"Science, Faculty of"@en .
"Physics and Astronomy, Department of"@en .
"DSpace"@en .
"UBCV"@en .
"Riggin, Michael Thomas"@en .
"2010-02-01T23:12:18Z"@en .
"1974"@en .
"Doctor of Philosophy - PhD"@en .
"University of British Columbia"@en .
"An analysis of the most commonly used type of Ion Cyclotron Resonance (ICR) spectrometer is given. Though the equations of motion of an isolated ion in the ICR geometry are extremely non-linear, it was found possible to decouple the longtitudinal oscillations due to the trapping potential from the cyclotron motion by exploiting the fact that the cyclotron frequency is very much greater than the trapping frequency. A previously unsuspected dependence of the cyclotron frequency and drift velocity of an ion on its spatial coordinates was discovered and experimentally investigated. The distribution of energies for ions at resonance with an applied r-f electric field is also discussed and improved techniques for the study of energy dependent cross-sections are proposed. Conventional ICR techniques were used to estimate collision frequencies of sodium and potassium ions in helium and argon gases. These experiments yield information about the d.c. drift mobility, in the zero field limit, of the alkali ions in inert gases and are discussed in terms of various models of the ion-atom interaction potential. A crossed beam arrangement was used to obtain preliminary estimates of low energy rate constants for both asymmetric and symmetric resonant charge transfer between alkali ion-atom pairs."@en .
"https://circle.library.ubc.ca/rest/handle/2429/19535?expand=metadata"@en .
"ANALYSIS OF ION CYCLOTRON RESONANCE by MICHAEL THOMAS RIGGIN B . S c , U n i v e r s i t y of Waterloo, 1969. M.Sc, U n i v e r s i t y of Waterloo, 1970. A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN. THE DEPARTMENT OF PHYSICS We accept t h i s t h e s i s as conforming t o the re q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA MARCH, 1974 In presenting th i s thesis i n p a r t i a l fulf i lment of the requirements for an advanced degree at the Universi ty of B r i t i s h Columbia, I agree that the Library s h a l l make i t freely avai lable for reference and study. I further agree that permission for extensive copying of th i s thesis for scholar ly purposes may be granted by the Head of my Department or by h i s representatives. I t i s understood that copying or publ ica t ion of th i s thesis for f i n a n c i a l gain s h a l l not be allowed without my wri t ten permission. Department of The Univers i ty of B r i t i s h Columbia Vancouver 8, Canada Abstract An a n a l y s i s of the most commonly used type of Ion Cy c l o t r o n Resonance (ICR) spectrometer i s given. Though the equations of motion of an i s o l a t e d i o n i n the ICR geometry are extremely n o n - l i n e a r , i t was found p o s s i b l e to decouple the l o n g t i t u d i n a l o s c i l l a t i o n s due to the trapping p o t e n t i a l from the c y c l o t r o n motion by e x p l o i t i n g the f a c t that the c y c l o t r o n frequency i s very much greater than the tra p p i n g frequency. A p r e v i o u s l y unsuspected dependence of the c y c l o t r o n frequency and d r i f t v e l o c i t y of an ion on i t s s p a t i a l coordinates was discovered and experimentally i n v e s t i g a t e d . The d i s t r i b u t i o n of energies f o r ions at resonance w i t h an a p p l i e d r - f e l e c t r i c f i e l d i s also discussed and improved techniques f o r the study of energy dependent c r o s s - s e c t i o n s are proposed. Conventional ICR techniques were used to estimate c o l l i s i o n frequencies of sodium and potassium ions i n helium and argon gases. These experiments y i e l d i n f o r m a t i o n about the d.c. d r i f t m o b i l i t y , i n the zero f i e l d l i m i t , of the a l k a l i ions i n i n e r t gases and are discussed i n terms of various models of the ion-atom i n t e r a c t i o n p o t e n t i a l . A crossed beam arrangement was used to obtain p r e l i m i n a r y estimates of low energy rate constants f o r both asymmetric and symmetric resonant charge t r a n s f e r between a l k a l i ion-atom p a i r s . ( i i i ) TABLE OF CONTENTS Section Page Preface v Acknowledgements v i L i s t of Figures v i i L i s t of Tables x i 1. I n t r o d u c t i o n 1 2. P o t e n t i a l s and F i e l d s i n the ICR C e l l , and Equations of Motion f o r an I s o l a t e d Ion. 8 3. General Method of S o l v i n g the Equations of Motion and Finding the Quasi-Cyclotron Frequency 20 4. Simple Expressions f o r x ( t ) , y ( t ) and the D r i f t V e l o c i t y 31 5. The Trap O s c i l l a t i o n s 38 6. Another Way to Express the Results 40 7. Influence of the Trap O s c i l l a t i o n s 42 8. The ICR Line Shape 56 9. Influence of the C e l l P o t e n t i a l s on the S p a t i a l D i s t r i b u t i o n of the Ions 64 10. Ad Hoc. 'Average Ion Model' f o r Studying the S p a t i a l D i s t r i b u t i o n of Ions i n an ICR C e l l 68 11. D i s t r i b u t i o n o f Ion K i n e t i c Energies i n an ICR C e l l 80 12. Experimental Apparatus: I o n i s a t i o n by E l e c t r o n Bombardment 92 13. E f f e c t of the E l e c t r o n Beam on ICR 97 ( i v ) S ection Table of Contents (Continued) Page 14. C o n t r o l o f the Ions' P o s i t i o n i n the I o n i s e r 111 15. Determination of the I n i t i a l D i s t r i b u t i o n of Energies i n the ICR System 123 16. Experimental Apparatus: Surface I o n i s a t i o n 134 17. Non^Reactive C o l l i s i o n s : D i s c u s s i o n o f C o l l i s i o n Frequencies and Ionic Energy D i s t r i b u t i o n Functions 147 18. D i s c u s s i o n of Ion-rAtom I n t e r a c t i o n 170 19. A Crossed Beam Experiment 177 20. Summary 214 Appendix 1 D r i f t Between D i f f e r e n t Regions of an ICR C e l l 216 Appendix 2 E f f e c t of Magnetic F i e l d Modulation on ICR 220 Appendix 3 B i - P a r t i c l e C o l l i s i o n s 226 Appendix 4 Moments of Energy D i s t r i b u t i o n Function 238 Appendix 5 The R e l a t i v e V e l o c i t y D i s t r i b u t i o n Function 240 References 245 Glossary of Symbols 249 (v) PREFACE Portions of t h i s t h e s i s are i d e n t i c a l i n content to manuscripts which have been accepted f o r p u b l i c a t i o n i n the Canadian Journa l of Physics. These papers are e n t i t l e d : 'Analysis of Ion C y c l o t r o n Resonance': by T.F. Knott and M. R i g g i n . 'Theory of Ion C y c l o t r o n Resonance': by M, Bloom and M. R i g g i n . 'Dependence of ICR on E l e c t r o s t a t i c P o t e n t i a l s ' : by M, R i g g i n and I.B. Woods Two s e c t i o n s (17 and 18) of the t h e s i s have been submitted f o r p u b l i c a t i o n under the t i t l e : 'ICR C o l l i s i o n Frequencies of A l k a l i Ions i n Rare Gases': by M. R i g g i n . The MKS system i s used throughout t h i s t h e s i s except f o r q u a n t i t i e s that are t r a d i t i o n a l l y expressed i n d i f f e r e n t u n i t s . T r a d i t i o n a l symbols are used f o r most q u a n t i t i e s , and t h i s leads to some d u p l i c a t i o n . To avoid confusion a g l o s s a r y of symbols i s included at the end of the thesis.. ( v i ) ACKNOWLEDGEMENTS The author i s pleased to acknowledge the guidance of Myer Bloom who supervised t h i s work. His i n f e c t i o u s good nature made working i n h i s lab a pleasure. The very capable t u t o r i n g o f Tom Knott and E r i c Enga was a l s o of considerable b e n e f i t to the author. D.LI.Williams was more than generous i n sharing magnet f a c i l i t i e s . ( v i i ) LIST OF FIGURES Figure Page 1. A Schematic Diagram of the ICR C e l l 3 2. E l e c t r i c F i e l d ; E (y,d) - E^ (o, o) 12 3. E l e c t r i c F i e l d ; E^ (o, z) - E (o, o) 14 4. E l e c t r i c F i e l d ; (o, z) 16 5. Contours of Constant E f f e c t i v e Magnetic F i e l d 29 6. Contours of Constant D r i f t V e l o c i t y 36 7. A P l o t of the Instantaneous Quasi-Cyclotron Frequency as a Function o f Z ( t ) 44 8. A P l o t of the Quasi-Cyclotron Frequency as a Function o f the Trapping O s c i l l a t i o n Amplitude 53 9. A P l o t of the D r i f t V e l o c i t y as a Function of the Trapping O s c i l l a t i o n Amplitude 55 10. The r - f E l e c t r i c F i e l d as a Function of the Trapping O s c i l l a t i o n Amplitude 59 11. The Eow Pressure ICR Line Shape 63 12. The Amplitude Averaged P o t e n t i a l 71 13. P o s i t i o n Dependence of the Amplitude Averaged E l e c t r i c F i e l d Gradient 74 14. P o s i t i o n Dependence of the Amplitude Averaged E l e c t r i c F i e l d 77 15. D i s t r i b u t i o n of Energy of the C y c l o t r o n Motion at Low Pressures 85 16. D i s t r i b u t i o n of Energy of the Trapping O s c i l l a t i o n 90 17. The E l e c t r o n Beam Filament 95 ( v i i i ) F igure L i s t of Figures (Continued) Page 18. The Dependence o f Magnetic F i e l d at Maximum Absorption on V^-V^ and the E l e c t r o n Beam Current 99 19. The Dependence of the Magnetic F i e l d at Maximum Absorption on V^ , and the E l e c t r o n Beam Current 101 20. The Dependence o f the Line Width on V -V~2 107 21. Dependencesofethe Line iWidth on V ^ i d a t :\u00E2\u0080\u00A2' .. 109 22. TheoDependencevof v'the MagneticvFieldgat-Maximum Absorption on V^-V^ i n the Source Region of the C e l l 114 23. The E f f e c t of the Source Trapping P o t e n t i a l on the Line Width 117 24. The V a r i a t i o n of the Average Ion P o s i t i o n i n the Analyser wi t h the Source Trapping P o t e n t i a l 119 25. The Dependence of the Line Width on V 1+V 2 i n the Source Region 121 26. Energy D i s t r i b u t i o n of P a r t i c l e s Formed by Molecular D i s s o c i a t i o n 129 27. The F r a c t i o n of Ions C o l l e c t e d by the Trap 132 28. The Primary A l k a l i Oven and I t s Mount 137 29. A Side View of the A l k a l i Oven and the ICR C e l l 139 30. The Dependence of the Line Width on the P o t e n t i a l of the Surface I o n i s e r 142 31. A Comparison of \"^K + and ^ A r + Resonances 146 32. A Pressure Broadened ICR Line 151 33. The ICR C o l l i s i o n Frequency o f K + and Na + Ions as a Function o f Helium Gas Pressure 153 ( i x ) Figure L i s t of Figures (Continued) Page 34. The ICR C o l l i s i o n Frequency of K + and Na + Ions as a Function of Argon Gas Pressure 155 39 + 35. The R e l a t i v e I n t e n s i t y of the K S i g n a l as a Function of Argon Pressure 158 36. The E f f e c t of C o l l i s i o n s on the Energy D i s t r i b u t i o n of the C y c l o t r o n Motion 162 37. The F r a c t i o n a l Energy Spread of the C y c l o t r o n Motion as a Function of Neutral P a r t i c l e Pressure 165 38. The ICR C o l l i s i o n Frequency of Na + i n Argon as a Function of the Ions' Average Energy 169 39. The Secondary A l k a l i Oven 178 40. A F i t of Theory and Experiment of a C o l l i s i o n 39 + Broadened K Resonance 186 41 + 39 + 41. The C o l l i s i o n Frequencies of K and K Ions i n Potassium Metal Vapour 188 42. The C o l l i s i o n Frequency of 2^Na + i n Potassium Metal Vapour 191 23 + 43. The Area Under the Na Resonance as a Function of the Vapour Pressure Inside the Secondary Oven 194 41 + 44. Area of the K Resonance as a Function of the Vapour Pressure Inside the Secondary Oven 201 39 + 45. V e l o c i t y Dependence f o r Charge Exchange Between K and 3 9 K 206 3 9 + 41 + 46. Dependence of Double Resonance of K and K on Secondary O s c i l l a t o r Frequency 209 47. The Double Resonance S i g n a l as a Function of Magnetic F i e l d 212 Figure L i s t of Figures (Continued) Page A2.1 The E f f e c t of Magnetic F i e l d Modulation Amplitude on the Apparent Line Width of the ICR Line 223 A2.2 The E f f e c t of Magnetic F i e l d Modulation Amplitude on the ICR S i g n a l I n t e n s i t y 225 A3.1 A C o l l i s i o n i n V e l o c i t y Space 229 A3.2 The One Body Equivalent of a B i - P a r t i c l e C o l l i s i o n 237 A5.1 The R e l a t i v e V e l o c i t y Vector 242 ( x i ) LIST OF TABLES Table Page 1. C o e f f i c i e n t s i n the Power Series Expansion of the E l e c t r i c F i e l d , E_ (y, z) 18 2. Hierarchy of Frequencies i n the ICR Spectrometer 46 3. Comparison o f Experimental and T h e o r e t i c a l D.C. D r i f t M o b i l i t i e s 174 4. D.C. D r i f t M o b i l i t i e s of A l k a l i Ions i n Rare Gases 176 - 1 -1. I n t r o d u c t i o n Ion Cyclotron Resonance spectroscopy (ICR) i s a w e l l e s t a b l i s h e d technique f o r the study of i o n - n e u t r a l c o l l i s i o n s [Wobscall et a l . , 1963] and other gas phase phenomena which r e q u i r e the r e t e n t i o n of charged p a r t i c l e s i n a w e l l defined region f o r r e l a t i v e l y long periods of time. B r i e f l y , the p r i n c i p l e of ICR may be explained as f o l l o w s . Ions, of mass m and charge q, o r b i t near a frequency = q/m B about a s t a t i c magnetic f i e l d B. When an o s c i l l a t i n g e l e c t r i c f i e l d w i t h frequency near i s a p p l i e d i n the plane of the c y c l o t r o n o s c i l l a t i o n the ions absorb energy from the e l e c t r i c f i e l d . This resonant absorption of energy may be detected d i r e c t l y using techniques s i m i l a r to those used i n Nuclear Magnetic Resonance [Abragam, I960]. A t y p i c a l ICR apparatus [Baldeschwieler, 1968; Beauchamp, 1967] i s shown i n F i g . 1, together w i t h the coordinate system ( I n s e r t F i g . 1) which we w i l l adopt f o r t h i s t h e s i s . Ions, normally produced by e l e c t r o n bombardment of an ambient gas, are e x t r a c t e d from the region i n which they are produced (the source region) by perpendicular e l e c t r i c and magnetic f i e l d s . The magnetic f i e l d i s a p p l i e d i n the -z d i r e c t i o n and an e l e c t r i c f i e l d , i n the -y d i r e c t i o n , r e s u l t s from p o t e n t i a l s and V 2 a p p l i e d to the upper and lower p l a t e s of the c e l l ( F i g . 1) r e s p e c t i v e l y . Since the i o n s ' motion i s unconstrained In the d i r e c t i o n of the magnetic f i e l d , they must be trapped between two suitably, biased e l e c t r o d e s (a p o s i t i v e b i a s V T to trap p o s i t i v e i o n s , negative f o r negative ions) o r i e n t e d i n the x-y plane. A f t e r production i n the source region the ions d r i f t i n the x - d i r e c t i o n under the combined i n f l u e n c e of the s t a t i c and magnetic f i e l d s i n t o the analyser or resonance region where they are detected v i a a change i n l e v e l of o s c i l l a t i o n - 2 -F i g . 1: A schematic diagram of the ICR spectrometer. The i n s e r t i n the r i g h t hand corner shows the co-ordinate system and the dimensions used i n the t e x t . - 4 -of an o s c i l l a t o r used to generate the e l e c t r i c f i e l d w i t h which the ions are brought to resonance [Robinson, 1959]. Hence there are three motions of the i o n s , a r a p i d c y c l o t r o n o s c i l l a t i o n i n the x-y plane, a net d r i f t w i t h 2 speed |(ExB)|/B i n the x d i r e c t i o n and p e r i o d i c o s c i l l a t i o n i n the p o t e n t i a l w e l l formed by the t r a p s . The ICR device then i s a type of radio-frequency mass spectrometer of f a i r l y good mass r e s o l u t i o n at low pressures and s u f f i c i e n t s e n s i t i v i t y to detect 1 to 10 ions per cubic centimeter [Beauchamp, 1970]. When operated i n a \" t r a p p i n g \" mode, by p l a c i n g e l e c t r o d e s at the ends of the d r i f t regions of the c e l l [Mclver, 1970] , ions may be r e t a i n e d i n the c e l l f o r s e v e r a l seconds. This f e a t u r e , combined w i t h the good s e n s i t i v i t y of ICR, allows measurements of cross s e c t i o n s f o r p h o t o d i s s o c i a t i o n of p o s i t i v e molecular ions [Dunbar, 1971], as w e l l as f o r photodetachment of e l e c t r o n s from negative ions [Smyth and Brauman, 1972]. Of course, the ICR spectrometer may be used f o r more conventional types of mass spectrometry such as the determination of r e l a t i v e abundances of i o n i c species from e l e c t r o n i o n i s a t i o n of gases and r e l a t i v e y i e l d s of products from i o n -molecule r e a c t i o n s . These a t t r a c t i v e features of t h i s device are enhanced by i t s a b i l i t y to s e l e c t i v e l y a c c e l e r a t e d i f f e r e n t i o n i c species to greater than thermal energies. Thus i t i s p o s s i b l e to do double resonance experiments by monitoring the change i n the ICR s i g n a l of one i o n , say C +, which r e s u l t s from e x c i t a t i o n to n o n - e q u i l i b r i u m v e l o c i t i e s of a d i f f e r e n t i o n A +. This type of experiment can be used, i n p r i n c i p l e , to measure the r a t e constants, k of r e a c t i o n s / + k + [1.1] A + B y C + D - 5 -as a f u n c t i o n of energy since the average energy of A can be c a l c u l a t e d i n terms of the s t r e n g t h of the e l e c t r i c f i e l d a p p l i e d at resonance w i t h the c y c l o t r o n frequency of A + and the time d u r a t i o n f o r which t h i s f i e l d i s a p p l i e d . In a given gaseous system a p a r t i c u l a r i o n C + may have s e v e r a l d i f f e r e n t precursor ions such as A + so that monitoring the ICR s i g n a l s t r e n g t h of C + and sweeping a probing o s c i l l a t o r over a wide range of frequencies w i l l r e v e a l a l l of the parent ions of C + through a change i n i t s p o p u l a t i o n . This type of experiment may r e v e a l the r e a c t i o n channels i n very complex ion-molecule systems. The ICR method i s p o t e n t i a l l y u s e f u l at i o n energies s u f f i c i e n t l y low (5tens of ev^ that the ions are contained i n a c e l l of reasonable s i z e ( s e v e r a l cmi) at standard l a b o r a t o r y magnetic f i e l d s . The ICR apparatus i s cheaper, more compact and e a s i e r to operate than other standard techniques such as merged beams, e t c . used i n t h i s energy range. However, these advantages are o f f s e t by the f a c t that there are s e v e r a l general aspects of the o peration of ICR devices that have not been f u l l y i n v e s t i g a t e d . For example, instru m e n t a l a r t i f a c t s have hampered k i n e t i c s t u d i e s of i o n molecules and charge t r a n s f e r r e a c t i o n s at the low energies a c c e s s i b l e to ICR. Loss of ions from the c e l l [Goode et a l . , 1970], i l l - d e f i n e d d r i f t v e l o c i t i e s [Smith and F u t r e l l , 1973] and inhomogeneous e l e c t r i c f i e l d s [Huntress et a l . , 1971] are a few of the problems encountered. To overcome these problems, t r a n s i e n t ICR experiments [Dunbar, 1971; Huntress,1971], pulsed techniques f o r d r i f t time measurement, [Smith and F u t r e l l , 1973] and d i f f e r e n t ICR geometries [Clow and F u t r e l l , 1971] have been developed. I t seems to t h i s author that these experimental attempts at improvement of the ICR device can b e n e f i t from a d e t a i l e d a n a l y s i s of the ion motion i n the ICR f i e l d c o n f i g u r a t i o n . - 6 -In t h i s t h e s i s we wish to discuss as e x p l i c i t l y as p o s s i b l e the complicated features of the ICR experiment and to present an approximate a n a l y s i s of some of the important p r o p e r t i e s of the ICR spectrometer. The a n a l y s i s i s based h e a v i l y on a w e l l defined treatment of the n o n - l i n e a r equations of motion of an i o n which are solved using an expansion of the c e l l p o t e n t i a l to the f o u r t h order i n the y coordinate. Previous treatments 2 of the equations of motion have included terms \"to y , at most, i n which case the equations are l i n e a r and the c y c l o t r o n and trapping motions are 4 r i g o r o u s l y decoupled. This i s untrue when y terms are considered. However, we w i l l e s t a b l i s h a procedure f o r o b t a i n i n g approximate decoupled equations of motion which, though n o n - l i n e a r , may be solved i n terms of Weierstrass e l l i p t i c f u n c t i o n s . From t h i s a n a l y s i s i t i s found that both the frequency at which the ions o r b i t the magnetic f i e l d and the d r i f t v e l o c i t y are dependent on the i o n s ' s p a t i a l coordinates. Using the a n a l y s i s o u t l i n e d i n Sections 2 to 6 of t h i s t h e s i s we c o n s t r u c t , i n Sections 7 to 10, an ensemble appropriate to the mechanism of i o n production and to the p o t e n t i a l c o n f i g u r a t i o n of the ICR c e l l . Various p r e d i c t i o n s of our model of the i o n i c motions are experimentally i n v e s t i g a t e d i n Sections 12 to 16. In non-resonant techniques of studying i o n - n e u t r a l r e a c t i o n s energy s e l e c t i o n i s obtained by a l l o w i n g the ions to pass across a w e l l defined p o t e n t i a l d i f f e r e n c e , i n which case the f i n a l energy d i s t r i b u t i o n i s independent of the i n i t i a l s t a t e . In Section 11 we f i n d from an e x p l i c i t energy d i s t r i b u t i o n that t h i s i s not true when ions are prepared by c y c l o t r o n heating i n the ICR geometry. Various experimental techniques f o r improving the energy r e s o l u t i o n are proposed i n t h i s t h e s i s , but are i n a p r e l i m i n a r y stage of development. - 7 -Nevertheless we are able to report p r e l i m i n a r y estimates of charge t r a n s f e r r a t e constants f o r near thermal c o l l i s i o n s of potassium and sodium ions w i t h potassium atoms. C o l l i s i o n frequencies of potassium and sodium ions w i t h helium and argon gases are a l s o reported. 2. Potentials and Fields in the ICR Cel l , and Equations of Motion for an Isolated Ion. Ordinary methods of solving Laplace's equation in rectangular coordinates (Churchill, 1941, p. 114) give the two-dimensional potential inside the ICR ce l l shown in Fig. 1: , \u00C2\u00BBk cosh[(2k+l)(^y/a)] [2.1] V(y,z) = V T - - E {[2V T - (V +V ) ] 1 TT k=o 2k+l 1 cosh[(2k+l)(TTb/2a)] \u00E2\u0080\u00A2sinh[(2k+l) (iry/a)] . -(V rV 2) / cos[(2k+l)Oz/a)] sinh[C2k+l) (7rb/2a)] The ele c t r i c f i e l d i s easily obtained from E_ = - VV. A power series expansion i s the most useful form for E=(0,E ,E ) \u00E2\u0080\u0094 y z in the equations of motion. Written in this way, E^(y,z) is [2.2] E (y,z) = E e y m z 2 n L J y w ' ' m,n J J m,n=o where [2.3] \u00E2\u0080\u00A2We (-) ( vVa) v ,- >k r o i i i w , .-,m+2n r ~\ a d ' E (-) [ (2k+l) fir/a) ] e (ro even) = 4 , \u00E2\u0080\u0094 ^ J ^ ' J J m ' n m! (2n)! k sinh[ (2k+l) (^b/2a) ] e ( . odd) . 4 I H t [ P M ) W . l l l ' a m,n \u00E2\u0080\u009E i r? \"> i * m\" ^ n j - cosh[(2k+l) (irb/2a)] have written V = V T - (V1+V2)/2 and V d = for the two combinations o f the a p p l i e d p o t e n t i a l s that appear i n eq. [2.1]. Using V-E_ = 0 i t i s simple to show th a t r n n r -v v ( m + l ) m 2n+l [2.4] E (y,z) = - Z - 7 =\u00E2\u0080\u0094fr- 6 , 7 2 L J zKJ * n (2n+l) m+l,n^ m,n=0 k J ' A u s e f u l r e c u r s i o n r e l a t i o n f o r the c o e f f i c i e n t s f o l l o w s from VxE_ = 0 (or d i r e c t l y from eqs. [2 . 3 ] ) : r? = 2n(2n-l) 1 J m+2,n-l (m+2)(m+l) em,n* With t h i s , we can express a l l the e i n terms o f e r m,n m,o F i n a l l y , the p o t e n t i a l can be w r i t t e n as a power s e r i e s u s i n g the c o e f f i c i e n t s e : m,n [2.6] V t y . O - V - J ( j L , e y-V\" m,n=0 where i s the p o t e n t i a l at the centre o f the c e l l [2.7] v c . v T - (4/\u00C2\u00BB) \u00C2\u00BB t tj ( 2 k t l ) J ^ t t a.Dfab/a.)]\u00E2\u0080\u00A2 1 I t i s i n t e r e s t i n g to compare the e l e c t r i c f i e l d c a l c u l a t e d from the f i r s t few terms o f eqs. [2.2] and [2.4] with t h a t given by Beauchamp and Armstrong (1968) f o r the c e l l geometry i n F i g . 1. They would say tha t [2.8] E = - (2V d/b) + 4 V T ( y / a 2 ) E z = - 4 V T ( z / a 2 ) This form i s not r e a l l y c o r r e c t f o r a r e c t a n g u l a r c e l l , but i s more acceptable than the choice o f McMahon and Beauchamp (1971), which does not s a t i s f y Maxwell's equations. The r e s u l t s o f both c a l c u l a t i o n s are - 10 -shown i n F i g s . 2, 3 and 4 f o r three cases o f i n t e r e s t . The c e l l dimensions are a = 0.025 m, b = 0.014 m, and the a p p l i e d p o t e n t i a l s V = 0.5 v o l t s , = 0.5 v o l t s , which are t y p i c a l o p erating values i n our experiments. (We w i l l use these values i n a l l numerical i l l u s t r a t i o n s i n t h i s t h e s i s . ) Eqs. [2.8] do not give even the r i g h t q u a l i t a t i v e z-dependence of E , which we w i l l see i s q u i t e important i n understanding the resonance c o n d i t i o n i n ICR c e l l s . The d i f f e r e n c e between the c o r r e c t E (y,0) and E z ( 0 , z ) , and eqs. [2.8] i s a l s o s i g n i f i c a n t . Values o f t h i s f i r s t s i x e are shown i n Table 1, f o r a v a r i e t y o f m,o ' c e l l geometries. The f i r s t l i n e g i v e s the l e a d i n g term i n E^ from eqs. [2.8] f o r comparison--as one e x p e c t s , i t agrees w e l l w i t h the more complete c a l c u l a t i o n o f e f o r f l a t c e l l s (b/a << 1) . r o,o Now t h a t we have complete and u s e f u l expressions f o r the e l e c t r i c f i e l d i n s i d e an ICR c e l l , we can t u r n to the equations of motion f o r an i s o l a t e d i o n o f mass m and charge q: [2.9] p = q ( E + v x B) B i s a homogeneous magnetic f i e l d p o i n t i n g i n the negative z - d i r e c t i o n , so t h a t t h i s becomes [2.10] x = -cocy [2.11] y = u x + K E (y,z) c y \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 [2.12] z = < E z ( y , z ) - 11 -F i g . 2: Ey(y,0) - E^(0,0) c a l c u l a t e d from the f i r s t three terms i n the power s e r i e s f o r E^Cs\u00C2\u00B0lid curve). For t h i s and a l l other i l l u s t r a t i v e c a l c u l a t i o n s , we take a = 0.025 m, b = 0.014 m and V = V, = 0.5 v o l t . t d The dashed curve i s the f i e l d used by Beauchamp and Armstrong C1968). The s o l i d curve agrees w e l l w i t h f i e l d s c a l c u l a t e d from eq. [2.1] to twenty-five terms over the region of the c e l l shown i n the diagram. - 13 -F i g . 3: E (0,z) - E (0,0) c a l c u l a t e d and compared as i n F i g . 2. - 15 -F i g . 4: E ( 0 , z ) , c a l c u l a t e d and compared as i n F i g . 2. 2 ro i o M - 17 -Table 1: Values of the c o e f f i c i e n t s e^ q i n the power s e r i e s expansion of E^(y,z) f o r a v a r i e t y of c e l l geometries, expressed i n dimensionless form, fe a m / V ) . The i l l u s t r a t i v e c a l c u l a t i o n s ' mo ' J i n t h i s paper are based on b/a = 0.56. The r e c u r s i o n r e l a t i o n , eq. [2.51 can be used to generate other low-order e . For n b m,n comparison, we i n c l u d e \" p a r a l l e l - p l a t e c a p a c i t o r \" values f o r 2a/(V^b) i n the f i r s t l i n e of the t a b l e . TABLE 1 b/a = 0.1 0.25 0.56 2a/(bVd) -20.0 -8.0 -3.57 -2.0 \u00E2\u0080\u00A21.0 < eoo / vd ) a -20.0 -8.00 -3.52 -1.67 -0.35 (e 1 Q/V t)a +4.88x10 -4 +0.75 +4.78 +4.38 +1.08 (e 2 ( )/V d)a- -2.36x10 -3 -0.035 -3.24 -5.74 -1.68 (e 3 0/V t)a^ -0.049 -19.8 -22.87 +1.69 < e40 / Vd ) a\" -0.650 +1.85 +32.55 +9.86 -1.197 < e50 / Vt ) a +15.55 +155.9 +5.915 -20.69 -0.489 - 19 -where we have abbreviated u>c = q B/m [2,13] tc = q/m for convenience. Eqs. [2.10] and [2.11] are combined by integration to give [2.14] y + a) c 2 y = +u,y o) + < E y(y,z) The ion is i n i t i a l l y at (x y z ) where i t has velocity fx ,y ,z ). ' o o o J o o o These equations describe three types of superimposed motion, which are coupled by the complicated y and z dependence of E_: 1) A d r i f t in the positive x-direction due to B_ and the non-zero average value of E . y 2) Oscillation in the z-direction described by eq. [2.12] (trap oscillation) 3) Quasi-cyclotron motion about the drif t i n g centre in the x-y plane, (described by eqs. [2.14] and [2.10]). The last of these is especially interesting because the frequency of this motion is detected in ICR experiments. The d r i f t motion is also important, a s i t is involved in line-width computation, and in most irate constant measurements. The next job is to investigate the coupling of these motions, and simplify\" the equations so that we can solve them in closed (but approximate) form. - 20 -3. General Method of Solving the Equations of Motion and Finding the Quasi-cyclotron Frequency. The really troublesome feature of the equations of motion for y and z (eqs. [2.14], [2.12]) is their coupling through _E (y,z). If we keep only the leading term in F. , however, the trap o s c i l l a t i o n is harmonic, with frequency to = eio ' while the frequency of the quasi-cyclotron o s c i l l a t i o n is approximately m . An ion with a mass of 40 amu has w m ^ 10^ sec \ while u ^ 2 x 10^ sec ^ , so that OJ_ << w . T c ' T c This means that the ion does not move very much in the trap (z-direction) while i t goes through a complete period of i t s quasi-cyclotron motion in a plane perpendicular to B_. The electric f i e l d in which this high frequency motion takes place can therefore be parameterised by z, the trap coordinate, while the trapping f i e l d can be averaged (at each value of z) over the rapidly varying y-coordinate of the ion. So, we can effectively uncouple the equations of motion given in the last section. Eq. [2.14] becomes, with the help of eq. [2.2] for E y ( y , z ) , \" 2 0 0 m [3.1] y + oj y = o) fx + u y ) + < E (m+1) e y m=U where [3.2] e = f e z m vm+lJ n mn n=0 - 21 -The trap motion is described by [3.3] z + 2K Z (n+1) t z 2 n + 1 = 0 n n n=0 [3.4] t =-r= T T T ^ TT\" ^ Cm + 1) e T < ym >\u00E2\u0080\u00A2 L J n (2n+l)(2n+2) <- J m+1 ,n : m=0 ' The average,< y m >, has to be calculated at each value of the trap coordinate, so that the t are functions of z. n Let us begin by solving eq. [3.1] for y(t). This is s t i l l an unpleasant-looking equation, since i t contains terms which are not linear in the dependent variable. The anharmonic terms cannot be ignored, in general, because they may produce an amplitude and i n i t i a l - p o s i t i o n dependent frequency s h i f t . Fortunately there is a general-- although little-known-- theory for the solution of equations with at most a cubic term in y. The f i r s t step in finding the solution to eq. [3.1] is integration by the substitution y = y (dy/dy) to get [3.5] y 2 = y 2 + 2u fx + to y .) (y - y )- w 2 (y 2 - y 2) 1 J ' Jo c o d o w 'o c 3 : o ~ \"2 , m+1 m+K + 2K E e fy - y ) A m w }o ' m=0 Truncation of the series at m = 1 gives just the harmonic approximation used by Beauchamp and Armstrong (1968), although the form of the fields (and so the values of the coefficients) are different in their work. - 22 -It is possible to keep terms up to and including m = 3 i f we are willing to have the solutions in terms of functions which are more complicated-looking and less familiar than the trigonometric, functions used to solve the harmonic approximation. These are known as Weierstrass e l l i p t i c functions. Generally speaking, an e l l i p t i c function is any function of a single complex variable that is doubly periodic in that variable. It must be single-valued and analytic in the f i n i t e plane, except at poles (which are i t s only singularities). Just as a function for which f(w + 2 n f 2 ) = f(w) (n an integer) is called singly periodic with half-period Q, a doubly periodic function is one for which the relation f(w + 2ntt + 2mQ') = f(w) holds for integer n and m. The double periodicity means that i t is enough to know i t s behaviour inside a parallelogram in the complex plane. If this region has only a double pole at the origin, the e l l i p t i c function is called a Weierstrass e l l i p t i c function. Whittaker and Watson (1927) have an excellent discussion of e l l i p t i c functions (ch. XX), while Southard (1968) gives an extensive summary of their properties. The solution of eq. [3.5] truncated at m = 3, as discussed by 2 Whittaker and Watson (1927, p. 452f), is outlined below. Let (ds/dw) =f(s) where f(s) is a quartic having no repeated factors; 4 3 2 [3.6] f(s) = a Q S + 4a^s + 6a 2s + 4a 3s + a^. Then for f(a) = 0, [3.7] w = / S d^fCO]\"' 5 may be inverted to express s as a function of w [3.8] s = a + I f'Ca)[G)Cw; g ^ ) - I f'Ca).]\"1 G'O'' j ^2* S3) 1 S the Weierstrass e l l i p t i c function formed from the invariants of the quartic(eq. [3.6]): 2 [3.9] g 2 = 3a 2 + a Q a 4 - 4a :a 3 \u00E2\u0080\u00A2t 2 2 [3.10] g 3 = - + 2 a i a 2 a 3 + a Q a 2 a 4 - a ^ - a \u00C2\u00B1 ^ The discriminant of the quartic [5.11] A = g 2 3 - 27g 3 2 turns out to be a useful guide to approximations for the solution and i t s r e a l half-period, O J 2 . The direct integration of eq. [3.5] gives [3.12] t = fV dsffCs)]\"*5 , yo where f(s) i s given by eq. [3.6] with - 24 -a l = K e 2 / 2 [3.13] a 2 = C-wc2 + 2/oj, as i t is related to the phase shift in the harmonic approximation. Eq. [3.8] is the solution to the uncoupled, anharmonic approximation i f we make the identification w = t + /oo and use the invariants (g and g^) formed with the coefficients from eqs. [3.13]. The real half-period ( O J 2 ) \u00C2\u00B0f the Weierstrass e l l i p t i c function is related to the angular frequency (OJ) of the quasi-cyclotron motion by [3.15] OJ = ir/o^. In Southard (1969, p. 649) the invariants are expressed in terms of the real half-period and a complete e l l i p t i c integral of the f i r s t kind - 25 -(we use u for i t s modulus). If A < 0 and u << 1, we find that and 2 To find u ^ , which is the f i r s t goal, we have to express g^, g and A in terms of the coefficients of the quartic function. Using eqs. [3.9], [3.10] and [3.13], we have [3.19] g 2 = ^ c o / { l - 4 j _ [ e i + 3e 2 Y Q + 6 e 3 ( Y O 2 - R Q 2 ) ] } c t 3 - 2 0 ^ h = k 1 ~ 6 - T t e l + 3 e 2 Y o + 6 e 3 ^ Y o 2 + 2 R o 2 ^ ] c and [3.21] A = 1 ^ ( R ^ ) 2 where o [3.22] Y = y + (x /u ) 1 \"J o 'o v o cr and - 26 -[3.23] R = (x 2 + y 2fyu L J o o \u00E2\u0080\u00A2 o c R q is the cyclotron radius of an ion in the magnetic f i e l d alone, where i t s frequency would be io . We already know that ^ C 7 T/ a ) c) > s o when the energy of the ion is expressed in electron volts and a l l other quantities in MKS units [3.24] u = -(4/3)(e 3/ K 2B 4) \u00E2\u0080\u00A2 (K.E.) For typical ions (mass = 40 amu, K.E. = (1/40) eV) in our I C R c e l l (with B = 0.78 tesla) we find that A < 0, because < 0. u is small -7 enough (y 2 x 10 ) that we can ignore i t in the relations between the invariants and the real half period. Thus, from either eq. [3.19] or eq. [3.20], the frequency of the quasi^cyclotron motion is [3.25] (0 = u>c[l - CK / u c 2 )Ce 1 + 3e 2Y Q + ee^J)] The fractional error in the frequency sh i f t , AOJ = OJ ^ u^, due to ignoring the term of order y is less than 0.1% for the typical case considered above. If A > 0, which i t w i l l be for cells with a < b, a similar 2 parameterisation in y 1 = 4y leads to the same expression for the quasi-cyclotron frequency. - 27 -The frequency of the applied rf f i e l d in the analyser section of an ICR c e l l is usually fixed, for technical reasons, and the magnetic f i e l d varied instead. A lower resonance frequency at fixed f i e l d means a higher f i e l d at fixed frequency. Hence, [3.26] AB = (1/u )(e. + 3enY + 6e7Y 2) L J w c 1 2 o 3 o ' is the change in resonance f i e l d for a given x , y and z. Averaged o o o over a l l (equally probable)values of x , AB depends e x p l i c i t l y on the i n i t i a l ion position (y ) and implicitly on the trap coordinate (z) through eq. [3.2]. This result is quite different from the harmonic calculation, which gives AB independent of y^ and z. Ions are produced over an extended region of the ICR c e l l , usually along y 'v o for (-a/2) < z < (a/2), and move toward the negative drift- plate as they leave the negative space-charge region of the electron beam. In Fig. 5, we show contours of AB for isolated ions near the centre of the c e l l . It is obvious that there are large effects due to variation of y Q (for the analyser region) and z (the trap o s c i l l a t i o n ) . The consequences of these are discussed in following sections. The expression for AB is accurate as long as the last term in eq. [3.26] is small compared to the f i r s t , which means that [3.27] y o \u00C2\u00AB Ce 1 0/3e 3 ( ))^ In our case, we are restricted to y << 0.01 m, approximately. - 28 -\u00E2\u0080\u0094 ^ Fig. 5: Contours of 'AB = constant (units are 10 Tesla) in the (y Q,z) plane. The bold lines are given by the theory of section 3 (field expansion to 3rd order y = 0); the rest is from the theory of section 6, using terms up to k = 25 in eq. [2.1]. The c e l l parameters are the same as for Fig. 2; the ion mass is 40 amu, and B = 0.780 Tesla (a = 1.867 x 10 6/sec). - 30 -By using e l l i p t i c functions, we have solved the simplified equations of motion for the quasi-cyclotron frequency. This i s possible because the y and z motions in an ICR c e l l have very different frequencies, which means that the more rapid component of motion in the complicated el e c t r i c f i e l d can be considered as taking place at a constant value of the more slowly varying coordinate. As long as this is true, and the potential i s well represented by a quartic in y, we have a trustworthy expression for the quasi-cyclotron frequency. In the next section, we tackle the problem of finding simple expressions for x(t) and y(t). - 31 -4. Simple Expressions for x(t), y(t) and the Drift Velocity. We wish to find useful expressions for the quasi-cyclotron and d r i f t motions of an ion in the ICR c e l l , for two reasons. The d r i f t velocity is an important ingredient in line-width calculations, and a knowledge of the quasi-cyclotron motion figures into the averaging procedures used in the interpretation of resonance experiments. In the last section we saw that Weierstrass e l l i p t i c functions can be used to solve simplified equations of motion for the y-coordinate - but eq. [3.8] is not very useful for practical calculations. Given the conditions under which we have solved for the quasi-cyclotron frequency, however, the motion in the x-y plane can be expressed in terms of trigonometric functions. The f i r s t step is to find the root of f(y)= 0 (given by eqs. [3.6] and\" [3.13]) near y = y . We find that [4.1] f(y) = y Q 2 + 2u ) c 2Y o(y-y o) (y2-yQ2) 2 2 + 2K[e Q(y-y o) + e^y -yQ ) * e 2 C / V > + e 3 ( v 4 - y o 4 ) ] . The root at a = yQ-\u00C2\u00A3 1 S computed by setting the f i r s t three terms in the Taylor series expansion of f about y Q equal to zero: f(y Q-e) = 0. Thus a b i t of algebra yields \u00C2\u00A9 [4.2] a = y o - [f'(y Q) - f ( a ) ] / f\"(y Q) - 32 -with [4.3] f (oO = [ ( f ( y 0 ) ) 2 - 2f ( y o ) f \" ( y 0 ) ] % Neg lec t ing terms o f order (Aco/wc) << 1, we f i n d that [4.4] f'Cy ) = 2u [x + Ke (y )/w ] 1 J KJ o c o y o c [4.5] f ' ( a ) = 2o3c { [ x o + Ke y(y o)/(o c] 2 + y / } ^ and [4.6] f\"(a) = - 2to2 where we use the abbrev ia t ion 2 3 [4.7] e (y ) = e + 2e,y + 3e\u00E2\u0080\u009Ey + 4e\u00E2\u0080\u009Ey y w o o 1 o 2 o 3 o f o r the y-component o f the e l e c t r i c f i e l d c a l c u l a t e d to t h i r d order in y , the i o n ' s i n i t i a l p o s i t i o n , o Now, us ing the r e l a t i o n between Weierstrass funct ions and Jacobi e l l i p t i c funct ions given by Southard (1968, p. 649), we f i n d that [4.8] 6>(w) = (co 2 /6 ) [ l +1 * + c n f r \" 2 1 - cn(uw Terms o f order y can be ignored i n to, where we a l ready know that they have smal l e f f e c t . S u b s t i t u t i n g from eqs [4.6] and [4.8] i n t o the general s o l u t i o n , with w = t + cj>/a> gives us [4.9] y (w) = a + (f'(a)/2co2) [1 - cn(4*|y)] An approximate expression for cn(u|u), valid for small u, is y [4.10] cn(u|u) = cos u + j (u - sin u cos u) sin u. The fractional error due to neglecting the term proportional to p in [1 - cn(u|u)] is of order u. We already have said that p \u00C2\u00AB 1 is required for our approximations, so the very simple result is that 'Xj [4.11] cn(u y) ^ cos u. Using eqs. [4.2] and [4.3] for a and f 1 (a) in eq. [4.9], the final r e s u l t i s [4.12] y(t0 = y n + -K [f* CvJ - - ' (c) cos (at + The phase angle is chosen so that y(p) = y Q: cos = f' (y )/f' (a). Now i t is simple to get x(t) by integrating eq. [2.10], 2 2 [4.13] x C t ) - x Q + {(1 - X q - - E y \u00E2\u0080\u00A2 1 OJ OJ J - (rr) [sin(wt + - sin ]. 2OJ There are just four independent i n i t i a l values of the coordinates (XQ, y ) and their derivatives (x , y ) involved in the expressions for x(t) and y(t). Both f' (a) and are defined in t&rms of these. The value of the trap coordinate (z) that appears implicitly in e (y^) is a free parameter at this point, as is explained in section 3. - 34 -The motion described by eqs. [4.12] and [4.13] is similar to the motion of a charged particle in crossed E_ and B_ fi e l d s . However, the d r i f t velocity, v D, which is given by the term in curly brackets in eq. [4.13] i s not just (E_ x B)/|B_| , although i t i s approximately that i f we neglect terms of order (ACJ/OJ). Then [4.14] v D = - e y(y Q)/B shows that the electric f i e l d must be evaluated at the i n i t i a l y-coordinate of the ion and the local value of the trap coordinate, z. Fig. 6 shows contours of constant v D, in the same way that A B was plotted in Fig. 5. The quasi-cyclotron motion takes place about the drif t i n g centre, which moves with velocity v^. In this frame, i t is e l l i p t i c a l (with the ratio of semi-axes equal to OJ^/OJ) . This is the same result as is obtained in the harmonic theory, .although we must.,remember that OJ is. position-dependent -when we take the higher order terms into account. In this section, we have obtained solutions for the motion of an isolated ion in the plane perpendicular to the static magnetic f i e l d . While the main features of this motion are similar to the results of the harmonic treatment, they are different in subtle but significant detail. In particular we find a previously unsuspected dependence of the quasi-cyclotron frequency and d r i f t velocity on the position of the ion in the c e l l . The assumptions needed to simplify the equations of motion and solve them in useful form are simple and well satisfied: (1) The trap o s c i l l a t i o n i s much slower than the quasi-cyclotron o s c i l l a t i o n . (2) The y-component of the electric f i e l d is given accurately by cubic terms in y. ? (3) The parameter u = (1/48) (q/m)E^''(0)CR0/wc) is very small. - 35 -Fig. 6: Contours of = constant C u nits are m/sec) for the same situation as in Fig. 5. o o o o o o o o in cs> r\u00C2\u00BB- co G) o ^ O J - 37 -Briefly, these conditions let us uncouple the equations of motion, terminate the power series expansion for the potential, and solve the resulting equations for the quasi-cyclotron motion in simple closed form. As we have previously stressed, the d r i f t velocity and quasi-cyclotron frequency depend on the coordinates of the ions. In order to relate the observed line width and resonance f i e l d to these quantities we must perform appropriate averages over a l l ions and their motions in the cell'. In particular their z dependence must be averaged over the unobserved trap o s c i l l a t i o n , which we examine in the next section. - 38 -5. The Trap Oscillations The separation of variables in the equations of motion leads us to the equation for z: n; n I2 I 2 A o v * r 2 n + 2 2n + 2> |>.1J z - z +2tcE t fz -z 1 = 0 n=0 n \u00C2\u00B0 where [5.2] t = -jrx\u00E2\u0080\u0094T-T-TX\u00E2\u0080\u0094I (m+1) e , < y > L J n (2n+l)(2n+2) n v J m+l,n J m=0 ' The average of y m has to be taken over one period of the quasi-cyclotron osc i l l a t i o n at a particular value of z, because i t s frequency and amplitude depend on z, by eqs. [3.25], [3.2] and [4.12]. Inspection shows us that < y m > is a function of z 2, so when eq. [5.1] is solved by the general method outlined i n Section 3, we find the coefficients of the quartic to be a Q Z - K e N / 6 a l = 0 [5-3] a 2 * - K ( e 1 0 + 2 e 2 0 y o + 3 e 3 ( ) y o 2 ) / 6 a 3 = 0 ' 2 2 4 a. = z +2<(t z +t,z ) 4 o v o o 1 o J - 39 -These lead to the solution for z(t) as in Section (3) and (4): [5.4] z(t) = [ Z Q 2 + {zo/uT)2]h cos CuTt + where [5.5] o>T2= <[e 1 0 + 2 e 2 0 y o + 3 e 3 0 y o 2 ] u^ , i s the frequency of osci l l a t i o n of an isolated ion in the trapping f i e l d , which is shown in Fig. 4 for y = 0. These expressions are correct i f the parameter 1 * 2 4 [5.6] y\" = Cg) < e n Z Q /U>T ) is much less than one. Assuming that the ions form a thermal population, this may be related to the parameter (y) governing the approximations made in understanding the y-motion of ions. [5.7] y\" ^ (i)(a) c/ai T) 4 y _2 Thus, u\" ^ 2 x 10 , which is sufficiently small for us to accept the approximate expression for z(t). - 40 r 6. Another Way to Express the Results The equations of motion which we have worked with up to now use an expansion of the electric f i e l d about the origin. In the results for w,(o_, and v,^ , we found expressions for E' fy ) and E fy ) , taken to ' T D 1 y ' o y o second or third order in y . This fact makes i t interesting to expand E about y = v in a Taylor series , and use the same tactics to solve y Q the equation of motion that follow when we truncate the series at the 3 (y~yo) term. The results are [6.1] u = (\u00C2\u00BB ( l - [E1 Cy ,z) + (x /oi DE1 1 (y ,z) 1 J c ( _ 2 L y w o ^ o c y 'o 2w j 1 c '2 \" 2 3x + 2y / 2OJ \" Y c for the frequency of the quasi-cyclotron osci l l a t i o n . With the s ame assumptions that we made in section (3), and noting that< Xq> = o for thermal ions, we have the very simple result that [6.2] AB = ^ \u00E2\u0080\u0094 E' (y ,z) 2oj y Jo c This supposes that [6.3] K E \" \u00C2\u00AB (yQ,z) \u00C2\u00AB 1, which is essentially the criterio n u \u00C2\u00AB 1. Physically, this means that - 41 -the second order part of the time-averaged variation in the energy associated 1 - 1 1 1 4 with the elec t r i c f i e l d , (p -)^ E y RQ> is much less than the energy 1 2 ? of the orbital motion, m^u R 2 c o For the d r i f t velocity, we find [6.4] V d = _ E (y o,z)/B. The trap o s c i l l a t i o n can, of course, be analysed in much the same way. The average of E^ over one quasi-cycloti-on period is approximated by E (y Q,z), and then expanded about z = 0. V\u00C2\u00ABE_ = 0 relates the f i r s t derivatives of E and E , so that y z* [6.5] to2 = KE y(y o,0) We see that the relation [6.6] co 2 = OJ 2 + to 2 c T which holds rigorously in the harmonic approximation (Beauchamp and 2 2 Armstrong, 1968) is really true only when w << to and z = 0. This i s another example of the small but important differences between that approach and the more complete treatment given here. - 42 -7. Influence of the trap oscillations The motion of ions in the ICR c e l l is conveniently described in terms of several characteristic frequencies. The ions may be pictured as precessing e l l i p t i c a l l y in the x-y plane (\"quasi-cyclotron motion\") about an instantaneous center, which oscillates along the z-axis (\"trap oscillation\") and drif t s in the x-direction. Strictly speaking, the quasi-cyclotron frequency is expressed in terms of the i n i t i a l value of y. Since the theory outlined previously implicitly assumes that the amplitude of the cyclotron motion is small, and since oj(y,z) is a slowly varying function of y and z, we may identify y with the centre of the cyclotron motion. This characteristic angular frequency to(y,z) of the quasi-cyclotron motion, varies with z over a range of frequencies Aoj(y,z) << OJ . That this inequality is very well satisfied may be seen from Fig. 7 in which o)(y,z) is plotted as a function of z for three different values of y for Ar + ions under typical operating conditions. The quasi-cyclotron frequency is modulated due to the trap oscillations at an angular frequency OJ^ which usually satisfies the inequalities [7.1] Aoj(y,z) \u00C2\u00AB (JJT << OJ as may be seen from Table 2. Finally, as a result of the d r i f t of the ions through the resonance region of the ICR c e l l , the ions spend a f i n i t e time T = &/v in a c e l l of length \u00C2\u00A3, giving rise to a resonance line width u>j which usually f a l l s between Aoj(y,z) and O J t. In this section, we use the inequality [7.1] to derive an approximate expression for the ICR frequency for ions having a well-defined value of y and which oscillate between z = \u00C2\u00B1z . The m effect of a distribution of y and z is described in the next section. J m For convenience of notation we re-write Eq. 2.1 in the form - 43 -Fig. 7: A plot of - u)(y,z) as a function of z for three different values of y for Ar + ions. The c e l l parameters are a = 0.025 m., b = 0.014 m., VT = V l = 0 , 5 v o l t s \u00C2\u00BB a n d V2 = \" \u00C2\u00B0 * 5 v o l t s - u c / 2 7 r i s 3 0 0 k H z corresponding to a magnetic f i e l d of about 0.780 Tesla. r- 45 r. Table 2. Values of characteristic frequencies of Ar ions in an ICR c e l l under the operating conditions of Figure 5. TABLE 2 6 Cyclotron Frequency = K B 1.885 x 10 (sec Trapping Frequency toT(0,0) = ^-j E^(0,0) 6.80 x 10^ (sec. T \u00E2\u0080\u00A2 T T - ^ 5.566K E (0,0) | _ Line Width 1 y ' 1 p o c . J , ,\u00E2\u0080\u009E IT. , , , , . , s to, = i 8.25 x 10 (sec. (Full Width at half height) % to l c Frequency Spread to(0,0) - to(0,0.35a) - [E 1 (0,0.35a) - E'(0,0)] 2.82 x 10 3 (sec\" wc y y - 47 -[7.2] V(y,z) = V T - I A^y) cos t(2k+l) ^ ] k=o where o ,_ nk cosh[(2k+l)^-] sinh[(2k+l) i y-] cosh[(2k+l)~] A \" sinh[(2k+l)-^J za 2a When the condition w c >> w T is satisfied, we can approximate the y-component of the ion motion over a time t << 2ir/u) by the expression, [7.4] y(t) = y +A c o s [ w ( y , z ) t + tf>] 3. where the amplitudes y and A and the phase factor are determined by the i n i t i a l conditions and w(y,z) i s defined by [6.1] in terms of the instantaneous center (y,z) of the quasi-cyclotron motion. If an ICR experiment were carried out over a time much less than 27r/u)^ ,, the ICR spectrum would consist of a superposition of lines centered at frequencies distributed over a range Au corresponding to the spatial distribution of ions in the c e l l . However, each line would, under such conditions, have a width much greater than OJ^ , so that the frequency spread AOJ would be undetectable according to [7.1]. In practice, of course, ICR experiments are usually performed over a time much greater than 2ir/^ so that the effect of the trap oscillations must be taken into account. - 48 v . The effect of the trap oscillations is to modulate \u00E2\u0080\u00A2 0) >> \u00E2\u0080\u0094 z c dz which is well met in the ICR apparatus. Using Eqs. [6.1], [7.2], [7..3] and r , - 3 V y \" \" 2' [7-6] 2 rfc 0 0 d A, (y) ft ... u>(y,z(f))dt' = u t - I \u00E2\u0080\u0094 V - cos[(2k+l)^-I]dt' c 2u ,L , 2 , c k=o dy 1 o a Using Eq. [7.6] i t i s possible to define an average quasi-cyclotron frequency as a function of y and z^ by numerical methods. F i r s t l y , the equation of motion for z(t) may be integrated for any given ICR c e l l potential parameters and the maximum amplitude of osci l l a t i o n z . Then, i f m - 49 -the integral in [7.6] is evaluated numerically at long times (w^t >> 1) i t w i l l be well represented as a linear function of time plus small oscillations. The linear factor which multiplies t is the \"average quasi-cyclotron frequency\" for (y,z^). Instead of presenting numerical results of such calculations, we choose to evaluate [7.6] in terms of a simplified description of the trap oscillations. We now evaluate [7.5] and [7.6] in the harmonic approximation for the trap osc i l l a t i o n s , i.e. [7.7] z(t) = z cos out m i This i s quite a good approximation near y=o, where the decoupled equation of motion for z i s almost linear. An accurate representation of z(t) for y ~ -b/4 would require the introduction of harmonics of w^ . The generalization of the results to be derived below to include higher harmonics is straightforward, but tedious, and is not presented here. Substituting [7.7] into [7..6] and using the well known expression (Watson, 1962; page 22) involving Bessel functions oo [7.8] cos(acosg) = J (a) + 2 Y ( - l ) n J 0 (a) cos2n8 o i zn n=l we obtain [7.9] where o)(y,z(t'))dt' = [u - n ( y , z J ] t - I , n m sin(2noJrrt) c o m u ^ n o j m 1 o n=l T ,2, n \u00C2\u00BB d'A.(y> (2k+l ) T T Z [7,10] , 2 n ( y , z m ) \u00C2\u00AB ( - l ) n ^ I \u00E2\u0080\u0094V- J 2 n [ ^ ] c k=o dy - 50 -As may be seen from the definition of fi\u00E2\u0080\u009E and Table 1, 2n ' 2n ^ AOJ S \u00E2\u0080\u0094 << 1. Therefore, substituting [7.9] into [Z.5] and expanding to f i r s t order in these small Quantities enables us to write [7.11] y ( t ) = y + A { C O S [ C J (y,z ) t+ ] + sin[w (y,z )t+] Y sin(2noj mt)}. a o m o m L. nco_ T n=l T where [7.12] % ( y , z m ) = U c - ^ ( y , z m ) We identify w 0 ( y 5 z m ) with the average frequency of the quasi-cyclotron motion, the average being taken over the trap oscillation in the harmonic approximation. The ICR spectrum for ions having a ve r t i c a l position y and an amplitude of oscillation in the trap should consist of a main resonance line at ui (y,z ) and small intensity s a t e l l i t e lines at the o J ' m 3 frequencies ^ ( y * ^ ) - 2noj^ ,, n = 1,2 We wish to emphasize that, though the harmonic approximation was used in treating z ( t ) , the general expression for the potential was s t i l l used in calculating u>(y,z) . The d r i f t velocity of the ions in the c e l l i s given by Eq. [6.4], which may be treated in a manner similar to the quasi-cyclotron motion. The ions d r i f t in the x-direction at an average velocity v n ( v > z m ) which depends on y and z , with small oscillations at the frequencies 2nw . The average - 51 -velocity is given in the harmonic approximation by Eqs. [6.4], [7.2], [7.7] and [7.8] to be 0 0 dA, (y) irz [7.13] v D(y,z m) - - i J - \ - J 0[(2k +1)-^] k=o The variation of u^Y*2^ a n c* V D ^ ' z m ^ w l f c n Y ari^ z m i s shown, respectively, in Figures 8 and 9. - 52 -.8: A plot of a) -to (y,z ) as a function of z , the maximum amplitude c c o m m of oscillation in the z direction, for three different values of y. The harmonic approximation for the trapping oscillations has been used. A l l parameters are the same as in Fig. 5. 1 I I L_ I L 2 4 6 '8 10 12 Z m (I0\"3M) - 54 -Fig. 9; A plot of the average d r i f t velocity as a function of z^ for three values of y under the same conditions as in Figure 5. - 56 -8. Line Shapes We have derived an expression for the ICR frequency OJ (y,z ) of an - J o m ion at a v e r t i c a l position y and trap oscillation amplitude z^. Suppose that N ions pass through the resonance region of the ICR c e l l per second and that a fraction p(y 5z m) dydz m of these are between y and y+dy and have maximum oscillation amplitude between z and z +dz . Then the power absorption in the ICR c e l l for an m m m r r oscillating f i e l d at a frequency oo^ is given by i 4 ! ' a / 2 [8.1] = N J Z dy d z m p(y>0 e(w, ;y,z ) , i m m 1 m b 'o 2 where \u00C2\u00A3(oj^;y,zm) i s the energy absorbed by a single ion characterized by (y,z m) in passing through the c e l l in a time x = SL/v^iy ,z^) . The ensemble averaged line shape I(OJ^) should not be confused with the line shape of an ion having a well-defined (y,z m), which is proportional to e (OJ^ ;y, z^) . It i s convenient to write [8.2] e C v y . z J = e r e s.(y> z m> G { V W o ( y ' Z m ) } where e (y.z ) i s the energy absorbed by an ion at resonance as i t res. m O J J traverses the c e l l and G{OJ -OJ (y.z )} is an unnormalized line shape function l o r n r which s a t i s f i e s G(o) = 1. In our discussion we have assumed thus far that each ion has a constant (y> z m) throughout i t s passage through the c e l l . - 37 -This implies that the time between collisions of the ions with the molecules of the background gas is much greater than T. In this low density, collisionless regime, i t is easy to show that ( B u t r i l l , 1969) 2 ^ 2 2 q x [8.3] e (y,z ) = ^ res. m 8m and Sin (\u00E2\u0080\u0094=-) [8.4] . GCfio,') = , 6<-' = \"i- % Cy,zm) ( ~ 2 ~ ) \ where i s the average magnitude of the y-component of the r-f f i e l d for iy an ion characterized by (y\u00C2\u00BB z m)\u00C2\u00BB t n e average being taken over the trap oscillations. Thus, 610, and T a l l depend on y and z^ in [8.3] and [8.4]. In order to calculate > t n e effective radio-frequency e l e c t r i c a l c i r c u i t must be specified. Usually, the trap potential i s at r-f ground, i.e. (V-jJ r_j = 0. Experiments could be performed with ^ i ^ r _ f = ~ ^ 2 ^ r - f ^ t ^ V r cos w t, but i t seems to be more customary to put (V,) r or (V.) r-f 1 r 1 r-f 2 r-f equal to zero. In any case, once the r-f potentials are specified, the co-efficients [^(y) ] r _ f analogous to the d-c co-efficients of [7.3] may be calculated and the value of Versus z^ for two values of y and three different rf voltage configurations. The values of E^/V^ a r e given for a = 0.025 m. and b = 0.014 m. RF 0 V RF O 0 0 0 0 0 V, RF R F v RF Case (I) Case(2) Case(3) 1.6 1.4 CVJ 1.2 o u. 1.0 >* - 0.8 E N ^ 0.6 cr. L U 0.4 0.2 0 -0.2 Case(l) y=-b/4 Case(3) y = 0 i 6 . 8 10 Z m 0CT 3M) 12 - 60 -each value of z. If the average recoil energy of the ions and the thermal energy of the background gas atoms or molecules from which the ions are produced are i^uch less than qV^, the ensemble of ions produced is well approximated by [8.5] pJy>zJ =4 s(y-y') o> m a ^where 6(y-y') i s a Dirac 6-furiction. As we shall discuss in Section 9, this spatial distribution of ions w i l l be preserved as the ions d r i f t from the source to the resonance region only i f certain stringent experimental conditions are met. Under these conditions, the line shape is given by 2N [8.6] I C ^ ) = ^ fa/2 o dz m e ( U l ; y ' > Z m ) It i s easy to see that [8.1] and [8.6] are each convolutions of two line shape functions. One of these shape functions is G(OJ) given by [8.4]. The other is the distribution of frequencies u (y'_,z ), each weighted in [8.6] by dz 2Ne (y',z ) , which would be obtained from the ions in the small region res \" m a between z and z +dz for very long c e l l s . In a long c e l l the condition m m m Am (y',z ) T >> 1 o m is s a t i s f i e d , where Aai (y',z ) i s the spread of frequencies associated with ' o \u00E2\u0080\u00A2 m * \u00E2\u0080\u00A2 the distribution of z . m Our numerical calculations indicate that under typical operating conditions, the width 1/T due to f i n i t e transit time of the ions in the c e l l - 61 -is somewhat larger than the spread due to the spatial distribution of the Au (y' ,z ) x << 1. o m Some typical line shapes predicted by [8.6] are shown in Fig. 11, but comparison with experiment is le f t to Section 16. The most striking features of the theoretical absorption spectra of Figure 11 are: 1. The frequency of maximum absorption and the line width are functions of y'. A detailed discussion of the properties w i l l be given in Section 10 in terms of a simplified \"average ion\" model. 2. The lines are not symmetric, the degree of asymmetry being dependent on y'^ . Such asymmetric lines have been observed by many experimenters, though they have not been discussed extensively in the literature. We find that the sign and nature of the asymmetry are well explained by our theory. In Figure 11, the absorption sideband in the high frequency Clow field) side of the resonance is noticeably larger than that on the opposite side for y' = -b/4. When V T is decreased to smaller values the asymmetry decreases. It is common practice in many laboratories to eliminate the asymmetry empirically by using small trapping voltages. The origin of the asymmetry lie s in the asymmetric distribution of frequencies associated with the distribution of trap oscillation amplitudes. However, the asymmetry is sometimes enhanced by the fact that the average d r i f t velocity is also a function of z . The plots of Figure 11 include both m of these effects as well as the variation of with z . A l l ly m these influences are included in Eqs. [8.2] - [8.6]. Calculations show, however, that the influence of <^2y > o n t' i e a s 3 a n m e t r y i - s generally less important than the other factors. 62 Fig. 11: The ICR line shape as given by Eq. 8.6, for three different values of y' and the same parameters as in Fig. 5. The bottom 6 \u00E2\u0080\u0094 scale shows a sweep of frequency, OJ^, with O J c = 1.885 x 10 sec. while the top scale shows a sweep of OJ c (or f i e l d , B) with OJ = 1.885 x 106 sec? 1. POWER ABSORPTION ( ARB. UNITS) - 64 -9. Influence of the Cell Potential on the Spatial Distribution of the Ions As discussed in Section 8, the ICR characteristics are influenced by the spatial distribution function of the ions in the resonance region, denoted by p(y,z m) in [ 8 . 1 ] . Usually, the ions are produced in a source region by an electron beam- and then d r i f t slowly into the resonance region. The electron beam, i f i t is sufficiently intense, gives rise to an inhomogeneous electric f i e l d which may contribute appreciably to the potential .energy of the ions when they are produced. The ions then d r i f t far enough away from the electron beam, while s t i l l in the source, that the electron beam contribution to the potential can be neglected. One can then define a two-dimensional source spatial distribution function P s(y s> z m s)* For sufficiently weak electron beams that i t s contribution to the c e l l potential is everywhere negligible, p (y ,z ) can be adequately approximated by [ 8 . 5 ] , but more s s ms generally i t i s necessary to take into account the motion of the ions in the potential of the electron beam to estimate p (y ,z ). r s w s ms In some experiments, i t has been found desirable to use different d r i f t potentials (V^,V2) and/or trapping potentials (V^) in the source and analyzer or other regions of the ICR c e l l . For example, Clow and Futrell (1970) introduced a reaction region between the source and analyzer. Primary ions were f i r s t excited by applying a resonant electric f i e l d in the form of a pulse in the source region and the effects of charge exchange reactions - 65 -were monitored in the analyzer or resonance region. By using a d r i f t f i e l d in the reaction region of magnitude much smaller than those in the source and detector regions, the probability of charge exchange reactions in the source and detector regions was minimized. In order to obtain the resonance region spatial probability distribution pCy.z^) from P s ( v s \u00C2\u00BB z m s ) \u00C2\u00BB ^ s necessary to integrate the equations of motion of the ions in the inhomogeneous electric f i e l d between the source and resonance regions to obtain y(y ,z ) s 3 s ms and z (y ,z ). Then, we can write a formal expression for p(y,z ) in terms m s ms r m of p (y ,z ) as follows s s ms [9.i] p(y> z m) v\ ra /2 d y s b S dz <5(y-y(y ,z )) 6(z -z (y ,z )) p (y ,z ) ms s ms m m 3 s ms r s w s ' ms o In a similar way, one can generate P s ( v s > z m s ) from the i n i t i a l spatial distribution function of the ions in the ionising electron beam i f the potential of the electron beam is known. The problem of integrating the equations of motion of the ions as they d r i f t from one part (region 1) of the c e l l to another part (region 2) i s complicated for the general case. It is possible to give simple solutions for two special limiting cases of experimental interest. We shall c a l l these cases the fast d r i f t and the adiabatic d r i f t limits, respectively. Expressed in terms of the separation s, of the two regions, the average angular trapping frequency OJ^ and average d r i f t velocity between the regions, these limits can be defined by inequalities w^s/v^ << 1 and w^,s/vD >> 1, respectively, i.e. in the fast d r i f t limit, the time to d r i f t from one region to the other is much less than a single trapping oscillation, while the - 66 -opposite is true in the adiabatic d r i f t limit. In each case, we assume that many quasi-cyclotron oscillations occur during the time of d r i f t between regions 1 and 2, i.e. u 0 s / v n > : > 1< The implication of the last approximation is that the d r i f t always occurs along equipotential surfaces. The particular path along these surfaces i s different for the fast d r i f t and slow d r i f t limits, while the paths for d r i f t speeds intermediate to these limits should l i e between the paths for the limiting cases. In the following discussion, we denote the values of (y,z ) bv (y ,z .,) and (y~,z ~) for regions 1 and 2, m x ml 2. mi respectively. The fast d r i f t l i m i t : From the preceding discussion, i t is clear that [9-2] Zml Zm2 and [9.3] v C y ^ ) = V (y 2,z r a 2) which, for this case, are sufficient to determine Y2^ yl' ZmP a i U* Zm2^ yl , ZmP' The adiabatic d r i f t l i m i t ; The well-known condition satisfied by adiabatic mechanical processes i s that the action integrals remain constant (Born, 1969) Assuming, that the trap oscillations are decoupled from the quasi-cyclotron motions, this condition gives [9.4] ml Z l d z l = Jm2 z 2dz 2 or equivalently [9.5] ml t V ( y l ' Z m l ) \" V ( y \"> ' 2 i ) ] 2 d z Jo i ' r [V(y 2,z m 2) - V ( y 2 , z 2 ) ] 2 d z 2 - 67 -which, combined with the energy conservation equation [9.3], is sufficient to determine y 2 ( y i ' z r a i ) a n d z m 2 ^ y l , Z m l ? ' 2 Practical considerations: Under typical operating conditions v^ = 10 m./sec. and OJ^ - lO^sec.\"*\" as may be seen from Table 1. Thus, the characteristic _3 distance which defines the two limiting cases is s^ = v^/u^, ~ 10 m. In experiments such as those of Clow and Futrell (1970) in which different trapping and d r i f t voltages are used in different regions of the c e l l , the distance over which the potential varies appreciably is determined by \"end effects\". The c e l l geometry dictates then that this distance is of the order -2 of the smaller of a or b, i.e. typically, of the order of 10 m. Thus, the adiabatic d r i f t limit should normally be applicable in this type of situation. On the other hand, in taking into account the influence of the potential due to the ionising electron beam on the d r i f t of the ions, i t would appear -3 that for typical electron beams (diameter :10 m.), the d r i f t rate may be intermediate between the fast and adiabatic limits. A calculation of the effect of d r i f t between two regions in the adiabatic d r i f t limit has been carried out in Appendix 1 for the harmonic potential approximation. It would be straightforward to carry out numerical calculations of the influence of d r i f t on the spatial distribution of the ions for more r e a l i s t i c potentials, but the available experimental results do not seem to warrant such an effort at this time. In the next section, we shall develop a simplified, ad-hoc model which we have found useful in taking into account the effect of ICR potentials on the spatial distribution of the ions. It should be noted, however, that the calculation in Appendix 1 indicates a considerable dispersion of the ions in the y-direction. - 68 -10. Ad-Hoc \"Average Ion Model\" for Studying the Spatial Distribution of Ions in an ICR Cell We have shown in previous sections that the ICR properties of an ensemble of ions depend on their spatial distribution. The ions are usually produced in the source with the relatively simple distribution of Eq. [8.5] corresponding to a well defined vertical position y in the c e l l and a uniform distribution of trap oscillation amplitudes z . It seems, from the considerations of Section 9 and Appendix 1, that as the ions d r i f t from the source into the resonance region of the c e l l , an appreciable dispersion of the ions in the y-direction may result. Even for the simple distribution [8.5], however, the line shapes are not simple in the collisionless regime. For example, a marked asymmetry of the ICR line shape was predicted in Section 8. In view of the complexity of numerical calculations of the ICR line shapes, we have looked for a simplified, but r e a l i s t i c model with which to probe the spatial distribution of the ions in the c e l l . The two most easily measured line shape parameters are the frequency of maximum power absorption a) and the l i n e width defined in Section 7. Both of these parameters, as 1 -2 well as the l i n e shape, are influenced by changes in the c e l l potentials V T, V^, and in a manner which depends on the spatial distribution of the ions. Thus, study of the dependence of OJ and Wj on the c e l l potentials can give information on the positions of the ions in the c e l l , thus serving as a useful diagnostic tool for ICR spectrometers. In our model, we replace the ensemble of ions described by the distribution function p(y,z m) by an '.'average ion\" at a vertical, position y - 69 -. having a potential energy corresponding to the average potential energy of the simplified distribution function [8.5]. The average potential energy is given by [10.1] _b 2 a/2 dzm V ' ' ^ and V(y',z ) is the average potential energy of an ion characterized by m (y',z m) in the harmonic approximation as is given by [7.2], [7.7] and [7.8] to be TTZ [10.2] VCy'^) = V T - I A^Cy') J Q I C2k+l)-p] k=o Substituting [10.2] into [10.1], we obtain [io .3] = v T - I \*\(y) k=o where [10.4] a . = k TT(2k+l) ( k + % ) T T J Q(p)dp is shown as a function of y in Fig. 12 for several different potential configurations. It is convenient for our purposes to express exp l i c i t l y in terms of the c e l l potentials and the position y of the \"average ion\". V -V V +V [io.5] = v T + f 1 ( y ) v d + f 2 ( y ) ( v T - v a ) ; v d = , v a = - \u00E2\u0080\u0094 ^ - 70 -Fig. 12: The position dependence of the averaged potential for five different values of V1-V\"2 with V T = 0.5 volt and V;L+V2 = 0. Similar curves for arbitrary V.. , V\u00E2\u0080\u009E and V may be obtained from Eq.[10.3], where, using [7,2] and I10.3J oo (-l) ka sinh[(2k+l)-^] [10.6] f 1 ( y ) = + - I k \u00C2\u00A3 1 TT i 2k+l . r . .-.Nirb-, k=o sxnh[ (2k+l)\u00E2\u0080\u0094] Za and \u00C2\u00BB (-1) a cosh[(2k+l)^y-] [10.7] f ( y ) = - \u00C2\u00B1 J k \u00C2\u00A3 2 TT . 2k+l , r , 0 , ..viib, k=o cosh[ (2k+l)-^\u00E2\u0080\u0094] za Similarly, the average quasi-cyclotron resonance frequency __^(y,z^)> is given by 2 \u00C2\u00B0\u00C2\u00B0 d A, (y) [10.8] = a) - ^ - I a. ^\u00E2\u0080\u0094 o c 2u> , L k \u00E2\u0080\u009E 2 c k=o dy = \" c - K [ g 1 ( y ) v d + g 2 ( y ) ( v T - v a ) ] 3E K \u00E2\u0080\u00A2 y c 2co 3 y c. J Clearly, [ 1 0 - 9 1 \u00C2\u00ABi - --sr \u00E2\u0080\u0094 i \u00E2\u0080\u00A2 1 -1>2 c dy The amplitude averaged electric f i e l d gradient is shown in Fig. 13. Often, the resonance magnetic f i e l d is measured keeping the frequency fixed. This is given by [10.10] B e f f = B + g ]_(y)V d + g 2 ( y ) ( V T - V a ) where is identified with the magnetic f i e l d corresponding to maximum ICR absorption intensity. Following the same procedure for the d r i f t velocity [2.13], which enters \"2 - 73 -Fig. 13: Position dependence of the averaged electric f i e l d gradient, OE /9y>. As in Figs. 5 and 6, V T = 0.5 v and V1+V2 = 0. - 75 -df (y) df (y) [ 1 0 ' 1 1 ] < V = E T [ - i y - V d + - d y - ^ T - V a ) ] - =i-The amplitude averaged electric f i e l d is shown in Fig. 14. Effect of a change in the c e l l potential We now consider the effect of a change in an ICR c e l l voltage parameter on the resonance characteristics of the \"average ion\". Suppose, for example, that a small change AV i s made in the V keeping V, and V constant. Then, 1 I C S the ratio of the change i n resonance f i e l d &^-e\u00C2\u00A3f t o V^ , i s given, in the limit AVT -> 0, by [10.12] ^ e f f d_\u00C2\u00A3l + _ y ^ f l ] ^ ^ 3V\u00E2\u0080\u009E, JV ,V. S 2 v y ; 1 d dy ^ T a'dy J V3V,/V ,V, T a d 17 \u00E2\u0080\u00A2 ' T a d The f i r s t term in [10.12] i s associated with the change in B due to the y local change in \u00E2\u0080\u0094 j L- produced by AV , while the second term i s due to the dy l shift in B e\u00C2\u00A3\u00C2\u00A3 resulting from the displacement of the ions as they follow the changed equipotentials due to AV^. Since the ions d r i f t along equipotentials (see Section 9), the \"average ion\" satisfies the condition 9y_ can be obtained from [10.5]. In order to do so, however, i t is necessary to that is a constant for different parts of the c e l l . Thus, Orn -)\u00E2\u0080\u009E 9 VT V a specify the manner in which the change in V^ changes in the source and 3y resonance region. Let us i l l u s t r a t e this remark by calculating Crtr\")\u00E2\u0084\u00A2 v d VT d'a for two different voltage configurations in the ICR c e l l . Case 1. The potentials V^, V,, V are taken to be identical in the source r T d a and resonance region. Assume that the ions are produced in the source at y=0 by an electron beam which perturbs the potential in the immediate v i c i n i t y - 76 -Fig. 14: Position dependence of the averaged d r i f t electric f i e l d . As in Fig. 5 V T = 0.5 v, and V1+V2 = 0. - 78 -of the ionising region by a constant amount A . Let y be the vertical position of the average ion in the resonance region. Since the ions d r i f t along equipotentials between the source and resonance region we obtain the following relationship from [10.5], [10.13.] V T + f 1 ( 0 ) V d + f 2 ( \u00C2\u00B0 ) ( V V a ) + A = VT + f l ( y ) V d + f 2 ( y ) ( W This gives the result required to complete [10.12] f\u00E2\u0080\u009E(0) - f.(y) [10.14] Cav V , V ~ df, df T d' a 1 9 V \u00E2\u0080\u0094 - + (V -V ) \u00E2\u0080\u0094 -d dy T a'dy Case 2. Suwnose that the c e l l potentials in the resonance (V_. V > V ) and ^ - I d a S \"S s source (V , V , V ) regions are independently controlled. Then, instead of i -Gl a [10.13],the d r i f t along equipotentials gives the relation [10.15] +. f 1 ( 0 ) V d + f 2(0)(V*-V a) + 'A = vj + f x ( y ) V d + f 2(y)(V^-V^) r Then, a variation of with no change in any other of the c e l l potentials gives the result [10.16'] ) l + f 2 ( y ) r r df, df_ 3 V T V V a V r \u00E2\u0080\u0094 ^ + (V r-V r)\u00E2\u0080\u0094 2-T d a Vd dy ^ T a ;dy v s v s.v s V d* a With the substitution of either [10.14] or [i-Oj-6] , or an expression for ) appropriate for the ICR c e l l under consideration, we can interpret the \" 3VT - 79 -variation of B with Vm. Similar results are derived for the variation of err 1 3 r r or with any of the c e l l voltages. The kind of behaviour predicted ef f d is illustrated below by substituting [10;1A] into [10-J-2] (Case 1) dg dg t i o , W C ^ > v , v - s 2(y) + S - S - j ^ [f 2 ( 0 ) - f 2 ( y ) ] \u00E2\u0080\u00A2 Therefore, the slope of a B e\u00C2\u00A3\u00C2\u00A3 versus V T plot is expected to be constant both for large and small V^ ,, but i t should be different in each limit i f y / 0. - s o -i l . D i s t r i b u t i o n of Ion K i n e t i c Energies i n an ICR C e l l An important a p p l i c a t i o n of ICR i s the study of the energy dependence of the s c a t t e r i n g and charge exchange c r o s s - s e c t i o n s f o r c o l l i s i o n s between ions and n e u t r a l atoms or molecules. In order to i n t e r p r e t the ICR experiments, i t i s necessary to know the d i s t r i b u t i o n of i o n energies over which the t h e o r e t i c a l , energy-dependent c r o s s - s e c t i o n s must be averaged. The ions are produced w i t h some s o r t of s p a t i a l and energy d i s t r i b u t i o n both of which depend on the production mechanism. They then o s c i l l a t e i n the trap and d r i f t from the source to the r e a c t i o n and anal y z e r regions of the c e l l . During t h i s d r i f t i n t e r v a l , i t i s p o s s i b l e to change the i o n energy by a p p l y i n g a resonant r - f e l e c t r i c f i e l d , both the time over which the ions are s u b j e c t to ICR and the r - f e l e c t r i c f i e l d amplitude being c o n t r o l l e d by the experimenter. In t h i s s e c t i o n , we c a l c u l a t e the f i n a l d i s t r i b u t i o n of i o n k i n e t i c energies r e s u l t i n g from trap o s c i l l a t i o n and the a p p l i c a t i o n of an ICR r - f f i e l d under the f o l l o w i n g assumptions. 1. The i o n motions along the z-axis and i n the x-y plane are independent of one another. 2 . The t r a p o s c i l l a t i o n s may be approximated by simple harmonic motion. 3. The r - f f i e l d , averaged over the trap o s c i l l a t i o n , i s uniform. 4. The i n i t i a l d i s t r i b u t i o n of ions i s given by Maxwell-Boltzmann d i s t r i b u t i o n f u n c t i o n s f o r motions p a r a l l e l and pe r p e n d i c u l a r to the z - a x i s . In an experimental s i t u a t i o n assumptions 1 to 3 are v i o l a t e d to a c e r t a i n extent by the complicated p o t e n t i a l s t r u c t u r e of the ICR c e l l . Assumption 4 i s v i o l a t e d when the ions are produced by molecular d i s s o c i a t i o n . - 81 -However, no discussion of the energy resolution of the ICR spectrometer is available in the literature for even the idealized case presented here. We denote the kinetic energy associated with motion along the z-axis and perpendicular to i t by E,f and Exy respectively at a time t after the ions have been produced. The total time-dependent kinetic energy E^is given by [ 1 1 . 1 ] \= E \u00E2\u0080\u009E + E x Since the para l l e l and perpendicular motions are independent according to assumption 1 , the distribution function for E{i and is the product of the individual distribution functions, i.e. [ 1 1 . 2 ] P(E\u00E2\u0080\u009E,E A)dE 1 ( dE^ = Pj-CE^dE,, PjL(EJL)dEJ_ We wish to calculate the functions E,(E\u00E2\u0080\u009E) and PA(Ej_) given that the distributions of i n i t i a l energies \u00C2\u00A3\u00E2\u0080\u009E and e\u00C2\u00B1 are Maxwellian, as stated in assumption 4 . If the i n i t i a l effective temperatures associated with e(| and e A are T \u00E2\u0080\u009E and T X , respectively, then _ 111 [ 1 1 . 3 ] P o | (( e\u00E2\u0080\u009E) = ( 7 7 T T - ) 2 -and kT [ 1 1 , 4 ] P o x ( E x ) = where P (e\u00E2\u0080\u009E) and P (s.) are normalized distributions for one and two on \" ox degrees of freedom, respectively. - 82 -Effect of ICR on the energy distribution: We f i r s t consider the calculation of ( E ^ ) . Suppose that after the ions are produced they are subjected to ICR for a time t in an r-f electric f i e l d of amplitude E^. Then, for ions having an i n i t i a l momentum whose projection i n the x-y plane makes an angle y with the (rotating) r-f f i e l d ( B u t r i l l , 1969) [11.5] E = E + 2(E e ) 2u + ^ J - m m J. \u00E2\u0080\u00A2* where [11.6] m 2 2 2 q E 1 t 8m and u = cosy We assume that y is randomly distributed between 0 and TT , from which i t is easily shown that the distribution function for u is given by [11.-7] p(y) i r(l-y )' -1 < y ^ +1 Eqs. [11.5] and [11.7] enable us to define a distribution function G(E\u00C2\u00B1,z\u00C2\u00B1) such that G(E_L,e_L)dE_L is the fraction of ions having kinetic energies between E^and EJ. + dEj_ for those ions having a given i n i t i a l kinetic energy e x , [11.8] G(E A, e j_) P(y{EJL,eJL}) TT[4E e - ( E . - E -e ) ]' m X X m X from which ^(E^) i s calculated as follows. + [11.9] W = _ G ( E j . > e x ) P o x ( e j - ) d e i - 83 -The lower and upper limits arise from the limits on u in [11.7] and are easily shown to be [11.10] e \u00C2\u00A3 = E A + E m \u00C2\u00B1 2(E mE A) Making the substitution [11.11] % = 2(E mE J/ 5w+ E x + E m Eqs. [114], [11.8] and [11.9] give the result E.+E _ m v kTx 2(EE ) 2 .[11.12] p A ( E l ) B ! _ e I Q [ k T j_\" ] wnere [11.13-] J-o(p) = J Q ( i p ) = \ +1 \u00C2\u00A3-pw \u00E2\u0080\u00A2 YT d w -1 (1-wV i s the zero'th order Bessel function of imaginary argument. The distribution function Pj_(EA) i s plotted as a function of E^ in Figure 15 for different values of E /kT, . It reduces, as expected, to the form of P (E.) for m X oj. E =0, since I (0) =1, while for the limit E /kT, \u00C2\u00BB 1, the asymptotic form m o m i of I Q(p) (see, e.g. Watson, 1962; page 204) may be used to obtain the result ( W 2 \u00E2\u0080\u00A2>\u00E2\u0080\u00A2 m j \" 4E kT, [11,14] P ^ E J = (^~rf2 e m , E m \u00C2\u00BB k T j m JL This Gaussian function, peaked at E = E , has a relative f u l l width J. m - 84 -Fig. 15: Plots of the distribution function for the energy associated with the cyclotron motion for three values of the average energy imparted to the ions by the r-f electric f i e l d . - 86 -at half-maximum given by [11.15] = (8ln2) 2 (\u00E2\u0080\u0094) m Thus, i f an ensemble of ions i s prepared with average energy one hundred times greater than i t s i n i t i a l energy spread kT^, the fractional width of the f i n a l distribution i s predicted to be - 0.235, which is very high. We conclude that the standard method of exciting ions to higher energies in ICR spectrometers is not capable of giving good energy resolution on i t s own, even i f the Inhomogeneities in the r-f electric f i e l d are ignored. The reason for this i s that ions produced with i n i t i a l momentum components parallel to the r-f ele c t r i c f i e l d are speeded up, while those with anti-parallel components are slowed down. Since the method of changing the ion energies involves the acceleration of the ions for a fixed time, the i n i t i a l energy spread i s amplified. By contrast, in those systems in which the change in ion energy i s produced by passing the ions across a given potential difference, a l l the ions acquire the same change of energy regardless of the i n i t i a l state. Effect of trap osc i l l a t i o n s : An ion produced at position z in the trap with a momentum component P z 2 such that P z = 2me1| w i l l o s c i l l a t e between \u00C2\u00B1z^ given by [.11.16] V(z ) = V(z) - V(0) + e\u00E2\u0080\u009E E V + e m __* e\u00C2\u00BB> Poll d \u00C2\u00A3 \u00C2\u00BB dv V2 + {V_ + E I {(v-l)} In [-o v One energy region of interest for the case V >> kT- is the region o H-E J ( >> kT\u00E2\u0080\u009E. For this case, Eq.'[ll-. 23] reduces, approximately, to [11.22] with e\u00E2\u0080\u009E =0. A plot of this distribution of energies is given in Figure 16 Although P(E K ,0) diverges as E\u00E2\u0080\u009E \u00E2\u0080\u00A2> 0, this logarithmic singularity i s integrable and thus presents no problem when P(E,|,e(() i s used to calculate an ensemble average. Included in Fig. 17 as an insert is a plot of f (E1) = J P(EJ( ,0)dE|(, 3 Jo which i s the fraction of ions having E|( ^ E 1 . In the f i r s t part of this section we have derived a result for the distribution cf ion kinetic energies due to the influence of the r-f electric - 89 -Fig. 16: Plots of the distribution function P(E(|,0) for two different values of the trapping well depth, V . The insert shows f^(E'), the fraction of ions with energy E ^ E'. - 91 -f i e l d under resonant conditions, while in the second part the distribution of energies due to the trapping oscillations has been calculated when kT(|<< Often, the ions are produced with an average kinetic energy ~ V q . Under such conditions, i t is clear that a certain fraction of the ions w i l l escap from the trap. A calculation of this fraction i s given in Section 15-- 92 -12. Experimental Apparatus : Ionisation by Electron Bombardment The ICR device used in these experiments is divided into three separate d r i f t regions as is common practice [Clow and F u t r e l l , 1970]. Ions are produced in the source region by an electron beam which traverses the c e l l in the direction of the magnetic f i e l d (-z direction; see Fig. 1) after passing through a small hole, about 0.003 m. in diameter, situated in the centre of the source trapping plate. Next to the source region is a reaction region through which the ions d r i f t unperturbed by the detection rf or the ionising electron beam. The purpose of this region is to decouple ion production in the. source from the detection r f . In the third region, the analyser , ions are detected by rf excitation of their quasi-cyclotron o s c i l l a t i o n using conventional resonance techniques [Robinson, 1958]. The d r i f t plates in the analyser also serve as the tank c i r c u i t of a Robinson osc i l l a t o r of standard design with rf level continuously variable over a range from 5 to 200 mv . peak to peak. Best ICR signals were obtained with the oscillator connected to the bottom d r i f t plate, a l l other electrodes being at ac ground. The d r i f t and trapping plates are 0.014 m. and 0.025 m. apart, respectively, in a l l regions. The source, reaction, and analyser region are 0.025, 0.035 and 0.061 m long in the direction of the ions' d r i f t (x-direction, see Fig. 1) and are e l e c t r i c a l l y insulated from one another by thin mica strips. Likewise the trapping electrodes in the source are separated from the trapping plates which serve both the reaction and analyser regions, so that, the - ^ source trapping potential may be varied independently of that in the other two regions. A l l electrodes are constructed of electropolished stainless - 9 3 -steel about 0.001 ra.thick and held in position by G.E. lucalox poly-crystalline alumina rods. The ionising electrons are produced by thermionic emission from a hot wire filament, mounted in a boron nitride holder, and are accelerated through a collimating grid and across the c e l l by negative biasing on the -4 filament. The negatively biased grid is a thin plate, about 5 x 10 m. thick, having a 0.002 m.diameter hole through which the electrons pass before entering the c e l l . A schematic diagram of the filament mount and grid i s shown in Fig. 17. A similarly constructed holder containing the electron collector i s mounted on the trap opposite the filament mount. Both filament and collector mounts are supported on the lucalox rods which hold the c e l l electrodes in place. Wire mesh grids, 67% transparent, are spot welded across the holes in the trap, through which electrons enter and leave the c e l l , to prevent penetration of electric fields associated with the filament, grid and collector voltages. Tungsten, iridium, and rhenium wire filaments of varying diameters were used in the experiments reported here, and the results were found to be independent of the type of metal used. The large static magnetic f i e l d B constrains electrons to small cyclotron r a d i i at normal electron energies (15 to 100 ev). Furthermore B minimizes secondary electron emission due to electrons striking metal c e l l electrodes [Farnsworth, 1925] and therefore the electron beam should be relatively well defined spatially near y = 0 in the c e l l . From the dimensions of the filament and grid aperture we expect the electron beam -4 to be about 1 x 10 m. thick and about 0.002 m. wide i n i t i a l l y and to diverge slightly due to coulombic repulsion as i t crosses the c e l l . - 94 \u00E2\u0080\u009E Fig. 17: A schematic diagram, to scale, of the filament and it s mount, with the grid plate not shown. Further construction details are given in the text. The electron beam is indicated by the arrow. 96 A 15-inch Magnion electromagnet with a 2-s inch gap was used. At about 0.9 Tesla the magnetic f i e l d gradient across the ICR c e l l was less than -3 -5 10 Tesla/m. With this homogeneity ICR line widths of about 8 x 10 Tesla were observable. B was measured with an NMR proton probe or a rotating c o i l calibrated by the NMR probe, whichever was convenient. The system was \u00E2\u0080\u00948 evacuated by a cold trapped CVC-PMCS-4 o i l diffusion pump to about 3 x 10 torr. Normal operating pressures, controlled by a variable leak valve, were about 3 x 10 ^ torr, well within the collisionless regime for the ions under consideration. The ICR experiment offers many different parameters for modulation in a phase sensitive detection scheme. Magnetic f i e l d , source d r i f t potential and source trapping potential were a l l modulated at various stages of this work. The results were found to be independent of these modes of operation. - 97 -13. Effect of Electron Beam on ICR In these experiments ion cyclotron resonance is monitored by fixing the detection os c i l l a t o r frequency and sweeping through the resonance with B. In practice the ICR absorption intensity, line width and magnetic f i e l d at maximum signal are dependent on the trapping and d r i f t potentials in the source, reaction and analysing regions. To simplify quantitative analysis we f i r s t consider the case in which the same trapping potential, V T, and d r i f t voltages, V and V^, on the top and bottom plates respectively,are used in a l l regions of the c e l l . If the c e l l potentials are different i n the various c e l l regions they w i l l be denoted by [V,] , 1 source [ v x] a T l a i y S e r s ^ l ^ reaction e t c-> b u t w h e n a 1 1 regions have the same electrode voltages the notation V etc. w i l l be used. With this notation the d r i f t e lectric f i e l d results from the potential V^-V^ while the average potential in the c e l l is mainly determined by and the average potential of the d r i f t plates, (V +V\"2)/2. In Figs.[18] and [19] we show the variation of B \u00C2\u00A3 f with V^-V^ and V T respectively, from which i t is clear that B ^ for A r + ions i s dependent on the electron beam current, I . The line width also exhibits a dependence e on I . This i s i n agreement with Smith and Fut r e l l (1973) who find that the residence times of the ions in the source are strongly dependent on the electron beam current. No dependence of B ^ or the line width on the grid or collector potentials i s observed although we detect a slight dependence on electron energy at high currents. The electron beam can influence the ions detected in the analyser in several ways. Fi r s t we w i l l discuss the effects of electron space charge - 98 -Fig. 18: The variation of ^ e^> t n e magnetic f i e l d at which ICR occurs, with V 1~V 2 for several different ionising currents. The oscillator frequency was 346.16 kHz while V +V\u00E2\u0080\u009E = 0 and V = 0.5 volt. 0.9055 LS 0.9050 0.9045 l e ~ 10 amp J e ~ 5 X 1 0 amp \u00E2\u0080\u0094I \" 1 \u00C2\u00BB 3.0 3.5 4.0 [VOLTS] - 100 -Fie. 19: The variation of B \u00E2\u0080\u009E, with V\u00E2\u0080\u009E for several different electron beam \u00E2\u0080\u0094 c eti i currents. The oscillator frequency was 346.16 kHz, while V l + V 2 = \u00C2\u00B0 a n d V1~ V2 = 1 , 0 V 0 l t \" 0.906 0.904 A < i 0.902 -CO a m p a m p 0.900 H \u00E2\u0080\u00947\u00E2\u0080\u0094 1.0 i r 2.0 \u00E2\u0080\u0094 | \u00E2\u0080\u0094 3.0 T 1 4.0 v T [VOLTS] - 102 -[Beauchamp, 1967; Woods et a l , 1973], Ions are produced uniformly across the z-axis of the c e l l near y = 0 (Fig, 1). The negative space charge depression due to the electron beam distorts the potential at which the ions are formed so that as positive ions leave the source where they are influenced by electrons they move downward-in the c e l l in such a way as to conserve their average potential energy; that i s , for our choice of coordinates positive ions move to negative values of y. In addition their amplitude of oscillation in the trap w i l l be somewhat altered, but this w i l l be discussed in Section 14. When the ions reach the analyser the effect of the electron beam is negligible but here instead of the ions being near y = 0, the centres of their quasi-cyclotron orbits are distributed about some average y satisfying the condition; [13.1] [] . = [] + A L J analyser source where A is the potential due to the electron beam and is the average potential energy of the ensemble, the average being performed f i r s t over one period of the trapping oscillation and then over a l l possible amplitudes of oscillation. Thus the space charge of the electron beam alters the spatial distribution of ions which changes the average quasi-cyclotron frequency of the ions. If we interpret as corresponding to the average quasi-cyclotron frequency of the ion ensemble then 3E-[13.2] B = B + <\u00E2\u0080\u0094Z> eff 2u) 9y c as we have seen in Section 10. However in writing this equation, i t must be emphasized that we - 103 -1) neglect the dependence of the line shape on the inhomogeneous rf e l e c t r i c f i e l d with which the ions are detected, 2) assume that the distribution of amplitudes z i s not altered during the passage of the ions through the complicated potentials inside the c e l l , 3) assume that a l l ions have a common y coordinate, A) assume that the quasi-cyclotron frequency and d r i f t f i e l d of the average ion are equivalent to the ensemble average frequency and d r i f t f i e l d respectively. Equation [13.2] predicts a f i e l d dependent shift (to the f i r s t order) in the resonance f i e l d of the ion ensemble due to the ele c t r i c f i e l d gradient inside the c e l l . The frequency of the tank c i r c u i t of the Robinson o s c i l l a t o r i s also shifted by the dispersion which usually accompanies power absorption in these devices [Anders, 1967; Hughes and Smith, 1971], This l a t t e r frequency s h i f t varies rapidly i n the v i c i n i t y of the ICR signal maximum, is a function of the number of ions in the c e l l and results in distorted l i n e shapes. No such frequency pulling was observed in the experiments reported here, although shif t s in the os c i l l a t o r frequency as large as 15 hz have been detected in our laboratory and have been reported by others [Anders, 1967] Because the spread in the ions' quasi-cyclotron frequencies due to their trapping o s c i l l a t i o n i s somewhat smaller than the width, in frequency units, of the ICR absorption as determined by the transit time of the ions through the analyser, i t i s often a good approximation to write, 5.566|^ I [13.-3] K = - o T T l ' V 1 2 - 104 -where Bj is the f u l l width at half maximum intensity of the absorption line. It is important to note that i f we interpret the position y as the average y coordinate for the distribution of ions y = y, then a knowledge of either B ... or B, is sufficient to determine y. ef f % J Eqs. [13.1], [13.2] and [13.3] may be used to discuss semi-quantitatively the dependence of B e \u00C2\u00A3 j on 1^ shown in Figs. [18] and [19]. The space charge depression due to the electron beam has been estimated by Haeff [1939] as \u00C2\u00BB I b [13.4] A = -4.79 x 10 6 w ^ o ~ where W is the width of the beam, E Q is the energy of the electrons in electron volts and A is the space charge potential in volts. Eq. [13.4] neglects complicated edge effects near the trapping electrodes [Morse and Feshbach, 1953; Pg. 1241] but is reasonably accurate near the c e l l centre. For c e l l potentials of V T = 0.5v., = 0.5v. and V 2 = -0,5v. and an electron current I * 5 x 10~6amp. with E Q = 30ev, Eqs. [13.1] and [13.4] give A = 0,22v, that + A = -0.09v., taking from Fig. 12. This corresponds to a position y ~ -0.0033m. in the analyser region of the c e l l and roughly accounts for the variation of B g \u00C2\u00A3 \u00C2\u00A3 with V^-V2 and V^ , of Figs. 18 and 19 respectively for I = 5 x 10 ^ amp. Because of the approximate nature of Eq. [13.4] and uncertainties in measuring the space charge depression of the potential may be underestimated somewhat in the preceding discussion. However, with this qualification we see that the space charge depression can be neglected - 105 -i f A << or when I << 10 6 amp. f o r 30ev e l e c t r o n s . T h i s means that e B should be independent of the e l e c t r o n current when I i s l e s s than about e r r e -10 ^ amp. which i s not normal ly the observed case wi th our system. The ampli tude of o s c i l l a t i o n i n the trap z i s a lso s e n s i t i v e to the space m r charge through i t s dependence on z . T h i s r e s u l t s i n a non-uni form d i s -t r i b u t i o n of z m ' s and y ' s fo r which the equations presented i n t h i s s e c t i o n are not p r e c i s e . A l s o , s t r a y e l e c t r o n s may be d i s t r i b u t e d over the e l e c -t rodes i f the c e l l sur faces are s l i g h t l y d i r t y from pump o i l vapour or s i m i l a r r e s i d u e . The p o t e n t i a l i n s i d e the c e l l i s then cons iderab ly d i s t o r t e d r e s u l t i n g i n compl icated non-uni form s p a t i a l d i s t r i b u t i o n s of the i o n s . We have, indeed , found that c l e a n i n g and baking the c e l l l e s -sens the dependence of B r , on I , but that p r e c i s e agreement between e f f e experiment and the theory presented here occurs only at very low e l e c t r o n c u r r e n t s , even under best c o n d i t i o n s . As we have s ta ted b e f o r e , by measuring we are able to e s t i -mate y f o r the i o n ensemble and s i n c e measurement of Bj a l s o y i e l d s y , we have an independent check on the est imate obtained from %e\u00C2\u00A3f F i g - 2-0 shows B, as a f u n c t i o n o f V . - V - w i t h V\u00E2\u0080\u009E and V-+V,, he ld cons tan t . The % 1 2 T 1 I s o l i d curve i s the v a r i a t i o n of Bj p r e d i c t e d by Eq. [13 .3] us ing va lues of y obta ined from the B \u00E2\u0080\u009E versus V..-V,, curve which i s shown by the i n s e r t J e f f 1 2 in the bottom r i g h t hand corner o f the f i g u r e . The dashed curve i s obta ined from Eq . [13 .3 ] ' assuming a constant e l e c t r i c f i e l d i n s i d e the c e l l ; i . e . , = ( V 1 - V 9 ) / b [Beauchamp and Armstrong; 1969]. F i g . 21 y Li. e x h i b i t s the v a r i a t i o n of B __ wi th V m when V, and V\u00E2\u0080\u009E are h e l d cons tan t . e r r T L i . Here the s o l i d l i n e shows B c a l c u l a t e d from B, (B, i s shown In the e f f h % i n s e r t ) and the dashed l i n e i s that p r e d i c t e d by Beauchamp and Armstrong [1969]. In both F i g s . [ 2 0 ] and [21] the agreement between theory and - 106 -Fig. 20: The variation of the ICR line width, Bj , with V -V, for V =0.5 volt and V^+V2 = 0 in a l l regions of the c e l l . The solid line i s the theoretical line width predicted by Eq, [3.6J while the dashed line is the line width predicted by assumption of a constant electric f i e l d , E =(V -V\u00E2\u0080\u009E)/b. The insert shows the variation of ' y 1 2 B with V,-V\u00E2\u0080\u009E from which the solid line was obtained. In this eff 1 2 ~9 experiment the electron current was 4.5 x 10 amp. with a mean electron energy of about 30 ev. and an oscillator frequency of 329.51 kHz. - 108 -Fig. 21: The variation of B \u00C2\u00A3f\u00C2\u00A3 with showing the behavior predicted from measurement of the line width (solid curve) and that predicted by Beauchamp and Armstrong [1969] (dashed lin e ) . In this experiment _g the electron current was 2.3 x 10 amp. In a l l d r i f t regions of the c e l l V^-V2 = 1.0 v. and v-|+^ 2 = ^' T' i e o s c l \u00C2\u00B1-'- a t : o r frequency was 329.54 kHz. \ 110 experiment i s q u i t e good. We have there fore demonstrated the v a l i d i t y of Eq . [-13.2] 5and [13.3] i n the low current regime when the harmonic appro-ximation f o r the p e r i o d i c t rapping o s c i l l a t i o n i s used to obta in the mot iona l l y averaged e l e c t r i c f i e l d and i t s grad ient wi th respect to y . The s p a t i a l d i s t r i b u t i o n of the ions i n the ICR c e l l i s a l t e r e d by the e l e c t r o n beam, and t h i s i s mani fested by a dependence of and on I . Measurement of B as a f u n c t i o n of V. . -V\u00E2\u0080\u009E and/or V\u00E2\u0080\u009E may be used e e t i I l i to est imate the s p a t i a l d i s t r i b u t i o n of the i o n s . I f there i s no space charge d e p r e s s i o n o f the p o t e n t i a l i n the source and y - 0 , B ^ ^ i s i n d e -pendent o f V j - ^ ( F i g . 1 3 ) . For a c e l l wi th geometry a = 0.025 m. and b = 0.014 m. , B _ f v a r i e s l i n e a r l y wi th V having a s lope of 0.5242 K / U -4 -(10 t e s l a / v o l t ) when y = 0 i n the a n a l y s e r . These va lues d i f f e r somewhat from the r e s u l t s o f Beauchamp and Armstrong [1969] s i n c e a r e a l i s t i c p o t e n t i a l f o r the ICR geometry has been used to take account of the t rapp ing o s c i l l a t i o n s - I l l -14. Control of the Ions' Position i n the Analyser In the preceding section we considered the variation of B, and B \u00E2\u0080\u00A2 ' h eff with V and V^-V^ when the potentials on the d r i f t and trapping plates were the same in a l l three regions of the c e l l . In some experimental situations i t may be advantageous to operate the ICR c e l l with different electrode potentials in the various c e l l regions. However, i t is important to note that changing the c e l l potentials can alter the spatial distribution of the ions in the c e l l and thus alter their average d r i f t time through the apparatus. In this section we wish to show how the position of the ions along the y-direction in the analyser can be controlled by adjustment of the d r i f t and trapping potentials in the source. Once again Bj and B w i l l be used to estimate the ions average position y. As mentioned previously .the d r i f t electric ..field ..results from V^-V^ while the average potential of the d r i f t plates is (W^+V^)/2 so the average potential of the source region may be adjusted with either [V 1 or J T source [V..+V\u00E2\u0080\u009E] . When an ion moves from the source to an analyser with a 1 2 source different set of electrode potentials, i t s new amplitude of trapping os c i l l a t i o n and position y in the analyser depends on the nature of the transition between the two regions. There are two limiting cases, a sudden transition and an adiabatic one. In the sudden transition both and the potential V(y,z ) remain constant, (see Eqs.\" [9.2], and [9.3]) but for most cases of interest in ICR, the transition between two regions of differing electrode potentials i s nearly adiabatic; that is to say the ions undergo several trapping oscillations during the transition. For this case the ratio of the z 's before and after the transition is equal to m - 112 -the fourth root of the ratio of the trapping well depth before and after the transition. This change in the amplitude of oscillation in the trap is not a particularly drastic one, being dependent on the fourth root of the c e l l potentials, so i t i s often a good approximation to ignore any change in the levels of os c i l l a t i o n and consider only the change in y via Eq. [13.. 1]. However we must proceed with caution in using this approach, since, for example, not only does y of the ion swarm change during a transition but so does the distribution of y's. In particular, ions with large levels of oscillation w i l l undergo a larger change in y than tho.se with small z 's. , \u00E2\u0080\u00A2 m Ions are produced in the source region with y ~ 0, where the amplitude averaged potential is independent of V^-V2 (Fig. 12.) and hence the spatial distribution of the ions should be independent of [Vn-V\u00E2\u0080\u009E] i f the other c e l l potentials are not altered. There-1 2 source fore, B r j. and B, w i l l also be independent of [V,-V\u00E2\u0080\u009E] . This i s eff ^ 1 2 source verified in-Fig. 22\"where the upper curve shows B r r versus [V,-V\u00E2\u0080\u009E] \u00C2\u00B0 v eff 1 2 source with other c e l l potentials: [V,+V_] = [V,+V\u00E2\u0080\u009E] , [v,+v\u00E2\u0080\u009E] . = 0 1 2 source 1 2 analyser = 1 2 reaction ^T\"'source '\u00E2\u0080\u00A2^reaction ^T^analyser \u00C2\u00B0- 5 v and [V..-VJ . . = [V - V j . = 1.0 v. 1 2 analyser 1 2 reaction Also shown in Fig. 22- i s the result of varying V^-V2 in a l l regions of the c e l l for the same electron current. As in Figs.18 and 20 the variation of B e\u00C2\u00A3j with V^-V2 results from both a spatial rearrangement - 113 -Fig. 22: A comparison of B versus [V -V\u00E2\u0080\u009E] and B versus V -V\u00E2\u0080\u009E, \u00E2\u0080\u0094 \u00C2\u00B0 r eff 1 2 source eff 1 2 0.859-1 0.858 H < i C Q 0.856 H 0.855 H Beff v s . [ V i - V 2 reaction V ! + V2=0.0 V T = 0 . 5 V Beff vs. V1-V2 Vl+ V 2 = 0 V T = 0 . 5 v source V , - v 2 ] analyser 1.0 2.0 3.0 V R V 2 [VOLTS] \u00E2\u0080\u00941 4.0 - 115 -of the ions and a local variation of the motionally averaged electric f i e l d gradient with [V,-V\u00E2\u0080\u009E] ., which causes a shift of the quasi-cyclotron e 1 2 analyser . frequency for off-centre (y ^ 0) ions. Fig. 23 shows the dependence of B on [V m] while the 6 v eff T Jsource insert shows B, versus this potential. Increasing [Vm] while % ^ & T source keeping a l l other electrode potentials constant, forces the ions to more positive y's in the analyser hence decreasing both the amplitude averaged electric f i e l d and electric f i e l d gradient of the ensemble. This accounts for the decrease in both B, and B with increasing [V m] . Fig.24-*2 eff T source shows y versus [V m] estimated from B in Fig.23, J T source eff Fig. 25 shows B, versus [V.,+V\u00E2\u0080\u009E] and the insert shows B versus & % 1 2 source eff this same parameter. Proceeding as in section 13 we use the. measured values of to obtain an estimate of y f rom. Eq. [13. 2^]? and then use Eq. [13. 3J.-to obtain B1 shown by the solid line in the figure. Agreement between the measured and predicted values of B^ is quite good at positive values of [V..+V\u00E2\u0080\u009E] but is very poor for [V..+V-] <0. There are two possible 1 2 source 1 2 source explanations for this behavior. F i r s t l y for negative [V..+V\u00E2\u0080\u009E] the ions are v 1 2 source forced to negative y's in the analyser where the potential is no longer harmonic and the approximation, z(t) = z^ coscot i s not a valid one. If this i s the case then we cannot use the harmonic approximation to evaluate the averages required by Eqs. [13.2]- and [13.3] and a tedious analysis of the line shape i s required to obtain a reliable y. A second possibility i s that in moving towards the negative d r i f t plates the ions undergo a transition to regions of higher trapping well depth. As we have already discussed this restricts their amplitude of osci l l a t i o n in the trap since the well - 116 -Fig. 23; The effect of the source trapping potential on B^^^ and B^. The potentials in the other regions of the c e l l are given in the text. The solid curve shows B ._ calculated from B, which i s shown in eff \ _g the insert. The electron current was 7.5 x 10 amp. while the oscillator frequency was 329.89 kHz. - 118 -Fig. 24: The variation of the average ion position in the analyser, y, with the source trapping potential. In this case y was calculated from the line width, shown as a function of [V ml in Fig. 11. T source - 120 -F i g . 25: The d e p e n d e n c e o f B, a n d B c c ( i n s e r t ) o n [V,+Vj . The \u00E2\u0080\u00945 r h e f f 1 2 s o u r c e o t h e r c e l l p o t e n t i a l s w e r e : [ V N - V . ] -i = [ V , - V J = [ V , - V 0 ] = 1.0 v o l t 1 2 a n a l y s e r 1 2 r e a c t i o n 1 2 s o u r c e 1 2 a n a l y s e r 1 2 r e a c t i o n [ V s o u r c e = [ V T ] a n a l y s e r = [ V T ] r e a c t i o n = \u00C2\u00B0 ' 5 Volt' _g T h e e l e c t r o n c u r r e n t w a s 7 . 5 x 1 0 a m p . a n d t h e o s c i l l a t o r f r e q u e n c y w a s 329.89 kHz. - 122 -depth increases with decreasing y. Indeed by restricting the range of averaging over in Eqs. [13.2]\" and [13.3]; we are able to obtain consistency between the measurements of B r r and B, as shown by the J eff dashed line in Fig. 13. However, in order to f i t the measured values of B, and B at [V,+V\u00E2\u0080\u009E] = -l.Ov. we must choose y =\u00E2\u0080\u00A2 -0.0045 m. and Jg eff 1 2 source restri c t z to values ^ 0.1a which is considerably smaller m than one would expect in the adiabatic transition limit. It is therefore likely that the agreement between the measurements of B and B i implied J \u00C2\u00B0 eff -2 by the dashed line i s fortuitous and that the real truth of the matter l i e s in a combination of the two p o s s i b i l i t i e s expressed above. - 123 -15. Determination of .-the I n i t i a l Distribution of Energy in'the ICR System The i n i t i a l distribution of kinetic energies of ions formed by electron impact depends on the nature of the electronic transition from which the ions result. Ions formed by dissociation of molecules may have kinetic energies several hundreds of times greater than kT. This energy results from conversion of internal energy of the parent molecule into kinetic energy of the daughter particles. On the other hand ions formed by electrons from atoms or molecules without dissociation have kinetic energies very near to that of the corresponding neutral particles (~ 3/2 kT) due to the small ratio, of the electron mass to the atomic mass. Those ions formed in the ICR c e l l with i n i t i a l energies very much greater than the trapping well depth may escape from the system, but since the ions are formed with equal a p r i o r i probability i n each dz in the range - a/2 ^ z - a/2 and the potential distribution inside the c e l l i s known, the fraction of ions retained by the traps may be calculated. We now wish to calculate this fraction which we w i l l denote by f. An ion from a mono-energetic ensemble of ions whose i n i t i a l velocity makes an angle 6 with the z axis has an energy [15.1] e\u00E2\u0080\u009E = ^ m v 2 = E cos 29 II 2 z associated with i t s z motion. We assume that the potential V(z) in the z direction i s well represented by the approximation of Section 11 V(z) = V(0,0) + V (2 z/a) 2 o - 124 -where V = V_ - V(0,0) i s the trapping well depth. Further, i f the f i r s t o T 3 assumptions outlined in Section 11 are valid, then those ions that are formed with e\u00E2\u0080\u009E - V - V(z) w i l l hit the c e l l walls and presumably be lost o from the system. It i s therefore possible to define a cut-off angle by V - V(z) ^ [15.2] cos 6 = (-2\u00E2\u0080\u0094 ) 2 c so that those ions with 8 < 0^ escape from the trap. Those ions with 6 > 6 c w i l l remain in the c e l l i f the magnetic f i e l d in the -z direction i s suff i c i e n t l y large to ensure that the ions' cyclotron radii are much less than b/2. If the i n i t i a l velocity distribution is isotropic, then the probability of having 6 between 6 and 8 + d8 in the interval 0 ^ 6 - TT/2 is sin9d6. Note that since motion in the +z direction is equivalent to motion in the -z direction insofar as kinetic energy i s concerned, the intervals 0 ^ 9 ^ n/2 and TT/2 ^ 6 < n are equivalent. There are two regimes of V q which we must consider for ions of energy E, V ^ E and V - E . There is a 8 for each value of V(z) so that the fraction o o c of ions F ( V ) , produced with potential V(z) = V which satisfies V Q - E i s given by [15.3] F(V) = 1 -e r c sin8d8 o V -V , = (-2\u00E2\u0080\u0094)* V E ' But in Section 11 the probability that an ion is formed with potential between V and V+dV i s given by [Eq.11.20]. Thus the total fraction of ions - 125 -collected for V ^ E is o V \u00E2\u0080\u00A2 o [15.4] f = F(V)g(V)dV J o In the case that V - E a l l ions formed at potentials V - V -E are o r o captured, FCV) = 1, and the fraction of mono-energic ions collected by the trap is [.15.5] f v ~ E V w w to r o v -v j g(V)dV + (-^r-)^g(V)dV o ^ V -E o V -E , V j , V - E , o o It i s now only necessary to average f over the distribution of i n i t i a l energies, g(E), to obtain the total fraction of ions collected at a given V . When ions are formed by molecular fragmentation their distribution of o . kinetic energies depends on the nature of the electronic energy state of the molecule before and after the dissociative transition. For these ions a general g(E) cannot be defined. However, g(E) for ions formed without dissociation i s well approximated by a Maxwell-Boltzman distribution r n /T*\ /4E~ ,1.3/2 -E/kT .[15.6] g(E) =J\u00E2\u0080\u0094 (\u00E2\u0080\u0094) e 126 where I g(E)dE = 1. The t o t a l f r a c t i o n of ions c o l l e c t e d by the traps foi ^ o t h i s Maxwell ian d i s t r i b u t i o n of i n i t i a l k i n e t i c energies i s therefore [15.7] (kT) -3 /2 V r o U - E / k T 1 6 2 fV - E \" V + cos -1 [V - E o dE + - E / k T ^ e dE Making the s u b s t i t u t i o n / E / V = s i n <*>/2 and using the r e l a t i o n rz r 3 c o s * s i n 2 v + d * = ^ (|)v r(v+i) i (3) [Gradshteyn and Ryzhik , 1965; Pg. 482], Eq. [15,7] becomes, a f t e r some a l g e b r a i c man ipu la t ion , irV j - V /2kT V V [15.8] = ( ^ e \u00C2\u00B0 [ 1 ^ ) + I 0 ( ^ ) ] where I and 1^ are Besse l func t ion with imaginary arguments. Eq. [l5.8~] gives the dependence of < f > on the trapping w e l l depth for a thermal popula t ion of i ons . Since the ICR s i g n a l i n t e n s i t y i s p r o p o r t i o n a l to the number dens i ty of ions i n the c e l l one can obta in informat ion about the i n i t i a l d i s t r i b u t i o n of v e l o c i t i e s from the dependence of the s i g n a l s trength on using r e l a t i o n s such as [15.8] - 127 -The energy distribution for ions formed by dissociation of a molecule is usually discussed i n terms of an energy level diagram of the type shown in Fig. 26. The energy distribution of the fragmented particles may be of two forms providing the transition of dissociation does not violate the Franck-Condon principle [Massey 1969; Chapter 12], In Fig. 26 a transition of type A results from excitation to a repulsive state of the molecule possibly giving rise to fragments with large kinetic energies. Type B transitions occur upon excitation of the molecule to a bound state for which part of the Franck-Condon region of the ground state l i e s above the energy of i n f i n i t e separation of the molecule's constitutents. The form of g(E) arising from these two types of transitions i s shown at the right in Fig. 26. Dissociative transitions which violate the Franck-Condon principle are also possible [Massey, 1969; Chapter 12] resulting in a third type of energy distribution of the fragmented particles. This type of transition w i l l not be considered here. The ICR signal intensity i s directly proportional to the number density of ions and thereby reflects their i n i t i a l velocity distribution. Thus, to estimate this distribution we need only study the dependence of the ICR signal on the trapping well depth, or v^. In order to avoid alteration of the c e l l potentials or the ionising electron beam the magnetic f i e l d was modulated for these experiments. In this care the observed signal i s the derivative of the absorption and the strength of the - 128 -Fig. 26: Energy distribution of particles formed by molecular dissociation. The lined section shows the Franck-Condon region for the ground state of the molecule and two possible transitions leading to dissociation are indicated by the arrows marked A and B. AOH3N3 DINOaiD313 - 130 -line is best represented by the peak to peak intensity which we may write [ B u t r i l l , 1969] [15.9] A = C T 2 PP where x is the average d r i f t time of the ions through the analyser, and C is a constant depending on the ionic mass and the level of the rf electric f i e l d . Eq.[15.9] neglects the effect of rf inhomogenity and the complicated dependence of the ICR line shape, through x and w e\u00C2\u00A3\u00C2\u00A3> o n the spatial distribution of the ions. In changing the .potentials of the ICR c e l l the spatial distribution of ions in the c e l l i s altered as i s the average d r i f t time through the c e l l . Thus A is a function of V\u00E2\u0080\u009E through both x and . Also, as we show in PP T Appendix 2 the line width and signal strength are dependent on b /\ where m -2 b is the f i e l d modulation amplitude and B. is the line width as defined m h previously in Eq. [13.3]. Since B^ changes when x varies i t is necessary to correct the measured signal strengths for the effect of f i e l d modulation and we have, [15.10] = C A (B, ) 2 pp -5 2 where the dependence of A on x is cancelled by multiplying by the PP experimentally determined Bj , and C contains a normalizing factor to cancel the dependence of A on b /B, . pp m % The solid lines in Fig. 27 shows versus V Q for a Maxwell-Boltzman distribution of i n i t i a l velocities in the one case (Ar +) and a gaussian distribution centred at 8.6 ev with 5.0 ev f u l l width at half maximum. - 131 -Fig. 27: Theoretical calculation of the fraction of ions collected by the traps as a function of the well depth (solid line) together with normalized experimental results for argon ions (\u00E2\u0080\u00A2) and protons from dissociation of H 0 (\u00E2\u0080\u00A2). V0 [VOLTS] - 133 -This latter case approximates the experimentally determined distribution of energies for protons formed by dissociation of molecular hydrogen by 75 ev. electrons [Dunn and Kieffer, 1963] but neglects the slight anisotropy of the real g(E) [Dunn, 1962]. Also shown in Fig. 27 are the experimental values of found by further normalizing the ICR signal strength to one value of V Q on the appropriate theoretical curve. The upper points are for Ar*~ ions and the bottom for protons. Normally we find that Ar + and ions with similar velocity distributions are lost from our system at higher trapping potentials than is predicted by Eqs. [15.5] to [15.8]. However, this is not surprising in view of,the complex dependence of the ICR signal on the electrostatic potentials, and other incalculable factors. But, at least a qualitative statement may be made. Ions from a thermal population are easily confined in the ICR c e l l at low trapping potentials while ions with large i n i t i a l kinetic energies require larger values of to trap an appreciable fraction of the originally formed ions. Using the technique outlined here, i t is possible to obtain a qualitative understanding of the distribution of.energies in the ICR apparatus. - 134 -16. Experimental Apparatus: Surface Ionisation . In the preceding sections we have shown that an ionising electron beam distorts the electrostatic potential of the ICR c e l l so that the spatial distribution of the ions becomes poorly defined. While the analysis' used here yields estimates of the average - position of the ions, there are obvious advantages to having the ions near the geometric centre of the c e l l . Since biasing the source trap greater than that in the analyser dees not sufficiently restrict the maximum amplitude of trapping o s c i l l a t i o n , i t seems best to inject the ions into the ICR apparatus from an external source (e.g. a mass spectrometer), 0 r to cross the ion beam with a well colliminated neutral beam of the particles being investigated, thereby selecting a relatively well defined energy range. Experiments of this type using a hot wire ioniser for ion production have been performed in our laboratory. When a neutral atom or molecule with ionisation potential I strikes a surface with work function >I. If T is the temperature of the surface, then the ratio of positive ions to s atoms re-emitted is [Zandberg and Ionov, 1959] { 1 6 m l ] ^ = A e ^ / k I s n a where A is a dimensionless coefficient often set equal to one. Because of their high work function, hot tungsten or platinunr filaments ( ~ 4.5 and 4.1 ev, respectively) are ideally suited for surface ionisation of the a l k a l i metals and a small class of molecules. - 135 -The c e l l used for experiments with a hot wire differed from that described in Section 12 only in that the source region and the electron beam were removed and replaced with a stainless steel oven. This is shown in Fig. 28 while a schematic side view of the oven and ICR c e l l is shown in Fig. 29. The oven is mounted on a stainless steel post attached to a rod which supports the ICR c e l l . The receptacle of the alka l i metal is 0.021 m. long and 0.005 m. in diameter. Upon heating, the atoms effuse from the oven through a channel -4 4 x 10 m. in diameter and 0.005 m. long. The heating filament was inductively -4 wound from 3 x 10 m. (O.D.) tungsten wire and passed through two holes in a ceramic insulator which was inserted into a 0.003 m. diameter hole at the front of the oven, very near to the effusion channel. Thus the front was the hottest part of the oven to prevent pile-up of metal atoms at the effusion hole. Two straps were attached to the sides of the oven (Fig. 28), but insulated from i t , to support a tungsten filament about 0.025 m. in front of the effusion channel. This filament (7 x 10 ^ m. O.D.) is less than 5% of the width of the ICR c e l l in length (i.e. about 0.001 m\u00C2\u00BB), so that ions are created very near to the geometric centre of the c e l l . The wire was biased slightly positive to cancel space charge due to electrons and prevent them from leaving the metal surface. The average position of the ions in the ICR c e l l is altered by this potential but i t s precise influence on the energy and spatial distribution is not yet known. In any case i t has been experimentally demonstrated that the spread in energies of ions leaving a hot tungsten surface is about kT g where T g is the temperature of the surface [Zandberg and Ionov, 1959]. -. 136 -. Fig. 28: The a l k a l i oven and i t s mount together with the ionisation filament. Filament Oven Mount < - 138 -Fig. 29: A side view of the alkali oven and the ICR c e l l . Trapping Electrode ( V T ) Top Drift Electrodes (V,) thermocouple Thermocouple Analyser Region \ / Bottom Drift Electrodes ( V 2 ) Tungsten Filament Heater ; Reaction Region Alkali Oven Copper Washer I dm - 140 -By varying the biasing on the filament the average energy of the ions formed there was changed. Thus both the amplitude of oscillation in the z-direction and the height of the ions in the y-direction w i l l change with the filament biasing resulting in a variation of the averaged electric f i e l d that the ions experience in the analyser. F i g . 30 shows the dependence of line width on the potential of the hot wire. At higher potentials Bj i s smaller since the ions move to positive y's where their \" is smaller. The y exact dependence of B^ on is very complicated since the manner in which the ions' total average energy is shared between potential energy (y position in the cell) and kinetic energy (oscillation in the trap, cyclotron oscillation) is not yet completely known. For small biasing voltages, however, one expects low trapping oscillations and an ion beam that is relatively well defined spatially. The temperature of the oven was measured with a copper-constantan 39 + thermocouple. Excellent ICR signals were obtained from K ions at an oven temperature of about 70\u00C2\u00B0C. corresponding to a potassium vapour pressure of about 4x10 torr. inside the oven with the filament a dull red in colour (~1400\u00C2\u00B0K.). The efficiency of ionisation of sodium on tungsten is consider-ably smaller than that of potassium [Datz and Taylor, 1956] , but adequate 23 + ICR signals of Na were obtained with an oven temperature near 150\u00C2\u00B0C. \u00E2\u0080\u00946 (the vapour pressure of sodium at 150\u00C2\u00B0C. is approximately 7.9x10 torr-[Nesmeyanov, 1963]) and a slightly hotter filament. The temperature was also monitored at the end of the ICR c e l l near the detection region and was found to vary appreciably from room temperature only when the ion oven was heated above 300\u00C2\u00B0C. Thus the experiments reported here were performed at about 293\u00C2\u00B0K. - 141 -Fig. 30: The line width of K ions formed by surface ionisation on a hot tungsten wire as a function of the biasing on the wire, ^. In this experiment the oscillator frequency was 404.34 kHz. and the ICR c e l l potentials were V 1~V 2 = 1.0 v., V 1+V 2 = 0 v. , and V T = 0.2 v. (\u00E2\u0080\u00A2) and V T = 1.0 v. (\u00E2\u0080\u00A2). - 143 -As we have mentioned i n Se c t i o n 8, most ICR researchers f i n d that the s i n g l e resonance l i n e s e x h i b i t an asymmetry i n which the s i d e bands on the high f i e l d (low frequency) s i d e of the maximum are suppressed. One p o s s i b l e explanation f o r t h i s asymmetry was suggested i n Section 8. The model, based on the p o s i t i o n dependence of the q u a s i - c y c l o t r o n frequency, p r e d i c t s that ICR l i n e s should be n e a r l y symmetric i f y > 0 but should e x h i b i t the experimentally observed asymmetry i f y < 0. A l s o , since the asymmetry i s suggested to r e s u l t from the r a t h e r l a r g e e l e c t r i c f i e l d g r a d ients i n the ICR c e l l , the asymmetry should be l e s s prominent i f the ions have, on average, small amplitudes of o s c i l l a t i o n i n the t r a p . Therefore i t i s i n t e r e s t i n g to compare low pressure l i n e shapes using a hot w i r e i o n i s e r and an e l e c t r o n beam to the theory o u t l i n e d i n S e c t i o n 8. However, such a comparison must n e c e s s a r i l y be q u a l i t a t i v e f o r s e v e r a l reasons. F i r s t l y the theory of S e c t i o n 8 assumes a s i n g l e unique p o s i t i o n along the y axis f o r a l l ions i n the system, a uniform d i s t r i b u t i o n of amplitudes of o s c i l l a t i o n along the z a x i s , and the harmonic approximation f o r the ion motion i n the trapping d i r e c t i o n . A l l of these assumptions are c e r t a i n to be v i o l a t e d i n a r e a l c e l l , so exact comparison between theory and experiment i s u n l i k e l y . Secondly, extensive m o d i f i c a t i o n s of the spectrometer are required to convert from i o n i s a t i o n by a hot w i r e to ion production by e l e c t r o n bombardment, and s i n c e the s p a t i a l d i s t r i b u t i o n of the ions i n the c e l l i s dependent on c e l l o r i e n t a t i o n and c l e a n l i n e s s , only q u a l i t a t i v e comparisons between the l i n e shapes of the two techniques can be expected. 39 + In F i g . 31 (a) we show the K resonance as a f u n c t i o n of the b i a s i n g on the hot w i r e i o n i s e r . The asymmetry of the l i n e s i s most prominent at low - 144 -bias voltages for which the ions are expected to be in the lower part of the c e l l . As in Fig. 30 the lines at low are broader. In Fig. 31 (b) 40 + 39 + a typical Ar resonance shows a more marked asymmetry than the K lines in qualitative accordance with theory. - 145 r-39 + 40 + Fig. 31: A comparison of K and Ar resonances. The c e l l parameters were V T = 0.5Svi,, ^1 = -V\"2 = \u00C2\u00B0- 5 v-39 K + Q t 0 J , / 2 7 r = 4 2 3 k H z (i) Vf,| \u00E2\u0080\u00A2 0.6 v (vi) Vf,|=0.05v ( b ) *\u00C2\u00B0Ar + at w\/2ir=Z30kH - 147 -17. Non-Reactive Collisions: Discussion of Collision Frequencies and Ionic Energy Distribution Functions In the ICR spectrometer the free motion of the ions in the crossed electric and magnetic fields is interrupted by collisions with atoms or molecules in the background gas. This results in a pressure dependence of the ICR absorption spectra from which i t is possible to determine the ion-neutral collision frequency defined by [Beauchamp, 1967] nM [17.1] K = ^ 7 m+M d o where m and M are the ionic and neutral masses respectively, n is the neutral number density, a, is the momentum transfer cross section and v is 3 d o the relative velocity of the colliding pair. The brackets in the above equation indicate an averaging of a^vQover the ion-atom relative velocity distribution function. In this section we report measurements of E, for sodium and potassium ions in argon and helium gases. These systems are particularly simple since both the alkali ions and the inert gas atoms have closed electronic shells, hence charge exchange between ion and atom in the bi-particle collision is unlikely. At the end of this section the velocity dependence of \u00C2\u00A3 is discussed and a crude velocity distribution for ions undergoing elastic collisions with neutrals is derived. However, f i r s t let us discuss the theoretical ICR line shape for ions undergoing non-reactive collisions. When the r-f electric f i e l d used to detect the ICR signal is uniform over the spatial distribution of the ions, B u t r i l l [1969] has shown that the rate of change of energy at time t of an ion that has moved freely under the influence of the crossed electric and magnetic - 148 -fields from t to t is o dE. E 2 q 2 sin 6oj(t-t ) X 1 o [ 1 7 , 2 ] dt\" = 4m where 6OJ = OJ -OJ, , OJ is the frequency of the ion at maximum absorption o 1 o intensity and OJ^ i s the detector frequency. If the average ion undergoes no collision during the time that i t is in the analyser then t = 0, but i f the neutral particle density is such that collisions between ion and neutral are possible then t is the time of the last c o l l i s i o n . At time t dE, in the analyser there is a distribution of -j-j^\u00E2\u0080\u0094 's associated with the distribution of t 's resulting from collisions [Bloom, 1971]. The probability - C t - t O ) / T C that an ion moves from t to t without collision is e while the o fraction of ions that undergo collision in time dt is dt Ix . Therefore 6 o o c the instantaneous power absorption is - ( t - t j / x dE t dE, V L V c -t/x dE, X 6 dt +n e C [ - \u00C2\u00B1 1 _ , dt x o o dt t =0 o J o c o where n is the number of ions in the analyser and x (=\u00C2\u00A3 \"S is the mean o c time between collisions. The f i r s t term of Eq. [17.3] accounts for a l l those ions which undergo collisions between 0 and t, and the second those that move freely for this interval. Substituting Eq. [17.2] into [17.3] yields a well known expression for the instantaneous power absorption [Comisarow, 1971; Dunbar, 1971; Huntress, 1971]; 2 2 dE. q E nn -t/x , . , r - , ^ . -i X n 1 o r c / r . . cos 6oJt s , -1, [17.4] *`^ = o T~ t e (6tL) S l n 6 a ) t ; ^ + Tr- ^ o 4m(ooj +x ) c c - 149 -. If we denote the average time which an ion spends\u00E2\u0080\u00A2in the analyser of the ICR c e l l by T then the average power absorption is giyen by 2 2 -2 2 q E (x -6o> )cos 6u)T [17.5] A(6co) = ~ \u00E2\u0080\u0094 2 \u00E2\u0080\u0094 { [ \u00E2\u0080\u0094 C -2 2 4m (x +6O> ) T T + 6W c c -1 , 2 - 2 x So) sin 6u)x - T / T 6OJ - X 0 C \u00E2\u0080\u00A2, C , C X , - 2 -2^ , 2 ] 6 + , 2 - 2 + T } \u00E2\u0080\u00A2 T + <5u) O U ) + T C c c The assumption of an average d r i f t time of the ions through the analyser region is of course a crude one i f there is a large dispersion of the positions of the ions in the ICR c e l l . However, the model has met with some success [Huntress, 1971] previously and should be a valid one, particularly for our geometry where the ions are produced with restricted amplitudes of oscillation in the trapping well. It therefore seems reasonable to estimate x from the low pressure ICR absorption line, so that i t is a simple matter to estimate x c for a given ion-atom pair from pressure broadened ICR lines. The ICR c o l l i s i o n frequencies g were obtained by a least squares f i t of Eq.[17.5] to experimental spectra. Fig. 32 shows one of the best f i t s of experiment to theory while Figs. 33 and 34 show estimates of ? for Na+ and K+, obtained with very low r-f electric f i e l d s , as functions of helium and argon pressures. When the average energy gained by an ion between collisions is very small the velocity distribution is nearly Maxwellian with the same temperature as the neutrals and the ICR co l l i s i o n frequency is simply related to the zero f i e l d d.c. mobility K(o) of the ions [Ridge and Beauchamp, 1971] - 150 -Fig. 32: A f i t of an experimental K resonance CO) to Eq. [17,5] -4 (solid line). The average d r i f t time was taken as 8.08x10 sec, -4 and an argon pressure of 2.75x10 torr. was measured. The noise level of the Robinson oscillator was less than the size of the dots representing experiment on the diagram. POWER A B S O R P T I O N CARB. UNITSU - 152 -Fig. 33: The collision frequency \u00C2\u00A3 = T c as a function of pressure for Na + and K + in helium gas. Typical ICR c e l l parameters are = 0.5v, V 2 = -0.5v, V T = 0.15 to 0.5v with a bias on the ionization filament from O.lv to 0.7v. 6000 4000 2000 I 2 3 PRESSURE (xlO~ 3torr) - 154 -Fig. 34: The collision frequency of K and Na as a function of argon gas pressure. Experimental parameters are the same as for Fig. 33. - 156 -A theoretical discussion of the relations among E,, K(o) and is given in Appendix 3 and the influence of various ion-atom interaction potentials on K(o) is discussed in Section 18. The solid lines in Fig. 33 and 34 were calculated from Eq. [17.6] using the experimental results of Tyndall et a l . as given by Massey [1971]. Agreement between the ICR measurements and the d.c. mobility experiments are within the present errors in measurement of pressure [about 10%], which was measured with a Bendix G IC-017-2 ion tube on a CVC-GIC-111A control consul. Both ion tube and consul were calibrated using a McLeod gauge. In this experiment the ion flux is independent of the pressure of the background gas unlike those experiments in which the ions are produced by electron bombardment of the neutrals. Thus i t is a simple matter to compare the ICR signal amplitude as a function of pressure with the theoretical prediction of Eq, [17.5]. This is done in Fig. 35 were we plot the relative 39 + intensity of the K signal at resonance versus argon pressure, the solid line being obtained from Eq. [17.5], Measurement of the ICR signal intensity at resonance can, in principle, yield as much information as measurement of the line shape [Huntress, 1971] especially at high pressures where the analysis is particularly simple. Plots such as Fig. 35 \u00E2\u0080\u00A2 give a useful check of our experimental determination of \u00C2\u00A3 from the line shape. We now wish to extend the discussion of the ionic energy distribution given in Section 11 to include the effect of collisions. This i s of interest since no such discussion is presently available in the literature nor is there available an expression for the average ion energy which adequately bridges the gap between the low pressure regime and the steady state limit where the average energy gained by the ion between collisions is equal to that lost to the neutrals through collisions. - 157 -Fig. 35: The relative intensity of the K signal as a function argon pressure. The solid line is obtained from Eq. [B. - 159 -If the i n i t i a l energy distribution of the ions' motion in the x-y plane is represented by a two dimensional Maxwellian, with an i n i t i a l temperature , then as we have shown in Section 11, the energy distribution function after the ions have been subjected to ICR for a time t-t during J o which no collisions occur is given by -(E A + E;)/kT A 2 ( E , H m i ' [17.8] P A(E A) = ^ \u00E2\u0080\u0094 I o i - \u00C2\u00A3 f - ) -x where I is the Bessel function with imaginary argument, o E' = -5-^ (t-t T = n ( t - t ) , 0 < t ^ t m 8m o 1 o ' o and is the amplitude of the r-f electric f i e l d . Now, assuming that the average time between collisions is T , the fraction of ions per element of energy d E j L that have undergone no collision - t / T C at time t under resonant conditions is just e [P, (E,)] _ . The t \u00E2\u0080\u0094U probability that an ion which underwent a col l i s i o n at t undergoes no o c collision between t and t is e while the fraction of ions which o undergo collisions in time dt is dt /T . so that at time t the fraction of o o c ions per element of energy dE^ whose last collision occured between t and - ( t - t )/T dt \u00C2\u00B0 t +dt is e [P.(E,)] . t may take on a l l values between o o A x t ^ 0 T o 3 o c and t, so a generalized distribution function for ions undergoing collisions may be written in the form, -t/x - ( V E ^ / k T 2(Ej^E )^ [17.9] P A ( E j L , V t) = e C ^ ^ : \ ( ~ ^ - ) - V T g ,t - ( t - t ) / , - 2 ^ - V 2 ( ^ H E ^ \u00C2\u00A7 L dt e 0 C e 8 I ( 2 - ^ L \u00C2\u00B1 - ( t - t ) ) kT T I o o kT o g c J o g - 160 -. 2 where E = nt \u00E2\u0080\u00A2 The integral in the above expression may be evaluated in terms of Generalized Hypergeometric Series, but the result is not very tractable and w i l l not be given here. The f i r s t term in Eq. [17.9] represents the contribution to the distribution function of those ions which do not undergo collisions in time t while the second term accounts for those ions which do. In the steady state limit, where the average energy gained between collisions is equal to the average energy lost to the neutral particles, Huntress[1971] has shown that the average energy of an ion is different by a factor (m+M)/2m from the total energy gained between collisions. Thus in the second term of Eq. [17.9] E 1 is replaced by (m+M)E'/2m. Also, in the limit of x ^ -> \u00C2\u00B0\u00C2\u00B0 the ions must m r ' m c come to equilibrium with the neutral gas so T , the i n i t i a l ion temperature, is replaced by T , the gas temperature, for those ions undergoing collisions. P. (E , x , t)dE, is the fraction of ions whose cyclotron energy f a l l s c between E\u00C2\u00B1 and Ej_ + dE^ after being subjected to a resonant r-f electric f i e l d for a time t in a medium where the average time between collisions with the background gas is x . In this constant mean free time model x ^( = 5) c c is assumed independent of the ion energy (i.e. x^ is assumed independent of t and t ). P (E,, x , t) is shown as a function of E, for several different o * c J\" values of t / x in Fig. 36 for the special case m=M. As we would expect P A(E A, x , t) reduces to PA (E^) when x ^ = 0 and to a two dimensional Maxwellian when x ^ ->\u00C2\u00B0\u00C2\u00B0. Also shown in the insert of Fig. 36 for t / x = 2.5 is c c r E f(E') = P,(E, , x , t)dE , the fraction of ions whose energy E o - 161 -Fig. 36: The ICR energy distribution function Pi,(EJ>, x , t) versus E x for several different values of x \"*\"t. P i(E., x , t) was computed c \u00E2\u0084\u00A2* c 2 numerically from Eq. [17.9] with m=M, E = nt = 1.0 ev and m kT, = kT = 0.025 ev. The insert shows the fraction of ions with E, < E' for the case t/x = 2.5. - 163 -th In Appendix 4 the n moments of P.(E. , x , t) are calculated. This is of interest for two reasons the f i r s t being that i t gives insight into the effect of the energy distribution on measurements of energy dependent cross sections; i.e. i t gives a feeling for oEx when a is expanded in powers of E x . Secondly, quantitative information on the spread of energies in the ICR apparatus is given by (E x - (EJ_) ) Z/Ex the fractional width of the distribution, which is shown by the solid lines in Fig. 37. The distribution P, (E, , x , t) which we have discussed above results when ions are prepared by subjecting them to resonant r-f for a fixed time t, such as in double resonance experiments [Clow and F u t r e l l , 1970]. If after passing through a region of high r-f for a time t the ions enter a region in which they experience no cyclotron heating, they w i l l evolve toward equilibrium with the gas and after a time t' in the new region the distribution function becomes where the primes indicate parameters in the region with no applied r-f. It is obvious from Eq. [17.10] that the ions very rapidly relax to thermal velocities i f C x p ^ is large. Often the ions are detected and heated by the same oscill a t o r , in which case the distribution function evolves, because i t is a function of time, during the course of the measurement. In this circumstance a crude measure 2 2 5--of the fractional spread in energies is given by [t - ( t ) ] 2 / < E _ L > t where the brackets < > indicate a time average over a l l t in the range 0 \u00C2\u00A3 t \u00C2\u00A3 x, x being the time that the average ion spends under the influence [17.10] P i C E x , x c , t, xj,, t 1 ) = e P L C E X , T c, t) + (1-e - 164 -F i g . 37: The f r a c t i o n a l energy spreads of the d i s t r i b u t i o n P j^CE^ , T , t) -1 4 as a funct ion of pressure p , for T t = 1.25 x 10 p. The s o l i d 2 2 ^ 2 l i n e s show [E. - E . ] 2 / E , for two d i f f e r e n t values of E = nt M * m 2 2 J' while the dashed l i n e s show [ < E A > . - t] 2 / < E J > > t for values of 2 E = TIT . F R A C T I O N A L E N E R G Y S P R E A D - 166 -of the heating r - f . From the dashed lines in Fig.37, this fractional spread in energies i s seen to be considerably greater than the width of Pi (E, , T , t) at low pressures. j\u00C2\u00BB J. c It should be emphasized that only the two-dimensional motion in the x-y plane has been considered here, neglecting the effect of the trapping oscillations in the z-direction. If the ions are produced with thermal energies at z - 0 in the trap then P(E^) is i n i t i a l l y a one-dimensional Maxwellian and ET= -r- kT + Ej_ i f the temperature of the gas and the ions are the same. The effect of collisions is not only to damp out the trapping oscillations but also to i n i t i a t e them in the presence of ICR by conversion of part of the energy gained from the r-f between collisions into motion in the z direction. A more r e a l i s t i c ICR energy distribution than that considered here must account for both of these effects. This has been done by Whealton and Woo [1971] for ions moving under the influence of a time independent e l e c t r i c f i e l d in the absence of a magnetic f i e l d . Fig. 38 shows the collision frequency, at constant pressure, or Na with argon neutrals as a function of ^ , which was altered by varying the r-f level of the detector oscillator and the value of x used in Eq. [A4.5] c was obtained from measurements made at small E^. If the ions are produced by electron bombardment with a l l ' possible amplitudes of oscillation in the trapping potential, one whould use the amplitude averaged r-f electric f i e l d (Section 8) to calculate t. However in this experiment the ions are i n i t i a l l y produced at the geometric centre of the ce l l and the spatial distribution of the ions can be controlled to a certain extent by the bias voltage of the filament, so that the r-f electric f i e l d amplitude at y=0, - 167 -z=0 was used to estimate E ^ . This of course is an over-simplification of the problem of calculating the average electric f i e l d strength. From Fig. 38, x ^ ' i s seen to be independent of < E A > t to about 1.0 ev (160 mv peak to peak r-f voltage; E^ = 11.5 ev) at which point a decrease in the total ion current collected at the end of the analyser was observed, indicating a loss of ions from the system. A similar decrease in the total ion current was noted at about the same r-f level in the collisionless regime. The total amount of r-f energy required to expand the ions cyclotron radii 2 2 2 to a value b/2 is considerably larger than either or = q E^x /(8 m) the latter being the average energy after a time x of those ions undergoing no collisions. This phenomena has been observed by many other workers [Beauchamp and Ridge, 1971; Goode et a l , 1971; Clow and Futrell, 1971], Since under most experimental conditions there is a considerable spatial dispersion of the ions in the y-direction of the ICR c e l l and since in the collisionless regime half of the ions have energies greater than the average 2 2 energy (Section 11), a loss of ions at energies less than mb w^ /8 is to be expected. This effect also could be a manifestation of the anharmonic nature of the electric fields inside the c e l l and may even be a result of coupling between the cyclotron and trapping oscillations at large cyclotron amplitudes. The collision frequency of Na + in argon is seen to be relatively independent of average ion energy. Similar results were obtained for K + in helium and argon and Na + in helium. Because the ICR col l i s i o n frequencies were obtained from a least squares f i t of the central portion of the experimental absorption spectra, they are a weighted average of \u00C2\u00A3 ( E A) over the off resonance energy distribution function and the effective energy of the ions for the measurements of Fig. 38 is lower than [Dunbar, 1971]. - 168 -Fig. 38: The ICR collision frequency of Na T in argon as a function of at two different gas pressures. The spread in energy at \u00E2\u0080\u0094^ -1 -1 resonance with t = 0,61 ev for = 3460 sec. is given by the solid bar at the bottom of the plot. 9000f 8000 7000 6000 5000 4000 3000-2000 1000-Ar Pressure - 4x io~4 torr Ar Pressure - 2.1 x 10 4 torr approximate spead in E x at < E x>t = 0.610 v J I I L J L J 1 0.2 0.4 0.6 t (ev) 0.8 1.0 1.2 170 18. Discussion of Ion-Atom Interaction At low energies the dominant interaction between an ion and a neutral arises from the well-known polarization attraction of the ion to the dipole induced on the atom by- the ion i t s e l f . This interaction i s described by the potential 2 118.1] V(r) \u00C2\u00BB - -SSL, 2r 4 where a i s the p o l a r i z i b i l i t y of the neutral atoms and r is the ion-atom separation. For the above potential a ,v is independent of v and the do o mobility takes a particularly simple form [Dalgarno, 1958] [18.2] K- = ^ *m.=.2hi o va where u is the reduced mass of the colliding pair, a' i s the p o l a r i z i b i l i t y 3 expressed i n a^, N is the neutral gas density at atmospheric pressure at the temperature of measurement and N is Loschmidt's number. Thus K' is o -1 2 measured i n (Volt sec.) cm. (PMU) 2. In many systems Eq. [18-^ 2] adequately accounts for the observed dc mobilities. However, for the a l k a l i ion-inert gas case the experimental mobilities of Tyndall et a l . [Massey, 1969] are consistently larger than indicated by a pure polarization attractive force and therefore.other.forms of the potential will.be.discussed. The effect of short range repulsion on the mobility has been approximated by assuming a purely hard sphere c o l l i s i o n and also by combining the polarization attraction and the hard sphere repulsion in the following manner [Langevin, 1905] - 171 -, 2 [18. 3'] VCr) - - r > D = oo r ^ D where D, the c o l l i s i o n diameter is the sum of the atomic and ionic r a d i i . \u00E2\u0080\u00948 Hasse and Cook [1931] have included a r repulsive term in the potential but usually both models f a i l to account for observed dc mobilities. More recently Mason and Schamp [1958] have calculated c o l l i s i o n integrals of the form ^ v ^ using a potential which includes ^ \u00E2\u0080\u009412 a r attractive term and a r repulsion as well as the polarization attraction. They write \ e J(1+Y) Q [18.4] V(r) = ^ e|(l+ Y)C-^) 1 2 - 4 Y ( ^ ) 6 - 3 ( l - y ) ( ^ ) 4 4 2 where 3 ( l - v ) E r = e a, r is the value of r at minimum V(r) and y is an m m adjustable parameter which determines the importance of the r ^ term relative to the r ^ attraction. The r ^ term in Eq. [18.4] accounts for a charge induced quadrupole attraction and the London dispersion energy while the -12 short range repulsion is entirely represented by the r term. By varying the two adjustable parameters in Eq.[18,4] Mason and Schamp [1958] were able to f i t their derivation of K' to the temperature dependence of the experimental mobilities and obtain values of r , e and y. On the other hand Patterson[1972] defined the c o l l i s i o n diameter D to be the value of r for which the interaction potential is zero. Thus using an experimental D the parameters r^ and e, and hence K' may be calculated - 172 -f o r any Y. Dymerski et a l . have used Patterson's technique to est imate the Mason-Schamp p o t e n t i a l parameters f o r anions in numerous molecular gases. We have a lso used th is procedure to c a l c u l a t e t h e o r e t i c a l K ' s f o r the Mason-Schamp p o t e n t i a l wi th Y=0, 0.25 as w e l l as f o r the Hasse-Cook p o t e n t i a l . Table 3 compares experimental and t h e o r e t i c a l d r i f t m o b i l i t i e s f o r these d i f f e r e n t models of the i n t e r a c t i o n p o t e n t i a l . Table 4 g ives the parameters of Eq. [18 .4 ] which lead to the best agreement between theory and experiment. On the b a s i s of these ICR experiments there seems no reason to suspect sys temat ic e r ro rs i n the measurements of T y n d a l l et a l , as suggested by Dalgarno et a l . [1958] and an ion-atom i n t e r a c t i o n p o t e n t i a l l i k e that of Mason and Schamp i s required to account fo r the m o b i l i t i e s of the a l k a l i ions i n i n e r t gases. Poor energy r e s o l u t i o n of the ICR spectrometer as w e l l as lack of p r e c i s e t h e o r e t i c a l in format ion on the e f f e c t of a v e l o c i t y dependent on the ICR l i n e shape prevents a d e t a i l e d study of the energy dependence of the ICR c o l l i s i o n f requenc ies . Never the less a f i r s t order energy d i s t r i b u t i o n f o r ions at resonance with an a p p l i e d r - f e l e c t r i c f i e l d was - der ived and the average ion energy obtained f o r a l l regimes of p ressure . It i s hoped that the treatment of the ICR l i n e shape given i n th is s e c t i o n w i l l lead to a bet ter understanding of the i n f l u e n c e of v e l o c i t y dependent rate constants on the ICR l i n e shape. - 173 -Table 3. A comparison of the effect of different ion-atom interaction potentials on the d.c. d r i f t mobility. TABLE 3 EXPERIMENTAL THEORETICAL DC Drift Mobility ICR0 Polarization^ 4-Power e Hard Sphere Lanagevin^ Mason-S champ 4-12 Power Hassd-Cook8 4-8 Power 39 + K In Argon 11.7a , 11.8 b 11.4 10.8 49.6 12.2 12.9 14.4 39 + yK In Helium 41.0 a 40.7 30.5 58.7 34.7 ' 46.8 58.4 23 + Na In Argon 11. 5a 11.6 10.8 66.6 12.1 11.6 13.2 23 + Na In Helium 41.9a 42.8 30.5 80.9 35.1 45.4 57.5 a Tyndall et a l . as recorded by Massey [1969] b James et a l . [1973] c This work- \" d Dalgarno et a l . [1958] e Patterson [1972] f Lanagevin [1905] , Hasse* and Cook [1931] g Coefficient of the higher power obtained by setting V(D) = 0 h a given by Landolt-Barnstein [1950], ionic rad i i from Seitz [1940, Pg.93] o o atomic diameters of helium and argon taken as 2.18 A and 2.6 A respectively [Mason and Schamp, 1958] - 175 -Values of the parameters in the Schamp-Mason potential that lead to best agreement between theory and measured d.c. mobilities. r 176 -TABLE 4 r (A) D(A) e(ev) K'(cm2(PMU)^/v 39 + K In Argon 0.25 3.0 2.63 0.13 11.8 39 + K In Helium 0.25 2.76 2.42 0.02 41.0 23 + Na In Argon 0 2.6 2.27 0.14 11.6 23 + Na In Helium 0.25 2,35 2.06 0.04 40.7 T-. 177 r 19. A Crossed Beam Experiment As we have already mentioned, an important application of ICR involves the study of rate constants for charge exchange or ion-molecule reactions. The measurement of the cross section cr(vQ) as a function of average velocity of the ions is of particular interest, but for such investigations i t is necessary to use as well-defined an ion velocity as possible. In ICR we measure not a cross section but a rate constant '^gfvllv >,.so in order to extract information about a i t is necessary to have a thorough knowledge of the relative velocity distribution of the interacting particles. Unfo-rtunately we saw in Sect. 10 that the ensemble of ions is quite complicated since i t involves a large distribution of trapping oscillation amplitudes. Furthermore, the result of resonant r-f is to amplify the i n i t i a l spread of ion velocities. In Sect. 16 a possible method of restricting the trapping oscillation amplitudes was proposed, but there appears to be no obvious method for eliminating the large spread in energy of the ion beam when ion cyclotron heating in the manner of double resonance experiments is used. However using a background gas seems not to be the best means of introducing a target for the ion beam. Somewhat better energy selection might be achieved by intersecting the ion beam at a well-defined height in the c e l l with a secondary neutral atom beam. After traversing the c e l l the neutral beam could be removed from the system by a suitable cold trap, and since the beam need not be confined in the magnetic f i e l d , the total atom flux could easily be monitored. In our laboratory we have started such a project, and although not complete, we will present preliminary measurements here. - 178 -F i g . 39. The secondary a l k a l i oven and i t s mounting c o l l a r cr UJ I - 180 r-A second a l k a l i oven (secondary oven, Fig. 39) was mounted at the end beam. To facilate i t s removal the secondary oven was mounted in a collar which contained the heating elements. Four screws in the bottom of the collar forced the oven into thermal contact with the heaters. The collar was mounted in the same manner as the primary oven. In these preliminary experiments charge exchange rate constants of a l k a l i metal ions with a l k a l i atoms in the secondary atom beam were estimated from c o l l i s i o n broadening of single resonance absorption lines. In order to obtain a relatively intense -4 -4 atom beam a large effusion o r i f i c e (3.5 x 10 m radius and about 5 x 10 m long) was used. Two systems were studied; one in which both secondary and primary ovens were charged with potassium and the other in which the primary oven contained sodium and the secondary potassium. Rate Equations and Method of Analysis: Before we specialize to the particular cases mentioned above let us consider the reactions; which are characterized by rate constants k'(=) and k\"(=). The ' o o number densities of atoms A and B are n^ and n^ respectively and the corres-ponding ion currents are N. and NR. The rate equations for these reactions are, i 'of the c e l l opposite the oven (primary oven, Fig. 28) used to generate the ion [19.1] + k * + A + B \u00C2\u00A3 A + B + k\" + B + A * B + A [19.2] dN A dt and [19.3] dN A dN B dt dt - 181 -The f i r s t term on the right hand side of Eq.[19.2] accounts for a loss of A + ions due to charge transfer to B atoms and the second represents a gain of A + ions from collisions of B + ions with neutral A atoms. The solution of Eq. [19.2] is Nn. k\" ,\u00E2\u0080\u009E . Nn k\" [19.4] N A(t),[N A(o) - + ^ k\u00E2\u0080\u009E] e\" * nA +- nB> + ^ ,\u00E2\u0080\u009E where N^(\u00C2\u00B0) is the value of N A ( t ) at t = 0 and N (= N^Co) + Ng(o)) is the current of ions. N D(t) may be obtained from Eq.[19.4] by interchanging A D and B, and k' and k\". To calculate the instantaneous power absorption at time t we follow the procedure outlined briefly in Sect. 17, bearing in mind that for reactions [19.1] the ion populations change with time in accordance with Eq.[19.4]. Consider ion A +, for which the co l l i s i o n frequency (i.e. the inverse of the mean time between collisions) for momentum transfer to both atoms A and B is and the co l l i s i o n frequency for charge exchange with atom B i s \u00C2\u00A3'. The fraction of A + ions that undergo charge exchange between t Q and t + d t Q ; is 5' d t and the fraction of B + ions that convert to A + ions in this time o interval is d t where i s the col l i s i o n frequency for charge transfer between B + and A. At time t the total number of A + ions that have their o momentum randomized in d t is o [19.5] T N A (t Q) K d t Q + T N B (t o) 5\" d t Q T again being the average d r i f t time in the x- direction. Note that T is a number of particles since is a current. The probability that an ion A + moves without c o l l i s i o n for a time t - t is e ^ t o ^ ^ +^ J , so at time t o ' the total contribution to the instantaneous absorption by those ions that have undergone collisions in the time interval 0 < t < t is - 182 -[19.6] (N Act 0) 5 + N b c V 5 \u00C2\u00BB ) e - c t - y c ^ ) d t t where i s g i v e n by Eq , [17.2] . To obta in the t o t a l instantaneous power absorbed at time t , A C t , 6u>), we must add to Eq . [19.6] the power absorbed by that f r a c t i o n f o f A + ions that move without c o l l i s i o n from time J o t = 0, so that o ' [19.7] A ( t , Su) = T C N A c y e \u00E2\u0080\u00A2 \u00C2\u00AB B cto) e \u00C2\u00BB ) d t + T fo V\u00C2\u00B0> ^ t =b o where the zero of time i s the time at which an ion enters the ana lyser . Fur ther , note that the i n t e g r a t i o n over t i n the above i s equivalent to i n t e g r a t i o n er x , the d is tance along the ax is of the ana lyser , s ince x = Y D t Q where ov V Q , the d r i f t v e l o c i t y , i s assumed constant f o r a l l ions and independent of the c o l l i s i o n f requenc ies . Eq . [19.7] may be used to obta in the t o t a l power absorp t ion , [19.8] A (fio) = - A ( t , 6 w ) d t . To proceed fu r ther i n our a n a l y s i s i t i s necessary to s p e c i a l i z e to the processes CEq. [19.9] to [19.12]) which we wish to inves t iga te here , U \u00E2\u0080\u009E k l 3% t 4i;,+ [19.9] [19.10] [19.11] [19.12] 3 V + 4 1 K - K + K 4 1 K + + 3 9 K J 4 1 K + 3 9 K + 23 M + 39., 3 23.. a 39,.+ Na -\u00C2\u00BB+ K ->\u00E2\u0080\u00A2 Na + K 3 V + 2 3 N a + 4 3 9 K + 2 3 N a + 39 We will also study charge exchange between like isotopes of K. These reactions may be written in the form applicable to Eqs. [19.9] - [19.2] since the a l k a l i atoms have a single s-electron in their outer shell, and the corresponding ions have closed shell configurations. Symmetric resonant charge transfer occurs when A and B are identical and the energy of the transferred electron i s the same in both atoms. Thus, reactions [19.9] and [19.10] are equivalent to symmetric 41 resonant charge transfer only i f the ionisation potentials of the K and 39 K atoms are identical. There are two main isotope shifts [Stacey, 1966] resulting from the f i n i t e mass of the nucleus and the overlap of the wave function of an s-electron with the nucleus. Neither of these effects have been completely investigated either theoretically or experimentally for elemental masses less than 60 A.M.U, but they are presumably very small. To 39 41 a f i r s t approximation the ionisation potentials of K and K are equal (4.339 ev), so at the typical thermal velocities considered here, the rate constants k^ and k\u00C2\u00A3 associated with processes [19.9] and [19.10] are the same. 39 + 41 + With k, = k\u00E2\u0080\u009E the current of both j>yK+ and K ions is independent of the time 1 2 K R spent in the ICR c e l l and Eq. [19.7] takes the form, A + + (B + + e\") \u00E2\u0080\u00A2* (A+ + e\") + B +. That i s , we picture a single electron which may be on either nucleus A + or B+ and neglect a l l other electrons in the atoms. This model is particularly [19.13] A (t,6u>) = NA(o) T [ e - ( t - t Q ) U+V) d t o o + e - 184 -Using Eq. [19.8] we get 2 2 2 2 q E N Co) CC. -5w ) cos 6a)T % s u sin6u>T - T E [19.14] A CM = 2 2 {[ - T 2 T l - 2 \u00E2\u0080\u0094 ] e 4mCC +<5o) ) \u00E2\u0080\u00A2*+ 5to \u00C2\u00A3 + Su) c c c 2 - e 2 6a> c , + g 2 ,2 + V } + 6 to c where C C = C + C . Eq. [19.14] i s identical to Eq. [17.5] with T^,\" 1 C=\u00C2\u00A3) 41 + replaced by E, . This equation gives the ICR line shape for K ions moving 39 through a vapour primarily composed of K atoms. We can neglect charge 39 + 41 exchange and momentum transfer between K and the K atomic isotope since i t comprises less than 7% of the total natural abundance by mass of potassium. If we make this approximation Eq. [19.14] also gives the ICR line shape of 39 + K ions with E, and C \u00C2\u00A3 appropriately redefined. In addition to N^Co) Eq. [19.14] contains only two unknown parameters T and \u00C2\u00A3 c which must be determined from experiment. On the basis of a single experiment the relative contributions of E, and to ? c cannot be estimated. As in Sect. 17 we choose to estimate T from the ICR line shape in the collisionless regime, E,^ = 0 and then to use this value of T in Eq. [19.14] to f i t theoretical and experimental lines at higher neutral particle densities, thereby obtaining an estimate of E,^. A typical f i t of a pressure broadened 39 + 41 + K resonance to Eq. [19.14] is shown in Fig. 40. Fig. 41 shows \u00C2\u00A3 for K 39 + 41 39 and K ( ^ c and ^ c respectively) as a function of the square root of \, the number density, n g 2 of potassium atoms inside the secondary a l k a l i oven. The significance of the dependence of E,^ on n s 2 w i l l be discussed after a discussion of the experimental and theoretical line shapes for Na+ 39 ions colliding with K atoms. - 185 -Fig. 40. A typical f i t of the theoretical (solid line) and experimental 39 + (points) ICR absorption line shapes for K ions colliding with a beam of potassium atoms. The number density inside 21 -3 the secondary oven was 4.2 x 10 m , with an average dr i f t _3 time of 1.0 x 10 sec. The Robinson oscillator frequency was 420.15 kHz and the r - f level in this.and-: subsequent experiments was such that E < 0.1 ev. POWER ABSORPTION [ARB. UNITS] - 187 -41 + 39 + Fig, 41. The ICR col l i s i o n frequency \u00C2\u00A3 c of K and K ions plotted as a function of the square root of the number density inside the secondary oven, n . (xio 1 0 m \" * 2 ) - 189 -Consider Eqs. [19.11] and [19.12]. In this case there i s an energy 23 + 39 23 39 + defect between the systems Na K and Na K of 0.8 ev since the 23 39 ionisation potential of Na (5.138 ev) is greater than that of K. Hence for reaction [19.12] to take place 0.8 ev must be transfered to the system (i.e. reaction [19.12] is endothermic) from the relative translational motion of the colliding particles, and for thermal velocities k^ = 0. Sodium ions 39 + moving at thermal velocities in potassium metal vapour give rise to K ions since reaction [19.11] is exothermic and k^ is non-zero. With k^ = k\" = 0 Eqs. [19.4] and [19.7] yield, [19.15] A(t,5a0 = N A(o)x { * h e \"? , t o e - C 5 + 5 ' ) ( t - t 0 ) d t dt ^ o + e o (C+C')t rdE CdtPt = o J Substituting Eqs. [19.15] and [17.2] into Eq. [19.8] we get, N (o)q 2E 2 [19.16] A(SaO = \u00E2\u0080\u0094 {F (\u00C2\u00A3 , \u00C2\u00A3 ) + f, (1-e K )} 4m (5io +? ) ^ where P -^+5')t 2 F,u,c'r= [ o o ccsu+o-sa ) C 0 S s^t -6ob:(2\u00C2\u00A3+\u00C2\u00A3') sin 6 np) is a function of n g through \u00C2\u00A3' and of n through N (-T'). The significance of the theoretical P A curve is discussed in the text. C ( n s , n p ) [ARBIT. UNITS] - 195 -41 39 where the subscript!, refers to the K isotope, 2 to K and v^ 2 is the relative velocity of the ion atom pair. A l l parameters in Eq. [19.19] are defined in Appendix 3. The f i r s t term represents elastic collisions and the second charge transfer. Using the factorization procedure outlined in Appendix 3 for simple elastic collisions allows us to re-write Eq. [19.19] as; [19.20] where [19.21] ( A , L3t ' coll -415 < V j >f-4-V < v x > f m. (1-P 1 2) (1-cos 6) bdb)v 1 2f (Vj) F (v_2)dv_1dV2 and [19.22] = < V12 2 n (1-P 1 2) (1-cos 0) bdb > n 3 9 41. P ] 2 f ( V j ) F (V 2) v 2 bdbd.edVjdVg = < v 1 2 . 2n: P 1 2 bdb > n 3 9 n^g is the number density of potassium -39 atoms seen by the ions. The brackets < > again indicate an average over the relative velocity distribution of the ion-atom pairs. This distribution is discussed- in Appendix 5. 41 + Eq. [19.22] defines the c o l l i s i o n frequency for charge exchange between K 39 41 and K, so Eq. [19.21] and [19.22] together give a formal definition of appearing in Eq. [19.13], An analogous equation to [19.22] defines the c o l l i s i o n 23 ? 23 + 39 frequency E, for charge transfer from Na to K which appears in Eq. [19.16]. 39 + 39 The co l l i s i o n term for K moving in K is [19.23] L3 t J c o l l (n +D 2 ( 1 - P 2 2 ) ( v 2 ' - v_ 2> 2 2f(v 2)F(V 2)bdbd edv 2dV 2 + ( ( v p c - v 2) P 2 2 v 2 2f(v 2)F(V 2)bdbdedv 2dV 2} -. 196 r. i 39 + where (v 2 ) is t n e velocity of K following charge exchange. When charge transfer occurs between an ion and i t s parent atom the ion appears to be scattered through an angle n-0 (see Fig. A3.1) and integrating (y_2')c - Y_2 over \"e gives [Beauchamp, 1967] [19.24] and this leads directly to [19.25] 39 39 39 \u00C2\u00A3 = \u00C2\u00A3 + \u00C2\u00A3* c n39 { < V22 ( I T + < v 2 2 (TI (1-P 2 2)(1-cos 0)bdb)> o P 2 2 (1+cos 6)bdb) o n39 11 { < V22 C (1-cos 0) bdb + 2 P 2 2 cose bdb)>} Eq. [19.25] defines the co l l i s i o n frequency \u00C2\u00A3 \u00C2\u00A3 appearing in Eq. [19.16] for 39v+ . . K moving in i t s parent gas. 41 39 \u00C2\u00A3 c and \u00C2\u00A3 c from Eq. [19.21], [19.22] and [19.25] di f f e r because when different isotopes collide the momentum of the fi n a l ion i s uncorrelated with the orientation of the rotating electric f i e l d associated with i t s ICR since the momentum of the i n i t i a l ion has no such correlation. This is not true for collisions between like isotopes. Thus there is a persistence of momentum 39 + 39 39 41 in the K K system that results in \u00C2\u00A3 c being somewhat smaller than \u00C2\u00A3_c, -as wi l l be shown using a well known model of symmetric resonant charge transfer The co l l i s i o n frequencies in the above expressions are expressed in terms of P \u00E2\u0080\u009E , the probability of charge transfer from ion i to atom j , which in general depends on the impact parameter b and the relative velocity v^ ... The cross section for charge exchange then is [Rapp and Francis, 1962]. 119.26] a (v) = 2 n c P. . (b, v. 0 bdb 1J i j - 197 -For symmetric resonant charge transfer CA+ + A ->\u00E2\u0080\u00A2 A + A+) an expression for P \u00E2\u0080\u009E may be derived from an analysis of the c o l l i s i o n complex AA+ treated as a one electron problem iFirsov, 1951]. The non-stationary state describing the c o l l i s i o n can be expressed in terms of the symmetric and antisymmetric stationary states of the single electron orbitals. The difference in energies of these anti-symmetric and symmetric states depends only on the separation + 2 2 2 2 r of A and A + e where r = b + v ^ t , t being the time measured from - \u00C2\u00B0\u00C2\u00B0 to \u00C2\u00B0\u00C2\u00B0. By choosing a semi-empirical wave function for A + + e Rapp and Francis [1962] find that P^ oscillates rapidly between 0 and 1 for b < b^ and is sma-11 for b > b^ so replacing P \u00E2\u0080\u009E by ^ for b <_ b^ yields [19.27] a \ = /fT b c Jj 1 where \u00C2\u00A3 n D (J_) b 3 / 2 c l + ! o j e -yb 1/a o ^ ^ a V:. 1 Y b i 6 ' o i j '1 I is the ionisation potential of the atom and y = 1/13.6. This theory assumes rectilinear motion of the ions and is valid only at f a i r l y high velocities. At low velocities the ionic orbits are not rectilinear but an ad hoc model of the charge exchange may be used to calculate a . At large impact parameters and low velocities the potential between the ion and atom is dominated by the polarization attraction and one speaks of two types of collisions; orbiting collisions in which the incident ion orbits the target atom and grazing collisions. The c r i t i c a l impact parameters b Q such that a l l collisions with b < b Q result in orbiting is [Gioumousis^and iStevenson, 1958] A 2 L ^ [19.28] b Q = ( 4 q a 2) ii. . v. . - 198 -where u . . and v.. are the reduced mass and relative velocity of the colliding 1J i l pair and a pure polarization attraction has been assumed. If b Q > b^ i t i s customary [Beauchamp, 1967] to take [19.29] P 1 2 = P 2 2 = \ , b < b Q = 0 b > b \u00E2\u0080\u0094 o Substituting Eq. [19.29] into Eqs. [ 19 .21 ] , [19.22] and [19.25] gives [19.30] * \ % \u00E2\u0080\u009E 3 9 ,(lf and A-, 2 , mn 2 . [19.31] 4 \ % n_. C n ( ^ + 2.21 n C 3 ^ ) c 3 9 \ 2 m l + m 2 \u00C2\u00BB12 a being the polarizability of the neutral potassium atom. The f i r s t term 41 in Eq. [19.31] is the col l i s i o n frequency for charge exchange \u00C2\u00A3 1 and the second is 4 1 \u00C2\u00A3 for the pure polarization potential [Dalgarno et a l , 1958] . 41 41 39 39 This crude theory indicates that \u00C2\u00A3 , \u00C2\u00A3 - \u00C2\u00A3 c and \u00C2\u00A3 c are in the ratios 2.1 : 1.1 : 1 .0, to be compared with experimental ratios of about 1.9 : 0.9 : 1.0 obtained from' Fig. 4 1 . In view of the crude nature of the theory and the accuracy of this experiment the agreement seems quite good. 23 + 39 For the case of asymetric charge transfer from Na to K we might assume thattthe cross section is the same as the cross section for orbiting collisions [McDaniel, 1969, Pg. 72 ; Groumousis and Stevenson, 1958] and calculate that ] 19.32] 2V = n_ q 2n \"* M 23 23 39 23 where y 2 ^ is the reduced mass of the Na+ K system. Wecthus expect \u00C2\u00A3' 39 and ? c to be in the ratio of 2.3 : 1.0 but experimentally find that 23 39 *\J \u00C2\u00A3'/ \u00C2\u00A3c <\j 2 . 9 . The reason for this rather poor agreement of experiment with Eq. [19.31] is not known, but may l i e in the experiment. Nevertheless, 199 23 39 i t is significant that \u00C2\u00A3' > \u00C2\u00A3 . Effusion: To this point we have not attempted to make a quantitative estimate of the rate constants for the two processes studied here, since this requires knowledge of the neutral particle number density. The rather striking dependence of the co l l i s i o n frequencies on the square root of the number density inside the secondary oven also requires some discussion. Atoms from the secondary oven traverse the c e l l , and a certain fraction are ionised on the hot wire placed in front of the primary oven. The ICR signal strengths are a measure of the total flux of atoms f a l l i n g on the ioniser. It is easy 41 + to show that the area of the K resonance is given by 2 p 2 [19.33] CCn , n ) = q . 1 ft N. (o) T L J ^ s' p\"^ 4mx A v ' 41 which is independent of the col l i s i o n frequency E,^ since the total number 41 + of K ions does not change with position in the c e l l at thermal velocities when both secondary and primary ovens are charged with potassium. Thus 41 + N. (O)T , the number K ions formed at the ioniser is dependent on both n and n . But, the quantity [19.34] R(n s) = G(n s, n p) - C(0, n ) 41 + where C(0, n ) is the contribution to the area of the K resonance of the primary beam, is proportional to the flux of atoms from the secondary oven. Fig. 44 shows that R(n s) is linear with n g 2 , a strong indication that the atomic number density seen by the ioniser is proportional to n g 2 . While this behavior of R( n s) indicates why the experimentally determined col l i s i o n h frequencies are proportional to , i t does not yield an absolute number density inside the c e l l . It is therefore necessary to investigate the process - 200 -Fig. 44. ^'-ns-' a s a f u n c t i o n \u00C2\u00B0f the square root of the number density inside the secondary al k a l i oven. ^Cng) is proportional to the flux of atoms from the secondary oven as explained in the text. - 202 -of effusion of atoms through an oven o r i f i c e . Depending on the ratio of the mean free path X of the atoms in a tube to the dimensions of the o r i f i c e , several different types of molecular flow can be distinguished. For a cylindrical tube of length L q and radius c, true effusion occurs only i f X \u00C2\u00BB L Q> C [Lew, 1967], and in this case the peak intensity and total flow rate are proportional to the pressure behind the source. When X is comparable with the length of the tube the peak intensity of the beam is not proportional to the pressure behind the source since collisions between atoms in the beam are then important. The atomic \u00C2\u00B0 18 -1 diameter d of potassium is 4.76 A indicating a mean free path A ^ 10 n g I . 2 ( = l / ( / 2 n d n s ) ) - This means that for at least a portion of the density range spanned in these experiments collisions between atoms in the source played a role in defining the emergent atomic beam. Giordmaine and Wang [1960] have studied molecular flow through tubes both experimentally and theoretically. They find that the peak beam intensity i s [19.35] I(o) = ^ Ja n * 8 d L 2 5 o while the total flow rate from the source is proportional to n^' In Eq.[19.35] v & is the average velocity inside the oven source not in the beam. Since the ioniser is aligned with the secondary oven's effusion tube, the dependence of the area of the ICR lines on n g 2 (see Fig. 43) is probably explained by Eq. [19.35]. Furthermore i f the ions :in the primary beam are relatively well defined spatially near the centre of the c e l l , they w i l l see a secondary atom flux that is proportional to n s 2 . This might explain why the co l l i s i o n frequencies are dependent on n g 2 also. It should be noted, however, that the half width of the angular distribution of particles in the co l l i s i o n -203 dispersed atomic beam is also proportional to n s 2 (recall that the total flow rate is proportional to n g) and i f the ions are distributed over a considerable fraction of the atomic beam they will see an average atom flux that is proportional to n g. One further cautionary note is that the theory of Giordmaine and Wang is usually only applied to very long tubes (i.e. L Q>>C) while for the oven used in these experiments L q is about a factor of 2 larger than c. We assume that the ion beam is homogeneous having a circular cross section of area A . At large densities in the source the angular distribution of the atomic beam may be approximated by [Giordmaine and Wang, 1960]. 3/2 a [19.36] 1(8) \u00C2\u00A3 1(0) cos and for small A the atom flux across A is P P I = 2n a sin @ I (0) d \u00C2\u00A3.211 \u00C2\u00BB 2 I (9) where @h?i-s Oann angle measured from the axis of the effusion tube and @^ i s the half angle subtended by A . is assumed small. The number density at a distance L from the source oven is now just I [19.37] n(L) = - a \u00E2\u0080\u0094 -A v P a ( c \u00E2\u0080\u0094 J n c 2 I g L\" 8d L ^ i T -2 S The number density obviously varies with distance from the source so we -2 -1 replace L in Eq. [19.37] by(L' L\") where L' is the distance from the - 204 -effusion tube to the ICR c e l l and L\" is the distance to the end of the analyser. Substituting for values of the parameters in Eq. [19.36] we obtain n^g \u00C2\u00A3 4.31 x 10 ^ n s 2 . for potassium atoms. Since the slope of 39 _8 3/2 ? c versus 2 plot in Fig. 41 i s 3.1 x 10\" m /sec. we obtain a rate -15 3/ -9 3 constant of 7 x 10 m sec. (7 x 10 cm /sec.) for symmetric resonant 39 + 39 charge transfer between K ions and K atoms. In Fig. 45 this rate constant is compared with other measurements by Kushnir et al [1959] who measured the attenuation of a potassium ion beam in a neutral potassium background. The solid line in Fig. 44 shows the theoretical prediction [Rapp and Francis 1962] of Eq. [19.27]. This line differs slightly from that published by Rapp and Francis, due to the use of a slightly different ionisation potentials. Also shown by the dashed lines in Fig. 45 is the value of a c v ^ obtained from Eq. [19.30] using two different values of the polarizability [Landolt-Bornstein, 1953]. The error bars on the point 2 J' represent estimates, of the errors in measurement of c / L Q 2 in the one case, and the kinetic temperature of the ions in the other. Double Resonance Double resonance experiments were also performed on the two a l k a l i ion-atom systems using the crossed beam arrangement, but these did not meet with much success. In order to selectively heat either ionic species 41 + 39 + ( K or K in this case), a secondary oscillator (Wavetek Model 114) was connected to the positive d r i f t electrode in the reaction region. The double resonance experiments consisted of fixing the magnetic f i e l d and 41 + monitoring the single resonance signal of K with the fixed frequency Robinson oscillator attached to the bottom plate of the analyser region. When the secondary oscillator was swept through the cyclotron frequency of - 205 -Dependence of the rate constant for charge exchange between 39 + 39 K and K on relative velocity. - 207 -39 + 41 + K ions a change in the K ICR signal was detected due to the coupling 39 + 41 + of K to K via reaction [19.9]. However we found that both the sign and magnitude of this double resonance signal werea functions'of the magnetic f i e l d as indicated in Fig. 46 which shows the change in the ICR single 41 + resonance signal of K as a function of the secondary oscillator frequency for several different values of the magnetic f i e l d . A qualitative explanation of the above phenomena might be as follows. 39 + In cyclotron heating the K ions we change both their average energy and 39 + spatial distribution. Changing the average energy of K increases the 41 + rate constant k^ hence increasing the number of K ions which leads to an 41 + increase in the K signal. On the other hand changing the spatial 39 + 41 + distribution of the K ions changes the spatial distribution of the K ions due to charge exchange reactions, resulting in a change in their average 41 + quasi-cyclotron frequency. This shift of the resonance condition of K 41 + also leads to a change in the K signal i f the magnetic f i e l d i s constant, 41 + but does not reflect a change in the net number of K ions detected. To 41 + pursue thismmatter somewhat more quantitively let us assume that the K 39 + ICR line shape is the same when K are irradiated as when they are not. 39 + 41 + Then in the absence of K cyclotron heating the K signal is [19.38] = qi G(B - B e \u00C2\u00A3 f) where B is the magnetic f i e l d , B e f\u00C2\u00A3 the magnetic f i e l d at maximum intensity 39 + and G(B - B ~_) is a shape factor so that G(0) = 1. When K is irradiated 41 + the K signal strength is [19.39] S 2 = Q2 G(B - B' f f) where B' the f i e l d at maximum intensity, i s different from B because eff ' eff 41 + of the spatial rearrangement of the K ions. The ICR double resonance signal is just the difference between $ 2 and S^, - 208 -Fig. 46 The double resonance signal AS for reaction [19.9] plotted 41 + as a function of B and t o 2 / 2 n . The K resonance was monitored with the Robinson oscillator and is indicated as a function of B. - B by the base line of the double 1 eff J resonance signals. AS was obtained by sweeping a heating oscillator (frequency denoted by i o 2 / 2 n ) through resonance 39 + with K . The Robinson oscillator frequency was 420.15 kHz. and the number density inside the secondary a l k a l i oven was 20 -3 39 + 6.2 x 10 m . The fin a l energy of the K ions was about E_ = 0.7 ev. 210 [19.40] S 2 - S 1 = Q 2 GCB - B\"eff) Q1 G CB B e \u00C2\u00A3 \u00C2\u00A3) Expanding G(B - B^ \u00C2\u00A3 \u00C2\u00A3) i n a Taylor's series about B - B g \u00C2\u00A3 \u00C2\u00A3 we obtain, [19.41] S 2 - S x = Q 2 [G(B - B e f f) \u00E2\u0080\u00A2 CB e \u00C2\u00A3 f - B; \u00C2\u00A3 \u00C2\u00A3) G'(B - B e \u00C2\u00A3 \u00C2\u00A3) + C \u00E2\u0080\u0094 \u00E2\u0080\u0094y G\"(B - B )+....] 21 e r - Q l G(B - B e \u00C2\u00A3 \u00C2\u00A3) (Q2 - Q l)GCB - B e \u00C2\u00A3 \u00C2\u00A3) - (B e \u00C2\u00A3 \u00C2\u00A3-B; \u00C2\u00A3 \u00C2\u00A3)G'(B - B e \u00C2\u00A3 \u00C2\u00A3 ) + -If we now normalize S to 1 then = 1 and S S [19.42] 2 ^ 1 = AS = Q G(B - B e \u00C2\u00A3 \u00C2\u00A3) + ( B e \u00C2\u00A3 \u00C2\u00A3 - B; \u00C2\u00A3 \u00C2\u00A3)(1 + Q) g'(B - B e \u00C2\u00A3 \u00C2\u00A3) where Q = '()Q2 - Q-^ )/Q^ . This expresses the double resonance intensity in terms of the single resonance signal and it s derivative, G' (B - B g \u00C2\u00A3 \u00C2\u00A3) = dG/dB, higher derivatives having been neglected. Q and ( B g \u00C2\u00A3 \u00C2\u00A3 - B^ \u00C2\u00A3 \u00C2\u00A3)(1 + OJ are parameters which contain information on the variation of the rate constant 39 + with the average energy of K as well as the spatial rearrangement of the ions by the secondary oscillator. Fig. 47 shows a f i t of AS versus B - B g \u00C2\u00A3 \u00C2\u00A3 (Eq. [19.42]) to the experimental results. Similar results were obtained at higher irradiating amplitudes where f i t s of the experiment to theory require higher derivatives of G. Using the theory developed in this section and plots such as Fig. 47 39 + i t should be possible to extract the dependence of k^ on K average velocity, provided that the r - f level of the secondary oscillator is maintained below 39 + the ejection threshold of K and i f the higher derivatives in Eq. [19.41] are small. We wi l l not attempt this analysis here, but these results on the energy dependence of reactions [19.9] to [19.12] wi l l be reported at a later date. - 211 -Fig. 47. A f i t of the experimental double resonance signal AS (c) for reaction [19.10] to Eq. [19.42] represented by the solid line. The line shape factors G(B - B^^) and 41 + G'(B - B^^) were obtained directly from the K single resonance line shape indicated in the previous figure. The analysis yields the parameters Q = 0.187 and B g \u00C2\u00A3 \u00C2\u00A3 -B' = 1.96 x 10\"4 Tesla. eff - 2 1 3 -A dependence of the sign A S on B has not been reported in the literature and appears to be unique to our apparatus. Similar results were obtained in the sodium-potassium system with the same Robinson oscillator, and the large separation between and 2 0 0 kHz) in this case seems to eliminate the possibility of beating between the oscillators. The same effect was noted when the secondary oscillator was applied to the top electrode of the analyser region, quite far removed from the ioniser. It is thus quite probable that the explanation offered here is the correct one, and the phenomena reflects the non-uniform number density in the beams. The analysis of this section has neglected spatial variation of the atomic flux from the secondary oven, so the results are admittedly crude ones. Furthermore estimation of the number density inside the ICR c e l l is based on a theoretical calculation, not on an absolute calibration, although the theory was crudely tested by weighingg the amount of metal plated out on a target. However, i t seems reasonable to expect more reliable results after the effusion from the ovens is studied more thoroughly. This may be done using spectroscopic techniques. - 214 -20. Summary In this thesis we have developed a theory of Ion Cyclotron Resonance for typical cells of rectangular cross section. The effect of inhomogeneous electrostatic fields on the dynamical motions of the ions was investigated in some detail using an expansion of the electric f i e l d to the third power of the y co-ordinate. An ensemble to specify the spatial distribution of the ions as they d r i f t through the complicated fields was developed. An explicit energy distribution function was derived for ions at resonance with a uniform r-f electric f i e l d . It was found that the i n i t i a l spread in energy of the ions was amplified by such resonant r-f fi e l d s . The ionic energy distribution is also broadened by the large distribution of trapping oscillation amplitudes. Production of the ions with small amplitudes of oscillation at the bottom of the trapping well has obvious advantages, and we have investigated one possible method of doing this. The hot wire ioniser is relatively easy to woperatieh but can only be used for a very small class of molecules. However, there seems no reason why the ions cannot be produced from a well collimated molecular beam which is made to cross an electron beam at the geometric centre of the c e l l . Better energy selection might also be expected if a second molecular beam crosses the ion beam at a well defined cyclotron radius. This secondary beam might be incident along the x-axis as in the experiments reported here or along the z direction in which case the particles need not be neutral. We believe that the experiments performed here, although in a preliminary stage of development, indicate that such techniques are feasible. It should be emphasized however that the theory of ICR presented here is based on a linearized model of the ionic motions in which the cyclotron motion is rigorously/ decoupled from the - 215 -trapping oscillation. At high r - f levels when the ion cyclotron radius becomes an appreciable fraction of the c e l l dimensions, this is no longer true, and the ionic motions and the manner in which energy is shared between the trapping and cyclotron amplitudes are probably quite complex. Of course the study of the energy dependence of cross sections for charge transfer and ion molecule reactions is not the only application of the ICR device. Its most attractive feature is i t s a b i l i t y to guide an ion beam at very near thermal energies in a well defined direction. Thus i t s major use is in the determination of thermal energy rate constants for ion molecule reactions. It is probably more reliable to determine the short range part of the interaction which determine these rate constants by temper-ature dependent studies rather than by their dependence on average ion energy. At near thermal energies the precision with which rate constants may be determined is dependent on the accuracy with which the pressure and average dr i f t time are measured. It is hoped that the calculation of the average dr i f t velocity and the treatment of the ICR line shape given here wi l l be of use to workers in this f i e l d . - 216 -Appendix 1 Drift of Ions Between Different: Regions of an ICR Cell Consider two regions 1 and 2 of an ICR c e l l of the type shown in Figure 1, which are characterized by c e l l parameters a^, b^ and , and (1) (1) (2) (2) voltages V (y^,z^), V and V (y2>z2)> v>p \u00C2\u00BB respectively. Suppose that an ion has a ver t i c a l position y^ and a reduced trap oscillation amplitude P- = 2z ,/a . After drifting from I to 2, the ion has a vertical position y\u00E2\u0080\u009E X ml _L 2. and reduced trap oscillation amplitude p\u00E2\u0080\u009E = 2z 0 / a 0 . As has been shown in Section 4, the values of and p2 can be obtained in a straightforward way in terms of y^, p^ and the c e l l dimensions and voltage parameters for the limiting cases of fast d r i f t and adiabatic d r i f t . In this Appendix we derive explici t expressions for y^ and p^ in the adiabatic d r i f t limit for potentials in regions 1 and 2 given by the harmonic approximation, i.e. [Al.l] V ( i )(y.,p^) = V ( i )(y.) + [ V ^ - V ^ ( y . ) ] p ] 2 where [A1.2] V ( i ) ( y . ) = V ( i )(y.,0) and z . [A1.3] p! = - i - (Note that 0 ^ p! < 1) i Then, Eqs. [9.3] and [9.5] for the adiabatic d r i f t limit may be written for this case as [A1.4] V ( 1 ) ( y 1 , p 1 ) = V ( 2 ) ( y 2 , p 2 ) - 217 and IA1.5] [V<\u00C2\u00BB-V<1><,1)r1pj[- lV<\u00C2\u00BB-V\u00C2\u00AB\y2\u00C2\u00BBV2 2 respectively. Substituting fcr from [A1.5] into [A1.4] and [Al.l] gives the equation 2 2 9 2 2 [A1.6] u 2 - U 1 P 1 U 2 + ( u ^ - u - A ) = 0 where [A1.7] u2\u00C2\u00B1 = V< i } - V ( i )(y.) and [A1.8] A = V< 2 ) - Vr^1} The only physically allowed solution to [A1.6] i s [A1.9] u 2 = \ + [ u 2 ( l - \ p 2 ) 2 + K]k By squaring [A1.9] and using [A1.7], this solution may be written in the form [ALIO] V ( 2 ) ( y 2 ) = V ( 1 ) ( y ] ) + u 2 p 2 ( l - \ p 2 ) { l - [1 + 2 - \ 2 2 ] H } . u l ( l - - P l ) Special Case: A \u00C2\u00AB u 2 = - V^Cy ). For this case of a small difference between the trap voltages of 1 and 2, - 218 -2 expansion of [A1.10] in powers of A/u^ gives the result A p 2 [Al.ll] V ( 2 ) ( y 2 ) = V ( 1 ) ( y ] L ) 2(1- 2 p p It may be seen from calculations such as these that the influence of different potentials in different regions of the c e l l can give a substantial dispersion of the beam in the y-direction. For example, i f one identifies A in [ A l . l l ] and [A1.8] with the effect of the ionising electron beam on the c e l l potential and i f i t s effect is to displace the potential near y^ by a constant amount A, so that [AI.12] v^ 2 )-v^ 1 } : v ( 2 ) ( y ; L ) - v ( 1 ) ( y ; L ) = A then the approximately linear variation of V ^ ( y ) for small changes in y, i.e. [A1.13] V ( 2 ) ( y 2 ) = V ( 2 ) ( y ; L ) + A(y2-Vl) + - -= V a ) ( y \u00C2\u00B1 ) + A + A(y 2- y ]_) + - -taken together with [ A l . l l ] gives [A1.14] y_-y 2 n I 2 (1- 2Pi) This result predicts a large dispersion of the beam [(Ay) /(Ay) . ~ 2], max min - 219 -since ions are produced with uniform probability in the range 0 ^ ^ +1. Note that since p2^P^_ ^-s Proportional to the fourth root of the ratio of potential well depths in regions 1 and 2 (see Eq. [A1.5])., the dispersion in the reduced maximum trapping oscillation amplitude produced by adiabatic d r i f t i s not expected to be as large as the dispersion in y. - 220 -Appendix 2: E f f e c t of Magnetic F i e l d Modulat ion On ICR S igna l s In the constant e l e c t r i c f i e l d approximation the low pressure ICR l i n e shape i s g iven by [ B u t r i l l , 1969] \u00E2\u0080\u00A2 2 * i , 2 . 7 8 3 sin oB \u00E2\u0080\u0094 ? 7 ft \"3 [A2.1]- A(B) = Z ' ^ 5 (2.783 5B/B, ) 2 '2 where B, i s the a b s o r p t i o n l i n e width defined by Eq.[13.3] and 6B(= B-B ,.,,) -2 err i s the d i s t a n c e a long the magnetic f i e l d axis from the centre of the l i n e . The l i n e shape i n E q . [ A 2 ; l ] has been normalized to u n i t y , [A2..2] A(B) dB = 1 I f we modulate the magnetic f i e l d wi th a smal l p e r i o d i c f i e l d , b cos ur> f and r m m sweep through the resonance at ^ e \u00C2\u00A3 \u00C2\u00A3 then t [A2.3] B( t ) = B (t) + b cos u:t-m m. where B'(t) i s the s lowly v a r y i n g a p p l i e d magnetic f i e l d . Under these cond i t i ons the output of a phase s e n s i t i v e detector i s p r o p o r t i o n a l to the c o e f f i c i e n t of the f i r s t harmonic term i n the F o u r i e r expansion of the resonance l i n e shape [Smith, 1964]. Therefore r e w r i t i n g Eq. [A2.1] s i n 2 2 * ( 6 B ' + b cos tot) [A2.4]- A(B) = 2 _ H ( U | 3 _ + _ b c o g w 2 B. m m h 2.783 v \u00E2\u0080\u0094 - \u00E2\u0080\u0094 ) a cos . nuo t TTBJ l n m H. n=0 - 221 -where K B ' = B 1 (t) - B We thus see that the quantity of interest is [A2.5] a = -1 IT f 1 T sin^(3 + a c o s 6 ) a , a - \u00E2\u0080\u0094 - cosOdG -TT ( 6 + \u00C2\u00AB c o s 6 ) In the above equation 6B' 3 = 2.783 and 2.783 b /Bi m -2 co t m As i s well known a^, the output of the p.s.d., i s an approximation to the derivative of the absorption line, but the nearness of a-^ to the real derivative depends on the amplitude of the magnetic f i e l d modulation. At very low values of b , a, is a quite adequate representation of the m 1 derivative of the true line shape but when b^ i s large the observed signal may be much broader than the real line. In Ion Cyclotron Resonance experiments the signal intensity i s often of interest and this too is a function of the modulation amplitude. Thus there are two parameters that must be examined as a function of b . The f i r s t i s the observed m peak to peak line width Bpp\u00C2\u00BB o r the separation between the extrema of a, , and the other is the value of a, at i t s maximum ( a j . 1 1 1 p In Fig.[A2\".l] we show B /B, versus b /B, obtained numerically from PP ^ m % J Eq. [A2.5] while Fig.[A2.2] shows ( a x) > normalized- to i t s maximum value, as a function of b /B, . It is interesting to note that maximum signal m h \u00E2\u0080\u00A2 intensity i s obtained for b /B, - 0.68 but undistorted line shapes occur m % only for b /B, < 0.2. J m *5 - 222 -Fig. A2.1; The effect of magnetic f i e l d modulation amplitude b on the apparent line width B of the low pressure ICR absorption PP r derivative. C N C N 00 CQ \ o E o o T O C O T q C N C N O W d d a - 224 -Fig. A2.2: The effect of magnetic f i e l d modulation amplitude on the ICR signal intensity. - 226 -Appendix 3 : Bi-Particle Collisions The effects of collisions on the motions of ions through a gas of uniform density are treated in this appendix. We.discuss both elastic collisions and resonant charge exchange and in the case of simple elastic collisions establish a formal relationship between E , the collision frequency, and K(o), the dc mobility. The influence of the ion-atom interaction potential on the momentum transfer cross-section and the transport properties of gases is also discussed. The outline presented here follows closely that of Beauchamp [1967]. From the Boltzmann equation i t has been shown that the time rate of change of some property x(v_^) of ions moving under the influence of external forces in a neutral gas is given by [ A l l i s , 1956] a q(n ) 9x(v ) [A3.1] f r[(n +) J = \u00E2\u0080\u0094 <(E(t) + v.xB). 1 \u00E2\u0080\u00A2> dt ^ l i f m. \u00E2\u0080\u0094 \u00E2\u0080\u0094 i \u00E2\u0080\u0094 3v. f l \u00E2\u0080\u0094 i + I j r [x(v!) - x(v.)] f(v.)F(v.)v.. bdbdedvdV - i - i - i - J i j where V. is the velocity of neutral atom j , v. the velocity of ion i , v.. the - J - i i j magnitude of v.-V. and F(V.) the three dimensional Maxwellian velocity - i \u00E2\u0080\u0094j - j distribution characterizing the neutrals j . f(v_^) is the velocity distribution of ion i and is normalized to (n+)^., the ion number density. In terms of the velocity distribution discussed in Section 17, f(E.) = (n +). E P (E.-E(|) P\u00E2\u0080\u009E(EI()dEM o 1 2 where E_^ = \u00E2\u0080\u0094 a n d P(l (E\u00E2\u0080\u009E) must include the effects of energy transfer, via - 227 -collisions between the cyclotron and trapping oscillations. The brackets < >\u00C2\u00A3 in Eq. [A3.1] indicate an average over f(v^). In a collision the ion interacts with a neutral, their velocities changing from v^ and V to v\ and respectively. The scattering parameters are b, the i n i t i a l impact parameter, e the scattering azimuth and 0, the angle through which the relative velocity vector v.-V. rotates on co l l i s i o n . Thus the f i r s t term on the right hand side of Eq. [A3.1] accounts for the rate of change of (n +)^ xCv^) due to externally applied electric and magnetic fields while the second term represents the rate of change of (n +)^ x(v^) due to collisions. The equation of motion of the average ion i is found by setting x(v^) = and, i f 9(n +) /9t = 0 9 9v. [A3.2] \u00E2\u0080\u0094 \u00E2\u0080\u0094 i - = 3L. E(t) + SL. : x B + ( \u00E2\u0080\u0094 i ) 9t i i ~~ c o l l 9v. where (\u00E2\u0080\u0094\u00E2\u0080\u0094-) accounts for the effect of collisions on . For ions i dt c o l l \u00E2\u0080\u0094 l r colliding with neutrals i and j there are five elastic or charge exchange reactions which alter ^ . These are \u00E2\u0080\u0094 i f (1) elastic collisions and charge exchange between ions i and their atomic parents, i + + i ->\u00E2\u0080\u00A2 i + + i + i + i + (2) the above two processes for i + ions with neutrals j , i + + j + i + + j .+ , . \u00E2\u0080\u00A2 ..+ i + J -> i + 3 (3) charge transfer between ions j and neutrals i .+ , . . ..+ J + i \u00E2\u0080\u00A2* 1 + J - 228 -Fig. A3.1; A collision in velocity space, elastic collision. V and v are not changed by an - 230 -The two processes in (1) above cannot be experimentally distinguished from one another. The second of (2) results in a decrease and (3) an increase in the population of ions i . If P is the probability of charge exchange between ion i and neutral j on collision then [Beauchamp, 1967] 3 v . [ A 3 - 3 J <;>r>coii \u00E2\u0080\u00A2 < 4 [ j (v!-v.)(l-P..)f(v.)F(V.)v.. bdbdedv.dV. \u00E2\u0080\u0094 i \u00E2\u0080\u0094 i i i \u00E2\u0080\u0094 i \u00E2\u0080\u0094 l i i \u00E2\u0080\u0094 l \u00E2\u0080\u0094 i [(v!) - v.] P.. f(v.) F(V.) v,. bdbde.dv.dV. \u00E2\u0080\u0094 i c \u00E2\u0080\u0094 i i i \u00E2\u0080\u0094 l \u00E2\u0080\u0094 x i i \u00E2\u0080\u0094 l \u00E2\u0080\u0094 l + I J*1J (v.- v.)(l-P..)f(v.)F(V.) v.. bdbdedv.dV. - i - i I J - l - j i j - i \u00E2\u0080\u0094 j v. P.. f(v.) F(V.) v.. bdbdedv.dV. - i i j - i - j i j - i - j + I Kzi)c p - i X-\u00C2\u00B1 f(Z-) F(Y_.) v.. bdbde.dv.dV.] j^2_J J J i J j i / where Cv\) c is the value of v^ following charge exchange and the reactions listed above correspond to sucessive terms in Eq. [A3.3]. For elastic collisions ( P\u00C2\u00A3^ =0) between ions of mass m with neutrals of mass M Eq. [A3.3] becomes (dropping the subscripts i and j) [A3.4] = f 3t c o 1 1 n+ (v'-v) f(v) F(v) V q bdbdedvdV where V q = ||v-V| . v, V, V Q , 9 and V ( = [mv + MV] / [m+M]) , the velocity of the centre of mass are indicated by Fig. A3.1 from which i t is seen that [A3.5] 2TT (v'-v) de = - 2-rr \u00E2\u0080\u0094 (1 - cos0)(v-V) - 231 -The momentum transfer cross section is defined as [McDaniel, 1964] [A3. 6] a, = 2TT d (1 - cos6) bdb and the collision term becomes [A3.7] (!) -M TE'coll \" n+(m+M) j ^ Y o d ^ since the average of in Eqs. [A3.5] and [A3.4] over F(V) vanishes. We 9v cannot evaluate unless the velocity dependence of o.v is known, but 9t c o l l d 0 > fortunately the momentum transfer cross section i s inversely proportional to V q i f the interaction potential between ion and atom depends on the inverse fourth power of the distance between their centres. For this polarization potential and others that are not strongly velocity dependent we may write, [A3.8] v 9 t ' c o l l -M (m+M)n+ (JX. dv^ o.v f(v) F(V) dvdV d o \u00E2\u0080\u0094 \u00E2\u0080\u0094 -M n (m+M) - f \" d o = - 5f where the dependence of f and F on n + and n respectively has been extracted from the average over ^ v >. The effect of elastic collisions i s therefore to introduce a damping term into the equation of motion for the average ion, - 232 -This equation has been solved for the ICR f i e l d configuration to yield precisely the same line shape as derived in Section 17. The analysis of that section requires that \u00C2\u00A3(or a.v ) be independent of v . This is true d o o only when the polarization force dominates the ion-atom interaction. -4 Terms other than the r term in the interaction potential result in a velocity dependence of a ^ v Q a n d become more important as the average velocity of the ions with respect to the neutrals becomes large. Alternately, we may say that the polarization force dominates the ion-atom interaction at low ion velocities. Therefore, the approximation in Eq.[A3.8] is best satisfied when low rf levels are used to detect the ions or when the neutral gas density is sufficiently high that the average energy gained by an ion between collisions is small. We have already seen that under these conditions [A3.10] f ( v ) = ( T T ^ ) 3 / 2 E - M V 2 / 2 K T and substituting into we obtain d o , (mv2+MV2) < ? r \u00E2\u0080\u0094 -\u00E2\u0080\u0094 \u00E2\u0080\u00A2 -2kT rA3 H I r = r\u00E2\u0080\u0094^ (A\"11) \ 3 which reduces to [A3.12] 5 = m+M IT y with g = < \u00E2\u0080\u0094 ) * v o . e a.v dvdV d o ,00 O 3 -g 2 g ad e &-.dg In d r i f t tube experiments the dc mobility is measured directly from the time of flight of a pulse of ions across a region of constant electric f i e l d - 233 -and pressure [Albritton et a l , 1967]. From analysis of the shape of the arrival times of the ions at a suitable counter, the longitudinal and transverse diffusion constants of the ions under study may be determined. Experiments such as these have been extensively investigated theoretically so that to date they provide the most reliable estimates of the ion-atom interaction. Kihara [1953] calculates that at low electric f i e l d strengths the average component of ionic velocity in the direction of the applied electric f i e l d is [A3.13] \u00E2\u0080\u0094SL-jr- E, = K(o) E^ z 16 n p Q. dc dc where E d c is the electric f i e l d and S is a collision integral with form? [A3.14] n = (M-)* 5 _g2 \u00C2\u00B0 d g q dg o The dc mobility in the limit of zero f i e l d is thus [A3.15] K(o) = ~r \u00E2\u0080\u0094 3 \u00E2\u0080\u0094 \u00E2\u0080\u00A2 16 n y It is clear that the dc mobility is simply related to the ICR collision frequency o n l y . i f - i s relatively constant and may be taken outside the integrals in Eqs. [A3.12] and [A3.14]. The relation [A3.16] K'(o) = -V mt,'-is therefore seen to apply only at very low ionic velocities. When the ionic velocity is not well approximated by a Maxwellian Eq. [A3.16] is not - 234 -valid since the operational definition of \u00C2\u00A3(Eq. [A3.8]) requires modification. The extension of ICR measurements to high rf electric f i e l d strengths therefore requires further theoretical development. Now let us consider a single elastic collision in detail. A two body collision is dynamically equivalent to the one-body collision problem in which a hypothetical particle of mass y = mM/m+M approaches a fixed scattering centre with impact parameter b; and velocity V q equal to the relative velocity of the two interacting particles. The one body collision is illustrated in Fig. A3.2. The distance from the hypo-thetical particle y to the scattering center is r and the force f i e l d of the scatterer is represented by a potential V(r). The effect of V(r) on y is to change i t s direction by an angle 9. 1 2 Far from the scattering center y has energy ^ yv^ so conservation of energy requires that 2 [A3.17] 4 v 2 = ikiir 2 + - ^ r + V(r) 2 0 2 2yr 2 where the right hand side of the equation is the energy of y in the presence 2 2 of V(r). The term J /2yr in the above expression represents the rotational kinetic energy of the system. Since = mr x r we may rewrite Eq. [A3.17] 1 2 1 2 2 2 2 [A3.18] 4 n v = ^yr + yv b /2r + V(r) l o l o The angle i> gives the orientation of r with respect to v and the value \u00E2\u0080\u0094 \u00E2\u0080\u0094o of $ for which r is a minimum is denoted by From Fig. A3.2 i t is clear that the trajectory of P is symmetric about $ and that - 235 -[A3.19] 6 = TT - 2$ The angle through which the relative velocity v^ i s rotated during a collision is completely specified by $, a quantity that is easily calculated \u00E2\u0080\u00A2 d e b 2 by noting that = = bv /r , from the constancy of angular momentum, and dt o [A3.20] j - \u00C2\u00B1\u00C2\u00A3 [1 - b i j % d b 2 2 y yv r o At the angle of closest approach =0, r = r^ and 0 is given by r r [A3.21] f a ^ d r = -dr a \u00E2\u0080\u009E , 2 (b/r ) dr ri _ 2 v ( r ) _ bV 2 2 2J yv r o Eqs. [A3.12], [A3.15] and [A3.21] relate K(o), 5 and a . For a given interaction potential the distance of closest approach r and $ may be ct calculated as functions of v and b. The average of l-cos6 over a l l possible o r impact parameters then yields as a function of V q . The appropriate average over the relative velocity distribution f i n a l l y yields the desired transport properties. Although this procedure is purely classical, Vogt and Wannier [1954] have shown that a quantum mechanical description of the polarizatic potential is similar in most respects. L o n - 236 -Fig. A3.2: The one body equivalent of a bi-particle c o l l i s i o n . - 238 -Appendix 4 : Moments of the Energy Distribution Function In section 17 an approximate energy distribution for ions subjected to resonant r-f for a time t was derived in which the possibility of collisions of the ions with a background gas was included. The n moment of this energy distribution may easily be calculated in the following manner; \u00E2\u0080\u00A2\u00E2\u0080\u0094- ,00 [A4.1] = E ^ P x ( E A , T , t)dE L ^1 - t / x c -Em/kT _ V k T ^ < i o ( - w t \" > d E J . + o J~ kT kT , dp e P e g n(m+M 2 2 I r\u00E2\u0080\u0094\u00E2\u0080\u0094\u00E2\u0080\u0094 2^T T c P | r\u00C2\u00BB -E./kT. 2 T d E ^ e ^ 8 I C\lm C E J I O E The integrals over E^ in the above expression may be expressed in terms of Whittaker function M^ ^ using the standard form [Gradshteyn and Ryzhik, 1965; Pg. 720] f 0 0 V~T a r(y+v+k 1 .2. 2 [A4.2] J o s a\"'8 I 2 v (26/?)ds - 7 5 ^ - 6\"1 eS ' 2\u00C2\u00BB(pj*X\u00E2\u0080\u009E ^ where T(k) is the Gamma function. The Whittaker functions resulting from substitution of Eq. [A4.2] into Eq. [A4.1] are transformed to degenerate hypergeometric functions $; - 239 -so that for n=0,l,2 Eq. [A4.L]- takes the following simple forms [A4.4] n=0 : E x = P X(E A, T t)dE x- 1 _ -t/x -t/x n=l : E. = CkT.+E )e + kT (1-e 1 x m g c. , (m+M)\u00E2\u0080\u009E 2 . } + 2m\" n T c Y ( 3 ' t h n=2 : Er = \"t / T c 2 e ( k T x ) Z (m+M) nx 2m kT C ) Y ( 5 ' t / T c } g (nrt-M)nx -t/x + 4 (-STwT ) Y ( 3 ' t / T c ) + 2(1 - e C) g where yCk, t/x ) is an incomplete Gamma function. When the same osc i l l a t o r is used to both heat and detect the ions, the experiment i s actually performed over a time x where x i s the total time the ions remain i n the resonant r - f . Thus the time average of E x is [\u00E2\u0080\u00A2A4.-5] = -J- t x rx E A dt -x/x X = kT + kT (e C - 1) \u00E2\u0080\u0094 g g T -X /X X T)T + kT (1 - e C) \u00E2\u0080\u0094 + \u00E2\u0080\u0094 Y ( 3 , X/X ) X X c n(m+M) x c ex 2m x 'o Y ( 3 , t/x c) dt and t i s similarly obtained from E x, - 240 -Appendix 5 : The Relative Velocity Distribution In this appendix we wish to calculate the relative velocity distribution for ion-atom pairs over which the rate constants measured in ICR experiments are averaged. As in Appendix 3 the ion velocity is denoted by v^, that of the atom by V. and the relative velocity by v.. = v. - V.. Angles (6., e.) \u00E2\u0080\u00943 J J \u00E2\u0080\u0094 I J \u00E2\u0080\u0094 i \u00E2\u0080\u0094 j 6 i ' iJ and (0_., \u00C2\u00A3j) define the orientation of v_^ and with respect of an arbitrary co-ordinate system and angles 6' and e' give the orientation of v^ with respect to V_. - y_\u00E2\u0080\u009E as shown in Fig. A5.1. The relative velocity distribution H(y_\u00E2\u0080\u009E) is given in general form by [M. Bloom, private communication]. [A5.1] H ( V = 4? - r2n sin ( 3 'de de' F (V. yv. dV . , 3 3 3 ) d df 4n SI: 1 V . f(v.)6(v. - V. - v..) m. The ionic distribution of speeds is assumed to be Maxwellian. 2 _a. v. [A5.2] f(v.) = A. e \" 1 with A. = 4 (nap and a \u00C2\u00B1 = - ^ r -i subject to the restrictions which are discussed in previous sections. It is further assumed that f(v.) is independent of (0.,e.) for any (8., e.). i 3 3 i i From Fig. [A5.1] the relation [A5.3] (V. + v. . ) 2 = V. 2 + v.} - v.. V. cos 6' - J - i ] 1 i l i l 1 is easily obtained and substituting into Eq. [A5.1] yields IA5.4J n sineHei \u00E2\u0080\u009E -a.(V.2+v... -2V.v..cos 0')dV. w 2 T? fir 1-11-., l'l 1 V.\" .F (V:) e J fV. *\u00E2\u0080\u00A2 . \u00E2\u0080\u00A2 . J 1 1 J J 3 0 u 2v. . a . i j i -a. (v. .2+ V.2-* F(V.) V. e 1 1 3 J sinh (2V.v..a.)dV. 1 1 i i l i 3 - 241 -A5.1 The relative velocity vector s N - 243 -There are two types of distributions F(V\) that are of interest here. Case 1. If the atomic distribution function is Maxwellian, analogous to Eq. A5.2, then 2 rai a j , -v. . ( J \u00E2\u0080\u0094 ) TT ai ai 3/2 P 1 J a, + a-i [A5.5] H(v. 0 = 4 ( n a l a J ) ' 6 1 3 !1 a^ + aj This distribution function describes the relative speeds when the target of the ion beam f i l l s a well defined spatial region. /It has been used in Appendix 3 to obtain Eq. [A3.11] from [A3.10].. Case 2. The velocity distribution in a molecular beam is not Maxwellian. The probability of an atom emerging from an oriface in an oven is proportional to i t s velocity, and as a result the velocity distribution inside a beam is [Ramsey, 1969] . - a. V. [A5.6] F(V.) = A! V. e 3 J 1 1 1 where A! = 2 a.2 and a. = M./2kT.. 1 1 1 1 1 Now, the relative velocity distribution becomes [A5.7] Hfv..) = A A i a.v. .2 oo , .2 i A.A. - 1 11 -fr\u00C2\u00AB ^' z 1 ~1 Q \u00E2\u0080\u00A2* - 1 1 e i j 2a. v. . l i i c.~ \u00E2\u0080\u009E -(d.+a.)v V. 2 e 1 3 3:; \u00E2\u0080\u00A2\u00E2\u0080\u00A2:sirih(2aiVjv.jDdVj o In the crossed beam experiment reported in Sect. 19, the number density inside the ICR c e l l was controlled by varying the temperature of the secondary ion oven, thus changing the velocity distribution in the atomic beam. It is therefore important to understand the dependence of measured co l l i s i o n frequencies on a_., and since Itlhese c o l l i s i o n frequencies are averages of a rate constant v. . a over Hfv. .\") the moments of Hfv. .\") are of interest, i j ij i j If a cross section a is proportional to v j j 0 then the corresponding c o l l i s i o n frequency is proportional to n+3 Hfv..) v.. dv... 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Soviet Phys. - Uspekhi (English Transl.) 2, 255, - 249 -GLOSSARY OF SYMBOLS Symbol Definition Page a Distance between trapping electrodes. 3 (aQ,a^,a2>SiT)>SLA) Coefficients of quantic in Weierstrauss calculation. 22 (a^)^ Maximum of f i e l d modulated ICR absorption derivative. 221 A(6io), A(6B) ICR absorption signal assuming uniform electric f i e l d . 149 A(t, 6OJ) Instantaneous power absorption. 181 A^ (y) y dependent term in the expansion of V(y, z) 47 A^ A convenience parameter; the normalization constant of th the velocity distribution function of the i compo-nent in the system, 240 A Area over which ions are distributed. 203 P A Distance between extrema of the ICR absorption PP F derivative, 130 Ar (\u00C2\u00A3,,\u00C2\u00A3.') Argon Atom. i . ' \u00E2\u0080\u00A2 * * , r s . : \u00E2\u0080\u00A2 jnisrv.'^ b Distance between d r i f t electrodes in the ICR c e l l . 3 b. Distance between d r i f t electrodes,in the i * \" * 1 c e l l I 216 region. b Impact parameter in a two particle c o l l i s i o n . 227 Impact parameter for orbiting collisions. 197 b^ Impact parameter such that P \u00E2\u0080\u009E = 0 for a l l b>b^ in Firsov theory of resonant charge exchange. 197 b Field modulation amplitude. 220 m B Static Magnetic Field. 3 B e \u00C2\u00A3\u00C2\u00A3 Magnetic f i e l d at maximum power absorption. 72 - 250 -Definition E*eff in the presence of an irradiating r - f electric f i e l d . Half width at half height (in magnetic f i e l d units). of the low pressure ICR absorption. Line width of the absorption derivative. Radius of effusion aperature. See Abramowitz and Stegun [1969, pg. 587]. Area under the ICR absorption line. Strengths of the derivative of the ICR absorption. Atomic diameter. Collision diameter; sum of atomic and ionic r a d i i . Expansion coefficient of the electric f i e l d . Sum of Z 2 n e over n. m,n I n i t i a l kinetic energy of an ion. Total kinetic energy of ion species i . Static electric f i e l d vector. Static electric f i e l d in d.c. mobility experiments. Energy gained by an ion from resonant r - f in a time Energy gained by an ion from resonant r - f . in a time Energy gained by an ion from resonant r - f in a time t-t . o Total time dependent kinetic energy, y component of two dimensional E_. Z component of two dimensional E_. i n i t i a l kinetic energy in the Z d 1 rc - i . Liji - 251 -Symbol Definition Page Amplitude (Z m) averaged (y, z). 75 Amplitude (Z m) averaged electric f i e l d gradient. 72 Kinetic energy associated with the cyclotron motion. 81 E Kinetic energy associated with the trapping oscillation. 81 E' An energy such that E(), , <_ E-'. 89 Ej_ Average of E x at time t. 239 t Time average of E x. 239 E Q Average electron energy. 104 ^1* ^rf' 1' e l e c t r i c f i e l d amplitude. 57 f Fraction of monoenergic ions collected by the trap. 125 f Fraction of ions that have moved for time t without o co l l i s i o n . 181 f(s) A quartic polynominal in s. 22 f averaged over a l l i n i t i a l ionic energies. 126 f ^ (y), f2Cy) Parts of A^ (y); convenience functions. 72 f (v^) Distribution of velocities v^ of ion species i . 226 F (y\) Distribution of velocities V_. of neutral species j . 226 F CV) Fraction of ions with potential V collected by the traps. 124 C S J S ' ) Part of the ICR absorption, used for convenience only. 189 g Reduced relative velocity (dimensionless). 232 g (E) Maxwell-Boltzmann distribution of ionic energies. 125 g (V) Distribution of ionic potentials in the trapping well. 87 g^, g^ Invarients of the quartic f ( s ) . 23 g1(y) > g 2(y) Convenience function; g i(y) = C'2^)\" 1 fV(y). 72 - 252 -Symbol Definition Page G (6u), G (6B) Unnormalized line shape function. 57 H (v. .) i l Distribution of relative velocities. 240 i A subscript only. 72 I Ionisation potential 134 I a Atomic flux through area A . 203 I e Ionising electron current. 104 V : i Bessel functions with imaginary argument. 83 ICR ensemble averaged power absorption. 60 i m Atomic flux as a function of angle. 202 j A subscript only. 226 j Angular momentum in relative coordinate system. 234 n v 1 th n order Bessel function with real argument. 49 k A subscript only. 47 .^k, k', k l k2 k \" ) k3 V Rate constants. 179 K Symbol for potassium atom. 150 K CO) D.C. d r i f t mobility at zero f i e l d . 149 K' Reduced zero f i e l d d.c. d r i f t mobility. 170 I C Length of analyser region. 46 L Distance from the source of atomic beam. 203 L', L\" Distances to the ends of the analyser from the atom source. 203 L o Length of the effusion tube. 202 m A subscript. 8 Cm, i r u ) .\"Ionic masses. 1 Electronic mass. - 253 -Symbol Definition Page (M, M . , m.) Neutral masses. 147 J 1 Mj. (p) Whittaker function. 238 \i,v n A subscript. 8 fn, n., n. , n , , .^ 147 j A Neutral number density. V n39 5 n Number of ions in the analyser. 148 o J (h +)^ Number density of ion species i . 226 n^ Number density inside the primary oven. 192 n^ Number density inside the secondary oven. 184 n + Positive ions emitted by surface ioniser. 134 n Atoms emitted by surface ioniser. 134 a J (N, N., ND) Ion current. 60 A ri Na Sodium atom. 152 p^ Ion momentum. 10 p Neutral gas pressure. 164 P (y-> General spatial distribution of ions. 56 P 0 (y, Z m) A special case of p (y, Z m) 60 P^ _. Probability of charge transfer between ion i and neutral j . 230 P (Ey) Distribution function of E^. 81 P() CE,),) Distribution function of E\u00E2\u0080\u009E 81 P,i(E'P)l) Special case of P|( (E,),) 89 P \u00C2\u00B1 (E^) Distribution function of in the limit of zero pressures. 81 P\u00C2\u00B1(Ex,i ^,t) Pj_(E^) generalized to include the effect of non-reactive collisions. 159 Symbol Definition Page q Ionic charge, equal in magnitude to the electronic charge in this thesis. 1 ^1' ^2' ^ Intensity of ICR signals with andv.withputJirradating oscillator. 207 r Ion-atom separation. 170 r a Distance of closest approach of reduced mass to scattering centre in a bi-particle c o l l i s i o n . 237 r Value of r at the minimum of the ion-atom interaction m potential.. 171 R Cyclotron radius. 26 o ' Rfn ) Contribution to the area of the ICR resonance of the v s' secondary oven. 199 s The argument of fCs) the quartic polynominal. 22 s Separation between two ce l l regions. 65 S^, ICR absorption with and without irradition by the secondary oscillator. 207 t Time. 24 t Time of a momentum randomizing c o l l i s i o n . 148 o to t Sum of CJ (m+1) over m; used for convenience, n 3 m+l,n ^ J ' T^, T.. Temperature of ion component i and neutral component j . 240 Tg Neutral gas temperature. 160 T s Temperature of an ionising surface. 134 T x Temperature associated with the two dimensional cyclotron motion. 81 T|( Temperature associated with the one dimensional trapping motion, 81 O 'jjt+ Average velocity in the direction of the electric f i e l d in the d.c. mobility experiment. 233 V [y, z), V Two dimension potential at (y, z) inside the ICR c e l l . 8 V. Velocity of neutral i before c o l l i s i o n . 229 -1 V! Velocity of neutral j after c o l l i s i o n . 229 -1 V Q Trapping well depth. 124 V^ Positive d r i f t potential. 1 V 2 Negative d r i f t potential. 1 V^ Trapping potential. 1 V^-j. Bias potential on the surface ioniser. 140 V d (V rV 2)/2 8 V t V T-(V 1 +V 2)/2 8 V Potential at the centre of the c e l l . 9 c Amplitude (z ) averaged potential. 69 V (V,+V9)/2 average potential of the d r i f t plates. 69 V(r) Ion-atom interaction potential. 170 - 256 -Symbol Definition Page w An arbitrary parameter. 22 W Width of the electron beam. 104 (x, y, z) Spatial coordinates inside the ICR c e l l . 2 x::,(t) ,y(t) ,z (t) Spatial coordinates of an ion as a function of time. 31 (X ,y ,z ) In i t i a l coordinates, 19 o J o o ^ (X, y, z) Components of velocity of an ion in the ICR c e l l . 10 (Xyiy^7ZQ)'t; I n i t i a l velocities 19 (X, y, z) Components of acceleration of an ion. 10 y Amplitude of y(t). 47 cl Y q I n i t i a l amplitude of y(t). 25 y Average of y position of the ion ensemble. 102 Z m Amplitude of oscillation in the trap. 49 a Zero of the quartic f(y). 31 a Atomic Polarizability 170 Inverse temperature parameter in distribution of velocities. 240 Integral over Bessel function, may be expressed in terms of Strauve functions. 69 -4 -6 Y Ratio of r fto' r term in V(r). 171 Y Angle between r-f electric f i e l d and i n i t i a l velocity. 82 Y (n> t) Incomplete Gamma function. 239 r(n) Gamma function. 238 <5o) Distance along the frequency axis from the maximum of an an absorption line. 57 ^Cy^y') Dirac delta function. 60 A Discriminant of the quartic f ( s ) . 25 - 257 -Symbol Definition Page A Space charge depression 104 ABj Shift of an ion's cyclotron frequency (expressed at a magnetic field) due to the electric f i e l d gradients. 27 Aco Spread in quasi cyclotron frequency resulting from the modulation of co by the trapping oscillation. 42 e Deviation of y Q from a the zero of r (y). 31 (e,e'-,e^) Azimuthal angles. 241 e(w^;y, Z ) Power absorbed by an ion at y with trapping amplitude Z . 56 m e (y, z ) Energy absorbed at resonance by an ion at y with res w ' m b J 3 3 trapping amplitude Z . 56 Energy associated with T^. 81 e Energy associated with T,( 81 + zx Limits on due to y. 82 E, Collision frequency associated with the momentum transfer cross section. 147 \u00C2\u00A3\", \u00C2\u00A3 ' Collision frequencies for charge transfer. 180 K Charge to mass ratio, q/m. 19 p. '\u00E2\u0080\u00A2 \u00E2\u0080\u00A2> ..'-tjPhase angle of y(t). 24 ^\u00E2\u0080\u00A2rienuin \" Orientation of r_ with respect to v^. 237 $ The value of at the position of closest approach of p to the scattering centre in a bi-particle c o l l i s i o n . .237 $(y,v,z) Degenerate Hyp.engeometr.ic..Function. 238 - 258 -Symbol Definition Page y \u00E2\u0080\u009E Reduced mass of particles i and j . 198 y Reduced mass in an elastic c o l l i s i o n . 170 y ( = cos y ) used for convenience only. 82 ( y , y ' , y \" ) Moduli of Jacobi E l l i p t i c function used in \u00E2\u0080\u00A2i Weierstrauss calculation. 25 ti q 2 E2/8 m. 159 f!P Weierstrauss E l l i p t i c function. 23 (p., p') Reduced trapping amplitude and z coordinate respectively. 216 Momentum transfer cross section. 147 a c Charge exchange cross section. 196 a An arbitrary cross section. 147 ?2 A parameter used for convenience of notation in the expansion of w(y,z). 49 n Solid angle, 240 0 Angle between the i n i t i a l velocity of an ion and the z axis. 123 6 c Cut off angle for 6 such that a l l particles with 6 >_ 6 ^ are trapped in the c e l l , 124 6 Angle through which the relative velocity vector is rotated in velocity space by an elastic c o l l i s i o n . 229 ( 8 ^ , 6 ' ) Angles specifying orientation of and y_\u00E2\u0080\u009E respectively. 242 0 Angle of a particle in a beam with respect to the axis of the effusion o r i f i c e , 203 Oj Half angle subtended by A^ a distance c from the effusion source, 203 - 259 -Symbol Definition Page T Mean d r i f t time of the ions through the analyser. 149 T ' Mean d r i f t time of the ions from the ioniser to the analyser. 192 T . Mean time between collisions. 148 c w(y, z) Instantaneous quasi cyclotron frequency of an ion. 26 wo ^' ^mJ Quasi-cyclotron frequency of an ion at y with trapping oscillation amplitude Z . 50 cj c Cyclotron frequency of an ion in a uniform electric and magnetic f i e l d . 1 Detector oscillator frequency. 57 u>2 Secondary oscillator frequency. 208 co-j. Trapping oscillation frequency. 20 u n Frequency of modulation. 220 0^ 2 Half period of the Weierstrauss E l l i p t i c function. 24 oil ICR line width (in frequency units) at half \" 2 maximum in the collisionless regime of pressure. 46 \u00E2\u0080\u00A2x. ! v i ' - \" n c ; r 0 -.: ' ICR '-i.b-Ol'pt "@en .
"Thesis/Dissertation"@en .
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"For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use."@en .
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"Resonance"@en .
"Analysis of ion cyclotron resonance"@en .
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