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 It i s not p r a c t i c a l to confine the i o n i z a t i o n region .to the small volume from which ions are withdrawn.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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Ion optics of the mass spectrometer ion source Naidu, Prabhakar Satyanarayan 1965
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Title | Ion optics of the mass spectrometer ion source |
Creator |
Naidu, Prabhakar Satyanarayan |
Publisher | University of British Columbia |
Date Issued | 1965 |
Description | The ion beam transmission efficiency of the ion source is an important factor in determining the sensitivity of a mass spectrometer. Vauthier (1955) has shown for a simple source that the transmission efficiency is very low. The present thesis examines the transmission efficiency of a more complex source. The first part of the thesis deals with the ion optical properties of a multiple slit ion source. The region of ion withdrawal has been sketched by computing the ion trajectories passing through the exit slit. It was found that for the more complex source the region of ion withdrawal is also much smaller than the total ionization space. It is not practical to confine the ionization region to the small volume from which ions are withdrawn. The effect of a source magnetic field has been taken into account. The perturbation of the trajectory due to the field is small, and therefore the mass discrimination due to the source magnetic field is imperceptible for heavy ions unless the field is of the order of a few webers/m². The multiple slit ion source produces a divergent ion beam, only a small fraction of which penetrates the exit slit. Obviously a system producing a beam converging at the exit slit to a narrow parallel ribbon will be most efficient. In order to devise such a system a theory of the inverse problem of particle motion is developed in the second part of the thesis. A procedure was found to determine a potential distribution required to guide a group of particles along a set of prescribed paths. There are two important limitations to the choice of paths: (a) there are certain paths which are not complete; that is a particle following such a path is turned back at certain points which we call mirror points. (b) The particles which do not satisfy the initial conditions of uniform energy and direction may deviate considerably from their projected paths leading to what we have called an unstable situation. 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 oscillatory paths, the system of damped oscillatory paths is stable. |
Subject |
Spectrometer Mass spectrometry Ions |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-09-21 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0053357 |
URI | http://hdl.handle.net/2429/37530 |
Degree |
Doctor of Philosophy - PhD |
Program |
Geophysics |
Affiliation |
Science, Faculty of Earth, Ocean and Atmospheric Sciences, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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