATIONS ARE VALID IF( ( IMAXPT-NPT/2.LT.NBEG).QR.! IMAXPT+NPr/2.GT.MFI \ l l l C GOTO 11 KPT=0 DO 210 1=1.NPT.IMC KPT=KPT*1 MREC=IMAXPT-NPT/2+1 FP(KPTI=IDATAXIMRECI PK 1KPTI= IDAT AY (MR EC I 210 CONTINUE C REFIT AROUND TOP OF PEAK CALL PL0F I 3 , KPT, SIG.SSt. TRUE. » I F (P (4 ) .EO.O.O I GOTO 37 YMXI1>=FP(11 YMX(21 = FPIKPT) PMAX(NPKI=XMAXtP.YMX.IREP.IGRAOI C REFIT WITH MORE POINTS IF FIT GIVES AMBIGUOUS EXTREMA I F I I R E P . E O . i l GOTO 424 IF( IREP.GE.3) GOTO 425 NPT=NPT*2 GOTO 425 426 WRITEI6.427I 427 FORMAT I/ IOX, ' * * * WARNING AMBIGUITY STILL EXISTS ***•1 424 YMAXI NPK) =YMX 11 I WRITE 16.22241 P I 11 , PI 2 I .YMAXINPK I ,P I3 I .P I 4) .PMAXINPKI 2224 FORMAT (///• CUBIC F IT PI 11 = • , IPE 16.9, 3X, • PI 21 = •. 1PE 1 6. 9, C 3 X , ' Y EXTREMUMIVOLT S I = ' , 0PF6 .0 , / , 12X ,•P I 31=•,1PE16.9, C 3 X . ' P I 4 I = ' . 1PE16.9 .3X, ' X E XTREMUMI VOL TS I = • . OPF 6. 0/1 PREC1S=SQRTI SS/ FLOAT I KPT I I SOS=FLOAT(KPT I * | P K I 1 » * * 2 * P K ( K P T1**2 +PKIK PT/21* *21/3.0 WRITEI6.205I SOS.SS 205 FORMATI10X,•SUM OF SOUARES ' ,G15.5,4X, C 'RESIDUAL SUM OF SOUARE S •, G 15. 5/I IPT=IMAXPT C TEST FOR PEAK CLIPPING IF 11PK IKPT/2) .LT .32768.01 .OR. IPK IKPT/21.GT.0 .011 RETURN 212 WRITEI6.213I 213 FORMATI/ IOX, ' * * * WARNING CLIPPED PEAK * * * • / » RETURN 37 WRITEI6,38I 38 FORMAT I/ IOX , ' * * * ERROR P14I=0 PROBALY CLIPPED PEAK ' , C * * * • / ! RETURN i t WRITEI6.12I 12 FORMAT I/ IOX, ' * * * ERROR INVALID ARRAY LOCATION: SCAN TOD' C . " SMALL * * * • / » RETURN END C C FUNCTION XMAXIP.YMX.IREF, IGRADI LOGICAL* l U.V DIMENSION PI4 I ,YMXI21 C SOLV FOR PEAK MAXIMUM ROOT=PI 3 1 * * 2 - 3 . 0 * P I 4 I * P ( 2 I IF I ROOT.L T.0.0 I GOTO 214 XMAX1= I -P13 ) * SORT IROOT) l/ (3 .0 *P (4 l I XMAX2 = I -P(3 I -S0RT (ROOT I 1/(3.0*P (411 C IGNORE AMBIGUITY FOR FIRST FIT USE MOST PROBABLE VALUE IF I IREP .EO.OI GOTO 90 C TEST FOR AMBIGUITY OF SOLUTION U=l IYMXI I I .LT.XMAX1I .AND. IYMXI2 I .GT.XMAX1I I V=((YMXI1).LT.XMAX2I.AND.IYMXI 2 I.GT.XMAX2II IF ( (U.AND..NOT.VI .OR. I .NOT.U.AND.VI I GOTO 90 C RESET RESCAN FLAG TO FLAG AMBIGUITY AND LIMIT NO. OF RESCANS IREP=IREP+1 WRITEI6.91I 91 FORMAT I/ IOX, ' * * * AMBIGUOUS SOLUTION TO OUADRATIC * * * • / ! I F I I REP .E0 .3 I GOTO 93 RETURN 90 IREP=1 93 YMAX1=0.0 YMAX2 = 0. 0 C CALCULATE HT. OF PEAK EXTREMA DO 92 1=1.4 YMAXI=YMAX1*IXMAX1**( 1-1) I *P( I » YMAX2=YMAX2+(XMAX2** I I - t l l *P I I I 92 CONTINUE C SELECT MOST PROBABLE VALUE FOR EXTREMA: TAKEN AS LARGEST C VALUE FOR +VE GOING PEAK AND SMALLEST VALUE FOR C -VE GOING PEAK ITEST=1 IFIYMAX2.LT.YMAX1I ITEST=-1 I F ( ( I TEST.EO.-I I .AND. I IGRAD.EO. I l l XMAX = XMAX2 IF( I ITEST.EO. l I.AND.I IGRAO.EO. 1) I XMAX = XMAX1 I F ( ( I TEST.EO.- I I•AND. I IGRAO.EO.-I I I XMAX=XMAX1 IF 11 ITEST.EO. l I .AND. I IGRAD. EO.-11 I XMAX = XMAX2 YMX(1I=YMAX1 IFIXMAX. E0.XMAX2I YMX(1I=YMAX2 RE TURN 214 WRITEI6.215I 215 FORMATI/ IOX. ' * * * WARNING NEGATIVE SOUARE ROOT * * * • ! YMXI11=0.0 RETURN END C C C THIS SUBROUTINE LOCATES THE XOVER FROM THE INTERSECTION OF C THE DATA AND THE BASELINE SUBROUTINE XOVERIXOVRI DIMENS ION X I1001,Y(100 I .P I (2 ) .YMAX I2 ) .Y7( 1001.YF( 1001 INTEGER*2 I0ATAXI2000I, IDATAYI2000 I COMMON/OIGDAT/IDATAX.IDATAY COMMON/LINE/Y,X.YF,YD COMMON/SCAN/IPT, NSTEP,BASE C3MM0N/PEEK/TRIG, PRE CIS COMMON/FL AGS/1 END.NBEG.NPTS.NF IN. NT OT C3MM0N/LINF IT/P l . SDA. SOB 31 IXPT=IPT C SET ITERATION AND FILTER FLAG KOUNT=0 C SCAN DATA FOR APPROX. X3VER. USE EXPANDED DATA FIELD TO C ALLOW FOR DIFFERENCES BETWEEN RAW INOISYI DATA AND FITTED C INO NOISE) DATA Yl 1 I =1 DATAY I IXPT) IXPT=I XPT+NSTEP C CHECK FOR END OF SPECTRUM 84 IFI IXPT+NSTEP.GE.NFINI GOTO 106 YI2I=IDATAY( IXPTI S L 0PE=Y ( 2 ) - YU ) C TEST SIGN OF SLOPE 33 IFISLOPEI 78,59. 72 C CHECK IF POINTS STRADDLE XOVER 72 I F I [Y l 11 .LE .BASE I .AND. IY I 2 I .GE .BASE I I GOTO 85 IXPT=IXPT+NSTEP Yl l l = Y ( 2 l GOTO 84 78 I F d Y l l l . G E . B A S E I . A N D . ( Y I 2 I . L E . B A S E ) ) GHTO 85 I XPT=I XPT+NSTEP Y l 1 ) = Yt 2 I GOTO 84 85 KSTART=IXPT-3*NSTEP/2 LFIN=IXPT+NSTE P/2 C READ IN CATA AROUND XOVER 15 K=0 INC=1LFIN-KSTART)/100 +1 DO 86 N=KSTART,LFIN,INC K = K + 1 X|K) = IDATAXINI YIKI = 1 DATAYlNI 86 CONTINUE NPT = K IFIKOUNT.EQ.OI GOTO 861 C APPLY ISIMPLE) DIGITAL FILTERING TO XOVER X COORDINATES C IF ILTER APPLIEO TO X PTS. TO REMOVE GLITCHES. F ILTERING Y C PTS. HAS NO EFFECT ON PRECISION OF FIT :POSENER N=NPT-1 NPT=NPT-2 DO 862 J = 2,N X ( J - l ) = ( X I J - l l+XI JI+XI J + l I 1/3.0 862 CONTINUE C FIT DATA TO STRAIGHT LINE AND FIND INTERCEPT ON X-AXIS 861 CALL LSFINPTI AX=IBASE-P1( II I / P K 2 I I F I KOUNT .E0 . i l GOTO 19 LPT=NPT-1 l l I F ISLOPE.NE.0) GOTO 10 59 WRITEI6.58I 58 FORMAT I/ IOX , ' * * * ERROR TRIGGER SLOPE=0 CLIPPED P E A K ? ' . C * * * • / ) RETURN C CALCULATE RECORD NO. FOR APPROX XOVER 10 I AXPT= I FIX I FLOAT I NPT I* IAX-X I 11 I / I XINPT I-XI 1) 11 +KSTART C REFIT DATA SYMMETRICALLY ABOUT XOVER KOUNT = KOUNT + l LFIN=IAXPT+NSTEP/2 KSTART = IAXPT-NSTEP/2 INC=NSTEP/100 + 1 GOTO 15 19 XOVR=AX C RESET IPT TO PREVENT OVERSHOOT IPT=LFIN-NSTEP WR ITE I6,2225 I P111J ,SDA,P112 I,SDB,XOVR 2225 FORMAT!/'XOVER FIT I NT ERCEPT = •, 1 PE 16 . 9 , •+/-•, 1PE1 3. 7, C3X , ' SLOPE =' , 1 P E 1 6 . 9 , ' * / - ' , 1PE 13.7.3X,•X OVER I VOLTS» = ' C 0 P F 6 . 0 / I RETURN 106 WRITE(6,107I IPT 107 FORMAT I/ IOX, ' * * * WARNING END OF F ILE ENCDUNTES ED AT ' C I P T = ' , I 4 , ' * * * • / ! IEND=1 RETURN END C C C WITH ACKNOWLEDGEMENT TO J.MAYO FOR WRITING THE ORIGINAL C ALGORITHM FOR THIS SUBROUTINE SUBROUTINE READATI KERR.KEND.NPTSI DIMENSION CALX I50 I .CALY I50 I LOGICAL* 1 B U F F E R I 3 . 5 l l . R Y T E 5 ( 5 . 5 l l . D A T E ( 6 I . D D A T F ( 4 l IN TEGER*2 IDATAXI2000I, IOATAYI20001 , 1HUFI 2,126>.LENGTH INTEGER*2 TEST/0/.DD/2/.XV/32677/ INTEGER FRE0I51I ,I DATE/0/ EQUIVALENCE I 8YTE5 11 I . IBUF I 11 I EQUIVALENCE I BUFFER! l l .FREOI111 EQUIVALENCE! IDATE.ODATEI 11 I CDMMON/DIGDAT/ I DA TAX.I DATAY COMMON/CALDAT/CALX.CALY.NP M = 0 LSTART=0 KEND=0 LINE=0 NREC=0 2 LA ST=LINE C LOAD BINARY DATA INTO INTEGER*2 ARRAY 1 CALL READ! I BUF ,LENGTH,0 . L INE .4 , £100 ) C REJECT LEADER RECORD OF ZEROES IFtLSTART.EO.Oi LSTART=LINE/1000 IF ILINE.EO.10001 G0T3 2 C TEST FOR START OF CALIBRATION TABLE I F I ! I BUF! 1. 11.EO .TEST I .AND . I I BUF I 2. l» . EO. TEST I C.AND.(L INE.NE.1000)1 K EN D= I I F I K E N 9 . E 0 . i l RETURN I FULAST .EQ .O I . AND . I L INE .NE . i l ) G3TD 201 IFILINE.NE.LAST+10031 GOTO 200 C TEST FOR ERRONEOUS RECORDS 201 IFILENGTH.NE.252 I GOTO 560 GOTO 620 560 XLINE = L I N E * . 0 O l WRITEI6.12001 XLINE 1200 FORMAT! " * * * ERROR IN LINE NUMBER ' . F 4 . 0 I KERR=1 GOTO 2 620 N= L ENGTH/4 C SEPARATE X AND Y POINTS AMD LOAD INTO INTEGE R*2 ARRAY DO 600 1=1.N C REVERSE X POINTS IDATAX(M+II=IBUF(1,I)+XV C DIVIDE BT TWO TO ALLOW USE OF 1*2 VARIABLES IDATAY IM + 11= IBUF 12. I l/DD C ANTI SYMETTRI SIZE DATA IFI I OATAYI M M ) .L T.OI I DAT AY I M* I I = I OATAYI MH l * XV 600 CONTINUE NREC=NREC*l M=M+63 GOTO 2 200 CONTINUE N=LENGTH/5 IFIN.EO.OI GOTO 210 C TEST FOR ERRONEOUS CALIBRATION TABLE IFILENGTH.NE.N*5I G3T3 120 03 220 1=1.N C SUBBROUTINES TO SEPARATE X AND FREQUENCY POINTS C (SEE U.B.C. CHARACTERI CALL MOV EC ( 3.BYTE5I 1. I I. BUFFER! I .1 I ) CALL MOV EC(2.BYT ESI 4.I I. I BUF ! I . I I) 220 CONTINUE C CONVFRT H.P BINARY TO DECIMAL CALL BINARY!N.BUFFER) FTEST=-1 NP = 0 C REMOVE ERRONEOUS CALIBRATION POINTS OO 230 1=1,N I F1FREQ I I I . EQ.0 ) GOTO 240 I F IFREOI I I .EO.FTEST I GOTO 240 NP=NP+1 IF INP.GT.50 I GOTO 243 CALX (NP I= IBUF (1 ,1 l *XV C RESTORE HIGH ORDER BIT TO FREQUENCY CALYINPI=(FREQ(I 1*1.OE+061*10.0 FTEST=FRE0(I I 240 CONTINUE 230 CONTINUE GOTO 241 243 WRITEI6.242I 242 FORMAT I/ IOX , ' * * * WARNING MORE THAN FIFTY CAL I B . P T S . ' C * * * ' / ) C READ DATE ETC. 241 CALL READ IDATE . LENGTH,0 , L INE . 4 , £100 ) LFIN=LINE/1000 DDATEI4I = DATE11 I IDAY=IDATE DDATEI4I=DATE(2) MONTH=IDATE DDATEI4 l *DATE(3 I IYEAR=IDATE DDATEI 4(=DATE( 61 I SPEC« IDATE NPTS=NREC*63 WRIT El 6 ,130) IDAY,MONTH,I YEAR,I SPEC.NPTS,LSTART,LFIN 130 F O R M A T I 5 X . I 2 , ' / • , 1 2 , ' / • , I 2 . 2 X . ' S P E C N O . ' , I 3 , 2 X , C 'NPTS=* , 14 ,10X , * L INE NOS. ' , I 3 . ' TO ' , 131 LAST«L INE * 1000 NREC=0 RETURN 100 CONTINUE KEND-2 RETURN 210 WRITEI6,16001 1600 FORMAT I'— * * * ERROR LENGTH OF FREQUENCY TABLE IS ZERO. KERR=1 RETURN 120 WRITEI6.121I 121 FORMAT I/ IOX , ' * * * ERROR IN CALIBRATION TABLE * * * • / ) KEND=1 RETURN ENO C C C THIS IS A SIMPLE LINEAR LEAST SQUARES FITTING PROGRAM SUBROUTINE LSF IN I IMPLICIT REAL*B IA-H.O-Z I REAL XI 1001 , Yl 1001 , PI I 21 ,YF I IOC I .YD 11001 , SDA, SOB RE AL*8 P11I2) COMMON/LINE/Y,X,YF,YD COMMON/LINFIT/Pl,SDA.SDB 3 SYY=O.OD+00 SXX = O.OD+00 SXY=0.0D+00 SX = O.OD-00 SY=O.OD + 00 DO 10 I = 1.N A= DBL EI X I I ) ) B=DBLEIYI I ) ) SX=SX+A SY=SY*B SXX=SXX+A*A SYY=SYY*B*B SXY=SXY*A*B 10 CONTINUE P= DFLOATIN) XAVE = SX/P VA VE = SY/P CXY=IP*SXY-SX*SYl/P CXX = IP*SXX-SX*SX1/P CYY=(P*SYY-SY*SYI/P P 1 U 2 I = CXY/CXX P i l l l l = ( SY-P11I2 l * SX I/P S D S = I C Y Y - P 1 1 I 2 I * P 1 1 I 2 I * C X X ) / I P - 2 . 0 D « - 0 0 I SDB=SNGL(OSORTISDS/CXXII SDA = SNGL(DSORTISDS/l 1. OD*-00/P+XA VE* XAVE /CXX I I I P l ( l ) = S N G L l P l l l l I I P l ( 2 l = S N G L ( P l H 2 l l RETURN END C c C MODIFIED VERSION OF THE ORTHOGONAL POLYNOMIALS FITTING C PROGRAM FROM THE U.B.C COMPUTING CENTRE ISEE U.B.C. CURVE! SUBROUTINE PLOFI K, M. S IGMA, SS ,LK I DIMENS I ON XI100) .Y I IOO I ,A I 4 I .B I 4 I . S I 4 I . P I 4 I .YMAX12 I DIMENSION SIGMAI4I.YFI IO0I.YDI 1001 .WTI 100) RE AL*8 PP11000.2 I,DD.BD,ALPHA.BETA LOGICAL LK .LL COMMON/LINE/Y.X. YF.YD COMMON/FIT/A.B.P.S.YMAX IF IK.LE.M-21 GO TO 8 WRITEI6.9) 9 FORMAT!' K > M - 2 ' l STOP 8 DO 87 1=1,M 87 WTIII=1.0 88 CONTINUE 89 W=0.0 DO 1 L=l ,M PPIL.11=0.0 W=W+WT1LI I PP IL . 21=1.0 SS=0.0 BI 11 = 0.0 LL = LK KSAVE=K IP2= 1 DO 4 1=1,K IP1=IP2 0 IP2=3-IP1 DD=0.D0 DO 22 L=l.M 22 DD = DD+Y I L )* PP IL, IP2)*WT(L) SI I )=OD/W [ F U . E Q . l ) GO TO 24 64 S IGMAI I -1 I = S I I I ,X ,Y ,M,A,B.S ,WTI I F I I . EO .2 ) GO TO 24 IF ILK) GO TO 24 IF I LL (GO TO 51 I F I S IGMA I1 -1 ) . LT .S IGMA I1 -2 ) IGO TO 24 LL = .TRUE. KSAVE = I-2 GO TO 24 51 IFISIGMAI I - l ) .GE.0.1*S IGMAI 1-21 I GO TO 24 LL=.FALSE. KSAVE=K 24 DD=O.DO DO 21 L = l,M 21 CD=DD*X(L ) *PP (L , I P 2 l * P P I L , I P 2 I*WTILI A l I 1=DD/W DO 2 L=1,M 2 P P I L , I P l ) = ( X ( L l - A I I I l * P P ( L . I P 2 I - B I I ) * P P ( L . I P II DD=O.D0 OO 23 L=l ,M 23 0D=DD*PP (L . I P I I * PP ( L , I P1 I *WT I L I W1=DD IF I l + l .GT.K IGO TO 5 BI1+1)=W1/W 5 W=W1 4 CONTINUE DD'O.DO DO 25 L - l . M 25 DD=DD+Y(LI *PP<L. IP l l *WTIL I S( K * l l =OD/W SIGMA<KI=SI IK*1.X,Y.M,A,B,S,WTI I F IK .E0 .1 IGO TO 49 IF ILL I GO TO 49 IF IS IGMAIK) .GE.S IGMAI K-1) )KSAVE = K-1 49 K=KSAVE C COMPUTE FITTED Y-VALUES FROM A.B.S USING RECURSION. DO 32 1=1,M BB=S1K*1) B l = S(KI + I X l I l -A IK I l *BB IF IK.EQ.1 IGO TO 33 KM l = K - l 31 B2=S(KM1I+(XII l - A I KM l ) )*B1-BI KM t+11*88 BB=B1 B1 = B2 KM1=KM1-1 IF IKMl.GT.OIGO TO 31 33 Y F ( I ) = B l YD 1 1 l = Y I I l - Y F I 1 1 W=YD(I l *YDI I I 32 SS = W+SS C COMPUTE COEFFICIENTS OF X * * J ' S KPl=K+l PP (1 .1 )=1 .0 PPI 1,2I=-AI II PP (2 ,2 )=1 .0 DO 199 1 = 1,KP1 199 P I I I =0 .0 P I l l = S t 1 l + S ( 2 > * I - A I I I I P I 2 I = SI 21 I F I K . L E . l I G O TO 212 J=3 IP2 = l 201 IP l= IP2 IP2 = 3 - I P l J M l = J - l JM2=J-2 ALPHA=-AI J - l ) BETA= -B I J - l I I F I J . E 0 . 3 I GO TO 203 DO 202 I =2 , JM2 202 P P I I , I P l l = P P I I - l , I P 2 l * A L P H A * P P 1 1 .1P2I *BETA*PPI I.I PI I 203 P P I 1 . I P 1 l = A L P H A * P P I l , I P 2 I + B E T A * P P I I . I P l ) PPI JMI, IP1)=PP( JM2.IP2 ) + ALPHA*PPI JM1 ,1 P2 I P P ( J . I P l I = PP IJM1, IP2) BD=S(Jl 03 208 1=1.J DD=P( II 208 PI I)= DD+BD*P PI I. I PI I J=J + 1 I F I J . L E . K P l ) G 0 TO 201 212 RETURN END C C C FUNCTION SUBROUTINE FOR PLOF FUNCTION SI I I .X.Y.M.A.B.S.WTI DIMENSION XI l l . Y l l l . A l II .BI l l . S I I I.WTI1) SI=0.0 DO 100 J= l .M BB=S(II B l = S I I - l l * I X I J I - A ( I - l l ) * B B IFI I .E0.2I GO TO 99 IM2=I-2 200 B2=S ( IM2 I+ IX I J I -A I IM2 I ) *B1 -B I IM2 * l I * BB BB=B !• B1=B2 IM2=IM2-1 IF I IM2.GT.0 I GO TO 200 99 BB=Y I J I -B l 100 SI = S I *BB*BB*MT(J I S I=SI/(M-I) RETURN END C C C ACKNOWLEDGEMENT TO J.MAYO FOR SUPPLYING THIS SUBROUTINE C ASSEMBLER SUBROUTINE TO CONVERT H.P. BCD TO DECIMAL CSECT CONVERT HP TO BINARY ENTER R12 LINKAGE OVERHEAD L R3.0IR1I GET ADDRESS OF LENGTH L R3.0IR3I GET VALUE OF LENGTH LR R5. R3 IN IT IAL IZE LOOP COUNTER LA R6.3 IN IT IAL IZE INPUT FIELD LENGTH LA R7.4 IN IT IALIZE OUTPUT FIELD LENGTH MR R2.R6 NUMBER OF BYTES IN 8UFFFR L R2.4IR1) GET ADDRESS OF BUFFER LA R4.0I R3.R2) IN IT IAL IZE INPUT FIELD POINTER LR R3.R5 GET LENGTH OF BUFFER SLA R3. 2 OUTPUT BUFFER LENGTH IN BYTES DECODE EOU * DECODE NEXT FIELD SR R 4 . R 6 POINT TO INPUT FIELD SR R3.R7 OUTPUT FIELD OFFSET MVC P A C K E D + 4 ( 3 ) » 0 ( R 4 I MOVE TO WORD BOUNDRY MVI PACKED+T.X 'CO' INSERT A •-• IN SIGN ZONE TR PACKED+4I31.HP TRANSLATE TO BCD L RO.PACKED+4 LOAD INPUT FIELD SRL R0 .4 RIGHT JUSTIFY NUMBER ST RO,PACKE0*4 STORE IN WORK AREA CVB RO,PACKED CONVERT T3 BINARY ST R0,0(R3,R2I STORE IN OUTPUT FIELD BCT R5.DECOOE TEST IF DONE EXIT 0,,MF = FS RETURN CODE 0 PACKED DC 0 ' 0 ' * . 0 . 1 . 2 . 3 . 4 . 5 . 6 . 7 . 8 . 9 . A . 8 . C . D . E . F HP TO BCO HP DC X ' 00010203FFFF0405FFFFFFFF06070809» 0. DC X ' 101112131F I F14151F1F I F1F16171819 ' 1. DC X ' 202122232F2F24252F2F2F2F26272829 ' 2. DC X«303132333F3F34353F3F3F3F36373839« 3. DC X ' FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF> 4. DC X ' F FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF• 5. DC X ' 404142434F4F44454F4F4F4F46474849 ' 6. DC X•505152535F5F54555F5F5F5F56575859 ' 7. DC X ' F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F ' 8 . DC X ' FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF< 9. DC X ' F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F ' A . DC X ' F FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF• B. DC X ' 606162636F6F64656F6F6F6F66676869 ' C. DC X* 707172737F7F74757F7F7F7F76777879 ' D. DC X ' 808182838F8F84858F8F8F8F86878889 ' E. DC X ' 909192939F9F94959F9F9F9F96979899 ' F. . 0 . 1 . 2 . 3 . 4 . 5 . 6 . 7 . 8 . 9 . A . B . C . D . E . F EJECT RO EOU 0 RI EOU I R2 ECU 2 R3 EOU 3 R4 EOU 4 R5 EOU 5 R6 EOU 6 R7 EOU 7 R12 EOU 12 END BINARY ro l -128-APPENDIX I I Raw Line-Width e t c . Data Line-Width, vs. Temperature P l o t s T-.vs .n/T P l o t s -129-Key to Tables T(C) Temperature °C T(K) Temperature K VISC V i s c o s i t y of Toluene i n cP VIS/T Viscosity/Temperature ALPHA" a" term (see s e c t i o n 2.5) TAUC x 2 i n pS 22 3 AVOL*10**22 Apparent Volume x 10 cm (assuming simple hydrodynamic model a p p l i e s ) . +/- Denotes ± ; e r r o r s MW Freq. Microwave Frequency - 1 3 0 -*3Cu ( I D b i s idimethyldithiocarbainate> i n Toluene Temp. Hidth P o s i t i o n -H8 'Fr.eq. 60. 3 5.210 7. 310 14.960 25. 560 3276.320 3192.950 3111.430 3033.960 9.0325632 9.0258150 9.0326242 9.0326529 Temp. Width P o s i t i o n AH Free* -32... 1 3.600 3273.940 9.031820 4.540 3191^:^20 9.1031^98 12.460 303^ 07 O S . 031955 55.6 4.620 3276.000 9.0326405 -25.9 3.530 3274.950 9.035891 6.450 3192.810 9.0326147 4.380 3193.090 9.C35907 12.710 3112.230 9.0325813 6.600 3113.380 9.035924 22.680 3035.630 9.0325594 11.010 3036.290 9.035950 51.6 4.510 3275.680 9.0322371 -22.1 3.570 3274.690 9.036189 6.040 3192.460 9.0322399 4.310 3192.970 9*036141 11.080 3111.880 9.0322371 6.400 3113.590 9.036140 21.600 3033.970 9.0322905 10.500 3035.150 9.036140 50.7 4.240 3275.570 9.0326319 -19.5 3.550 3274.250 034940 5.790 3192.650 9.0325564 4*340 3192.660 S.x^35086 10.850 3112.260 9.0325708 : 6.380 3113.190 9.035162 19.160 3032.570 9.0325527 10.260 3035.850 9.035247 45.6 3.990 3275.210 9.0323887 5.37 0 3192.390 9.0323610 10.120 3112.050 9.0323591 15.530 3033.500 9.0323677 -16.4 3.560 327 4.280 9.035899 4.300 3192.840 9.035898 6. 190 < 3113.570 9.035897 9.480 3036*690 9.035885 44. 8 3. 910 3276. 300 9.0357628 5. 130 3193.590 9.0357685 8.720 3113.230 9.0357714 18.180 3035.540 9.0357714 -11.0 3.650 3273.830 9. 035571 4.340 3192.540 9.035622 5.960 3113.350 9.C35617 9.290 3036.500 9.035641 42.4 4.100 3274.900 9.0322437 5.120 3192.140 9.0321960 8.820 3111.810 9.0321293 15.740 3033.640 9.0320358 -5.7 3.710 3273.540 9. 035814 4.350 3192.600 9. 035800 5.960 3113.770 9.035797 8.110 3037.130 9.035795 42.2 3.830 3274.920 9.0323868 5.020 3192.270 9.0323896 8.370 3111.880 9.0323925 16.570 3035.380 9.0324020 -0. 1 4.020 3272.390 9.035575 4.640 3192.150 9.035582 5.900 3113. 950 9.C35586 8.060 3037.750 9. 035605 37.4 3.680 3275.920 9.0362940 4.680 3193.530 9.0362911 7.940 3113.360 9.0362873 13.270 3035.720 9.0362835 4.9 4.010 3271.680 9. 032516 4.550 3191.230 9.032164 5.850 3112.820 9.032513 8.160 3036.480 9.032515 35.0 3.630 3275.660 9. 0360041 4.620 3193.390 9.0360022 7.300 3113.300 9.0360107 13.100 3035.860 9.0360146 9.0 4.370 3271. 440 9. 035601 4.970 3191.800 9.035594 6.000 3114. 100 9.035594 7.8 50 3038.430 9.0355 96 - 1 3 1 -ft3cu ( H ) b i s fdimethyldithiocarbamafce) •^^i;it^•:T6iA^ege;•:iG:b!R^^: Temp, width P o s i t i o n MB Freq, Temp.;8idtii Position MH Freg 14.6 4.220 3270.920 9.0324602 4.780 3190.960 9.0324459 5.§70 3112.980 9.0324326 7.610 3037.04 0 9.0324202 50.0 5.620 3269.190 9.034877 6.060 3190.760 9.034871 6.780 3114.360 9.034873 7.630, 3040. 050 9i 034704 20.6 4.790 3270.630 9.0352030 5.280 3191.470 9.0352011 6.360 3114.210 9.0352001 7.830 3038.920 9.0351610 55. 0 5.950 3268.780 9.034791 6.350 3190.240 9.034790 7.240 3114.480 9*034785 7.5790 304«.24O9i 034786 25.2 4.550 3270.020 9.0321360 5.130 3190.560 9.0321445 6.080 3113.110 9.0321312 8.100 3037.630 9.0321178 55. 5 6.03 0 3 267.64 0 9.031674 6.400 3189.300 9.031663 7.300 3111.500 9.031662 7.1800 3039. 100<i-&* 031662 30.6 5.230 3269.980 9.0350885 5.700 3191.590 9.0350790 6.500 3114.350 9.0350742 7.870 3039.390 9.0350666 59.4 6.020 3268.450 9.034720 6.640 3190.640 9.034713 7.210 3114.550 9.034707 8.810 3040. 350 9. 034702 34.1 4.940 3269.290 9.0319319 5.470 3190.320 9.0319386 6.590 3113.180 9.0319433 7.620 3038.120 9.0319462 64.4 6.4 70 3268.150 9.034616 6.870 3190.370 9.034622 7.670 3114.600 9.034629 9,130 3040.510 9.034635 35. £ 4.960 3270.280 9,0350952 5.590 3191.040 9.0350838 6.400 3114.180 9.0350885 7.700 3039.180 9.0350990 39.5 5.640 3269.160 9.0349646 6.210 3190.910 9.0349607 6.830 3114.430 9.0349588 8.190 3039.940 9.0349541 40.6 5.180 3269.950 9.0350456 5.770 3191.170 9.0350409 6.540 3114.360 9.0350370 7.610 3039.470 9.0350332 44.8 5.570 3268.430 9.0318460 5. 910 3189. 970 9. 0318336 6.850 3113.350 9.0318222 8.390 3038.690 9.0317669 45.0 5.340 3269.570 9.0349646 5.900 3191.040 9,0349607 6.680 3114.420 9.0349588 7.990 3039.720 9.0349588 -133-***** CUC11 )-63 BIS DIMETHYL DITHIOCARBAMATE IN TOLUENE ***** T(C( -60.3 -55. 6 -51.6 -50.7 -45.6 -44.8 -42.4 -42.2 -37.4 -3 5.0 -32. 1 -2 5.9 -22. 1 -19.5 -16. 4 -11.0 -5.7 -0. 1 4.9 9.0 1 4.6 20.6 2 5.2 30.6 34. I 3 5.6 39.5 40.6 44. 8 45.0 5 0.0 55.0 55.5 59.4 64.4 TIKI 212.9 217.6 221.6 222. 5 22 7.6 228.4 230.8 231.0 235.8 238.2 241.1 247.3 251.1 253.7 2 56. 8 262.2 267.5 273.1 278.1 282. 2 287.8 293.8 298.4 303.8 307.3 308. 8 312.7 313. 8 VIS/T 0. 1377E-.01 0. 1161E-01 0.1012E-01 0.9825E-02 0.8351E-02 0.8148E-02 0. 7581E-02 0.7536E-02 0.6567E-02 0.6147E-02 0. 5690E-02 0.4864E-02 0.444 IE-02 0. 4181E-02 0.3899E-02 0. 3471 E- 02 0.3114E-02 0.2792E-02 0.2544E-02 0.2365E-02 0.2148E-02 0. 1946E-02 0. 18 10E-02 0. 1668E-02 0. 1584E-02 0. 1550E-02 0. 1467E-02 0. 1445E-02 v i s e 2.9322 2.5261 2.2431 2. 1860 1.9006 1. 8611 1.7497 1.7409 1.5484 1.4643 1.3719 1.2029 1.1151 1.0608 1.0014 0.9101 0. 8330 0.762 5 0.7076 0. 6673 0.6181 0. 5718 0.5402 0.5067 0.4869 0.4787 0.4587 0.4533 ALPHA" I . 113+/-1.455 0.987+/-0.666 0.957+/-0.257 0.969+/-0.210 1.045*7-0.199 1.062+/-0.216 1.123*7-0.252 1.131+/-0.257 1 .285+/-0.275 1.366+/-0.261 1.473+/-O.232 1.708+/-0.164 1 .848+/-0. 129 1.944+/-0. 102 2.062+/-0.088 2.276-+/-0.077 2.481+/-0.093 2.701+/-0.105 2.910*/-0. 124 3.092+/-0.141 3.337-+/-0.158 3.61 5+/-0.148 3.822*7-0.121 4.084+/-0.087 4.249+/-0.053 4.318+/-0.038 4.508+/-0.020 4.558*7-0.027 TAUC(PS) 137.0+/-13.6 118.5+/- 6.2 104.7+/- 2.4 101.8+/- 1.9 86.6+/- 1.8 84.5+/- 2.0 78.3+/- 2.3 77.7+/- 2.3 66.7+/- 2.4 61.7+/- 2.3 56.2+/- 2.0 46.1+/- 1.4 41.0+/- l . l 37.8+/- 0.8 34.4+/- 0.7 29.4+/- 0.6 25.6+/- 0.7 22.5+/- 0.7 2 0.4+/- 0.8 18.8+/- 0.9 17.1 + /- 1.0 15.5+/- 0.9 14.4+/- 0.7 13.1+/- 0.5 12.3+/- 0.3 12.0+/- 0.2 11.2+/- 0. I 11.0+/- 0.1 AVQL*10**22 1.373 1.409 1.428 1.431 1.432 - 1.431 1.426 1.424 1.401 1.385 1.36 4 1.308 1.2 73 1.249 1.219 I. 171 1.137 1.113 1.105 1.098 1. 102 1.099 1.100 1.08 4 1.074 1 .072 1.055 1.052 318. 0 0. 1364E- 02 0.4337 4.762+/-0.055 10.2+/- 0.3 1.034 318.2 0. 1360E- 02 0.4328 4.769+/-0.068 10.2+/- 0.4 1.035 323.2 0. 12 73E-02 0.4114 5.003+/-0.079 9.6+/- 0.4 1.039 328.2 0. 1194E- 02 0.3918 5.21 7+7-0. 055 9.5*7- 3.3 1.098 328.7 0. U86E- 02 0. 3899 5.241+/-0.046 9.5+/- 0.2 1.107 332.6 0. 1130E-02 0.3758 5.386+/-0.059 10.1+/- 0.3 1.230 337.6 0. 1064E- 02 0.3591 5.572+/-0.239 11.8+ /- 1.3 1.538 CU C11) -63 BIS DIMETHYL DI THIQCflRBRMFITE IN TOLUENE X X X X X X X X X I 1 1 1 1 1 1 1 • n o 1 1 1 0 . J 1 0 . 0 2 0 . 0 3 0 . 0 4 0 . 0 5 0 . 0 6 , 0 . J 7 0 . 0 8 0 . 0 9 VIS/T (X10-1') -1 35 -6 3 C u ( i i ) - b i s l d i e t h v l d l t h i o c a r b a a a t e ) i n Toluene Temp. Width P o s i t i o n MH Freg. -51.5 5.201 3276.682 9.0374490 7.341 3193.943 9.0375036 15.352 3114.092 9.0375530 31.048 3034.434 9.0 376234 -44.9 4.442 3276.275 9.0376380 6.293 3193.811 9.0376324 12.883 3113.865 9.0376156 23.735 3036.320 9.0375688 -40.9 4.177 3276.006 9.0377592 5.843 3193.771 9.0377390 11.797 3113.479 9.0376970 20.800 3036.256 9.0376700 -36.4 ; 3.848 3275.588 9.0375472 5. 279 3193.502 9.0375459 10.028 3113.777 9.0375473 18.362 3036.740 9.0375484 Temp. .Width P o s i t i o n MM Freq 29. 1 3.584 3271.117 9. 037693 4.181 3191.838 9. 036914 5.408 3114.604 9. 036905 7.560 3039.242 9. 036895 39.2 3.750 3270.396 9. 0367 89 4.345 3191.689 9. 036786 5.466 3114.852 9. 036779 7.286 3039.934 9. 036772 49.2 3.974 3269.750 9. 036680 4.617 3191.381 9. 036667 5.558 3114.992 9. 036658 7.027 3040.441 9. 036652 57.6 4.229 3269.121 9. 036561 4.805 3191.178 9. 036549 5.659 3115.051 9. 036543 7.133 304 0.900 9. 036539 -31.2 3.652 3275.244 9.0375185 4.888 3193.334 9.0375228 8.850 3113.631 9.0375202 16.311 3036.326 9.0375152 -20.2 3.355 3274.424 9.0374260 4.338 3193.051 9.0374260 7.377 3113.785 9.0373928 12.800 3036.639 9.0373878 -10.4 3.276 3273.770 9.0373543 4. 101 3192.787 9. 0373581 6.528 3113.928 9.0373536 10.856 3036.936 9.0372601 -0.2 3.265 3273.074 9.0372761 4.016 3192.572 9.0372708 5.919 3114.057 9.0372657 9.687 3037.756 9.0372601 9.0 3.283 3272.457 9.0371300 3.987 3192.320 9.0371630 5.648 3114.275 9. 0371564 8.408 3038.268 9.0371497 19.7 3.464 3271.725 9.0370532 4.057 3192.086 9.0370517 5.539 3114.467 9.0370414 7.754 3038.855 9.0370318 - 9 £ l -***** CUMII-63 BIS DIETHYL T<C» T(K) 44.9 228.3 40.9 232.3 36.4 236.8 31.2 242.0 20.2 253.0 10.4 262.8 -0.2 273.0 9.0 282.2 19.7 292.9 29.1 302.3 39.2 312.4 49.2 322.4 57.6 330.8 VIS/T 0.8173E-02 0.7255E-02 0.6387E-02 0.5558E-02 0. 4249E-02 0.3428E-02 0.2797E-02 0.2365E-02 0. 19 75 E-02 0. 1706E-02 0. 1473E-02 0.1286E-02 0. 1155E-02 v i s e 1.8659 1.6852 1.5125 1.3451 1. 0749 0.9008 0.7637 0.6673 0.5784 0.5157 0. 46 02 0.4147 0. 3822 IN TOLUENE ***** ALPHA" 0.444*/-0.711 0.613+/-0.523 0.749*7-0.321 0.876+7-0.142 1.120+/-0.153 1.391*/-0.1*7 1.690+/-0.086 1.935+/-0.054 2 .198+/-0.086 2.430+/-0.118 2.730+/-0.081 3.080+/-0.079 3.361+/-0.132 TAUCIPS1 AV0L*10**22 127.7*/- 6.6 2.158 110.6+ /- 4.8 2.104 95.3*/- 2.9 2.060 81.5+/- 1.3 2.025 60.9*/- I .3 1.978 47.5+/- 1.2 1.915 36.7*/- 0.7 1.810 29.4+/- 0.4 1.717 23.5+/- 0.6 1.6*5 20.0+/- 3.8 1.623 17.0+/- 0.5 1.594 14.3+/- 0.5 1.540 13.1+/- 0.7 1.563 o ry CUUI)-63 BIS DIETHYL DITHIOCARBAMATE IN TOLUENE O H ZD a -CE-X X X X X x>< 1 1 1 1 1 1 1 1 1 I I 1 0.0 0.01 0.02 0.J3 0.04 0.J5 0.06 0.07 0.08 0.09 0.1 0.11 0. VIS/T (XIO-1 ) -139-* 3 C u ( I I ) b i s f d i - n - p r o p y l d i t h i o G a r b a m a t e ) i n Toluene Temp, Width P o s i t i o n HW Freg., 41.0 5.685 3278.656 9.0434765 8.230 3196.135 9.0434795 16.415 3116.873 9.0434939 31. 3 3038.3 9.0434991 35. 5 4.978 3278.256 9.0434745 7.072 3196.064 9.0434776 14.266 3116.889 9.0434656 27.351 3040.768 9.0434525 31. 1 4. 569 3277.893 9.0434590 6.343 3195.881 9.0431440 12.319 3116.754 9.0433336 24.556 3039.666 9.0433516 26.2 4. 207 3277. 434 9.0432950 5.839 3195.658 9.0432882 11.000 3116.584 9.0432858 21.797 3040.072 9.0432843 20.0 3.858 3276.973 9.0432213 5.281 3195.473 9.0432151 10.059 3116.174 9.0432100 20.280 3039.666 9.0432050 10.7 3.499 3276.326 9.0421034 4.675 3195.178 9.0430927 8.392 3116.121 9.0430885 14.218 3039.746 9.0430823 0.7 3.298 3275.596 9.0429763 4.247 3194.881 9.0429593 7.197 3116.268 9.0429454 13.692 3040.299 9.0429351 10.0 3.235 3274.859 9.0428189 4.082 3194.584 9,0427997 6.495 3116.316 9.0427862 10.944 3040.447 9.0427745 19.5 3.173 3274.152 9.0426557 3.937 3194.301 9.0426411 6.125 3116.484 9.0426281 9.622 3040.635 9.0426168 30.2 3.190 3273.395 9.0424845 3.892 3194.012 9.0424582 5.751 3116.672 9.0424456 8.654 3041.217 9.0424379 Temp. Width P o s i t i o n MW Freg 39.3 3.244 3272.766 9. 042312 3.908 3193.762 9. 042300 5.527 3116.844 9. 042292 8. 378 3041.750 9. 042283 49.7 3.354 3272.029 9. 042205 3.960 3193.537 9. 042191 5.390 3117.031 9. 042180 7.743 3042.342 9. 042169 60.7 3.500 3271.230 9. 042031 4.089 3193.270 9. 042016 5.400 3117.195 9. 041999 7.402 304 2.965 9. 041985 81.2 3.862 3269.814 9. 041749 4.420 3192.715 9. 041706 5.406 3117.441 9. 041677 7.114 3043.967 9. 041654 3-1 -50.0 -40.0 -30.0 -20.0 -10.0 0.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90 TEMP C -142-UJ h-(X 2; cr co cr u o >-O ce a . cn CD co tn 1 X X — r ~ 0'0> 002 r~ Q'081 —I— •031 — r ~ -021 O'PGT onHi n-oe 0-03 c o s O'O -143-6 3 C u ( i D b i s ( d i - n - b u t y l d i t h i o c a r b a m a t e ) i n Toluene Temp. Width P o s i t i o n MW Freq. Temp. Width P o s i t i o n MW Freg. -30.2 5.792 3277.980 9. 0440645 51.0 3.416 3272.186 9. 043130 8. 148 3196.154 9. 0440805 4.159 3193.553 9. 0431 17 16.379 3117.717 9. 0441023 5.893 3117.002 9. 043106 28. 970 3042.590 9. 0441246 8.929 304 2.342 9. 043097 -24. 8 5. 255 3277.566 9. 0441026 60,9 3.4 90 3271.3 93 9. 043000 7.313 3195.977 9. 0441036 4.160 3193.2 97 9. 042981 14.321 3117.141 9. 0441065 5.801 3117.096 9. 042967 27.411 3041.904 9. 0441070 8.312 3042. 816 9. 042952 -20.4 4.865 3277.193 9. 0440614 80.0 3.615 3270.203 9. 042676 6.645 3195.727 9. 0440527 4.2 97 3192.818 9. 042645 12.916 3116.545 9. 0440500 5.617 3117.285 9. 042615 23. 36 2 3040.248 9. 0440467 7.873 3043.826 9. 042594 -15.0 4.510 3276.840 9.0439865 6.152 3195.602 9.0439819 11.783 3116.646 9.0439805 21.655 3040.371 9.0439752 -9.8 4.285 3276.451 9.0439235 5.715 3195.363 9.0439248 10.591 3116.570 9.0439269 19.005 3040.430 9.0439283 -0.3 3.888 3275.809 9.0438501 5.116 3195.115 9.0438460 9.109 3116.566 9.0438378 16.091 3041.031 9.0438318 9.3 3.628 3275.090 9.0437384 4.702 3194.809 9.0437174 7.868 3116.520 9.0437064 14. 106 3041.238 9.0437028 19.0 3.462 3274.465 9.0436255 4.450 3194.541 9.0436121 7.199 3116.680 9.0436020 12.130 3040.842 9.0435945 29.5 3.408 3273.674 9.0434862 4.275 3194.176 9.0434685 6.615 3116.707 9.0434530 10.871 3041.410 9.0434409 40.6 3.387 3272.900 9.0433248 4.129 3193.889 9.0433060 6.232 3116.793 9.0432897 9.777 3041.895 9.0432772 * * * * * C U I I I I - 6 3 BIS DIBUTYL DITHIOCARBAMATE IN TOLUENE * * * * * T<C» T( Kl V I S / T v i s e ALPHA" TAUCIPSI A V 0 L * 1 0 * * 2 2 - 3 0 - 2 2 4 3 . 0 0 . 5 4 1 7 E - 02 1.3163 1 .C87+/ -1 .083 1 5 5 . 5 + / - 10 .2 3.962 - 2 4 . 8 2 4 8 . 4 0 . 4 7 3 6 E - 0 2 1.1763 0 . 7 7 6 + / - 0 . 4 7 8 1 4 1 . 0 * / - 4 . 5 4 .110 - 2 0 . 4 2 5 2 . 8 0 . 4 2 6 8 E - 02 1.0791 0 . 7 5 3 + / - 0 . 2 9 5 1 2 7 . 8 * / - 2 .8 4 . 134 - 1 5 . 0 2 5 8 . 2 0 . 3 7 8 1 E - 02 0. 976 3 0 . 8 5 9 * 7 - 0 . 2 1 5 1 1 1 . 9 + / - 2 .0 4 .087 - 9 . 8 2 6 3 . 4 0 . 3 3 8 5 E - 02 0 .8916 1 . 0 0 8 * / - 0 . 2 0 8 9 8 . 2 + / - I .9 4 .004 - 0 . 3 2 7 2 . 9 0 . 2 8 0 3 E - 02 0.764 9 1 . 2 2 5 * / - 0 . 1 7 6 78 . 5+ / - 1.6 3.868 9.3 282. 5 0. 2352E - 02 0.6645 l . 3 3 0 * / - 0 . 1 2 l 6 5 . 2 + / - 1.1 3.827 19 .0 292 .2 0 . 1 9 9 7 E - 02 0 .5836 1 . 4 0 3 + / - 0 . 1 6 4 5 5 . 7 + / - 1.4 3.850 2 9 . 5 3 0 2 . 7 0 . 1 6 9 6 E - 02 0.5132 1 . 5 5 4 * / - 0 . 1 7 1 4 6 . 8 + / - 1.4 3.815 4 0 . 6 313 . 8 0. 1445E-02 0.4533 I . 8 1 2 + / - 0 . 0 2 8 3 7 . 7 + / - 0 . 2 3.599 5 1 . 0 3 2 4 . 2 0. 1257E- 02 0 .4074 2 . 0 4 6 + / - 0 . 0 7 5 30.6+ / - 0 . 6 3.368 6 0.9 334. 1 0 . 1 1 0 9 E - 02 0. 3707 2 .165+ / -0 .041 2 7 . 2 + / - 0 . 3 3.388 8 0 . 0 353 .2 0 . 8 9 0 0 E - 03 0 . 3 1 4 4 2 . 4 4 7 + / - 0 . 0 8 5 2 2 . 1+/- 0 . 6 3.434 CU(11)-63 BIS DIBUTYL DITHIOCARBflMRTE IN TOLUENE 9-1 X X J> i X o X X I 1 1 1 1 1 1 1 1 1 1 1 0.0 0.01 0.02 0.u3 0.04 0 .U5 0.0C 0.07 0.0B 0.09 0.1 0.11 0.12 VIS/T (X10-1 ) -147-6 3 C u MI) b i s ( d i - n - h e x v l d i t h i o G a r b a m a t e ) i n Toluene Temp, Width P o s i t i o n MW Freg, Temp. .Width P o s i t i o n MW Freg -10.4 5. 931 3276.936 9.0442176 39.3 3.472 3273.213 9.043662 8.705 3195.969 9.0442249 4.610 3194.188 9.043637 16.875 3118.580 9.0442132 8. 144 3 11 7. 230 9. 043653 34.863 3044.123 9.0442304 14.417 3043.154 9.043640 -6.2 5.474 3276.598 9.0441831 49.6 3.367 3272.547 9.043559 7.918 3195.793 9.0441834 4.361 3193.930 9. 043585 16.165 3118.299 9.0441966 7.444 3117.281 9.043561 30.802 3042.779 9.0441800 12.597 3043.150 9.043523 0.3 4. 922 3276.039 9.0441246 59. 9 3.297 3271.729 9.043331 6.S78 3195.533 9.0441184 4.179 3193.516 9.043353 14. 522 3117.752 9.0441367 6.673 3117.348 9,043338 26.386 3042.010 9.0441182 11.417 3043.496 9.043337 5.0 4.659 3275.736 9.0440793 70.0 3.320 3271.307 9.044044 6.569 3195.318 9.0440609 4.066 3193.510 9.044053 13. 167 3117.688 9. 0440850 6.385 3117.703 9. 044041 24.900 3041.300 9.0440847 10.215 3043.807 9.044021 9.4 4.379 3275.594 9.0448929 80.1 3.275 3270.340 9.043075 6.137 3195.350 9.0448765 4.046 3192.957 9.043036 11.382 3117.502 9.0448713 6.061 3117.547 9.043046 21.961 3043.195 9.0449400 9.807 3044.088 9.043026 10.1 4.365 3275.357 9.0440253 6.111 3195.096 9.0440046 11. 915 3117.580 9.0440221 22.448 3042.951 9.0440195 80.8 3.291 3270.520 9.043819 4.057 3193.201 9.043832 6.051 3117.779 9.043814 9.626 3044.287 9. 0437 97 13.8 4.187 3275.328 9.0448619 5.809 3195.244 9.0448417 10.998 3117.369 9.0448694 20.684 3042.621 9.0448806 90.4 3.319' 3269.852 9.043618 4.044 3192.893 9.043589 5.837 3117.846 9.043591 9.005 3044.734 9.043591 19.7 3.S71 3274.680 9.0439002 5.455 3194.807 9.0439238 10.211 3117.217 9.0439108 19.007 3042.301 9.C438880 100.7 3.378 3269.137 9.043394 4.0 38 319 2.617 9. 043377 5.740 3117. 949 9. 043388 8.482 3045.355 9.043376 24.5 3.810 3274.564 9.0448619 5.125 3194.887 9.0447796 9.561 3117.482 9.0447734 17.315 3042.572 9.0447500 30.4 3.694 3273.896 9.0437855 4.928 3194.432 9.0437577 9.000 3117.264 9.0437638 15. 998 3042. 854 9.0437467 - 8 M -***** CU(II)-63 BIS DIHEXYL TCC) T ( K) VI S/T vise -6.2 267. 0 0.3145E-02 0.8398 0.3 273.5 0.2771E-02 0.7579 5.0 278. 2 0.2540E- 02 0. 7065 9.4 282.6 0.2348E-02 0.6635 10. 1 283.3 0.2319E- 02 0.6571 13.3 287. 0 0.2177E-02 0.6247 19.7 292.9 0. 1975E-02 0.5784 24. 5 297. 7 0.1830E- 02 0.5448 30.4 303.6 0. 1673E-02 0.5079 39. 3 312. 5 0.1471E- 02 0. 4597 49.6 322. 8 0.1280E- 02 0.4131 59.9 333.1 0.1123E-02 0. 3741 70.0 343.2 0.9958E-03 0.3418 80. 1 353.3 0.8891E-03 0.3141 80.8 354.0 0.8823E- 03 0.3123 90.4 363. 6 0.79 70E- 03 0.2898 100.7 373.9 0.7186E- 03 0.2687 IN TOLUENE ***** ALPHA" 0.236+/-0.6 12 0.36 7+/-0.503 0.443*7-0.355 0.529+/-0.253 0.535+V-0.245 0.6U+/-0.212 0. 7 l l * / - 0 . 191 0.809+7-0.168 0.921*/-0.150 1.088+/-0.167 1.262*/-0.159 1 .408*7-0.042 1.533+/-0.130 1.683*/-0.157 1.693*/-O.153 1.843+/-0.021 2 .012*7-0.057 TAUC(PS) AVOL*10«*22 167.9*/- 5.8 7.369 144.7* /- 4.7 7.211 130.7*/- 3 .3 7.106 119.!•/- 2.4 7.003 117.5*/- 2.3 6.991 109.0*/- 2.0 6.914 97.5*/- 1.8 6.815 89.2*/- I .5 6.733 80.5+/- 1.4 6.643 69.4*/- 1.5 6.512 58.7+/- 1 .4 6.338 50.1 + /- 0.4 6. 157 43.3+/- 1.1 6.001 37.9*/- 1 .3 5.877 3 7.6*/- 1.2 5.875 33.5*/- 0.2 5.800 28.6+/- 0.4 5.491 - 1 5 0 -73 _ l I— CE s: <x C D cr o >-UJ x Q C O C D C O I o 002 f-C'CSl r~ r~ O'Ofrl O'OOl 3HH1 —r~ rroe 0 -09 0TJ> a'Q2 (TO -151-* 3 C u ( I I ) b i s ( d i - n - o c t v l d i t h i o c a r b a i a a t e ) i n Toluene Temp, 10.5 Width P o s i t i o n MW Freg. Temp. Width P o s i t i o n F r e q 15.6 5.522 3275.768 8.001 3195.650 15.406 3118.227 47.331 3034.094 5.058 3275.402 7.400 3195.400 13.699 3118.164 44.699 3036.234 4.853 3275.053 6.846 3195.242 13.919 3118.123 26.858 3043.258 25.0 4.549 3274.717 6. 364 3195.098 12.458 3118.363 27.376 3044.551 20.0 9.0450538 9.0450368 9.0450352 9.0450527 9.0449698 9.0449833 9.0449756 9.0449559 9.0449332 9.0449607 9.0449566 9.0449362 9.0449029 9.0449214 9.0449071 9.0448859 30.1 4.318 3274.383 9.0451775 6.060 3195.096 9.0451926 12.089 3118.629 9.0452549 24.151 3043.469 9.0452433 30.5 4.304 3274.328 9.0448438 6.042 3194.959 9.0448587 11.818 3118.346 9.0448614 21.221 3043.842 9.0448383 35.0 4.172 3274.021 9.0451307 5.723 3194.813 9.0451582 11.315 3118.143 9.0451471 19.536 3043.131 9.0451358 39.3 4.002 3273.711 9.0447683 5.516 3194.693 9.0447941 10.397 3117.846 9.0447817 19.809 3044.105 9.0447606 48.6 3.775 3273.035 9.0447024 5. 115 3194.359 9.0446818 9.190 3118.135 9.0447045 19.374 3044.225 9.0446871 48.7 3.718 3273.006 9.0443458 5. 115 3194. 334 9.0449599 9.103 3117.818 9.0449536 16.830 3044.521 9.0449375 59.1 3.562 3272.268 9.044565 4.778 3194.045 9.044586 8.451 3117.879 9. 044579 13.9 86 304 4.469 9.044558 69.0 3.463 3271.578 9.044467 4.469 3193.758 9.044479 7.728 3117. 967 9. 044475 13.378 3044.275 9.044454 78.8 3.371 3270.873 9.044356 4.304 3193.465 9.044362 7.070 3118.031 9. 044365 12.581 3044.949 9.044342 89.4 3.335 3270.070 9.044345 4.197 3193.066 9.044324 6.521 3118.092 9. 044272 10.842 3045.201 9.044228 99.0 3.331 3269.320 9.044058 4.130 3192.803 9.044020 6.273 3118.230 9.044066 10.512 3045.957 9.044050 ***** CUIII1-63 BIS DIOCTYL DITHIOCARBAMATE IN TOLUENE ***** TtC) TIKI VIS/T vise ALPHA" TAUCIPSJ AVDL*10**22 30.1 303.3 0. 1680E- 02 0.5097 0.441+/- 0.323 12 0.6+/- 3.0 9.908 3 0.5 303.7 0. 1670E- 02 0.5073 0.443+/- 0.300 119.7+/- 2.8 9.891 3 5.0 308.2 0. 1564E- 02 0.4820 0.455+/- 0.211 110.0*/- 2.0 9.714 39.3 312.5 0. 1471E- 02 0.4597 0.538+/- 0.28 7 101.5*/- 2.6 9.522 42.6 315. 8 0. 1405E- 02 0.443 8 0.618+/- 0.291 95.4+/- 2.7 9.374 48.7 321.9 0. 1295E-02 0.4168 0.799+/- 0.235 8 5.4*/- 2. 1 9.107 59.1 332.3 0. 1134E- 02 0.3769 1.084*/-0.166 71.8+/- 1.5 8.733 69.0 342.2 0. 1007E-02 0.3448 1.261*/-0.166 62.2+/- 1.5 8.521 78.8 352.0 0. 9018E- 03 0.3174 1.340*/- 0. 069 5 5.0*/- 0.6 8.425 89.4 362.6 0. 8053E- 03 0. 292 0 1.402*/-0.279 48.8+/- 2 .4 8.364 99.0 372.2 0.7307E-03 0.2720 1.572+/-0.215 43. 1+/- 1. 8 8. 139 CU(11J-63 BIS DIOCTYL DITHIOCflRBflMfiTE IN TOLUENE CE-' I X 9H X X X X 1 1 1 1 1 1 1 1 1 I I 0.0 0.01 0.02 0.03 0.04 0.05 0.06 , 0.07 0.0E 0.09 0.1 O.li VIS/T [xiir1) -156-* 3 C u ( I I ) b i s ( p y r o l l i d i n e - N ^ g a r b o d i t h i o a t e V i n l o l u e n e Temp. Width P o s i t i o n MW Freq. Temp.:Width P o s i t i o n MW Freg 55.0 5.366 3274.166 9.0362120 8.315 3191. 172 9. 0362074 17.426 3110.992 9.0361999 34.433 3033.244 9.0362565 9.0 3.125 3269.338 9.036080 3.941 3189.357 9.036075 5.764 3111.346 9.036068 8.828 3035.422 9.036065 49.5 4.655 3274.082 9.0366363 7,059 3191.314 9.0367845 15.221 3111.375 9.0366024 26.918 3031.371 9.0366278 19,5 3.318: 3268.592 9*035984 4.012 3189. 107 9. 035978 5.610 3111.592 9.035971 8.143 3036.086 9.035968 45.0 4.198 3273.439 9.0366318 6.340 3190.975 9.0366471 12.681 3110.824 9.0366687 25.280 3031.371 9.0366818 28.8 3.490 3267.922 9.035902 4. 168 3188. 855 9. 035896 5.542 3111.746 9.035892 7.7/19 3036. 590 9.035891 40.8 3.872 3273.080 9.0366495 5.700 3190.725 9.0364890 11.298 3110.688 9.0363947 22.967 3033.602 9.0363396 39.4 3.729 3267.137 9.035796 4.350 3188.623 9. 035787 5.581 311 1. 955 9. 035777 7.496 3037.234 9.035769 35.2 3.571 3272.463 9.0360455 5.146 3190.430 9.0360582 9.877 3110.678 9.0360693 18.504 3033.426 9.0360807 49.1 ; 3.961 3266.428 9.035658 4.557 3188. 303 9. 035650 5.682 3112.053 9.035643 7.432 3037.715 9.035636 29.7 3.348 3272.090 9.0361111 4.742 3190.260 9.0361284 9.052 3110.750 9.0361258 15.654 3033.590 9.0361240 56.4 4.195 3265.834 9.035553 4.762 3188.066 9.035536 5.862 3112.227 9.035529 7.424 3038.115 9.C35527 19.3 3.130 3271.406 9.0361738 4.271 3190.078 9.0361826 7.567 3110.811 9.0362654 13.077 3033.666 9.0362685 8.8 3.059 3270.586 9.0361930 4.094 3189.746 9.0361913 6.660 3111.090 9.0361859 10.815 3034.357 9.0361853 0.2 3.038 3270.037 9.0361509 3.969 3189.615 9.0361460 6,143 3111.248 9.0361414 9.745 3034.854 9.0361333 ***** CU<m-63 BIS PYROLLIDINE CARBODITHIOATE IN TOLUENE ***** TICI TCK) VIS/T vis e ALPHA" TAUCIPS) AV0L*10**22 -4 5.0 228.2 0.8198E-02 1.8708 -0.402+/-0.575 141.9*/- 5.4 2.389 -40.8 232.4 0.7233E- 02 1.6811 -0. 111+/-0.382 121.8*/- 3.6 2.325 -35.2 238. 0 0. 6181E-02 1.4711 0.299*/-0.274 99.7*/- 2.5 2.228 -29.7 243.5 0.5348E-02 1.3023 0.647+/-0.351 82.8+/- 3.2 2. 138 -19.3 253.9 0. 4162E-02 1.0568 I. 090+/-O.450 60.9*/- 4.0 2.020 -8.8 264.4 0.3316E-02 0.8767 1.324*7-0.306 47.8*/- 2.6 1.991 -0.2 273.0 0. 2797E-02 0.7637 1.478+/-0. 160 40.4*/- 1.3 1.993 9.0 282.2 0.2365E- 02 0.6673 1-693+/-0.097 33.6*/- 0.7 1.963 19.5 292.7 0.1981E- 02 0.5799 2.021+/-0.037 26.6+/- 0.3 1.857 28.8 302.0 0. 1714E-02 0.5175 2.345+/-0.043 21.8*/- 0.3 1.756 39.4 312.6 0.1469E- 02 0.4592 2.661+/-0.012 18.7+/- 0.1 1.754 49.1 322.3 0. 1288E-02 0.4151 2.912+/-0.097 17.4*/- 0.6 1.870 56.4 329.6 0.1173E- 02 0. 3866 3.248+/-0.057 15.2+/- 0.3 1.789 w. ZDo. 9H a CU(II)-63 BIS PYROLLIDINE CflRBODI THIOFITE IN TOLUENE Ul CD X X X X* X X X 1 1 1 1 1 1 1 1 1 1 1 1 0.0 0.01 0.02 0.03 0.04 0.05 0.0E 0.07 0.08 0.09 0.1 O . l i 0.12 VIS/T (X10-1 ) -159-» 3 C a (II) b i s r h e x a m e t h v l e n e i m i n e - N - c a r b o d i t h i o a ^ ^ Temp. Width P o s i t i o n Freg, -56.0 6*. 148 8.911 19.421 38.999 3277.961 3194.846 3114.682 3037,418 9,0373510 9.0373490 9.0373394 9.0373356 -50.3 4.902 3277.180 9.0372323 7.058 3194.352 9.0372028 14.125 3113.945 9.0371908 29.372 3035.977 9.0371865 Temp. /Width P o s i t i o n flW Freg 19.8 3.307 3271.990 9. 036150 3.956 3192.277 9. 036144 5.388 3114*635 9*036139 7*923 3038,;951 9.036134 29.7 3.397 3274.297 9.036048 3.990 3192.033 9.036038 5.304 3114.770 9.036032 6.918 3042.866 9,036027 -45.4 4.462 3276.691 9.0371417 38.6 3.535 3270.621; 9,035934 6.177 3194.227 9.0371582 4.113 3191.840 9.035923 12.395 3114.057 9.0371665 5.288 3114*959 9.035919 23.953 3036.492 9.0371487 7.180 3040.004 9.035912 -40.0 3.996 3276.230 9.0370948 5. 532 3193. S75 9.0370839 10.698 3113.92 8 9.0370781 20.274 3036.207 9.0370713 -36.2 3.737 3275.982 9.0370353 5.260 3193.857 9.0370284 9.792 3113.898 9.0370212 19.029 3036.652 9.0370177 48.0 3.721 3270.021 9.035821 4.262 3 191.596 9.035800 5.307 3115.070 9.035579 6.932 3040.514 9.035792 58.0 3.948 3269.139 9.035306 4.476 3191.174 9.0 35294 5.381 3115.104 9.035284 6.952 3040.883 9.035277 •31.1 3.553 4.772 8.782 16.215 3275.580 3193.732 3113.932 3036.914 9.0370220 9.0370158 9.0370124 9.0370118 -19.8 3.286 3274.686 9.0364024 4.320 3193.209 9.0364038 7.339 3113.898 9.0364017 12. 554 3036.873 9.0364014 -10.4 3.190 327 3.930 9.0363481 4.053 3192.967 9.0363449 6.420 3114.025 9.0363394 10.984 3037.260 9.0363190 -0.2 3.149 3273.254 9.0361824 3.931 3192.742 9.0361735 5.972 3114.156 9.0361628 9.427 3037.652 9.0361530 10.2 3. 204 3272.482 9.0360669 3.931 3192.445 9.0360553 5.585 3114.365 9.0363888 8.545 3038.363 9.0360183 ***** ClMII»-63 BIS HEXAMETHYLENE TICI TIKI 40.0 233.2 36.2 237.0 31.1 242.1 19.8 253.4 10.4 262.8 -0.2 273.0 10.2 283.4 19.8 293.0 29.7 302.9 38.6 311.8 48.0 321.2 58.0 331.2 VIS/T 0. 7068E-02 0.6352E-02 0.5544E-02 0.4210E-02 0. 3428E-02 0.2797E-02 0. 2315E-02 0.1972E-02 0. 1690E-02 0.1486E-02 0.1307E-02 0. 1150E-02 vise 1.6483 I.5054 1.3422 1.0668 0.9008 0.7637 0.6562 0.5777 0. 5120 0.4632 0.4198 0.3808 IN TOLUENE ***** ALPHA" 0.366+/-O.220 0.451+/-0.249 0.654+/-0.221 1.126+/-0.095 1.383+/-0.059 1.577+/-0.109 1.783+/-0.132 2 .041+/-0.061 2.331+/-0.055 2.538+/-0.053 2.686+/-0. 132 3.023+/-0.148 TAUCIPSI AVOL*10**22 108.5+/- 2.0 2.120 97.9+/- 2.3 2.127 84.0+/- 2 .0 2.093 59.5+/- 0.8 1.953 46.7+/- 0.5 1.883 37. 8* /- 0.9 1.864 30.7+/- 1 .0 1.829 24.3+/- 0.4 1.700 18.8+/- 0.3 1.536 16.4+/- 0.3 1.525 16. 1+/- 0.8 1.704 13.7+/- 0.8 1.650 4. <X-CU(II)-63 BIS HEXAMETHYLENE CARBODITHIOATE IN TOLUENE X X X X x x x X X X Q I 1 1 1 1 1 1 1 1 1 1 I I 0 . 0 0 . 0 1 0 . 0 2 0 . 0 3 0 . 0 4 0 . 0 5 0 . 0 6 0 . 0 7 0 . 0 B 0 . 0 9 0 . 1 0 . 1 1 0 . VIS/T (X10-1 ) -163-* 3Cu (II) b i s { o c t a m e t h y l e g i m i h e - N ^ G a g b o d i t f e i o a t e j art Toi^ieBe Temp. Width P o s i t i o n Freq. Temp. Width P o s i t i o n Freq 39.0 5.222 3280.098 9.0450939 7.314 3197.500 9.0450352 13.711 3117.764 9.0449961 32.822 3041.504 9.0449634 34.1 4.674 3279.611 9.0449266 6.586 31S7.250 9.0449409 12.684 3118.088 9.0449382 24.152 3040.867 9.0449557 29.9 4.367 3279.293 9.0449715 5.854 3197.154 9.0449583 11.076 3117.557 9.0449362 21.475 3040.455 9.0449767 39.4 3.422 3274.264 9.044637 4.048 3195.367 9.044599 5.444 3118.412 9. 044555 7.6 31 3043.324 9. 044524 48.5 3.583 3273.566 9.044446 4.106 3195.0 80 9.044459 5.342 3118.514 9.044441 7.499 3043. 961 9. 044418 58.7 3.688 3272.893 9.044342 4.247 3194.875 9.044329 5.364 3118.738 9.044311 7.258 3044.420 9.044324 24.9 4.068 3278.863 9.0448316 5.443 3197.039 9.0448757 10. 170 3117.566 9.0449394 19.083 3040.666 9.0452346 19,2 3.893 3278.582 9.0450634 5,079 3196.928 9.0450886 9.183 3117.443 9.0450822 16.443 3040.945 9.0451048 10. 1 3.601 3277.934 9.0450873 4.545 3196.727 9.0450735 7.807 3117.723 9.0451265 13.582 3041.084 9.0451376 0.4 3.430 3277,162 9.0449694 4.248 3196.410 9.0449537 6.902 3117.691 9.0449717 11.457 3041,201 9.0449849 8.6 3,314 3276.553 9.0449391 4.105 3196.178 9.0449108 6.270 3117.844 9.0449310 10.036 3041.635 9.0449198 19.1 3.305 3275.793 9.0448562 3.963 3195.893 9.0448256 5,852 3118.010 9.0447649 8.962 3042.281 9.0447418 77,6 4.142 3271,389 9,044163 4.616 3194.273 9.044159 5.550 3119.039 9.044123 7.071 3045.623 9. 0441*64 30.2 3.359 3274.953 9.0446955 4,014 3195.580 9.0446715 5. 582 3118.240 9,0446766 8.254 3042.746 9;0446686 — b 9 l -***** CU(II»-63 BIS OCT AMETHYLENE T(C> T( Kl VIS/T vise -34.L 239.1 0.6000E- 02 1.4346 -29.9 243.3 0.5375E-02 1.3079 -24.9 248.3 0.4747E- 02 I. 1787 -19.2 254.0 0.4153E- 02 1.05 48 -10. 1 263.1 0.3406E-02 0.8962 -0.4 272.8 0.2808E- 02 0.7660 8.6 281.8 0. 2381E-02 0. 6710 19.1 292.3 0. 1994E-02 0.5829 30.2 303.4 0.1678E- 02 0.5091 39.4 312. 6 0.1469E-02 0.4592 43.5 321.7 0.1298E- 02 0.4176 5 8.7 331. 9 0.1140E-02 0.3783 77.6 350.8 0.9138E-03 0.3206 IN TOLUENE ***** ALPHA" 0.469+/-0.297 0.560+/-0.240 0.740+/-0.180 0.972*7-0.090 1.297+/-0.022 1.522*7-0.043 1.656*7-0.021 1.806+7-0.074 2.016+/-0.109 2.226*/-0.098 2.419*/-O.104 2.593+/-0.198 3.205+/-0.151 TAUC«PS» AVOL*10**22 128.7*/- 2.8 2.961 113.6*/- 2.2 2.917 97.8+7- I .7 2.845 82.7+/- 0.8 2.749 63.9*/- 0.2 2.591 49.9*/- 0.4 2.454 40.8*/- 0.2 2.363 33.0*/- 0.6 2.282 26.7*/- 0.8 2.196 22.6*/- 0.7 2.129 1 9.8* /- 0.7 2.102 17.7+/- 1 .2 2.141 13.5+/- 0.8 2.043 - 1 6 6 -z LU T3 - J LU c n n CD oe cr LU o X I — LU 5: c c o to on ro CD i X X ZD O — r ~ C 9 — r ~ — r ~ CO! r-O'CSl r-0TJ3I COM r-n -osr cooi onbi — r ~ C Q 8 ca -167-6 3 C u (II) b i s idodeGvlmethyleflimine— I f-carboditM f Temp. Width P o s i t i o n MW Freq. -20.1 5.344 3277.492 9.0436733 7,727 3195.834 9,0436430 15.147 3117.293 9.0436675 30.715 3041.537 9.0436597 -16.0 5.000 3277.160 9.0436342 7.043 3195.777 9.0436424 13.614 3116.988 9.0436373 26.425 3040.650 9.0436242 -10.5 4,536 3276.633 9.0435917 6,328 3195.492 9.0435952 12.239 3117.061 9.0435760 22.827 3042.307 9.0435957 -4.4 4.208 3276.221 9.0435436 5.853 3195.322 9.0435298 11.379 3116.975 9.0435524 21.0 3040.543 9.0435433 0.6 3.968 3275.889 9.0434854 5.427 3195.238 9.0435052 1.0.208 3116.715 9.0434927 19.007 3040.541 9.0434932 10.0 3.628 3275.201 9.0434175 905 3194.865 9. 0434033 8.756 3116.8079.0433825 16,123 3041.207 9.0433995 Temp. Width P o s i t i o n H» Freg 60. 2 3.217r 3271i. 596 9.042719 3,972 3193.492 9*042726 5.832 3117.297 9.042703 9.117 3043, } 97 9, 042710 79.0 3.321 3270.197 9.042453 4,. 025 3193; 010 9* 042417 5.528 3117,652 9. 042403 8.046 3044.180 9.042400 19.5 3.434 3274.467 9.0432966 4.512 3194.590 9.0432732 7.825 3116,824 9.0432822 12.939 3041,375 9.0432812 28.3 3.304 3273.803 9.0431713 4.220 3194.330 9.0431707 7.095 3116.891 9.0431487 ; 12.160 3041.809 9.0431663 40.3 3.230 3272.936 9.0430305 4. 101 3193. 994 9.0436025 6.587 3117.051 9.0430071 10,830 3042.365 9,0430050 50.0 3.258 3272.268 9.0428733 4.004 3193.779 9.0428628 6.182 3117.152 9.0287704 9,573 3042.840 9.0428500 ***** CU(11>-63 BIS DODECYLMETHYLENE CARBODITHIOATE IN TOLUENE ••*** TIC) TIKI VIS/T vise ALPHA" TAUC(PS) AV0L*10**22 -16.0 257.2 0.3865E-02 0.9941 0.414+/-0.382 141.0+/- 3.6 5.036 -10.5 262.7 0.3435E- 02 0.9023 0.560+/-0.285 123. 5+/- 2.6 4.964 -4.4 268. 8 0.3034E- 02 0. 8157 0.643+/-0.263 108.8+/- 2 .4 4.951 0.6 273.8 0.2755E-02 0.7544 0.717+/-0.201 98.3+ /- 1.9 4.925 10.0 283.2 0.2323E- 02 0.6580 0.919+/-0.152 80.1+/- 1 .4 4.759 19.5 292.7 0.1981E-02 0.5799 1.159+/-0.134 64.3+/- 1.2 4.482 28. 3 301.5 0.1726E- 02 0.5205 1.342+/-0.159 53.5+/- 1.4 4.282 40.3 313. 5 0.1451E- 02 0.454 8 1.498+/-0.114 45.0+/- 0.9 4.287 50.0 323.2 0.1273E- 02 0.4114 I .591+/-0.013 40.4+/- 0. 1 4.382, 60.2 333.4 0.1119E- 02 0. 3731 1.758+/-0.039 3 3.3+/- 0.3 4.112 79.0 3 52.2 0.899 8E-03 0.3169 2.054+/-0.061 25.8+/- 0 .4 3.957 CUUJ)-63 BIS D3DECYLMETHYLENE CARBCIDITHlOflTE IN TOLUENE i— o.i 0.0 I— 0.01 0 . 0 2 I— 0 . 0 3 0 . 0 4 0 . 0 S 0 . U 6 VIS/T (X3 0-1 ) 0 . 0 7 —1 0 . 0 8 -1 0 . 0 9 0.11 - 1 T 1 -6 3 C u ill)- b i s (S~methYl-n-OGtadeGyldithiocaj:bafflat;e)i i i n Toluene Temp. Width P o s i t i o n MW Freq. -26 .3 8.965 3275.467 9.0369488 11.478 3194,336 9.0369621 25.091 3117.021 9.0369773 99.999 3013,023 9.0369932 Temp. Width P o s i t i o n MW Freq 46.9 3,599 3269.871 9.035947 4.518 3191.123 9.035932 7.158 3114.428 9,035921 11.713 3039.703 9. 035912 •21.5 7.347 3275.307 9.0369633 10.318 3193.912 9.0369592 22.110 3116.260 9.0369550 49.385 3033.598 9.0368904 56.5 3.568 3269.160 9. 035947 4.422 3190.762 9,035932 6.727 3114.555 9.035921 10.669 3040,174 9,0359 12 -15 .9 6.546 3274.742 9.0368385 9.104 3193.488 9.0368324 18.560 3115.297 9.0368265 37,350 3039.158 9.0368169 -9 .9 5.823 3274.322 9.0367767 8.045 3193.156 9.0367699 16.689 3114.951 9.0367688 31.420 3041.639 9.0367663 51.3 3,567 3269,717 9.036476 4,471 3191.160 9. 036454 6.885 3114.678 9.036443 11.142 3040.277 9.0364 37 61.4 3.551 3268.916 9.036257 4.356 3190.865 9.C36242 6.470 3114.775 9.036219 10.217 3040,748 9.036212 -5 .4 5.444 3273.854 9.0367177 7.546 3192.859 9.0367128 14.683 3114.895 9.0367077 29.632 3040.305 9.0367022 68.7 3.591 3268*396 9,036130 4.301 3190.615 9.C36134 6.195 3114.793 9.036135 9.620 3040,807-9v036137 -0 .2 4.989 3273.471 9.0366117 6.925 3192.715 9.0365962 13.685 3114 .406 9.03659 06 26.013 3039.297 9.0365862 78.0 . 3.654 3267%662 9.036012 4.314 3190.418 9.036011 6.072 3114.916 9,036010 8.997 30*1,373 9*036010 10.0 4.411 3272.668 9.0364998 5.965 3192.328 9.0364908 11.418 3114.240 9.0364839 19.893 3039.063 9.0364800 88.9 3.676 3266.564 9.034970 4.379 3189.758 9. 034953 5.877 3 X W , 7 9 3 9.034943 8.517 3041.725 9.034934 18.8 4.064 327 1.934 9.0363822 5.453 3192.031 9.0363681 9,854 3114.199 9.0363572 17,598 303 8.973 9.0363491 98.3 3.785 3265.855 9.034762 4.473 3189,436 9iC334744 5,858 3114,941 9.034730 8.210 3042. 166 9. 034717 29.2 3,825 3271. 182 9,0362390 4,966 3191.682 9,0362309 8.609 3114.285 9.0362226 15.141 3039.492 9.0362161 38.4 3.702 3270.496 9.0361160 4.720 3191.377 9.0361002 7.706 3114.322 9.0360892 13. 167 3039.369 9.0360774 ***** CLH 11 J-63 B IS N-METHYL TIC) TIKI VIS/T v i s e -9.9 263.3 0. 3392E- 02 0.8931 -5.4 267. 8 0. 3095E- 02 0. 82 89 -0.2 273.0 0. 2797E- 02 0. 7637 10.0 283.2 0. 2323E- 02 0.6580 i a. a 292.0 0. 2004E-02 0.5851 29.2 302.4 0. 1703E- 02 0.5151 38.4 311.6 0. 1490E- 02 0.4642 46.9 320.1 0. 1326E- 02 0.4245 51.3 324.5 0. 1252E- 02 0.4062 56.5 329.7 0. 1171E- 02 0.3862 61.4 334.6 0. 1103E- 02 0.3690 68.7 341. 9 0. 1011E- 02 0.345 7 78.0 351.2 0.9098E-03 0.3195 88.9 362. 1 0. 8095E- 03 0. 2931 98.3 371. 5 0. 7358E-03 0.2733 DITHIOCARBAMATE IN TOLUENE ***** ALPHA" TAUCIPS1 AV0L*10**22 0.268*/-O.267 172.9+/- 2.5 7.037 0.269+/-0.245 157.6+/- 2.3 7.029 0.446+/-O.263 139.2*/- 2.5 6.869 0.935+/-0.293 106.9+/- 2.7 6.352 1.236+/-0.390 86.5+/- 3.6 5.959 1.425+/-0.259 70. 1+/- 2.3 5.685 1.523+/-0.109 60.1+/- 1.0 5.565 1.637+/-0.142 52.2+/- 1.2 5.429 1.717+/-0.137 48.2+/- 1.2 5.320 1.834+/-0.099 43.6+/- 0.8 5.135 1.945+/-0.038 39.5+/- 0.3 4.944 2.097+/-0.055 34.2+/- 0.4 4.673 2.238+/-0.038 29.8*/- 0.3 4.526 2.332+/-0.242 27.9+/- 1.8 4.753 2.588+/-0.040 23.1+/- 0.3 4.338 C U ( J I ) - 6 3 B I S N-METHYL OCTADECYL DITHI0CHR3HMRTE I N TOLUENE X X X X X X X X X* X _) 1 1 1 1 1 1 1 1 r 1 i l 0.0 0.01 0.02 0 .03 0 .04 0.05 O.Of 0.07 0.08 0 .0? 0.1 0.11 0 . V I S / T (X10" 1 ) -175-APPENDIX I I I The molecular volumes of the dithiocarbamates were c a l c u l a t e d from van-der-Waals r a d i i and group volumes taken from Bondi (110). The volumes c a l c u l a t e d were compared w i t h volumes from c r y s t a l l o g r a p h i c data where a v a i l a b l e . The data used was as f o l l o w s Group o Volume -CH 3 22.7 - C H 2 - ( l i n e a r ) 17.0 -CH 2 - ( c y c l i c ) 16.6 =N- 7.2 c s 2 50.1 Toluene 98.8 Cu 11.5 The volumes c a l c u l a t e d are given i n t a b l e 7.2. C r y s t a l l o g r a p h i c volumes, V , where c a l c u l a t e d as f o l l o w s (65) 3 Vc = "2j- Volume u n i t c e l l ) / ( N o . of molecules i n u n i t c e l l ) j the o r e s u l t s ( i n A^) are Dithiocarbamate van-der-Waals V o l . V c r e f s. Me2NH 220 220 I l l Et 2NH 288 300 60 Pr 2NH 356 380 112 P y r o l l i d i n e 261 266 113 For h i g h l y symmetric molecules the van-der-Waals volume i s a reasonable approximation. However f o r the long chain dithiocarbamates -176-some difficulties arise; firstly there will be 'dead volume' (65); toluene molecules cannot get between the chains and hence these molecules will appear larger to toluene; and secondly the conformation of the molecules in solution is unknown making i t very difficult to estimate the 'dead volume'. This problem is most serious with the dioctylamine and dihexylamine derivatives. With the aid of space f i l l i n g models and assuming the conformation in solution is that of minimum steric hinderance between the substituent chains, it was estimated that the dihexylamine and dioctylamine deri-vatives can exclude the equivalent of two and four toluene molecules respectively; that is a 'dead volume' correction of 200 and 400 A^ , respectively, has to be made for these molecules. On the whole the probes used in this study look like rectangular slabs so we need a consistent method of converting the dimensions of the 'slab' to an 'equivalent ellipsoid'. This was done here by choosing the constraints that the lengths of the symmetry axes of the 'slab' and the ellipsoid are in the same ratio and that the 'slab' and the ellipsoid have the same volume; i.e. if a,b,c are the dimensions of the 'slab' and a',b',c' are the semi-axis lengths of the corresponding ellipsoid, then, 4 ^ (a'b'c*) = abe where a' = ka, b' = kb, c' = kc -177-su b s t i t u t i n g we get k J = 3/4TT hence i f we use the length of the dithiocarbamate i n Perrins and 3 the s l i p equations we have to multiply a by k, i . e . 3/4ir for a . The dimensions a,b,c were found with the a i d of space f i l l i n g models and c r y s t a l l o g r a p h i c data. In cases where the molecules are not p e r f e c t l y rectangular (see I below) a weighting procedure based on cross-sectional areas was adopted. CH, CH, Probable conformation of Cu(Et 2DTC) 2 i n s o l u t i o n Q H 2 C H 2 cL QH 2 k C H , The length of the dithiocarbamates i s usually well.defined; using t h i s , the cross-sectional area obtained from models and the constraint that the cross-sectional area cannot change one can estimate the remaining 'rectangular' dimension. Schematic of molecule • - / r -! x i s required b X equivalent 1 rectangular dimension 4 • » a - - --178-Cross-sectional area A (from models) is n A = 4(a"b") + ab n Cross-sectional area of equivalent rectangle We have the constraint that A^ = A hence, x = [4(a"b") + ab]/a 

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