Science, Faculty of
Physics and Astronomy, Department of
DSpace
UBCV
Enga, Eric
2011-05-25T23:59:16Z
1969
Doctor of Philosophy - PhD
University of British Columbia
Two experiments are described. One is the successful observation of the resonant deflection of a beam of neutral potassium atoms at a frequency of 7.2 Mhz, in agreement with the predictions of the theory of the Transverse Stern-Gerlach (TGS) experiment. The other is a proposal for a charged particle Stern-Gerlach experiment, which is based on an extension of the TSG experiment to time independent, inhomogeneous magnetic fields having the form
[formula omitted]
If the field B₁(r) is well chosen, the charged particle trajectories are confined in a stable beam by the resulting Lorentz forces for motion generally along the z axis. This is, in fact, the principle of strong focusing which is now widely used in accelerator design. But in such a system it is also possible to satisfy the criterion for a TSG experiment, since in a frame of reference moving with a particle in the z direction, the field B₁(r) is rotating in time.
https://circle.library.ubc.ca/rest/handle/2429/34896?expand=metadata
OBSERVATION OF THE TRANSVERSE STERN-GERLACH EFFECT IN NEUTRAL POTASSIUM AND AN ANALYSIS OF'-A. CHARGED . PARTICLE STERN-GERLACH EXPERIMENT BY ERIC ENGA B.A.Sc., University of B r i t i s h Columbia, 1962 M.A.Sc., University of B r i t i s h Columbia, 1966 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of PHYSICS We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA December, 1969 In presenting t h i s t h e s i s in 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 tha permission 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. Depa rtment The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8, Canada Date ABSTRACT Two experiments are described. One i s the successful observation of the resonant d e f l e c t i o n of a beam of neutral potassium atoms at a frequency of 7.2 Mhz, i n agreement with the predictions of the theory of the Transverse Stern-Gerlach (TGS) experiment. The other i s a proposal f o r a charged p a r t i c l e Stern-Gerlach experiment, which i s based on an extension of the TSG experiment to time independent, inhomogeneous magnetic f i e l d s having the form B(r,z) = B Qk + B ^ r l e 1 ^ I f the f i e l d B^(f) i s well chosen, the charged p a r t i c l e t r a j e c t o r i e s are confined i n a stable beam by the r e s u l t i n g Lorentz forces f o r motion generally along the z axis. This i s , i n f a c t , the p r i n c i p l e of strong focusing which i s now widely used i n accelerator design. But i n such a system i t i s also possible to s a t i s f y the c r i t e r i o n f o r a TSG experiment, since i n a frame of reference moving with a p a r t i c l e i n the z d i r e c t i o n , the f i e l d "B\ (r) i s r o t a t i n g i n time. - i i i -TABLE OF CONTENTS Page ABSTRACT i i LIST OF FIGURES i v ACKNOWLEDGEMENTS v i i CHAPTER 1 Introduction 1 2 C l a s s i c a l Theory of the Neutral P a r t i c l e Trans-verse Stern-Gerlach Experiment 3 Experimental Procedure 14 4 Experimental Results ... 21 5 A Proposal for a Charged P a r t i c l e TSG Experiment... 31 6 The Tr a j e c t o r i e s of an Ion with Zero Magnetic Moment i n a DC H e l i c a l Quadrupole :44 7 The Tr a j e c t o r i e s of an Ion with Non-Zero -Magnetic Moment i n a DC H e l i c a l Quadrupole f o r Weak Lorentz Forces 53 8 The Influence of the Solenoid F i e l d on the T r a j e c t o r i e s , and the Stern-Gerlach E f f e c t f o r Large Lorentz Forces i n a DC H e l i c a l Quadrupole ... 67 9 Experimental Considerations 87 10 Concluding Remarks 97 BIBLIOGRAPHY 99 APPENDIX A The H e l i c a l Quadrupole Magnetic F i e l d 101 B The Derivation of the T r a j e c t o r i e s of an Ion i n a H e l i c a l Quadrupole F i e l d Consisting of Orthogonal E l e c t r i c and Magnetic F i e l d s Superimposed on a Homogeneous F i e l d and on which an Additional Radial Force i s Introduced 105 C An Ion Gun Design Suitable f o r Producing Small Diameter, Well Collimated Ion Beams . 112 - i v -LIST OF FIGURES Figure Page 2-1 Schematic p l o t of a component of the Stern-Gerlach force versus time i n the impulse approximation .... 6 2-2 Representation of the r - f f i e l d i n the ro t a t i n g co-ordinate system (x, y, z). It has been assumed that (f) i s i n the x-y plane and makes an angle - <f> with respect to the x-axis. i s the compon-ent of J along the e f f e c t i v e f i e l d at t = 0 6 2-3 I l l u s t r a t i o n of the o r i e n t a t i o n of B-^(r) = G(x-iy) at a p o s i t i o n r = x+iy 3-3 E l e c t r i c a l c i r c u i t of the quadrupole wires showing d e t a i l s of the p a r a l l e l resonant c i r c u i t used 11 3-1 Schematic diagram of the o v e r a l l system (not to scale) 1 5 3-2 Drawing of the d e f l e c t i o n system with construction d e t a i l s omitted 18 19 4-1 (a) Plot of the percentage increase i n beam in t e n s i t y at the detector as a function of p o s i t i o n i n the x-y plane due to the focus-ing e f f e c t of the d e f l e c t i o n system i n a large, uniform-intensity beam 26 (b) Contour map of the focused "pip", as con-structed from F i g . 4-la. The numbers on the contours are percentage increases i n beam i n t e n s i t y due to the r o t a t i n g quadrupole f i e l d (ordinate of F i g . 4-la). The dotted l i n e s are interpolated contours between widely separated points 27 4-2 Beam i n t e n s i t y p r o f i l e for undeflected narrow beam. 28 4-3 Relative change of the beam i n t e n s i t y pattern for a narrow beam at resonance. The undeflected beam p r o f i l e i s shown i n Figure 4-2 29 4- 4 Plot of the beam i n t e n s i t y near the beam center as a function of f i e l d B f or f i x e d frequency and d i f f e r e n t amplitude o? the r - f f i e l d and i t s gradient 30 5- 1 (a) A h e l i c a l quadrupole wire system 33 - V -Figure Page (b) A quadrupole wire system showing the electrodes necessary to produce an e l e c t r i c f i e l d orthogonal to the magnetic f i e l d ^3 5-2 An i r o n core quadrupole, showing the electrodes necessary to produce an e l e c t r i c f i e l d orthogonal to the magnetic f i e l d 36 5-3 (a) I l l u s t r a t i o n of the frequency vectors i n the r o t a t i n g co-ordinate system (x, y, 1). B^(r) i s i n the x,y plane and makes an angle -<j> with respect to the x axis. Jg Is- the component of J along the e f f e c t i v e f i e l d 40 (b) An i l l u s t r a t i o n of the vector Jg s i n 0 i n the (x, y, z) co-ordinate system at the time t ... 40 7-1 A pl o t of t r a j e c t o r i e s from equation (7-6) . The dotted t r a j e c t o r i e s are those obtained when the adiab'atic condition i s s a t i s f i e d as the p a r t i c l e crosses the axis. The s o l i d t r a j e c t o r i e s are those obtained when the sign of Jg changes as the p a r t i c l e crosses the axis 63 7-2 T r a j e c t o r i e s s i m i l a r to those of F i g . 7-1 63 7-3 T r a j e c t o r i e s s i m i l a r to those of F i g . 7-1 64 7-4 T r a j e c t o r i e s s i m i l a r to those of F i g . 7-1 65 7-5 An i l l u s t r a t i o n of a system of stops which would give 100% p o l a r i z a t i o n of the emerging beam 66 7- 6 A somewhat simpler system to that i l l u s t r a t e d i n F i g . 7-5 which would give p a r t i a l p o l a r i z a t i o n of the inner beam 66 8- 1 A chart of the regions where the roots Rj and R2 are r e a l or complex. The l i g h t regions indicate that both are r e a l , and the shaded regions indicate that one or both are complex 71 9- 1 I l l u s t r a t i o n of the dimensions of a h e l i c a l l i n e and a h e l i c a l s t r i p 88 9-2 I l l u s t r a t i o n of the dimensions of a rectangular wire h e l i c a l winding 88 9-3 Plot of minimum power versus wire radius for a rectangular copper wire h e l i c a l winding at room temperature, from equation (9-11) 91 - v i -Figure Page A - l I l l u s t r a t i o n of the o r i e n t a t i o n of the f i e l d B,(f) = -G(y+ix) at the point r = x+iy 102 A-2 Plot of the factor C(q) defined by equation (a-8).. 104 C-l Diagram of the ion gun described i n appendix C 118 C-2 Typical performance of the ion gun described i n appendix C. V i s the p o t e n t i a l of the i o n i z i n g chamber 119 - v i i -ACKNOWLEDGEMENT It i s a pleasure to acknowledge the very great assistance and enthusiasm of Professor Myer Bloom, who has guided t h i s project to i t s present stage. Special thanks are due him for h i s help during the writing of t h i s d i s s e r t a t i o n . I s i n c e r e l y thank Dr. Hin Lew, who contributed much of the design for the neutral p a r t i c l e experiment, and who was most h e l p f u l with advice on many experimental matters. The experimental work received e s s e n t i a l support from the Physics machine shop, e s p e c i a l l y from Doug Stonebridge. I wish to thank Peter Flainek f o r his technical assistance, and the members of the D i v i s i o n of Pure Physics, National Research Council, f o r t h e i r help with the portion of the experiment which was performed there. Special thanks are due my wife, S y l v i a , for her support during t h i s work. This research was supported by grants from the National Research Council of Canada. CHAPTER 1 Introduction In the conventional Stern-Gerlach experiment a beam of atoms having magnetic moment u i s passed through a time-independent inhomo-geneous magnetic f i e l d B ( f ) . For s i m p l i c i t y , suppose that u = yYiJ, where Jfi i s the angular momentum and y i s the gyromagnetic r a t i o , and that B(r) consists of a large homogeneous part B q oriented along the z axis and an inhomogeneous part B^(?) with | (?) | << | B* | . The atoms acquire momentum while i n the region of the inhomogeneous f i e l d because of the Stern-Gerlach force F = (u . V ) B ( f ) . Since J precesses about the z axis at the Larmor frequency CJ q = -yH > the only time-independ-ent contribution to F arises from J . Therefore, the net change i n momentum for t >> u ^ i s proportional to J . One of the most important and best-known r e s u l t s i n modern physics i s that J_ i s found experimentally to take on only the di s c r e t e values M = - J , - J + 1, ,+ J . We can say that the Stern-Gerlach experiment provides us with a method for preparing a spin system i n any one of the dis c r e t e quantum states M. A few year ago i t was shown t h e o r e t i c a l l y (Bloom and Erdman 1962) .that a more general form of the Stern-Gerlach experiment may be defined, using time-dependent inhomogeneous magnetic f i e l d s , i . e . - 2 -B ( f , t ) = B Q + B ^ r ) cos wt d - l ) T h i s g e n e r a l i z e d form o f t h e S t e r n - G e r l a c h e x p e r i m e n t was named t h e " T r a n s v e r s e S t e r n - G e r l a c h " e x p e r i m e n t (TSG) because i t was p r e d i c t e d t h a t f o r (JJ = C J o , 2 J + 1 d e f l e c t e d beams s h o u l d a l s o be o b s e r v e d , t h e quantum number M' = - J , - J + 1, ... + J a s s o c i a t e d w i t h each o f t h e s e d e f l e c t e d beams i n t h e TSG e x p e r i m e n t b e i n g a s s o c i a t e d w i t h J^, t h e p r o j e c t i o n o f J along;„;the x a x i s o f a c o o r d i n a t e system r o t a t i n g w i t h a n g u l a r v e l o c i t y a j Q about t h e z a x i s . T h i s a r i s e s from t h e f a c t t h a t , i f t h e o s c i l l a t i n g inhomogeneous m a g n e t i c f i e l d a t a g i v e n p o s i t i o n i s s p l i t i n t o two r o t a t i n g f i e l d s , one o f them i s synchronous w i t h t h e p r e c e s s i n g J v e c t o r a t . r e s o n a n c e . Thus, t h e c u m u l a t i v e changes i n momentum a r e p r o p o r t i o n a l t o J . In t h i s t h e s i s we d e s c r i b e two e x p e r i m e n t s . One i s t h e s u c c e s s f u l o b s e r v a t i o n o f t h e r e s o n a n t d e f l e c t i o n o f a beam o f n e u t r a l p o t a s s i u m atoms a t a f r e q u e n c y o f 7.2 Mhz, i n agreement w i t h t h e p r e d i c t i o n s o f th e t h e o r y o f t h e TSG e x p e r i m e n t . The r e s u l t s o f t h i s e x p e r i m e n t have a l r e a d y been p u b l i s h e d (Bloom, Enga, Lew 1967). The o t h e r i s a p r o p o s a l f o r a c h a r g e d p a r t i c l e S t e r n - G e r l a c h e x p e r i m e n t , w h i c h i s based on an e x t e n s i o n o f t h e TSG ex p e r i m e n t t o t i m e - i n d e p e n d e n t , inhomogeneous f i e l d s have t h e form I f t h e f i e l d B ( r ) i s w e l l c h o s e n , t h e ch a r g e d p a r t i c l e t r a j e c t o r i e s z B ( r , z ) = B k + B,(f)e o 1 ^ J iwz (1-2) - 3 -w i l l be confined i n a stable beam by the r e s u l t i n g Lorentz forces f o r motion generally along the z axis. This i s , i n f a c t , the p r i n c i p l e of strong focusing which i s now widely used i n accelerator design. But i n such a system i t i s also possible to s a t i s f y the c r i t e r i o n f o r a TSG experiment, since i n a frame of reference moving with the p a r t i c l e i n the z d i r e c t i o n , the f i e l d B^(r) i s r o t a t i n g i n time. In analogy to the TSG experiment, i t appears now to be possible to observe the 2J + 1 beams i n such an experiment. The c l a s s i c a l theory of the neutral p a r t i c l e TSG experiment i s presented i n Chapter 2. The experimental procedure and the apparatus are described i n Chapter 3, while the experimental measurements and t h e i r i n t e r p r e t a t i o n are given i n Chapter 4. We develop the extension of the TSG experiment to time independent, space varying f i e l d s i n Chapter 5 and propose a Stern-Gerlach experiment for charged p a r t i c l e s which i s developed i n Chapters 6 through 9. In Chapter 6 we consider the beam behavior f o r the Lorentz forces only. In Chapter 7 we introduce the Stern-Gerlach force i n the l i m i t that the Lorentz force i s comparible to, or weaker than, the Stern-Gerlach force, and examine the beam t r a j e c t o r i e s . The case when the Lorentz force i s larger than the Stern-Gerlach force i s treated i n Chapter 8. In Chapter 9 the experimental problems of t h i s experiment are examined and the r e s u l t s of some preliminary experimental work developing a suitable ion beam are presented. CHAPTER 2 C l a s s i c a l Theory of the Neutral P a r t i c l e Transverse Stern-Gerlach Experiment A neutral atom of mass m and magnetic moment IT having i n i t i a l p o s i t i o n r(0) and momentum p(0) i s assumed to in t e r a c t with the magnetic f i e l d B(r,t) given by equation (1-1) . The p o s i t i o n and momentum of the atom at any time t are given by r ( t ) = f(0) + 1 p ( f ) d t ' (2-1) o p(t) = p(0) + f F ( f ) d t ' (2-2) ' o where the instantaneous Stern-Gerlach force F(t) i s given by F(t) = Y^("(t) • V ) B ( f ( t ) ,t) (2-3) while the time dependence of J ( t ) i s governed by the equations dJ/dt = yJ x B ( f ( t ) , t ) (2-4) Solution of equations (1-1), (2-1)-(2-4) gives r ( t ) , p ( t ) , and J ( t ) as a function of r ( 0 ) , p(0), t and J ( 0 ) . In many cases, i f t i s - 5 -made s u f f i c i e n t l y large, there i s a c o r r e l a t i o n between r ( t ) and J ( t ) , so that measurement of the i n t e n s i t y d i s t r i b u t i o n i n the beam enables one to draw conclusions about the allowed values of J . It i s d i f f i c u l t to solve equations (1-1), ( 2 r l ) - ( 2 - 4 ) for a general B ( f , t ) . The ess e n t i a l features of the Stern-Gerlach experiment can be found by using the "impulse approximation", which, i n f a c t , i s a very good approximation for the experimental arrangement used here and described i n the next chapter. Impulse Approximation r(t)=r(0)=r=constant i n the region B 1 ( f ) ^ 0 In t h i s approximation the change i n momentum Ap due to the Stern-Gerlach force i s calculated, assuming that the displacement of the atom i n the region of the inhomogeneous f i e l d i s n e g l i g i b l e . This change i n momentum can be measured by allowing the atoms to undergo free f l i g h t for a time T between the inhomogeneous f i e l d region and the detector, so that the displacement due to the Stern-Gerlach force i s re l a t e d to the momentum change by Ar = (T/m) Ap ( ? - ^ Now, i f one calculates any component F^(t) of the Stern-Gerlach force, using equations (1-1), ( 2 - 3 ) and ( 2 - 4 ) one obtains a plo t such as that shown i n Fi g . 2-1 i n which F^(t) i s made up of a constant term plus other terms which o s c i l l a t e s i n u s o i d a l l y with time. For times that are long compared with the period of o s c i l l a t i o n T q of F ^ ( t ) , Ap^(t) i s proportional to the time-averaged value of F^, which i s the constant term, i . e . - 6 -L. > "time Figure 2-1. Schematic p l o t of a component of the Stern-Gerlach force versus time i n the impulse approximation. Figure 2-2. Representation of the r f f i e l d i n the ro t a t i n g coordinate system (x ,7,z). It has been assumed that B^(r) i s i n the x,y plane and makes an angle -<j> with respect to the x-axis. i s the component of J along the e f f e c t i v e f i e l d at t = 0. - 7 -( A p ( t ) ) t > > t » t<F(t)> (2-6) o The component of J which i s quantized i n each of the 2J + 1 deflected beams i s found by examining the dependence of <F(t)> on J . For the conventional Stern-Gerlach experiment, OJ = 0 and i t i s obvious from equations (1-1), (2-3), and (2-4) that 3.B, (r) ( A p ( t ) ) t > y t , t ^ J z (2-7) so that J z i s quantized. For w ^ 0, th i s i s no longer true, as w i l l be seen below. We f i r s t write down equation (2-4) i n the impulse approximation a f t e r making the transformation +i/iit J = J ± i J = J e (2-8) ± x y ± • Jx> Jy> and J z are the components of J i n a coordinate system rotating with angular v e l o c i t y At about the z axis . dJ +/dt = ;i(AojJ + + l/2w 1 +J z) ±i(w l zCos ojtJ + - l/2w 1| + u ) (2-9) dJ z / d t = - l / 4 i ( J + a ) 1 - j'_a)1 + ) - l / 4 i ( J + c j 1 _ e 2 l u t - J * _ w 1 + e " 2 l W t ) (2-10) where - 8 -AOJ = (JJ-CU , a) = - Y B (2-11) o o ' o ^ 1 co1 = - Y B 1 ( f ) 3 a i 1 ± = u l x ± i c o l y (2-12) It i s well known (Winter 1955) that the term involving a)^z cos cot gives r i s e to resonance e f f e c t s at io = w /2n, while the terms i n u, e+^ l a ) t o 1± give resonances at u o/(2n+l), where n i s an integer. These multiple quantum e f f e c t s are important when the in e q u a l i t y << B q i s not s a t i s -f i e d . As we s h a l l demonstrate i n Chapter 4, a large resonance i s indeed observed i n the TSG experiment near CJ = COq/2 f o r large values of B ^ . Replacement of the Linear O s c i l l a t i n g F i e l d by a Rotating F i e l d We now assume that the value of B ^ i s s u f f i c i e n t l y small that the influence of the terms which are e x p l i c i t l y time dependent i n equations (2-9) and (2-10) i s n e g l i g i b l e , so that these terms may be dropped. This corresponds to assuming that = 0 and to replacing the magnetic f i e l d B (r) cos tot by (1/2)B ( r ) e 1 ( 0 t , i . e . , one of amplitude ( 1 / 2 ) B ( r ) and r o t a t i n g with angular v e l o c i t y co about the z axis. Of course, a l i n e a r o s c i l l a t i n g f i e l d i s decomposed into two oppositely r o t a t i n g f i e l d s , but only the one that rotates i n the same sense as the spin precession i s e f f e c t i v e near resonance for small B-^ (see, f o r example, Ramsey 1963, p. 146). It i s well known that under these conditions the vector J = (J x,Jy,J_) precesses about the e f f e c t i v e f i e l d at an angular frequency u = [(Aco) 2 + 1/4 i ^ 2 ] 1 / 2 (2-13) 9 -a? is oriented at an angle © w i t h respect to the z axis, as illustrated in Fig. 2-2 for an r-f . field oriented in the x-y plane in the laboratory frame at an angle -<}> with respect to the x axis at the position r. Representing this by a complex number, V r ) = B l x ( f ) + i B l y C ? ) ' ( 2 _ 1 4 ) = B^rje"^ the angle 6 i s given by tan 6 =. 1/2 — - — = - T ~ (2-15) o We specify the i n i t i a l conditions for J j i n terms of a coordinate system f i x e d i n the r o t a t i n g frame. i s the component of J at t = 0 along the e f f e c t i v e f i e l d , which has polar angles (6, -cb) as shown i n Fig. 2-2. and are the components along the axes having polar angles (1/2TT + 0, -<£) and (1/2TT, 1/2'TT - <j>) , r e s p e c t i v e l y . In terms of these i n i t i a l conditions, the solutions to equations (2-8), (2 r9), and (2-10) are as follows: J + ( t ) = J_*(t) = {l/2(l+cos6) (J 1+iJ 2)exp[i(co+u e)t]+l/2(l-cos0) (J^-i^^xptifio-io^ t J+JgSinOexpfiwt) } exp(-i<j>), (2-16) J z (t)= -l/2sin0 [ ( J 1 + i J 2 ) e x p ( i w e t ) + ( J 1 - i J 2 ) e x p ( - i c o e t ) ] + J 3 c o s 6 (2-17) - 1 0 -Using equations ( 2 - 3 ) and ( 1 - 1 ) , the Stern-Gerlach force may be written 9 B ( ? ) 3 B (r) 8 B ( ? ) F(t) = v * [ J x ( t ) ~ ± r - * J y ( t ) - 3 ^ — + J J t ) Jcos.t ( 2 - 1 8 ) Special Case In the experiment to be described in the next two chapters the oscillating f i e l d was produced by a four-wire system, as shown in Fig. 2 - 3 . Near the center of symmetry of the four wires (r << R ) , the f i e l d is well approximated by B^r) = G(x-iy) = Gr* ( 2 - 1 9 ) where G is the gradient of the f i e l d having dimensions gauss/cm and is given by G = 0.8 — ( 2 - 2 0 ) R o where I is the current in each wire in amperes and R q is the distance in centimeters from the center of symmetry to the center of each wire. The orientation of this f i e l d for a position r = x+iy is shown in Fig. 2 - 3 . Using eqs. ( 2 - 1 8 ) and ( 2 - 1 9 ) , i t is seen that F(t) = YfiGJ.(t)costot ( 2 - 2 1 ) - 11 -- 12 -From equations (2-16) and (2-21), we see that F(t) consists of terms which o s c i l l a t e at frequencies OJ , 2to, and 2oo ± UJ . In addition, there i s one time-independent term which gives the time-averaged force to be <F(t)> = 1/2 yfiGJ 3 sine e 1* (2-22) This expression contains the following information. (a) The time-averaged force i s proportional to the value of the compon-ent of J along the e f f e c t i v e f i e l d i n the ro t a t i n g frame. It i s t h i s component of J which i s quantized i n the generalized Stern-Gerlach experiment on neutral atoms. It may be noted that t h i s component of J i s independent of time i n the inhomogeneous f i e l d region. (b) The change of momentum Ap i s i n the r a d i a l d i r e c t i o n i n the x-y plane. Since i s quantized, a c i r c u l a r beam i s decomposed into 2J + 1 ring s . The same e f f e c t i s obtained f o r the conventional Stern-Gerlach experiment f o r t h i s geometry (Beenewitz and Paul 1954). (c) The dependence of Ap on frequency i s contained i n the factor sine = 2 [ ( A c o ) 2 + W l 2 / 4 ] 1 / 2 Therefore, when the r o t a t i n g f i e l d approximation i s v a l i d , the change i n momentum i s maximum at the Larmor frequency and i s an even function of Aoo. It may be noted that the conventional Stern-Gerlach experiment for t h i s geometry ( B Q = 0 = OJ) i s also described by the theory. In addition, one can say from the argument presented i n going from equation (2-21) to equation (2-22) that, i n order to e s t a b l i s h • - 13 -quantization of J along the e f f e c t i v e f i e l d , the time spent by the atoms i n the inhomogeneous f i e l d region must be much greater than the period of precession i n the e f f e c t i v e f i e l d , i . e . , t >> ^. F i n a l l y , the r e s u l t s given here may be compared with the quantum mechanical c a l c u l a t i o n of Bloom and Erdman (1962) for J = 1/2. A potassium atom having the average v e l o c i t y i n a t y p i c a l beam experiment _ g has a de Broglie wavelength much less than 10 cm, while the c o l l i m a t i o n system used i n t h i s experiment (and other t y p i c a l beam experiments) _3 l o c a l i z e s the atoms to not less than 5 x 10 cm. Thus, quantum mechanics i s only needed to describe the angular momentum properties of atoms. When the spin functions associated with d i f f e r e n t l o c a l i z e d wave packets (i n momentum space) i n the quantum mechanical c a l c u l a t i o n (Bloom and Erdman 1962, equations (28)-(36)) are examined for a r b i t r a r y w, i t i s seen that they correspond to spins quantized along the e f f e c t i v e f i e l d . For J = 1/2, there are only two such spin states. However, at the low f i e l d s at which the present experiment was done, the nuclear spin, I = 3/2 i s strongly coupled to the electron spin J to form a t o t a l angular momentum F = 1,2. in weak external f i e l d s , these s p l i t into eight Zeeman components with f i v e d i s t i n c t e f f e c t i v e magnetic moments M g y /4,M = 2, r J O r 1, 0, -1, -2, where y i s the Bohr magneton and g i s the Lande g factor O J 2 of the S.. , ground state of potassium. CHAPTER 3 Experimental Procedure The main aim of t h i s experiment was to detect the predicted trans-verse Stern-Gerlach resonance and to check the theory of chapter 2. Since measurement of the resonance frequency to an accuracy of a few percent was s u f f i c i e n t f o r t h i s purpose, i t was possible to keep the design of the equipment r e l a t i v e l y simple. A schematic diagram of the o v e r a l l system i s shown i n F i g . 3-1. A potassium beam was chosen since potassium atoms are easy both to produce and to detect. The beam was produced by heating potassium metal to 300°C i n an i r o n oven, with a hole of diameter 0.005 cm to form the beam. This temperature corresponds to a vapor pressure for potassium of about 1 Torr. A sin g l e f i l l i n g of the oven was s u f f i c i e n t to l a s t f or 60 hours of continuous operation. Micrometer adjustments were provided to p o s i t i o n the oven i n the plane transverse to the beam. The c o l l i m a t i o n was done with a hole placed 31.4 cm from the oven. Any one of three d i f f e r e n t sized holes of diameter 0.17 cm, 0.0125 cm, 0.005 cm, r e s p e c t i v e l y , could be brought into p o s i t i o n by admustment of a single micrometer. The center of the d e f l e c t i o n system, which was 10 cm i n length, was placed 15.6 cm from the co l l i m a t i n g hole, and the detector opening was 196 cm from the co l l i m a t i n g hole. The long f r e e - f l i g h t region of 136 Detector aperture Free Flujkt Region. Magnet Electron Muliiplier Pre- Amp nal Solenoid Quadrupole Were Deflector Pittances in Cffn-hmetfers Hot t« Sc«l« Tootked Chopper Figure 3-1. Schematic diagram of the o v e r a l l system (not to s c a l e ) . about 180 cm which t h i s provided had the advantage of increasing the si z e of the deflec t i o n s and correspondingly reducing the tolerances necessary for the detector p o s i t i o n i n g and opening. The detector was a tungsten hot-wire i o n i z e r . The ions were c o l l e c t e d on an electron m u l t i p l i e r a f t e r crude mass analysis. The mass spectrometer was r e a l l y not necessary but was used because i t was an in t e g r a l part of the i o n i z e r which was already on hand. The detector opening was rectangular and was formed by four knife-edged jaws which were each positioned by a micrometer. The hot wire was 0.018 cm i n diameter and was placed 25 cm from the detector opening. The beam was chopped at 30 c.p.s. with a toothed wheel driven by a synchronous motor. The output of the electron m u l t i p l i e r was amplified with a narrow-band 30 c.p.s. preamplifier, with a gain of 100, and then fed into a commercial l o c k - i n a m p l i f i e r (Princeton Applied Research Model JB4). With the K oven at about 300°C a beam collimated with the 0.005 cm _9 diameter hole gave a maximum output current of 10 A from the electron m u l t i p l i e r . With an estimated gain of 10^ for the m u l t i p l i e r , t h i s meant that the incident K beam corresponded, a f t e r i o n i z a t i o n , to an -14 ion current of about 10 A (neglecting losses i n the mass spectrometer). Commercial tungsten wire, even the so- c a l l e d undoped kind, contains large amounts of K impurity and an i n i t i a l t r i a l showed that the noise from the wire would swamp any expected resonance signal with observation time constants of a few seconds. Hence i t was necessary to go to a potassium-free wire grown from W(C0)^, according to the recipe-described by Greene (1961). With such a wire and the maximum usable oven temperature of 300°C - 17 -we obtained a signal-to-noise r a t i o f o r the d i r e c t beam of 30 to 1 with a 3-second time constant on the l o c k - i n a m p l i f i e r . The vacuum system was b u i l t p a r t l y of brass and p a r t l y of nonmagnetic s t a i n l e s s s t e e l . Three o i l d i f f u s i o n pumps with l i q u i d nitrogen cooled b a f f l e s were used, one i n each of the two end chambers and one i n the f r e e - f l i g h t region. The three regions could be i s o l a t e d from each other _7 by valves. A pressure of 3 x 10 Torr was obtained, except i n the oven -7 region, where the pressure was 8 x 10 Torr. The d e f l e c t i o n system consisted of a solenoid and a quadrupole wire system, as i l l u s t r a t e d i n F i g . 3-2. Four push rods at each end of the solenoid allowed a three dimensional p o s i t i o n i n g of the en t i r e assembly within the vacuum envelope. The solenoid had overwound ends and gave a measured homogeneity of 1 part i n 600 over the 10-cm center region. The method used to match the very low resistance of the quadrupole wires to the nominal 50-ohm output of the r - f . source was to make the quadrupole wires part of a p a r a l l e l resonant c i r c u i t , as shown i n F i g . 3-3. The c i r c u i t used had an impedance of 120 ohms at a resonant f r e -quency of 7.22 Mhz. The maximum current used i n the wires was 11.3 A (peak). This corresponds to a d r i v i n g voltage across the c i r c u i t of 65 V r.m.s., and a t o t a l power d i s s i p a t i o n of 35 W i n the c i r c u i t . To absorb t h i s heat, the wires were cemented with epoxy into accurately machined brass blocks, and the s i l v e r e d mica capacitor was glued to a copper plate i n thermal contact with the r e s t of the system. This assembly was a l l mounted i n a water-cooled copper tube on which the solenoid was wound. cjuadrvpale w i r e 7 tf -toning co i l Solenoid 2 10 cm 2 0 cm If c o p p e r u M v e S (*M€>ftW<?) e<fOall<j s p a c e d ow a 2 m*r\ d i q . c i r c l e F i g u r e 3-2. Drawing o f t h e d e f l e c t i o n system w i t h c o n s t r u c t i o n d e t a i l s o m i t t e d . o lnpo"t tao orm5 ai 7.2 * 18 fcWG copper wtrcs [O.S cr* "tolal lervej-fk eacU <j(i/ed -to brass backing blocks \ Silver wuca, leads O.tcw IOA^ rf -tuning coi I # 8 A W G copper 2-tonus, IcmdiV*,. KO i Figure 3-3. E l e c t r i c a l c i r c u i t of the quadrupole wires showing d e t a i l s of the p a r a l l e l resonant c i r c u i t used. - 20 -The brass backing blocks produce image currents which act as a second quadrupole system with twice the ^relative spacing of the wires. These image currents reduce the f i e l d gradient due to the wires by about 25%. The maximum wire currents used (11.3 A (peak)) corresponded to a gradient of approximately 680 gauss/cm, as calculated by using equation (21), taking the image currents into account. To f a c i l i t a t e the alignment of the quadrupole wires with respect to the beam, two holes 0.073 cm i n diameter, one at each end of the solenoid, were f i x e d r e l a t i v e l y accurately on the axis of the quadrupole system. These holes could then be located and the d e f l e c t i o n system centered, using the beam i n t e n s i t y as the i n d i c a t o r . A c r y s t a l c o n t r o l l e d o s c i l l a t o r (Heathkit SB-400) d r i v i n g a one-kilowatt l i n e a r a m p l i f i e r (Heathkit SB-200) was used for the r - f . power source. The d e f l e c t i o n of an atom having i n i t i a l v e l o c i t y v p a r a l l e l to the -2 z axis i s Cv , where the constant C i s obtained from equation (2-22)(at resonance, put s i n 6 = 1 ) and equation (19.3) of Kusch and Hughes (1959). _2 The average d e f l e c t i o n i s found by f i r s t showing that (v ) = m/(2kT), using the v e l o c i t y d i s t r i b u t i o n function of the beam. For our system T = 39 573°K. For K i n a small f i e l d B i t i s appropriate to c a l c u l a t e the d e f l e c t i o n of an atom i n a spin state Mp = 1, since the s t a t i s t i c a l weights of the states M = 2, 1, 0, -1, -2 are 1, 2, 2, 2, 1, r e s p e c t i v e l y . r In order to compare with experiment, we give here the r e s u l t of 0.011 cm f o r the average d e f l e c t i o n for a gradient of 430 gauss/cm. CHAPTER 4 Experimental Results The t h e o r e t i c a l predictions which have been v e r i f i e d are contained i n eq. (2-22). The s i g n i f i c a n c e of t h i s equation i s discussed immediately following i t i n Chapter 2. To v e r i f y the existence of the r a d i a l force f i e l d , a large diameter beam collimated only by the 0.073-cm hole i n the d e f l e c t i o n system was produced. This beam was of e s s e n t i a l l y uniform i n t e n s i t y over a diameter of 0.3 cm at the detector. Since the expected most probable d e f l e c t i o n was less than 0.03 cm, no large change would occur i n the i n t e n s i t y p r o f i l e except near the point corresponding to the o r i g i n of the r a d i a l force f i e l d . Here an increase i n i n t e n s i t y should occur with a radius approximately equal to the average d e f l e c t i o n , r e s u l t i n g i n a sort of "pip" at the center of a uniform f i e l d . This i s due to those atoms which are forced inwards and converge on the center area. This focusing e f f e c t i s a consequence of any d e f l e c t i o n system that produces a constant r a d i a l force f i e l d , and i s i d e n t i c a l , f o r example, with that obtained i n a conventional quadrupole focusing experiment (Bennewitz and Paul 1954) . This "pip" or focused spot was explored i n considerable d e t a i l by moving the detector aperture, previously described, over the area of the pip. The aperture dimensions were set at 0.010 x 0.010 cm. The r e s u l t s are shown i n F i g . 4-la. In F i g . 4-la each curve represents a step-by-step scan of the beam - 22 -f i e l d i n the v e r t i c a l d i r e c t i o n . The d i f f e r e n t curves correspond to d i f f e r e n t h o r i z o n t a l settings of the aperture. Each point on the curves was obtained by measuring the beam i n t e n s i t y f i r s t with the d e f l e c t i n g f i e l d o f f ( s t r i c t l y speaking, with the solenoid f i e l d o f f so that the quadrupole f i e l d was o f f resonance) and then with the d e f l e c t i n g f i e l d on. The percentage increase i n i n t e n s i t y i s plo t t e d along the ordinate of the f i g u r e . From t h i s family of curves a rough contour map of equal i n t e n s i t y l i n e s may be constructed, as shown i n F i g . 4-lb. Neglecting fo r the moment the secondary peaks i n Figs. 4-la and 4-lb, i t may be noted, f i r s t , that the focusing e f f e c t i s approximately c y l i n d r i c a l l y symmetric; secondly, that i t occurs completely within a diameter of about 0.1 cm, with no other observable fl u c t u a t i o n s within the 0.3-cm diameter c i r c l e , over which the whole beam extended; and t h i r d l y , that the h a l f width at h a l f maximum of the focused spot f o r a gradient of 430 gauss/cm i s about 0.025 cm, which i s i n agreement with the order of magnitude calculated i n a previous section. The center of the focused spot i s assumed to be the p o s i t i o n of the symmetry axis of the quadrupole wires i n the plane of the detector. The secondary peaks shown i n Figs. 4-la and 4-lb probably a r i s e from some aberration i n the focusing system. They almost certainly'do not a r i s e from any s p a t i a l r e s o l u t i o n of the beam into i t s various 'com-ponents since the beam i s broad compared to the d e f l e c t i n g power a v a i l a b l e . The focusing e f f e c t was then studied on a very f i n e beam. With a collimator hole 0.005 cm i n diamter, the undeflected beam p r o f i l e i s shown i n F i g . 4-2. - 23 -The f u l l width at h a l f i n t e n s i t y i s about 0.030 cm. The e f f e c t of turning on the quadrupole f i e l d i s as shown i n F i g . 4-3, the r e s u l t s having been obtained and p l o t t e d i n exactly the same manner as those i n F i g . 4-la. One can see that the curves are roughly symmetrical about the v e r t i c a l = -20 x (0.001 cm) p o s i t i o n and that there i s a strong focusing action at v e r t i c a l = -20 x (0.001 cm) and h o r i z o n t a l = -15 x (0.001 cm). This p o s i t i o n coincides with the center of the focused spot represented i n Figs. 4-la and 4-lb. In other words, i t i s the p o s i t i o n of the symmetry axis of the quadrupole wires at the detector. Since the zeros of the axes i n F i g . 4-3 correspond to the center of the undeflected beam, the quadrupole wires and the undeflected beam were aligned to about 0.02 cm i n 180 cm (the f r e e - f l i g h t length) or 1 part i n i o 4 . As shown i n F i g . 4-3, there i s a depletion of atoms at about .0.-025 cm from the d e f l e c t o r axis (as measured i n the plane of the detector). Father out i n the wings of the beam there i s a p o s i t i v e change i n i n t e n s i t y again, i n d i c a t i n g that some atoms have been thrown outwards from the center of the beam. These are defocused atoms, i . e . , atoms which, i n the r o t a t i n g frame, have e f f e c t i v e magnetic moments i n the x d i r e c t i o n of opposite sign to those of the focused atoms. I f that i s so, then we have resolved the beam into two components. Apparently, our low d e f l e c t i n g power, coupled with the Maxwellian d i s t r i b u t i o n of v e l o c i t i e s i n the beam, has precluded our observation of a l l f i v e of the expected components. It should be pointed out that, because of the preliminary nature of the present experiment, no attempt has been made to achieve the ultimate i n r e s o l u t i o n or l i n e shape. Fi g . 4-4 shows the f i e l d dependence of the resonance. The - 24 -detector p o s i t i o n and opening were held f i x e d with respect to the beam, i n a p o s i t i o n that had a good signal-to-noise r a t i o . The strength of the quadrupole f i e l d B^, at constant gradient, acting on the beam was varied by moving the d e f l e c t i o n system and holding the beam f i x e d , i n other words, by sending the beam down the d e f l e c t i o n system at d i f f e r e n t distances from the zero axis. The f i e l d gradient was changed by changing the r - f . voltage drive across the system. The curves were obtained by holding the frequency f i x e d at 7.22 Mhz and changing the solenoid f i e l d (B, ) i n di s c r e t e steps, and reading the output meter of the l o c k - i n a m p l i f i e r . The undeflected beam i n t e n s i t y was p e r i o d i c a l l y checked to ensure that the system was not d r i f t i n g . These curves are i n agreement with eq. (2-22) . The resonance maximum occurs f o r B.. = a) /Y = 10.4 gauss, the f i e l d f o r a Larmor o o ' • precession frequency of 7.2 Mhz, and the l i n e width f o r each curve, given by the p l o t of sin/; 6 versus B q , i s approximately equal to B^- NO attempt has been made, however, to make a d e t a i l e d t h e o r e t i c a l f i t of the observed l i n e shape. The double quantum resonance which appears i n two of the curves of Fig. 4-4 i s discussed t h e o r e t i c a l l y i n chapter 2. It occurs f o r large B^ (10 to 16 gauss) as predicted, but i n t h i s experiment a large B^ only occurs i n the region of the quadrupole f i e l d where the higher-2 2 order terms of the expansion B^ = G-^(xi - yj) + G^(x - y ) i + ... are becoming important. The e f f e c t s of these higher-order terms have been neglected i n the theory, as have been the components of B^ p a r a l l e l to B , which occur near the ends of the wires. o' The s h i f t of the double quantum resonance towards lower f i e l d s which i s apparent i n the two curves i s probably a Bloch-Siegert s h i f t - 25 -(Ramsey 1963, p. 222), which results from the use of a linearly oscillating f i e l d B instead of a true rotating f i e l d . i - ; 4 «> ti --o i CD Si ct T o t a l u n i f o r - r w intensify haim LuUiW a t delector-Grad ient = *4-50 gauss / cw D e t e c t opening 0.01 on* * O-OI c**\ = 0 - 3.C Figure 4-la. ON -W -*o -ao O -v20 +«K> +60 Vert ica l position o f <Jet*e+or .001 c m ) (a rb i t rary ^ e r o p o s i t i o n ) Plot of the percentage increase i n beam i n t e n s i t y at the detector as a function of po s i t i o n i n the x,y plane due to the focusing e f f e c t of the d e f l e c t i o n system i n a large, uniform-intensity beam. - 27 -rlori^orftal post*. ( . 0 0 1 cm) Figure 4-lb. Contour map of the focused "pip", as constructed from Fig. 4-la. The numbers on the contours are percentage increases in beam intensity due to the rotating quadru-pole f i e l d (ordinate of Fig. 4-la). The dotted lines are interpolated contours between widely separated points. - 28 -C arbt"tror^ -$era position) Figure 4-2. Beam intensity profile for undeflected narrow beam. - 29 -Vertical position of detector relative "to center of deflected beam X . 001 cm Figure 4-3. Relative change of the beam intensity pattern for a narrow beam at resonance. The undeflected beam profile is shown in Fig. 4-2. I I I I 1 I L_i I 1 1 1 1 0 0.2. 0.«f O.fc O-fc «.0 1-2. 1-4 1-8 SO Solenoid current C a w P s ) Figure 4-4. Plot of the beam i n t e n s i t y near the beam center as a function of the f i e l d B f o r fixed frequency and d i f f e r e n t amplitude of the r f f i e l d and i t s gradient. CHAPTER 5 A Proposal f o r a Charged P a r t i c l e TSG Experiment The major obstacle to performing a conventional Stern-Gerlach experiment on charged p a r t i c l e s i s that the Lorentz force associated with a charged p a r t i c l e moving i n a magnetic f i e l d i s f a r larger than the Stern-Gerlach force. The spread of v e l o c i t i e s inherent i n any io n i c beam would give r i s e to an uncertainty i n the Lorentz force which would undoubtedly mask the e f f e c t of the Stern-Gerlach force. When the neutral p a r t i c l e TSG experiment was proposed i t seemed l i k e l y that i t would be adaptable to charged p a r t i c l e s , since i t would be possible to send a charged p a r t i c l e beam p a r a l l e l to the large homogeneous magnetic f i e l d and to observe resonant displacements i n the plane perpendicular to t h i s f i e l d due to an o s c i l l a t i n g inhomogeneous magnetic f i e l d . Some of the d i f f i c u l t i e s associated with one sp e c i a l type of charged p a r t i c l e TSG experiment have already been discussed (Bloom and Erdman 1962; Ra s t a l l 1962; Byrne 1963). In adapting the neutral p a r t i c l e TSG experiment to charged p a r t i c l e s our preliminary thoughts were that i n t h i s geometry the amplitude of the ion beam o s c i l l a t i o n s produced by the o s c i l l a t i n g Lorentz force would be inversely proportional to the frequency of the o s c i l l a t i n g magnetic f i e l d , so that by using a high enough frequency these o s c i l l a t i o n s would not se r i o u s l y impair the s p a t i a l r e s o l u t i o n of the beam or the - 32 -Stern-Gerlach e f f e c t . In general terms these conclusions were correct, although at the time i t was not r e a l i z e d that t h i s system i s one of a broad category of strong focusing systems. Preliminary work was begun on the extension of the ac quadrupole system, which was used i n the neutral p a r t i c l e TSG experiment, to charged p a r t i c l e s . Much of the work at t h i s stage went into producing an ion beam, but several prototype ac quadrupoles were also b u i l t . At the outset, i t was r e a l i z e d that two major problems would have to be over-come to u t i l i z e the ac quadrupole f or a TSG experiment with i o n i c beams. The f i r s t problem was to reduce the e l e c t r i c f i e l d produced by the voltage drop along the wires. This e l e c t r i c f i e l d was s u f f i c i e n t l y strong, f o r the low energy beams which must be used, to produce very large o s c i l l a -tions i n the beam. It was r e a l i z e d much l a t e r that t h i s e l e c t r i c f i e l d could be u t i l i z e d to good advantage to cancel the magnetic Lorentz force o s c i l l a t i o n s , but th i s did not eliminate the problem of reducing i t without s i g n i f i c a n t l y reducing the o s c i l l a t i n g magnetic f i e l d . The second problem was to cool the quadrupole wires s u f f i c i e n t l y well that very high f i e l d gradients could be produced. Most of the work which went into the ac quadrupole was centered on these two problems; however, the a u x i l l i a r y problems of generating high power RF (the designs were based on an RF power input of 4000 watts), grounding and sh i e l d i n g i t , and matching the impedances, also took considerable time. In the middle of t h i s development i t was r e a l i z e d that a much simpler system could be b u i l t using time independent currents i n a h e l i c a l quadrupole, such as i s i l l u s t r a t e d i n F i g . 5-la. This idea led - 33 -Figure 5-la. A h e l i c a l quadrupole wire system. Figure 5-lb. A quadrupole wire system showing the electrodes necessary to produce an e l e c t r i c f i e l d orthogonal to the magnetic f i e l d . - 34 -to the extension of the TSG experiment to space varying fields as mentioned in Chapter 1, and f i n a l l y we realized that we had been working a l l along within the general concept of strong focusing systems. Several months later we added an electric f i e l d orthogonal to the magnetic f i e l d in the quadrupole, as illustrated in Fig. 5-lb, which serves the important function of neutralizing the major portion of the Lorentz force, for a particular beam velocity. When the dc helical quadrupole is used, rather than the ac quadru-pole, the following important simplifications occur. There are no skin-depth or induced current phenomena, so that additional electrodes for control of the electric f i e l d may be inserted into the quadrupole without attenuating the magnetic f i e l d . Also, the f u l l cross-section of the quadrupole wires carries the current so that the resistance and the power dissipated are both less than for the ac case. In fact, i t should be appreciated that since the magnetic f i e l d gradient and resistance are each inversely proportional to the cross sectional area of the wires, and since the cooling capacity varies at least in proportion to the area, one can make this system as large as necessary without having the cooling capacity lag behind the power required to maintain a given f i e l d gradient. Finally, i t is much easier to produce, shield and ground, and match impedances, for dc currents than high frequency ac currents. So far, we have restricted this discussion to a helical quadrupole formed from current-carrying wires, since this is a form of construction which is simple and straight forward, and which has the important advantage that the additional electrodes which are inserted to produce the orthogonal electric f i e l d may be made concentric with the quadrupole - 35 -wires, as i l l u s t r a t e d i n F i g . 5-lb. Such a construction should give a very high degree of f i e l d orthogonality. However, a h e l i c a l quadrupole f i e l d may also be produced using iron-cored magnets, as i l l u s t r a t e d i n F i g . 5-2. In t h i s case the electrodes producing the orthogonal e l e c t r i c f i e l d must be placed between the i r o n poles, and the ortho-gonality achieved may not be very good. F i n a l l y , i t i s possible that super-conducting quadrupoles can be developed which would s i g n i f i c a n t l y increase the magnetic f i e l d gradients which can presently be attained. In Chapter 2 we have given the c l a s s i c a l theory of the transverse Stern-Gerlach experiment for a neutral p a r t i c l e . It i s clear that the conclusions presented there regarding the Stern-Gerlach force may be c a r r i e d over d i r e c t l y to the charged p a r t i c l e case, since i t i s not necessary to consider the Lorentz force unless one wishes to compute trajectories-.That i s , a neutral p a r t i c l e or a charged p a r t i c l e , i f placed i n the same f i e l d configuration, w i l l experience the same Stern-Gerlach force. In the following, we e s s e n t i a l l y repeat much of Chapter 2 for the p a r t i c u l a r case of a p a r t i c l e moving generally i n the z d i r e c t i o n i n a time-independent h e l i c a l quadrupole f i e l d . The Stern-Gerlach force i s given by: F S G = (u -y)B(f,z) (5-1) or F S G x = Kh +Uyh + U z l l ) B x ^ F c r = (U V - + u | - + U ! - )B (5-2b) SGy 1 x 3x y 3y z 3z J y K J Figure 5 -2 . An i r o n core quadrupole, showing the electrodes necessary to produce an e l e c t r i c f i e l d orthogonal to the magnetic f i e l d . - 37 -F = (u TT—,+ u — + u — — ) B SGz ^ x 8x y 3y z 3z ; z In the h e l i c a l quadrupole B(f,z) i s given by: B(r,z) = B I + B } + B t v ' 1 x y • o B = G (x s i n 2wz - y cos 2coz) B^ =-Gg(y s i n 2toz + x cos 2UJZ) B = constant o Therefore, s u b s t i t u t i n g equation (5-3) into (5-2) y i e l d s FSGx = ^B^ Ux S ^ n 2 - Z ~ u y c o s 2 ~ z ^ ^SGy = ~^B^Ux ° O S 2 - Z + U y S^" n 2 — z ^ F c r = 0 SGz If we rewrite equation (5-3) i n the form: - 38 -then i t i s obvious that i n a coordinate frame moving i n the z d i r e c t i o n at v e l o c i t y V q , the f i e l d B(f,z) w i l l consist of a component (?) r o t a t i n g with the angular v e l o c i t y 2ojyo, and a solenoid f i e l d B Q. We now define a coordinate frame *yi,y which i s centered on the p a r t i c l e and which rotates i n time at the angular frequency co about the z axis, such that the angular v e l o c i t y of the component B^(r) about the z axis i n the frame x,y i s zero. Since i n general the p a r t i c l e may have an angular frequency :;coT i n time about the z axis ( a r i s i n g from the Lorentz forces, f or example) we have the r e s u l t that; co = 2cov - coT (5-7) where co^ = angular frequency i n time of the p a r t i c l e about the z axis In the coordinate frame "x/y, J precesses about the e f f e c t i v e f i e l d at an angular frequency 2 2 1/2 u)e = [(Aco) / + ] ^ (5-8) where Aco co-co = -YB, -YB^r) (5-9) (5-10) - 39 -The effective f i e l d is oriented at an angle 0 with respect to the z axis (as illustrated in Fig. 5-3a) where: u l tan 0 = - -=• (5^11) Aw If, as in Chapter 2, we take as the component of J along the effective f i e l d at t = z = 0, and then calculate the time-average of the Stern-Gerlach force, i t is clear that only the components of wi l l yield a cummulative force, since only is synchronous with the f i e l d (r) (This is shown explicitly in Chapter 2). Noting from Fig. 5-3b, that; B u = yn\J7 sine + terms oscillating at u> (5-12a) B u = yfiJ sine ^ — + terms oscillating at Ui (5-12b) Y 3 IBJTOI we can substitute equation (5-12) into (5-4) to yield, for times large compared to — : e < FSGx > = ^ J 3 S i n 9 l G B l F < F S G / = Y M 3 S i n S l G B l r This may be written < FSG > Y t f J 3 sine | GB| - (5-13) Figure 5 -3a . Illustration of the frequency vectors in the rotating coordinate system {%,y*,z~). ^ ( F ) is in the x,y plane and makes an angle -<j> with respect to the ~ axis. is the component of J along the effective f i e l d . u 2 Figure 5-3b. An i l l u s t r a t i o n of the vector sin 0 in the (x,y,z) coordinate system at the time t. - 41 -with s i n e = 77—2 v72 (5"14) Therefore the time averaged Stern-Gerlach force i s r a d i a l , with a magnitude that depends on the s i n 6 f a c t o r noted above. This r e s u l t i s e s s e n t i a l l y the same as equation (2-22) i n Chapter 2, except for a fa c t o r of 2 which enters because the l i n e a r o s c i l l a t i n g f i e l d of the Chapter 2 system i s decomposed into two r o t a t i n g components which have 1/2 the magnitude of the o r i g i n a l f i e l d . We have discussed the detection of the neutral p a r t i c l e Transverse Stern-Gerlach e f f e c t i n Chapters 3 and 4, and indi c a t e i n t h i s Chapter that other detection techniques may be more sui t a b l e for the charged p a r t i c l e case. We can suggest four methods of detection as follows; 1) Resonant De f l e c t i o n Method The method used for the neutral p a r t i c l e case may also be used f o r charged p a r t i c l e s . This method involves a resonance i n the Stern-Gerlach force v i a the s i n 6 factor (see equation 5-14), and the detection of a consequent change i n the i n t e n s i t y d i s t r i b u t i o n of the beam. The resonance i n the case of the h e l i c a l quadrupole f i e l d could be produced by f i x i n g v , thus f i x i n g the apparent angular frequency of the h e l i c a l quadrupole f i e l d i n the r e s t frame of the p a r t i c l e s , and then tuning the precession of the magnetic moment to t h i s frequency using the longitud-i n a l magnetic f i e l d B q. As we s h a l l see l a t e r , the width of the resonance i s anticipated to be very large. The fa c t that the l o n g i t u d i n a l f i e l d also produces a d i r e c t e f f e c t on the beam i n t e n s i t y d i s t r i b u t i o n makes the resonant d e f l e c t i o n method much more d i f f i c u l t than f o r the neutral p a r t i c l e case. 2) Computational Method The computational method involves a d e t a i l e d comparison of the experimental beam i n t e n s i t y d i s t r i b u t i o n as a function of the various experimental parameters with that computed from the equations of motion f o r the charged p a r t i c l e s . It would be necessary to pick out those features of the i n t e n s i t y d i s t r i b u t i o n which are associated with the small Stern-Gerlach force. 3) Comparison Method In t h i s method the i n t e n s i t y d i s t r i b u t i o n of beams of s i m i l a r ions having d i f f e r e n t magnetic moments would be compared and the e f f e c t s c h a r a c t e r i s t i c of the Stern-Gerlach force exhibited. For example, 4 + He ions i n the ground state would have two magnetic states correspond-- h ing to magnetic moments of approximately ±y^, where y g i s the magnetic moment of the electron. On the other hand, ^He+ would have 3 values for the p r o j e c t i o n of the magnetic moment along the external magnetic f i e l d i n low magnetic f i e l d s because of the e f f e c t of the nucleus of spin 1/2. The magnetic moments of these states would be approximately y , o, and -y , with p r o b a b i l i t i e s , 1/4, 1/2, and 1/4, re s p e c t i v e l y . 4) The Rabi Magnetic Resonance Method The Rabi method (Ramsey 1963) consists of sending the ions through a system composed of a " p o l a r i z e r " , a "depolarizer" and an "analyzer". The p o l a r i z e r and analyzer f o r charged p a r t i c l e Stern-Gerlach experiments would be h e l i c a l quadrupole systems which each produce a c e r t a i n i n t e n s i t y d i s t r i b u t i o n at the output given a c e r t a i n d i s t r i b u t i o n of i n t e n s i t i e s and - 43 -spins at the input. I f the d i s t r i b u t i o n of spins at each p o s i t i o n in, the beam were changed i n the region between the p o l a r i z e r and analyzer, the i n t e n s i t y d i s t r i b u t i o n at the output of the analyzer would also be changed. Transitions between d i f f e r e n t spin states can be produced by a depolarizer i n which the spins undergo magnetic resonance. For charged p a r t i c l e s the depolarizer would consist of a lon g i t u d i n a l time independent f i e l d and a transverse r f . f i e l d , both of these magnetic f i e l d s being homogeneous. When the usual magnetic resonance conditions are s a t i s f i e d , the p r o b a b i l i t y of a p a r t i c l e undergoing a t r a n s i t i o n from one spin state to another i n the depolarizer i s large. When the beam passes from the p o l a r i z e r through the depolarizer to the analyzer, i t i s necessary to s a t i s f y conditions of "adiabatic passage" (Abragam, 1961) i n order to preserve the d i r e c t i o n of quantization of the spins when the magnetic resonance conditions i n the depolarizer are not s a t i s f i e d . It would seem that of the four methods described above the Rabi method i s the best one. This method d i f f e r s from the other three i n that a change i n the i n t e n s i t y d i s t r i b u t i o n of the beam i s achieved by varying a parameter (the frequency of the r f . f i e l d i n the depolarizer) which has no appreciable d i r e c t e f f e c t on the i n t e n s i t y d i s t r i b u t i o n , but which e f f e c t s the i n t e n s i t y only by changing the populations of the spin states. CHAPTER 6 The T r a j e c t o r i e s of an Ion with Zero Magnetic Moment i n a DC H e l i c a l Quadrupole In the following chapters we are attempting to f i n d out whether a h e l i c a l quadrupole system can be used to separate spin states i n a Stern-Gerlach experiment. A basic t h e o r e t i c a l requirement f o r such a study i s the c a l c u l a t i o n of reasonably precise expressions f o r the p a r t i c l e t r a j e c t o r i e s . The d i r e c t approach, of fi n d i n g the solutions to the equation of motion of an ion with a non-zero magnetic moment i n a h e l i c a l quadrupole system i s complicated because the equation i s non-linear. It has not, to our knowledge, been solved. We have, howev developed approximate, but useful solutions which are based on the c h a r a c t e r i s t i c s of the average Stern-Gerlach force given by equation (5-13). In t h i s chapter we are only considering the t r a j e c t o r i e s of ions with zero magnetic moment. For t h i s case the equation of motion may be solved exactly i n the a x i a l region, and we have done t h i s i n appendix B. In Chapter 7 we have developed solutions f o r a charged p a r t i c l e with non-zero magnetic moment which are v a l i d i n the l i m i t that the Lorentz force i s comparible to the Stern-Gerlach force. In t h i s l i m i t i t i s a v a l i d approximation to decouple the r a d i a l motion from the - 4 5 ' - -tangential motion. The radial equation of motion which results is linear and is solved in Chapter 7. In Chapter 8 we have developed approximate solutions based on the similarity between the radial nature of the average Stern-Gerlach force given by equation (5-13) and a force of the form 0i, where $ is a constant. We show that the coefficient $ may be chosen to represent the cummulative effect of the Stern-Gerlach force over a range of z which must be specified. The substitution of the force $r. for the average Stern-Gerlach force reduces the equation of motion to a linear equation, and the solutions to this equation, evaluated at the end point of the arbitrary interval over which $ represents the cummulative effect of the Stern-Gerlach force, yields for each interval a point of the trajectory. In the rest of this chapter we are only considering the motion caused by the Lorentz force with the solenoid f i e l d absent, since an understanding of this motion is very helpful before considering the additional effect of the Stern-Gerlach force. We show that this motion has a very simple character in the two extreme regions of operation. For very weak forces the motion is almost planar with a sinusoidal amplitude. For strong forces a magnitude of the Lorentz force is defined beyond which the motion is divergent. This may be called the point of in s t a b i l i t y , since the beam "blows up" quite rapidly beyond this point. At this point the trajectories are closely approximated by diverging spirals, with the radius growing linearly with distance along the optic axis, and the spiral pitch synchronous with the helical winding. Between the two extremes, the trajectories are bounded and periodic - 46 -with four characteristic frequencies, which reduce to two dn the helical frame of reference defined in appendix B. The motion is rather like a corkscrew twisted about the optic axis in this region. In appendix B, we give the derivation of the equation of motion for a helical quadrupole which includes both magnetic and electric f i e l d gradients (Gg and Gg) and a solenoid f i e l d (B q). As we have mentioned, a radial force term (§) is also included, but for the rest of this chapter we are considering only the case for $ = 0. Except for the inclusion of $ and B q, these equations are discussed quite extensively in the literature. G. Salardi et a l . (1968) studied the device as a lens to collect and focus charged particles. K.J. Le Couteur (1967) studied particle guiding in helical multipole fields and the quadrupole in particular. He also considers the higher order terms which are important in the off-axial region, and shows that the device can be bent into a cir c l e and s t i l l confine a beam. L.C. Teng (1959) studied the device in the axial region. A.M. Strashkevich et a l . (1968) shows that at r e l a t i v i s t i c velocities i t is possible to make the device somewhat achromatic by adding the equivalent of the term G^ described in appendix B. N.I. Trotsyuk (1969) shows that the device can be used to focus atoms and molecules under certain conditions. Note that the electric and magnetic fields are orthogonal to one another, and as usual under this condition, the respective forces are colinear. For this reason the gradient terms appear in a single parameter © : © = — 9 — (G E + v oG B) (6-1) mv o - 47 -In most of the l i t e r a t u r e t h i s point i s not e x p l i c i t l y noted, since the concern i s usually only with the magnetic f i e l d , because of i t s strong focusing properties f o r energetic p a r t i c l e s . For our purposes t h i s i s an important point because by choosing G £ to cancel the term V QGg we can make © a r b i t r a r i l y small without s a c r i f i c i n g the magnetic f i e l d gradient, which of course produces the Stern-Gerlach e f f e c t . The solutions i n appendix B are s t r i c t l y v a l i d only for small displacements from the optic (z) axis. Pearce (1969) has shown by numerical i n t e g r a t i o n that for displacements less than L/20, where L i s the h e l i c a l step length, the equations are v a l i d . When the displacements are greater than L/20, the a x i a l magnetic f i e l d becomes appreciable, non-l i n e a r terms appear i n the equations of motion, and the system becomes more strongly focusing. I f , i n appendix B, we put B Q = 0, the term W reduces to w, and i f , i n addition, $ = 0, we can write the t r a j e c t o r i e s , equations (b-22), as: ojy x = x cosu„z + -=- y sinw,z + I cosco0z-cosu), Z I - o ^ 2 ttl yo *1 © —*o -2 + [ — since, z- — sinco.z] * (6-2a) cox 0) . " O r -i y = y C O S U K Z - —=• x sinto„z + I C O S O J , z-cosco 0zl — o -1 OJ2 o -2 L -1 -2 -^o -1 + [ — sinco, z- sinco„z] (6-2b) 0 a "I <^ " 2 where - 48 »1 2 2 co + © ^2 2 2 co - © co = 2TT_ L (6-2c) If © i s small, we can rewrite these equations by expanding co^ and co„ to « co + 1/2 ©/co -\ „ « 1 (6-3) |©/c/ Then applying simple trigonometric transformations, equation (6-2) takes the form: 2 & o . . © z x = x cosco„z + y smco nz + smcoz s m r—-- o — 2 J o -1 .ris. "" 2co 2cox (_• z + coscoz s i n •=— (6-4a) 2OJXO © z y = y coscoz - x smco„z - smcoz s m -:— - *° " 0 " 2 © " 2 a 2coyQ © z + coscoz s i n -=— (6-4b) for © / c o 2 << 1 Transforming to the laboratory frame; - 49 -x = xcoswz - ysinwz y = ycosojz + x sinuz (6-5) ~~ © z © z 2O)X Q © z x j? x cos — + y s m ^ — + s m o 2oj ' o 2OJ ^ (6-6) © z © z 2 uy y « y cos •=— + x s m - — + o 2co o 2OJ - y o sm I f x , y are small, i . e . ; 2cjr | r j « | - ^ ° | (6-7) 0 where 1 r = f x + y o 1 ^ o ' c 2' then t h i s motion i s nearly planar, and we can describe i t by the radius r ; (Note 1) 2_r © z r = s i n •=— + r (6-8) © 2^ Note that taking the l i m i t as © 0 i n equation (6-6) , one gets simple free f l i g h t , as expected: Note 1: r as used here i s . s t r i c t l y a one dimensional coordinate rather than [ f | , since both p o s i t i v e and negative values are allowed. - 50 -X = X + X z 0 (6-9) y = y0 +.v We can make the following observations. The trajectories are 2 bounded provided © < to . For very small © , the motion is modified from simple free flight to a long wavelength sinusoidal motion, with a half period given by: z , = 2m (6-10) As (Q> becomes larger, the motion becomes more complicated with two characteristic frequencies, or modes, co^ and a^. In this region one must be careful about the relative importance of the axial (r = 0) and off-axial particles when making generalizations. Generally, when discussingvthe focusing properties of these systems, only the axial particles are considered. This involves the implicit assumption that: cor |r | « | ^ | (6-11) © In many beam handling experiments this is a good assumption, but in this experiment we are considering beams with -4 r « 10 m o r <a 10"3 o From equation (6-11), this means that we are only just i f i e d in - 51 -generalizing to a x i a l p a r t i c l e s i f i L » IO" 1 m (6-12) We introduce the useful dimensionless parameter "a": a = -j (6-13) t o Then the condition of equation (6-12) for a t y p i c a l step length L = 2 x -2 10 m, becomes; a << (6-14) As w i l l l a t e r be shown, t h i s corresponds to a rather small value of © . 2 F i n a l l y , we consider what happens N when © = OJ J. i . e . , a = 1. 2 Taking the l i m i t i n equation (6-2), as © ->w , we get; s i n /2 tjz • s i n tl)Z O r . — -| x = x + y = — + x =— + — [l-cos/2 coz] ~ 0 ° /2 ° /2 0) = -cox z + y cos /2~ coz + — rcos/2" coz - 11 (6-15) — o ' o "* i£ ~ . r/2 sin/2 0)Z + y [ = z] J o L co J f o r a = 1 Comparing terms, we note that the dominant term becomes; - 52 -r x » 0[r + — 1 L o to r y « -((D X + * )z + 0[r + — ] (6-16) v— o ' o L o t o J ^ ^ fo r r (<o,x + y )z| >> 0[r + — ] (6-17) ^ o o 1 L o to J For the values quoted e a r l i e r , i . e . r « 10 4m o r « 10 o -2 L «s 2 x 10 m (6-18) Condition (6-17) becomes z » 3 x 10 3m (6-19) Under these conditions i t can be seen that the motion i s very soon a simple s p i r a l i n the laboratory frame, with the s p i r a l synchronous with1' the h e l i x and the radius growing l i n e a r l y with (z) . CHAPTER 7 The T r a j e c t o r i e s of an Ion with Non-Zero Magnetic Moment i n a DC H e l i c a l Quadrupole f o r Weak Lorentz Forces In t h i s chapter an approximate equation of motion i s developed for the r a d i a l motion of an ion with non-zero magnetic moment i n a dc h e l i c a l quadrupole system which i s v a l i d when the Lorentz force i s comparable to, or weaker than, the Stern-Gerlach force. In the l i m i t that the Lorentz force i s zero((B> -> 0), the expressions y i e l d the exact t r a j e c t o r i e s of a p a r t i c l e experiencing the average Stern-Gerlach force given by equation (5-13) . In order f o r (Q) to be small while maintaining a large Stern-Gerlach force, orthogonal magnetic and e l e c t r i c f i e l d s must be u t i l i z e d as shown i n F i g . 5-lb. In t h i s way the Lorentz force; F = q (E + v x B) can be made a r b i t r a r i l y small f o r a p a r t i c u l a r v e l o c i t y , while maintain-ing a large magnetic f i e l d gradient. From a p r a c t i c a l point of view i t i s probably not desirable to reduce the Lorentz force much below a few percent of i t s value f o r the magnetic f i e l d alone, since t h i s would require an extremely monochromatic beam and also a very high mechanical tolerance i n the apparatus. - 54 -In Chapter 6 i t was shown that the t r a j e c t o r i e s i n a h e l i c a l quadrupole i n the absence of the Stern-Gerlach force are nearly planar f o r small © and near a x i a l p a r t i c l e s . Under these conditions the r a d i a l dependence of the t r a j e c t o r i e s i s given by equation (6-8); (Note 1) 2ji°\ 2sF0 ©_z_ + r , G ( 7 _ 1 } s i n 2co o \ v J — < K 1 CO where r(z=0) = r J o dr, • j—(z=0) = r d z v J o This equation i s the so l u t i o n to the equation of motion; i- 2 d 2 r ,_ H = m V o ~2 (7-2^ dz where; F„ = -mv 2 ( ^ — ) 2 ( r - r ) (7-3) H o 2co o J In the region i n which equation (7-1) i s v a l i d , we can approximate the Lorentz force i n the h e l i c a l quadrupole by (equation 7-3) . I f the average Stern-Gerlach force (equation 5-13) i s added, a one dimensional equation of motion i s obtained which y i e l d s the desired Note 1: r as used here i s s t r i c t l y a one dimensional coordinate rather than | r | , since both p o s i t i v e and negative values are allowed. - 55 -r a d i a l dependence of the t r a j e c t o r i e s i n t h i s region: d 2 r ® 2 , U 5 l G B l ® 2 + ( ) r = 3 _ _ + (__) r ( 7 - 4 ) dz 2 2(j) 2E 2co o where = Y ^ J 3 s i n 6 , (Note 2) ^ = l ( s i g n r)^(Note 3 ) ( 7 - 5 ) The s o l u t i o n to equation ( 7 - 4 ) i s expressed i n terms of the i n i t i a l displacement r and the i n i t i a l r a d i a l v e l o c i t y r as follows; r o o u | G | <Q> z 2(or <0) z r = r + ( ^ ) Z 3 [1 - cos - 5 - ] + - s i n ( 7 - 6 ) It should be kept i n mind that these solutions are v a l i d approxi-mations only f o r ; Note 2: In Chapter 5 we have shown that the Stern-Gerlach force w i l l be r a d i a l l y inwards or outwards depending on the o r i e n t a t i o n of the magnetic moment with respect to the e f f e c t i v e magnetic f i e l d i n the system. The r a d i a l sense of t h i s force w i l l not change i f the adiabatic condition i s not v i o l a t e d . However, i n t h i s one dimensional formulation the adiabatic condition can be v i o l a t e d i n the a x i a l region since the e f f e c t i v e f i e l d goes to zero as the p a r t i c l e crosses the a x i s . Thus, associated with J 3 i s a p r o b a b i l i t y that i t may change i n t h i s region i f the adiabatic condition i s v i o l a t e d . Note 3 : The f a c t o r 3 must be inserted because i n t h i s one dimensional representation the sign of the Stern-Gerlach force depends on the sign of r . - 56 -2cof I ' d « ' — 0 ' (7-7) T < K 1 Note that i n the l i m i t of © -> 0, we obtain the exact t r a j e c t o r i e s for the average Stern-Gerlach force alone: |G | r = r + r z + — \ r z ; <0> = 0 (7-8) o o 4E ^ ' This would apply to a neutral p a r t i c l e i n a magnetic h e l i c a l quadrupole, or to a charged p a r t i c l e when the Lorentz force cancelation i s complete for orthogonal e l e c t r i c and magnetic f i e l d s . A f o c a l length i s defined by; r ( z f ) = r o (7-9) From equation (7-6); 2tor I i - o r << © z f Er (0) ° © tan — = - — n T T - r /jx (7-10) CO Also note from equation (7-6) that a l l p a r t i c l e s are refocused f o r : © z 2o 2m; n = 1, 2, 3 (7-11) Since t h i s f o c a l point i s independent of u^, no Stern-Gerlach e f f e c t - 57 -would be observed. From equation (7-8) the f o c a l length f o r © = 0 i s ; 4Er z f = - u ^ r - o = 0 (y-12) This f o c a l length i s only defined f o r u^ < 0. D i f f e r e n t i a t i n g equation (7-10) y i e l d s the change of f o c a l length (Az^) for a change i n u^; (7-13) Or, expanding about u^ = 0; a) |G |Au Az = ^ f - (7-14) 1 Er O o From equation (7-6) the maximum excursion, r , occurs f o r z = z , n max m where; o , * E © 2u> . -1 , o z = tan ( m © S U 3 l G B l < ( 7 " 1 5 ) From equation (7-6), i f u^ = 0 the f o c a l length i s ; 2lT(±) I I - I - - I I r-o © - 58 -From equation (7-6), i f u^ = 0 the focal length i s ; u3 = ° 2 IT CO o 2uf « 1 (7-16) Then equation (7-14) can be written: Az- i G B l L A U5 Er a 4TT f o o (7-17) Differentiating equation (7-6) yields the change in radius for a change in u : Ar | G R | L AU, r(uv=0) 2TT aEf 3 o tan 4cb (7-18) From equations (7-17) and (7-18) i t can be seen that the relative effect on the trajectories for a change in u^ scales approximately as; IS|L_ aE In Chapter 9 these parameters are discussed from the experimental point of view, considering such things as the power requirements and cooling capcities needed to produce a given value of G D , and the D energy range and collimation range in which a beam might be produced. - 59 -This j u s t i f i e s , to some extent, the values chosen for the trajectories in Fig. 7-1, but, of course, only a successful experiment is the fina l j u s t i f i c a t i o n . Clearly, one desires large [Gg|L, but small aE. Chromatic aberation is present in these systems. Both the para-meters "E" and "a" change i f the energy changes. We have; T2 fZEp (G c + Jm GD) (7-19) 2 ^E ~B 8TT E Differentiating, this becomes; ^ " . ^ [ 3 ^ . ( 1 1 . !] ( 7 . 2 0 ) a E L 1 / . 2 „ d m a J 1 IOTT E ' If the term in the brackets is zero, the chromatic effect of "a" vanishes This is the condition for minimum chromatic aberration in the system. [ 2 ? N m It is useful to solve for the product L J — from equation (7-20) Aa with — = 0, in the following form; cl ^ m 16TT2 (L/a)(G R/E)q ^ ~ 2 l ) This represents the condition for minimum chromatic aberration for given (L/a) and (Gg/E).. Note that equation (7-6) may be written in terms of the ratios i (L/a) and (|GB|/E), i.e.; 7 u 3 r = r Q + (L/ays~- (|G B|/E)[l-cos(a/L)T7z] + (L/a)«ro (7-22) sin(a/L)irz - 60 -Figures 7-1 to 7-4 are plot s of the t r a j e c t o r i e s obtained from equation (7-6) f o r various values of (L/a) and (|Gg|/E). Each fig u r e shows s i x t r a j e c t o r i e s , obtained from the three i n i t i a l conditions • -5 -4 -3 r = 0, r =10 ,10 ,10 , and two values of the magnetic moment o o • u_ = +u , and -u . 3 o o The condition f o r minimum chromatic aberration (equation 7-21) i s i n order, f o r figures 7-1 to 7-4; L j ^ = 23.7; 15.8; 7.9; 1.6. This w i l l i l l u s t r a t e that i t i s very d i f f i c u l t to s a t i s f y the condition f o r minimum chromatic aberration, e s p e c i a l l y i f the product LG B ) i s large, which i s the case when the Stern-Gerlach e f f e c t i s _2 large. For example, i n F i g . 7-1, i f we make L = 2 x 10 m, then a He + beam would require an energy of about.02 eV to s a t i s f y t h i s condition. It i s s t i l l harder to s a t i s f y the condition f o r the other f i g u r e s , although p a r t i c l e s of larger mass than He + would help. As an example of the chromatic e f f e c t i n these examples, consider F i g . 7-2. From equation (7-20) we obtain: , — - 11.5 (7-23) a E K J In t h i s case the dominant chromatic aberration occurs f o r the "a" terms. + -2 For a He beam, "thermal" energies correspond to E -7.4 x 10 eV (300°C); f o r a 2 eV beam t h i s gives; - 61 -Putting t h i s into equation (7-23) we have f or the parameters used i n Fi g . 7-2; — »* 50% a This points out the fac t that the chromatic e f f e c t s due to a thermal spread i n the beam energy w i l l cause large t r a j e c t o r y modifications. In t h i s case they are as large as those caused by changing "a" by 50%. Obviously, the chromatic e f f e c t s must be c a r e f u l l y considered i n a d e t a i l e d design. Some v e l o c i t y s e l e c t i o n might prove u s e f u l , but i t seems more l i k e l y that i t would not be necessary i n many cases since such a large Stern-Gerlach e f f e c t can be obtained. Figures 7-5 and 7-6 i l l u s t r a t e a simple system of stops which could be placed at the ex i t of the Stern-Gerlach p o l a r i z e r region to obtain beams pol a r i z e d i n either of the two senses shown schematically i n the diagrams. Obviously, other systems of stops can be devised, depending on the degree of p o l a r i z a t i o n which i s required. The conditions depicted i n F i g . 7-5, for example, which involve a hollow converging ion beam would give 100% p o l a r i z a t i o n f o r both p o l a r i z a t i o n senses, whereas removing the r e s t r i c t i o n of having a hollow beam produces 100% p o l a r i z a t i o n for one sense of p o l a r i z a t i o n but only p a r t i a l p o l a r i z a t i o n for the other. An inherent advantage which these systems possess i s that the ion beam i s guided by the combined Lorentz and Stern-Gerlach forces at a l l - 62 -times. This means that perturbing forces which are bound to be present are not n e c e s s a r i l y serious, since a small beam displacement from the optic axis w i l l be compensated f o r by an a d d i t i o n a l focusing force. By extending the strong focusing Lorentz force beyond the p o l a r i z i n g region on both ends one can also pre-focus and post-focus the beam. If v e l o c i t y s e l e c t i o n was required t h i s could be accomplished i n a section of the h e l i c a l quadrupole p r i o r to the p o l a r i z i n g region by using the v e l o c i t y dependence of the f o c a l points. In Chapter 9 the experimental problems of achieving a high magnetic f i e l d gradient Gg, and a low beam energy E, are considered. From F i g . 9-3, i f we assume E = 2 eV, i t seems very easy to produce a value Gg/E = 500 gauss/cm/eV f o r apertures of a few mm and step length L = 2 cm. It i s much harder to produce Gg/E = 5000 gauss/cm/eV, but i t should be possible i f the aperture i s as small as possible and a long step length i s used. Pulsed operation i s very sensible for producing high f i e l d gradients since i t lessens the fundamental r e s t r i c t i o n imposed by the l i m i t e d cooling capacity of the system. Any value of L/a may be produced within the range of "a" (0 < a < 1). Of course, i t i s necessary to have very good orthogonality between the e l e c t r i c and magnetic f i e l d s i f very small values of "a" are used. - 63 -Figure 7-1. A plot of trajectories from equation (7-6). The dotted trajectories are those obtained when the adiabatic condition is satisfied as the particle crosses the axis. The solid trajectories are those obtained when the sign of J3 changes as the particle crosses the axis. Figure 7-2. Trajectories similar to those of Fig. 7-1. 1.61 IZ .8 Radius Q-.61 •r s • Optic Avcis ttie+ers 6 / S / / V 1.2 1 .6 . Figure 7-3. Trajectories similar to those of Fig. 7-1. Figure 7-4. Trajectories similar to those of Fig. 7-1. - 66 -Entrance ^rture ^tf/ Rtrtn. collector passes Complete l|j polarcjccl inner beam ^ ^ D i s k collector passes Completely polarised ow+er beam convc/gm^ unpblav-i^ ecf ior\ beam Figure 7-5. An il l u s t r a t i o n of a system of stops which would give 100% polarization of the emerging beam. / . Km/) collector passes partly potanjed inner beam Dr-sk. collector passes Completely polarised outer beam Sold corvuera »nj polarised i o r \ beam Figure 7-6. A somewhat simpler system to that illustrated in Fig. 7-5 which would give partial polarization of the inner beam. CHAPTER 8 The Influence of the Solenoid F i e l d on the T r a j e c t o r i e s , and the Stern-Gerlach E f f e c t f o r Large Lorentz Forces i n a DC H e l i c a l Quadrupole In t h i s chapter we wish to compare the influence of the solenoid f i e l d B q on the t r a j e c t o r i e s to that of the Stern-Gerlach force, to determine, f o r one thing, i f i t i s possible to u t i l i z e the resonant nature of the s i n 9 term i n the average Stern-Gerlach force (Equation 5-13). We also wish to estimate the Stern-Gerlach e f f e c t when the Lorentz force i s large compared with the Stern-Gerlach force i n the dc h e l i c a l quadrupole system. Our conclusions are rather negative. It i s doubtful that the resonant e f f e c t of the s i n 0 term can be detected by simply sweeping the f i e l d B q on and o f f the resonance condition (sin 0 = 1 ) because the t r a j e c t o r i e s are modified too much by the r e s u l t i n g change i n the Lorentz force. It i s also doubtful that the Stern-Gerlach e f f e c t i s usable when large Lorentz forces are present i n the dc h e l i c a l quadrupole. We show i n t h i s chapter that the Stern-Gerlach e f f e c t i s generally small, f o r t h i s case, and that to detect i t would require a p r o h i b i t i v e l y stable and mechanically accurate apparatus, and a very monochromatic beam. In the l i g h t of Chapter 7, i t appears that the key to a successful experiment i s the near c a n c e l l a t i o n of the Lorentz force using orthogonal magnetic and e l e c t r i c f i e l d s . - 68 -For very small values of " r " the Stern-Gerlach force w i l l dominate the motion, since the Lorentz force goes to zero on the optic axis, but i t should be remembered that the Stern-Gerlach force i s at a l l times very small. In equations (8-16), (8-17), and (8-18) we have expressions f or the radius r * which marks the t r a n s i t i o n between the region of space i n which the Lorentz force deominates and that i n which the Stern-Gerlach force dominates. I f the Stern-Gerlach force dominates, the Stern-Gerlach e f f e c t w i l l be large, but as we w i l l show, the radius r * u s u a l l y i s extremely small. Outside of the radius r * , i n the Lorentz force dominant region we have made a simple approximation to the equation of motion for a non-zero magnetic moment to estimate the minimum Stern-Gerlach e f f e c t . Although the average Stern-Gerlach force i s of a very simple form, being r a d i a l and of constant magnitude, a non-linear equation r e s u l t s when i t i s incorporated into the equation of motion. From equation (5-13) we have; < F S G > = ^ * J 3 S i n 9 I S I) 7 (8-1) we define; K = YnJ 3 sine|G B| (8-2) Following the development of the equation of motion i n appendix B, the r e s u l t i n g ; equations when <F c ; r> i s included are - 69 -2 - l A 2 , _ l x - 2Wy - (w + © - -A ) + — V " ) x = 0 ~ i v 4 mv 2 -,o mv r o ?.+ 2Wx - (W2 - © - j(|p>2 + -V }2 = 0 o mv r ( 8 - 3 ) mv r o The presence of the £/r term makes these equations non-linear. The basis of most of this chapter is the simple approximation of replacing 1/r by l/<r>, where <r> is a constant, representing a radius averaged over the length of the trajectory of the particle. This replacement reduces the equations ( 8 - 3 ) to the linear equations ( b r l 3 ) by making the identification: $ = 5/<r> ( 8 - 4 ) If we put <r> = r m a x > this amounts to replacing the Stern-Gerlach force by a force everywhere smaller, and w i l l certianly give a minimum for the Stern-Gerlach effect. We cannot justify this approximation any further. It seems very reasonable, and since the results are not very encouraging we are not inclined to carry the analysis any further at this time, since an experiment based on the alternative presented in Chapter 7 appears to be very promising. Nevertheless, this kind of analysis should be quite helpful in conjunction with computor calculations of the trajectories. We believe that a computor used alone may not give as much insight as approximate solutions such as these. It is helpful to do some preliminary analysis of the equations (b-13) which we w i l l use as the basis for our approximate solutions. - 70 -These include the term $, and can be solved exactly. If both the roots R^' and R^' are r e a l , the t r a j e c t o r i e s are bounded, and the solutions are as given i n equation (b-22) . Note that the term $ can change the f o c a l length, the maximum excursion, or cause a r o t a t i o n about the optic axis. I f one or both of the roots become complex, the t r a j e c t o r i e s are no longer bounded, since divergent terms appear i n the solutions. F i g . 8-1 indicates the regions where the roots R^' and R2' are r e a l or complex. The l i g h t regions indicate that both are r e a l , and correspond to bounded solutions, and the shaded regions indicate that one or both are complex, corresponding to divergent solutions. This f i g u r e i s derived i n the following way. The roots may be written, from appendix B: (8-5) or; W(l - ab - a J 1 - 4b/a ) 1/2 (8-6) where; - 71 -Figure 8-1. A chart of the regions where the roots R 1 1 and R 2' are r e a l or complex. The l i g h t regions i n d i c a t e that both are r e a l , and the shaded regions indicate that one or both are complex. - 72 -b = f / Q (8-8) i qB 2 [*/* - ] ( 8- 9) (G„ + v G c) L V / M 4m The i n s i d e r a d i c a l i s i m a g i n a r y f o r ; o < a < 4b -4b < a < o (8-10) The l i n e , a = 4b (8-11) marks t h i s boundary i n F i g . 8-1. The e n t i r e r o o t may be z e r o . P u t t i n g e q u a t i o n s (8-6) e q u a l t o z e r o y i e l d s ; R 1 I = 0 f o r a = b <; 1/2 a = ITb b * " 1 / 2 R • = 0 f o r a = - ^ r b > 1/2 2 1-b a = ^ b , -1/2 (8-12) (8-13) These l i n e s a r e drawn i n F i g . 8-1. The symmetries i n t h i s system can be u t i l i z e d . They show, f o r example, t h a t t h e l o w e r h a l f o f F i g . 8-1 i s r e d u n d a n t . The t r a n s f o r m a t i o n ; - 73 -y w © "* I X -> -W (8-14) leaves the equations of motion (equation b-13) invarient. This is also true, of course, for the solutions. Since the interchanging of the x and y axis corresponds to a rotation of v/2 about the optic axis, or a translation of L/4 along the optic axis; the transformations W -»» -W and © - © are equivalent to one another, except for a rotation or translation. Also, a helical f i e l d has the property that the transformations V Q - V q and co -to are equivalent, except for possible rotations about, or translations along, the optic axis. Since the roots R-^' and R^' do not change for any of these transformations, the solutions can differ only in the i n i t i a l conditions which they represent. For a beam with cylindrical symmetry no change would be noticed for the above transformations, since a l l i n i t i a l conditions are represented. Thus the lower half of Fig. 8-1 is redundant since i t represents one of the above transformations. Referring to Fig. 8-1, a line defined by "a" = constant crosses three boundaries between the bounded and unbounded solution regions, provided |a| < 2. At each boundary, where a root or radical is zero, we can solve for $ from equation (8T9) , (8-11), (8-12) and - 74 -(8-13). This value of $ defines a radial force which is the transition between a region in which the radial Lorentz force dominates and a region in which the $ force dominates. Since these relations are independent of z, we can equate: $* = £/r* (8-15) where the asterisk denotes that a root or radical is zero. This relation defines a radius, which marks the transition in space between the region in which the Lorentz force dominates and that in which the Stern-Gerlach force dominates. This means that near the zero of the roots, the Stern-Gerlach force can precipitate or delay the onset of the divergent behavior within this radius. Using equation (8-15) to define r*, and solving for §* from equations (8-9), (8-11), (8-12), and (8-13) we obtain; E(a 2W 2 + e V ) mv o for the line a = 4b in Fig. 8-1, and; r * = ^ — (8-17) 1 • I 1 7 2E((l-I)aW2 + l / 4 ( ^ ) 2 ) ) £ o r the line a = in Fig. 8-1, and; r * (8-18) 3 , qB 2E(-(1 + 4- )aW + 1/4 (—) Z) a mv o - 75 f o r the l i n e a = i n F i g . 8-1 The Solenoid F i e l d B o The i n f l u e n c e of the so l e n o i d f i e l d B on the Stern-Gerlach e f f e c t o can be estimated from the equations i n appendix B. B q appears i n the roots of the equation of motion i n the terms, (equations b-14 and b-16); qB W = OJ + Tr-2- (8-19) — 2mv o ^ 0 , 2 1/4 ( — ( 8 - 2 0 ) 2 mv mv o o In terms of the dimensionless parameters "a" and "b" of F i g . 8-1, these are; q(G + v G ) a = ® / W Z = ^ — O 2 1 ) W mv o qB 2 b = f / ® = ( G t Y G ] b / q - - i i - 3 (8-22) E o B C l e a r l y , B q becomes the dominant term i n equation (8-22) when; ^ 2 , , , I > l * / q l (8-23) and, i n equation (8-19), B q i s the dominant term f o r qB 2m^ T I > it ^ o - 76 -The Stern-Gerlach force can be approximated by replacing $ with 5/<r>, where <r> is approximately the aperture size of the apparatus, as outlined at the beginning of this chapter. For a beam of particles described by; He+ ions E = 2eV yftJg = U q (Bohr magneton) (8-25) L = 2 x 10"2 m and putting sine = 1 -4 <r> =10 m equation (8-23) becomes; B q 2 > 9.6 x 10"8|GB| (MKS) (8.26) or, for G D = 100 gauss/cm; D |B | > 3.1 gauss (8.27) and equation (8-24) becomes; | B J > 2600 gauss (8.28) It can be seen from equation (8-27) that rather small values of B w i l l dominate the Stern-Gerlach effect. This seems to preclude o v using a swept B q mode for detecting the effect, as we w i l l further il l u s t r a t e . The reason for including the solenoid f i e l d B q i s , of course, to maximize the Stern-Gerlach effect via the term sin 9 (eqn. 5-14) . To il l u s t r a t e , we can calculate B to maximize sin 6, and also determine - 77 -the resonance width, for the case with; _2 L = 2 x 10 cm G„ = 1000 g/cm E = 2eV He+ = beam ion ( g _ 2 g ) YftJ3 = u Q -4 r = 6 x 10 m max ^ « 2OJV 1 L - o These values could apply, for example, to the trajectories drawn in Figs. 7-1 to 7-3. Putting Aco = 0 to maximize sin 6, yields, from equations (5-7) and (5-9); B Q = -- 278. g^ss (8.30) The half-width at half-maximum of sin 9 i s ; (Aco),/= ±/3to, (8-31) It i s , of course, determined by the maximum radius of the particle -4 trajectory, and for the example cited, T m a x = 6 x 10 m ; we have; B n(r) = G Dr (eqn. 5-6) 1 max B max n J (8.32) = 60 gauss Therefore the half-width i s ; - 78 -(Aw). , ^J-=- ~ 100 gauss (8-33) Comparing equations (8-30) and (8-33), we can conclude that the resonance is essentially centered about B =0, and that this broad ' o resonance would be d i f f i c u l t to distinguish from the trajectory changes produced by the Lorentz forces for changing B , as noted earlier. It is possible to design the apparatus to enhance the resonance effect in order to simplify detection. This could involve chosing an ion with a smaller y, increasing u>, increasing E, decreasing r , or m£ix decreasing Gg. These steps a l l seem to reduce the overall effect, so the compromise which must be made, sacrificing the size of the signal in order to sharpen the resonance, must be decided entirely in terms of signal to noise with whatever detection scheme is proposed. We have treated the case for small © in Chapter 7, now we consider the remaining range of © using the approximation outlined in the introduction to this chapter. The easiest way to do this is to put G„ = B =0. This does not leave out much information for the r E o practical reason that the magnetic f i e l d gradient G alone is sufficient to make © large. The only reason for introducing G„ is to reduce © , to minimize the effects of a large © on the Stern-Gerlach effect. The effect of B q has been considered separately in this chapter by noting when i t becomes the dominant term in the expressions f and W. The simple magnetic helical quadrupole B = - G C = 0 o E - 79 -The term "b" has a simple interpretation for this case. From equation (8-9) we have; b = 0 (8-34) This is just the ratio Of the radial force term ($r) to the Lorentz force (qGgVQr). If $ represents the Stern-Gerlach force, then one would expect "b" to be rather small. If we take, as typical values, the experimental parameters of equations (8-25), and put $ = £/<r>, then; Some simple estimates of the effects on the trajectories due to different values of "b" can be made. Differentiating equation (8-6) we get, for constant "a"; b ~ 6 x 10 -5 (8-35) The region of interest is clearly close to b = 0 in Fig. 8-1. l ' /l-4b/a - a) (8-36) ( /l-4b/a 2 + a) If we consider only changes about b = 0,'and find the relative change by dividing through by the root R ' or R ' we get; - 80 -A R l ' Ab 2-a. , _ n -R7- T {T^> B -0 (8-37) A D ! fl 2 Ab 2+a. _ n = " — b " ° Except for the regions near a = ±1, these terms are very small since Ab is very small. Using the approximation $ = £/<r>, and differentiating equation (8-34) A b = ( Y f t sin 8 N qv <r> ' 3 G. n o B Using the parameter values of equations (8-25), and putting yft-AJ 2u , we obtain: o b % 10 4 (8-39) From equations (8-37) and (8-39) i t is obvious that the trajectory 4 changes that this w i l l cause wi l l be of the order of 1 part in 10 , unless "a" is very near ± 1. 4 If we suppose that the focal point changes by 1 part in 10 due to the Stern-Gerlach effect, then the most obvious d i f f i c u l t y in detecting i t is the chromatic aberration of this system. If we take as a f i r s t approximation, that the focal length varies as (equation 6-10); 0 2Trcomv f © q GB then the velocity of the beam and hence i t s energy must be constant to 4 better than about 1 part in 10 . This result seems typical of operation in the region between very small "a" (a ^ 0) and large "a" (a = ±1). That i s , a small effect is present, but to detect i t requires a nearly monochromatic beam. From equation (8-17), for a - 1/2, yf^J^ = U Q , E = 2eV, B Q = 0, we obtain; r 2 * % 10 m Thus, only within this very small radius would we expect a large Stern-Gerlach effect for the operating region just discussed, which -4 is consistent with our conclusion that at a radius of 10 m, only a very small effect is present. Now consider the regions very near a = ±1. If we put a = ±1, we note that for small "b", the roots given by equation (8-6) can be expanded. Consider the case for a = 1 then; R 1 I = w/b (8-41) R2' * u/2-3b 2 also 6 _ (from equation b-19) 2 © -co (from equation b-15) Put x = y = 0 o J o Then for b > o, the roots are real, and we can simplify equation (b-22) with the above terms: - 82 -• sin yzuz o , A - — x - x " — + — (coswvb: z - cos/2 wz) 0 javT a " ~ - (8-42) x y — (cos/ET (JJZ - cos/2" ojz) + — f/2~ sin/2~ OJZ - S^ N' /^" a " - a /F with 4b << 1 a = 1 B =0 o b > 0 Since /b" <</2, the focal length of these equations is very near; z £ - — 2 - (8-43) If we put $ = 5/<r> to represent the Stern-Gerlach force over this focal distance, we can estimate "b" as in equation (8-35), for "typical" parameter values such as those in equation (8-25) . Taking; -4 b = 10 L = 2 x 10 2 m (8-44) we obtain from equation (8-43) z £ - 1 meter The maximum excursion of the equation (8-42) is determined mainly by the factor; K r — — (8-45) m a x /bco - 83 -I f we again put $ = £/<r> to represent the Stern-Gerlach force over the range of one f o c a l length (z,0, then r w i l l be somewhat 6 - r max greater than <r>. A conservative estimate i s to put r = <r>. Then 6 r max we can say, from equation (8-45), that a l l p a r t i c l e s entering the system within the angle given by: * 0 = < r > ^ (8-46) w i l l be focused within the distance z^ (equation 8-46). Using the values of equation (8-44). y = TT x 10" 4 (8-47) This establishes that a cone of p a r t i c l e s with b >o w i l l be focused within a reasonable distance i n t h i s system. Consider now the p a r t i c l e s with b < o. The root R^' i n equations (8-41) i s imaginary, so that the solutions (8-42) w i l l be diverging. Computing; 2 - 2 2 r = x + y we have from equation (8-42) : . 2 - • . 2 2 f O • 2. 2S in / 2o iZ ^ O ,. ,2 r r — r r 2 r = ( - j — + y Q ) 2 — — 2 ( c o s h i r l b l z + c o s *^k z 0). U) -- 2coshto,/ibT'z cos/2coz) - X Q y o — sin/2o)Z . (coshco/|b |' z o 9 (8-48) 2/2y o 2 ^ *o 2 -cos/2coz)- ? sin/2 coz sinh/|b |coz + —~ s±nyi: /i^ Tcoz 0) / jb j " ~ " - u |'b| X Q y Q 2x y + o sinh ;2 / |b I toz 2 s i n h l / l b I - z cos/fuz — vTb I u / | b |' 84 where /|bj < < ^ b < o a = 1 B =0 o For small z this is oscillatory at / ^ U J Z , but as z gets larger, the oscillating terms become less important and the dominant term becomes; y r 55 1/2 — — sinh /fbT" O J Z (8-49) If we make the conservative estimate that; r = <r> = r aperture and i f we put $ = ?/<r> to represent the Stern-Gerlach force over a distance equivalent to the focal length for particles with b > o; v.jb \£ then equation (8-49) gives a conservative estimate of the minimum angle which a particle may enter the system with in order to be expelled beyond the radius of the aperture in the distance z^. Using the parameter values of the previous example, except that -4 b = 10 , we obtain from equation (8-49) y > 5 x 10"5 J o This is a very conservative estimate, and i t indicates that most -4 particles with b = -10 would be expelled from this system out to -4 the radius of 10 meters within a distance of one meter of f l i g h t . This example has shown that i t might be possible to operate a particle separation from the Stern-Gerlach force. An obvious problem is to maintain the a = 1 condition with sufficient precision. For example, noting in Fig. 8-1, that the slope of the line ' = 0 at a = 1 is 1, we can estimate that "a" must be constant to at least the order of "b", i f the apparatus is to separate the b < o particles in a -4 diverging mode, and focus the b > o particles. If b ^ 10 , then "a" 4 must be constant to about 1 part in 10 or better. We have, from equation (8-21); Thus both the gradient Gg, and the beam velocity V q must be constant 4 to about 1 part in 10 or better. We can arrive at the same conclusions from equation (8-17) which for this case can be written; magnetic helical quadrupole in the a = 1 region and achieve a qG a 2 w mv o * B (8-50) r 2 qv Q(l-l/a) G B Putting a = 1 yields r^* = °°, which confirms the result that a large effect can be expected for the a = 1 region, and indicates that at the point of instability for the Lorentz force alone (a = 1), any - 86 - • small additional radial force is sufficient to dominate the motion, in the sense that i t w i l l precipitate or delay the instability, for any r. Equation (8-50) also yields the same sta b i l i t y criterion, for i f -4 we put = 10 m, the average radius which we have considered, we can -4 solve for "a" to obtain a - 1 + 10 . Thus "a" must be constant to 4 about 1 part in 10 or better, i f the Stern-Gerlach force is to dominate -4 out to the radius of 10 m. CHAPTER 9 Experimental Considerations In this chapter some general remarks are made concerning the design of helical quadrupoles and some justification is given for the choice of the parameter values used in the examples of Chapters 6 to 8. To begin with we w i l l determine the relations between step length, aperture size, gradient and minimum power for a helical quadrupole system. Let R q be the radius of a helix, let L be the step length, and L^ the total length as shown in Fig. 9-1. Then the total length of the helical line is L. ,„; HT' L T J l + q2/4 1 L, HT (9-1) 4TTR o q L (9-2) If d is the width of a helical strip, (Fig. 9-1) then D, the width measured longitudinally i s ; (9-3) If four wires are to be wound then the width of each wire cannot exceed Figure 9-1. Illustration of the dimensions of a helical line and a helical strip. Figure 9-2. Illustration of the dimensions of a rectangular wire helical winding. - 89 -d* r R ° x (9-4) 2 \\ + q2/4 For a fixed amount of power (P), the maximum magnetic f i e l d gradient (Gg) is obtained i f the depth of the wire (B) is 2/3 the inner radius (Rj) of the wire, for fixed wire width (d) (see Fig. 9-2), assuming rect-angular wire is used. To show this we have, from appendix A; I /P G D a — T a ^—^ (9-5) R o where R is the radius to the center of the wire and R is the wire o resistance. Assume d is constant, and R^ is constant, then; G R a — =- (9-6) (R: + B/2r dGB differentiating, we find - r ^ — is a maximum for; ClD B = 2/3Rj (9-7) The minimum resistance occurs when B satisfies equation (9-7) and "d" satisfies the equality in equation (9-4). For a single helical winding of solid, rectangular, copper wire, this resistance is 4(1.7 x 10"8)L (1 + q2/4)A R . = ^ (MKS) (9-8) min _, 2 v ' v 1 TTR o where A is a " f i l l i n g factor" to allow for insulation on the wires. 90-For the case of wires with a f i n i t e width "d", the factor s i n <^ must <P be included in the expression for Gg, from appendix A; G b = C(q) ^ 8 X l 0 2 ~ 7 ^ (MKS) (9-9) ^o where (j> = uod When the space between the wires is very small, is at i t s smallest value which i s ; = .63, $ =TT/2 (D = L/4)1 (9-10) From equations (9-9) and (9-10), for conditions of minimum resistance; 1 2 2 4 (3.94 x 10 )G_ R R . P = B ° M I N (MKS) (9-11) C (q) To i l l u s t r a t e , consider systems similar to those used as examples in Chapters 7 and 8. Typical values are; -2 -2 -2 L = 2 x 10 m, 4 x 10 m, 6 x 10 m Gg = 1000 g/cm L T = 1 m The radius R q may chose to minimize the power, consistent with the aperture required for the beam. Fig. 9-3 is a plot of P . from r mm equation (9-3) versus R q for the values given above. - 91 -|0 I 1 1 1 1 a 3 4 * 6 Ro mm Figure 9-3. Plot of minimum power versus wire radius for a rectangular copper wire helical winding at room temperature, from equation (9-11) . - 92 -Quite clearly, the relation between the aperture size, pitch length, and the power required to produce a given gradient Gg is an important consideration in design. For a given system, the maximum gradient (Gg) w i l l be determined by the cooling capacity. The value chosen for most of the examples in Chapters 7 and 8 was G D = 1000 gauss/ D cm. This is a very conservative value. It can easily be produced as illustrated in Fig. 9-3 for aperture sizes consistent with the beam 4 size. It should be possible to increase Gg to at least 10 gauss/cm in a practical system; which would apply, for example, to the trajectories plotted in Fig. 7-4, for a 2eV beam. Increasing the Stern-Gerlach force (i.e., by increasing GD) has the important additional advantage, besides increasing the Stern-Gerlach effect, that the slow ions w i l l be less subject to stray fields and " d i r t " effects, because of the focusing effect of this force. The Ion Beam From the theory of Chapters 6 through 8 i t is evident that the ion beam is a very important consideration in designing the apparatus. Foremost, i t is necessary to have a beam which is small in cross section and well collimated, secondly, i t should be as intense, pure, and monochromatic as possible, consistent with a very low energy. Obviously, a compromise must be made between these requirements. For the purpose of giving examples in Chapters 7 and 8 we assumed a beam energy of 2eV. With care, i t seems possible to achieve such a beam with the required size and collimation, and an intensity measured at least in tens of particles/sec. It should be kept in mind,however, - 93 -that the Stern-Gerlach effect i s approximately proportional to GD/E D (see, for example, equation 7-23) so that reducing the beam energy is equivalent to increasing the magnetic f i e l d gradient by a like amount. A large literature exists on ion sources, but most of this work has been to develop intense sources of fast ions, and typically the energy spread is of the order of a volt. An intense source of slow Argon ions has been reported by F. Hushfar et a l . (1967). They obtain 14 2 a beam intensity of 10 particles/cm sec at an average energy of 2eV using a plasma source and a novel extraction system which allows both electrons and ions to be simultaneously extracted in a space charge neutralized beam. However, the energy spread appears to be large, about a volt, which is typical of most plasma sources. Two good sources of slow ions adaptable to a wide variety of ions, are photoionization and electron bombardment sources. Of the two, photoionization can produce the most monochromatic beam. Weissler et a l . (1959) using a well defined beam of ultraviolet radiation obtained from a Seya-Namioka grating monochromator could produce ions at 9 eV with an energy spread of .04 eV. However, the yield of ions is much lower than that attainable with electron bombardment. It is useful to give a brief analysis of the electron bombardment source as a source of ions for this experiment. The rate of ion production is given approximately by the formula; R = n a(v )n v x ° n s (9-12) o e e e 3 cm sec where - 9 4 -number density of atoms the velocity dependent ionization cross section in square centimeters 2 the number of electrons/cm /sec If we consider the ionization of He to He+, with a source pressure -4 of 5 x 10 Torr, then; n = o a(v ) ^ eJ n v = e e . 13 3 n = 1.6 x 10 atoms/cm o -17 2 a(v ) = 3.3 x 10 cm 6 (W.E. Lamb, Jr., and M. Skinner, 1950) 16 2 Typical electron densities are from 1 x 10 electrons/cm /sec 16 2 with a "weak" source up to 38 x 10 electrons/cm /sec in a "strong" source such as that of Plumlee (1957). If we assume n v = 5 x 10^^ electrons/cm 2/sec, then e e R = 2.6 x 10 1 3 ions/cm3/sec (9-13) We now omit a l l details of ion extraction and focusing, and simply assume that the ions a l l originate in a small sphere which radiates ions at the rate R into the solid angle 4TT . If we take the diameter of this sphere to be equal to the diameter of the beam, then we would expect a beam of diameter .050 cm, and a divergence f = = _3 10 to have an intensity of; -3 2 2.6 x 10 1 3 x 4/3ir (.025)3 x ( 2 X^°—^— - 540 ions/sec (9-14) This performance was achieved at a beam energy of 4eV in the ion gun described in appendix C. The rate of ion production given in equation (9-13) is typical. Lipeles (1966) reports this value in an ion source similar to that of Novick and Commins (1958). With a beam diameter of about .4 cm and a divergence of * q = 0.6, he obtains a beam intensity of 3 x 10^ ions/sec. at a beam energy of 10 eV. He also measured the energy spread to be almost .4 eV or 4% for his beam. His beam intensity, compared with that of equation (9-14) illustrates the very large decrease in total particles/sec which one must expect when the same source is used to produce a smaller diameter and better collimated beam. A very serious problem with slow ion beams is the rapid f a l l in intensity below an energy of some 10 eV. For example, the ion gun described in appendix C would produce a beam about two orders of magnitude more intense at 10 eV than at 4 eV beam energy, as illustrated in Fig. C-2. Much of this is due to the collimating effect of the lens system, which increases the effective solid angle at the ion source at higher beam energies. However, at a beam energy of 3 eV, the ion count was equal to the metastable background count, some two orders of magnitude below the intensity at 4 eV. This very rapid f a l l (beam cut off) is typical of these sources and is apparently due to stray fields. Our own experience was that the cleanliness of the aperture next to the ionizer exit aperture had a pronounced effect on the beam cut-off. This aperture is presumably intercepting a large flux of electrons and ions from the ionizer and could conceivably contaminate very quickly. We have tentatively concluded that an ion beam at 2 eV - 96 -with an intensity of some tens of particles/sec can be produced with an electron bombardment source in a clean vacuum system; and with the required size and collimation for the examples given in Chapter 7. However, we believe that an apparatus could be designed to polarize a much larger and more intense beam, but this would require a major effort of engineering. 9 CHAPTER 10 Concluding Remarks This thesis has added essentially two new ingredients for consideration to the conditions for which a charged particle generalized Stern-Gerlach experiment can be successfully performed, from those previously reported (Bloom and Erdman 1962; Rastall 1962; Byrne 1963; Bloom, Enga and Lew 1967). These new considerations are the extension of the Transverse Stern-Gerlach experiment to time independent, space varying inhomogeneous magnetic fields given by equation (1-2), and the introduction of an electric f i e l d E(r) which is orthogonal to the magnetic f i e l d B^(f) (equation 1-2) in the plane transverse to the general particle motion. The development of these new considerations in Chapters 5 through 9 indicates that the experiment is considerably simplified from that previously envisaged, and that no fundamental di f f i c u l t y remains to the successful execution of a charged particle Stern-Gerlach experiment. The most f r u i t f u l application of such an experiment would probably be the precision measurement of the low-lying energy levels of a wide variety of ions and molecular ions, which are of great interest in chemistry and astrophysics. Very few methods have proven useful for such measurements. We believe that the generalized Stern-Gerlach - 98 -experiment w i l l now allow the study of such ions in the same general way that the Stern-Gerlach experiment has been applied to the study of atoms and molecules. Another application is the construction of a polarized ion source for use in nuclear physics. Such sources as are presently in use ionize after atomic state selection, whereas this experiment makes possible the state selection after ionization. This difference may prove to be important in future designs. A short review of polarized ion sources now in use is given by Drake (1967). - 99 -BIBLIOGRAPHY Abragam, A. 1961. Principles of Nuclear Magnetism (Oxford University-Press, London). Bas, Von E.B. and Gaug, H. 1968. Z. Angew, Math Phys. (Switzerland), Vol. 18, No. 4, 557. Bennewitz, H.G. and Paul, W. 1954. Z. Physik 139, 489. Bloom, M. and Erdman, K.L. 1962. Can. J. Phys. 40, 179. Bloom, M., Enga, E. and Lew, H. 1967. Can. J. Phys. 45_, 1481. Byrne, J. 1963. Can. J. Phys. 41_, 1571. Drake, Jr., C.W. 1967. Atomic and Electron Physics, Vol. 4 pp. 226-257 (Academic Press, New York and London). Greene, E.F. 1961. Rev. Sci. Instr. 32_, 860. Hanszen, K.-J. and Lauer, R. 1967. Focusing of Charged Particles, Vol. I, pp. 251-307, (Academic Press, New York and London). Hushfar, F., Rogers, J.W. and Webb, D. 1967. Rarefied Gas Dynamics Fifth Symposium, Vol. II, pp. 1427-1442 (Academic Press, New York). Kusch, P. and Hughes, V.W. 1959. Handbuch der Physik, Vol. 37 (Springer Verlag), pp. 1-172. Lamb, Jr. W.E. and Skinner, M. 1950. Phys. Rev. 78_, 539. Le Couteur, K.J. 1967. Plasma Physics, Vol. 9, 457. Lipeles, M. 1966. Thesis, Columbia University (unpublished). Lippert, W. and Pohlit, W. 1952. Optik 9_, 456. Lippert, W. and Pohlit, W. 1953. Optik 10_, 447. Novick, R. and Commins, E.D. 1958. Phys. Rev. I l l , 822. Pearce, R.M. 1969. University of Victoria, Private communication. - 100 -Pierce, J.R. 1954. Theory and Design of Electron Beams, second edition. (De. Van Nostrand and Co., New Jersey). Plumlee, R.H. 1957. Rev. Sci. Instr. 28, 830. Ramsey, N.F. 1963. Molecular Beams (Oxford University Press, reprinted from f i r s t edition, 1955) . Rastall, P. 1962. Can. J. Phys. 40, 1271. Salardi, G., Zanazzi, E. and Uccelli, F. 1968. Nuclear Instr. and Methods, 5£, 152. Strashkevich, A.M. and Trotsyuk, N.I. 1968. Soviet Physics - Technical Physics, Vol. 13, no. 3, 384. Teng, L.C. 1959. Helical Quadrupole Magnetic Focusing Systems, ANAL-55 February. Trotsyuk, N.J. 1969. Soviet Physics - Technical Physics, Vol. 13, No. 7, 944'. Watson, G.N. 1962. Theory of Bessel Functions, Second Ed. (Cambridge University Press). Weissler, G.L., Samson, J.A.R., Ogawa, M., and Cook, G.R. 1959. J. Opt. Soc. Am. 49_, 338. Winter, J. 1955. Compt. Rend. Acad. Sci. 241, 375. Zankel, K. 1968. Nuclear Instr. and Methods 65, 322. - 101 -APPENDIX A The Helical Quadrupole Magnetic Field The magnetic f i e l d of a quadrupole wire system near the center of symmetry is well approximated by; B^f) = -G(y + ix) (a-1) at the position r = x + iy, where; G = 0.8 -Kj gauss/cm (a-2) R o with, I = current in each wire in amperes, R q = distance in centimeters from the center of symmetry to the center of each wire. The orientation of this f i e l d is shown in Fig. A-1. This f i e l d is similar to the f i e l d illustrated in Fig. 2-3, except that the orientation of the axes x,y with respect to the quadrupole wires is different. If this system is twisted to form a helical quadrupole, as illustrated in Fig. 9-1, the f i e l d w i l l rotate with z at twice the rate which the wires rotate with z, i.e., the f i e l d is represented in the axial region by a f i e l d of the form; B ^ z ) = -GB(y + i x ) e l 2 - z (a-3) G B = C(q)G (a-4) 2TT (a-5) - 102 -Figure A - l . Illustration of the orientation of the f i e l d = -G(y + ix) at the point r = x + iy. - 103 -L = helical step length (a-6) 4?rR The factor C(q) can be calculated from the paper of Le Couteur (1967), who solves for the general helical multipole case and the helical quadrupole in particular. He also considers the higher order terms 3 0(r ), which are lef t out in the above relation. We give; C(q) = l/2q 2K 2(q) + l/4q 3K 1(q) (a-8) where K is a modified Bessel function of the seconddkind. n In order to make the identification with Le Couteur we give the relation (Watson 1962, Eqn. 3, pg. 79) zK '(z) =-vK (z) - zK ,(z) (a-9) V V v-1 The factor C(q) is plotted in Fig. A-2. Equation (a-3) may be written with a homogeneous solenoid f i e l d (B ) superimposed along the z axis. Bf r,z) = B x i + + B_H (a-lOa) B = G D(x sin 2tdz - y cos 2uz) (a-lOb) X D — B^ = -Gg(y sin 2_z + x cos 2wz) (a-lOc) B z = B Q (a-lOd) - 104 -1.0 .9 .8 . 7 .5 .4-3 r Z .1 0 2 3 Figure A-2. Plot of the factor C(q) defined by equation (a - 105 -APPENDIX B The Derivation of the Trajectories of an Ion in a Helical Quadrupole Field Consisting of Orthogonal Electric and Magnetic Fields Superimposed on a Homogeneous Field and on. which an Additional Radial Force is Introduced From appendix A we have the magnetic f i e l d of a helical quadrupole; B(f,z) = B x i + B yj + Bzk (b-la) B^ = Gg(x sin 2ojz - y cos 2_z) (b-lb) By = -Gg(y sin 2u>z + x cos 2wz) (b-lc) B = B z o where; 2TT _= — L = helical step length G D = magnetic gradient D The corresponding electric helical quadrupole f i e l d which is orthogonal to B"(r,z) in the x,y plane can be found by putting; thus; E(r) . B(r) =0 with B = 0 (b-2) z E(r) = E I + E J + E 1 < (b-3a) E = G p(y sin 2oiz + x cos 2wz) (b-3b) E = G (x sin 2wz - y cos 2uz) (b-3c) y 106 E = 0 (b-3d) z where; G„ = electric gradient b It is very useful to include in these solutions a radial force term F which we define as: r F = $r (b-4) r $ is a constant, and is left arbitrary in this derivation. It is discussed in Chapter 8 , where this additional force term is used. The equation of motion which we wish to integrate i s ; j 2— _ F ^ = q/m(v x B(f) + E(f)) + -± (b-5) dt This may be written as; A2 a x dt TT = q/m(E + v B - v B ) + - x (b-6a) 2 ^ ^ x y z z y m ,2 =-%r = q/m(E + v B - v B ) + - x (b-6b) 2 n v y z x x z m dt J d 2z Q | = q/m(vxB - v Bx) (b-6c) dt We make the restrictions that; - 107 -then; where; dz v = - j — = v <te constant (b-7b) z dt o dx dx dz » 0 . v = - n r = j — J T - = V X (b-8a) x dt dz dt o K J v = % = v y (b-8b) y dt o7 v x = (b-8c) y - % Cb-M) i t follows that; d x 2 v x (b-8e) dt 2 A- = v 2y (b-8f) dt ° ,2 -^4 = 0 (b-8g) dt With these restrictions equation (b-6c) drops out. Substitution of equation (b-8) into (b-6) yields; x = -X-TT (E + v B - v B ) + - ^ 5 - x (b-9a) 2 v x y o o y J 2 v J mv J J mv o o y = q 0 (E + v B - v B ) + =• y (b-9b) y 2 % y o x x o 2 1 K J mv y mv o o - 108 -We wish to transform these equations to a set of axes (x,y) which rotate at the ratewz, i.e., x = x cos wz + y sin wz (b-lOa) y = y cos wz - x sin wz (b-lOb) differentiating with respect to z, we obtain; x = x cos wz + y sin wz + wy (b-lla) y = y cos wz - x sin wz - wx (b-llb) and differentiating again; x - 2wy - w2x = x cos wz + y sin wz (b-12a) § 2 y + 2wx - w y = y cos wz - x sin wz (b-12b) Substituting for x and y from equation (b-9) , equation (b-12) can be expressed after some manipulaton in the following form; x - 2Wy - (W2 + © + f)x = 0 (b-13a) y + 2Wx - (W2 - © + f)y = 0 (b-13b) where qB ; o mv o 109 •fj- - l/4(—y (b-16) 2 vmv ' v J mv o o Equation (b-13) can be combined into a single fourth order, linear, homogeneous equation; T + 2(W2 - f)x* + (W2 + © + f) (W2 - © + f)x = 0 (b-17) Rz Solutions of the form e are assumed. Four roots R are obtained; R: = (-W2 + f +J© 2 - 4W 2f) 1 / 2 = (-W2 + f + 6 ) 1 / 2 (b-18a) R2 = (-W2 + f - 6 ) 1 / 2 (b-18b) R3 = -R1 (b-18c) R4 = -R2 (b-18d) where; 6 = J®2 - 4W2f (b-19) The solutions can be written as; x = C ^ l 2 + C 2e R2 Z + C 3 e " R l Z + C 4e" R2 Z (b-20a) W2 + © + f - R l 2 W2 + © + f - R 2 2 Z -2WR1 C l 6 1 + -2WR2 C 2 6 2 W2 + 0+£-R 2 W2 + © +f-R 2 : i_ c e l : — C e 2 -2WR, 3 -2WR„ 4^ (b-20b) - 110 -The constants, C, can be obtained in terms of the i n i t i a l conditions Denoting values at z = 0 by subscript o, we obtain the following from equations (b-10) and (b-11); x = x -o o y = y -o J o (b-21) x = x + oiy J -o o ~J o y = y - O J X Lo 1o - o The equations(b-20), together with their f i r s t derivatives, evaluated at z = 0 using the values from equations (b-21), give four equations which can be solved for the four unknown C's. For the case where the four roots are a l l pure imaginary so that they may be written in the form R = iR', where the prime denotes a real number, then the solutions to equation (b-17) can be written as follows, where the constants have been solved for in terms of the i n i t i a l conditions given by equation (b-21) . x 01.72 y sinR 1 'z x = —[(cosR 1'z+cosR 2'z)+ = (cosR-j^ ,z-cosR 2'z)]+ 2— [ 0 ) ( ^—; sinR 'z „, .,,2 A Atx sinR 'z sinR 'z x sinR 'z . 2 2W(o)W-W + © - f j - a © , 2 1 _ Q r 1 R 2 ' J L 6 R2' R l ' 2 R l ' sinR„'z „„,2 ^ sinR_'z sinR 'z • Wr n , n , , 2 2W-© f 2 1 )]+ y T-[COSR1 'z-cosR9'z] R2* 6 1 R2' " R y 0 0 1 Z (b-22a) - I l l -sinR 'z sinR 'z 2 ^ „, ~. sinR 'z sinR.'z - LaC R I < + R 2 - ) n 5 R^ [cosR^ ' Z+COSR2' z + ~ — ®-(cosR^ ' Z-COS.R2 ' z) ] W •^ •[cosR^ ' z-cosR2' z] sinR ' z sinR 'z 0 2 .-^sinR 'z sinR 'z P 1 z zW + z 1 L R~' + ~~RT5 + 6 1 RT1 RT1 J J (b-22b) - 112 -APPENDIX C An Ion Gun Design Suitable for Producing Small Diameter, Well-Collimated, Ion Beams The ion gun described in this appendix was developed specifically to produce small diameter, well-collimated ion beams using an electron bombardment ion source. The major inovation from standard designs such as given by Novick and Commins (1958) or Plumlee (1957) is the incorpora-tion of the ring focus electron gun, which is well suited to ionizing small diameter beams because of i t s geometry. A diagram of the ion gun is given in Fig. C-l. The ring focus electron gun has been used by others for various purposes but i t has not, to our knowledge, been incorporated into an ion gun suitable for adaption to this experiment. Bas and Gauz (1968) have analyzed i t s design in some detail. Zankel (1968) has calculated electrode shapes to maintain a coaxial current flow by compensating for end effects, although our design does not include this compensation. The advantages of the ring focus design are threefold. It is a very compact, simple structure, with a cylindrical geometry which matches the beam geometry and reduces the various f i e l d asymmetries to a minimum. It u t i l i z e s an electrostatic focusing system which is self-focusing at a l l bombardment energies. Electrostatic focusing in the electron gun has the advantage that i t does not affect the ion beam as would a magnetic focusing system. Finally, the ring focus design produces a very intense electron beam in the axial region where the ionizing events are most useful. - 113 -The major problem with the ring focus design is the magnetic f i e l d produced by the filament heater current which deflects the electrons so that they do not enter the axial region where the ionization is to occur. For our purpose this problem was overcome by using "on-off" heater current with a filament having a large thermal mass. The useful beam is produced during the "off" period, which may be a large fraction of each cycle. The detector may be gated in synchronism with the heater current to increase the signal to noise. This "on-off" cycle has the additional advantage that the magnetic f i e l d from the filament current lead-in wires does not affect the ion beam, so that these wires may be placed conven-iently to simplify construction. The lens system consists of a Pierce lens (Pierce 1954) to i n i t i a l l y collect and focus the ions, three single aperture lens, and a three element unipotential lens (Lippert and Pohlit, 1952 and 1953), followed by a collimating aperture. The Pierce lens was chosen because i t s focal length and aberation constants may be closely calculated. The Pierce lens, together with the unipotential lens, may be designed to correct to f i r s t order for chromatic aberration, and partly correct for spherical aberration. They may also be designed as a crude monochromator by making the system very chromatic. Referring to Fig. C-l for an explanation of the symbols, we give the following design formulae; The ion gun w i l l focus the beam at in f i n i t y i f ; - 114 -where; [SCg-i,) + 3b1 + 4h] (c-2) 4[h + b' + g - i x ] b ' f S C g - i ^ + 4h] B = " 16h[h + b + g - i ] ( C _ 3 ) 4 d l 1 ' t >n x i = I F-d- ( c - 4 ) The variation of the focal point (z £) of the unipotential lens as a function of the spread in beam energy (AU q) is given by; (Hanszen and Lauer 1967) AU A z f = - c c h ^ -T- (c"5) where C^^00) is the chromatic aberration constant of the unipotential lens. For achromatic operation of the ion gun we put; Az f = - A i 3 (c-6) where Ai^ is the variation in the image point of the lens preceding the unipotential lens as a function of the spread in beam energy. We give the formula; .2 .2 Ai = (b')' -^2. [ 1 ( ± n ^ T f " ) + J-1 Cc-7) 3 V ( h - i 2 ) 2 ( l ^ f ^ g - i ^ 2 >1+S)± 2 ± 3 where; f 2 - 4h IXzVM ( c . 9 ) f3 = " f- (C"10) 7~ ( c- u ) ^ ( g - i j ) 2 • F W The minimum chromatic aberration constant for a unipotential lens i s ; C c h O ) = -3 I (c-13) The condition for achromatic operation, equation (c-6), and equation (c-7), yield for this particular case the unipotential lens length (£); ^ 2 ^ ) CC-14) (h-i 2) z (1+S) ' f jCg-ijr ( l * S ) f 2 r3 In the normal operating range (S > 0) we have f 2 , f^ < 0, while f^< 0 for a l l S. Such values yield I > 0 which indicates that achromat operation is possible in the normal operating range i f the gun dimensions are chosen to give a reasonable value of "I". The ion gun which we have built has the following nominal dimensions in inches, - 116 -V = 0.20 d 1 = 0.25 g = 0.075 h = 0.160 b' = .158 d = .020 o z = 1.7 c The unipotential lens has the nominal dimensions; a = 0.176 b = 0.176 d = 0.088 I = 0.44 for which; z f = 0.145 C c h « From equation (c-1), this ion gun should focus at in f i n i t y for; V = -3V or V = (l/2)v In practice, i t is very d i f f i c u l t to build the ion gun accurately enough to expect precise agreement between experiment and theory. For our gun we obtained the best results by operating the unipotential lens " - 117 -somewhat below (V), with V ~ -2V. Note that the condition V = (1/2^ / is an unfavourable one because the positive ions experience only a very weakly accelerating electric f i e l d on exit from the ionizer. Fig. C-2 gives typical results for a He+ beam with a pressure in -4 the ionizer of approximately 5 x 10 Torr (He). These results were obtained by directing the beam through a field-free region 50 cm long between the ion gun exit aperture and four knife-edge jaws which could be independently moved to explore the beam intensity. The ions were collected by an electron multiplier (Bendix, model 306) . The -7 pressure in the f i e l d free region was 5 x 10 Torr. The ion gun was constructed of Type 304 stainless steel with alumina insulators. - 118 -Electrode s«. (pafetftw) Ft lament Lomjer chamber (~l00v) (v) Pusher \ Filament focys / Spherical Anode (~V) \ (~IOOv) / / ( V ) Unipotential Let ) t Uhipo+eKtial Lew Untpofervfial L«r\s focal point Wid plane Figure C - l . Diagram of the ion gun described in appendix C. - 119 -Average Number of Ions per sec. (H*+) £ - 7 * 1 0 * A t Beam dioenjence at hal-f wave, intensity Beam dia*we"ter at Source » .050 cm 3 4- £ 6 7 9 V c : Ion Beam £ner<j\j ( e V ) Figure C-2. Typical performance of the ion gun described in appendix C. V is the potential of the ionizing chamber.
Thesis/Dissertation
10.14288/1.0084805
eng
Physics
Vancouver : University of British Columbia Library
University of British Columbia
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Graduate
Stern-Gerlach experiment
Potassium
Observation of the transverse Stern-Gerlach effect in neutral potassium and an analysis of a charged particle Stern-Gerlach experiment
Text
http://hdl.handle.net/2429/34896