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Ion optics of the mass spectrometer ion source Naidu, Prabhakar Satyanarayan 1965

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The University of B r i t i s h Columbia FACULTY OF GRADUATE STUDIES PROGRAMME OF THE FINAL ORAL EXAMINATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY of PRABAKAR NAIDU B.Sc. (Hons.), Indian I n s t i t u t e of Technolo Kharagpur (Calcutta), 1960 M.Tech., Indian I n s t i t u t e of Technology, Kharagpur (Calcutta), 1962 MONDAY, JULY 12TH, 1965, AT 10:30 A.M. IN ROOM 206, CHEMICAL ENGINEERING BUILDING COMMITTEE IN CHARGE Chairman: J„ R. MacKay External Examiner: A. J . H. Boerboom FOM - Laboratory for Mass Separation Amsterdam, The Netherlands C. W. Clark F. W„ Dalby G. P. Erickson J . A. Jacobs R. D. Russell K. 0. Westphal ION OPTICS OF THE MASS SPECTROMETER ION SOURCE ABSTRACT The ion beam transmission e f f i c i e n c y of the ion source i s an important factor i n determining the sen-s i t i v i t y of a mass spectrometer. Vauthier (1955) has shown for a simple source that the transmission e f f i -ciency i s very.low, .The present ..thesis examines the transmission e f f i c i e n c y ol a more complex source. The f i r s t part of the thesis deals with the ion op t i c a l properties of a m u l t i p l e , s l i t ion source. The region of ion withdrawal has been sketched by com-puting the ion t r a j e c t o r i e s passing through the exit s l i t . It was found that for the more complex source the region of ion withdrawal i s also much smaller than the t o t a l i o n i z a t i o n space. It i s not p r a c t i c a l to confine the i o n i z a t i o n region .to the small volume from which ions are withdrawn. The ef f e c t of a source mag-netic f i e l d has been taken into account. The pertur-bation of the tr a j e c t o r y due to the f i e l d i s small, and therefore the mass discrimination due to the source magnetic f i e l d is' imperceptible for heavy ions unless the f i e l d is- of the order of a. few webers/m^, The multiple s l i t ion source produces a divergent ion beam, only a small f r a c t i o n of which penetrates the exit s l i t . : Obviously a system producing a beam converging at the exit s l i t .to,..a narrow p a r a l l e l ribbon w i l l be most efficient'.' ' In order" to devise such a system a theory of the inverse problem of p a r t i c l e motion i s developed i n the s.ecpnd. part of the t h e s i s . A procedure was 'found to determine a potential d i s t r i -bution required to guide a group of, p a r t i c l e s along a set of prescribed paths. There are two important l i m i t a t i o n s to the choice of paths: (a) "there are c e r t a i n paths^ which are not complete; that i s a p a r t i c l e following such a path i s turned back at c e r t a i n points which we c a l l ' mirror' points. (b) The p a r t i c l e s which do not s a t i s f y the i n i t i a l conditions of uniform energy and d i r e c t i o n may deviate con-siderably from t h e i r projected paths , leading to what we have.called an unstable s i t u a t i o n . Fortunately the complete-paths are stable, and the i n -complete paths are unstable, Of the two types of convergent paths studied, namely, exponentially de-creasing and damped o s c i l l a t o r y paths, the system of damped o s c i l l a t o r y paths i s stable.,. GRADUATE STUDIES F i e l d of Study: Geophysics Advanced Geophysics Radioactive and is o t o p i c processes i n Geophysics Modern Aspects of Geophysics Related Studies: Wave propagation J„ C. Savage Electromagnetic theory G. M. Volkoff D i g i t a l computer programming C„ Froese Introduction to quantum mechanics F, W. Dalby J. A. Jacobs R„ D, Russell J,-A..Jacobs PUBLICATIONS NAIDU, P . S. T e l l u r i c f i e l d and a p p a r e n t r e s i s t i v i t y o v e r an i n c l i n e d n o r m a l f a u l t . (To a p p e a r i n C a n a d i a n J o u r n a l of E a r t h S c i e n c e . ) NAIDU, P . S , , K . 0. WESTPHAL. The i n v e r s e p r o b l e m o f p a r t i c l e m o t i o n and i t s a p p l i c a t i o n . (Read a t C a n a d i a n A s s o c i a t i o n o f P h y s i c i s t s a n n u a l m e e t i n g h e l d i n J u n e 1 9 6 5 ) . The University of B r i t i s h Columbia FACULTY OF GRADUATE STUDIES PROGRAMME OF THE FINAL ORAL EXAMINATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY of PRABAKAR NAIDU .Sc., (Hons.), Indian I n s t i t u t e of Technolog Kharagpur (Calcutta), 1960 M.Tech., Indian I n s t i t u t e of Technology, Kharagpur (Calcutta), 1962 MONDAY, JULY 5TH, 1965, AT 10:30 A.M. IN ROOM 206, CHEMICAL ENGINEERING BUILDING COMMITTEE IN CHARGE Chairman: I. McT. Cowan C. W. Clark F. W. Dalby G. P. Erickson J. A, Jacobs R. D. Russell K. 0. Westphal External Examiner: A. J . H. Boerboom F0M - Laboratory for Mass Separation Amsterdam, The Netherlands ION OPTICS OF THE MASS SPECTROMETER ION SOURCE ABSTRACT The ion beam transmission e f f i c i e n c y of the ion source i s an important factor i n determining the sen-s i t i v i t y of a mass spectrometer. Vauthier (1955) has shown for a simple source that the transmission e f f i -ciency i s very low. The present thesis examines the transmission e f f i c i e n c y of a more complex source. The f i r s t part of the thesis deals with the ion op t i c a l properties of'a multiple s l i t ion source. The region of ion withdrawal has been sketched by com-puting the ion t r a j e c t o r i e s passing through the exit s l i t . It was found that for the more complex source the region of ion withdrawal i s also much smaller than the t o t a l i o n i z a t i o n space. It i s not p r a c t i c a l to confine the i o n i z a t i o n region to the small volume from which ions are withdrawn. The effect of a source mag-netic f i e l d has been taken into account. The pertur-bation of the t r a j e c t o r y due to the f i e l d i s small, and therefore the mass dis c r i m i n a t i o n due to the source magnetic f i e l d i s imperceptible for heavy ions unless the f i e l d i s of the order of a few webers/m^. The multiple s l i t ion source produces a divergent ion beam, only a small f r a c t i o n of which penetrates the exit s l i t . Obviously a system producing a beam converging at the exit s l i t to a narrow p a r a l l e l ribbon w i l l be most e f f i c i e n t . In order to devise such a system a theory of the inverse problem of p a r t i c l e motion i s developed i n the second part of the t h e s i s . A procedure was found to determine a potential d i s t r i -bution required to guide a group of p a r t i c l e s along a set of prescribed paths. There are two important l i m i t a t i o n s to the choice of paths: (a) there are c e r t a i n paths which are not complete; that i s a p a r t i c l e following such a path i s turned back at c e r t a i n points which we c a l l mirror points, (b) The p a r t i c l e s which do not s a t i s f y the i n i t i a l conditions of uniform energy and d i r e c t i o n may deviate con-siderably from t h e i r projected paths leading to what we have c a l l e d an unstable s i t u a t i o n . Fortunately the complete paths are stable, and the i n -complete paths are unstable. Of the two types of convergent paths studied, namely, exponentially de-creasing and damped o s c i l l a t o r y paths, the system of damped o s c i l l a t o r y paths i s stable. GRADUATE STUDIES F i e l d of" Study: Geophysics Advanced Geophysics Radioactive and is o t o p i c processes i n Geophysics Modern Aspects of Geophysics Related Studies: Wave propagation J . C. Savage Electromagnetic theory G. M. Volkoff D i g i t a l computer programming C. Froese Introduction to quantum mechanics F. W. Dalby J. A. Jacobs R. D. Russell J . A.-Jacobs PUBLICATIONS NAIDU, P. So T e l l u r i c f i e l d and apparent r e s i s t i v i t y over an i n c l i n e d normal f a u l t . (To appear i n Canadian Journal of Earth Science.) NAIDU, P. S., K, 0. WESTPHAL. The inverse problem of p a r t i c l e motion and i t s a p p l i c a t i o n . (Read at Canadian As s o c i a t i o n of P h y s i c i s t s annual meeting held . i n June, 1965 .) ION OPTICS OF THE MASS SPECTROMETER ION SOURCE by PRABHAKAR SATYANARAYAN NAIDU B.Sc. (Hon.), I n d i a n I n s t i t u t e of Technology, Kharagpur, ( C a l c u t t a ) , i960 M.Tech., I n d i a n I n s t i t u t e of Technology, Kharagpur, ( C a l c u t t a ) , 1962 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of GEOPHYSICS We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA May 1965 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study, I f u r t h e r agree that per-m i s s i o n f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood that copying or p u b l i -c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n permission* Department The U n i v e r s i t y of B r i t i s h Columbia, Vancouver 8, Canada Abstract The i o n beam transmission e f f i c i e n c y of the i o n source i s an important f a c t o r i n determining the s e n s i t i v i t y of a mass spectrometer. Vauthier ( 1 9 5 5 ) has shown f o r a simple source that the transmission e f f i c i e n c y i s very low. The present t h e s i s examines the transmission e f f i c i e n c y of a more complex source. The f i r s t part of the t h e s i s deals with the i o n o p t i c a l proper t ies of a m u l t i p l e s l i t i o n source. The region of i o n withdrawal has been sketched by computing the ion t r a j e c t o r i e s passing through the ex i t s l i t . I t was found that f o r the more complex source the region of i o n withdrawal i s also much smaller than the t o t a l i o n i z a t i o n space. I t i s not p r a c t i c a l to confine the i o n i z a t i o n region to the small volume from which ions are withdrawn. The e f f e c t of a source magnetic f i e l d has been taken into account. The per turbat ion of the t r a j e c t o r y due to the f i e l d is smal l , and therefore the mass d i s c r i m i n a t i o n due to the source magnetic f i e l d i s imperceptible f o r heavy ioris unless 2 the f i e l d i s of the order of a few webers/m . The mul t iple s l i t i o n source produces a divergent ion beam, only a small f r a c t i o n of which penetrates the e x i t s l i t . Obviously a system producing a beam converging at the e x i t s l i t to a narrow p a r a l l e l r ibbon w i l l be most e f f i c i e n t . In order to devise such a system a theory of the inverse problem of p a r t i c l e motion i s developed i n the second part of the t h e s i s . A procedure was I i found to determine a po t e n t i a l d i s t r i b u t i o n required to guide a group of p a r t i c l e s along a set of prescribed paths. There are two important l i m i t a t i o n s to the choice of paths: (a) There are ce r t a i n paths which are not complete; that is, a p a r t i c l e following such a path i s turned back at c e r t a i n points which we c a l l mirror points. (b) The p a r t i c l e s which do not s a t i s f y the i n i t i a l condi-tions of uniform energy and d i r e c t i o n may deviate considerably from t h e i r projected paths leading to what we have c a l l e d an unstable s i t u a t i o n . Fortunately the complete paths are stable, and the incomplete paths are unstable. Of the two types of convergent paths studied, namely, exponentially decreasing and damped o s c i l l a t o r y paths, the system of damped o s c i l l a t o r y paths i s stable. i i i TABLE OP CONTENTS ABSTRACT LIST OP FIGURES ACKNOWLEDGMENT CHAPTER I SENSITIVITY OF MASS SPECTROMETER 1.1 Introduct ion 1.2 Ion o p t i c s of the i o n source 1.3 Statement of the problem and scope of work CHAPTER II MULTIPLE SLIT ION SOURCE Synopsis 2.1 Equation of t r a j e c t o r y 2.2 Method of s o l u t i o n 2.3 Subproblems 2.4 Global a n a l y s i s 2.5 E f f i c i e n c y of ion beam transmission CHAPTER III THE INVERSE PROBLEM OF PARTICLE MOTION 3.1 3 .2 3 .3 3.4 CHAPTER IV Synopsis Theory of inverse problem E f f i c i e n t ion source S t a b i l i t y a n a l y s i s Electrode c o n f i g u r a t i o n SUMMARY AND CONTRIBUTION OF THE PRESENT WRITER 4.1 Summary 4 . 2 Contr ibut ions of the present wri ter PAGE i v v i i i 1 - 8 9 - 7 2 73 - 104 105 - 109 i v Table of Contents (continued) APPENDIX A APPENDIX B BIBLIOGRAPHY PAGE 110 - 111 112 - 1 1 4 115 - 116 LIST OF FIGURES PAGE F i g . 1.1 Approximate sketch of the m u l t i p l e s l i t ion source 5 F i g . 1.2 Mathematical model of the m u l t i p l e s l i t ion source 6 F i g . 2.0 Right-handed coordinate system used throughout the work 10 F i g . 2.1 Geometry of various subproblems 18 F i g . 2.2 Conformal mapping of the w^> o plane into the whole of the z-plane by Schwarz-C h r i s t o f f e l method 20 F i g . 2.3 T y p i c a l ion t r a j e c t o r i e s 22 F i g . 2.4 Representative ion t r a j e c t o r y 24 F i g . 2.5 P o s i t i o n of cross-overs vs . s t a r t i n g point 25 F i g . 2.6 Second cross-over v s . r e p e l l e r f i e l d 26 F i g . 2.7 Domain of convergence of s e r i e s expansion 28 F i g . 2.8 P a r a x i a l region 29 F i g . 2.9 Conformal mapping of the w > o plane in to the whole of z-plane by S c h w a r z - C h r i s t o f f e l method 31 F i g . 2.10 S o l u t i o n of n o n - l i n e a r equation 33 F i g . 2.11 Repel ler pla te 36 F i g . 2.12 Repel ler plate p o s i t i o n , cross-over points 37 F i g . 2.13 Entrance s l i t and e x i t s l i t 40 F i g . 2.14 E f f e c t of the e x i t s l i t on the angle of divergence 41 F i g . 2.15 D e f l e c t o r p la tes kept at d i f f e r e n t poten-t i a l s 43 F i g . 2.16 P o t e n t i a l d i s t r i b u t i o n around d e f l e c t i n g pla tes : 44 F i g . 2.17 D e f l e c t i o n of beam 45 L i s t of Figures (continued) F i g . 2.19 Angle of i n c l i n a t i o n of the c e n t r a l t r a j e c t o r y measured at a distance of +25 u n i t s F i g . 2.20 Magnetic f i e l d normal to the plane of paper Page F i g . 2.18 D e f l e c t i o n of c e n t r a l t r a j e c t o r y measured at a distance of +25 u n i t s 47 48 50 F i g . 2.21 Ion t r a j e c t o r i e s i n the presence of the magnetic f i e l d 52 F i g . 2.22 E f f e c t of magnetic f i e l d 53 F i g . 2.23 E f f e c t of magnetic f i e l d on the p o s i t i o n of cross-over 54 F i g . 2.24 Conformal mapping of w > o plane onto the whole of the z-plane 56 F i g . 2.25 T y p i c a l ion t r a j e c t o r i e s 64 F i g . 2.26 Ion withdrawal s t r i p . No r e p e l l e r f i e l d . 66 F i g . 2.27 E f f e c t of r e p e l l e r f i e l d . Ion withdrawal s t r i p 67 F i g . 2.28 E f f e c t of r e p e l l e r f i e l d . Ion withdrawal s t r i p s 68 F i g . 2.29 Repel ler f i e l d v s . p o s i t i o n of withdrawal s t r i p s 69 F i g . 2.30 Experimentally observed r e l a t i o n s h i p between ion current and r e p e l l e r voltage 70 F i g . 2.31 E f f e c t of r e p e l l e r f i e l d on the shape of the s t r i p s 71 F i g . 3.1 Converging paths 74 F i g . 3.2 A x i a l p o t e n t i a l f o r Y » e x p ( - X ^ ) 82 P i g . 3-3 A x i a l p o t e n t i a l f o r converging paths Y » e x p (- l> x n ) 8 3 F i g . 3.4 System I : p o t e n t i a l d i s t r i b u t i o n f o r converging paths 87 P i g . 3.5 System I I : p o t e n t i a l d i s t r i b u t i o n f o r damped o s c i l l a t o r y paths 88 v i i L i s t of Figures (continued) Page Pig. 3. 6 System I: f i r s t fundamental solution 94 Pig. 3. 7 System I: second fundamental solution 95 Pig. 3. 8 System I: perturbation functions 96 Pig. 3. 9 System I: perturbation functions 97 Pig. 3. 10 System I I : f i r s t fundamental solution 98 Pig. 3. 11 System I I : second fundamental solution 99 Pig. 3. 12 System I I : perturbation functions 100 F i g . 3. 13 System I I : perturbation functions 101 Pig. 3. 14 The r e l a t i v e merit of multiple s l i t ion source and the newly proposed source 103 v i i i Acknowledgment The w r i t e r w i s h e s t o e x p r e s s h i s s i n c e r e t h a n k s t o D r s . R. D. R u s s e l l and K. 0. Westphal who s u p e r v i s e d t h e r e s e a r c h , and t o D r s . W. P. Slawson and A. B. L. W h i t t l e s who made a v a i l -a b l e c e r t a i n i n f o r m a t i o n r e g a r d i n g t h e mass s p e c t r o m e t e r of t h i s l a b o r a t o r y . Thanks a r e a l s o due t o Dr. G. P. E r i c k s o n f o r r e a d i n g t h e m a n u s c r i p t c r i t i c a l l y . P r o f e s s o r J . A. J a c o b s made i t p o s s i b l e f o r t h e w r i t e r t o come t o t h e U n i v e r s i t y o f B r i t i s h C o lumbia, and has made the s t a y h e r e b o t h p r o f i t a b l e and e n j o y a b l e t h r o u g h h i s s t i m u -l a t i n g l e c t u r e s and t h r o u g h p e r s o n a l i n t e r e s t i n h i s s t u d e n t s . The w r i t e r w i s h e s t o thank Dr. C. W. C l a r k o f t h e Mathe-m a t i c s Department f o r h i s h e l p f u l d i s c u s s i o n on t h e p r o o f of the i n v e r s e p roblem. The w r i t e r was a l s o c o n s i d e r a b l y bene-f i t e d f r o m s t i m u l a t i n g d i s c u s s i o n s w i t h J . A l l a r d . The w r i t e r g r a t e f u l l y acknowledges t h e h e l p r e c e i v e d f r o m the s t a f f a t t h e Computing C e n t r e o f t h e U n i v e r s i t y of B r i t i s h C o l u m b i a . Mr. John B l e n k i n s o p has k i n d l y p r o o f r e a d the m a n u s c r i p t . The work was f i n a n c e d t h r o u g h t h e N a t i o n a l R e s e a r c h C o u n c i l o f Canada g r a n t s t o D r s . R. D. R u s s e l l and W. F. Slawson. The w r i t e r would l i k e t o e x p r e s s h i s g r a t i t u d e f o r a N a t i o n a l R e s e a r c h C o u n c i l S t u d e n t s h i p . 1 Chapter I S e n s i t i v i t y of mass spectrometer 1.1 Introduct ion The mass spectrometer has become an important t o o l i n the study of g e o l o g i c a l problems, as w e l l as i n many other branches of physics and chemistry. No matter what p a r t i c u l a r a p p l i c a t i o n with which the mass spectrometrist i s involved he Is in teres ted i n c e r t a i n fundamental proper t ies of h i s i n s t r u -ment. S p e c i f i c a l l y he i s concerned with i t s a b i l i t y to separate adequately ion beams of d i f f e r e n t r a t i o s and i t s a b i l i t y to produce ion beams of s u f f i c i e n t i n t e n s i t y f o r accurate measure-ment. This t h e s i s i s concerned more d i r e c t l y with the second c o n s i d e r a t i o n . The s e n s i t i v i t y of a mass spectrometer depends l a r g e l y upon the s e n s i t i v i t y of the measuring system and the e f f i c i e n c y of the ion source. The shot noise represents a fundamental l i m i t a t i o n of the ion beam magnitude which can be measured, but p r a c t i c a l considera t ion may also present f u r t h e r l i m i t a t i o n s . I t i s very much more d i f f i c u l t to s p e c i f y the fundamental l i m i t a t i o n s of the ion source e f f i c i e n c y . To a casual observer the e f f i c i e n c y of the source would appear to be p a r t i c u l a r l y poor. For the mass spectrometer operated i n t h i s laboratory one molecule out of a m i l l i o n molecules reaches the magnetic analyser tube i n form of an i o n , which compares wel l with the f i g u r e s quoted by Mayne (I960) as 10~4 to 10~7. It Is n a t u r a l to look more deeply in to the loss of such a large f r a c t i o n of the sample. 2 The e f f i c i e n c y of the source may be thought of as a pro-duct of two fa c t o r s : the f i r s t Is the i o n i z a t i o n e f f i c i e n c y , that i s , the f r a c t i o n of neutral molecules ionized by the electron beam; and the second Is the ion beam transmission e f f i c i e n c y , that i s , the f r a c t i o n of beam ultimately enter-ing the magnetic analyser tube. To Increase the i o n i z a t i o n e f f i c i e n c y , we may increase either the i o n i z i n g electron cur-rent or the p r e v a i l i n g gas pressure. Unlimited electron cur-rent, however, cannot be obtained from an electron gun without impairing the s t a b i l i t y or reducing the l i f e of the filament. On the other hand, the p r e v a i l i n g gas pressure cannot be i n -creased i n d e f i n i t e l y without enhancing the scattering of the ion beam by the neutral gas molecules and causing other problems. The electrons may be concentrated i n a ti g h t beam by a uniform magnetic f i e l d i n the d i r e c t i o n of the electron beam and thus the scattering of the beam can be minimized. Since the electrons s p i r a l around the magnetic l i n e s of force, the e f f e c t i v e path length of an electron i s increased. The ef f e c -ti v e path length i s given by where L and L 1 are respectively the normal path length i n the absence of magnetic f i e l d and the e f f e c t i v e path length In the presence of magnetic f i e l d , and V r and V z are respectively the transverse and lon g i t u d i n a l v e l o c i t y components. As men-tioned i n Appendix A i t Is questionable whether the Increase i n path length i s s i g n i f i c a n t . 3 The i o n o p t i c a l system of the iaton source consis ts of a stack of p a r a l l e l p la tes c a r r y i n g long but narrow, p a r a l l e l , coplaner s l i t s ( f i g . 1.1). The f u n c t i o n of the system i s to draw the ions from the ion chamber, to accelerate them to the i desired energy, and to i n j e c t them into the magnetic analyser tube as a narrow nondiverging beam. Such ions , according to Vauthier (1955), are drawn from a narrow s t r i p , whose width i s equal to the ex i t s l i t ( f i g . 1 . 2 ) . Consider the i o n source of the mass spectrometer of t h i s labora tory . The width of the region of i o n i z a t i o n (roughly equal to the width of the e lec t ron beam) i s about 0 . 5 cm. ; the width of the ex i t s l i t i s 0.01 cm. The e f f i c i e n c y of ion beam transmission, therefore , i s equal to 0 . 0 2 . , 1 .2 Ion o p t i c s of the source (review): The basic design of the ion source used i n modern gas source mass spectrometry has not changed since i t s f i r s t appearance i n 1947, then designed by N i e r . The ion source used i n t h i s labora -tory , which i s a vers ion of the Nier source, i s shown i n f i g . 1 . 1 . The ion o p t i c s of the complete system i s not known, except the approximate analyses of the s p e c i a l i z e d versions given by Jacob ( 1 9 5 0 ) , Vauthier ( 1 9 5 5 ) , Die tz ( 1 9 5 9 ) , and Boerboom ( i 9 6 0 ) . Jacob (1950) considered the ion o p t i c s of a system c o n s i s t -ing of a cathode, entrance p l a t e , and anode p l a t e . He measured the a x i a l p o t e n t i a l of the system i n an e l e c t r o l y t i c tank ( E i n -s t e i n and Jacob, 1 9 4 8 ) ; the data according to him could be represented, c lose to the cathode, by 4 where V i s the p o t e n t i a l , A and K are the constants ; and away from the cathode by V - A «-xf (. < A c losed s o l u t i o n of the p a r a x i a l t r a j e c t o r y equation (see page 15) where the prime denotes d i f f e r e n t i a t i o n with respect to x , was given by him. The method besides being approximate cannot be used over a wide range, because the approximation of the a x i a l p o t e n t i a l by an exponential f u n c t i o n i s not v a l i d over such a wide range. Vauthier (1955) considered a s ingle plate with a narrow s l i t and uniform f i e l d on i t s r i g h t hand s i d e . In the p a r a x i a l equation (1.1), he dropped o f f the f i r s t term and the rest he integrated numerica l ly . Obviously such a method i s only approximate. Die tz (1959) considered a system c o n s i s t i n g of a cathode, entrance p l a t e , and d e f l e c t o r p l a t e s . He measured the p o t e n t i a l d i s t r i b u t i o n i n the system from an analog model on conducting paper and drew the t r a j e c t o r i e s us ing the t r i g o n o -metric method (Jacob, 1951). In regard to the use of conducting paper, i t has been the experience of the present w r i t e r (Naidu, 1962) that the p r e c i s i o n of p o t e n t i a l measurement i s l i m i t e d to about ±-5$. Furthermore, the trigonometric method requires the p o t e n t i a l d i f f e r e n c e between the two consecutive r e f r a c t i n g surfaces to be small , which i s d i f f i c u l t to a t t a i n , p a r t i c u l a r l y close to the s l i t , where the gradient i s very l a r g e . Hence, to meet the condi t ions of the method, we have to take a large number of small steps, which, i n e v i t a b l y , introduces a large 5 5000 V sooo v —" Repel ler pla te < Filament o.o4" s l i t 4500 V 0.o l2 n s l i t -1800 V D e f l e c t o r pla tes 0 V .004" s l i t (Magnetic analyser tube) F i g . 1.1 Approximate sketch of the m u l t i p l e s l i t i o n source used i n the mass spectrometer of t h i s laboratory (adopted from J . S. S tacey 's t h e s i s , 1962). 6 < — j © P i g . 1.2 Mathematical model of a m u l t i p l e s l i t i o n source. Plates extend to 0 0 A . 1 . 2 . 3 . 4 . . I o n i z a t i o n space . Repel ler p la tes . Entrance s l i t p la te . D e f l e c t o r p la tes . E x i t s l i t p la te Magnetic f i e l d normal to the plane of paper 7 numerical e r r o r . L a s t l y , Boerboom considered a system of three p l a t e s , but he ignored while computing the ion t r a -j e c t o r i e s the perturbat ion f i e l d due to the s l i t s . Fur ther -more, he has not discussed the ion beam transmission e f f i -c iency of the source. We s h a l l compute the ion t r a j e c t o r i e s i n the ion source ( f i g . 1.1) which i s f a r more complex than any d i s c u s -sed by the previous authors. The basic problem i s to f i n d what f r a c t i o n of the t o t a l number of ions formed i n the i o n i -za t ion space (A i n f i g . 1.2) f i n d s i t s way through the e x i t s l i t . T h i s , n a t u r a l l y , demands a complete study of the system. 1.3 Statement of the problem and scope of the present work. In the previous sect ions we have attempted to out l ine the f a c t o r s l i m i t i n g the s e n s i t i v i t y of a mass spectrometer. Poor i o n beam transmission appears to be a main l i m i t i n g f a c -t o r . The broad purpose of the present work i s to study the process of i o n beam transmission, i n the hope that such a study may contr ibute to the design of a bet ter source. With t h i s i n mind, we set f o r t h the f o l l o w i n g problems: (a) To study the ion o p t i c a l proper t ies of the sys-tem shown i n f i g u r e 1.2; thereby to maximize the parameters f o r optimum e f f i c i e n c y . (b) To consider an inverse problem; that i s , hav-ing s p e c i f i e d the beam shape f o r maximum e f f i c i e n c y , to ob-t a i n the p o t e n t i a l d i s t r i b u t i o n required to guide the ions along the s p e c i f i e d paths. The mathematical theory of particle motion througa a system 8 of s l i t s representing an average source i s developed i n the second chapter. The p o t e n t i a l d i s t r i b u t i o n i n the system i s obtained a n a l y t i c a l l y , arid the t r a j e c t o r y equation i s integrated numerically. The theory of the inverse problem and i t s application to ion beam transmission are presented i n the t h i r d chapter. A new method of estimating the e f f e c t of the thermal energy i s developed. The l a s t chapter summarizes the main contributions of the present th e s i s . 9 Chapter II Multiple s l i t ion source Synopsis The motion of charged p a r t i c l e s through a system of p a r a l l e l and coplaner s l i t s i n the presence of crossed electromagnetic f i e l d i s considered. The equation of a tr a j e c t o r y i n non-parametric, time independent form i s derived from basic considera-tions. The approximations leading to a l i n e a r paraxial equation are c l e a r l y stated. The solution of the paraxial t r a j e c t o r y equation Is approached i n two d i f f e r e n t ways; that i s , (a) part-by-part approach and (b) global approach. The part-by-part approach provides a physical insight into the problem and i t i s r e l a t i v e l y simple but les s precise; on the other hand, the global approach, though more precise, i s mathematically involved. F i n a l l y the e f f i c i e n c y of the ion beam transmission i s discussed i n the l i g h t of ion o p t i c a l properties of the system. 2.1 Equation of traj e c t o r y The force exerted by an e l e c t r i c and magnetic f i e l d Hi on a charge q and mass m moving with a v e l o c i t y V i s given by where we assume V « c , and where c i s the v e l o c i t y of l i g h t . m - ^ ( ? + n7x i f ) ( 2 * 1 ) Let us consider the motion i n the x,y-plane, where I E , (2-2) and t = K (3-* (2.3) 1 0 Right-handed co-ordinate system U 3 e d throughout the present work. 11 subscr ipts X ; ^ } and £ i d e n t i f y the x , > , and £ components of the f i e l d r e s p e c t i v e l y ( f i g 2.0). On s u b s t i t u t i n g equations (2.2) and (2.3) in to (2.1), we obtain the system of equations X =• fy* ( E a + (2.4) and ^ = K ( E K C . x ) (2.5) where the d o t ( . ) i n d i c a t e s d i f f e r e n t i a t i o n with respect to time t . We may now express the equations (2.4) and (2.5) i n non-parametric form, i.e., independent of time t . M u l t i p l y i n g (2.4) with <*i and (2.5) with ^ , i n t e g r a t i n g , and adding we f i n a l l y obtain <£. AV = i w ( x 2 4 ^ ) (2 . 6 ) where ^ ^ v i s the net change i n the p o t e n t i a l energy of a p a r t i c l e plus the i n i t i a l energy. From (2 . 6 ) we obtain where the prime stands f o r the d i f f e r e n t i a t i o n with respect to x. But and s u b s t i t u t i n g in to (2.5) we obtain 12 or 1 + - ^ + A V ( , 4 ^ * ) S m 0 (2.7) The above equation i s the basic equation describing the motion of a charged p a r t i c l e under the influence of a two dimensional e l e c t r i c f i e l d and a magnetic f i e l d along the z-axis. When E»z=0, the equation reduces to Equation (2.8), f o r the case of an e l e c t r i c f i e l d , was derived by Euler (1773) using the v a r i a t i o n a l p r i n c i p l e . For the sake of s i m p l i c i t y we s h a l l express E x and E-y i n terms of the p o t e n t i a l difference . , A v - Vo -V -r € where Ve i s the p o t e n t i a l at a* , the s t a r t i n g point and £ the i n i t i a l energy divided by the charge. Since and 37 £* = Ti. The new symbol ^ a c t u a l l y refers to the energy of the p a r t i c l e ( i t has a l l the properties of a p o t e n t i a l function), and we hence continue to c a l l i t p o t e n t i a l . Introducing these changes into the equations (2.7) and (2.8), we obtain / / (2.9) 13 and (2.10) Paraxial approximations: Equation (2.9) may be considerably s i m p l i f i e d i f we r e s t r i c t ourselves to the paraxial region, that i s , the region close to the x-axis. I f the p o t e n t i a l § i s symmetric about the x-axis, we may write The above equation has to s a t i s f y the Laplace equation. On substituting (2.11) into the Laplace equation and l e t t i n g the c o e f f i c i e n t s of (jm C^-o- a n ')% o we obtain «K*>>J= ^ : ' ^ ^ + ^ . c - i j r x « % i 2 , 1 2 ) where the superscripts on denote the order of d i f f e r e n t i a t i o n with respect to x. Equation (2.12) may be looked upon as the Taylor's series expansion of c? around the x-axls. S u f f i c i e n t l y close to the axis, we may truncate the series to f i r s t two terms, v i z : Therefore and f i n a l l y ^ a." 14 We f u r t h e r assume that 0 + y ' 2 ) - ^ 1 , since ^'<<1 . I n t r o -ducing these approximations in to equation (2.9), we o b t a i n * 2 * ? o y + * . ' y + ^ o / / ^ + s * / ¥ « i . * 0 (2.13) and when 6^ =. o A 6? G ^" ^ #07' + «S>0"^ = o (2 .14) Equations (2.13) and (2.14) contain the p h y s i c a l and geo-m e t r i c a l parameters; i n order to reduce them to as few as p o s s i b l e , we introduce the dimensionless q u a n t i t i e s x , Y j and <J> defined by ac =. oo X where w and k are the constants with the dimensions of length and p o t e n t i a l r e s p e c t i v e l y . We f i n a l l y obtain the dimensionless form and A*.Y" + *„v+ -O ( 2 A 6 ) where the quantity u> &%J~K i s dimensionless . I t i s i n t e r e s t i n g to note some of the proper t ies of equation ( 2 . 1 6 ) : (a) since the equation i s independent of k , the t r a j e c t o r y i s independent of the actual magnitude of the p o t e n t i a l , but dependent on the geometry of the p o t e n t i a l d i s t r i b u t i o n . Equation 2.13 was f i r s t derived by Boerboom (1957) using v a r i a t i o n a l c a l c u l u s . 15 (b) i f we replace X by - x equation (2.16) remains invariant. P h y s i c a l l y speaking, a p a r t i c l e may t r a v e l i n a positive.or negative d i r e c t i o n ; such an ambiguity i n d i r e c t i o n w i l l play an important role i n the next chapter. 2.2 Method of solution A closed solution of the paraxial equation can be obtained only f o r a few simple forms of the po t e n t i a l "I. : f o r example, when the p o t e n t i a l i s of the form given by =. cx x^-t- b x -t- C (2.17) Without going into the d e t a i l s , the solution of equation (2.16) may be given by(shibata ?•i960) y - A cos/a t -t- 6 Sin 1 where £ =. -p=r In ( J A X l t b x t C + x/^ + ^ t - ^ • Sin-' ( A * ' " ) Unfortunately, the po t e n t i a l even f o r a single s l i t , not to speak of a system of s l i t s , cannot be represented by (2.17); therefore, the above method i s of l i t t l e use and we have to resort to numerical integration. One could, of course, approximate any po t e n t i a l over a s u f f i c i e n t l y small range by (2.17) and then apply the closed solution given above. But such a procedure, besides being as time-consuming as a numerical integration, i s d e f i n i t e l y a d i s -advantage over the numerical integration, i n p a r t i c u l a r i n studying the motion of p a r t i c l e s which do not s a t i s f y the para-x i a l conditions. 16 The Runge-Kutte method, though not the most accurate one— the question of e r ror i s a d i f f i c u l t one—is nevertheless a stable method, and that Is what was mostly needed f o r our work. The question of round-off e r rors i s discussed i n Appendix B. Before turning to the study of the ac tual s l i t system, an Important point that was so f a r neglected by the previous workers (Vauthier , 1955) i s discussed below. I f a p a r t i c l e were to s tar t at o from res t , that i s $ e =. o ; i t f o l l o w s from the equation (2.16) that the d i r e c t i o n Y^°) along which the p a r t i c l e moves and Y<°) > t n e distance from the ax is , are no longer independent of each other . Prom (2.l6) i t fo l lows that Y ; ( o) = - Z L . \CO) £' that i s , the d i r e c t i o n of motion i s f i x e d f o r a given value of V ( o ) . T h i s leads to the f o l l o w i n g complications which was overlooked by the previous workers. For numerical i n t e g r a -t i o n the equation (2.16) i s u s u a l l y wr i t ten i n the form Y"c X ) = - J - § " C X ) Y + % / ( > 0 Y / 2 $ t * ) (2.18) Using a s tep-by-step method, s t a r t i n g at X - 0 , the r ight hand side becomes ^ , an indeterminate form. T h i s d i f f i c u l t y was overcome by Vauthier by assuming $/-*) very small but not zero . The present author has found that the above assumption leads to considerable e r r o r . The indeterminateness, however, can be removed by applying L 1 H o s p i t a l ' s r u l e . 17 If a. magnetic f i e l d i s present, an additional complication arises, which i s discussed i n Appendix B. The numerical computations were c a r r i e d out on the IBM 7040 at the Computing Centre of the University of B r i t i s h Columbia. 2.3 Subproblems After having derived the t r a j e c t o r y equation and having discussed the necessary numerical methods to solve the equation we proceed to analyse the mathematical model of a multiple s l i t source ( f i g 1.2). We adopt two d i f f e r e n t approaches, namely: (a) The analysis by parts, that i s , the model i s considered part by part, i n p a r t i c u l a r , the e f f e c t of each part on the ion t r a j e c t o r y . (b) The global analysis, that i s , the system i s considered as a whole. Although the analysis by parts i s not as rigorous as the global analysis, i t provides a considerable insight into the problem.^ The global analysis, on the other hand, i s much more accurate but leads, as we s h a l l see l a t e r , to a system of non-linear algebraic equations, the solution of which i s a problem i n i t s e l f . Since the physical understanding of the e f f e c t of each part upon the ion t r a j e c t o r y i s more important than the rigour i n the analysis we have emphasised on the analysis by parts. We s h a l l now divide the problem into f i v e parts and treat each one of them as a separate subproblem. (a) Entrance s l i t plate with uniform f i e l d on both sides of the plate ( f i g 2.1a). 18 E i o (e) 0 17z E -e (e) P i g . 2.1 Geometry of various subproblems. 1 9 (b) Entrance s l i t plate and r e p e l l e r plate ( f i g 2 . 1 b ) . (c) Entrance s l i t and ex i t s l i t plate ( f i g 2 . 1 c ) (d) Deflector plates ( f i g 2 .Id) (e) Magnetic f i e l d i n the source region ( f i g 2 . 1 e ) In case of each one of the above subproblems we f i r s t obtain the pote n t i a l d i s t r i b u t i o n using the method of conformal transfor-mation, and then integrate the paraxial equation. Subproblem (a) A convenient method of solving a two dimensional boundary-value problem i s by the method of conformal transformation. Under a conformal transformation, the Laplace equation and homogenous boundary conditions remain inva r i a n t . A mapping function that maps the u-axis of the w-plane into two semi-i n f i n i t e plates i n the z-plane (see f i g 2 . 3 f o r d e t a i l s of mapping) i s given by £ - K t J + ( 2 . 1 9 ) where P i s constant. If the s l i t width i s i-1 (I=/HT ) , we obtain from ( 2 . 1 9 ) The boundary value problem now reduces to f i n d i n g the solution of the Laplace equation i n the w-plane, which s a t i s f i e s the boundary conditions: V - o on the u-axis V —> E, Re ( f u) W — * co V _> £, U ( P/u) 20 •+ E. V = 0 J s. V = 0 Z = X + < Y W = U + ' V \ \ « %o. -^L . -1 0 1 F i g . 2.2 Conformal mapping of the w y o plane onto the whole of Z-plane by Schwarz-ChristofFel method. 21 Consider a f u n c t i o n V ( " , v - ) as defined by V ( u , v ) = R< ( > ( e i w + i i ) ) ( 2- 2°) VJ ' we see that the above f u n c t i o n s a t i s f i e s the Laplace equation and the boundary c o n d i t i o n s . To map t h i s p o t e n t i a l onto the z -plane , we note that equation (2.19) may be expressed as and - te ( 2- t l ^ - ^ 1 ) • On s u b s t i t u t i n g i n (2.20) we obtain A l l a r d and R u s s e l l (1963) considered a s i m i l a r problem (but assumed o ) and gave numerical values of p o t e n t i a l d i s t r i -bution around the s l i t . However, they d i d not express the p o t e n t i a l d i s t r i b u t i o n i n a convenient form as i n equation (2.21). I f we l e t Y= o , we obtain the a x i a l p o t e n t i a l V«(x) = { E, (x •+Jx-+0-25 ) t-i E a ( x -J*^7s) (2.22) We are now i n a p o s i t i o n to integrate the p a r a x i a l equation not ing 3?Cx) = \ ( f l ) - V t f (*) . Two cases have been i n v e s t i g a t e d : ( i ) E z = o ; £ , ^ o (exact value i s not necessary) , that is , a p a r t i c l e under the inf luence of a c c e l e r a t i n g f i e l d o n l y . ( i i ) Ei*ro,E,*cO that i s , the p a r t i c l e i s under the inf luence of the r e p e l l e r f i e l d and the a c c e l e r a t i n g f i e l d . A few t y p i c a l t r a j e c t o r i e s f o r case ( i ) of the p a r t i c l e s s t a r t i n g 1 0 -Ion t No re ca jec tor ie s e l l e r f i e s t a r t i n g s Id . point ; 2 = 0 ° ' 5 Ent] y •anee s l i t L 5 \ J 15 20 25 30 35 X 4 0 - 0 . 5 F i g . 2.3 23 from rest are shown i n f i g . ( 2 . 3 ) . The general shape of the t r a j e c t o r i e s i s i l l u s t r a t e d i n f i g . ( 2 . 4 ) . I t has, as shown, two cross-overs x c , and *c 2 , the p o s i t i o n s of which depend upon the s t a r t i n g plane fCXo) and the r a t i o £ - I / E , . F i g -ures (2 .5) and (2 .6) i l l u s t r a t e the r e l a t i o n s h i p between the p o s i t i o n of cross-overs and the parameters x c , the s t a r t -ing plane, and £ , 7 E ( > the r a t i o of r e p e l l e r f i e l d to the a c c e l e r a t i n g f i e l d . We see from f i g . (2 .5) that , when most of the ions (*o<-3j) are focused a short distance away from the s l i t . The p o r t i o n of the t r a j e c t o r y from the second cross-over onwards may be approximated by a quadratic f u n c t i o n where ro i s a constant depending upon * 0 and E^/H , , and >s i s the slope of the t r a j e c t o r y at the second cross-over and also a f u n c t i o n of xc and E*-/e, . I t may be i n t e r e s t i n g to note that equation (2.23) i s also the equation of t r a j e c -tory In a uniform f i e l d ; therefore , i t appears that the e f f e c t of the s l i t width does not extend beyond a distance equal to a few s l i t widths. In the presence of , however, the ions are no longer focused a short distance from the s l i t , but, depending upon e V / E , , over a considerable length along the a x i s . (See sect ion 2.5 f o r f u r t h e r d i s c u s s i o n . ) The approximations made to l i n e a r i z e the trajectory equa-t i o n g i v i n g the p a r a x i a l equation are u s u a l l y very r e s t r i c t i v e and are val id only within the close neighborhood cf the a x i s . Most 24 E 2 + E i F i g . 2.4 Graphical i l l u s t r a t i o n of a representative t r a j e c t o r y . The p o s i t i o n of cross-over X C l and X c ^ depends upon the s t a r t i n g plane P(XQ) and E g / E ^ . 11 Po St (X s i t l o n o artlng p 0 f c ross -Dint. F i r s t c I over vs.! j ross-over X ~ \ >> 0 Second cross-ov er) • F i r s t cross-c ver Sec ond crof s-over 1 x C 2 ) 0 -2 ^ 6 -10 -12 -14 -16 I n i t i a l p o s i t i o n of a p a r t i c l e X Q F i g . 2.5 2 6 T J ^ H <D <D > - H r> r, | KJ 0) O rA U H O 0.) •tt <D f d o < ffi o ^ 1 1 1 c H) iii CO > • • o Jx! o CO H CO aaAo-ssoao j o uoxq.fsoj 27 authors seem to be s a t i s f i e d with the approximations without s t a t i n g how f a r from the axis one can go without v i o l a t i n g them. But we f e l t i t necessary to sketch the region around the axis where the p a r a x i a l approximation w i l l lead to an error l e s s than a c e r t a i n predetermined value . The p o t e n t i a l d i s t r i b u t i o n around the axis i s approxi-mated by a ser ies The ser ies i s found to be convergent w i t h i n a domain bounded by a curve A i n f i g . (2.17). I f we take the f i r s t two terms only , as was done i n the p a r a x i a l approximation, the region where the f i r s t two terms are adequate i s shown by curve B i n f i g . (2.7). To test the net e f f e c t of the various approxi-mations, and thus to determine the region of v a l i d i t y , we approximate the exact equation (2.16) to the increas ing degree of s e v e r i t y : (a) Exact equation (b) Taking the f i r s t two terms (c) Non- l inear p a r a x i a l equation * $ . Y " . D O + Y'V ( 2 . 2 4 C ) and (d) L inear p a r a x i a l equation (2.24d) P i g . 2.7 - Domain of convergence of ser ies expansion. 30 The above equations were integrated numerically and the t r a -j e c t o r i e s were compared f o r a simple case dealt with i n t h i s sub-problem. I t was found that, i f a p a r t i c l e were to start anywhere within the shaded region i n figure ( 2 . 8 ) , the r e s u l t i n g t r a j e c t o r y i s within 5$ of the exact t r a j e c t o r y obtained from equation (2.24a). Indeed, the l i n e a r paraxial equation gives better r e s u l t s as compared to the other two approximate equations i n the paraxial region. Subproblem (b) In the previous subproblem we have considered a single s l i t and a 'uniform' f i e l d on either side ( E, the accelerating f i e l d on the right hand side, and Ej. the r e p e l l e r f i e l d on the l e f t ) . Here we s h a l l consider a r e p e l l e r plate and the entrance s l i t plate as shown i n f i g 2.1b. I t i s i n t e r e s t i n g to know the e f f e c t of the r e p e l l e r plate, p a r t i c u l a r l y when i t i s close to the entrance s l i t plate, l e , when d i s small; and how i t d i f f e r s from that of the uniform f i e l d ( i e , d->oo ), which we considered i n the previous subproblem. In order to obtain the p o t e n t i a l d i s t r i b u t i o n , we use, as before, the method of Schwarz-Christoffel. The transformation function i s given by to + (2.25) where the constants IP and a are given i m p l i c i t l y by (2.26) 3 1 \ )• c ! v 0 A V C fr d 'v A -> E Z = X + 1 Y vr = u + 1 v • v _S 2 __CC «_ _AAJ _ _ S i - 1 -a 0 a 1 F i g . 2 . 9 ^ Conformal mapping of the w )>o plane onto the whole of Z-plane by Schwarz-ChrlstofFel method. jj^ = - 1> Cw- i ) C w+0 ( w+aj" '^u-cv)" 1 32 I t i s i n t e r e s t i n g to note that , i f we l e t d -**> , ^ 0 and p , then the transformation f u n c t i o n (2.25) reduces to (2.19) as expected. And also as , a ~ * [ a n d *° "*a > equation (2.25) reduces to which i s just the l i n e a r t ransformation. To solve the system of equations (2.26), we s h a l l e l iminate p between them and rewrite Let '? ^ d" and V| - a The above equation becomes * _ 2 ( a ^ , ^ . ^ ] (2.27-) » + 1 The above equation may be solved i n a number of ways; f o r instance, Newton-Rapson, Regula F a l s i , e t c ; but here we s h a l l adopt a d i f f e r e n t method which does not require us to guess at the i n i t i a l approximation of the root . We s h a l l express (2.27'$ i n the form of a d i f f e r e n t i a l equation, v i z . : ,2 8 v 1 J and integrate numerical ly with the i n i t i a l values ( 34 n. o ( t h i s f o l l o w s from the f a c t that as d -* 0 0 / * -> 0 ). The s o l u t i o n i s shown i n f i g u r e (2.10). Having obtained the transformation f u n c t i o n , we proceed to solve f o r the poten-t i a l d i s t r i b u t i o n , f i r s t i n the w-plane and then map onto the z - p l a n e . The boundary condit ions to be s a t i s f i e d are V - V 0 0 ^ 1 / - = . o , l u \ > a V - V 0 + . A V <?n o / = 0 , I IA I N< a The l a s t boundary c o n d i t i o n f o l l o w s from the f a c t that i n the z-plane Prom the transformation f u n c t i o n (2.25) i t f o l l o w s that To f i n d the p o t e n t i a l d i s t r i b u t i o n i n the w-plane, we con-sider the f o l l o w i n g cases and superpose them l a t e r : (a) VV = Ve on the u - a x i s and uniform f i e l d i n the w>0 plane 35 (b) V b - AV the u - a x i s . on the u - a x i s n )> -a and zero on the rest of (c) V c - - A v on the u - a x i s , u>a and zero on the u - a x i s Vc = ( t a n ML- ^ Therefore , the net p o t e n t i a l i n the w?0 plane i s given by V - - V , + v b ^ v f ( 2 . 2 8 ) . w + a Now, i n order to map the p o t e n t i a l d i s t r i b u t i o n onto the z -plane , we have to Invert the transformation f u n c t i o n (2.25)> which i s equivalent to the process of s o l v i n g i t as a nonlinear equation f o r a given value of z . I t i s then expressed as a system of two nonlinear equations Z C k L IA - a (/ +<Tv J and Y - -2 2<A L- 1/2. (2.29a) (2.29b) Since M - OA = o i t f o l l o w s that x-^=o j u s i n g these as i n i t i a l c o n d i t i o n s , we can solve the above system of equations by the 1 Re] (a) >eller p l a B = V / E ! ; e / f i e l d =- 0 . 0 2 , d = - 1 0 \ \ \ (b) Y (c) (d) E2 / E ! =• 0 EZ/E-L = 0 B - 0 . 0 , 0 0 2 , d-» .0 CO N X \ \ v\ k \\ 0 . 5 (c). 1 k • 9 " J.—" H q 1 ""' vV^v^, \ \ « ! ^ — J 2 T ~ P i g . 2 . 1 1 38 method we have devised f o r (2.27') • However, since we are, in teres ted i n the a x i a l p o t e n t i a l only (paraxia l approximation), i t i s s u f f i c i e n t to consider (2.29a) and l e t AA — which may be integrated numerical ly with the i n i t i a l condit ions vX - v = o . We are now i n a p o s i t i o n to integrate the p a r a x i a l t r a j e c t o r y equation. The computer was programmed to accept the values of d and * V E ( A V r e p e l l e r f i e l d , and £ a c c e l e r a t i n g f i e l d ) , then to compute the constants P and a , and to i n v e r t the transformation f u n c t i o n at each step. A few of the computed t r a j e c t o r i e s are shown i n f i g (2.11). The t r a j e c t o r i e s (a) and (b) are drawn to show the e f f e c t of proximity of the r e p e l l e r p l a t e ; t r a j e c t o r y (a) i s drawn f o r d=-\o and t r a j e c t o r y (b) f o r d —* c~ keeping the r e p e l l e r f i e l d the same, i e , =• c-ooi The mere presence of the r e p e l l e r pla te without any r e p e l l e r voltage ( A V = - D ) seems to a f f e c t the t r a j e c t o r y of the p a r t i c l e as shown by c and d i n the same f i g u r e . T h i s , of course, we would expect, because the r e p e l l e r plate w i l l d i s t o r t the p o t e n t i a l d i s t r i b u t i o n . The r e l a t i o n between the p o s i t i o n of the cross-overs ( x C l and xCg) and the r a t i o A v / k i s shown i n f i g u r e 2.12. The r e l a t i o n s h i p i s very s i m i l a r to that when the plate i s at i n f i n i t y (compare with f i g 2.6) 39 To summarize, we see that the e f f e c t of a r e p e l l e r v o l t -age i s i n no way much d i f f e r e n t from that of a r e p e l l e r f i e l d ( i . e . when d = - °° ) unless i t Is close to the s l i t (d <^10). I f d i s very small (d —> 0) , the e f f e c t of the entrance s l i t plate i s considerably masked by the r e p e l l e r p l a t e , and i n the end, when d = 0, there i s no per turbat ion f i e l d at a l l . The p r i n c i p a l e f f e c t of the r e p e l l e r f i e l d i s to s h i f t the cross-over along the x - a x i s , thus to produce a l i n e a r l y d i f f u s e d focus which, i n the absence of a r e p e l l e r f i e l d , would have been sharp. Indeed, i t i s easy to adjust the r e p e l l e r f i e l d such that some of the p a r t i c l e s are focused just i n f r o n t of the e x i t s l i t . I f somehow a l l molecules are i o n i z e d w i t h i n a small space, we could , by adjust ing the r e p e l l e r f i e l d , focus' a l l the ions l n f r o n t of the e x i t s l i t (see sec t ion 2.5 f o r f u r t h e r d i s c u s s i o n ) . Subproblem (c ) : In t h i s subproblem we s h a l l consider the e f f e c t of the ex i t s l i t , p a r t i c u l a r l y on the angle of divergence of the beam. The geometry of the problem i s i l l u s t r a t e d i n f i g . (2.13). Since L >^ > 1, the mutual coupl ing between the s l i t s i s n e g l i g i b l e ; therefore , each s l i t may be considered separately . The p o t e n t i a l d i s t r i b u t i o n i n the three d i f f e r e n t regions, using (2.21), may be expressed by region II 40 Entrance s l i t width 1 E x i t s l i t width S L - 4 0 P i g . 2.13 Entrance and E x i t s l i t p la tes , 150 " E f f e c t of the e x i t s l i t -on the angle of divergence (X 0 = -4 ) E x i t s l i t width (S) Pig. 2.14 42 region I I I (2 .30) (centre of co-ordinates at the centre of the entrance s l i t . ) The p a r a x i a l t r a j e c t o r y equation i s now integrated where the a x i a l p o t e n t i a l <3? i s to be obtained from (2 .30) depending upon the r e g i o n . The angle of divergence of the beam f o r the d i f f e r e n t values of the ex i t s l i t i s shown i n f i g . (2 .14) f o r a p a r t i c l e s t a r t i n g at X 0 = - 4 , Y Q = 1.0 . For any other p a r t i c l e , s t a r t -ing at a height , say Y Q = <L , the same r e l a t i o n s h i p holds true except that the v e r t i c a l scale i s to be m u l t i p l i e d by a . The slope of the t r a j e c t o r y that just passes through the ex i t s l i t - - g r a z i n g trajectory-—can be determined from the f i g u r e s ( 2 . l 4 a ) and (2 . 14) . F i r s t l y , one determines Y 0 of the grazing t r a j e c t o r y f o r a given e x i t s l i t width (s) from f i g u r e ( 2 . l 4 a ) and f i n a l l y , using f i g u r e ( 2 . 1 4 ) , the slope of the t r a j e c t o r y . Subproblem (d): In t h i s subproblem we s h a l l consider the d e f l e c t i o n of the beam passing between two p l a t e s , which are kept at d i f f e r e n t p o t e n t i a l s ( f i g . 2 . 15) . We s h a l l assume a uniform f i e l d on both sides of the p l a t e s . A narrow monoenergetic beam enters the gap from l e f t to r i g h t . F i r s t of a l l , we solve the boundary value problem to determine the p o t e n t i a l d i s t r i b u t i o n . It i s again convenient to use the method of conformal transformation 43 P i g . 2.15 D e f l e c t o r places kept at d i f f e r e n t p o t e n t i a l s . Beam passes from l e f t to r i g h t . P o t e n t i a l d i s t r i b u t i o n around the d e f l e c t i n g places D e f l e c t i o n of beam Eg/fc-L = 1.0 AV'/E-L = 0.5 0.5 F i g . 2.17 46 which i s i l l u s t r a t e d i n P i g . (2 .2 ) . The boundary condit ions to be s a t i s f i e d by the p o t e n t i a l f u n c t i o n are V - V z on ' w <o , V = o V = V, o n l A > d , V-0 V - » E ft«(Pw) w V — > E R e ( p /u) . w —> o Such a f u n c t i o n i s given by J ] * J (2.31) where P = - % P o t e n t i a l d i s t r i b u t i o n may be mapped onto the z-plane using transformation f u n c t i o n (2.19). The p a r a x i a l approximations are no longer v a l i d because the p o t e n t i a l d i s t r i b u t i o n i s not symmetric about the axis (since V^\JZ ); therefore , we s h a l l have to integrate the exact equation (2.10). We may s i m p l i f y the p o t e n t i a l d i s t r i -but ion considerably by l e t t i n g E-j_ = E 2 > and thus we have the d e f l e c t i n g f i e l d given by n U 1 (2.32) which i s i l l u s t r a t e d i n f i g . (2 .16) . Except i n the immediate v i c i n i t y of the gap, the f i e l d i s normal to the radius vector and i n v e r s e l y p r o p o r t i o n a l to the r a d i a l d i s t a n c e . Introducing the above approximation, the t r a j e c t o r y equation may be written **Y"+ Y'- ^T7[ -° AV I 1*1 >° (2.33) 1 ?o». / (Min.) / A / o / a 60' / igle of i n c l i n a t i o n r i o n t r a j e c t o r y b +25 u n i t s 2 1 / 0 60' -1 AV/E-L -2 120' F i g . 2.18 D e f l e c t t r a j e c t ion of the ovj at +25 c e n t r a l u n i t s V • 1.0 -E 2 / E 1 = 1.0 L /o 1 =1 -1.0 P i g . 2.19 49 The equation (2.33)> though simple, i s not very u s e f u l when we have to trace the t r a j e c t o r y across the gap. We have, therefore , preferred to integrate the exact equation. A beam o r i g i n a l l y p a r a l l e l to the x -axis i s d e f l e c t e d by the p l a t e s as shown i n f i g . ( 2 . 1 7 ) . I t should be noted that the beam s t i l l remains p a r a l l e l a f t e r i t s passage through the gap. The t o t a l d e f l e c -t i o n from the axis i s l i n e a r l y r e l a t e d to A V / E ( f i g . 2 . l 8 ) . One may compute the t o t a l d e f l e c t i o n at any distance X(X ) 1) f o r a given value of ^ v / L from the f i g u r e s (2.17) and ( 2 . l 8 ) ; as f o r example, suppose we wish to f i n d the d e f l e c t i o n at a distance X = 10 and A V / E . - 1. From f i g . (2.17) we measure the d e f l e c t i o n at X = 10 ( A V / £ = - 0 . 5 ) . P lot t h i s point on f i g . ( 2 . l 8 ) and draw a l i n e j o i n i n g to the centre of co-ordinates , thus g i v i n g a new r e l a t i o n between the d e f l e c t i o n and 6 V / E . Using the new r e l a t i o n , i t i s easy to determine the t o t a l e f -f e c t f o r any value of ^ V e . . S i m i l a r l y , we can determine the angle of d e f l e c t i o n at any point X from f i g u r e s (2.17) and ( 2 . 1 9 ) . Subproblem (e) : In t h i s subproblem we s h a l l consider i o n t r a j e c t o r i e s i n the presence of a magnetic f i e l d . A magnetic f i e l d of the order of a few hundred gauss i s of ten used to guide the i o n i z -ing e l e c t r o n beam i n the source r e g i o n . We s h a l l assume a uniform magnetic f i e l d normal to the plane of a paper. The p a r a x i a l t r a j e c t o r y equation taking in to account the magnetic f i e l d i s given by z $ Y "+ <$,'Y ' -r <£>„"Y + \W !~I±\ = O V l< rvi 0 50 © Entrance. s l i t © © Pig.' 2: 20 © Magnetic f i e l d normal to the plane of the paper (coming out) 51 (2.34) (see p . 14) where 8* stands f o r the magnetic f i e l d , and w i s equal to the entrance s l i t width. The presence of the magnetic f i e l d introduces an a d d i t i o n a l term into the p a r a x i a l equation and makes i t inhomogeneous. If Y-^  and Yg are the fundamental solut ions of the corresponding homogeneous equation, the p a r t i -cular s o l u t i o n may be given i n terms of Y-, and Y9 as f o l l o w s : Therefore , the e f f e c t of the magnetic f i e l d i s given by (2.35). Since 6?- i s constant, i t may be taken outside the i n t e g r a l s i g n ; and hence the e f f e c t i s p r o p o r t i o n a l to the f i e l d strength. In p r a c t i c e , however, to obtain the p a r t i c u l a r s o l u t i o n , i t i s convenient to integrate the equation with the homogeneous i n i -t i a l condit ions ( i . e . y(o)-\{/co)=. o ). There appears to ex is t a c e r t a i n d i f f i c u l t y as pointed out before to i n i t i a t e the numerical s o l u t i o n . The inhomogeneous part i s s ingular at the s t a r t i n g point because q> = o at the s t a r t i n g p o i n t . Y i r* —— *• (2.35) where A ( Y 0 Y g Y , Y v - Y ^ Y ' J -1 Ion t ra je presence magnetic : (e lec t ron j t o r i e s i n >f source ' i e l d the 0. 5 /yL0.Jj / / 0 'm 2 w/ra2 . 6 11 16 2 1 ^ ^ 26 31 36 X -c .5 P i g . 2 .21 - 1 . 0 E f f e c t of D z = 1 w (e lectron magnetic sber/vn2-) f i e l d ^ q \ l 6 11 16 21 26 3 l / / / / ^ P i g . 2.22 I I ro i H O P o s i t i o n of cross-over vjn o Effect field o: of seco: Df raagne i the po id cross i ca cr O H* H-< c+ O CD H" t-j O P H" OS ro ro ro CO CR 13 CD ct 4^ O •t) H> CD H & . U 1 55 By v i r t u e of the i n i t i a l condit ions ( y to) - Y ' ( O ; = . o )', the f i r s t two terras vanish, but the l a s t term becomes i n f i n i t e . I f , however, \*-Xc\ X —> Xp the i n t e g r a l i s convergent as shown by the Cauchy test (see Appendix B ) . T h i s , however, does not help to i n i t i a t e the numerical s o l u t i o n . We have to adopt a p h y s i c a l viewpoint : when the v e l o c i t y of the ion i s small ( $ —? o ), the force due to the magnetic f i e l d w i l l be small , hence one may ignore the inhomogeneous part when i n i t i a t i n g the numerical s o l u t i o n . The equation (2.34) was integrated numerical ly f o r the f o l l o w i n g values of the constants: 2 g> = .1, .5 webers/m E = 10^ vol t /meter w = I O " 3 meter \ = 1.7592 x 10^ coloumb/kilogram (electron) The t r a j e c t o r i e s are shown i n f i g . (2.21). The p a r t i c u l a r s o l u -t i o n of (2.34) f o r Qir - 1 w/v>1 and other constants as above i s i n f i g . (2.22). T h i s f i g u r e may be used to compute the ef -f e c t of the magnetic f i e l d f o r any other set of constants (e .g . f o r PbCch 3 )^ ion i t may be obtained by m u l t i p l y i n g by a f a c t o r 0 . 0 0 ^ 5 ) . An important question i s often asked regarding the mass d i s c r i m i n a t i o n due to the magnetic f i e l d i n the source r e g i o n . Consider two ions (isotopes) of masses and mt , so that 5 6 '0 AV i i ,v 2 i "Vn B 'V, JR' i o Z = x + y plane W = U + V s -c - q -b - p - a a p b q P i g . 2 . 2 4 Conformal mapping of w ) 0 plane onto the whole of the Z-plane 57 W~~ I t _ 1 £J?1 The t r a j e c t o r i e s of the two ions w i l l be separated by ( = 0 . 0 0 8 f o r P b * ° « a n d Pb50sC^3)£ i o n s ) . Nor-mally , the separation of heavy ions ( w » i ) i s very small , and c e r t a i n l y much smaller than the ex i t s l i t even f o r the f i e l d of the order of 1 weber /m 2 „ Moreover, the separation may be minimized by p l a c i n g the e x i t s l i t i n the neighborhood of *Y = 0 . In some cases i t may be p o s s i b l e to f i l t e r out the l i g h t e r ions by adjust ing the magnetic f i e l d . 2 . 4 Global a n a l y s i s : In the previous sec t ion we analysed the ion o p t i c a l system ( f i g . 1 . 2 ) consider ing each part separately . We now proceed to t reat the problem on a g lobal b a s i s . The primary task i s to f i n d the p o t e n t i a l d i s t r i b u t i o n w i t h i n the space bounded by the various p l a t e s . We assume as before the s l i t s to extend to 1 <*> i n the d i r e c t i o n of the y - a x i s . Boerboom (1957* 1959* i 9 6 0 ) considered a system of three such s l i t s and obtained the a x i a l p o t e n t i a l . The f o l l o w i n g treatment i s e s s e n t i a l l y based on h i s work and also gives a method of i n v e r t i n g the transformation f u n c t i o n . We s h a l l use the method of conformal mapping to solve the boundary value pro-blem. The mapping f u n c t i o n i s given i n the d i f f e r e n t i a l form as cK u) (*)+*)"' c^ +i0/' tw+b)'<:^  + <i)+,( o" V * o + l ( 2 . 3 6 ) 58 where a, k , b , ? , c and « are constants and s a t i s f y the r e l a t i o n Equation (2 .36) may be written as = K [ ( AC + 3 D ) + AC ) + AO f ££) + CC where (2.37) (2.38) Integrat ion of equation (2 .37) y i e l d s the transformation f u n c t i o n . + 6 D ' ( ^ ) ' n ( ^ ) ] ( 2 . 3 9 ) 59 There are six unknowns, and therefore we seek six independent equations connecting them. We can get three of them by i n t e -grating equation (2.37) about a , fc5 and c over an i n f i n i t e l y small semicircle. 1 » « C f t ' ^ J , H C « l - e l ) n M r - ^ Q = . - u ( b ^ - ^ X ^ - %L)C b K ^ ) 7T T t C ^ - c U ) ^ - ^ ( c - - ^ ) C ^ - f j ( c - - , v J (2.40) The remaining three r e l a t i o n s we obtain by integrating equation (2.37) across the three s l i t s . The three equations are t UJ , -J, c^-^K^^K^-^J (2.4i) After integration equations (2.41) become CP L K (i-b) ct - i ; J . (2.42) 6 0 w h e r e < =. ' A C + A ( 5 + e c ' - f S>D' A- Ac (£z£) A a, y - AD ( b r - ^) 2 b a n d a.cv 6 - 0 D' C£l±lJ ( 2 . 4 3 ) A C I n ( 2 . 4 0 ) t o m a k e t h e r i g h t h a n d s i d e r e a l a n d p o s i t i v e , we h a v e t o a s s u m e o f t h e t y p e w h e r e t i s a p o s i t i v e u n k n o w n c o n s t a n t . S u b s t i t u t i n g (2.38) i n ( 2 . 4 3 ) , we o b t a i n < - I - a = ( ft-i c) .1 - c d3-<U) = o-i ( 2 . 4 4 ) M a k i n g u s e o f ( 2 . 4 4 ) , we m a y w r i t e e q u a t i o n s ( 2 . 4 0 ) a n d ( 2 . 4 2 ) a s 61 (2 .45) The set of s ix nonlinear equations given by (2 .45) may be solved f o r s ix unknowns. The transformation f u n c t i o n (2 .39) may now be wri t ten as or b L v U-A n + a / (A. u+b / ' V u - c u + c J (2.46) 2- 0 - f - c ) V y> - j (2.47) Prom (2.46 and 2.47) i t fo l lows that the V" - a x i s i s mapped onto the x -axis and the u. - a x i s onto the y - a x i s . To f i n d the inverse transformation of (2.46) and ( 2 .47), we express these equations as a system of two ordinary equations with x and y as independent v a r i a b l e s and integrate them with the f o l l o w i n g 62 as i n i t i a l condit ions Having obtained the transformation f u n c t i o n , we proceed to solve the p o t e n t i a l problem f i r s t i n the w-plane and l a t e r map onto the z-plane using the inverse t ransformation. In the w-plane the p o t e n t i a l f u n c t i o n has to s a t i s f y the f o l l o w -ing boundary c o n d i t i o n s : V - o OK ( a, - «0 V - 0 * ( b, c) (see f i g u r e 2.24) To solve the boundary value problem, we consider only one of the boundary condi t ions at a time and assume V = o on the rest of the u - a x i s . These solut ions are then to be added to -gether to give the required s o l u t i o n . T h i s k ind of superposi-t i o n i s permissible f o r l i n e a r d i f f e r e n t i a l equation such as the Laplace equation. The s o l u t i o n i s then given by ' re L K - B I M V - \ u-a u + «JJ rt L I A - c U- b j K [_ * + 'C u-fb J 34 JT c - u t4 U J (2.49) T h i s Is v a l i d everywhere except at c e r t a i n points on the u-axis 63 ( i . e . <yt b 11, f, %) i , - a J - b J - t, -1 , - f, - | ). T h i s completes the formal s o l u t i o n of the boundary value problem. 2.5 E f f i c i e n c y of ion beam transmission: The present sect ion i s devoted to the drawing together of the r e s u l t s of the d i f f e r e n t subproblems discussed i n sect ion 2.3. We s h a l l a lso discuss the ion o p t i c a l proper t ies of the system, p a r t i c u l a r l y with reference to the ion beam transmis-sion e f f i c i e n c y , that i s , the f r a c t i o n of the beam u l t i m a t e l y enter ing the magnetic analyser tube. Since the s l i t widths are much smaller than the separa-t i o n between the two consecutive p l a t e s , the mutual coupling i s expected to be n e g l i g i b l e . The r e p e l l e r p l a t e , however, i s not always as f a r as i s required (d ^ 10) f o r minimum c o u p l i n g . The f o l l o w i n g d i s c u s s i o n , nevertheless , i s based on the assump-t i o n that the r e p e l l e r plate i s at a distance equal or greater than ten times the entrance s l i t width. Further , I t was not f e l t necessary to compute the ion t r a j e c t o r i e s . u s i n g the more precise p o t e n t i a l d i s t r i b u t i o n given by (2.49). The p r e c i s i o n thus gained would be unmeanlngful consider ing the e r rors i n t r o -duced by the p a r a x i a l approximations and the assumption of zero i n i t i a l energy. The t y p i c a l i o n t r a j e c t o r i e s i n the absence of a r e p e l l e r f i e l d are shown i n the f i g u r e (2.25). They have, i n general , two c r o s s - o v e r s . Furthermore, a l l the ions s t a r t i n g from rest are bound to pass through the entrance s l i t " * and cross the axis A p a r t i c l e w i l l not s t r i k e the p l a t e s because they are at a higher p o t e n t i a l than the surrounding space. I f , however, the p a r t i c l e has f i n i t e i n i t i a l energy, t h i s w i l l not be true f o r a l l p a r t i c l e s . T y p i c a l i< V e r t i c a l >n t r a j e c t scale exag sries gerated. 2 0 . = o 5 Ehtrar ce s l i t y E X E t i t s l i t I) s> 1U 20 30 -C .5 F i g . 2 . 2 5 65 at a short distance from the s l i t and thus produce a f a i r l y sharp f o c u s . The p o r t i o n of the t r a j e c t o r y to the r i g h t of the second cross-over may be approximated by a simple quadratic expression (2.23), which i s , i n c i d e n t l y , the i o n t r a j e c t o r y i n a uniform f i e l d . The t r a j e c t o r i e s are divergent ; hence only a few p a r t i c l e s are capable of passing through the ex i t s l i t which i s of ten much smaller than the entrance s l i t (about a tenth of entrance s l i t width) . By a process of ac tual observation whether a given t r a -jec tory s t a r t i n g at a point PCx 0 , ^ 0 ) passes through the e x i t s l i t kept at a distance of f o r t y s l i t widths, we have sketched i n f i g . (2.26) a region — a narrow h o r i z o n t a l s t r i p of width approxi -mately equal to the e x i t s l i t width. A p a r t i c l e s t a r t i n g any-where w i t h i n the region i s bound to pass through the ex i t s l i t , and a l l ions formed outside the region are n a t u r a l l y l o s t comp-l e t e l y . T h i s i s c a l l e d the area of ion withdrawal. However, the s i t u a t i o n i s somewhat d i f f e r e n t i n the pre -sence of a r e p e l l e r f i e l d i n the i o n i z a t i o n space. Now the second cross-over i s no longer sharp as before but l i n e a r l y d i f f u s e d along the a x i s . It i s n a t u r a l to expect to be able to adjust the r e p e l l e r f i e l d such that the cross-over point l i e s just i n the neighbourhood of the e x i t s l i t . Unfortunately , I t i s not p o s s i b l e to do so f o r a l l the ions as we see i n f i g . ( 2 . 2 7 ) . Only a small f r a c t i o n of ions coming from a c e r t a i n s p e c i f i c region of the i o n i z a t i o n space w i l l cross the axis near the e x i t s l i t . In f i g . (2 .29) we have given a r e l a t i o n between the 66 L P o cx o Exit sll in o • o i trawal str: .ler field o i — •• Ion with! (No repel ,/^> " T r ^ s V . V •0 ^ " ^ ^ rH ^  ^ I E f f e c t of r e p e l l e r f i e l d . Ion withdrawal s t r i p s . E2^1 0.0025 (Vert , scale = 10 x hor . scale) 0.5 -0.5 r-0.05 F i g . 2.27 P i g . 2.28 69 4 6 R e p e l l e r V o l t a g e P i g . 2.30 E x p e r i m e n t a l l y o b s e r v e d r e l a t i o n s h i p b e t w e e n t h e r e p e l l e r v o l t a g e a n d t h e i o n c u r r e n t . 71 E Z / E , = 0.0 F i g . 2.31 E f f e c t of r e p e l l e r f i e l d on the shape of the s t r i p s . 72 r e p e l l e r f i e l d and the p o s i t i o n of the s t r i p of ion withdrawal. For a given value of a r e p e l l e r f i e l d , , we see from f i g . ( 2 . 2 9 ) that there are two values of X0 , two s t r i p s of ion w i t h -drawal ( f i g . ( 2 . 2 7 ) ) . The shape of the s t r i p s i s shown i n f i g . 2 . 2 8 f o r f *7E, = 0 . 0 0 2 . Thus, i n the presence of a r e p e l l e r f i e l d the ions from a v e r t i c a l s t r i p are focused i n the neigh-bourhood of the ex i t s l i t . I f somehow the i o n i z a t i o n could be confined to the narrow s t r i p , we should be able to withdraw almost a l l the i o n s . But unfor tunately , i n p r a c t i c e , i t i s d i f f i c u l t to confine the i o n i -z a t i o n to such a r e s t r i c t e d space ( 2 to 3 mm). However, we can adjust the r e p e l l e r f i e l d such that the region of withdrawal corresponds to the region of high ion d e n s i t y . The peaks i n the experimentally observed r e l a t i o n between the r e p e l l e r f i e l d and the ion current ( f i g . ( 2 . 3 0 ) ) may be explained by the above con-c l u s i o n based on the t h e o r e t i c a l c o n s i d e r a t i o n s . The e f f e c t of the ex i t s l i t i t s e l f on the Ion t r a j e c t o r y , e s p e c i a l l y the change i n the angle of i n c l i n a t i o n as an ion passes through the e x i t s l i t i s found to be very small ( less than 1 0 $ ) . Also the e f f e c t of the magnetic f i e l d i s small , 2 unless the f i e l d i s of the order of a few webers/m . Fur ther -more, there w i l l be no percept ible mass d i s c r i m i n a t i o n due to the source magnetic f i e l d unless the f i e l d i s l a r g e . In summary, the ion beam transmission e f f i c i e n c y could be increased only i f we could confine the i o n i z a t i o n to a r e s t r i c t e d space and adjust the r e p e l l e r f i e l d a c c o r d i n g l y . 73 Chapter I II The inverse Problem of P a r t i c l e Motion Synopsis : In t h i s chapter we introduce the concept of the inverse pro-blem of p a r t i c l e motion, i n the usual problem we seek the t r a -jec tory of a p a r t i c l e , having been given the e l e c t r o s t a t i c poten-t i a l d i s t r i b u t i o n . In the inverse problem, we seek the poten-t i a l d i s t r i b u t i o n to guide a p a r t i c l e along a prescr ibed path. S t a r t i n g from the basic t r a j e c t o r y equation and Laplace equation we have shown that f o r any prescr ibed path a unique p o t e n t i a l d i s t r i b u t i o n can be obtained. Next, we general ize the problem: a unique p o t e n t i a l d i s t r i b u t i o n can be obtained f o r a set of prescr ibed paths s a t i s f y i n g c e r t a i n geometrical r e l a t i o n s . F i n a l l y , using the theory, two types of ion sources are proposed. The s t a b i l i t y a n a l y s i s , however, shows that only one of them i s s table , and therefore p r a c t i c a l . The l a s t sec t ion deals with the p r a c t i c a l r e a l i z a t i o n of the required p o t e n t i a l d i s t r i b u t i o n . 3.1 Theory of Inverse Problem: Usual ly we are given a p o t e n t i a l d i s t r i b u t i o n and required to f i n d the p a r t i c l e t r a j e c t o r i e s . But i n the most a p p l i c a t i o n s we desire the p a r t i c l e s to f o l l o w c e r t a i n paths; f o r Instance, c e r t a i n des i rable paths i n the ion source of a mass spectrometer are shown i n f i g . 3.1. This gives r i s e to the inverse problem: that i s , to f i n d a p o t e n t i a l d i s t r i b u t i o n to guide the p a r t i c l e s 74 Converging paths. The shaded s t r i p stands f o r the i o n i z a t i o n space. The ions are to be guided along a set of converging paths. Damped o s c i l l a t o r y paths. The ions are to be guided along the paths. The damped o s c i l l a t o r y paths are i n a way~ complementary to converging paths. For more explanation see text p . 85 . F i g . 3 .1 75 along the prescr ibed paths. In t h i s sect ion we are going to develop the basic theory of the inverse problem of p a r t i c l e motion. The author i s unaware of any previous work on t h i s problem except a suggestion, f i r s t made by Pierce (1954), that the p a r a x i a l equation (2.16) may be regarded as an equation f o r the p o t e n t i a l on the axis i f ^ and i t s d e r i v a t i v e s are g iven . The f o l l o w i n g are the two basic theorems i n the theory of the inverse problem: (a) There e x i s t s a p o t e n t i a l d i s t r i b u t i o n to guide a p a r t i c l e along any desi red path. (b) A group of p a r t i c l e s may be guided along a set of p a r a x i a l paths. The proof of the above theorems i s given below. We s h a l l confine ourselves to the two dimensional case; however, the arguments may be extended to the three dimensional case, too . In the t r a j e c t o r y equation <^  t ^ ' and y" are now known (or prescribed) funct ions of x. The t r a j e c t o r y equation and Laplace equation form a system of p a r t i a l d i f f e r e n t i a l equa-t i o n s . (The c o e f f i c i e n t s of $ x and fc? are funct ions of x o n l y . ) (3.1) I'D - Q (3 .2) 76 We have to solve the above system of equations f o r the poten-t i a l . The technique of s o l v i n g the equations i s as f o l l o w s : The equation (3.1) i s solved f i r s t , given the p o t e n t i a l § along the prescr ibed path, to obta in <^ >, ^ ^ and i n the neigh-bourhood of the path. Knowing Ox. and we can compute the normal d e r i v a t i v e ^ . The p o t e n t i a l cb and i t s normal d e r i v a t i v e const i tute the Cauchy problem. Now i n order to f i n d the p o t e n t i a l <^  everywhere i n the x, y -plane , we have to solve the Cauchy problem f o r (3.2). The p o t e n t i a l i s assumed to be a n a l y t i c as required f o r the uniqueness of the Cauchy problem (Hellwig, 1964). The f i r s t order p a r t i a l d i f f e r e n t i a l equation (3.1) i s equivalent to a system of ordinary d i f f e r e n t i a l equations d e f i n -ing the c h a r a c t e r i s t i c s of the equation (see Courant and H i l b e r t , p . 62). I f we Introduce a parameter arc length along the c h a r a c t e r i s t i c s , the system of ordinary equations becomes Ay _ - - _ 2 - (3.3) 7 d 5 |_v ~v' 2-where, as u s u a l , the prime i n d i c a t e s d i f f e r e n t i a t i o n with respect to x. Every surface generated by (3-3) i s an i n t e g r a l surface of (3.1). A unique surface, however, may be defined from the i n i t i a l value problem f o r (3.3). Let us define a space curve C by 77 p r e s c r i b i n g x ; , and c^> as a f u n c t i o n of a parameter t (arc length) such that the p r o j e c t i o n c0 of c on the x , y -plane coincides with the prescr ibed path of a p a r t i c l e . Now, i n the neighbourhood of c0 , we seek an i n t e g r a l surface t j ^ C x ^ ; which passes through C , that i s , a s o l u t i o n of (3.1) f o r which fciC-t) =. $ ( x it) , y tt)) holds i d e n t i c a l l y i n t . To solve the i n i t i a l value problem, l e t us draw through each point of C a c h a r a c t e r i s t i c , that Is , the s o l u t i o n of (3.3); t h i s i s poss ible i n a unique ,way w i t h i n a c e r t a i n neighbourhood. We thus obtain a f a m i l y of character-i s t i c s x - x C s , trj These curves w i l l generate a surface ^ C i , ^ i f , u s i n g the f i r s t two f u n c t i o n s , we can express s and "t i n terms of x and ^ To be able to do t h i s , we must show that the Jacobian A = • l l _ l l • ?2L i s non-vanishing . Using equation (3*3) we obtain A = y' yt. + x t which Is non-vanishing f o r a l l values of y/ . The geometrical 78 i n t e r p r e t a t i o n of the c o n d i t i o n that the Jacobian i s non-vanish-ing i s that at every point on c 0 the p r o j e c t i o n of the tangent d i r e c t i o n and c h a r a c t e r i s t i c d i r e c t i o n on the x ,y -plane are d i s -t i n c t . In f a c t In our case, since cwA tl =- -I \ A X.I tk<\A ^ the base c h a r a c t e r i s t i c and the path of a p a r t i c l e are orthogonal . We therefore conclude that f o r any simple * prescr ibed path a s o l u t i o n of equation ( 3 . 1 ) may be found i n the neighbourhood of the path; and therefore , , the normal gradient may be found. The p o t e n t i a l d i s t r i b u t i o n i n the x ,y -plane may be obtained as a s o l u t i o n of the Cauchy problem f o r equation ( 3 - 2 ) . T h i s es ta -b l i s h e s theorem I . We next proceed to the second theorem. We r e c a l l that In the s o l u t i o n of ( 3 . 1 ) as an i n i t i a l value problem, the p o t e n t i a l ^ was prescr ibed a r b i t r a r i l y along the p a t h of a p a r t i c l e . T h i s may now be chosen such that a group of p a r t i c l e s moves along a set of p a r a x i a l paths that are prescr ibed again a r b i t r a r i l y . The t r a j e c t o r y equation of the neighbourhood p a r t i c l e s i n the p a r a x i a l approximations i s given by Waters ( 1 9 5 8 ) : ( 3 . 5 ) * The path may not have any double p o i n t s . 79 where M. Is the normal distance of a neighbouring par-t i c l e from the c e n t r a l t r a j e c t o r y , and k the pad-iu-o of curva-ture of the c e n t r a l t r a j e c t o r y at a given p o i n t . In equation (3*5) we may assume that M- and K are known (or prescribed) funct ions and solve the equation f o r <^> . Thus, one may obtain § f o r any prescr ibed path of a p a r t i c l e . But unless the set of paths i s s u i t a b l y chosen,the p o t e n t i a l § w i l l be d i f -ferent f o r each i n d i v i d u a l path. The c o n d i t i o n that the set of paths must s a t i s f y so that a unique p o t e n t i a l f u n c t i o n may be obtained i s as f o l l o w s : Let M = c ( i - + a i - t ~ - t qxt-2+ -. a n !.:V1 ) (3 .6) where t i s arc length , and C a var iable constant - - by varying c one may obtain a ser ies of paths. If we subst i tute (3 .6) in to (3 .5)* we f i n d that the r e s u l t i n g equation does not contain the var iable constant C . Hence, a set of paths prescr ibed by varying C gives r i s e to a unique p o t e n t i a l f u n c t i o n >^ Since i n (3 .6) ex, , a , _ , 4^  -- o<r\ are a r b i t r a r y , the set of paths represented by (3 .6) i s quite general ; however, the f u n c t i o n (3 .6) must be continuous and must possess continuous d e r i v a t i v e s up to the second order with respect to * A simple case of considerable importance i s where the x-axis i s the c e n t r a l path. Then, the equation (3.1) and (3 .5) reduce to §y = o (3.7) 2 ^ " + Sly'-* o (3 .8) 80 I t may be noted that any p o t e n t i a l d i s t r i b u t i o n which i s sym-metric about the x -axis s a t i s f i e s (3.7). To obtain the p o t e n t i a l at any point fCx,v) t we solve the Cauchy problem, which i n t h i s case i s p a r t i c u l a r l y simple: Let cpcx) be the a x i a l p o t e n t i a l , then the p o t e n t i a l at any given point P i s given by (see Morse and Feshbach, p . 689) The Cauchy problem i s known to be s e n s i t i v e to the small f l u c t u a t i o n s i n <^>o and may give r i s e to large e r r o r s ; but since we are b a s i c a l l y in teres ted i n the p a r a x i a l region where the e r r o r s , i f any, w i l l not be l a r g e ; t h i s i s not a very serious drawback. 3.2 E f f i c i e n t ion source: The s e n s i t i v i t y of a mass spectrometer i s l a r g e l y l i m i t e d by the i n e f f i c i e n t ion source. Only one molecule out of a m i l -l i o n molecules reaches the analyser tube as an Ion. In the pre-vious chapter we have discussed the ion o p t i c a l proper t ies of the conventional source. There, we have seen that unless the i o n i z a t i o n i s confined to a very small region, we cannot improve the e f f i c i e n c y by merely adjust ing the various geometrical and f i e l d parameters of the source. We f e e l that the ent i re design of the source may have to be discarded i n favour of a more com-plex system. The question we now have to face i s how are we going to devise a new system that would have high e f f i c i e n c y . 8l One way Is to t r y a number of them u n t i l we h i t upon a r i g h t one. This i s obviously a tedious and uncer ta in process . The theory of the inverse problem, which we have developed i n the l a s t sec t ion , comes to our a i d i n obtaining a system that has the des i red p r o p e r t i e s ; thus, the long and laborious t r i a l and error method can be avoided. Imagine a set of paths slowly converging into a t h i n p a r a l l e l beam as shown i n f i g . (3.1) . I n i t i a l l y the paths are wide apart and therefore cover a large p o r t i o n of the i o n i -z a t i o n space. The paths i n f i g . (3.1a) are exponential ly con-verging while those i n f i g . (3.1b) are damped o s c i l l a t o r y . The c e n t r a l path Is however, the x - a x i s . We now apply the theory developed i n the previous sec t ion to f i n d the p o t e n t i a l d i s -t r i b u t i o n such that the p a r t i c l e s are guided along these paths. A c l a s s of converging paths may be represented by ^ =. cx e/x|o Q- b oc where ^ i s a var iable constant, and b and v\ are f i x e d constants . S u b s t i t u t i n g i n (3.8), we obtain (3.10) The equation reduces to a p a r t i c u l a r l y simple form f o r b = i and •n = i (3.11) •* We are consider ing two d i f f e r e n t systems having the same pro-p e r t i e s . I t w i l l be c l e a r l a t e r why we consider two systems. F i g . 3.2 8 3 FIG. 3-3 AXIAL POTENTIAL FOR CONVERGING PATHS Y= a exp ( -bx n ) 84 whose s o l u t i o n Is given by i~ (see Kamke, p . 412) A s e r i e s s o l u t i o n of (3.10) can be obtained f o r b=. I and r\ ~ T. . Then the equation reduces to (3.12) Consider the ser ies ^ 6 •=- a 0 -> c \ , x + x 1 ^ S u b s t i t u t i n g i n (3.12) and l e t t i n g the c o e f f i c i e n t s of *- and i t s higher powers go to zero, we obtain the recurrence r e l a t i o n which enables us to f i n d the constants c v ^ > a?> , ^ having been given fl„ and A , . The s e r i e s i s convergent (though the number of terms required to a t t a i n an accuracy w i t h i n 1. 0.001$ increases r a p i d l y f o r x ^ 2). For large i n t e r v a l s , that i s x)> 2, and f o r other values of b and 0 i t i s convenient (and e f f i c i e n t from the point of machine time) to solve (3.10) numer ica l ly . These so lut ions are shown i n f i g . (3.3). From f i g . (3-3) i t i s obvious that the so lut ions are os-c i l l a t o r y and unstable . The p o t e n t i a l along the path i s o s c i l -l a t o r y and hence a l t e r n a t i v e l y p o s i t i v e and negat ive . Let us imagine a p a r t i c l e s t a r t i n g at A i n f i g . (3.2) with a u n i t 85 i n i t i a l energy and moving along the a x i s . The p a r t i c l e i s ac-celerated u n t i l i t reaches the point [ 6 and thereaf ter d e c e l -erated u n t i l i t reaches the point C where i t s energy goes to zero . I t cannot proceed any f u r t h e r unless the sign of the charge on the p a r t i c l e i s changed. Since t h i s Is not p h y s i c a l l y p o s s i b l e , i t goes back to G and then to A . The point C may be c a l l e d a mirror p o i n t . There are as many mirror points as the number of zeros of equation (3.10). Therefore , the con-verging paths are not n e c e s s a r i l y a c c e l e r a t i n g path's nor complete paths, i . e . , f ree of mirror p o i n t s . Such a behaviour i s quite consistent with the proper t ies of the equation .of motion (Chapter I I , eq . (2.16). The equation (3.8) i s almost symmetric f o r and ^ . I t may be wri t ten as or (3.12) They would have been exact ly symmetrical but f o r the f i g u r e 2 i n the equation. By exact symmetry we mean, i f ^ ~ $ c *-) i n (3.12) and f^O i n (3.13), we would have obtained exact ly the same s o l u t i o n s . In any case the presence of the f i g u r e 2 does not completely destroy the symmetry f o r If i n eq. (3.9) 86 the t r a j e c t o r i e s w i l l be unstable o s c i l l a t o r y f u n c t i o n s ; but such t r a j e c t o r i e s are not of i n t e r e s t . On the other hand, i f § o - eoLf £ b ocT> ) we should obtain o s c i l l a t o r y t ra jec tor ies . Let us consider a simple f u n c t i o n $ •=. -ex f C b ) (3.14) The equation (3.8) reduces to whose s o l u t i o n i s given by (3.15) where A and 6 are a r b i t r a r y constants . Since § i s a mono-t o n i c a l l y i n c r e a s i n g f u n c t i o n , the r e s u l t i n g paths are both ac-c e l e r a t i n g and complete, that i s , without any mirror p o i n t s . Though the f i r s t system i s not of much p r a c t i c a l importance (because of the presence of the mirror points)' as compared to the second system with the damped o s c i l l a t o r y paths; i t i s i n t e r e s t -ing to consider both f o r comparison. Furthermore, we s h a l l show i n the next sec t ion that the f i r s t system i s e s s e n t i a l l y unstable . The Cauchy boundary value problem was solved as shown i n Section ( 3 .1 ) . The p o t e n t i a l d i s t r i b u t i o n i n the x ,y -plane i s shown i n f i g u r e s (3.4) and ( 3 -5 ) . A t y p i c a l i o n t r a j e c t o r y i s shown by the dashed curve i n each of these f i g u r e s . 0 0 FIG. >3-4 SYSTEM I POTENTIAL DISTRIBUTION FOR CONVERGING PATHS FIG. 3-5 SYSTEM II POTENTIAL DISTRIBUTION FOR DAMPED OSCILLATORY PATHS 89 3.3 S t a b i l i t y a n a l y s i s : The basic assumptions i n the theory of the inverse problem are that a p a r t i c l e s t a r t s with the same i n i t i a l energy and i n a d i r e c t i o n p r o p o r t i o n a l to the height of the s t a r t i n g p o i n t . These assumptions are, however, r a r e l y s a t i s f i e d i n a ' r e a l ' source, f o r the thermal energy spread i s bound to ex is t and also the p r o b a b i l i t y of emission i n a l l d i r e c t i o n s i s equal . The present sec t ion i s , therefore , devoted to the study of the e f f e c t s of more r e a l i s t i c i n i t i a l c o n d i t i o n s , condi t ions not s a t i s f y i n g the requirements of the theory. One consequence i s that some of the p a r t i c l e s may deviate considerably from t h e i r projected paths, l e a d i n g to an unstable s i t u a t i o n . The systems leading to unstable paths are i m p r a c t i c a l , and therefore , every proposed system must be c a r e f u l l y examined. The two cases we have studied here present an example of a stable and an un-stable system. The gross e f f e c t of the thermal energy spread has been considered by several authors: Pierce (1954), C u t l e r and Hines (1955), and K i r s t e i n e n (1963). Assuming the Maxwel-l i a n v e l o c i t y d i s t r i b u t i o n and equal p r o b a b i l i t y of emission i n a l l d i r e c t i o n s at the cathode, these authors attempted to f i n d the l o n g i t u d i n a l current densi ty d i s t r i b u t i o n at any plane ^ = const . But, such a method i s not very u s e f u l to study the s t a b i l i t y of a system because the s t a b i l i t y information could be obtained e a s i l y by consider ing just one p a r t i c l e instead of a whole ensemble of p a r t i c l e s as i n t h e i r method. We s h a l l , therefore , adopt a method due to Poincare (1905) and Moulton (1926) with c e r t a i n modif ica t ions to s u i t our needs. 90 The p a r a x i a l t r a j e c t o r y equation of an ion whose i n i t i a l energy d i f f e r s from the mean by £ i s given by (3.16) where ^ i s the s o l u t i o n of (3.16') where ^ 0 i s the path of a normal p a r t i c l e . We can express the s o l u t i o n of (3.16) i n terms of (3.17) where > ^ ^ ^ 7 ^ are the per turbat ion f u n c t i o n s . S u b s t i t u t i n g (3.17) in to (3.16) and l e t t i n g the c o e f f i c i e n t s of e and i t s higher powers tend to zero, we obtain a system of d i f f e r e n t i a l equations: 2$ (3.18) 91 A l l per turbat ion funct ions may be required to s a t i s f y the i n i t i a l condit ions a m rUl -O oJ: z - - o w h i l e the p r i n c i p a l s o l u t i o n s a t i s f i e s the given i n i t i a l c o n d i t i o n s . Let u„ and V0 be the fundamental so lut ions of (3.16 ' ) s a t i s f y i n g the i n i t i a l condi t ions u 0 s. | U 0 - z> V 0 - o < * | = a U 0 -+ b 1>c so that we may write 1 where cv and k are the i n i t i a l c o n d i t i o n s . The f i r s t per-turbat ion f u n c t i o n may be wri t ten as H - VC f * X y * ° d x + U 0 ( t i l * t: ( 3 . 1 9 ) where A ( U 0 , V 0 J =2^.(UoVe - Up u c) (u)^an,oKca^) To obtain t h i s r e f e r to Morse and Feshbach, p . 530; i n 5.2.19 l e t c, - c t - o , because of our i n i t i a l c o n d i t i o n s . 92 S u b s t i t u t i n g f o r ^ 0 i n (3.19) we obtain or •r U A . - a f - 2 Up Up (3.20) X 4- U 0 * -r U, ] (3.21) Compare each of the brackets on the r ight -hand side of (3.21) with (3.19). Each of them represents a per turbat ion func-t i o n corresponding to one of the fundamental s o l u t i o n s , U 0 and 'V'o ; hence we may write ^ , =• <^  U , -+• b 1>j where ^ , and i>, are the f i r s t order fundamental per turbat ion funct ions corresponding to the fundamental so lut ions and V~e r e s p e c t i v e l y . Fol lowing a s i m i l a r procedure, we can show that ^ = a. + b V-n f o r a l l values of Y\ . The fundamental per turbat ion funct ions may be obtained by s o l v i n g the equation (3.16') and (3.18) nu-m e r i c a l l y , t r e a t i n g them as a system of ( t i t I ) equations. T h i s 93 approach to f i n d i n g the per turbat ion funct ions overcomes the necessi ty of evaluat ing them f o r each set of i n i t i a l c o n d i t i o n s ; i n t h i s respect t h i s method i s superior to that of Poincare and Moulton. For the sake of s i m p l i c i t y l e t us spec i fy the normal i n i -t i a l condit ions as and the i n i t i a l energy as u n i t y ( C o) i J . The normal t r a j e c t o r y of a p a r t i c l e with more r e a l i s t i c i n i t i a l condit ions i s then given by n „ • ^ r ^ u . o + b f " , + a Z_ e h l i * + b ^ l 6 t W n (3-22) The fundamental so lut ions w.o } v e and the f i r s t three fundamental per turbat ion funct ions f o r both the systems are i l l u s t r a t e d i n f i g u r e s (3.6) to (3.13). These may be used to compute the t o t a l per turbat ion f o r any given set of more r e a l i s t i c i n i t i a l c o n d i -t i o n s . The fundamental per turbat ion funct ions f o r the system I increase monotonically i n the region where the p a r t i c l e s are being accelerated, and outside t h i s region they tend to o s c i l l a t e (not shown i n the f i g u r e s ) . Hence i t i s expected that the more r e a l i s t i c i n i t i a l condi t ions w i l l be a m p l i f i e d to such a stage where the p a r t i c l e s deviate so much from t h e i r projected paths that i t leads to what we have c a l l e d I n s t a b i l i t y . Therefore , 1.0 -0.5 Syst F i r s t fund ( uo em I amental so ) l u t i o n Co ruputed ( *Expecte d 0.5 1 .0 X 1.5 F i g . 3.6 9 6 9 8 P i g 3.11 101 102 the system I i s b a s i c a l l y unstable . The system II , on the other hand, i s character ized by damped o s c i l l a t i n g per turbat ion func-t i o n s ( f i g . 3.11 to 3.13) so that the more r e a l i s t i c i n i t i a l condi t ions are damped out, l eading to a stable system. The system I ,besides being u n s t a b l e , i s not u s e f u l as an ion source without an a d d i t i o n a l a c c e l e r a t i n g system as i t i t s e l f does not accelerate the Ions. T h i s , however, i s not necessary i n the system I I . I t may be noted that the f i e l d l n the i o n i z a t i o n space i s very weak; therefore we expect a small energy spread (apart from the thermal energy spread) i n the r e s u l t i n g i o n beam. 3.4 E lectrode c o n f i g u r a t i o n : There are i n p r a c t i c e at l eas t two ways to r e a l i z e the prescr ibed p o t e n t i a l d i s t r i b u t i o n : (a) set up a system of e lectrodes having the shape of the e q u i p o t e n t i a l l i n e s (planes In three dimensions), (b) set up the p o t e n t i a l d i s t r i b u t i o n on.the faces of a rectangular box as done by Orr (1963)• The dimensions of the system could be quite a r b i t r a r y . However, to s a t i s f y the p a r a x i a l approximations involved i n the theory (see page 9), we must r e s t r i c t the ac tual zone of i o n i z a t i o n to a s t r i p of height l e s s than u n i t y on any chosen s c a l e . Note that t h i s does not put any r e s t r i c t i o n s on the ac tual dimensions so long as we choose the proper sca le , but the p o t e n t i a l s have to be proper ly sca led . A l l p o t e n t i a l s given here are with Probable region of ion withdrawal i n the new source Region of ion withdrawal f o r c o r r e s p o n d i n g ^ mult iple s l i t source o o H O 00 0.01 cm V = 0 P i g . 3.14 The r e l a t i v e merit of the new source and the m u l t i p l e s l i t source. Compare the regions of ion withdrawal. 104 respect to the i n i t i a l energy, which was taken as unity f o r si m p l i c i t y . I f the source i s to be operated say at 100° when the average energy i s 0.01 vo l t , the potentials should 2 be scaled down by a fact o r of 10 . 105 Chapter IV Summary and contr ibut ions of the present wri ter 4.1 Summary: The s e n s i t i v i t y of a mass spectrometer i s a f u n c t i o n of the s e n s i t i v i t y of the current measuring system and of the ion source. The l a t t e r can be expressed as the product of the e f -j f i c i e n c y of i o n i z a t i o n and the e f f i c i e n c y of ion beam transmis-s i o n . The pre l iminary examination of the various sources of l i m i t a t i o n s has indica ted that the low e f f i c i e n c y of ion beam transmission may be a f a c t o r l i m i t i n g the s e n s i t i v i t y of the ion source. A d e t a i l e d mathematical a n a l y s i s of the i o n o p t i c a l pro-p e r t i e s of the source was undertaken with the aim of evaluat ing the e f f i c i e n c y of ion beam transmission. A mathematical model representing an average source—a stack of pla tes c a r r y i n g p a r a l -l e l and coplaner s l i t s—was i n v e s t i g a t e d . F i r s t we developed the equation of t r a j e c t o r y i n the general ized two dimensional e l e c t r i c and normal magnetic f i e l d . Series of approximations— p a r a x i a l approximations—were introduced to l i n e a r i z e the equa-t i o n of t r a j e c t o r y . Further , the i n i t i a l energy of the ions was assumed to be zero f o r the sake of s i m p l i c i t y . The problem was approached i n two d i f f e r e n t ways: (a) p a r t - b y - p a r t approach (b) g l o b a l approach 106 The p a r t - b y - p a r t approach, though not very p r e c i s e , provides considerable i n s i g h t in to the mechanism of i o n beam transmission. On the other hand, the g lobal approach i s f a r more precise but the p r e c i s i o n thus gained would be meaningless consider ing the errors introduced by the p a r a x i a l approximations and the as-sumption of zero i n i t i a l energy. The model was broken down Into several parts and the e f f e c t of each part on the ion t r a -jec tory was s tudied . The combination of r e p e l l e r pla te and the entrance s l i t plate plays an important, ro le i n the mechanism of ion beam t rans-m i s s i o n . A l l the ions formed w i t h i n the l i m i t s of these two pla tes pass through the entrance s l i t and form a divergent beam. If the r e p e l l e r pla te i s at a distance greater than ten times the entrance s l i t width ( & ^ 10), i t i s as i f being at i n f i n i t y and may be replaced by a uniform f i e l d , the r e p e l l e r f i e l d . In the absence of the r e p e l l e r f i e l d a divergent beam i s formed with i t s apex at a short distance (about one s l i t width) from the entrance s l i t . The width of the beam grows on and i s several s l i t widths (entrance s l i t width) at the ex i t s l i t ; hence only a small f r a c t i o n of the t o t a l beam can pass through the ex i t s l i t , which i s normally a tenth of the entrance s l i t and the rest of the beam i s , therefore , l o s t completely. We have shown that the ions which pass through the e x i t s l i t are drawn from a small h o r i z o n t a l s t r i p of width equal to the ex i t s l i t ( f i g . 2.26). When the r e p e l l e r f i e l d i s present, the s i t u a t i o n i s d i f -107 f e r e n t . The beam Is s t i l l divergent , but the apex i s not sharp but d i f f u s e d along the a x i s . By adjust ing the r e p e l l e r f i e l d , we can focus the ions from a c e r t a i n part of the i o n i z a t i o n space i n the neighbourhood of the ex i t s l i t . Such a s t r i p of ion withdrawal i s , f o r example, shown i n f i g . (2.28). The p o s i -t i o n of the s t r i p may be moved to and f r o by varying the r e p e l -l e r f i e l d according to the r e l a t i o n s h i p shown i n f i g . (2.27). The e f f e c t of the e x i t s l i t , p a r t i c u l a r l y on the angle of divergence of an ion t r a j e c t o r y as i t passes through the ex i t s l i t i s very small—about 10$ of the normal angle of d i v e r -gence i n the absence of the ex i t s l i t . The d e f l e c t i o n of the ion beam produced by the d e f l e c t i n g plates i s l i n e a r l y re la ted to the d e f l e c t i n g p o t e n t i a l ( i . e . , the p o t e n t i a l d i f f e r e n c e between the p l a t e s ) . I t i s found that , f o r small d e f l e c t i o n s , the shape of the beam remains unchanged a f t e r d e f l e c t i o n . The presence of the magnetic f i e l d i n the ion source region, i f the f i e l d i s weak (a few tens of gauss), i s merely to s l i g h t l y perturb the shape of the ion t r a j e c t o r y . In the weak f i e l d no percept ible mass d i s c r i m i n a t i o n i s observed. To increase the e f f i c i e n c y of the i o n beam transmission of the present source, we w i l l have to confine the region of i o n i z a t i o n to the region of ion withdrawal (2-3 mm wide s t r i p ) , which i n p r a c t i c e , i s very d i f f i c u l t to a t t a i n . However, we can maximize the e f f i c i e n c y by l e t t i n g the region of ion withdrawal f a l l exact ly over the region of highest ion d e n s i t y . 108 A system producing a divergent beam, as i s the present m u l t i p l e s l i t source, i s b a s i c a l l y i n e f f i c i e n t . On the other hand, a system capable of producing a convergent beam w i l l be more e f f i c i e n t . Instead of pursuing a t r i a l and error method u n t i l we h i t upon the desi red system, we posed to ourselves an inverse problem, that i s , given any path , to f i n d the poten-t i a l d i s t r i b u t i o n so as to guide a p a r t i c l e along the prescr ibed path. In t h i s connection we have proved the f o l l o w i n g two basic theorems: (a) There e x i s t s a p o t e n t i a l d i s t r i b u t i o n to guide a p a r t i c l e along any desi red path . (b) A group of p a r t i c l e s may be guided along a set of p a r a x i a l paths. A case of considerable importance i s where the c e n t r a l t r a j e c t o r y i s the x - a x i s . We have considered two types of con-vergent paths as shown i n f i g . (3.1 ) — exponent ia l ly converg-ing and damped o s c i l l a t o r y paths. Although, In p r i n c i p l e , we may choose the paths of any shape (but always paraxia l ) quite a r b i t r a r i l y , there are c e r t a i n l i m i t a t i o n s : (a) incomplete paths — a p a r t i c l e may be turned back at a c e r t a i n p o i n t . (b) unstable system — abnormal p a r t i c l e s , i . e . those not s a t i s f y i n g c e r t a i n i n i t i a l condi t ions of uniform energy and d i r e c t i o n deviate considerably from t h e i r projected paths. The two systems we have considered here i l l u s t r a t e the above 109 l i m i t a t i o n s . The paths shown i n f i g . (3.1a) are incom-plete and the system i s also unstable ; while the damped o s c i l l a t o r y paths shown i n f i g . (3.1b) are complete and the corresponding system i s s table , therefore p r a c t i c a l . 4.2 Contr ibut ions of the present w r i t e r : Fol lowing are the c o n t r i b u t i o n s of the present w r i t e r : (a) Complete a n a l y s i s of the ion o p t i c a l proper-t i e s of the mul t iple s l i t source. That the e f f i c i e n c y of the source i s low on account of the small region of ion withdrawal was shown f o r the f i r s t t ime. (b) Theory of the inverse problem and i t s a p p l i c a -t i o n to e f f i c i e n t ion source. T h i s i s c o n s i -dered to be the major c o n t r i b u t i o n . (c) Mathematical t o o l s such as the s o l u t i o n of c e r t a i n algebraic equations and the i n v e r s i o n of transformation funct ions expressing them as d i f f e r e n t i a l equations, and the development of a method to take in to account the e f f e c t of the thermal energy and random i n i t i a l d i r e c t i o n may also be considered as c o n t r i b u t i o n s of the present w r i t e r . 110 Appendix A 1. An example of ion source e f f i c i e n c y c a l c u l a t i o n : The f o l l o w i n g are the observations made on the mass spectrometer at the Geophysics Laboratory of the U n i v e r s i t y of B r i t i s h Columbia ( A . B . L . Whi t t les , personal communication): -11 / 8 T o t a l ion current . . . . 2.4 x 10 amp. (1.5 x 10 ion/sec) ft l 4 Mass flow 0.3 x 1 0 " ° gm/hr (2.4 x 10 moVsec) E f f i c i e n c y of the ion source 0.6 x 10 2. T h e o r e t i c a l l y expected i o n currents : i + = L * <r x p x i ' where V p o s i t i v e ion current L path length of e lec t rons }p gas pressure i " e l e c t r o n beam i n t e n s i t y >f Townsend's c o e f f i c i e n t of i o n i z a t i o n Reasonable estimates of these parameters are L- 2 cm. s = 10 electron/cm/mm Hg -4 10 mm Hg = 500 x 10" 6 amp 10"^ amp. I l l 3 . E f f e c t of the source magnetic f i e l d on the e l e c t r o n path lengths : Let and V Y be the v e l o c i t y components i n the z -and r - d i r e c t i o n s r e s p e c t i v e l y . 6 , • e • \/Y - v/". w where ft* i s the magnetic f i e l d i n the source region and Y*» the radius of the s p i r a l o r b i t . where e^L.' i s the e l e c t r o n path length i n time i n t e r v a l L' = /\/RlpaL where L i s the t o t a l path length . Since VY <^  v* f o r a wel l co l l imated beam, L/(_ - 1 and also the t o t a l path length Is Independent of the magnetic f i e l d . 112 Appendix B 1. Numerical er ror es t imat ion: The commonly used numerical method of s o l v i n g a set of ordinary d i f f e r e n t i a l equations Is a f t e r Runge (1895) and Kutte (1901). For t h i s method the bounds on the t o t a l e r ror committed at the end of each step, though d i f f i c u l t to evaluate an exact expression f o r , may be estimated from the data obtained at several consecutive steps (Scraton, 1964) or obtained by con-s i d e r i n g d i f f e r e n t step s i z e s . If at three consecutive steps, &y ( , and AY^ are computed with constant step s i z e , an estimate of e r r o r i s given by Scraton (1964) E r r o r = ( 1 0 & Y , + '9 Ay^ + AYa-3 h y[-h - 1 K 7^)^30 On the other hand, i f £> y, and ^Y, are the computed values a c e r t a i n point with step s izes W and zh r e s p e c t i v e l y , the approximate t o t a l e r ror Is given by Hildebrand.. (1956) E r r o r = | *Y,- A ~ \ | / 3 0 A few numerical experiments were c a r r i e d out to estimate the t runcat ion and round-off e r r o r s . The round-off er ror was estimated by c a r r y i n g out the computation with d i f f e r e n t number of s i g n i f i c a n t d i g i t s : 4, 8, 12 and 16 on a var iable word length machine, IBM 1620. The table below shows the r e s u l t s of the com-p u t a t i o n . 113 S i g n i f i c a n t d i g i t s Rela t ive round-off e r ror 4 8 12 16 The bulk of computation was, however, c a r r i e d out on the IBM 7040 with s ingle p r e c i s i o n , that i s , 9 d i g i t s . The round-off e r r o r , therefore , i s presumed to be l e s s than one percent . To obtain the above accuracy, l t i s not necessary to have the step size small everywhere but only close to the s l i t where the t r a j e c t o r y undergoes rapid changes, and away from i t we can take l a r g e r steps. The Runge-Kutte process allows the v a r i a t i o n of step size at any step; therefore , we have l i n e a r l y increased the step s i z e — the smallest step i s equal to 0.05 at the s l i t and increases to 2 at 40 u n i t s from the s l i t . 2. Mathematical complications introduced when the magnetic f i e l d i s taken into c o n s i d e r a t i o n : For numerical i n t e g r a t i o n of equation (2.15), we write i n the form 4$ 1$ 10" 5 $ 0 (reference zero) 114 where C = • U) • / 2<t-V m K Since at v = O and c^^Y -t <$'Y ' - o the above equation reduces to '/ c Y —> J- ^ r - —> c*> 2 - ^ i n other words, the equation has a s i n g u l a r i t y at X =X0 . Assuming, however, ±. a X a s X —» o , i t i s easy to see from the Cauchy 1 s convergence tes t that the i n t e g r a l i s convergent, but the numerical process cannot be i n i t i a t e d at that p o i n t . T h i s d i f f i c u l t y can be overcome by adopting a p h y s i c a l view—the force due to the magnetic f i e l d tends to zero as v e l o c i t y tends to zero ; therefore , we may drop of f completely the term containing the magnetic f i e l d when i n i t i a t i n g the numeri-c a l process . 115 Bibl iography A l l a r d , J . L . and R u s s e l l , R. D. (1963) The e l e c t r i c poten-t i a l i n the region of a t h i n s l i t . B r i t . J . A p p l . Phys . , V o l . 14, 8 0 0 . Boerboom, A. J . H. (1957) De Ionenoptjfk Van de Massaspec-trometer, Unpublished Ph.D. t h e s i s , R i j k s u n i v e r s i -t e i t te L e i d e n . Boerboom, A. J , H . (1959) Numerical c a l c u l a t i o n of the p o t e n t i a l d i s t r i b u t i o n i n i o n s l i t lens system I . Z. f u r Naturforschung, V o l . 14, 809. Boerboom, A. J . H. ( i960) Numerical c a l c u l a t i o n of the p o t e n t i a l d i s t r i b u t i o n i n ion s l i t lens system I I . Z . f u r Naturforschung, V o l . 15, 244. Boerboom, A. J . H. ( i 9 6 0 ) Numerical c a l c u l a t i o n of the p o t e n t i a l d i s t r i b u t i o n i n ion s l i t lens system I I I . Z. f u r Naturforschung, V o l . 15, 253. Boerboom, A. J . H. ( i 9 6 0 ) Modif ied ion s l i t lens f o r v i r t u a l v a r i a t i o n of s l i t width. Z . f u r Naturforschung, V o l . 15, 350. Courant, R. and H i l b e r t , D. (1953) Methods of mathematical p h y s i c s . V o l . I I , Interscience Publ ishers , Inc. C u t l e r , C . C . and Hines, M. E . (1955) Thermal v e l o c i t y e f -f e c t s i n e l e c t r o n guns. Ins t , of Radio Engineers P r o c , V o l . 43, 307. D i e t z , L . F . (1959) Ion o p t i c s f o r V-type surface i o n i z a -t i o n f i lament used i n mass spectrometer. Rev. Scien-t i f i c Instruments, V o l . 30, 235. E i n s t e i n , P. A. and Jacob, L . (1948) On some f o c a l proper t ies of an e l e c t r o n o p t i c a l immersion o b j e c t i v e . P h i l . Magazine, 39, 20. H e l l w i g , G. (1964) P a r t i a l d i f f e r e n t i a l equations, An i n t r o -d u c t i o n . B l a l s d e l l P u b l i s h i n g C o . , Chap. I I I . Hildebrand, F . B . (1956) Introduct ion to Numerical A n a l y s i s . McGraw-Hill Book Co. Jacob, L . (1950) Some cross-over proper t ies i n the e lec t ron Immersion o b j e c t i v e s . J . A p p l . Phys . , V o l . 21, 966. 116 Jacob, L . (1951) An i n t r o d u c t i o n to e l e c t r o n o p t i c s . Mathuen P u b l i c a t i o n s . Kamke, E. (1959) D I f f e r e n t i a l g l e i c h u n g e n Lo sung sine thoden und Losungen. Band I, Chelsea P u b l i s h i n g Company. K i r s t e i n e n , P. T . (1963) Thermal v e l o c i t y e f f e c t s i n a x l -a l l y symmetric s o l i d beams. J . A p p l . Phys . , V o l . 34, 3479. Mayne, K. I . ( i960) Stable isotope geochemistry and mass spectrometric a n a l y s i s , i n Methods i n Geochemistry. Edi ted by Smales, A. A. and Wager, L . R . , In ter -science Publishers Inc. Morse, P. M. and Feshbach, H. (1953) Methods of t h e o r e t i c a l p h y s i c s . V o l . I, M c G r a w - H i l l Book Co. Moulton, F . R. (1926) Methods i n e x t e r i o r b a l l i s t i c s . Dover P u b l i c a t i o n s I n c . , Chap.IV. Naidu, P. S. (1962) Studies on the t e l l u r i c f i e l d response of some g e o l o g i c a l inhomogeneities. Unpublished M. Tech. t h e s i s , Indian I n s t i t u t e of Technology, Khargpur, C a l c u t t a . Orr, D. (1963) Aberrat ions i n e l e c t r o s t a t i c quadrupoles. Nuclear Instruments and Methods, V o l . 24, 377» Pierce , J . R. (1954) Theory and design of e l e c t r o n beams. D. van Nostrand Company I n c . , Chapter I I . Poincare', H. (1905) Les methodes nouvelle de l a mecanique c e l e s t i a I, Chap. I I , G a u t h e i r - V i l l a r s , P a r i s . Scraton, R. E . (1964) Es t imat ion of t runcat ion er ror i n Runge-Kutte method and a l l i e d processes. The Com-puter Journal , V o l . 7, 246. Shibata , N. ( i960) On a s o l u t i o n of the p a r a x i a l ray equa-t i o n i n an a x i a l l y symmetrical e l e c t r o s t a t i c f i e l d . J . Phys. soc . Japan, 15, 200. Stacey, J . S. (1962) Method of r a t i o recording f o r lead isotopes i n mass spectrometer. Unpublished Ph.D. t h e s i s , U n i v e r s i t y of B r i t i s h Columbia. Vauthier , R. (1955) A p p l i c a t i o n s de l ' o p t l q u e des charge e l e c t r i q u e s a l a spectrometrie-de masse. Annales de Physique, V o l . 10, 968. Waters, W. E . (1958) R i p p l i n g of t h i n e l e c t r o n r ibbons . J . A p p l . Phys . , V o l . 29, 100. 

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