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Atomic beam polarized 3He+ ion source Vyse, Robert Norman 1970

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AN ATOMIC BEAM POLARIZED. 3He + ION SOURCE by ROBERT NORMAN VYSE B.A.Sc .,• U n i v e r s i t y of B r i t i s h Columbia, 1965 M.AcSc*, U n i v e r s i t y of B r i t i s h Columbia, 19&7 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 t h e s i s as conforming to the re q u i r e d standard , THE UNIVERSITY OF BRITISH COLUMBIA January, 1970 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an advanced degree a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r ag ree t h a p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y pu rposes may be g r a n t e d by t h e Head o f my Depar tment o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l no t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Depa r t m e n t The U n i v e r s i t y o f B r i t i s h Co lumbia Vancouver 8, Canada i ABSTRACT AN ATOMIC BEAM POLARIZED 3 H e + ION SOURCE A beam of polarized ions has been produced using atomic beam method techniques. This method has the a t t r a c t i o n of being capable of producing an ion beam with p o l a r i z a t i o n s up to 1 0 0 $• The p o l a r i z a t i o n of 3 g e beams presently produced by o p t i c a l pumping techniques i s of the order of 5#« The apparatus i s composed of three main sections, the atomic beam source consisting of a supersonic nozzle cooled to l i q u i d helium temperatures to produce a low v e l o c i t y atomic beam, the tapered hexapole magnet to s p a t i a l l y separate the p a r t i c l e s i n the two magnetic spin substates, and the electron bombardment i o n i z e r to produce $Ee+ ions from the neutral 3 g Q atomic beam. The low v e l o c i t y beam i s required because the nuclear magnetic moment of $Ee i s of the order of 1000 times smaller than the el e c t r o n i c magnetic moment used to separate beams i n conventional Stern-Gerlach magnets and to achieve a high i o n i z a t i o n e f f i c i e n c y . The measured i n t e n s i t y of the beam produced by the atomic beam source cooled to l i q u i d helium temperature was 1 x 1 0 x o atoms/sr-sec, the most probable v e l o c i t y was 3 1 0 m/sec, and the v e l o c i t y f u l l width at half maximum was 5"0 m/sec. The beam f l u x through the i o n i z e r increases by a factor of 1 . 3 when the hexapole f i e l d i s turned on, i n good agreement with the t h e o r e t i c a l l y expected increase. i i This increase corresponds to a p o l a r i z a t i o n of 65% of the atomic beam. A 12nA3jj e+ i o n beam was obtained corresponding to an i o n i z a t i o n e f f i c i e n c y of approximately 0»15%* i i i CHAPTER I CHAPTER II CHAPTER III CHAPTER IV TABLE OF CONTENTS Page INTRODUCTION 1 METHODS FOR PRODUCING POLARIZED 3rfe BEAMS 7 A. The Optical Pumping Method 7 B, The Atomic Beam Method 9 - PRODUCTION OF MOLECULAR BEAMS ^ A. Motivation for Development of Molecular Beams. B. Properties of the Molecular Flow Beam l l f C. Properties of the Nozzle Beam 1 5 ( 1 ) Gas Flow Through the Nozzle 1 7 ( 2 ) The Free J e t Expansion .... 18 ( 3 ) The Freezing Surface 2 3 (k) V e l o c i t y D i s t r i b u t i o n of P a r t i c l e s i n the Beam 2 6 ( 5 ) Intensity Available from Nozzle Beams 3 0 (i) Freely Expanding Jet Intensity.. 3 0 (ii) Beam Intensity Downstream from the Skimmer. 3 0 ( 6 ) Deviations from Ideal Behaviour 3 1 ( 7 ) Low Temperature Nozzle Sources and their Uses 33 - THE POLARIZED $ E e + BEAM SOURCE...... 3 7 A. The Low Temperature Atomic Beam Source.... 3 7 ( 1 ) General Description .. 3 7 i v Page (2) Thermal Transpiration Corrections . to Pressure Measurements 4 - 2 (3) Carbon Resistor Temperature , B. The Hexapole Magnet . C. The Elec t r o n Bombardment Ionizer 52 CHAPTER V - TECHNIQUES FOR MEASUREMENT OF ATOMIC BEAM INTENSITY AND VELOCITY 58 A. Measurement of the Atomic Beam Intensity.. 58 B. The Time-of-Flight Measuring Apparatus.... 59 CHAPTER VI - RESULTS OF STUDIES OF THE ATOMIC BEAM 66 A. The Atomic Beam Intensity. 66 B. The Atomic Beam Vel o c i t y 82 CHAPTER VII - POLARIZATION AND IONIZATION OF THE 3Re BEAM 8 8 A. The Trajectories of Atoms through the Hexapole Magnet 88 B. The Calculated P o l a r i z a t i o n of the Atomic Beam 90 C. The P o l a r i z a t i o n Measurement and Ion Beam Y i e l d 95 CHAPTER VIII -' CONCLUSIONS 99 A. Comparison with Other Sources of Polarized 3 H e + Ions 99 B. Measurement of the 3He+ Beam by the D(3He,P)l+He Reaction 1 0 1 C. Possible Improvements of the Polarized 3fle + Beam Apparatus 10h V Page (1) Improvements Increasing the Atomic Beam Intensity... 10M-(2) Improvements Reducing the Atomic Beam Ve l o c i t y 105" (3) Improvements i n Ionization E f f i c i e n c y and Extraction 106 (k) Overall System Improvement P o s s i b i l i t i e s by Changing Geometry... 1 0 7 (5) Improvements i n Vacuum System Reducing Background Ion Y i e l d 1 ° 9 v i APPENDIX A, APPENDIX B, APPENDIX C, APPENDIX D, APPENDIX E, APPENDIX F. INTENSITY FROM A FREELY EXPANDING JET.... Page 111 INTENSITY AND VELOCITY DISTRIBUTION OF PARTICLES IN THE JET AFTER PASSING THROUGH A SKIMMING ORIFICE , 113.. TRAJECTORIES OF PARTICLES PASSING THROUGH A HEXAPOLE MAGNET 120 1 . Equation of Motion of a Magnetic Dipole i n a A x i a l l y Symmetric Multipole F i e l d . . . . I 2 0 2. Trajectories of a Magnetic Dipole i n a , P a r a l l e l Hexapole Magnet l 2 ^ " 3 . Trajectories of a Magnetic Dipole i n a Tapered Hexapole Magnet 126 CALCULATION OF THE SIGNAL SHAPE FROM THE TIME-OF-FLIGHT APPARATUS 1 3 2 A LOW TEMPERATURE NOZZLE BEAM FOR A POLARIZED 3 H e + I O N SOURCE by R. Vyse, J.C. Heggie and M.K. Craddock reprinted from 6 t h I n t ' l Symposium on Rarefied Gas Dynamics. 2, Academic Press Inc. New York (1969) < 138 LOW TEMPERATURE ATOMIC 3 He BEAM FOR USE IN A POLARIZED 3fte+ ION SOURCE by ' R. Vyse, D. Axen and M.K. Craddock -Rev. S c i . I n s t r . In press BIBLIOGRAPHY 1*3 v i i LIST OF FIGURES Page 1 . Energy Levels of 3 n e Atoms i n an External Magnetic F i e l d (not to scale) 8 2 . General Arrangement of the Components of the Polarized ^He Apparatus showing the d i f f e r e n t i a l pumping required to handle the JRe gas flow 10 3« Nuclear P o l a r i z a t i o n of Singly Ionized 3f fe Atoms for Equal F i e l d s i n the Ionizing and Target Regions 1 2 h. Schematic Representation of Nozzle Beam , Source 5 . Discharge C o e f f i c i e n t vs. Reynolds Number based on Experimental Flow and V i s c o s i t y at Stagnation Conditions 19 6 . . Schematic Representation of Flow from an O r i f i c e into an Evacuated Region 2 0 7 . D i s t r i b u t i o n of Mach Number Along the Axis of Symmetry of the Expanding Je t 2 1-8 . Terminal Mach Numbers as a Function of Inverse Knudsen Number based on Stagnation Conditions at the Nozzle 2 5 9 . Schematic Representation of R a d i a l l y Expanding Flow through a Skimmer to a Detector. 2 7 1 0 . The Low Temperature 3 H e Atomic Beam Source... 3 8 1 1 . The Adjustable Nozzle-Skimmer Assembly ^ 3 1 2 . Conditions for Thermal Transpiration E f f e c t . . kh 1 3 . Thermal Transpiration Corrections to Measure-ments of Nozzle-Skimmer and Skimmer-Collimator Pressure "+6 1*+. A.C. Bridge Used to Monitor Resistance of Carbon Resis tor . Thermometer ^ 9 Input Power Induced Heating of Carbon Resistor Thermometer with Resistor at Near C a l i b r a t i o n of Carbon Resistor Thermometer... Dimensions of the Components of the Hexapole Magnet Magnetic F i e l d Strength i n the Region of the Pole Tips as a Function of the E l e c t r i c a l Current Through the Co i l s Measured Value of the Magnetic F i e l d Strength as a Function of the Radial Distance from Central Axis Schematic Representation of Ionizer and Ion Measurement Apparatus (A) D.C. Current Measurement (B) Chopped Current Measurement Two S l i t Chopper.. Ionizer Ion Y i e l d as a Function of Background Hydrogen Gas Pressure Schematic Representation of Time-of-Flight V e l o c i t y Measurement Equipment Phototransistor Reference Signal C i r c u i t . . . . . Schematic of Ion Gauge Signal C i r c u i t , Reference Signal and O s c i l l i s c o p e Display.... Broadening of Experimental Signal due to F i n i t e Width Shutter Function. Nozzle Beam Curve Includes Correction for a 2.5 cm Detector Length Assumed Intensity P r o f i l e of P a r t i c l e s Passing through the Chopper Opening (Shutter Function). Tangential Chopper V e l o c i t y = 35 m/sec Room Temperature Beam Intensity P r o f i l e s for a 0 . 2 cm Diameter Nozzle Room Temperature Beam Intensity P r o f i l e s as a Function of Nozzle-Skimmer Separation i x Page 30. Continuous Beam P r o f i l e Taken with Adjustable Nozzle-Skimmer Assembly 70 31. Beam Intensity P r o f i l e s for a 0.2 cm Diameter - Nozzle Operated at 77°K 72 32. Liquid Nitrogen Temperature Beam Intensity P r o f i l e s as a Function of Nozzle-Skimmer Separation 73 33- Beam Intensity P r o f i l e s as a Function of Nozzle-Skimmer Separation with the Nozzle . at Liquid Helium Temperature... 75 3*+. Beam Intensity P r o f i l e s for 0.2 cm Diameter Nozzle Operated at Liquid Helium Temperature. 37. The Uncorrected Data of F i g . 36 Divided by the Nozzle Pressure to V e r i f y Existence of Scattering 76 35. ^ e Beam In t e n s i t i e s as a Function of Nozzle Stagnation Pressure Po for Three Temperatures To. The dashed l i n e s passing through the o r i g i n are f i t s to the data a f t e r correction for scattering 78 36. Beam In t e n s i t i e s for 3f}e and ^ He Beams at Liquid Helium Temperature Before and After Correction for Scattering. Only the uncorrected experimental points are shown when the cor r e c t i o n i s small 79 80 38. Normalized Intensity P r o f i l e vs. V e r t i c a l Displacement from Beam Axis. Stagnation Pressure Po = k20 Torr Stagnation Temperature.To = 77°K 83 39. Typical ^Ee Time-of-Flight Spectrum for Liquid Helium Cooled Nozzle. Stagnation Pressure i s 36 Torr. Horizontal Time Scale i s 0 .5 m sec/div. The Upper Trace shows the Time Reference Light Pulse... 85 kO. Results of F i g . 39 Converted into a V e l o c i t y Spectrum. The curve shown i s a f i t of Eq. 17 to the experimental spectrum 86 Typical Trajectories of Focussed and Defocussed ^He Atoms passing through the Hexapole Magnet. The Source-Magnet Separation i s 15 cm and the F i e l d at the Magnet Pole Tips i s taken as 9000 Gauss. E f f e c t of Source-Magnet Separation on Intensity and P o l a r i z a t i o n of Atomic Beam. Nozzle at l+.2°K E f f e c t of Source-Magnet Separation on Intensity and P o l a r i z a t i o n of Atomic Beam. Nozzle at 7°K Schematic Diagram Indicating Relative •Location of Components Used i n Atomic Beam P o l a r i z a t i o n Measurement. Dimensions i n cm.. Change i n D i f f e r e n t i a l P i r a n i Detector Signal when Hexapole Magnet i s Turned On and Off Typical Ionizer Signals from Chopped Atomic Beam with Hexapole Magnet Turned On and Off. Horizontal Scale 0.5 ms/div. V e r t i c a l Scale 0.5 mv/div Enhancement Ratio of Ionized Beam as a Function of Magnet E x c i t a t i o n Current. Defining Diagram for Calcul a t i o n of Flow through Nozzle-Skimmer System D e f i n i t i o n of Certain Variables used i n Ca l c u l a t i o n of Flow through Nozzle-Skimmer Sys tern Schematic Diagram Showing Two Poles of a Ra d i a l l y Symmetric Magnet.. Schematic Diagram Showing the,Parameters used to describe the Tapered Hexapole Magnet. Rectangular Chopper Shutter Function. Rectangular Detector Response Function Geometry of Time-of-Flight Apparatus LIST OF TABLES Page 1. Detailed Summary of Heat Leaks into the Cryostat of the'Low Temperature Atomic Beam Source. Measurements were made after i n i t i a l construction of source and a f t e r extensive modifications necessary to reduce heat leak to the expected design values 39 2. Selected Trajectories of Focussed and Defocussed ~>lie Atoms Passing Through the Tapered Hexapole Magnet 89 3. Comparison of d i f f e r e n t Methods of Producing Polarized 3He + Ions... 100 ACKNOWLEDGEMENTS x i i I would l i k e to thank my supervisor, Dr. M.K. Craddock," for his continued i n t e r e s t throughout the course of the work described i n this thesis. I would also express my thanks to Dr. D. Axen, who o r i g i n a l l y b u i l t much of the ex i s t i n g polarized helium-3 beam apparatus, for his continuous i n t e r e s t and untiring help with many of the experimental measurements described i n this thesis. I would also thank Drs. White, Erdman and Warren for their i n t e r e s t and encouragement during the course of this work. I am extremely g r e a t f u l to the many technicians who have contributed to this work over a period of many years; i n p a r t i c u l a r , Messrs. D. Haines, C. Sedger and D. Stonebridge for their u s e f u l suggestions and excellent craftmanship. I wish to- thank the National Research Council for one bursary and three scholarships held during the course of this work. 1 CHAPTER I INTRODUCTION The study of the nature of nuclear forces and the part they play In various reaction mechanisms has been a subject of considerable i n t e r e s t for many years. The spin dependent nature of these forces has been a subject of more recent i n t e r e s t . In essence the spin dependence of the nuclear force implies that the force between two i n t e r a c t i n g nucleons depends both on the magnitude and o r i e n t a t i o n of the spin vector characterizing the magnetic moment of the p a r t i c l e s . Various expressions for this i n t e r a c t i o n have been postulated and are often represented i n nuclear models by p o t e n t i a l s ; a common example i s the spi n - o r b i t p o t e n t i a l Vso oC where i s the Pauli spin operator and i_ i s the o r b i t a l angular momentum operator. From an experimentalist's point of view the study of the nature of these forces i s only possible because of the a v a i l a b i l i t y of spin sensitive detectors, p o l a r i z e d targets or polarized beams. The spin of a p a r t i c l e i s measured r e l a t i v e to some reference axis, sometimes a magnetic f i e l d d i r e c t i o n . For spin i p a r t i c l e s such as protons, and 3He there are two nuclear magnetic spin substates mj = +-§• or m-j. = depending on whether the nuclear spin l i n e s i t s e l f up p a r a l l e l (spin up) or a n t i p a r a l l e l (spin down) to the d i r e c t i o n of the magnetic f i e l d a x i s . The p o l a r i z a t i o n P of an ensemble of such p a r t i c l e s i s given by •_ N(+i)-N(-£) P — . N(+£)+N(-£) where N(+£) i s the f r a c t i o n a l number of p a r t i c l e s i n the beam or target with their spins aligned p a r a l l e l to the magnetic f i e l d while N(-i) i s the f r a c t i o n a l number aligned a n t i p a r a l l e l . A beam or target with equal populations of both substates i s said to be unpolarized and has zero p o l a r i z a t i o n while a beam or target with a l l spins oriented i n the po s i t i v e d i r e c t i o n would have 100$ p o l a r i z a t i o n . Certain techniques have been developed for the study of spin dependent forces. Because of the strength of the spin-o r b i t force the reaction products of any reaction involving p a r t i c l e s with nonzero spin and angular momentum w i l l be p a r t i a l l y p o l a r i z e d . These polarized p a r t i c l e s can be collimated into a beam for use i n a subsequent reaction. Beams formed i n this manner are referred to as secondary beams. (p,p) e l a s t i c scattering can be used to produce a beam of polar-ized protons while the D(D,3R"e)n reaction can be used to produce a polarized neutron beam. Unfortunately these secondary beams frequently have inadequate p a r t i c l e fluxes to be of use i n many experiments which might be envisaged to study, for example, the spin dependence of a selected reaction mechanism. Certain reactions can also be used to analyze the p o l a r i z a t i o n of desired p a r t i c l e s . For example, the e l a s t i c scattering of polarized protons o f f carbon or helium r e s u l t s i n a l e f t - r i g h t asymetry about the incoming beam axis which can be related to the incoming proton p o l a r i z a t i o n . • The production of more intense beams of polarized p a r t i c l e s has become possible through the development of polarized i o n sources. These devices prepare beams of polarized p a r t i c l e s for i n j e c t i o n into accelerators. By i n j e c t i n g p o l a r i z e d p a r t i c l e s into the accelerator much higher beam i n t e n s i t i e s can be produced than by scattering or reaction techniques. Polarized proton and deuteron ion sources, which allow the preparation of intense beams with po l a r i z a t i o n s of near 100$ for the case of the protons, have been extensively developed and they are now available as commercial items. After the proton and deuteron one of the next simplest p a r t i c l e s that could e f f e c t i v e l y be used i n p o l a r i z a t i o n studies i s ^He, an atom with nuclear spin of £ and a zero electronic spin. Unpolarized JHe has been used extensively as both a target, and p r o j e c t i l e i n nuclear physics studies for many years. Many of the i n t e r e s t i n g p o s s i b i l i t i e s for research using 3He as a p r o j e c t i l e were discussed by Bromley and Almqvist (Br60) i n a lengthy review a r t i c l e i n i 960 . Since that time ^He induced reactions have been widely studied. Because of the strong spin dependence of the nuclear force none of these studies can be considered complete u n t i l the detailed e f f e c t s of spin have been c a r e f u l l y investigated. In an experiment with unpolarized beams and targets and with no observation of the p o l a r i z a t i o n of the outgoing p a r t i c l e s the detailed spin dependence of the reaction i s l o s t due to "the average over the magnetic quantum numbers of i n i t i a l and sum over the f i n a l states. Knowing the p o l a r i z a t i o n of some combination of incoming p r o j e c t i l e , target nucleus, or the outgoing reaction products can r e s u l t i n much more s p e c i f i c information on the de t a i l e d behaviour of the spin dependent forces. For this reason many previously studied i n e l a s t i c , s t r i p p i n g and pickup reactions may be suitable candidates for renewed study* G. C. P h i l l i p s (Ph66) has recently reviewed e x i s t i n g nuclear reaction studies using polarized ^Ee targets and beamso Studies at Rice Uni v e r s i t y using a polarized 3 He target i n the 3He(d,p)lfHe reaction (Ba65) have provided the. f i r s t i n d i c a t i o n that o p t i c a l model theories of s t r i p p i n g processes must include tensor forces,, I t i s clear that further studies with polarized beams and targets need be c a r r i e d out for other s t r i p p i n g reactions to further define their spin dependence. Polarized nuclei can also be used to help make assignments of spins and p a r i t i e s of excited states of nuclear systems. Again the Rice group has used a polarized 3ne target i n the study of the unstable nucleus ^ L i formed as an intermediate state i n the 3ne(p,p)3He reaction (Ba67) . They have determined information about the spins and p a r i t i e s of T = 1 states of the mass k system. In this reaction i t i s b a s i c a l l y impossible to deduce a unique set of phase s h i f t s i f only e l a s t i c scattering data with unpolarized beams and targets are available along with spin measurements of the scattered proton. However, i f spins of the 3ne are also measured the degenerate solutions can be rejected and a unique determination of phase s h i f t s becomes possible. Polarized 3ne beams are also of i n t e r e s t because i t i s the neutron that i s polarized; thus i n a sense one has available a variable energy neutron beam for reactions involving the neutron eg. ( 3 n e,pp), and perhaps (3He,n). Another i n t e r e s t i n g p o s s i b i l i t y for polarized 3 p e a s w e l l as a l l other polarized p a r t i c l e s i s the testing of conservation l a v s . The s e l e c t i o n of c r u c i a l experiments to test p a r i t y and time r e v e r s a l invariance i n s p e c i f i c reactions should be g r e a t l y aided by the a v a i l i b i l i t y of control over the p o l a r i z a t i o n of the incoming beam. Selection of these c r u c i a l experiments may be aided by the systematic c l a s s i f i c a t i o n by Moravcsik, Csonka and Scadron (M066) of a l l experimental p o s s i b i l i t i e s ' f o r reactions involving four nucleons of given spin. To perform many of these p o t e n t i a l experiments a polarized ^He beam i s required. To estimate the possible f l u x of p o l a r i z e d ^He attainable as a secondary beam, consider the e l a s t i c s c a t t e r i n g of 3He o f f Sle. P h i l l i p s and M i l l e r (Ph59) show this s c a t t e r i n g can r e s u l t i n near 1 0 0 $ p o l a r i z a t i o n of the He under c e r t a i n conditions. For a ^He bombarding laboratory energy of 5»2 Mev, the cross section for the production of 1 0 0 $ p o l a r i z e d ^He scattered at *+5° i n the center of mass frame i s 0 . 1 6 2 b/sr. I f a 1 0 0 pA^Ee beam bombards a ^He gas target 3 0 0 kev thick ( 5 a tin pressure and 1 . 2 cm long) and i f the s o l i d angle of the r e s u l t i n g beam i s r e s t r i c t e d to 1 0 " 2 sr then the f l u x of the r e s u l t i n g scattered beam i s 0 . 0 1 nA. I t i s u n l i k e l y a l l the above conditions could be achieved i n practise thus 0 . 0 0 1 nA might be a more reasonable estimate of the possible p o l a r i z e d ^He i o n current. This i n t e n s i t y i s too low to be of much use for many nuclear studies and thus we must turn to other techniques to produce more intense beams. Two methods have been proposed for the production of polarized 3 H e + ion beams. An o p t i c a l pumping technique has been su c c e s s f u l l y developed at Rice U n i v e r s i t y . At U.B.C. an atomic 6 beam method for the production of polarized ^Ee+ proposed by Warren, Axen and Klinger (Wa63) i s under development. I n i t i a l design, construction and preliminary testing of the U.B.C. ion source has been reported by Axen (Ax65). The development of the ion i z e r has been reported by Vermette (Ve6h) and the d i f f e r e n t i a l P i r a n i detector by Jassby (Ja6*0 • The present thesis w i l l discuss the detailed study of the atomic beam formation and i t s subsequent p o l a r i z a t i o n and i o n i z a t i o n . The contents of this thesis i s divided into the following chapters. In Chapter II a general d e s c r i p t i o n of the two techniques used to produce polarized 3jje+ ^ o n beams i s presented. In Chapter III the th e o r e t i c a l background of the low temperature nozzle source which i s e s s e n t i a l to the operation of the U.B.C. ion source i s reviewed. Chapter IV gives a d e t a i l e d d e s c r i p t i o n of the mechanical operation of U.B.C.'s ion source. Chapter V describes the techniques used to measure the i n t e n s i t y and v e l o c i t y of the atomic beam. The r e s u l t s of these measurements are presented i n Chapter VI. Chapter VII considers the e f f e c t the hexapole magnet has on the 3jfe t r a j e c t o r i e s and the r e s u l t i n g beam p o l a r i z a t i o n . Possible improvements to the polarized i on source are considered i n Chapter VIII. 7 CHAPTER II METHODS FOR PRODUCING POLARIZED 3 H 9 + BEAMS A. The Optical Pumping Method. The polarized ^Ee + ion source based on o p t i c a l pumping techniques, su c c e s f u l l y developed at Rice University, d e l i v e r s k pA of ions with a p o l a r i z a t i o n measured by a nuclear double scattering experiment of 0.05~to.01 (Ba68, Fi69). The emittance of the beam was estimated to be 1 cm»rad*ev^. The energy l e v e l s relevant to the o p t i c a l pumping scheme are shown on F i g . 1. A weak se l f - s u s t a i n i n g e l e c t r i c discharge i n very pure 3 n e gas, produced by a 50 MH zrf f i e l d around an o p t i c a l pumping c e l l , excites some of the ^S 0 ground state 3ne atoms to the 23S]_ metastable state. Right-hand c i r c u l a r l y p o l a r i z e d resonance l i g h t directed into the c e l l along the axis of the applied magnetic f i e l d produces^m ? = +1 t r a n s i t i o n s from the lower (mF = - 3 / 2 , - i ) 2 3 s ^ hyperfine sublevels to the 23 P 0 l e v e l s . Atoms i n the 2 3 p l e v e l s de-excite to the various 2 3 s 1 l e v e l s with nearly equal p r o b a b i l i t i e s . As this process i s repeated over many cycles atoms are removed from the negative mF hyperfine l e v e l s of the metastable atoms and are placed i n the p o s i t i v e mp l e v e l s , producing a p o l a r i z a -t i o n of the metastable atoms. The" negative m^  l e v e l s can be populated i n the same way using left-hand c i r c u l a r l y polarized l i g h t ; a p o l a r i z a t i o n of the opposite sign r e s u l t s . The p o l a r i z a t i o n i s transferred from the metastable atoms to the much more numerous l^S ground state atoms by means 4 x l 0 4 c m " 1 2 'S, 1.6 x lO 5 c m " 1 l u l l I ^ u 11 u 11 2 " S 23R < 9233cm -I 1/2 - 1 / 2 F= /2 0.22 cm" 1 F=3/2 < - l / 2 > - - I / 2 J 1/2 -1/2 1/2 - 1 / 2 3 /2 1/2 - 1 / 2 - 3 / 2 F i g . 1 Energy Levels of 3 n e Atoms i n an External Magnetic F i e l d (not to scale)* of spin exchange c o l l i s i o n s , and, under continuous i l l u m i n a t i o n from the pumping l i g h t , the ground state p o l a r i z a t i o n reaches an equilibrium value equal to that of the metastable atoms. The r f discharge produces both atoms i n the metastable state as well as the ions. The i o n p o l a r i z a t i o n comes into equilibrium with the atomic ground state p o l a r i z a t i o n because of a very large t-10 cm ) cross section for electron exchange via He -He c o l l i s i o n s . The ions are extracted from the o p t i c a l pumping c e l l by standard r f ion source techniques. The gas p o l a r i z a t i o n i n the o p t i c a l pumping c e l l was measured to be 0 .05to .01, which, within experimental error, i s the measured value of the ion p o l a r i z a t i o n . Thus i t appears possible to extract an ion beam with the same p o l a r i z a t i o n as the gas i n the pumping c e l l . As pol a r i z a t i o n s of approximately 60$ (Ga65) have been achieved i n gas samples under optimum conditions, further improvements i n the p o l a r i z a t i o n of the ion beam can be expected. The present p o l a r i z a t i o n appears to be l i m i t e d by the short dwell time i n the o p t i c a l pumping c e l l . This may be overcome by increasing the dimensions of the pumping c e l l , hence allowing the $Ee atoms to remain under the influence of the pumping r a d i a t i o n for a longer time. B. The Atomic Beam Method. The p a r t i c l e s i n each of the two possible nuclear spin substates i n an atomic beam of ^Ee can be separated by passing the neutral beam through an inhomogeneous magnetic f i e l d . This technique was suggested by Warren, Axen and Klinger (Wa63) He 3 atoms 1~ Polarized H e 3 atoms — P o l a r i z e d He 6 ions skimm collimator leybold 2heraeus HG45 6" CVC I O " CVC 8" four stage d i f ferent ia l pumping system F i g . 2 General Arrangement of the Components of the Polarized. ^Ee Apparatus showing the d i f f e r e n t i a l pumping required to handle the 3 n e gas flow. 11 as a method for producing a polarized ^Ee + ion beam. F i g . 2 shows a schematic view of this scheme. An atomic beam produced using a supersonic nozzle cooled to l i q u i d helium temperatures i s passed through the inhomogeneous magnetic f i e l d of a hexapole magnet. As atomic ^Ee has no electronic magnetic moment, those p a r t i c l e s with nuclear spin p r o j e c t i o n +•§• i n the d i r e c t i o n of the applied f i e l d are deflected towards the magnet pole pieces and subsequently removed from the beam while those of nuclear spin p r o j e c t i o n are deflected toward the central axis and focussed into the i o n i z e r . The very small size of the ^He nuclear moment and the need for a magnet of reasonable length producing conventional magnetic f i e l d strengths requires that the v e l o c i t y of the p a r t i c l e s entering the magnet be very low. In the o r i g i n a l proposal an atomic beam with a most probable v e l o c i t y of 180 m/sec was to pass through a tapered hexapole magnet 50 cm i n length. Under these conditions i t was calculated that an atomic beam could be prepared with near 100$ p o l a r i z a t i o n , that i s , a l l the p a r t i c l e s passing into the io n i z e r would be i n the nuclear spin p r o j e c t i o n s t a t e . Ioniza t i o n of the beam would be achieved with an electron bombardment type i o n i z e r . The p o s i t i v e ions so produced would be focussed and accelerated to produce a beam of polarized ^He + ions suitable for nuclear r e a c t i o n studies. The percentage p o l a r i z a t i o n of the ionized beam depends on the magnetic f i e l d strength present at the ioniz e r and the target. Axen (Ax65) has calculated the ^>Ee nuclear p o l a r i z a t i o n expected of s i n g l y ionized ^Ee atoms for equal f i e l d s i n the io n i z i n g and targe.t regions His r e s u l t s are shown i n F i g . 3. 3 6 9 H (kilogauss) F i g . 3 Nuclear P o l a r i z a t i o n of Singly Ionized 3^e Atoms for Equal Fields i n the Ionizing and Target Regions. For zero f i e l d i n the two regions a p o l a r i z a t i o n of 50% i s achieved and the p o l a r i z a t i o n increases u n t i l i t reaches almost 100$ with a f i e l d of about 9 KG i n both regions. Thus by i o n i z i n g and placing the target i n a region of high but not te c h n i c a l l y excessive magnetic f i e l d i t i s possible to obtain a beam of near 100$ nuclear p o l a r i z a t i o n . I t i s this very high p o l a r i z a t i o n which makes the atomic beam method of producing polariz e d ^He + ions so p o t e n t i a l l y a t t r a c t i v e i n nuclear physic experiments. 13 CHAPTER III THE PRODUCTION OF MOLECULAR BEAMS A. Motivation for Development of Molecular Beams. Atomic beams have been instrumental i n the advancement of many f i e l d s of physics. E a r l y experiments v e r i f y i n g the Maxwellian v e l o c i t y d i s t r i b u t i o n helped e s t a b l i s h the Kinetic theory of gases. The c l a s s i c s p l i t t i n g of a s i l v e r beam into two beams as i t passed through an inhomogeneous magnetic f i e l d was an early r e s u l t explainable by s p a t i a l quantization,a r e s u l t of quantum theory. Molecular flow beams, beams where flow through the opening i s c o l l i s i o n l e s s , can be f i n e l y collimated but the i n t e n s i t y a v a i l a b l e i s very low. For experiments requiring higher i n t e n s i t y Kantrowitz and Grey (Ka51) i n 1951 proposed a system which hopefully would r e s u l t i n considerable improvements of beam i n t e n s i t i e s and v e l o c i t y spread. Their proposal employing a supersonic Laval nozzle appeared to provide a means of increasing beam i n t e n s i t i e s by at l e a s t one order of magnitude and to allow s i g n i f i c a n t reductions i n the v e l o c i t y spread of the beam. Although o r i g i n a l l y proposed as a technique for the study of other p h y s i c a l phenomena, the nozzle beams r a p i d l y became the subject of intense study. This study came about due to the f a i l u r e of most prototype sources to perform i n the predicted fashion. Subsequent investigations have revealed much of the true behaviour of these nozzle sources and now the expected Ik behaviour of a given nozzle system can be predicted with reasonable confidence. The discussion of theory and res u l t s to be presented w i l l be directed towards a better understanding of the operation of the nozzle source used i n the production of a p o l a r i z e d 3 H e + ion beam. The production of this polarized i o n beam has been the main motivating force for the development work which has gone into the design of a low v e l o c i t y atomic beam source. The l a s t section i n this chapter w i l l summarize ex i s t i n g work on l i q u i d helium cooled nozzle sources and discuss possible uses for low temperature nozzle sources. B. Properties of the Molecular Flow Beam. Before discussing the nozzle source the basic properties of the molecular flow beam i . e . a beam formed by free molecular flow through an o r i f i c e w i l l be described. Molecular flow implies that the mean free path A of the gas p a r t i c l e s i n the source i s considerably larger than the diameter d and length of the o r i f i c e that i s , the Knudsen number Kn = V d i s much greater than unity. The operation of a molecular flow beam has been discussed (Sm55"} Ra5&) but w i l l be summarized below for completeness. The fate N at which molecules pass through an aperture of area A i s equal to the number of molecules h i t t i n g that area of the wall per second, and i s given by N = invA molecules/sec (1) where n i s the number of molecules per unit volume i n the source and v = 1 QkT i S the average speed, K i s Boltzman's constant, TTTn 15 T the oven temperature (°K), and m the mass of the gas molecule. The i n t e n s i t y I at a distance r from the oven source i s given by T - i Tiv A cos O molecules /cm*-Sec ( 2 ) T l " where 8 i s the angle between the radius vector r and the normal to the aperture. Eq. 2 can be rewritten for the case of centerline i n t e n s i t y i e . 0 = 0 i n terms of the t o t a l p a r t i c l e gas flow through the o r i f i c e N as The number of p a r t i c l e s passing through the opening with a given v e l o c i t y V follows from Maxwells law and i s given by T f \ J\t - ?T / V \ 5 p ~ V / ^ L dV ci - l -oms/cm - Sec X(v)dV - tto(z) 6 £ (If) where «L- J~^'is the most probable v e l o c i t y . The above equations represent the physical s i t u a t i o n providing A » d holds; however, as the oven pressure i s raised there i s a gradual t r a n s i t i o n from free molecular to viscous flow. In this t r a n s i t i o n region the i n t e n s i t y i s l i m i t e d by l a c k of c o l l i m a t i o n due to c o l l i s i o n s between molecules i n the beam which r e s u l t s i n a low I/N r a t i o and consequently requires excessive pumping capacity to remove the background gas for the useful i n t e n s i t y obtained. C. Properties of the Nozzle Beam. Kantrowitz and Grey (Ka5l) proposed a supersonic nozzle source for the production of molecular beams as shown i n F i g . k. 16 /////// h77/ nozzle F i g . k Schematic Representation of Nozzle Beam Source. The cross section of their nozzle was shaped to produce a flow with a predetermined Mach number M, equal to the r a t i o of the v e l o c i t y of mass motion W to the l o c a l v e l o c i t y of sound a. The core of the beam so produced would be extracted with a su i t a b l y shaped skimmer and collimator system with no anticipated i n t e r a c t i o n between the skimming apertures and the beam. The obvious advantages of such a system were the large increase i n i n t e n s i t y and considerable reduction i n v e l o c i t y spread of the beam produced when compared to an oven beam. The large i n t e n s i t y increase arises because of the high gas densities i n nozzle flows compared to the low gas densities i n molecular beam flows. The attainment of beams with a v e l o c i t y of mass motion i n excess of the l o c a l sonic v e l o c i t y r e s u l t s i n a v e l o c i t y d i s t r i b u t i o n considerably narrower than that achieved with oven beams. I t was soon discovered, however, that i t was not possible 17 to produce a beam of controlled properties, that i s a gas flow characterized by a predetermined Mach number using the t h e o r e t i c a l l y shaped nozzle nor was i t possible to extract the beam produced without interference between the skimming elements and the background gas. The nature of the beam a c t u a l l y produced i s discussed i n the following sections. (1) Gas Flow Through a Nozzle. The mass gas flow, G, through a nozzle for one-dimensional, f r i c t i o n l e s s adiabatic flow i s given by (Sh53)s where A = e f f e c t i v e cross s e c t i o n a l area of the nozzle. = r a t i o of s p e c i f i c heats, po and To are the stagnation pressure and temperatures upstream of the nozzle entrance, that i s the pressure and temperature of the gas i n the region where the flow i s e s s e n t i a l l y completely random. "One dimensional flow" means that the flow properties are assumed to be constant i n any plane perpendicular to the d i r e c t i o n of flow and hence applies to the case of a gas flowing through a nozzle of varying cross-s e c t i o n a l area. The e f f e c t i v e cross s e c t i o n a l area of the nozzle i s under c e r t a i n conditions less than the actual geometrical area because of viscous e f f e c t s . The r a t i o of e f f e c t i v e area to actual geometrical area i s known as the discharge c o e f f i c i e n t . Govers, Le Roy and Deckers (G069) give experimental values of 18 this c o e f f i c i e n t f o r helium gas passing through a nozzle of diameter d as a function of nozzle Reynolds number, R A-m N P - — ; — , based on experimental flow N and v i s c o s i t y ri at stagnation conditions. Their results for Helium are shown i n F i g . 5 at stagnation temperatures between 2 9 5°K and 120H-°K using a nozzle with d = 0.266 mm. Although the re s u l t s do not collapse onto a single l i n e they do show what f r a c t i o n of the t h e o r e t i c a l l y calculated flow w i l l be achieved under experimental conditions characterized by a given Reynolds number at temperatures of 2 9 5°K and 120 lf°K. Unfortunately no re s u l t s are available for temperatures near 7 ° K where our nozzle w i l l operate. In spite of the f a c t our nozzle w i l l operate at Reynolds numbers i n excess of 600 i t appears possible that the discharge c o e f f i c i e n t may be considerably below unity. This however presents no problems. (2) The Free Jet Expansion. Consider an o r i f i c e of diameter d as shown i n F i g . 6 with a near vacuum on one side and a pressure s u f f i c i e n t l y high on the other side that slows can be treated by continuum gas dynamics techniques, that i s A ^ d , on that side. The flow through such an opening (As66) forms a j e t as shown i n F i g . 6 defined by a b a r r e l shock and the Mach disk. The Mach disk and b a r r e l shock form the boundary between continuum and free molecular flow. In this "zone of s i l e n c e " the flow expands i s e n t r o p i c a l l y unaffected by the presence of background gas outside the j e t boundary and can be treated by methods of continuum gas dynamics. This expansion has been described 19 0 2 0 0 4 0 0 6 0 0 TT D 72. F i g . 5 Discharge C o e f f i c i e n t vs. Reynolds Number based on Experimental Flow and V i s c o s i t y at Stagnation Condition; 2 0 t h e o r e t i c a l l y for = l.h by Owen and Thornhill (0w52) using the method of c h a r a c t e r i s t i c s and confirmed experimentally by Reiss ( F e 6 3 ) and Sherman ( S h 6 3 ) . jet boundary F i g . 6 Schematic Representation of Flow from an O r i f i c e into an Evacuated Region. Their s o l u t i o n i s applicable to any j e t flowing into any external pressure i n that region bounded by the o r i f i c e and the f i r s t wavefront which r e g i s t e r s the existance of an external pressure outside the j e t . Askenas and Sherman (As66) extended Owen and T h o r n h i l l s 1 s o l u t i o n to gases with7f= I . 6 7 (eg. Helium). They suggest the following formula- for the centerline Mach number of a free j e t : 24 - ^ 1 . 6 7 K 18 UJ CQ « 5 • <s£ z x 12 u < K = i .4 6 — 1 1 i i o 0 " 4 8 12 S6 2 0 24 • - DISTANCE FROM NOZZLE EXIT (nozzle diameters) F i g . 7 D i s t r i b u t i o n of Mach Number Along the Axis of Symmetry of the Expanding J e t . 22 where x = distance from o r i f i c e along the c e n t e r l i n e j B , C and X Q are constants such that for Y = 1.6?; B = 3.26, C = 0.31 and £2, = 0.075. This three term formula i s accurate for X ^ d with maximum deviations from the c h a r a c t e r i s t i c data of^-g^ of M. The calculated values are shown graphically i n F i g . 7. I t i s i n t e r e s t i n g to note that these r e s u l t s are independent of the nozzle temperature and pressure. For adiabatic flow with d e n s i t i e s . s u f f i c i e n t l y high that methods of continuum gas dynamics may be applied, the l o c a l pressure and temperature p^ and T j of a beam, that i s the pressure and temperature measured i n a reference frame moving with the v e l o c i t y of mass motion W as the beam expands through a nozzle, may be expressed i n terms of the nozzle stagnation conditions and the l o c a l Mach number (Em58). This treatment gives and T , = . T o Pi -( 7 ) The l o c a l sound v e l o c i t y Q = (9) and the v e l o c i t y of mass motion W - M (10) In the i n e r t i a dominated region of the free j e t expansion Askenas and Sherman (As66) have shown that the density decreases along each streamline i n proportion to the inverse square of distance from an apparent source a distance x 0 down-stream from the actual o r i f i c e . They also show that the calculated angular dependence of the density f i e l d can be 23 represented by the simple formula (11) with an accuracy of about 3>% of n( r , o ) . That i s , their data from a method of c h a r a c t e r i s t i c s s o l u t i o n d i f f e r s less than 1% from the r e s u l t s obtained using Eq. 1 1 . The constant (j> depends on the s p e c i f i c heat r a t i o for the gas used; thus for d i s k XJJJ i s given by Askenas and Sherman (As66), as a function of the pressure r a t i o Po/Pbg across the o r i f i c e , by: where Pbg i s the background pressure i n the nozzle exhaust region. This formula gives reasonable agreement to experimental results (3) The Freezing Surface. The free expansion discussed e a r l i e r e s s e n t i a l l y transfers energy from the i n t e r n a l degrees of freedom of the molecules to directed mass motion. Thus we may consider the beam molecules to be characterized by some l o c a l temperature T, i n a l o c a l reference frame moving with the v e l o c i t y of mass motion. A c e r t a i n c o l l i s i o n frequency between the molecules of the beam i s required to support this continuous reduction i n the randomness of the motion of the molecules with respect to the l o c a l reference frame. When the gas density gets s u f f i c i e n t l y low and this minimum c o l l i s i o n frequency i s not maintained, the The distance from the o r i f i c e to the Mach (12) independent of the value of Y, i n the region 15 £ 0 / /Pba~ 17,000 . expansion to higher Mach number ceases and the system "freezes" at a given temperature. Once this freezing region i s reached the gas continues to expand r a d i a l l y , the i n t e n s i t y decreasing as but with no further change i n temperature. I t should be noted that the l o c a t i o n of this freezing surface i s not coincident with the Mach disk l o c a t i o n . Empirical determination of the terminal Mach number Mj. at which t r a n s l a t i o n a l freezing occurs based on nozzle stagnation conditions, has yielded the r e s u l t (An65). M t = l . - ' B ( K r O ^ ) ( 1 3 ) This r e s u l t was obtained for Argon beams at room temperature; subsequent i n v e s t i g a t i o n has shown that i t also holds f o r other gases. However, Abuaf et a l (Ab66) have shown that experimental values for helium f a l l below the r e s u l t s predicted using this equation; his r e s u l t s are shown i n F i g . 8 . Knuth (Kn6l+) calculated the l o c a t i o n of the freezing planes using the r e l a x a t i o n times for the c o l l i s i o n processes involved. He found for a monatomic gas , V = 5 / 3 that where Kn 0 = the Knudsen number based on stagnation conditions. Knuth's r e s u l t s are also shown on F i g . 8 . The agreement with experimental r e s u l t s for Argon i s pdor for low Knudsen numbers but improves for larger Kn Q• Naturally the t r a n s i t i o n between the continuum to t r a n s i t i o n and then free molecular flow i s not a sudden process so that the above formula and the model on which they are based only represents a f i r s t approximation , to the r e a l s i t u a t i o n which i s properly described only by a • Stagnation Conditions at the Nozzle. 26 complete s o l u t i o n of the k i n e t i c equations. (*+) V e l o c i t y D i s t r i b u t i o n of P a r t i c l e s i n the Beam. P a r t i c l e s . i n an expanding je t from a miniature supersonic nozzle can be considered to move with respect to a l o c a l reference frame which i s moving r a d i a l l y outwards from the nozzle with the v e l o c i t y of mass motion. This behaviour i l l u s t r a t e d i n F i g . 9 includes the presence of the freezing surface. For s i m p l i c i t y the res u l t s presented i n this arri the next section w i l l assume the freezing surface i s coincident with the skimmer opening. Because the r a d i a l density v a r i a t i o n i s the same on both sides of the freezing surface, the i n c l u s i o n of the freezing surface upstream of the skimmer w i l l have no e f f e c t oh the r e s u l t s obtained. The p a r t i c l e s as seen i n the moving frame have a Maxwellian v e l o c i t y d i s t r i b u t i o n characterized by some l o c a l temperature. In f a c t , c a r e f u l studies of the v e l o c i t y d i s t r i b u t i o n of nozzle beams that consider both the r a d i a l and angular dependence of the v e l o c i t y d i s t r i b u t i o n f i n d that the Maxwellian d i s t r i b u t i o n i s characterized by one temperature T// i n the r a d i a l d i r e c t i o n u^ and another Tj_ i n the u 2 and u^ dire c t i o n s (Fi67). At the o r i f i c e T„ = Tj_ but due to differences i n the rates for t r a n s l a t i o n a l r e l a x a t i o n the two temperatures progress to the d i f f e r e n t terminal values achieved when the expansion stops. Thus the v e l o c i t y d i s t r i b u t i o n function can be written F i g . 9 Schematic Representation of Ra d i a l l y Expanding Flow throug a Skimmer to a Detector. 28 • f ( U i ^ - ( - ( W i ) -T(u 3 N) d u , c l u i d"-2ir kT„ ex (15) This function represents the f r a c t i o n a l number of p a r t i c l e s i n the beam with v e l o c i t i e s u^, U£j i n some range duj, du2, du^, I f v e l o c i t y d i s t r i b u t i o n measurements are made only along the beam axis i t i s not possible to determine both T,( and T x and i t i s common to set T n = T j_ = T ; , thus r e s u l t i n g i n the s i m p l i f i e d v e l o c i t y d i s t r i b u t i o n -PO,) f (uO- f (u 5s) d u , d u x d u 3 = (^Ty* e*P[z^, fa-"'?  r U * } ] d M ' ( 1 6 ) The above equations give the v e l o c i t y d i s t r i b u t i o n of p a r t i c l e s at some point i n the beam. The f r a c t i o n a l i n t e n s i t y or d i f f e r e n t i a l i n t e n s i t y d i s t r i b u t i o n function, I(u) - d i ? 0 f p a r t i c l e s with a given v e l o c i t y a r r i v i n g at a detector depends on the r e s t r a i n t s imposed by the geometry of the complete nozzle system i n summing over a l l possible v e l o c i t i e s to obtain the t o t a l i n t e n s i t y of p a r t i c l e s at the detector. I f p a r a l l e l flow at the entrance to the skimmer i s assumed (one of Kantrowitz and Grey's o r i g i n a l assumption) Eq.B8 of Appendix B gives (B18) 3 - ^ > - w ) 2 where ^(u) : ^ e (B19) and l i s i s the centerline isentropic beam i n t e n s i t y expected from a free j e t expansion unaffected by the presence of a skimmer as discussed i n Section 3C and£\ s I s the half angle subtended by the skimmer with respect to the nozzle e x i t as shown i n F i g . 4-8. Now i f the r a d i a l flow divergence of the beam at the skimmer i s included i n c a l c u l a t i n g the i n t e n s i t y at the detector Eq. B8 of Appendix B gives (^~\'k Q ( u ) (B20) where g(u) i s as before and • ; G r M ^ { i - ^p[-( 21^yi- t o^0D] ( B 2 D now ? i 4 ^ J ^ ^ M 2 thus when the skimmer subtends a very small half angle 2^M 2 Sir\*(^?) 1 and Q-(u) PC ^ r= eo f tS-fa.A + Then — «* 3(11) as i n Eq. B18 where the flow at. skimmer i s d u assumed p a r a l l e l . When the skimmer subtends a large half angle 2 * r V ^ ( f ) y>\ a n d S A P [ - Z W 2 J s i n 2 ^ ) ^ 0 a n d thus modifing the v e l o c i t y dependence of Eq. B8 to ^ * 9 ( u V ( ^ M = ^ Z expL~^ ( U - \ N ¥ ] ( i 7 ) when the r a d i a l dependence of the flow i s considered for flows with appropriate Mach number and skimmer radius. This i s the form of the d i f f e r e n t i a l i n t e n s i t y d i s t r i b u t i o n function recommended 30 by Hagena and Morton (Ha67) (5) Intensity Available From Nozzle Beams. ( i ) Freely Expanding Jet. The case of the f r e e l y expanding j e t without a skimmer or collimator using Eq. 11 to represent the density f i e l d about the beam axis has been considered. As shown i n a simple c a l c u l a t i o n given i n Appendix A the expected f l u x density I on the beam axis i s I = 0.6N atoms/steradian - sec (A5) where N = G/m = p a r t i c l e flow through the nozzle given from Eq. 5» The equivalent expression for molecular flow beam i s I = N/TT atoms/steradian - sec (3) ( i i ) Beam Intensity Downstream from the Skimmer. The t h e o r e t i c a l l y expected beam i n t e n s i t y a f t e r the gas has passed through a skimmer i s discussed i n Appendix B. Two cases are considered: One assumes the conditions of the Kantrowitz and Grey model (Ka5l), namely: (1) isent r o p i c flow upstream of the skimmer ( 2 ) p a r a l l e l flow at the skimmer and (3) c o l l i s i o n l e s s flow downstream of the skimmer, while the other considers the case as suggested by Hagena and Morton (Ha67) where c r i t e r i o n ( 2 ) i n the Kantrowitz and Grey model i s modified to take into account the divergent nature of the flow at the skimmer. 31 The r e s u l t of this analysis i s (1) P a r a l l e l flow at skimmer entrance. I - 1 ,5 S i f ) V s ( ! Ar Vf) ' (Bl?) (2) R a d i a l l y diverging flow at skimmer entrance. I - I ^ C i - c o s V . e ^ S m * s ) ( B 1 2 ) Both Eqs. B12 and B17 are for the case of the detector-skimmer separation %77 / s , the nozzle-skimmer separation, and make use of the s i m p l i f i c a t i o n s afforded by assuming M > 3. Eq. B17 based on the Kantrowitz-Grey model predicts an i n t e n s i t y proportional to the skimmer area but i t i s clear that i t predicts unreasonably high i n t e n s i t i e s I > I i s i f the skimmer i s so large that ^ ) Sin V s y | . This unphysically high i n t e n s i t y i s due to the neglect of the divergent nature of the flow approaching the skimmer. Including the divergent nature of the flow i n the c a l c u l a t i o n r e s u l t s i n Eq. B12. This second equation no longer shows an unlimited increase of I with either M or sinc<s but predicts an i n t e n s i t y approaching I^g, the i n t e n s i t y one would expect i f no skimmer had been present at a l l . (6) Deviations From Ideal Behaviour. Experimental investigations have shown that the i n t e n s i t i e s predicted i n the previous section are not always achieved i n p r a c t i c e . This discrepancy between theory and experiment i s caused at l e a s t i n part by the assumption i n the th e o r e t i c a l c a l c u l a t i o n that (1) the beam i n unaffected by the presence of the skimmer and ( 2 ) the beam i s unaffected by the presence of background gas The e f f e c t of these i d e a l i z a t i o n s i s often a severe reduction of beam i n t e n s i t y below that expected from the previous a n a l y s i s . The presence of a shock wave at the skimmer entrance was expected to cause a reduction of the beam i n t e n s i t y due to the presence of the skimmer but the electron-beam flow-v i s u a l i z a t i o n photographs of McMichael and French (McM66) f a i l e d to detect any l o c a l b u i l d up of gas molecules upstream of the skimmer. The skimmer interference i s now postulated to occur downstream of the skimmer entrance i e . inside the skimmer, and to be caused by a cloud of low v e l o c i t y molecules whose creat i o n i s caused by molecules r e f l e c t e d o f f the inside of the skimmer. The skimmer degradation of the beam can be avoided by placing the skimmer i n a region where ^ I' ( A n 6 6 ). This M c r i t e r i a generally r e s u l t s i n large nozzle-skimmer separations. Scattering of the beam by background gas occurs i n a l l regions of the apparatus i f the pressures are s u f f i c i e n t l y high. The beam-intensity at a distance / from the source can be calculated using r \n _ \ -no-dX 2 > I o e (is) where I and I 0 are the attenuated and unattenuated beam i n t e n s i t i e s , n the l o c a l gas density and cr the scattering cross section. Scattering of the beam i n the nozzle-skimmer region i s 33 a s p e c i a l case, as the presence of the barrel shock p a r t i a l l y prevents the beam from scattering i n the region between the nozzle and the Mach disk l o c a t i o n . Scattering of the beam occurs downstream of the Mach disk and occurs under c e r t a i n conditions to a lesser extent upstream as well (Br66, An65b). Thus to reduce scattering i n the nozzle-skimmer region to a minimum the skimmer should be upstream of the Mach disk l o c a t i o n which can be predicted using Eq. 12. This condition usually c o n f l i c t s with the separation for minimum skimmer interference and hence a compromise s i t u a t i o n develops. Typical experimental i n t e n s i t y p r o f i l e s obtained as a function of nozzle-skimmer separation are shown i n F i g . 30. The i n t e n s i t y maximum i s the physical r e a l i z a t i o n of this compromise between background gas scattering and skimmer interference. Downstream of the maximum, background scattering dominates while upstream, skimmer interference dominates. At small nozzle skimmer separations the r e l a t i v e l y high beam i n t e n s i t y i s caused by the free j e t "popping" through the skimmer and expanding towards the collimator which now acts as a skimmer. With increased separation the j e t slowly returns to i t s normal l o c a t i o n and skimmer interference i s at i t s maximum hence the minimum i n beam i n t e n s i t y at Lfo ~ 5. (7) Low Temperature Nozzle Sources and their uses. Few experimental r e s u l t s e x i s t describing the nature of helium beams formed by nozzles cooled to l i q u i d helium temperatures. Becker, Klingelhofer and Lohse ( B e 6 l , Be62) report observing a condensed helium beam with an i n t e n s i t y of 3 k l 0 2 x l 0 1 9 atoms/sr-sec, a v e l o c i t y of 165 m/sec and a v e l o c i t y FWHM corresponding to a Mach number of about 80. These results were obtained with a 0.15 mm diameter nozzle operated at 7*K> Torr. They report no observations of an uncondensed beam at l i q u i d helium i n t e n s i t i e s . Zapata, Ballard and Cabrera (Za69) have recently made some measurements of i n t e n s i t y and v e l o c i t y d i s t r i b u t i o n of ^ e beams produced by a cryogenically cooled nozzle source. They report a peak i n t e n s i t y of 6x10^ atoms/ sr-sec and a beam v e l o c i t y of 320 m/sec. This corresponds to a nozzle temperature of 10°K. These results obtained with a 0.11 mm diameter nozzle operated at pressures up to 200 Torr are i n reasonable agreement with corresponding r e s u l t s presented l a t e r i n this work. No work has apparently been done with ^Ee nozzle beams cooled to l i q u i d helium temperature. The study of condensation fragments i n low temperature Argon beams has been c a r r i e d out by Milne and Greene (Mi67) using mass spectrometer techniques and Audit and Rouault (Au69) using electron d i f f r a c t i o n techniques. Both these experimenters were interested i n studying the intermolecular p o t e n t i a l acting between groups of Argon atoms. Milne used these experimental measurements to test t h e o r e t i c a l calculations for the concentration of dimers, trimers etc. i n a p a r t i a l l y condensed beam. Hopefully studies with helium beams sim i l a r to those made with argon beams w i l l show r e s u l t s attributal to quantum eff e c t s which would be present with helium at low temperatures but which would not have played a noticeable role i n the case of Argon. The use of both % e and ^He may r e s u l t i n i n t e r e s t i n g 35 differences because of the Fermi-Dirac and Bose Enstein S t a t i s t i c s obeyed by the two gases. Becker et a l (Be5 L, Be56, Be6l, Be62) i n their o r i g i n a l work with low temperature nozzle beams were interested i n studying condensation phenomena. P a r t i a l l y condensed beams formed from a mixture of the isotopes, ^>Ee and ^He, might be used to p r e f e r e n t i a l l y enrich the r e s u l t i n g beam with one of the isotopes. Such a technique at higher temperatures could possible be used i n the separation of l i g h t and heavy water vapour, a process of considerable commercial i n t e r e s t . Isotopes can be enriched i n the free j e t expansion without condensation because of the dependence of the v e l o c i t y components on the mass of the p a r t i c l e s . Knuth and Fisher (Kn68) have used Argon beams expanded from room temperature to measure v i s c o s i t y cross sections at temperatures as low as 10°K. Similar studies could be car r i e d out with helium beams s t a r t i n g from room, l i q u i d nitrogen or l i q u i d helium temperatures. Again the a v a i l a b i l i t y of beams of both ^He and Sfe allows comparison between the effects due to their d i f f e r e n t s t a t i s t i c s . As w i l l be mentioned l a t e r i n this •5 k work, the difference i n scattering cross section for JHe and He has been observed for jets expanded from a l i q u i d helium cooled nozzle. The a v a i l a b i l i t y of a low energy helium beam i s of i n t e r e s t i n the study of gas-surface i n t e r a c t i o n . Zapata, B a l l a r d and Cabrera (Za69) have constructed a l i q u i d helium cooled nozzle source to allow the study of the surface phonon 36 spectrum of a c r y s t a l by scattering an aerodynamic beam of p a r t i c l e s o f f a pure, i s o t r o p i c c r y s t a l . A low temperature helium beam may be of use i n the study of the free j e t expansion i t s e l f , i n p a r t i c u l a r i n the nature of the t r a n s l a t i o n a l r e l a x a t i o n effects which even for helium beams expanded from room temperature show a behaviour d i f f e r e n t from that of most other gases. 37 CHAPTER IV THE POLARIZED 3 H e + BEAM SOURCE A. The Low Temperature Atomic Beam Source. (1) General d e s c r i p t i o n of Atomic Beam Source. F i g . 10 shows the f u l l atomic beam apparatus. ^He gas precooled to l i q u i d nitrogen temperature passes through a cryostat f i l l e d with l i q u i d helium and subsequently flows through a nozzle, skimmer and collimator system thus forming -the atomic beam. The copper cryostat weighing 10 kg i s supported at the top by a single s t a i n l e s s s t e e l tube. The l i q u i d helium entering the cryostat passes through a transfer l i n e inserted through the center of this tube; the evaporating gas passes up inside the support tube through a heat exchanger used to precool the incoming ^He gas and then back to the helium recovery system. The bottom of the cryostat i s connected to the l i q u i d nitrogen cooled s h i e l d by a thi n s t a i n l e s s s t e e l bellows, acting as a d i f f e r e n t i a l pumping bulkhead. The design of the cryostat, i t s supports and connections along with the q u a l i t y of the vacuum surrounding the chamber, determines the heat leak to the l i q u i d helium and hence the evaporation rat e . The calculated and measured heat leaks of the various components of the cryogenic system i n d i f f e r e n t .stages of assembly are summarized i n Table 1. The heat leak was determined by measuring the rate of helium gas b o i l o f f from the cryostat. The reduction of the r a d i a t i o n heat load from-38 T O H H e R E C O V E R Y S Y S T E M — mitt"" 7Z N O Z Z L E A S S E M B L Y E J E C T O R L I Q U I D N I T R O G E N He I N L E T - 2 TL T R A N S F E R T U B E H E A T S H I E L D H E X A P O L E E T rzzo D I F F U S I O N P U M P r F i g . 10 The Low Temperature ^>Ue Atomic Beam Source, Table 1 Detailed Summary of Heat Leaks into the Cryostat of the Low Temperature Atomic Beam Source. System Condition Stripped cryostat (no bellow; s o l i d plates over a l l openings) Bellows between k°K cryostat and copper heat shield Radiation over 10" d i f f u s i o n pump (pump not operating) Radiation b a f f l e on Cu heat s h i e l d Pumping to Leybold pump Operation of large d i f f pump Pumping ports on Cu heat s h i e l d Total Measured Expected F i n a l l y heat . heat reduced leak leak to (watts) (watts) (watts) 0.25 0.3*+ 0.2 0.16 0.16 0.16 0.3 0.0 0 0.76 OoO 0.16 0.63 0.06 0.009 0.2 0.0 0.06 0.25 0.0 0 2.55 0.56 C 6 7 ho 300°K surfaces by painting 77°K ba f f l e s with high eraissivity aqua dag was e s s e n t i a l to the succesful operation of the cryostat. The heat load on the system was a c t u a l l y reduced by about 10$ when gas flow through the nozzle was introduced. This confirms the expected high e f f i c i e n c y i n excess of 98$ of the heat exchanger system and also indicates that the cold gas flowing i n the channel to the Leybold Ejector pump acted as a sink for a f r a c t i o n of the heat leak coming through the metal bellows there. The t o t a l measured heat leak for the system was 0.6 watts. With a cryostat capable of holding 6 l i t e r s of l i q u i d helium this allowed 7 hours operation between f i l l i n g s . Typical cool down and i n i t i a l f i l l i n g of the cryostat required 19 l i t e r s of l i q u i d helium. Precooling of the system to 77°K for 8 hours required about 80 l i t e r s of l i q u i d nitrogen while the continued operation of the system required 8 l i t e r s / hour. The vacuum pumping system was separated into four parts. One s e c t i o n was used to pump the nozzle-skimmer region; one to pump the skimmer-collimator region; one to pump the region surrounding the cryostat and magnet, and a fourth to pump the ion i z e r chamber. The pumping system for the nozzle-skimmer region consisted of a Leybold Hg h5 mercury ejector pump with a pumping speed of 4-5 l i t e r s / s e c at a pressure of 10"^ Torr and 20 l i t e r s / s e c at 10"^ Torr. This pump was capable of operating into a backpressure of 30 Torr for a closed r e c i r c u l a t i n g ^He gas system but was normally provided with a Welch l*+02 pump. The skimmer-collimator region was. pumped hi with a CVC 10" o i l d i f f u s i o n pump having a pumping speed of kOOO l i t e r s / s e c at pressures below 10"3 Torr. The surrounding region was pumped with 2 Heraeus 6" d i f f u s i o n pumps each having pumping speeds of 1500 l i t e r s / s e c . The i o n i z e r chamber was pumped with a CVC 6" d i f f u s i o n pump having a pumping speed of 1HO0 l i t e r s / s e c o The 10" and three - 6" d i f f u s i o n pumps were backed with a Stokes *+0 cfm rotary backing pump to maintain the necessary forepressure. The 10" and two Heraeus 6" pumps were provided with water cooled b a f f l e s , the Leybold pump was provided with a l i q u i d nitrogen cooled trap and the CVC 6" pump was provided with a freon cooled b a f f l e . Pressure measurements i n the nozzle-skimmer and skimmer-collimator regions were made by means of P i r a n i gauges attached to the end of long s t a i n l e s s s t e e l tubes with inside diameters of 2„51+ mm and 2.h mm res p e c t i v e l y . These tubes passed from the appropriate region through the required thermal and vacuum bulkheads to the outside of the main vacuum chamber. Since one end of the tube was normally at room temperature and the other at either 77°K or h.2°K corrections for thermal t r a n s p i r a t i o n e f f e c t s were necessary. These corrections w i l l be discussed i n d e t a i l i n the next section of this chapter. The pressure i n the main system was monitored with an i o n i z a t i o n gauge. I t was possible to i n t e r l o c k the ion gauge control box t r i p out c i r c u i t to a r e l a y c o n t r o l l i n g the d i f f u s i o n pump power to prevent damage to the d i f f u s i o n pumps should the pressure i n the system become excessive. The o i l d i f f u s i o n pumps used Dow Corning 705 S i l i c o n e f l u i d which proved to be a very r e l i a b l e and robust pump f l u i d withstanding, ' k2 without damage, many accidental exposures to atmospheric pressure. Nozzle input and nozzle stagnation pressures v/ere measured with 2 Wallace-Tiernan gauges; one 0-760 Tori* and the other 0-1+00 Torr range. For nozzle pressures i n excess of 1 Torr the thermal.transpiration corrections required were less than 3$ and were hence ignored. De t a i l s of the nozzle, skimmer, and collimator system i s shown i n F i g . 11. The nozzle-skimmer distance could be adjusted by revolving the skimmer-collimator carriage on a screw thread (26 threads per inch) cut on the main nozzle assembly. The gear on the end of the skimmer-collimator carriage could be connected by a gear chain system to a rod passing through the vacuum chamber w a l l . An O-ring seal allowed the rod to be rotated from outside the vacuum system. One complete revolution of this rod correcponded to 1 /8 of a r e v o l u t i o n of the skimmer-collimator carriage which increased or decreased the nozzle-skimmer separation by 0.00^8 inches. This externally adjustable system functioned only with the nozzle at room temperature as the 0-rings freeze at l i q u i d nitrogen temperature and below. Nozzles, skimmers, and collimators of'appropriate size and shape could be i n s t a l l e d on this framework. Precise alignment was achieved by mounting the framework i n a lathe and observing the various apertures with a telescope as the nozzle system was rotated. (2) Thermal Transpiration Corrections to Pressure > Measurements. © o-ring n ^moin nozile ossemoty 7J /•skimmer-collimofor carnage 64 cog g ear F i g . 11 The Ajustable Nozzle-Skimmer Assembly. ,-r kk When one end of a pressure sensing device i s at a temperature d i f f e r e n t from the other end i t i s necessary to correct the measured pressure for thermal transpiration e f f e c t s . F i g . 12 Conditions for Thermal Transpiration Effect, A t y p i c a l s i t u a t i o n i s shown i n F i g . 12 i n which two volumes and V 2 at temperatures and T 2 are connected by a tube of diameter d. I f the density i n the' volume i s such that A >~? d then the gas flow through any opening according to Eq. 1 i s JL proportional to nT 2. The steady state i s established when (19) now Ii - I i i V / z When c o l l i s i o n s between molecules predominate over c o l l i s i o n s against the walls, ie.A<^d, then the condition for equilibrium i s P]_ = P 2» Numerical values for 'the r a t i o P j / p 2 where T j < T 2 are presented by Roberts and Sydoriak (Ro56) for 3 j fe and ^ e . Their r e s u l t s are given as a function of the product p 2d which allows the necessary correction for tubes of varying sizes to h5 be e a s i l y determined. The length of the tube does not enter into these considerations. The appropriate r e s u l t s from their work r e l a t i n g to the measurement of pressure i n the nozzle-skimmer and skimmer-collimator regions are shown i n F i g . 13. At the low pressure end the dependence approaches the p, /T \^ i = / 1 \ r e l a t i o n s h i p expected from free molecular flow P 2 I V considerations while at the higher pressure end the dependence approaches the p-^  = p 2 dependence expected from continuum considerations. (3) Carbon Resistor Temperature Measurement. The temperature of the nozzle i n the atomic beam apparatus was measured with a 33-CL A l l e n Bradley r e s i s t o r i n one arm of a A.C. Wheats tone bridge operated at 1 K cycle as shown schematically i n F i g . l k 0 I t was possible to n u l l the bridge to - 1 a t l i q u i d helium temperatures. The n u l l resistance at near l i q u i d helium temperature as a function of o s c i l l a t o r input voltage i s shown i n F i g . 15. Normal operation of the bridge was i n the plateau region of this curve where the power disipat e d i n the r e s i s t o r was so small that i t d i d not influence the resistance of the carbon r e s i s t o r . The c a l i b r a t i o n of t h i s r e s i s t o r at room 295°K, l i q u i d nitrogen 77°K, and l i q u i d helium *t.2°K temperature i s shown i n F i g . 16. The resistance R of such a r e s i s t o r as a function of temperature T normally can be expressed by the r e l a t i o n s h i p (C152) - a ^ R -f-b ( 2 0 ) h6 nozzle-skimmer pressure correction skimmer-collimator pressure correction 2°K Pw • 300°K 1 200 300 P w (millitorr) F i g . 13 Thermal Transpiration Corrections to Measure-ments of Nozzle-Skimmer and Skimmer-Collimator Pressure. 180 — O ©-^ 170 — 160 ( • 150 - ; i ' ^ ; » 140 C . ! ° < -£ '20 U J cr 110 100 — 9 0 : ; r • H ! ? - . ]..:„_ L ' J 0 10 2 0 3 0 4 0 5 0 6 0 7 0 A M P L I T U D E SETTING on hp oscJIIotor 8 0 9 0 100 F i g . 15 Input Power Induced Heating of Carbon Resistor Resistor at Near Liquid Temperature. Thermometer with -r 1+8 k9 where a and b are suitable f i t t i n g constants,, The s t r a i g h t dotted l i n e i n F i g . 16 i s a l e a s t squares f i t to the experimental points with a = 0.4-5 and b = -2 .17. 1 nrrrx impedonce transformer I 1 scope F i g . Ik A.C. Bridge Used to Monitor Resistance of Carbon Resistor Thermometer. , B. The Hexapole Magnet. The tapered hexapole magnet, 50 cm long, used to separate the p a r t i c l e s i n the two possible nuclear spin substates of ^He i s shown i n F i g . 17. This magnet was o r i g i n a l l y designed by Axen (Ax65) to produce a near 100% separation of nuclear spin states for a Mach k nozzle source operating at 2 .2°K. The measured f i e l d strength i n the region of the pole tips as a function of the e l e c t r i c a l current through the c o i l s , and the measured value of the magnetic f i e l d strength as a function of the r a d i a l distance from the central axis are shown i n F i g s . .18 50 cms. . : J _J «_ 2-54 cms. Figure 17- Dimension of the Components of the Hexapole Magnet 51 to CO bO O H •H W a •H •H O •H +5 CU a to Pole Pieces P o s i t i o n of Probe 0 Current Increasing • Current Decreasing 20 T O oO E l e c t r i c a l Current i n Amperes "50 Figure 18. Magnetic F i e l d Strength i n the Region of the Pole Tips as a Function of the E l e c t r i c a l Current Through C o i l s . ?2 and 19 respect!vely. The average f i e l d gradient i n the region r = 2 . 5 t o r = 3 n i m i s 70 ,000 gauss/cm while the measured f i e l d at the pole tips Ho i s 9000 gauss. Ce The E l e c t r o n Bombardment Ionizer. The i o n i z e r used i n this experiment was an electron bombardment type i o n i z e r s i m i l a r to one b u i l t by Weiss (We6l) but employing side instead of a x i a l extraction. The development of the prototype io n i z e r i s described by Vermette ( V ^ ^ f ) . Vermette used filaments with rectangular cross section while the present i o n i z e r ' s filaments are c i r c u l a r i n cross section. A schematic end view of the i o n i z e r used i n this experiment i s shown i n F i g . 20. The filaments consist of 5 lengths of 0 . 0 1 0 " diameter tungsten wire. The active i o n i z a t i o n volume has a rectangular cross section 0 . 2 5 " wide by 0.22" high and a length of ^ . 75" . The ion current i s c o l l e c t e d on a plate (negatively biased) immediately at the side of the io n i z e r and was measured using a Hewlett Packard Model *+25A DC microvolt-ammeter. The e l e c t r o n i c s used are shown i n F i g . 2 0 A . The* above technique was suitable for measuring ion current yields by introducing c a l i b r a t i o n gases at known pressures into the i o n i z e r chamber. Measurement o f . i o n y i e l d s from the actual atomic beam was complicated by the necessity to separate the ion beam y i e l d from the r e s i d u a l background gas ion y i e l d . The lack of a bulkhead separating the i o n i z e r region from the magnet vacuum chamber and also excessive outgassing of the. i o n i z e r during operation was mainly responsible for this condition. These two d i f f i c u l t i e s were overcome by the i n s t a l l a t i o n of a beam chopper between the Magnetic Field in Kilogauss o o o Radial Distance from Central Axis in MMS. Figure 19 Measured Value of' Magnetic Fie l d Strength as a Function of the Radial Distance from Central Axis. grid - plot e separation ~ 0 2 2 f i l a m e n t widths = 0.25" f i l a m e n t terfgth =4.75" plate + 2 0 0 v 4 <+)—% grid i O O O ( filament ion collector hp AI-V ammeter 0.47 jjf scope signal F i g . 20 Schematic Representation of Ionizer and Ion Measurement Apparatus. (A) D.C. Current Measurement (B) Chopped Current Measurement F i g . 21 Two slit Chopper 55 end of the magnet and the entrance to the i o n i z e r . The motor and reference s i g n a l c i r c u i t used are i d e n t i c a l to those described i n chapter 5 for the t i m e - o f - f l i g h t velocity-measuring apparatus. . The chopper used i n the ion y i e l d measurement was e s s e n t i a l l y the same as that shown i n F i g . 21 except that the two s l o t s have been widened to 0.8 cm and lengthened to h cm. The electronics used i n making the ion beam current measurement employing the chopper system i s shown i n F i g . 20B. The i o n i z a t i o n e f f i c i e n c y of the i o n i z e r for hydrogen gas was determined by slowly r a i s i n g the background pressure i n the i o n i z e r chamber and measuring the r e s u l t i n g ion y i e l d . The r e s u l t s are shown i n F i g . 22. The pressure measurements were made with a Bayard-Alpert type i o n i z a t i o n gauge and were increased by a factor of 2.2 to allow for the r e l a t i v e s e n s i t i v i t y of the gauge to hydrogen and nitrogen, the gas for which the gauge i s c a l i b r a t e d . The operating conditions of the i o n i z e r during this measurement are summarized below. Plate and g r i d voltage 260 V Plate + g r i d current 600 mA The r e s u l t i n g i o n i z a t i o n e f f i c i e n c y of the i o n i z e r for hydrogen gas at room temperature i s 6.5 A/Torr. This y i e l d shown as a dashed l i n e on F i g . 22 compares favourably with the 6 A/Torr reported by Glavlsh (G168) of Auckland Un i v e r s i t y for a strong f i e l d a x i a l i o n i z e r . The University of B r i t i s h Columbia i o n i z e r has an i o n i z a t i o n volume of k.2 cm^ while the Auckland i o n i z e r has an e f f e c t i v e volume of 3«5 cm^. These volumes are relevant i n comparing i o n i z a t i o n y i e l d s of the 56 57 background gas but are not the relevant volume when considering actual beam i o n i z a t i o n e f f i c i e n c i e s . The l a t t e r efficiency-depends more on the active volume through which the atomic beam passes. The length of the i o n i z a t i o n zone, Ik cm for the Auckland ioni z e r and 12 cm for the U.B.C. io n i z e r , gives a better i n d i c a t i o n of the i o n i z i n g a b i l i t y . ' On the surface, the two ion i z e r s appear roughly comparable but the Auckland ioni z e r i s much superior i n at l e a s t two - respects, f i r s t i t i s a strong f i e l d rather then weak f i e l d i o n i z e r thus allowing s u b s t a n t i a l l y higher p o l a r i z a t i o n values and second i t s a x i a l design re s u l t s i n beam emittances much superior to the side extraction i o n i z e r . 58 CHAPTER V TECHNIQUES FOR MEASUREMENT OF ATOMIC BEAM INTENSITY AND VELOCITY A . Measurement of Atomic Beam Intensity. Measurements of atomic beam i n t e n s i t y were achieved using a d i f f e r e n t i a l P i r a n i detector. This detector, described by Jassby (Ja6*+), has a s e n s i t i v i t y of 1 . 0 ^ 0 . 1 x l O 1 ^ atoms/ If sr - sec -y&t-volt s i g n a l measured with a He oven source beam. Jassby shows that the r e l a t i v e s e n s i t i v i t y S of the detector for ^He and ^He depends on the r a t i o of s p e c i f i c heats C y and accomodation c o e f f i c i e n t s o£ of the two gases according to the following r e l a t i o n Se „ C v 4 W c °^4Ue (2D Se C v 3 H e <* - 3 H e At room temperature the s p e c i f i c heats of the two gases are i d e n t i c a l ; thus the r a t i o reduces to V _ ^ H e ( 2 2 ) Thomas, Krueger-and Harris (Th69) give an experimental value for the r a t i o cX^^ / c< 3^ e of accomodation c o e f f i c i e n t s on clean tungsten at 308°K as 1.088i.029. I t i s u n l i k e l y that the tungsten filaments used i n our detector operating at 560°K w i l l be p a r t i c u l a r l y clean; nevertheless this number indicates the approximate difference i n s e n s i t i v i t y of the gauge for ^Ee and He that should be expected. 59 Bo The Time-of-Flight Measuring Apparatus. The v e l o c i t y of the atomic beam was measured using a t i m e - o f - f l i g h t system. F i g . 23 shows the main components of this system. The chopper used i n this system i s shown i n F i g . 21 and consists of a 1*+ cm diameter disk 0 .8 mm thick with 2 s l o t s each 2 mm wide and 2 cm long. The re v e r s i b l e motor located inside the vacuum chamber i s capable of revolving the disk at 30,000 rpm. A trigger reference si g n a l i s provided by the phototransistor i n the c i r c u i t shown i n F i g . 2h, The length of the f l i g h t path from the chopper to the center of the i o n i z a t i o n gauge 2 .5 cm i n diameter and 2 .5 cm long was 57°^ cm. The low frequency e l e c t r o n i c noise on the s i g n a l was removed by the high pass f i l t e r shown i n F i g . 25 . The high frequency s i g n a l was displayed on a Tektronix 56h o s c i l l i s c o p e screen. The reference s i g n a l from the phototransistor was used to trigger the o s c i l l i s c o p e ; the time separation of the trigger and s i g n a l pulse was measured with the chopper rot a t i n g i n both clockwise and counterclockwise d i r e c t i o n s . Measurements i n both di r e c t i o n s were required to eliminate errors due to inaccurate mechanical alignment which resulted i n the phototransistor being triggered s l i g h t l y before or a f t e r the atomic, beam was chopped. The average of the two times and the measured f l i g h t path are then used to calculate the most probable v e l o c i t y of the beam. In Appendix D the shape of the t i m e - o f - f l i g h t s i g n a l S(t) i s shown to be r e l a t e d to the d i f f e r e n t i a l i n t e n s i t y d i s t r i b u t i o n function I(v) for the case of an i n f i n i t e l y short 60 chopper disk motor beam axis 0^photo transistor 2.5 2 . 5 cm i o n gauge F i g , 23 Schematic Representation of Time-of-Flight V e l o c i t y Measurement Equipment. I M 1.5 K -o 7.5 volts ^ output F P T - 100 F i g . 2*+ Photo transis tor Reference Signal C i r c u i t . shutter function and detector length by the re l a t i o n s h i p S(-t) - Cor\s+an+ _L X (V) ( D 1 ) LoKere V -' t For a given I(v) the f u l l width at half maximum FWHM of the s i g n a l S(t) i s used to obtain the Mach number M of the beam. | 1 I 1000 pf ' ion gouge I detector i 5 0 0 K L high pass f i l ter a ion gouge signal b re ference signal textronix 564 ~, . storage -Oinpufa scope Q in put b friggg Q D _____ light photo transistor F i g . 25 Schematic of Ion Gauge Signal C i r c u i t , Reference Signal and O s c i l l i s c o p e Display. The e f f e c t a f i n i t e width rectangular shutter function and detector length has on the sign a l shape S(t) are also considered i n Appendix D. The rel a t i o n s h i p between the F ¥ H M A t Q of the si g n a l S(t) with an i n f i n i t e l y thin shutter function and the FWHM A t for a rectangular shutter function of f i n i t e width T i s shown i n F i g . 26. The c a l c u l a t i o n o f _ t \\ J 1 I M i l 11! I I I I I 1 111 \ .1 .3 .6 1 3 6 10 F i g . 26 Broadening of Experimental Signal due to F i n i t e Width Shutter Function. Nozzle Beam Curve Includes Correction for a 2.5 cm Detector Length. ON 63 was done for a detector length of 2.51* cm. Thus with the aid of F i g . 26 the FWHM of the experimental signal A t was related to the th e o r e t i c a l s i g n a l A t and hence to a Mach number M of the d i f f e r e n t i a l i n t e n s i t y d i s t r i b u t i o n function I ( v ) , Also shown on this figure are some si m i l a r r e s u l t s obtained by Becker and Henkes (Be56b) for a d i f f e r e n t i a l i n t e n s i t y d i s t r i b u t i o n function a r i s i n g from an oven beam with a Maxwellian v e l o c i t y d i s t r i b u t i o n . In the c a l c u l a t i o n given i n Appendix D the shutter function i s assumed rectangular. Although the r e a l shutter function i s considerably more complicated, the consequence of the approximations r e s u l t i n g i n the assumption of a rectangular shutter function are shown to be small. With the time of f l i g h t geometry shown i n F i g . 23 the 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 time of a group of p a r t i c l e s passing through the chopper s l i t depends mainly on the i n t e n s i t y p r o f i l e of the atomic beam at the chopper and the s o l i d angle subtended by the ion gauge. The speed of r o t a t i o n of the chopper, the chopper s l i t width and other geometrical parameters are also considered i n determining the r e a l shutter function. From the geometry of the experimental arrangement only p a r t i c l e s eminating from the nozzle within a cone subtended by the extremities of the i o n gauge detector enter the detector. Hence, at the chopper l o c a t i o n , the beam has at most an e f f e c t i v e diameter of 5.3 mm. As i s shown i n F i g . 38 the actual measured beam p r o f i l e at the chopper l o c a t i o n had a FWHM of 7 mm. The i n t e n s i t y i n the centre 5.3 mm wide s t r i p varied about ± \ % f r 0 m the average of i t s value at the side 6k of this region and the peak value at the centre. Thus the beam i n t e n s i t y p r o f i l e can be assumed to be rectangular. This assumption Implies that the beam cross section i s not c i r c u l a r but i s i n f a c t square. Hagena, Scott, and Varma (Ha67b) have shown for the case of a shutter width equal to the width of the beam that this approximation had l i t t l e e f f e c t on the shape of the shutter function. With these assumptions and an average tangential chopper v e l o c i t y ' o f 35 m/sec, corresponding to our experimental conditions, the calculated i n t e n s i t y d i s t r i b u t i o n of p a r t i c l e s passing through the chopper as a function of time i s shown i n F i g . 27. _ ( ld iv .= 0 . 0 2 8 ms) F i g . 27 Assumed Intensity P r o f i l e of P a r t i c l e s Passing through the Chopper Opening (Shutter Function). Trangential Chopper V e l o c i t y = 35 m/sec. To apply properly the r e s u l t s of Appendix D this i n t e n s i t y d i s t r i b u t i o n must also be assumed approximately rectangular.in shape. As w i l l be described i n section 6B, the c o r r e c t i o n to the experimental s i g n a l as a r e s u l t of the f i n i t e chopper width i s s u f f i c i e n t l y small, less than 5%, that the 65 approximations made i n deriving the correction formula do not produce any s i g n i f i c a n t errors. 66 CHAPTER VI RESULTS OF STUDIES OF THE ATOMIC BEAM A. The Atomic Beam Intensity. The atomic beam source previously described has been tested under many varying conditions. Parameters such as the nozzle diameter d, skimmer diameter, nozzle-skimmer separation Xs, stagnation pressure p 0 and stagnation temperature T 0 have been varied and the r e s u l t i n g beam i n t e n s i t y I measured. Typical data are presented for.0 . 2 mm and 0.025 mm diameter nozzles. Beam i n t e n s i t i e s for a 0.2 mm diameter nozzle as a function of nozzle pressure for p a r t i c u l a r nozzle-skimmer separations and the nozzle at room temperature are shown i n F i g . 28. The skimmer and collimator were 0.6 mm and 1 mm i n diameter r e s p e c t i v e l y while the skimmer-detector separation was 15 cm. The i n t e r p r e t a t i o n of these contours i n the manner described i n Chapter I I I i s most c l e a r l y seen by considering cross sections obtained at fixed stagnation pressures and varying the nozzle-skimmer separation. Such a set of curves i s presented i n F i g . 29 for the room temperature data of F i g . 28. With the exception of one or two points the data points were f i t t e d with a smooth curve. The d i f f i c u l t y i n producing completely consistent data can be attributed to the lack of p r o f i l e s at a s u f f i c i e n t number of nozzle-skimmer separations, the d i f f i c u l t y of keeping the detector alignment exactly .the same between runs when the system was disassembled N O Z Z L E - S K I M M E R SEPARATION (nozzle d iameters) F i g . 29 Room Temperature Beam Intensity P r o f i l e s as a Function of Nozzle-Skimmer Separation. 69 and reassembled to vary the nozzle-skimmer separation, and the a c c i d e n t a l l y enforced use of a d i f f e r e n t nozzle for the measurements at a separation of 5 nozzle diameters. The room temperature data shown i n F i g . 30 was obtained with the adjustable nozzle-skimmer apparatus. The physical alignment of the detector remained fi x e d throughout the measurement and the nozzle-skimmer separation was adjusted from outside the vacuum system. The large number of data points can be f i t t e d with very smooth curves. Although the nozzle and skimmer used i n obtaining the data of F i g . 30 were approximately the same diameter as those used i n obtaining the data of F i g . 29 , they are not the same nozzle and skimmer. This accounts for the differences i n absolute I n t e n s i t i e s and nozzle-skimmer separations reported for the peaks and v a l l e y s , however, the general shape of the p r o f i l e s remains the same. The res u l t s shown i n these two figures i l l u s t r a t e the t y p i c a l dependence of beam i n t e n s i t y on nozzle-skimmer separation obtained with supersonic nozzle systems, namely, the large beam i n t e n s i t y at very short nozzle-skimmer separations followed by a reduction i n i n t e n s i t y to a minimum followed by a further increase i n i n t e n s i t y to a maximum and subsequent attenuation. This behaviour- i s due to the ef f e c t s described i n Chapter I I I . For very small nozzle-skimmer separations the beam passes through the skimmer with no i n t e r a c t i o n and then expands as a free j e t downstream of the skimmer. As the separation i s increased the beam i s influenced by the skimmer and eventually the "skimmer i n t e r a c t i o n " described previously takes i t s maximum e f f e c t . Further separation of the nozzle and skimmer o 0 ^ 5 t 3 NOZZLE P R E S S U R E P c o x O — x T0 = 295°K • A O ! 0 TORR 3 4 8 3 140 xx A O o o Po **** oo^* Co.-<3» o A A* O Ox ^O O P A A A 0< AA A A A & *A*6cP A A A O V : A A O * t3 4 2 4 ' • • • • • D^d • • o& 0 : E : °o • T 0 =77 A O 'K. 6 0 5 10 15 2 0 25 3 0 N O Z Z L E - S K I M M E R S E P A R A T I O N 35 L/D Fig.. 30 Continuous Beam P r o f i l e Taken with Adjustable Nozzle-Skimmer Assembly, 71 r e s u l t s i n a reduction of the gas density at the skimmer entrance. Reduced gas density at the.skimmer implies reduced skimmer i n t e r a c t i o n so that the beam i n t e n s i t y increases. As the separation gets even larger the skimmer becomes downstream of the Mach disk and background gas scattering becomes more and more s i g n i f i c a n t , eventually overwhelming any increase i n beam i n t e n s i t y due to reduced skimmer i n t e r a c t i o n . The appearance of the 'maximum maximorum' described by Campargue (Ca66) i s seen i n both F i g s . 29 and 30 where for increasing nozzle pressures the peak i n t e n s i t y grows slowly then reaches a maximum and thereafter f a l l s o f f . A s i m i l a r treatment of the l i q u i d nitrogen temperature data shown i n F i g , 31 gives the p r o f i l e s shown i n F i g . 32 . An examination of the data shows a behaviour si m i l a r to the room temperature data, though with much less prominent maxima, for a set of nozzle stagnation pressures considerably lower than i n the room temperature case. This "compression of contours" i s caused by the higher gas d e n s i t i e s , hence much smaller mean free paths and greater scattering for a given nozzle pressure at the lower temperature. Also the higher gas densities at the skimmer increase the skimmer i n t e r a c t i o n . Another e f f e c t which s t a r t s to become evident at higher pressures and lower temperatures i s beam formation from the skimmer or collimator openings. This e f f e c t causes increased beam i n t e n s i t y for increased stagnation pressures at large nozzle-skimmer separations. More w i l l be said about this while discussing the l i q u i d helium cooled beams i n the next paragraph. A s i m i l a r p l o t of i n t e n s i t y as a- function of nozzle-NOZZLE - SKIMMER SEPARATION (nozzle diameters) F i g . 3 2 L i q u i d Nitrogen Temperature Beam I n t e n s i t y P r o f i l e s as a Function of Nozzle-Skimmer Separation. skimmer separation using the nozzle stagnation pressure as a parameter i s shown i n F i g . 33 for the data displayed i n F i g . 3*+. This data was obtained with the nozzle cooled to l i q u i d helium temperature. The large scatter of points i s due to the great compression of the peaks and valleys into a nozzle pressure range of only a few Torr and the dominant role played by the formation of subsidiary beams from the skimmer and c o l l i m a t o r . With experimental r e s u l t s for so few nozzle-skimmer separations and because of the scatter of the observed points the consistent trends do not appear as i n the data points obtained at higher nozzle temperatures. The gradual r i s e i n i n t e n s i t y at large nozzle-skimmer separations i s consistent with a reduction of the gas flow past the skimmer and hence a r e s u l t i n g reduction of pressure i n the skimmer-collimator region and downstream of the collimator. The improved background pressure r e s u l t s i n improved transmission of subsidiary beams from the skimmer and collimator and the r e s i d u a l nozzle beam. The actual beam i n t e n s i t y versus nozzle pressure p r o f i l e s measured at a fix e d nozzle-skimmer separation are of considerably more i n t e r e s t i n this case. For the p r o f i l e at a separation of 19«5 nozzle diameters shown i n F i g . 3 L the beam i n t e n s i t y i n the region up to the f i r s t maximum which occurs at a nozzle pressure of 1 Torr i s attributed to a true nozzle beam.. The reduction of beam i n t e n s i t y for higher nozzle pressures i s attributed to the dominance of background gas scattering i n the nozzle-skimmer region. The subsequent increase i n beam i n t e n s i t y i s caused by the combined e f f e c t of a small r e s i d u a l nozzle beam along o 5 o O O "5 C/) _ _ A 0 nozzle pressure X I torr 0 2 torr EI 10 t o r r A 2 0 torr j 50 torr 1 1 10 2 0 3 0 NOZZLE-SKIMMER SEPARATION (nozzle-diameters) F i g . 33 Beam Intensity P r o f i l e s as a Function of Nozzle-Skimmer Separation with the Nozzle at Liquid Helium Temperature. . 77 with a subsidiary beam from the skimmer and collimator openings. In data obtained with a smaller nozzle, described i n the next paragraph, the pressure range over which the f i r s t peak occurs i s expanded considerably and the actual behaviour of the nozzle beam i s much e a s i l y determined. Comparison with t h e o r e t i c a l predictions w i l l be described i n d e t a i l i n that section. F i g . 35 shows res u l t s obtained with a 0.025 mm diameter nozzle at room, l i q u i d nitrogen, and l i q u i d helium temperatures. A 0.57 mm diameter skimmer and a 1 mm diameter collimator were used for these measurements. The nozzle-skimmer separation was set at 0.28 cm and the skimmer detector distance at 16 cm. F i g . 36 shows i n greater d e t a i l the r e s u l t s using both ^Ee and ^ e beams with the nozzle cooled to l i q u i d helium temperature. In both - figures the deviation from a l i n e a r r e l a t i o n with pressure i s due to background gas scat t e r i n g i n the nozzle-skimmer region. This can be checked by assuming the beam i n t e n s i t y I should be l i n e a r l y proportional to the nozzle pressure p 0 and the scattering pressure should be l i n e a r l y proportional to p 0 a l s o . F i g . 37 shows a p l o t of log \~p~0) v s Po* T ^ e observed l i n e a r decrease of i n t e n s i t y . with p 0 i s consistent with scattering of the beam according to I = I D e (18) where I 0 , the beam i n t e n s i t y with no s c a t t e r i n g , and n, the gas density i n the nozzle-skimmer region, are both assumed to be proportional to the nozzle pressure p 0? or i s the e f f e c t i v e scattering cross section and X the distance along the beam axi s . The difference i n the slopes for 3 j j e a n d He i n F i g . 37 indicates that the e f f e c t i v e scattering cross o <u v> I tn E o O >-_ _ < UJ CQ 4.4 4.2 4.0 3.8 3.6 3.4 3 .2 3.0 2 . 8 2.6 2.4 2 .2 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0 - i i © I I G) ' 0 A 0 / / / A / A / • / A V. a / A i/ / /A A 4 H e ROOM TEMP, a 4 H e L N 2 T O 4 H e L H e T © 3 He L H e T 0 1 0 0 2 0 0 3 0 0 4 0 0 5,00 6 0 0 7 0 0 8 0 0 N O Z Z L E P R E S S U R E P 0 (TORR) Fig. 3 5 0 10 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 P 0 (TORR) F i g . 3 6 Beam I n t e n s i t i e s for ^ Ee and ^ He Beams at Liquid Helium Temperature Before and After Correction for Scattering. Only the uncorrected experimental points are shown when the cor r e c t i o n i s small. F i g . 37 The Uncorrected Data of F i g . 36 Divided by the Nozzle Pressure to V e r i f y Existance of Scattering. 81 s e c t i o n for ^He is. 1 . 7 times larger than that of -% e. Both experimental v i s c o s i t y measurements and quantum mechanical ca l c u l a t i o n s also indicate a larger cross section for ^He. The d i f f e r e n c e i n cross section i s attributed to the d i f f e r e n t s t a t i s t i c s followed by 3 H e a n d t h Q a b s e n c e o f a n e a r stationary state i n the 3He 2 system (Bo51). The s t r a i g h t l i n e i n F i g . 36 was predicted using Eq. A5 assuming T 0 = 6.7°K, D = 0.025 mm and a 3ne beam. The dashed l i n e s shown i n F i g . 35 are the ^ e data, s i m i l a r l y corrected for scattering at room, l i q u i d nitrogen and l i q u i d helium temperatures. The dashed l i n e s are required to pass through the o r i g i n . The slopes of these l i n e s are i n the r a t i o 1 s 1.9 : 9 .5 . . Eqs. 5 and A5 predict the i n t e n s i t y should be d i r e c t l y proportional to the nozzle stagnation pressure p Q and i n v e r s e l y proportional to the square root of the nozzle stagnation temperature T 0. The straight l i n e s confirm the l i n e a r p 0 dependence; their slopes would correspond to temperatures 295, 82 and 3.3°K. However, v e l o c i t y measurements, discussed l a t e r , indicate temperatures of i 295, 77 and 7°K so that the t h e o r e t i c a l T Q^ dependence would appear to break do\^n at very low temperatures. Eqs. 5 and A5 p r e d i c t that the beam i n t e n s i t y should be i n v e r s e l y proportional to the square root of the atomic mass, so that the -'He i n t e n s i t y should be a factor J 3 larger than the ^ e i n t e n s i t y . The equality of the slopes for 3 H e A N D H ^Q L N F I G > 26 i s p a r t i a l l y a ttributable to the 9$ lower detector s e n s i t i v i t y expected for 3ne, as discussed i n Section 5A. There remains, however, a 6% discrepancy which could be due either to a s t i l l larger difference i n s e n s i t i v i t y for ^ Ee and ^He, or to an e f f e c t i n the nozzle system not taken account of i n the simple theory. A ^ e beam v e r t i c a l p r o f i l e taken 16 cm from the skimmer i s shown i n F i g . 38. It has a. f u l l width at half maximum (FWHM) of 7 mm (2.5°) and a f u l l width of Ih mm. The p r o f i l e was taken with a nozzle stagnation temperature and pressure of 77°K and L20 Torr r e s p e c t i v e l y . Gas flow through the 0.025 mm diameter nozzle i s estimated at 0.17 cc/sec (STP) f o r ^ He gas at p 0 = 50 Torr and l i q u i d helium temperatures. B. The Beam V e l o c i t y . A t i m e - o f - f l i g h t measuring apparatus was used to measure the most probable v e l o c i t y and v e l o c i t y d i s t r i b u t i o n of ^He and ^He beams produced using the cryostat at room, l i q u i d nitrogen and l i q u i d helium temperatures. For Sle at these temperatures the measured most probable v e l o c i t i e s were 1660, 850 and 270 m/sec re s p e c t i v e l y , using the 0.025 mm diameter nozzle and the ti m e - o f - f l i g h t geometry discussed i n section 5 B . Using the form of t h e . d i f f e r e n t i a l i n t e n s i t y d i s t r i b u t i o n given by Eq. 17, f i t s to the most probable v e l o c i t y determined the temperature of the nozzle to be 295, 77 and 7°K while s t i l l allowing a wide range of Mach numbers. At l i q u i d helium temperatures the ^Ee beam had a measured most probable v e l o c i t y of 310 meters/sec. The $Ee and ^He >-CO LU i -Q bJ N _! < DI O 1 i T i r i r 0 © \ .0 \ / 9 \ o o / J _ _ _ 0. J_ ' ' ' ' ' J -7 -6 -5 -4 -3 - 2 - 1 0 1 2 3 4 5 6 7 VERTICAL DISPLACEMENT (mm) F i g . 38 Normalized i n t e n s i t y P r o f i l e vs. V e r t i c a l Displacement from Beam Axis Stagnation Pressure Po = *f20 Torr Stagnation Temperature To = 77°K CO v e l o c i t i e s should d i f f e r only be a factor proportional to the square root of their masses eg. | lUi^ ^ = f 4" ' = 0.865. Experimentally a r a t i o of 270/310 = 0.87 was obtained i n good agreement with the expected dependence. The difference between 7 ° K and H-.2°K, the temperature of the l i q u i d helium, i s attributed td poor thermal contact between the nozzle and the cryostat and a thermal gradient due to the heat leak to the nozzle assembly. A temperature of 7 ° K i s consistent with carbon r e s i s t o r measurements of the nozzle temperature which indicated a temperature on the outside of the nozzle of 8 . 5°K. The measurement of the temperature using a carbon r e s i s t o r i s discussed i n Section HA 3 . A t y p i c a l 3He ti m e - o f - f l i g h t spectrum obtained with a chopper speed of 11,000 rpm i s shown i n F i g . 39. In this measurement the nozzle temperature was reduced s l i g h t l y by pumping on the helium r e s e r v o i r . A v e l o c i t y spectrum derived from the time spectrum i s shown i n F i g . ho. The experimental s i g n a l with a FWHM of 53 m/sec i s f i t t e d quite well by the dashed t h e o r e t i c a l curve obtained from Eq. 17 with the same FWHM and most probable v e l o c i t y using a stagnation temperature T 0 = 5 . 9 ° K and a Mach number M = 10. The chopper d i s t r i b u t i o n function shown' i n F i g . 27 has a FWHM T = 0.1^3 ms and the FWHM of the experimental sig n a l i s A t = 0.37 ms; thus the re s o l u t i o n R = = 2 . 6 . From F i g . 26 this corresponds to a broadening of A t - A t 0 = k% where At0 i s the width of the A t o i d e a l d i s t r i b u t i o n which would have been obtained had the shutter s l i t and the detector width been i n f i n i t e l y thin. The . ail. is. : ist. '., 1 I i l l . .. i = ? ' •'-•| : 1 , . / \ F i g . 39 T y p i c a l 3 H E Time-of-Flight Spectrum for Liquid Helium Cooled Nozzle. Stagnation Pressure i s 36 Torr. Horizontal Time Scale i s 0.5 m sec/div. The Upper Trace shows the Time Reference Light Pulse. # 1 98 w • • : as F i g . L 6 Typical Ionizer Signals from Chopped Atomic Beam with Hexapole Magnet Turned On and Off, Horizontal Scale 0.5 ms/dlv. V e r t i c a l Scale 0.5 mv/div. 1.0 £ 0.9 co 0.8 5 0 7 N I i 0.6 o z 0.5 0.4 0.2 o.ih : 0 _ L / / G O o I o ^ e exper imenta l resu l t s FWHM = 53 m/sec To= 5.9°K , M-10 [ FWHM=53 m/sec \ \ o 2 0 0 250 3 0 0 VELOCITY ( M/ SEC) 3 5 0 4 0 0 F i g . 40 Results of F i g . 3 9 Converted into a V e l o c i t y Spectrum. The curve shown i s a f i t of Eq. 17 to the experimental spectrum. co ON 87 approximations made here should not s i g n i f i c a n t l y increase this value of h%. Because of the small size of the re s u l t i n g c o r r e c t i o n i t w i l l be neglected. The experimental r e s u l t s shown i n F i g . ko were obtained with a nozzle stagnation pressure of 36 Torr. For this pressure and a stagnation temperature of 5.9°K the corresponding nozzle stagnation Knudsen number i s 1,8 x 10"^. Using Eq. 13 the terminal Mach number i s calculated to be 15 while the experimental data shown i n F i g . 8 indicates that for He at this Knudsen number the terminal Mach number would be 12, This i s i n good agreement with the experimental determination of M = 10 considering that the r e a l width of the measured d i s t r i b u t i o n i s a c t u a l l y s l i g h t l y narrower and as a r e s u l t the Mach number w i l l be s l i g h t l y higher. It should be noted that no i n d i c a t i o n of a condensed f r a c t i o n of the beam was observed i n the v e l o c i t y measurements. That i s , no second peak at a lower v e l o c i t y corresponding to a condensed f r a c t i o n of the beam appeared i n the ti m e - o f - f l i g h t spectrum. The appearance of a second peak has been described by Becker, Bier and Henkes (Be56) as being t y p i c a l of condensed beam formation. The good agreement between the measured and expected p o l a r i z a t i o n of the ^ He beam (as w i l l be mentioned l a t e r ) i s confirmation that at most a very small f r a c t i o n of the beam was condensed. 88 CHAPTER V I I POLARIZATION AND IONIZATION OF THE 3He BEAM Ac The T r a j e c t o r i e s of Atoms through the Hexapole Magnet. The development of the equations d e s c r i b i n g the t r a j e c t o r i e s of p a r t i c l e s through the hexapole magnet has been d e s c r i b e d by Axen (Ax65) and i s summarized i n Appendix C. The p o s i t i o n and slope of a focussed p a r t i c l e i n the tapered s e c t i o n of the magnet i s given by Eqs. C32 and C3*+ as a f u n c t i o n of the r a d i a l p o s i t i o n of the p a r t i c l e at the magnet entrance. The r a d i a l p o s i t i o n and divergence of the defocussed p a r t i c l e i s gi v e n by Eqs. C36 and C37. The p o s i t i o n and slope of the p a r t i c l e s i n the p a r a l l e l s e c t i o n of the magnet were c a l c u l a t e d using Eqs. C25 and C26 i n the case of the focussed p a r t i c l e and Eq. C27 i n the case of the defocussed p a r t i c l e . The i n i t i a l p o s i t i o n and slope of the p a r t i c l e as i t enters the p a r a l l e l s e c t i o n i s determined from the s o l u t i o n of the t r a j e c t o r i e s a t the end of the tapered s e c t i o n s . The hexapole magnet was designed f o r a beam w i t h a most probable v e l o c i t y of 175 meters/sec. As has been d i s c u s s e d i n the preceeding s e c t i o n the beam produced had a c o n s i d e r a b l y higher v e l o c i t y . T h e - c a l c u l a t i o n s of the p o s i t i o n and slopes of p a r t i c l e s a t c e r t a i n l o c a t i o n s i n the magnet f o r va r y i n g r a d i a l d e f l e c t i o n s at the magnet entrance and var y i n g s e p a r a t i o n of the magnet from the nozzle source are summarized i n Table 2. I n these c a l c u l a t i o n s the f i e l d H Q at the pole t i p s of the magnet was 9000 gauss. T y p i c a l t r a j e c t o r i e s of.the Table 2 Selected Trajectories of Focussed and Defocussed •^ He Atoms Passing Through the Tapered Hexapole Magnet FOCUSSED TRAJECTORIES R = r a d i a l d e f l e c t i o n S = slooe SOURCE-MAGNET SEPARATION P a r t i c l e Radial Deflection) 5 cm 15 cm 16 cm V e l o c i t y (m/sec) at entrance to magnet (cm) At End of Taper At End of Magnet At End of Taper At End of Magnet At End of Taper At End of Magne t 245 . 0 .025 0 . 0 5 0.075 0 , 1 0.125 R= 0.0914 S= 0.0039 i 0 . 1 8 2 8 0.0078 0.1717 0.0003 0.3433 0.0007 R=0.o44 S=0.00095 0 .087 0.0019 0.13 0 .0029. 0.17 0.0038 0.22 0.0047 0 .055 - 0 . 0 0 0 4 0.109 - 0 . 0 0 0 7 0.1644 -0 .0011 0.215 - 0 . 0 0 1 4 0.0425 0.0009 0.0847 ' 0 . 0 0 1 7 0.1274 0.0026 0.1698 0 .0035 0.212 O.0523 - o . o o o 4 0.1016 - 0 . 0 0 0 8 0.1539 -0.0012 0.2061 -0 .0016 0.255 310 0 .025 0 . 0 5 0.075 0 . 1 0.0946 0.0043 0.1892 0.0086 0.2070 0.0018 0 . 4 l 4 l 0.0037 0 . 0 4 6 0.0012 0.092 0.0024 0.138 O0OO36 0.185 0.00486 0.072 0.0002 0.1445 0 .0005 0.2168 • 0.0007 0 . 2 9 0 .001 0 . 0 4 4 7 0.0011 0.0893 0.0022 0.1340 0.0034 0.1787 0 .0045 0.0681 0.0002 0.1316 o .ooo4 0.2074 0.0006 0.2756 0.0008 DSFOCUi 3SED TRAJECTORIES 245 0 .025 0 . 0 5 0 .075 R=0.91 R = 0 . l 8 0.36 0 .54 0.16 0 . 3 3 0.50 ' 310 0.025 0 . 0 5 0.075 0.79 0 . 1 5 8-4 0 . 1 4 0 . 2 8 0 . 4 2 90 focussed and defocussed atoms for the case of a nozzle-magnet separation of 16 cm are shown i n F i g . kl. As can be seen, the magnet proves incapable of bringing p a r t i c l e s to a focus on the magnet axis i f they have a v e l o c i t y of 310 m/sec ir r e s p e c t i v e of the r a d i a l distance from the axis at which they enter the magnet except for a small number of p a r t i c l e s very close to the magnet ax i s . The magnet does succeed however i n keeping the t r a j e c t o r i e s of p a r t i c l e s which enter the magnet less than 1 mm from the axis within the pole tips although the t r a j e c t o r i e s are s t i l l diverging when they leave the magnet. These t r a j e c t o r i e s should be a good approximation to the paths the 3jje atoms produced from the nozzle source w i l l take through the actual magnet. B. The Calculated P o l a r i z a t i o n of the Atomic' Beam. The r e l a t i v e numbers of focussed and defocussed p a r t i c l e s passing through the hexapole magnet which subsequently pass through some s p e c i f i e d i o n i z a t i o n volume downstream of the magnet e x i t i s needed i n order to calculate the expected p o l a r i z a t i o n of the beam. This information could i n p r i n c i p l e be obtained from the atomic t r a j e c t o r i e s presented i n the previous section. This would involve c a l c u l a t i n g the t r a j e c t o r i e s for p a r t i c l e s with many d i f f e r e n t v e l o c i t i e s and i n i t i a l conditions then deciding whether or not a given t r a j e c t o r y passes through the magnet and into the i o n i z e r . Then the r e s u l t would have to be appropriately weighted for the e f f e c t of the v e l o c i t y d i s t r i b u t i o n of the p a r t i c l e s i n the beam and the s o l i d angle of p a r t i c l e s of a given v e l o c i t y F i g . kl Typical Trajectories of Focussed and Defocussed ~>He Atoms passing through the Hexapole Magnet.-..The Source-Magnet Separation i s 15 cm and the F i e l d at the Magnet Pole Tips i s taken as 9000 Gauss. 92 which would contribute to the f i n a l i n t e n s i t y . Glavish (GI67, G168) has written a computer program which does t h i s . His program takes into account a l l the possible t r a j e c t o r i e s the atoms can take and the v e l o c i t y d i s t r i b u t i o n of atoms i n the beam. This program was i n i t i a l l y used for calculations involving the focussed proton states for a beam with a Maxwellian v e l o c i t y d i s t r i b u t i o n . These calculations were used for optimizing the various system parameters for a polarized proton i o n source developed at the U n i v e r s i t y of Auckland. The program was modified at U.B.C. to calculate the t r a j e c t o r i e s of both the focussed and defocussed states of an atomic 3 He beam with the v e l o c i t y d i s t r i b u t i o n c h a r a c t e r i s t i c of a nozzle beam. The r e l a t i v e i n t e n s i t y and p o l a r i z a t i o n obtained inside an i o n i z e r aperture 1.0 cm i n diameter are shown i n Figs, H-2 and. 43 as a function of the distance of the magnet entrance from the nozzle source and the temperature of the nozzle. Calculations were done for magnet-ionizer separations of 0, 10, and 20 cm. No change i n the i n t e n s i t y and p o l a r i z a t i o n inside the 0 .5 cm radius i o n i z a t i o n volume i s predicted for these three separations. More w i l l be said of these r e s u l t s i n Chapter VIII. For the conditions under which the magnet was operated namely, nozzle temperature = 7°K, Mach number of beam = 10 and a magnet-source separation = 15 cm, Glavish's program has been used to calculate the r a t i o of the t o t a l f l u x of atoms into the i o n i z e r aperture with the magnetic f i e l d on to the 93 H e 3 T ' 4 . 2 ° K M-10 175 | facu&&e4- " " — — 150 — 125 -co c I 100 2 ^ P ' °? INTENSITY 1 (ari POLARIZATION P O Oi — 25 - • 0 1 1 I 0 5 10 15 S O U R C E - M A G N E T S E P A R A T I O N (cm) F i g . k2 E f f e c t of Source-Magnet Separation on Intensity and P o l a r i z a t i o n of Atomic Beam. Nozzle at 4-.2°K. 100-He6 T=7°K M=I0 Ionizer diameter = lcm Jfocussed C L 75-t z 50. S O U J V o - Cu 25-5 10 15 SOURCE-MAGNET SEPARATION (cm) F i g . **3 E f f e c t of Source-Magnet Separation on Intensity and P o l a r i z a t i o n of Atomic Beam. Nozzle at 7°K. t o t a l f l u x of atoms into the ionizer aperture with the magnetic f i e l d o f f . This r a t i o (R) i s equal to 1.33. The p o l a r i z a t i o n predicted for the conditions mentioned above i s 62$. C. The P o l a r i z a t i o n Measurement and Ion Beam Y i e l d . Two detection methods were used i n determining the p o l a r i z a t i o n of the 3 He nuclei a f t e r passing through the hexapole magnet. The f i r s t method, using the d i f f e r e n t i a l P i r a n i gauge detector, measured the change i n neutral beam i n t e n s i t y with the magnet turned on and o f f , while the second method measured the change i n ionized beam under similar circumstances with the magnetic f i e l d on and o f f . nozzle - 3 - -- 1 5 -m o g n e t 0 .6 _L_ T . . [ 0 5 6 ionizer b • 5 0 12 T - 2 0 I d i f f erent io i f Pironi d e t e c t o r fo708 cm F i g . kh Schematic Diagram Indicating Relative Location of Components Used i n Atomic Beam P o l a r i z a t i o n Measurement. Dimensions i n cm. The l o c a t i o n of the d i f f e r e n t i a l P i r a n i detector and the ion i z e r r e l a t i v e to the magnet while these measurements were made i s shown i n F i g . M+. Typical r e s u l t s of the f i r s t method are shown i n F i g . 45 where the e f f e c t of the magnetic f i e l d being on and of f i s shown. The increase of approximately 50% i n neutral beam i n t e n s i t y i s interpreted to indicate an e f f e c t i v e 96 p o l a r i z a t i o n of the 3ffe n u c l e i passing through the magnet although i t i s d i f f i c u l t to quote any.degree of accuracy to the absolute value of the increase. Typical r e s u l t s obtained using the ionizer to measure the change i n beam i n t e n s i t y with the magnetic f i e l d on and o f f are shown i n F i g . ^6. In this experimental measurement an increase i n beam i n t e n s i t y of $0% was observed. Because of the small number of ions produced from the atomic beam as compared to the large number produced from the background gas (lslOOO) the incoming atomic beam was modulated. The chopping system and associated detection electronics were described i n Section *+B. The r e s u l t s from 18 measurements, si m i l a r to the one shown i n F i g . k6y yielded a r a t i o R of the i o n current with the magnet on to the ion current with the magnet o f f of R = 1.32-0.1. The increase of ionized beam obtained as a function of the magnet e x c i t a t i o n current i s shown i n F i g . *+7. The r a t i o R increased slowly as the magnet current was increased. This data i s not inconsistent with the value of R reaching a plateau at a magnet current of 70 amps. This value of current corresponds to the value shown i n F i g . 18 at which the magnetic f i e l d begins to saturate. The measured ion current was 12 nA with the magnet on, the i o n i z e r emission current at 750 mA and a plate voltage of 350 V. The conditions under whic h this measurement was made were such that the estimated 3jje atomic beam at the entrance 18 —1 to the magnet was 0A5 x 10 atoms (sr - sec) and the most probable v e l o c i t y of the beam was 310 m/sec. Assuming 50$ of T I M E F i g . Change i n D i f f e r e n t i a l P i r a n i Detector Signal when Hexapole Magnet i s Turned On and Off. ^ 3 98 the atoms entering the magnet pass through the ionizer then the 12 nA ion current corresponds to an i o n i z a t i o n e f f i c i e n c y of 0.15#. a: 1.4 1.3 2 I 2 h < K I.I h 1.0 A Run a Ei Run b O Run c 0 ^ 6 -•Q • J L 0 20 4 0 6 0 8 0 MAGNET CURRENT (amps) F i g . h? Enhancement Ratio of Ionized Beam as a Function of Magnet E x c i t a t i o n Current. The experimental r a t i o R = 1.32-0.1 obtained by measuring the i o n y i e l d from the i o n i z e r i s i n good agreement with the t h e o r e t i c a l l y expected r a t i o R = 1.33 calculated using Glavish's program. This r a t i o i s equivalent to a calculated p o l a r i z a t i o n of the atomic beam of 62%. The actual p o l a r i z a t i o n of the s i n g l y ionized '^ He w i l l depend on the magnetic f i e l d strength i n the i o n i z e r . The expected p o l a r i z a t i o n of the ionized beam as a function of f i e l d strength assuming the p o l a r i z a t i o n of the atomic beam i s 100 i s shown i n F i g . 3« CHAPTER VIII CONCLUSIONS A. Comparison with other Sources of Polarized ^He + Ions. Three methods are presently used for the production of polar i z e d ^He + ions. The production of a polarized secondary beam following a nuclear reaction has been discussed through the example of e l a s t i c scattering of unpolarized ^He by ^He i n Chapter I. The Rice U n i v e r s i t y polarized i on beam i obtained by o p t i c a l pumping techniques has been discussed i n Section 2A. The U.B.C. polarized beam obtained by atomic beam techniques i s the subject of this thesis where the'results of availa b l e p o l a r i z a t i o n and ion current measurements are discussed i n Section 7C A comparitive summary of the p o l a r i z a t i o n and ion current available from these three techniques i s presented i n Table 3» The time required i n an experimental measurement to obtain a desired f r a c t i o n a l error p i n the measurement i s inversely proportional to P I. This figu r e of merit P I given i n Table 3 demonstrates the advantage of the ion source technique over the secondary beam technique i n producing a useful beam of polarized ^He. The figure of merit i s at l e a s t 3 orders of magnitude higher for the ion source techniques. Both the Rice University and U.B.C. ion p sources have s i m i l a r P I values but are characterized by low p o l a r i z a t i o n and high ion current i n the case of the Rice U n i v e r s i t y source and high p o l a r i z a t i o n and low current i n the case of the U.B.C. source. Both the beam p o l a r i z a t i o n and • Table 3 Comparison of d i f f e r e n t Methods of Producing Polarized 3jj e+ i o n s . Method of Production $ P o l a r i z a t i o n P Beam Current I Figure of Merit P 2I Secondary Beam eg. ^ e ^ R - e ^ H e ^ e 100$ .001 - .Olnf\ IO" 6 Optical Pumping Rice University Ion Source 5$ 3/xA .007 Atomic Beam U.B.C. Ion Source 65% P o t e n t i a l l y 100$ 12 nA .005 extracted ion y i e l d can be p o t e n t i a l l y increased i n both the Rice and U.B.C. sources. The p o l a r i z a t i o n of the Rice source might be increased as high as 60% and the U.B.C. source p o l a r i z a t i o n to 100$. Improved atomic beam formation and i o n i z a t i o n e f f i c i e n c y may r e s u l t i n an order of magnitude increase i n the U.B.C. ion current. B. Measurement of the ^>Ee+ Beam P o l a r i z a t i o n by the DC^He.P^He Reaction. The measurement of the atomic beam p o l a r i z a t i o n i s discussed i n Chapter VII and the theoreti c a l r e l a t i o n s h i p between the atomic beam p o l a r i z a t i o n and the ionized beam p o l a r i z a t i o n as a function of the magnetic f i e l d strength i n the i o n i z e r and target regions i s i l l u s t r a t e d i n F i g . 3» In spite of this t h e o r e t i c a l r e l a t i o n s h i p an experimental determination of the ionized beam p o l a r i z a t i o n would be desir a b l e . Although i n p r i n c i p l e the p o l a r i z a t i o n of the 3fj.e beam could be determined by measuring the asymmetry i n the ^He(3He, 3He)Ste r e a c t i o n ; the energy required for a s i g n i f i c a n t amount of P wave scattering i n order to have a non-vanishing L«S sp i n - o r b i t force i s inconvenient f o r preliminary ion source development work. A more suitable method i s to use an exoergic nuclear reaction to produce polarized spin £ p a r t i c l e s of s u f f i c i e n t energy to make p o l a r i z a t i o n analysis by e l a s t i c scattering possible. An appropriate reaction i s DC^HejP^He ( Q = 18.36 MeV). I f this reaction i s i n i t i a t e d by S waves and assumed to proceed through the J = 3/2 + resonance i n L i ^ then a simple r e l a t i o n s h i p exists between the p o l a r i z a t i o n of the incoming 3 He beam PC^He) and the p o l a r i z a t i o n P(p) of the protons which are emitted at an angle of 90° with respect to the d i r e c t i o n of the axis of p o l a r i z a t i o n of the incoming ^He p a r t i c l e s (Fi67) PCp>= - ( 2 / 3 ) P ( J » e ) (23) To measure the p o l a r i z a t i o n of the Rice University polarized ^He4* ion source the p o l a r i z a t i o n of the protons was measured through the l e f t - r i g h t asymmetry A of their e l a s t i c scattering i n a high-pressure (35 atmospheres) helium f i l l e d polarimeter. Details of the experimental apparatus used are discussed by Findley ( F i 6 7 ) . The asymmetry A i s defined as where N L i s the number of protons scattered to the l e f t side of the beam i n the polarimeter and the number scattered to the r i g h t and N L + N R = N, the t o t a l number of events. The polarimeter had an analyzing power of - 0 . 6 . Thus the measured experimental asymmetry A of the protons i s related to the p o l a r i z a t i o n of the ^Ee beam P(3R"e) by / A M The f r a c t i o n a l error / —j^- j i n the measured asymmetry i s equal to where N i s the t o t a l number of observed events. The asymmetry i n the case of the Rice p o l a r i z a t i o n measurement was determined to be 0.0227^0.0033 (Ba68). This 103 value i s an averaged asymmetry obtained by reversing the sign of the beam p o l a r i z a t i o n every 5 minutes to cancel errors due to detector s o l i d angle and e f f i c i e n c y asymmetries i n the polarimeter. To obtain this accuracy according to Eq. 25 3 - 4 j i k on a deuterium ice target at 300 KeV Findley reports a counting rate i n the polarimeter of 2 events per second. Thus the i r measurement should take approximately 13 hours; however, they report approximately 18-24 hours of active data taking was required to obtain adequate s t a t i s t i c s . The differ e n c e i n the two times may be the r e s u l t of background runs. , The time required to determine the p o l a r i z a t i o n of the U.B.C. i o n source can e a s i l y be estimated from Eq. 25 and the Rice group's experimental count rate. For a 3g e D e a m p o l a r i z a t i o n of 0.65,A = 0.26 (This assumes strong f i e l d i o n i z a t i o n ) . I f a ±20% measurement of the beam p o l a r i z a t i o n A A i s desired -j~ - 0.2 and Eq. 25 shows 350 counts are required. For weak f i e l d i o n i z a t i o n the p o l a r i z a t i o n i s reduced by one-half and the number of counts required increased to about 1500, I f the 3 H e w e r e accelerated to 600 KeV where the D(3He,P)1+He cross section i s a maximum instead,of the 300 KeV i n the Rice U n i v e r s i t y case, the cross section for proton production increases by a factor of 4 and the experimental count rate i n the polarimeter thus increases by a factor of 4. However, the U.B.C. i o n current i s approximately 10 nA as compared to the Rice source 3-4 nA. Combining the increase i n count rate required a t o t a l of 90,000 counts. For a ^He beam current of because of the higher ^Ee energy and the decrease due to lower beam current the expected count rate i s 2/100 counts per second. The measurement of the beam p o l a r i z a t i o n to -20f0 accuracy using, this count rate and strong f i e l d i o n i z a t i o n would require approximately 5 hours or using the quoted time of l8-2k hours for the Rice measurement our measurement would require 7-9 hours. Weak f i e l d i o n i z a t i o n would increase the time required for these measurements by a factor of C. Possible Improvements of the Polarized 3ne + Beam. (1) Improvements Increasing the Atomic Beam Intensity. One of the most e f f e c t i v e ways of increasing the ultimate ion y i e l d of the beam apparatus i s to increase the neutral atomic beam i n t e n s i t y . From the data shown i n F i g . 37 i t appears as i f considerable increases i n i n t e n s i t y can be achieved by reducing the scattering i n the nozzle-skimmer region. Reduced scattering i s achieved by increasing the pumping speed and hence reducing the background gas pressure i n this region. The pumping speed at the nozzle i s l i m i t e d by the long pumping channel from the nozzle to the Leybold pump and the r e s t r i c t i v e geometry i n the nozzle-skimmer region. A possible s o l u t i o n to these combined problems i s to remove the present nozzle-skimmer pumping system and to pump the nozzle-skimmer region with the 10" d i f f u s i o n pump presently pumping the skimmer-collimator region. The present skimmer-collimator region might be pumped either by the 10" pump also or by the pumps which presently pump the region downstream of the e x i s t i n g collimator. The discussion of the properties of 105 the nozzle operation given i n Section 3C shows that the collimator might best be eliminated. The ultimate improvement achieved by increased pumping speed may be l i m i t e d by the small size of the nozzle and skimmer apertures. However, i t does not appear unreasonable to expect an increase i n beam i n t e n s i t y of a factor 2 with r e l a t i v e l y simple modifications to the ex i s t i n g apparatus, perhaps even more with extensive modifications. (2) Improvements to Reduce the Atomic Beam Velocity. The reduction of the v e l o c i t y to that corresponding to a stagnation temperature of H-.2°K, the temperature of the l i q u i d helium i n the cryostat, i s expected with a better thermal attachment of the nozzle to the cryostat. The present nozzle i s not i n d i r e c t contact with the l i q u i d but i s attached by a brass bar to the bottom of the cryostat. This has proved to provide inadequate thermal contact as evidenced by v e l o c i t y measurements of the beam and carbon resistance thermometer measurements of the nozzle. I t i s proposed that a new cryostat would be designed with the nozzle a c t u a l l y surrounded by the coolant. This should reduce the nozzle temperature to H-.2°K. In f a c t recent work, subsequent to the measurements mentioned i n this thesis, with a modified cryostat has produced an atomic beam with a v e l o c i t y corresponding to l+.2°K. This confirms that the high v e l o c i t y measured previously was caused by poor thermal attachment of the nozzle to the cryostat. In the modified cryostat the nozzle i s now almost completely surrounded by l i q u i d helium thus ensuring adequate thermal contact. Lowering the temperature of the nozzle from 7 CK to nearer l+.2°K w i l l improve the p o l a r i z a t i o n . The improvement expected i n the calculated t r a j e c t o r i e s of the lower temperature p a r t i c l e s can be seen i n F i g . ^1 where the t r a j e c t o r i e s for both 310 m/sec (7°K) and 2k5 m/sec ( Lf .2 0K) p a r t i c l e s are shown. As shown i n this figure the 2h^ m/sec p a r t i c l e s can be focussed to a point on the beam axis whereas-the 310 m/sec p a r t i c l e s are s t i l l diverging at the magnet e x i t . The results shown i n F i g s . h2 and *+3 indicate that the p o l a r i z a t i o n w i l l increase from 62% with the 7°K beam to 8k% with the >+.20K beam. The d e s i r a b i l i t y of cooling the beam further to 2.2°K as was o r i g i n a l l y proposed (Wa63) i s very evident from the point of view of increased separation of the spin states and increased i o n i z a t i o n e f f i c i e n c y both of which depend on the v e l o c i t y of the atoms. (3) Improvements i n Ionization E f f i c i e n c y and Ion Extraction. The d i f f i c u l t i e s i n extracting a useful ion beam from the side of the i o n i z e r , as must be done with the present i o n i z e r , have not yet been f u l l y considered. I t w i l l be most sensible to reconsider an a x i a l ionizer because of the considerably increased ease with which the ions may be focussed into a useful beam afte r they leave the i o n i z e r . In constructing a new i o n i z e r care must be taken to insure minimum outgasing from the constituent components of the ionizer when the high ele c t r o n emissin current bombards the plate and causes general 107 heating of the apparatus. The present i o n i z a t i o n e f f i c i e n c y of 0.15$ could be improved both by reduced atomic v e l o c i t y and improved i o n i z e r design. Reducing the atomic v e l o c i t y increases the time the atoms stay i n the i o n i z a t i o n region thus increasing the i o n i z a t i o n e f f i c i e n c y . Improved ionizer design would allow higher electron bombardment currents hence higher i o n i z a t i o n e f f i c i e n c y . A new i o n i z e r should be of the strong f i e l d type i n order to take advantage of the s i g n i f i c a n t enhancement of the ultimate nuclear p o l a r i z a t i o n of the ionized beam as shown i n F i g . 3» (h) Overall System Improvement P o s s i b i l i t i e s by Changing the Geometry. Some increase i n beam i n t e n s i t y should, at f i r s t glance, be obtained by moving the source of atoms closer to the magnet entrance as the i n t e n s i t y of p a r t i c l e s entering the magnet i s proportional to 1/r . This would also produce a r e a l increase i n i on current i f the magnet were i n f a c t able to focus the increased number of p a r t i c l e s entering the magnet aperture. The d i f f i c u l t y i s that the p a r t i c l e s can be considered as coming from a point source; thus the ad d i t i o n a l p a r t i c l e s enter the magnet with an i n i t i a l divergence and r a d i a l l o c a t i o n such that the present magnet i s incapable of focussing the extra atoms into a useful beam. This e f f e c t i s evident i n Table 2 where the t r a j e c t o r i e s of atoms with varying r a d i a l displacements at the magnet entrance are tabulated for nozzle-magnet separations of 5 and 15 cm. For the increment size (0.025 cm) of the r a d i a l displacement used i n these calculations 5 p a r t i c l e s with a r a d i a l displacement up to 0.075 cm at the entrance to the magnet pass through the magnet when the source-magnet separation i s 15 cm. However, when the source-magnet separation i s reduced to 5 cm only atoms which enter the magnet with a r a d i a l displacement from the axis of less the 0.025 cm pass through the magnet without h i t t i n g the pole pieces. Hence moving the magnet closer to the source w i l l not r e s u l t i n a l l the a d d i t i o n a l p a r t i c l e s being focussed into a useful beam. The comparison between varying source-magnet separation i s shown more q u a n t i t a t i v e l y i n F i g . k2. The i n t e n s i t y I of focussed atoms passing through the ioni z e r annulus varies less than 3% when the source-magnet separation i s varied from 5 to 15 cm. Also shown on this figure i s the p o l a r i z a t i o n P or r e l a t i v e amounts of focussed and defocussed atoms as defined e a r l i e r which enter the io n i z e r annulus. The p o l a r i z a t i o n increases from a value of 73$ for a source-magnet separation of 5 cm to 84$ for a source-magnet 2 separation of 15 cm. The parameter P I i s of greater i n t e r e s t to the experimentalist as this figure i s inversely related to the length of time needed to obtain the same s t a t i s t i c s for a p a r t i c u l a r experiment with beams of varying i n t e n s i t y and p o l a r i z a t i o n . - The value of this parameter i s 20$ larger for source-magnet separations of 15 cm than for the 5 cm separation. As a r e s u l t of the above there appears to be no advantage i n moving the magnet closer to the source as any s l i g h t gain i n i n t e n s i t y through the io n i z e r annulus i s l o s t because of a l a r g e r reduction i n the p o l a r i z a t i o n of the atoms. 109 (5) Improvements i n Vacuum System Reducing Background Ion Yieldo The maximum acceptable pressure i n a given section of the atomic beam apparatus i s determined by the c r i t e r i o n that the beam should not suffer a s i g n i f i c a n t loss of i n t e n s i t y by scatt e r i n g as i t passes through the region of i n t e r e s t . At room temperature i t i s not too d i f f i c u l t to keep beam losses to 10%, This means that the mean free path of the gas must be 10 times the scattering length through which the beam must pass. Thus i f the chamber were 100 cm long a pressure of 10 Torr would be adequate to keep beam losses to less than 10$. At lower temperatures, i n p a r t i c u l a r at l i q u i d helium temperatures, the s i t u a t i o n i s considerably more d i f f i c u l t as the mean free path at a given pressure i s approximately 100 times smaller than at room temperature. The pressure requirement i n the io n i z i n g chamber i s set by the r a t i o of the number of polarized 3jje + ions produced from the atomic beam to the number of ions formed from the background gas i n the i o n i z i n g region. Naturally this r a t i o should be as large as po s s i b l e ; the exact size depends on the composition of.the background gas ions as i t may be possible to separate the useful polarized ^Ee+ ± o n s f r o m the background gas ions by means of momentum analy s i s . The present s i t u a t i o n of about 15 nA of polarized 3 H e + l o n s a n d 1 0 ^ o f background gas ions i s n a t u r a l l y i n t o l e r a b l e . Momentum analysis i s required to reduce contaminants i n the beam to a minimum so undesired lons would be eliminated. The atomic beam density 110 i n the i o n i z e r region corresponds to a pressure of approximately 10~7 Torr; the background gas pressure should be brought well below this as the volume of background gas available for i o n i z a t i o n i s larger than the volume of beam p a r t i c l e s . To improve the pressure i n the i o n i z a t i o n region a bulkhead should be i n s t a l l e d between the magnet and the ioniz e r chamber. This d i v i d i n g bulkhead would have only a small hole, the size of the magnet e x i t , located on the beam axis to allow the beam to pass through to the i o n i z e r . Such a system would reduce the flow of unpolarized -^Ee atoms from the magnet region and would allow e f f o r t s to improve the vacuum to be concentrated i n one s p e c i f i c area. Reduction of outgassing of the i o n i z e r can be achieved by rebuilding the ionizer with s t a i n l e s s s t e e l parts and by a reduction i n the number of wires and other materials which are subject to excessive outgassing upon electron bombardment. Although the vacuum requirements i n the magnet region are less severe than i n the i o n i z a t i o n region, 10"^ Torr would be s a t i s f a c t o r y , a pump should be provided on the magnet chamber to ensure low background gas pressures around the pole tips of the magnet. APPENDIX A INTENSITY FROM A FREELY EXPANDING JET The angular dependence of the Intensity from a f r e e l y expanding nozzle j e t i s given by Eq. 1 1 as 1(e) =K « s ^ ( § ^ ) ui) where 0 i s the angle between the radius vector to any point and the normal to the aperture and (j> i s a constant, (f> = 1.365 for Y = 1.67 The t o t a l integrated i n t e n s i t y over a l l angles must equal the t o t a l molecular gas flow N through the nozzle given by Eq. 5 i e . e-W Total f l u x = N = x ( e ) dn(e) (A2) where dJn(Q) - ITT s m 0 J © Thus N = COSHES)-n: " a 9 d e ( A 3 ) Performing the in t e g r a t i o n r e s u l t s i n N = _ K TX ( O . Z 6 5 ) or K = N / C2TT (0.265)] . = 0 . 6 N 1 ( 9 ) = : 0.6 N C O - ' ( f j ) " ( A » f ) Therefore the centerline i n t e n s i t y (0 = 0) can be written 112 center fine r N d l o m s (s+eraJ,Afi-sec)" ( A 5 ) 113 APPENDIX B INTENSITY AND VELOCITY DISTRIBUTION OF PARTICLES IN JET AFTER PASSING THROUGH SKIMMING ORIFICE The i n t e n s i t y and v e l o c i t y d i s t r i b u t i o n of atoms a r r i v i n g at a l o c a t i o n on the beam axis downstream from the skimmer can be calculated with reference to F i g . 4 8 . Assuming r a d i a l flow o r i g i n a t i n g at the nozzle with the v e l o c i t y of mass motion w i n the r a d i a l d i r e c t i o n and a uniform atomic density over that region cut out by the skimmer, the number of 2 p a r t i c l e s passing through some small rectangular area (da) located a distancei^/' X 5 along the axis can be found by summing the contributions to the f l u x from a l l elemental areas dA = 2TT X$ sir\ov doC on the spherical surface which characterizes the t r a n s i t i o n from continuum to free molecular flow. The f l u x of p a r t i c l e s passing through this area i s given by: where T\ i s the number density of p a r t i c l e s on the spherical surface and u Is their v e l o c i t y . The or i e n t a t i o n of U2 and UT i s shown i n F i g . 48 and u, da 3 (Bl) (B2) As the area (da) i s very small, only atoms with v e l o c i t y very close to u" where u" i s the v e l o c i t y vector from the point i n 2 question to the center of the area (da) as shown i n F i g . 49 w i l l contribute to the f l u x . F i g . H-9 D e f i n i t i o n of Certain Variables used i n C a l c u l a t i o n of Flow through Nozzle-Skimmer System. Thus the values of U]_, u 2 , and u^ may be written, defining & and <f> as i n F i g . 4-9, U ]_ = U COS O u 2 = U Sin 0 Cos <f> (B3) u 3 = u s i n e s i n f , Since i 1 ^ Xs, 0 and the small increment i n u^, u^ and u^ may be w r i t t e n as 116 where u i s allowed to take on a l l values from 0 toe<?Eq. B l can be written as: fl u \ ' ' 1 % now 2. and i \ also •n Xs = ^T/s /w (B5) Thus Eq. Bk becomes * r C ? (B6) 01 = co Integrating over angles (B7) .i * ^ , x - 4 2 % X t ^ i ^ - c - ^ * (B8> now integrating over v e l o c i t i e s u and r e c a l l i n g 117 2 & ° 5 £ D - e r K - ^ ^ C o s ^ ) ] (BIO) 7-where - 2 f e T -for Mach numbers rV\>39rf may be replaced by i t s assymptotic l i m i t 0 and er^C-^P w) and er-ff-J^Twcos CK<,) may be replaced by their assymptotic l i m i t - 1 . Eq. BIO thus s i m p l i f i e s to sin* is vito + ' W " ^ + C ° 5 ^ ^ ( B 1 1 ) for M > 3 I and Eq. B l l becomes However, i f i n Eq. B4 the r a d i a l divergence of the flow at the skimmer had been neglected ( i e . the flow i s assumed p a r a l l e l at the skimmer entrance) then Q = 0 jdf\ ~ f\$KiMMziz and Eq. Bh becomes as before X r . (Mf X 118 flc W = X / s IT sm'" oi, (u-w)' and -7) Hs W - -J-/s Thus /vv / „ . . , . A I r X ; . T T S"M V< / m V/z „ ~ u w U T T K T J (B15) Performing the i n t e g r a t i o n over u i n Eq. Bl4 res u l t s i n ^M 1 _^M2 2. z A z / fav^ (B16) f o r -[NA > 3 Thus reducing Eq. Bl6 to x"= x / s $ /AV S a * x * n ( B 1 7 ) 2 Eq. B15 can be written d2E where — = J i ' T T * n ' * s / m y/z ^ ( B 1 8 ) ^ u ? e - ^ ^ w ^ ( B 1 9 ) Eq. B8 can be cast into a s i m i l a r form dT = /J2_ _ _ y ^ q ( u ) CT(U) U (B20) where and q(u) i s as given by Eq. B19 120 APPENDIX C TRAJECTORIES OF PARTICLES PASSING THROUGH A HEXAPOLE MAGNET 1. Equation of Motion of a Magnetic Dipole i n a A x i a l l y Symmetric Multipole F i e l d . The energy of i n t e r a c t i o n ^ / between a magnetic f i e l d H and a magnetic dipole M i s /ty = - M « H . The c l a s s i c a l energy W follows from this W = - / / f / / / . / c o s 9 (Cl) \tfhere M has been replaced by the magnetic momentwhich i s at an angle 6? with respect to the magnetic f i e l d H . Letting the component of the magnetic moment i n the d i r e c t i o n of the applied f i e l d be equal tcyC-eff we can write Eq. C l W = -yO<e« I HI (C2) The force on a dipole of constant strength i s given by F = -grad W , , ( C 3 ) yUeff gradlHl Two poles of a r a d i a l l y symmetric multipole f i e l d are shown i n Fi g . 50. The z axis i s taken as the axis of symmetry and the f i e l d i s assumed to be constant i n this d i r e c t i o n , i e . = 0 . The determination o£ the magnetic f i e l d strength i n the plane perpendicular to this axis i s then a two dimensional problem. 121 entrance aperture source F i g . 50 Schematic Diagram Showing Two Poles of a R a d i a l l y Symmetric Magnet. The pertinent Maxwell Equations are: V - B = O i n free space j = D = M = 0 and the f i r s t two equations become: • V - H = O ( c * o V x H = 0 " (C5) Hence H can be described as the gradient of a scalar magnetic p o t e n t i a l (j) as H = -grad <j> using Eq. Ck v2<p ~ o ( C 6 ) (C7) which i s Laplace's equation. A general s o l u t i o n of Laplace's equation suitable for two dimensional geometry i s F(V) = <p (G8) where V •= X f 1% (C9) A general property of this s o l u t i o n i s that l i n e s of constant (p are perpendicular to l i n e s of constant IjJ . As <f> represents the l i n e s of constant magnetic p o t e n t i a l , the l i n e s of constant (JJ represent the l i n e s of force. Writing Eq. C6 i n vector form as where i and J, are unit vectors i n the x and y d i r e c t i o n , the square of the magnitude of H i s |Hl' = 3 x >1 using the Cauchy-Riemann equations S I = _ ^ the above expression becomes IH which using Eq. C8 can be written as 2 lb FCv) ^ 11 1 H a x (CIO) and H ax ( C l l ) Any polynomial i n V i s a s o l u t i o n of Laplace's equation. Thus the general s o l u t i o n given by Eq. C8 can be expressed as FM = f dn V " ( o i 2 ) Each possible multipole f i e l d configuration can be expressed by one term of this polynomial as F „ M = dn V " ( c i 3 ) In c y l i n d r i c a l c o o r d i n a t e s , ! and Q , the vector V = X +ly can be written as V - r ( cos e 4- i < \ ^ 6 ) and Eq. C12 may be written F(v) = £ |^| e r e thus , . * £ + S*^  now H = -grad <^  from Eq. C6 and i n c y l i n d r i c a l coordinates the expression for the gradient i S grad = £ h- + © J_ r 2e where £ and 6 are unit vectors. Thus — j For a single multipole f i e l d we may drop the summation and the r e l a t i v e azimuthal term Sn . H n i s r a d i a l where 7) 6 f Srx - P , TT y 2TT Y 3> TT^ In any case J r c o s ( n e 4 r ^ K V ) - D s i n ( r i © + S n ) | = | a n d . H r " ~ ' c c l 5 ) In terms of Eq. C3 and Eq. Cl5 the force on the dipole i s £ ^y(J<*K -n(-n-i) drs rA _ 2 ( c l 6 ) and i s always i n the d i r e c t i o n of r . The equation of motion of the dipole i s then F - mn y_r be 0 r Sid ^ ' MeU -n(-n-)) dn rA"Z (017) where m i s the mass of the dipole and t i s time. Those dipoles which have a negative component of the magnetic moment i n the d i r e c t i o n of the applied f i e l d H are accelerated towards the central axis of the r a d i a l f i e l d , while those with a p o s i t i v e component i n this d i r e c t i o n are deflected away from the central a x i s . 2. Trajectories of a Magnetic Dipole i n a P a r a l l e l Hexapole Magnet. The hexapole f i e l d i s represented by the general term of Eq. C12 F(v)-= o l 3 V 3 (C18) using Eqs. C8 and C9 and changing to c y l i n d r i c a l coordinates f(y) ~ cp t- 1 (p = J 3 r ? (cos 1® f <- <^ 5 a ) thus the l i n e s of constant magnetic p o t e n t i a l <p and the l i n e s of force 1^  are given by 0 - d5 r1 c o s 3e (019) and y = d 3 r 3 sin 3 0 (C20) 125 ' The parameters used to describe the hexapole magnet are shown i n F i g , 50, where a i s the distance from the source to the magnet entrance, r 0 i s the r a d i a l p o s i t i o n of the p a r t i c l e at the magnet entrance, r m i s the r a d i a l distance to the pole t i p s , Ho i s the strength of the magnetic f i e l d at the pole t i p s , and r and H denote the p o s i t i o n of the p a r t i c l e i n the magnet. In terms of these parameters, the magnitude of the magnetic f i e l d strength at any r a d i a l p o s i t i o n , as given by Eq. C15 i s l " l = C r J H o ( 0 2 1 ) The equation of motion of the dipole i n the hexapole f i e l d as given by Eq. C17 i s d V r + 2 / 6 e t f Ho r d t 7 " '771 TrX l e t t i n g V 7 (C22) the t r a j e c t o r y of the focussed p a r t i c l e s (those with a negative component of/Xeff i n the d i r e c t i o n of H) i s r = A Sin b.rj- f B cos b,-fc using z - ut this equation becomes and the slope of the trajectory i s I f the p a r t i c l e s enter the p a r a l l e l magnet with divergence S and radius r then the a r b i t r a r y constants are B = r and s s su A = . In terms of these constants Eqs. C23 and G2h become ;, \ ' ,L. ->\ r = s u sin(^) f r s cos(bjl) (C25) The s o l u t i o n of the d i f f e r e n t i a l equation describing the t r a j e c t o r i e s of the defocussed p a r t i c l e s i s b,t „ ~b,t .r = A, e B, e which using £ = ut can be written as b . * — r ^ A, e + B, e ^ I f the p a r t i c l e s have slope S d and p o s i t i o n r ^ at 2-= 0 then {(r, f u _ii)e u 4- ^ ( r d - < £ ^ ) e ~ - ^ (027) 3 . Trajectories of a Magnetic Dipole i n a Tapered Hexapole Magnet. The geometrical parameters, describing the tapered hexapole magnet are shown i n F i g . 5lo The magnetic f i e l d strength, Ho, at the pole tips i s assumed to be constant along the length of the magnet as should be approximately true i f the magnet i r o n i s saturated i n this region. This turns out to be a ,good approximation i n p r a c t i c e . 127 Z | J > — 2 ~ F i g . 51 Schematic Diagram Showing the Parameters used to describe the Tapered Hexapole Magnet. By s i m i l a r triangles i n F i g . 51, the pole tip radius r as a m function of Z i s given by r^~_ L± ^ (C28) At any p o s i t i o n 2, the f i e l d strength i n a plane perpendicular to the d i r e c t i o n 2 i s given by Eqs. C. 21 and C. 28. 1 HI = r l i i 1 Ho The c a l c u l a t i o n of the t r a j e c t o r i e s i s l i m i t e d to those cases for which the v a r i a t i o n of the magnetic f i e l d strength i n the 2 d i r e c t i o n i s s u f f i c i e n t l y small .so that the component of grad lfjI i n this d i r e c t i o n can be neglected. Thus for the case of the tapered hexapole f i e l d , the general equation of motion of the dipole as given by Eq. C17 becomes 128 \ TV) u x where the change of variable ~£ = ut has been made. Letting ^ • a _ Z/de-f-C Ho (C29) the above expression becomes ^ = t b: r 3 ^ ? t (C30) where the p o s i t i v e and negative signs r e f e r to the defocussed and focussed portions of the beam respectively. Within the above stated l i m i t a t i o n s , the s o l u t i o n of this equation should not d i f f e r s i g n i f i c a n t l y from the s o l u t i o n for the case of the p a r a l l e l magnet. The so l u t i o n of the negative part of Eq. C30 can be shown (Ax65) to be r--DZkCoS [ J b f J J ^ 2 ^ } (C31) The two a r b i t r a r y constants D and £ , determined by the boundary conditions that r = r Q and = at Z z £, are a n d R _ X f "^J 6 - - t a n ' ' j _ J 2, 129 using these two constants the r a d i a l positions of the dipole i n the tapered hexapole magnet at a p o s i t i o n 2 , as given by Eq. C31 as r z. 2r0 [~k>l + \ Cz "0 1 cos 4bl -/ -±J _ -fan"' } (C33) By d i f f e r e n t i a t i n g Eq. C32 with respect to 2 , the slope of the tr a j e c t o r y i s k •+ ^ ( ^ \ c o s ^ _ s/n ? (C34) 4 b i - l ( j The s o l u t i o n of the p o s i t i v e part of Eq. C30, corresponding to the tr a j e c t o r y of the defocussed p a r t i c l e s i s l+ Hbi + i ' i - > f 4 b i The a r b i t r a r y constants, and C^ ., as determined by the boundary conditions are C3 = r . 2* J~4b\ ( 2-, 1 + x M f e ' j H , 2, Substituting these two equations i n Eq. C35 the r a d i a l p o s i t i o n r of the defocussed p a r t i c l e s at p o s i t i o n "2 i n the hexapole magnet i s (C36) By d i f f e r e n t i a t i n g this expression with respect to the slope of the tr a j e c t o r y of the defocussed p a r t i c l e s at p o s i t i o n Z i n the hexapole magnet i s 4-Zr, b+$^}{2f - 0 f a \ (C37) Thus the t r a j e c t o r i e s of the p a r t i c l e s through a combined tapered and p a r a l l e l hexapole magnet are obtained for the case of the focussed p a r t i c l e s as follows. F i r s t the p o s i t i o n and slope of the p a r t i c l e s at the end of the tapered section are obtained using Eqs. C32 and C3*+. The p o s i t i o n and slope of the p a r t i c l e s i n the p a r a l l e l region are then calculated using Eqs. C25 and C 2 6 . A s i m i l a r proceedure for the defocussed p a r t i c l e s uses Eqs. C36 and C37 to calcu l a t e the p o s i t i o n and slope at the end of the tapered section and uses C27 to c a l c u l a t e the p o s i t i o n i n the p a r a l l e l region. 132 APPENDIX D CALCULATION OF SIGNAL SHAPE FROM TIME-OF-FLIGHT APPARATUS If the chopper allows only an i n f i n i t e l y short burst of p a r t i c l e s to s t a r t on the way to the detector and i f the detector i s i n f i n i t e l y short then the s i g n a l S (t) observed at the detector can be d i r e c t l y related to the d i f f e r e n t i a l i n t e n s i t y d i s t r i b u t i o n function K v j - j- of the p a r t i c l e s i n the i n i t i a l beam • s ( t ) = A " r ( v ) w h e r e V r t ( D 1 ) V and A " = constant since the detector s i g n a l i s proportional to the density D - — o f the p a r t i c l e s at the detector separated by a distance L from the source. Unfortunately the chopper takes a f i n i t e length of time to pass across the atomic beam p r o f i l e and as a r e s u l t a group of p a r t i c l e s with a f i n i t e time spread i s allowed past the chopper. Also the detector has a f i n i t e length so that the s i g n a l received w i l l be a sum of contributions from a l l sections of the detector. Consider the case where the chopper allows a rectangular burst of p a r t i c l e s such as shown i n F i g . 52 to enter the system. This d i s t r i b u t i o n w i l l be c a l l e d the shutter function. Also assume the detector has a length V and the response from a l l sections of i t are equal, that i s we have a rectangular detector response function shown i n F i g . 53• The geometry of the system i s now as shown i n F i g . 54. Consider some point Jl i n 133 F W H M = r T time t F i g . 52 Rectangular Chopper Shutter Function. i ' / v F i g . 53 Rectangular Detector Response Function. CHOPPER DETECTOR £'• F i g . 5\ Geometry of Time-of-Flight Apparatus. 1 3 > the d e t e c t o r a t some t i m e t ; the s i g n a l a r i s i n g f r o m p a r t i c l e s a t t h i s p o s i t i o n i n space and t i m e S C t , / ) i s g i v e n b y ; S(tj) = . f / ) " JV (D2) V> • V s i n c e a n i n c r e m e n t i n i n t e n s i t y i s g i v e n by = X (V) d v where n , ' • V = JU 1 (D3) V 2 = -t/fc-'t) p r o v i d e d - f r > T Summing o v e r c o n t r i b u t i o n s f r o m a l l e l e m e n t s o f the d e t e c t o r S ( t ) = J c U S(t,X) set) = [ u j A" IM av <"*>• T a k i n g the f o r m o f the d i f f e r e n t i a l i n t e n s i t y d i s t r i b u t i o n f u n c t i o n recommended b y Hagena and M o r t o n (Ha67) where A ' - con$4anV we have v i where A = c o n s t a n t m a k i n g a change o f v a r i a b l e X = v^p ( V- w ) ( 1 7 ) (D5) 135 (D6) we have 136 and making the change of variables — w hi - J ? 1 ^ ~ w a f t e r some s i m p l i f i c a t i o n •b A w TT i (-b-l) (eric -ti - er$c-tl) -a (erkt? -ertct," A vTfr (*-^(cr(ctl-erttt;y+ t(erf t [ -erf*/1)] (D8) now Thus S(t) reduces to 137 A computer programme was written to calculate the above function for various values o f f , iY and/, . For the fixed Jl \ and Hr used i n this experiment the e f f e c t •Z has on widening the FWHM of the observed signal 4 f as compared to the th e o r e c t i c a l s i g n a l &t0 i s summarized i n F i g . 26 where T E ' t . APPENDIX E BEPBINTED fSOM ith INTL. SYMPOSIUM Off MEFtEO GAS BYKABiCJ, t © 196? ACADEMIC PRESS INC, NEW YORK A LOW TEMPERATURE NOZZLE BEAM FOR A POLARIZED 3 H e + ION SOURCE R. Vyse, J.C. Heggie and M.K. Craddock Physics Department, U n i v e r s i t y of B r i t i s h Columbia Helium beam i n t e n s i t i e s obtained from nozzles cooled to 77°K and 4.-2°K are studied f o r p o s s i b l e use as an atomic beam source i n a po l a r i z e d JHe ion source. ' Po l a r i z e d ion sources require atomic beams of high i n t e n s i t y to counteract the very low i o n i z a t i o n e f f i c i e n c i e s obtainable by e l e c t r o n bombardment. Furthermore, 3 He atoms have only t h e i r small nuclear magnetic moment, so that Stern-Gerlach s p l i t t i n g of the two spin states requires an unreasonably long magnet unless the atomic v e l o c i t i e s are s u f f i c i e n t l y small. A nozzle of the s t y l e suggested by Kantrowitz and Grey^, cooled to l i q u i d helium temperatures, therefore appeared to be a pos s i b l e s o l u t i o n to the r e q u i r e -ments of high i n t e n s i t y and low v e l o c i t y . Many groups have studied the behaviour of atomic beams produced by small nozzles and have found s u b s t a n t i a l depar-tures from the i d e a l behaviour o r i g i n a l l y postulated. These departures are a t t r i b u t e d to the i n t e r a c t i o n ^ between the skimmer and the supersonic beam, and the background gas scattering-* taking place i n the region between the nozzle and the skimmer. Previous work using c r y o g e n i c a l l y cooled nozzles^ and helium gas, however, has been aimed more at examining condensation than at maximizing the beam in t e n s i t y . The atomic beam source has been b r i e f l y described by Axen^ and co n s i s t s of a D = 0.2 mm diameter nozzle and 0.4 mm diameter skimmer attached to a cryostat capable of sus-tained operation at 4.2°K. The d i f f e r e n t i a l pumping system simultaneously provides the vacuum required f o r the opera-t i o n of the atomic beam and the cryogenic system. With t h i s source we are examining the behaviour of an 939 V Y S E , H E G G I E , A N D C R A D D O C K atomic 4He beam with the nozzle at room, l i q u i d nitrogen and l i q u i d helium temperatures. In each case the skimmer and c o l l i m a t o r are cooled to the same temperature as the nozzle. Operation of the atomic beam source at room tem-perature using an 0.2 mm diameter converging nozzle shows the standard dependence^ of i n t e n s i t y on nozzle-skimmer separation measured i n nozzle diameters, L/D, ( F i g . 1). At 77°K measurements have so f a r been made at four d i f f e r e n t separations, and i n d i c a t e a s i m i l a r dependence over a com-pressed distance s c a l e . At 4.2°K measurements have been r e s t r i c t e d to two separations with the tubing nozzle des-scrfbed below. Figure 2 shows the e f f e c t of nozzle temperature (T 0) and pressure (P 0) on beam i n t e n s i t y f o r two f i x e d nozzle-skimmer separations. For these measurements the o r i f i c e consisted of a s e c t i o n of 0.0095" diameter tubing approxi-mately 10 nozzle diameters i n length. The behavour of the tubing nozzle i s s i m i l a r to that of a converging nozzle, as can be seen i n the i n s e r t of i n t e n s i t y v a r i a t i o n with nozzle-skimmer separation. The p e r i o d i c s c a t t e r i n e x p e r i -mental points i n the i n s e r t i s due to an alignment d i f f i -c u l t y . T y p i c a l nozzle exhaust chamber pressures (PgQ-mTorr) are shown i n brackets beside the relevant experimental p o i n t s . A s i m i l a r set of curves i s shown i n F i g . 3 f o r a 0.2 mm diameter nozzle made by p i e r c i n g a hole i n a piece of 0.001" brass shim shock. Again the t y p i c a l background pressures i n the nozzle exhaust chamber are shown. There i s evidence that the beam i s being attenuated by s c a t t e r i n g by background gas i n both the nozzle exhaust chamber and i n the region between the skimmer and c o l l i m a t o r . T y p i c a l pressures i n the skimmer-collimator region, P s c , measured 2 cm from the beam axis and corrected f o r thermal t r a n s p i r a t i o n , are tabulated i n Table 1 along with the es-timated r e s u l t i n g f r a c t i o n of beam observed ( l / I D ) f o r t y p i c a l points chosen from F i g . 3. The l / I 0 r a t i o was c a l -culated assuming a simple exponential s c a t t e r i n g r e l a t i o n -ship using a v i s c o s i t y based mean free path. U n c e r t a i n t i e s i n the e f f e c t i v e mean fr e e path for s c a t t e r i n g out of a beam at 4.2°K, and i n the pressure measurements, give r i s e to l a r g e r u n c e r t a i n t i e s s t i l l i n the attenuation because of i t s exponential dependence. The u n c e r t a i n t i e s quoted i n Table 1 are based on i 50% u n c e r t a i n t i e s i n mean fr e e path. The r a t i o of observed to t h e o r e t i c a l beam i n t e n s i t i e s , 940 S I X T H R A R E F I E D GAS D Y N A M I C S uncorrected for attenuation, i s plotted as a function of the Knudsen to Mach number ratio in Fig. 4 for the experimental points shown in the previous figure. The Knudsen number is based on calculated free stream conditions at the skimmer entrance and the Mach number is determined using the nozzle-skimmer separation and the method of characteristics solu-tion of Owen and Thornhill^. Considering the extent of the attenuation of the higher P 0 measurements at 4.2°K, the results of Fig. 4 show a considerable increase in beam in-tensity over what we might expect based on the results of Fenn and Deckers as indicated by the straight line. .REFERENCES 1. A.Kantrowitz, J.Grey, Rev. Sci. Instr. 22., 328 (1951). 2. J.B.Fenn, J.Deckers, 3rd Int. Symp. on Rarefied Gas Dynamics, JL, 497, Academic Press (1966). 3. J.B.Fenn, J.B.Anderson, 4th Int. Symp. on Rarefied Gas Dynamics, 7_, 311, Academic Press (1966). 4. E.W.Becker, R.Klingelhofer, P.Lohse, Z.Naturforschung, 17a, 432 (1962) 5. D.Axen, M.K.Craddock, K.L.Erdman, W.Klinger, J.B.Warren, Proc. 2nd Int. Symp'. on Polarization Phenomena of Nucleons, 94, Birkhauser, Basel (1966). 6. P.L.Owen, C.K.Thornhill, ARC R and M #2616 (1952). TABLE 1 Attenuation of Beam Between T D (°K) P o (Torr) p r sc 77 7 77 15 1.5 77 36 3.2 77 134 1.6 4.2 2 3 4.2 - 7i 7 4.2 20 2 4.2 32 3.5 (Torr) IO - 4 x 10"4 x IO" 4 x IO - 3 x 10~ 5 x 10 , x 10 , x IO" 4 0.92 + .05 0.88 + .08 0.72 + .1 0.25 + .15 0.45 + .15 0.15 + .1 0.0045 + .07 0.00007 + .01 941 K> 6 CC V) o CO ~o 4 X > b 3 (/) z z 5 i < U J CD NOZZLE PRESSURE P„ T 0=295°K T„=7 7°K D IOTORR o a 34 <° * 83 - c o 140 * o o o J I « I I I t I O S 10 IS 20 25 30 35 N O Z Z L E - S K I M M E R S E P A R A T I O N L / D Fig. 1, Typical Results from Atomic Beam Source IOO ISO N O Z Z L E P R E S S U R E TORR Fig. 2. Beam Intensity as a Function of Pressure (Tubular Nozzle) 942 SIXTH RAREFIED GAS DYNAMICS L/D = 5 — o TC= 295-K 3(7) -O-0-© ' o T0 = 77*K T„= «*K 50 100 I S O NOZZLE PRESSURE TORR 200 Fig. 3. Beam Intensity as a Function of Pressure 943 a. o X cc K X tf .01 .001 VYSE, HEGGJE, AND CRADDOCK i .\>. e I I 11 0.01 J I M i l l o.i _LL L O J L_J K N / M Fig. 4. The Ratio of Experimental to Theoretical Beam Intensity as a Function of Knudesen to Mach Number Ratio. 944 APPENDIX F LOW TEMPERATURE ATOMIC 3He BEAM FOR USE IN A POLARIZED 3 H e + I 0 N SOURCE R. Vyse, Do Axen and M.K. Craddock Rev. S c i . In s t r . In press 144 BIBLIOGRAPHY Ab 66 N. Abuaf, J.B. Anderson, R.P. Andres, J.B. Fenn, • D.R. M i l l e r , 5th I n t . Sym. on R a r e f l e l d Gas Dynamics, Academic Pr e s s , N.Y. (1966). An 65a J.B. Anderson, J.B. Fenn, Physics of F l u i d s 8, 78O (1965). An 65b J.B. Anderson, J.B. Fenn, 4th I n t . Sym. on R a r e f i e d Gas Dynamics, Academic Pr e s s , N.Y. (1965)-An 66 J.B. Anderson, R.P. Fenn, J.B. Fenn, Molecular Beams. Advances i n Chemical Physics V o l . X I n t e r s c i e n c e (19o6). As 66 H. Ashkenas, F.S. Sherman, 4th I n t . Sym, on R a r e f i e d Gas Dynamics, 85, Academic Pr e s s , N.Y. (1966). Au 69 P. A u d i t , M. Rouault, 6th I n t . Conf. on R a r e f i e d Gas Dynamics, Academic Press, N.Y. (1969)« Ax 65 D. Axen, Phd. Thesis, Univ. of B r i t i s h Columbia, 1965, unpublished. Ba 65 S.D. Baker, G. Ray, G. C. P h i l l i p s and G.K. Walte r s , Phys. Rev. L e t t e r s , I f , 115 (1965). Ba 67 S.D. Baker, D.H. McSherry, D.O. F i n d l e y and G.C. P h i l l i p s , B u l l . Am. Phys. Soc. (N.Y. Meeting, January, 1967). Ba 68 S.D. Baker, E.B. C a r t e r , D.O. F i n d l e y , L.L. H a t f i e l d , G.C. P h i l l i p s , N.D. S t o c k w e l l , G.K. Walt e r s , Phys. Rev. L e t t s . , 20, 738 (1968). Be 54 E.W. Becker, K. B i e r , Z e i t . Fur Naturforschg, 9a, 975 (1954). ' Be 56a E.W. Becker, K. B i e r , W. Henkes, Z e i t . Fur Physik, 146, 333 (1956). Be 56b E.W. Becker, W. Henkes, Z e i t . Fur Physik, 146, 320 (1956). Be 61 E.W. Becker, R. K l i n g e l h o f e r , P. Lohse, Z. Naturforschg, 16a, 1259 (1961). Be 62 E.W. Becker, R. K l i n g e l h o f e r , P. Lohse, Z. Naturforschg, 17a, 432 (1962). Bo 51 J . de Boer, E.G. Cohen, Physica 12? 993 (195D. 145 Br 60 D.A. Bromley and E. Almqvist, Reports on Progress i n Physics, Volume XXIII, A.S. Strickland, Ed. (The Physical Society, London, i 9 6 0 ) p. 544. Also D.A. Bromley and E. Almqvist, Chalk River Report, CRP-881, (AECL, Chalk River, Ontario, 1 9 5 9 ) . Br. 66 R.F. Brown, J.H. Heald, 5th Int. Sym. on Rarefied Gas Dynamics, Academic Press, N.Y. ( 19o6) . Ca 66 R. Campargue, 4th Int. Symp. on Rarefied Gas Dynamics, 279, Academic Press, N.Y. ( I 9 6 6 ) . C l 52 J.R. Clement, E.H. Quinnell, Rev. S c i . Inst. 2J_, 213 (1952) . Em 58 H.W. Emmons, Fundamentals of Gas Dynamics, Princeton University, Press (1958). Fe 63 J.B. Fenn, J . 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Varma, School of Engineering and Applied Science, Univ. of V i r g i n i a , Report # AST-4038-103-67u ( I 9 6 7 ) . Ja 64 D. Jassby, MSc. Thesis, Univ. of B r i t i s h Columbia (1964) unpublished. Ik6 Ka 51 A. Kantrowitz, J . Grey, Rev. -Sci. Inst., 22, 328, (195D. Kn 6h E.L. Knuth, U.C.L.A., Dept. of Engr., Report No. 6>+-53 ( 1 9 6 * + ) . Kn 68 E.L. Knuth, S.S. Fisher, J . Chem. Phys., J+8, l6?h (1968). McM 66 G.E. McMichael, J.B. French, Physics of F l u i d s , Ikl9 (1966). Mi 67 T.A. Milne, F.T. Greene, J . Chem. Phys., ^095 (1967).. Mo 66 M.J. Moravcsik, Proceedings of the 2nd International Symposium on P o l a r i z a t i o n Phenomena of Nucleons, ed. P. Huber and H. Schopper, Birkhauser, 1966, p. 159. Ow 52 P.L. Owen, C.K. T h o r n h i l l , Aeronautical Research Council Report and Memoranda #2616 (1952). Ph 59 G.C. P h i l l i p s , P.D. M i l l e r , Physical Review, 115. 1268 (1959). Ph 66 G.C. P h i l l i p s , Polarized Targets and Ion Sources, Proceedings of the International Conference on Polarized Targets and Ion Sources, Sac lay, France, Dec. 5-9, 1966, p. 215. Ra 56 N.F. Ramsey, Molecular Beams, Oxford Univ., Press (1956). Ro 56 T.R. Roberts, S.G. Sydoriak, Phys. Rev. 102, 30k (1956) Sh 53 A.H. Shapiro, The Dynamics and Thermodynamics of Compressible F l u i d Flow, V o l . 1 , Ronald Press, New York (1953). Sh 63 F.S. Sherman, 3rd Int. Sym. on Rarefied Gas Dynamics, Academic Press, N.Y. (1963). Sm 55 K.F. Smith, Molecular Beams, Methuen and Co. Ltd., London (1955). Th 69 L.B. Thomas, C L . Krueger, R.E. Harris, 6th Int. Sym. on Rarefied Gas Dynamics 1015, Academic Press, N.Y. (1969). Ve 6h C. Vermette, MA.Sc. Thesis, Univ. of B r i t i s h Columbia, (196*+) unpublished. Wa 63 J.B, Warren, W. Klinger, D, Axen, Prog, i n Fast Neutron Physics ed. G.C. P h i l l i p s , J.B. Marion,. J.R. Risser, Rice University Semicentennial Publications, 335 (1963). We 61 R. Weiss, Rev. S c i . Inst. 32, 397 (1961). Za 69 R.N. Zapata, R. Ba l l a r d , N. Cabrera, 6th Int. Sym. on Rarefied Gas Dynamics, Academic Press, N.Y, 997. 

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