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Polarization of electron impact light from helium Whitteker, James Howard 1967

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The University of British Columbia FACULTY OF GRADUATE STUDIES PROGRAMME OF THE FINAL ORAL EXAMINATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY of JAMES WHITTEKER B-.Sc. Carle ton University, 1962 FRIDAY, OCTOBER 6, 1967, AT 3:30 P.M. ROOM 301, HENNINGS BUILDING COMMITTEE IN CHARGE Chairman: B. N. Moyls F.W. Dalby M. McMillan B. Ahlborn H. Gush C E . Brion C.A. McDowell External Examiner: Prof„ Robert Krotkov University of Massachusetts Amherst Massachusetts Research Supervisor: F.W. Dalby POLARIZATION OF ELECTRON IMPACT LIGHT FROM HELIUM ABSTRACT The polarization of light from helium atoms excited by the impact of low energy electrons has been 3 3 ° measured for the spectral lines 2 P - 2 S (10,829 A) 3 3 ° and 3 P - 2 S (3889 A). An electron beam carrying a current of 10yU A was directed into helium gas at a -3 pressure of 4 x 10 torr or less. Polarization was measured as a function of electron energy in a range from the excitation threshold (approximately 23 electron 3 3 volts) to 50 e.v. For the 2 P - 2 S line, this work represents the f i r s t reported measurement of this type. There is special interest in the value of polarization near the excitation threshold. The theo-re t i c a l threshold polarization for both lines studied in this thesis is 36.6%. In the experiment of this 3 thesis, the observed polarization of the 2 P line rises to 21% near threshold, and by means of a curve f i t t i n g procedure may be extrapolated to 32 * 67„. The polari-3 3 zation of the 3 P - 2 S line rises to 117» and may be extrapolated to 15 1 3%. GRADUATE STUDIES Field of Study: Atomic Physics Elementary Quantum Mechanics Waves Electromagnetic Theory Nuclear Physics Plasma" Physics Spectroscopy Special Relativity Theory Molecular Spectroscopy Advanced Spectroscopy Advanced Quantum Mechanics Wo Opechowski J.C. Savage GiM. Volkoff \ . J.B. Warren L„ de Sobrino A.M. Crooker H. Schmidt F.W. Dalby A.J. Barnard H. Schmidt AWARDS 1958-61 International Nickel Co. Scholarship 1962-65 National Research Council Scholarship 1966 UBC Graduate Fellowship 1967 National Research Council Postdoctorate Fellowship POLARIZATION OF ELECTRON IMPACT LIGHT FROM HELIUM by JAMES HOWARD WHITTEKER B.Sc, C a r l e t o n U n i v e r s i t y , 1962 t 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 r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA October, 1967 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an advanced d e g r e e 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 a g r e e t h a t 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 p u r p o s e s 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 n o 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 . Depar tmen t o f P s, cs  The U n i v e r s i t y o f B r i t i s h Co lumb ia Vancouver 8, Canada Date Q f r f . £ ? / Of { 7  ABSTRACT The p o l a r i z a t i o n of l i g h t from helium atoms excited by the impact of low energy electrons has been measured for the spectral l i n e s 2 3P - 2 3S (10,829 A) and 3 3P - 2 3S (3889 A). An electron beam carrying a current of lOyuA was directed into helium gas at a pressure of 4 x 10 torr or le s s . P o l a r i z a t i o n was measured as a function of electron energy i n a range from the e x c i t a t i o n threshold (approximately 23 electron volts) to 3 3 50 e.v. For the 2~T - 2 S l i n e , t h i s work represents the f i r s t reported measurement of th i s type. There i s special interest i n the value of p o l a r i z a t i o n near the e x c i t a t i o n threshold. The th e o r e t i c a l threshold p o l a r i z a t i o n for both l i n e s studied i n t h i s thesis i s 3 6 . 6 $ . In the experiment of t h i s thesis, the observed p o l a r i z a t i o n of the 2 JP l i n e r i s e s to 21$ near threshold, and by means of a curve f i t t i n g procedure may be extrapolated to 32 1 6$. The 3 ' 9 p o l a r i z a t i o n of the 3 P - 2^S l i n e r i s e s to 11$ and may be extrapolated to 15 i 3^, TABLE OF CONTENTS CHAPTER PAGE I . INTRODUCTION . . . . . . . . . . 1 References and Footnotes f o r Chapter I . . . 9 I I . THEORY 11 2.1 I n t r o d u c t i o n . . . . . . . . 11 2.2 The C o l l i s i o n Process 14 2.3 The R a d i a t i o n Process 17 2.4 T h r e s h o l d P o l a r i z a t i o n 21 2.5 2 P P o l a r i z a t i o n C a l c u l a t i o n s 24 2.6 D e p o l a r i z a t i o n due to a Magnetic F i e l d . 25 2.7 P o l a r i z a t i o n as a F u n c t i o n of Angle . . 26 2.8 I n t e n s i t y as a F u n c t i o n of Angle. . . . 27 2.9 The E f f e c t of E l e c t r o n Beam D i s p e r s i o n on P o l a r i z a t i o n 28 References and Footnotes f o r Chapter I I . . . 29 I I I . EXPERIMENTAL DETAILS . . . . . . 30 3.1 Vacuum System 30 3.2 Helium Source . 34 3.3 E l e c t r o n Gun - Design and Op e r a t i o n . . 36 3.4 E l e c t r o n Gun - C o n s t r u c t i o n 39 3.5 E l e c t r o n Gun Mount 43 3.6 C o l l i s i o n Chamber 45 3.7 O p t i c s 47 3.8 P h o t o m u l t i p l i e r s 50 3.9 P h o t o m u l t i p l i e r C o o l i n g 53 i v CHAPTER PAGE I I I . 3 - 1 0 Signal Processing 5 3 3 . 1 1 Polaroid Turner 6 0 References and Footnotes for Chapter I I I . . . 6 3 IV. EXPERIMENTAL RESULTS 64 4.1 Data 64 4.2 Energy Scale 7 6 4.3 Experimental Sources of Error 7 8 References and Footnotes for Chapter IV. . . . 85 V. DISCUSSION OF RESULTS AND CONCLUSIONS. . . . . 8 6 5 . 1 P o l a r i z a t i o n Structure 8 6 5 . 2 Threshold P o l a r i z a t i o n 8 6 5 . 3 E x c i t a t i o n Curves 8 8 5 . 4 Conclusions 8 9 5 . 5 Suggestion for Further Work 9 0 References and Footnotes for Chapter V . . . . 9 1 APPENDIX I I I A. Properties of an Electron Beam. . . . 9 2 APPENDIX I I I B. The Pot e n t i a l i n a Region E l e c t r o s t a t i c a l l y Shielded by Grids 9 5 APPENDIX I I I C. Signal Processing Theory 1 0 0 APPENDIX IV A. P o l a r i z a t i o n of Light due to Optical Elements 1 0 3 APPENDIX V A. P o l a r i z a t i o n Model • 1 0 5 LIST OF TABLES TABLE PAGE I. Theoretical Threshold Polarizations . . . . „ . 22 I I . Comparison of C o l l i s i o n and Spin-Orbit Interaction Times . 23 I I I . Threshold Polarizations and Other Parameters Found by Curve F i t t i n g 8 8 LIST OF FIGURES FIGURE PAGE 1. Expected and Observed P o l a r i z a t i o n Curves. . . . 4 2. Energy Level Diagram for Helium 7 3. E x c i t a t i o n and Emission. 12 4. Vacuum System. . 31 5. Arrangement of Apparatus . . . . . 33 6. Ion Gauge Cal i b r a t i o n for Helium 35 7. Electron Gun . . . . . . . . 37 8. Electron Gun Mount, Cutaway View 44 9. "Vapour Degreasing" Method of Cleaning Vacuum Parts 46 •10. Evidence of Potent i a l Minimum 45 11. Focusing Properties of Optical System 49 12. Optical Transmission of Interference F i l t e r . . . 51 13. Optical Transmission of Interference F i l t e r . . . 52 14. Photomultiplier Cooling 54 15. Electronics: Block Diagram 55 16. Electron Gun Cathode Supply 57 17. Microammeter for Electron Beam 58 18. Device Used to Rotate Polaroid 6l 19. E l e c t r o s t a t i c Shield 95 20. E l e c t r o s t a t i c Shield . 96 21. P o t e n t i a l Inside Shield 97 22. RC F i l t e r of Phase Sensitive Detector 100 v i i FIGURE PAGE 2 3 - 2 6 . • P o l a r i z a t i o n Data for 3889A Line 6 5 - 6 8 2 7 - 3 3 . P o l a r i z a t i o n Data for 1 0 , 8 2 9 A Line 69-75 3 4 . P o l a r i z a t i o n as a Function of Pressure . . . 77 35- P o l a r i z a t i o n as a Function of Optical Aperture 79 3 6 . P o l a r i z a t i o n and Ex c i t a t i o n Curves for the 3 3S - 2 3P (7065A) Line . . . . . . . . . 8 l 37. Intensity Model. 105 3 8 . Mathematical Model of P o l a r i z a t i o n 108 ACKNOWLEDGEMENTS I wish to thank Professor F. W. Dalby for suggesting the problem and for supervising the research. I wish also to thank my wife for her valuable assistance i n the preparation of th i s thesis. This work was supported by The National Research Council of Canada. CHAPTER I INTRODUCTION Atomic l i n e r a d i a t i o n excited by electron impact w i l l , i n general, be polarized r e l a t i v e to an axis p a r a l l e l to the electron beam. A simple way to v i s u a l i z e the effect i s to think of the atom as a c o l l e c t i o n of charged p a r t i c l e s connected by springs. I f this atom i s h i t d i r e c t l y by a p r o j e c t i l e t r a v e l l i n g i n the z d i r e c t i o n , the atom w i l l tend to vibrate i n modes i n which the displacements of the e l e c t r i c charges are along the z axis. The l i g h t that i s radiated, then, w i l l be polarized to some extent along the z axis. I t turns out that the p o l a r i z a t i o n observed i s , i n f a c t , usually positive with respect to the z axi s , as one would expect from the description just given, although a minority of spectral l i n e s show negative p o l a r i z a t i o n . When we wish to speak quantitatively of the p o l a r i z a t i o n , we use the following d e f i n i t i o n for the degree of p o l a r i z a t i o n P. I f in i s the i n t e n s i t y of l i g h t with i t s e l e c t r i c vector i n the di r e c t i o n of the electron beam, and i f I x is the i n t e n s i t y of l i g h t with i t s e l e c t r i c vector i n the di r e c t i o n perpendicular to the electron beam, we have P - I" - I x I" - I x I 2 I t i s usually understood that the l i g h t i s viewed from a d i r e c t i o n perpendicular to the electron beam. I t i s very d i f f i c u l t to make detailed predictions of the p o l a r i z a t i o n of l i g h t due to electron impact. For a given spectral l i n e , the p o l a r i z a t i o n depends i n general on the d e t a i l s of the c o l l i s i o n process, and is therefore d i f f i c u l t to calculate. Even though the laws governing electron-atom c o l l i s i o n s at non-r e l a t i v i s t i c energies are completely known, the c a l c u l a t i o n of low energy scattering and e x c i t a t i o n cross sections at low energies i s extremely complex. By low energies i s meant energies less than a few hundred electron v o l t s , i n the region where the Born approxi-mation does not apply. In f a c t , even though there has been intensive work on electron-atom scattering i n the past few years the only type of electron-atom scattering that can be calculated accurately i s electron-hydrogen e l a s t i c scattering,"'" and to a 2 lesser extent, electron-hydrogen i n e l a s t i c scattering. For other cases, one can make approximations, such as the distorted wave approximation, that are somewhat better at low energies than the Born approximation, but which cannot r e a l l y be expected to pro-duce detailed, accurate cross sections."'" This implies that they also cannot be expected to produce detailed, accurate p o l a r i -zation curves. In spite of these d i f f i c u l t i e s , i t i s possible to make a unique prediction for the p o l a r i z a t i o n of a given l i n e very close to the e x c i t a t i o n threshold for that l i n e . This prediction I 3 depends only on angular momentum considerations, as w i l l be seen i n Chapter I I . The p o l a r i z a t i o n predicted at threshold i s the maximum possible for that spectral l i n e . Measurements of the p o l a r i z a t i o n of l i g h t due to electron impact were made for a number of atoms by several workers during the years 1925 to 1935- During the same period a number of measurements were made of a related phenomenon, the p o l a r i z a t i o n of l i g h t due to the absorbtion of resonance radiation. Of the electron impact experiments, the only ones that involved a detailed study of the p o l a r i z a t i o n as a function of the energy of the electrons were those by Skinner and Appleyard. They studied several l i n e s i n the mercury spectrum, and measured the p o l a r i -zation of l i g h t due to electrons ranging i n energy from 0 .5 or 1 e.v. above threshold to about 200 e.v. Typically t h e i r curves have the general appearance of the one shown i n Figure 1(B). With decreasing electron energy, the p o l a r i z a t i o n r i s e s i n magnitude to a maximum which occurs a few volts above threshold. Then as threshold i s approached more cl o s e l y , the p o l a r i z a t i o n drops toward zero. This l a s t effect i s one that i s contrary to theor e t i c a l expectations, which indicate a curve of the form shown i n Figure 1(A). In the past few years, there has been a r e v i v a l of Interest i n the subject due to the use of p o l a r i z a t i o n measurements as a means of detecting microwave t r a n s i t i o n s . Most of the recent measurements, however, have not been concerned with p o l a r i z a t i o n I 4 as a t o o l for other experiments, but have been done with a view to understanding the phenomena involved. This i s true also of the measurements described i n this thesis. In the past few years there have been p o l a r i z a t i o n measurements on atomic hydrogen, helium I I , l i t h i u m , sodium, and mercury.^ 0 100 e.v. 100 e.v. F i g . 1. Expected (A) and observed (B) p o l a r i z a t i o n curves. The result s of these experiments may be described b r i e f l y as follows: H: Measurements indicate that the p o l a r i z a t i o n remains f i n i t e near threshold, but do not allow a comparison of the threshold value with theory. This i s because the Balmer ex. data involve three unresolved l i n e s with unknown in t e n s i t y r a t i o s , and because the Lyman <x. results are very imprecise. He: The p o l a r i z a t i o n of most of the l i n e s varies rapidly near threshold. Curves of the form of Pigure 1(B) are common i n the work before 19&3, but do not appear i n results published since then, and i t appears that the effect i s a property of the experimental method rather than a property of electron-helium c o l l i s i o n s . Nevertheless, except i n the case 8 of some of the most recent work, the p o l a r i z a t i o n has not been observed to come close to the t h e o r e t i c a l value at threshold. In the case of the exception just mentioned, the p o l a r i z a t i o n i s observed to r i s e to the threshold value from a minimum that i s very close to threshold. He I I : The p o l a r i z a t i o n of one l i n e has been shown to increase monotonically with decreasing electron energy, but comparison with theory i s impossible. L i ^ , L i ' 7 , Na 2^: The polarizations of the resonance l i n e s have been found to r i s e monotonically with decreasing electron energy (as i n curve A) and to approach the predicted value at threshold, exactly as one would expect. In fact these measurements are good enough to check the previously published values of hyperfine structure and natural l i n e width. However, there i s a l i t h i u m l i n e for which the polar-i z a t i o n decreases close to threshold. Hg: The p o l a r i z a t i o n of the D line s i s f i n i t e near threshold, but the observations involve two unresolved li n e s with an unknown i n t e n s i t y r a t i o . These results indicate, I 6 however, that the e a r l i e r results of Skinner and Appleyard were wrong near threshold, probably as a r e s u l t of the low l i g h t i n t e n s i t i e s involved. The question to be answered then, i s that of the behavior of the p o l a r i z a t i o n curves near threshold. This i s where theory gives a d e f i n i t e answer, and t h i s i s also where measurements are d i f f i c u l t because i n t e n s i t i e s are low. The experimental work described i n t h i s thesis pursues that question for two l i n e s i n the helium spectrum. The energy l e v e l diagram for helium i s shown i n Figure 2. The l i n e s connecting d i f f e r e n t energy levels represent spectral l i n e s or multiplets for which the p o l a r i -zation due to electron impact has been measured. Double li n e s indicate those for which data appear i n t h i s thesis. The numbers at the top of the columns indicate the theoreti-c a l threshold polarizations from lin e s ( i n the case of singlet levels) or multiplets ( i n the case of t r i p l e t levels) o r i g i -nating i n these columns. These numbers apply to L -*• L - 1 t r a n s i t i o n s i n the case of upper P and D states, and L-* L + 1 t r a n s i t i o n s i n the case of upper S states. 1 i The largest polarizations have been observed with D- i J l i n e s . The 1P- *S l i n e s are predicted to have the largest polar-i z a t i o n s , but the observed polarizations are not high, probably because, except at very low pressures, some of the l i g h t i s due to e x c i t a t i o n by trapped resonance ra d i a t i o n . The 3 P-2 S -19- e.v. Figi 2. Energy Level Diagram for Helium. The ground state, which is not shown, is a 's state and is at zero electron volts. I 8 (3889A) t r a n s i t i o n i s the only one for which the p o l a r i z a t i o n did not, u n t i l recently, appear to approach zero near thresh-old. The primary concern of this thesis i s the p o l a r i z a t i o n of the 2 3 P - 2 3 S ( 1 0 , 8 2 9 A ) multiplet of helium. This multiplet is of p a r t i c u l a r interest because i t originates from a low ly i n g l e v e l that i s comparatively w e l l separated from neigh-bouring l e v e l s , and because some th e o r e t i c a l work has been done on i t . U n t i l now, no o p t i c a l measurements of p o l a r i -zation have been reported for t h i s m u l t i p l e t , although the alignment of the 2 ^ s l e v e l r e s u l t i n g from the 23p_2^S t r a n s i t i o n has been measured by an atomic beam method.9 The 3 3 P - 2 3 S ( 3 8 8 9 A ) multiplet i s also studied i n th i s thesis. In th i s case, the work i s not new, although the s t a t i s t i c a l accuracy i s somewhat higher than i n previous measurements. Because other workers have studied t h i s m u l t i p l e t , the results presented here also serve as a test of the experimental method. I References and Footnotes for Chapter I 9 1. A br i e f and l u c i d account of the state of low energy electron-atom scattering theory as of 1964 i s given by E. Gerjuoy i n Physics Today _l8, 24 (May 1965) . 2. P. G. Burke, H. M. Schey, K. Smith, Phys. Rev. 129, 1258 (1963). 3- References for t h i s early work are given by I. C. Per c i v a l and M. J . Seaton, P h i l . Trans. Ser. A 251, 113 (1958). 4. An account of thi s work i s given i n A.C.G. M i t c h e l l and M. W. Zemansky, Resonance Radiation and Excited Atoms (Cambridge University Press, London, 1961). 5- H.W.B. Skinner and E.T.S. Appleyard, Proc. Roy. Soc. (London) Ser. A 117, 224 (1927). 6. For example, W. E. Lamb and T. H. Maiman, Phys. Rev. 105, 573 (1957)• 7. References for recent p o l a r i z a t i o n measurements are l i s t e d below with respect to the atomic species studied. H: H. Kleinpoppen, H. Kruger, and R. Ulmer, Physics - Letters 2, 78 (1962). W. L. Fi t e and R. T. Brackmann, Phys. Rev. 112, 1151 (1958). He: R. H. McFarland and E. A. Sol t y s i k , Phys. Rev. 127, 2090 (1962). D. W. 0. Heddle and C. B. Lucas, Proc. Roy. Soc. (London) Ser. A 271, 129 (1963). R. H. Hughes, R. B. Kay, L. D. Weaver, Phys. Rev. 129, 1630 (1963). (8). R. H. McFarland, Phys. Rev. Letters 10, 397 (1963). (8). D.W.O. Heddle and R.G.W. Keesing, Proc. Roy. Soc. (London) Ser. A 2_5_8, 124 (1967) . E. A. So l t y s i k , A. Y. Fournier and R. L. Gray, Phys. Rev. 153, 152 (1967). R. H. McFarland, Phys. Rev. 15_6, 55 (1967). L i 6 , L I 7 , Ha 2 3; H. Hafner and H. Kleinpoppen, Zelt. Phys. 198, 315 (1967). Hg: H. G. Heideman, Physics Letters 13, 309 (1964). H. K. Holt and R. Krotkov, Phys. Rev. 144, 8 ? (1966). CHAPTER I I THEORY 2.1 Introduction The theory of the p o l a r i z a t i o n of l i g h t from atoms due to electron impact has been reviewed and developed by Percival and Seaton. They derive expressions for p o l a r i z a t i o n i n terms of the r e l a t i v e cross sections for e x c i t a t i o n to the various M L states, where M L i s the Z component of o r b i t a l angular momentum. They do not attempt to calculate the cross sections themselves. Such a ca l c u l a t i o n involves a l l the deta i l s of the c o l l i s i o n process and at best can be done only approximately. However, at threshold, one can predict the po l a r i z a t i o n without a detailed knowledge of the c o l l i s i o n process. This w i l l be discussed l a t e r . The theory developed by P e r c i v a l and Seaton i s r e s t r i c t e d to atoms which can be described i n terms of LS coupling, and which have zero o r b i t a l angular momentum i n their ground states, and i t i s also r e s t r i c t e d to dipole ra d i a t i o n . Otherwise i t i s quite general. The effect of hyperfine structure i s calculated, including the case i n which the hyperfine separation i s comparable with the natural l i n e width. (This i s of interest i n the case of atomic hydrogen.) In the following pages I s h a l l outline the theory that i s applicable to helium. Hyperfine structure w i l l II 12 therefore be ignored. The theory presented here w i l l follow P e r c i v a l and Seaton quite closely i n discussing the c o l l i s i o n process, but w i l l depart from them to some extent i n discussing the radia t i o n process. P e r c i v a l and Seaton use the tensor operator methods developed by Racah, while the theory presented here w i l l use the more pedestrian methods described by Condon and Shortley. The emission of radia t i o n due to electron impact i s considered to take place i n two d i s t i n c t steps: (i) the c o l l i s i o n process, i n which the atom i s excited from the i n i t i a l (usually ground) state to an excited s t a t e a n d ( i i ) the radia t i o n process, i n which the atom drops to state /f , with the emission of a photon. (See Figure 3) F i g . 3. E x c i t a t i o n and Emission. I I 13 These two processes take place on quite d i f f e r e n t time scales. The c o l l i s i o n process takes a time presumably of the -14 order of 10 sec. (the t r a n s i t time of the scattered electron at 0.1 volts) or less (for higher energies). The radiat i o n time i s much longer, 10"^  sec. for the li n e s studied in this thesis. There i s another time i n t e r v a l of importance, namely the time required to transfer angular momentum from the o r b i t a l state of an atom to i t s spin state, and vice versa. As an estimate of th i s spin-orbit interaction time, we take the inverse of the fine structure t r a n s i t i o n angular frequency, i . e . l/2nf. The shortest interaction time found i n t h i s way for helium i s 10~H sec., which occurs i n the 2 P state. Thus, to a good approximation, we can consider that, for the duration of the c o l l i s i o n process, the spin and o r b i t a l angular momenta are uncoupled. On similar grounds, we may assert.that there i s no spin-orbit i n t e r -action between the atom and the incident electrons. For the reasons given above, we can assume that spin and o r b i t a l angular momenta are separately conserved during the c o l l i s i o n , and that spin coordinates are not involved i n the c o l l i s i o n process. Since spin i s l e f t out of the picture during the c o l l i s i o n , the relevant quantities i n a description of the c o l l i s i o n process are the cross sections for e x c i t a t i o n to the various o r b i t a l angular II 14 momentum states. These states we l a b e l by their M L values, and we denote the cross sections by Q M . . In order to describe the rad i a t i o n process, however, we must describe the excited states of the atom i n the L S coupled scheme. We require, therefore, a r e l a t i o n between the e x c i t a t i o n cross sections i n the L S coupled scheme and the cross sections Q M l i n the uncoupled scheme. This i s what i s done i n the next section. 2.2 C o l l i s i o n Process The wave function of the scattered electron at a large distance r from the scattering centre has the form e 1^ 1" fp ( k ) , where fp (k) i s c a l l e d the scattering amplitude, and i s a function of the d i r e c t i o n ( k ). The subscript indicates the state into which the atom i s excited. The cross section for scattering into a s o l i d angle d u about d i r e c t i o n (k) i s proportional to jf^s (k) | ^ dw, and the t o t a l cross section for exc i t a t i o n into state / S i s proportional to ° | f^ (k) | dw. Now the re l a t i o n s h i p between the scattering amplitudes i n the /s=SUMj scheme and the/3 = S L M S M L scheme i s where t h e . C M s M L M j are vector coupling c o e f f i c i e n t s . I t follows that I I 15 (2) We can eliminate the mixed terms i n this expression by-applying our assumption that spin and o r b i t a l angular momenta are separately conserved during the c o l l i s i o n . These conditions are: ML + mi = 0 Ms + ms = Ms - mg (3) CO where m' refers to the incident electron and m to the scattered electron. In the following discussion l e t t e r s with a single prime refer to the i n i t i a l l e v e l ; unprimed l e t t e r s refer to the excited l e v e l ; and l e t t e r s with a double prime refer to the l e v e l after r a d i a t i o n has taken place. Refer to Figure 3- The r i g h t hand side of (3) i s zero because i n the i n i t i a l state, ML=0 by assumption. We have rat-0 because we define the z axis to be directed p a r a l l e l to the d i r e c t i o n of t r a v e l of the incident electron. I f we expand the scattering amplitude into spherical harmonics i n the following way, (5) we f i n d , by applying the conservation conditions, that vanishes unless Ms=Mg, M L=M°. II 16 Hence Therefore the cross section i s given by Q^LTMj ^ ^ f c ^ n ^ J Q s L n 5 M L ( 7 ) Now by our previous assumption, Q 5 L , M 3 H l i s independent of Ms, so we write Q M U = Q s L M f t M L (8) Furthermore, because of the a x i a l symmetry of the system, Q ^  i s independent of the sign of Mu. That i s , there i s nothing i n the system to favour l e f t or r i g h t hand rotations. Hence we write Q M L = Q | M J (9) This leaves us with Q s L T M ^ ^ ^ ^ n ^ J ysVr Q | M J ( 1 0 ) which i s the relationship we set out to f i n d . We s h a l l also require the r e l a t i v e p r o b a b i l i t i e s of ex c i t i n g the various fine structure l e v e l s . We have But M s h j ' s ' L ' J 2L+\ (12) This follows from the symmetry properties of the vector coupling c o e f f i c i e n t s . Therefore I I 17 Q 7 = £L±A C U ( a L + i)(as+ - i ) " L . (13) where we define As we might expect then, the cross section for e x c i t i n g a given fine structure l e v e l i s simply proportional to the s t a t i s t i c a l weight of that l e v e l . 2.3 The Radiation Process Our next task i s to f i n d the p o l a r i z a t i o n of the radi a t i o n that i s due to each JMj —• j'Mj t r a n s i t i o n . After that, we sum over Mj and Mj to obtain the p o l a r i z a t i o n of each spectral l i n e . F i n a l l y , we sum over J and' j" to obtain the p o l a r i z a t i o n of the mult i p l e t , which i s the quantity that i s measured. The degree of p o l a r i z a t i o n i s defined as T11 - T-'-We now put t h i s into a form that i s more convenient for calculations. Define I x - I y - I X l I z - T . \ . ' 1 - 1 * . + ! ? + I z Then we obtain p _ 2>Iz. - I The p o l a r i z a t i o n , then, i s determined by the r e l a t i v e values of I z and I , and t h i s i s what we now calculate. The emission rate A^ for l i g h t of a given p o l a r i z a t i o n /i i n the t r a n s i t i o n II 18 « // JMj —* J Mj i s proportional to the square of the t r a n s i t i o n moment. That i s , we have A (J ^ j - - * J " n j ) = K K I • 1-1P/4 i J " f l j ' > | % '. (16) where P i s the dipole moment operator p+, = 1Z K i s a constant c h a r a c t e r i s t i c of the multiplet. The Mj, MJ';dependence of (16) i s given by Condon and Shortley, The Theory of Atomic Spectra, (which w i l l be referred to as TAS) equations 9 1^1• Bearing i n mind the required quantities i n (15), and notic i n g that this i s a convenient point at which to sum over Mj,.we obtain A° (j M T->J ") - Y\ j ( J M T I P0 \S" M .> 1 * flfJMT->rWKI I C T M T I P / < | J - " M " > I % = n 7 x M«r ' "' (ir j K k v T p j J " )7* i % ( j " , J ; /) where the ( j j P j j ) are the reduced matrix elements of TAS 9^11, and the — ( J , J ) are the r e s u l t of summing the MjMj dependent part of the matrix elements oveTycc. They are Z ( j , J + l ) = ( j + l ) ( a J + 3) ZZ (J, J ) •= J (J + i) (!8) Z ( J , T - | ) = J U J - I ) (TA5 1*5) Equations (17) give the rate of emission i n the (J —»j') l i n e from one atom i n the state JMj. The rate of e x c i t a t i o n of II 19 atoms to the state JMj i s proportional to the ex c i t a t i o n cross section to that state, which i s S L J \ - % . Q J M = E (Cr-\-ji, nT) —' Q i M^l (io) The rate of radia t i o n i s proportional to the quantities n A ° ( J M J - » J " ) W j r b A ( 5 L J M T ) A ( S L T i V) ^ f e A ^ J is+r Q i M j | < J M j | R » l J " n j ) r 0 A ( j ^ - > P ) where A(SLJMj) i s the t o t a l emission rate from the state SLJMj. Next, to f i n d the rates of emission i n the spectral l i n e J—>J , we sum (19) over Mj to obtain 1 ( J - J ' ) - K - < ^ f ^ y £ L Qi«. i l<J!Pi^>r - <J'J") ( 2 0 ) where we have used (12) and (13). K has been replaced by K° i n order to accommodate such quantities as atom density and electron beam current, and A(SLJMj) which i s independent of J and Ms (TAS 13% ) . Equations (20) enable us to derive expressions for the po l a r i z a t i o n of l i g h t i n a given spectral l i n e by use of I I 20 equation (15). We r e q u i r e the M j dependence of ( j M j | P 0 j J " M j ) which i s given i n TAS 9 11 and the v e c t o r c o u p l i n g c o e f f i c i e n t s t a b u l a t e d i n TAS p. 76. In the case of helium s i n g l e t l i n e s , the procedure i s q u i t e easy, s i n c e the v e c t o r c o u p l i n g c o e f f i c i e n t s reduce to u n i t y , and we have J = L, J* = if . In the case of h e l i u m t r i p l e t l i n e s , however, there i s even more work to do, because we want the p o l a r i z a t i o n of the whole m u l t i p l e t . A f t e r summing equations (20) over M j , we must then sum over J and z" . T h i s can be done u s i n g the e x p r e s s i o n s f o r the J and J / 7 dependence of <J ;P| J*;, g i v e n i n TAS 1138. The r e s u l t s of these c a l c u l a t i o n s appear i n the f o l l o w i n g form: For upper S s t a t e s (L=0) P = 0 (21) For P —> S (L = 1—» L = 0) l i n e s or m u l t i p l e t s p _ (S(Qo — Q (oo\ h 0 Q o + h, Q , ^ j For D —* P ( L = 2 —> L = 1) l i n e s or m u l t i p l e t s p _ G (Q Q +- Q i - A Q z l h o Q o + h , Q , - i - h a . 0 x ' K 2 1 These formulae and n u m e r i c a l t a b l e s of the c o e f f i c i e n t s are g i v e n i n P e r c i v a l and Seaton, Tables 1 and 2. In the case of \ 3-S m u l t i p l e t s (the ones s t u d i e d in- the t h e s i s ) we have P _ 15 (Qo - Q 1 ) 1 oh\ I I . 21 2.4 Threshold P o l a r i z a t i o n In general, the cross sections Q j M t | are d i f f i c u l t to calculate, but one can assert that as the energy of the incident electrons i s reduced to approach the threshold value, Qi/Qo approaches zero. I f th i s i s so, the p o l a r i z a t i o n near threshold i s completely determined and i s 15/41 = 36.6$ i n the case of -P- -3 multiplets. The j u s t i f i c a t i o n for t h i s assertion i s as follows. The i n i t i a l state of the electron-atom system has i t s z component of o r b i t a l angular momentum equal to zero, and therefore this must be true of the f i n a l state also, under our previous assumptions. i.e . M L + m, = 0 (3) I f the energy of the scattered electron i s s u f f i c i e n t l y small, the scattered electron must be i n an S state, because otherwise i t s impact parameter would be impossibly large. I f this i s so, then we have m£ = 0 , and therefore M L = 0 also. Therefore Q,, /Q0 approaches zero. The predicted threshold polarizations for several types of t r a n s i t i o n s i n helium are shown i n Table I. In view of the fact that measurements have i n several cases f a i l e d to show a p o l a r i z a t i o n near threshold as high as predicted, i t i s worthwhile considering where the assumptions leading to the predictions may be vulnerable. Two possi-b i l i t i e s come to mind. I I TABLE I THEORETICAL THRESHOLD POLARIZATIONS Spectral Line Threshold P o l a r i z a t i o n 1S - 'p 0$ 1P - 'S 100$ 'D - 'P 1P - 50$ M u l t i p l e t 3S - 3P 0% 3p - 3S 36.6$ •a 3 JB - P 31.7$ ( i ) The range i n energies over which electrons are scattered i n pure S waves may be smaller than the energy width of the electron beam. The radius of an excited Helium o . . atom i s 5 or 10 A, (depending on the state). The impact parameter of a 0.1 e.v. electron i s o A, so the threshold prediction i s not dependable for energies much larger than t h i s . An electron beam from a thermionic cathode has, at best, an energy resolution of about 0.2 v o l t s . I t i s i n fact with t h i s consideration i n mind that the 2 P state was chosen for study. A helium atom i n an n=2 state i s smaller than one i n an n=3 or higher state. I I 23 ( i i ) The factor by which the spin-orbit time i s larger than the c o l l i s i o n time i s not overwhelmingly large, and requires closer examination. The numbers given i n Table I I are very rough, but they provide some idea of the size of the quantities involved. The c o l l i s i o n time i s calculated for the largest electron energy for which the scattered electron should be predominantly i n an S state, and i s compared with the spin-orbit interaction time. TABLE I I COMPARISON OF COLLISION AND SPIN-ORBIT INTERACTION TIMES Symbol Atomic Radius R Energy for Impact Parameter - R and - 1. E Velocity Corresponding to E v Frequency of 1 - 0 f . s . Transition • f Spin-Orbit Interaction Time 1/2 nf C o l l i s i o n Time R/v ~Cc Ratio T-is/z, 2 3P 3 3P o a 5A 12A 0.15 e.v. 0 . 0 3 e.v. 2.3x10^ cm./sec. 1.0x10^ cm./sec 10 9 2.8x10 8.1x10 H z -12 T 1 6x10 sec. 2x10 sec. 2 . 2 x l 0 " 1 5 sec. 1 . 2 x l 0 _ l i | sec 3 x l 0 3 2 x l 0 3 I I 24 The value R i s estimated from a plot of electron density i n the hydrogen atom. E and v are simply taken from the Bohr formula. The r a t i o of spin-orbit time to c o l l i s i o n time seems to be large enough to allow the figures i n the table to be adjusted by as much as a factor of 10 perhaps, and s t i l l leave the assumption intact that spin coordinates are not involved i n the c o l l i s i o n process. Notice, however, that e s p e c i a l l y for the 3~T l e v e l , i t i s possible that the threshold p o l a r i -zation holds up to only very small energies above threshold. 3 2.5 2 P P o l a r i z a t i o n Calculations Massey and Moiseiwitch have done a distorted wave 3 c a l c u l a t i o n of e x c i t a t i o n cross sections to the 2 P l e v e l of helium. They make predictions for the p o l a r i z a t i o n of l i g h t 3 3 i n the 2 P 2 - 2^S l i n e . In order to compare these predictions with experimental p o l a r i z a t i o n s , i t i s necessary 3 3 toconvert them to polarizations of the 2 P - 2 S multiplet. Fortunately, there i s a one to one correspondence between P and Q i / Q , 0 5 and this i s e a s i l y done. The con-version formula i s c-( hz+h/) / h, hf - H, u: v " G - ( h . » h . ) ,7 ( a 5 ) h» h,' — h, h/ I I 25 where P i s the multiplet p o l a r i z a t i o n and P/ Is the l i n e p o l a r i z a t i o n . The values obtained are plotted along with the experimental data i n Figure 28. 2.6 Depolarization due to a Magnetic F i e l d The p o l a r i z a t i o n of l i g h t from an atom depends on the r e l a t i v e populations of the magnetic states of the excited l e v e l . A magnetic f i e l d directed at r i g h t angles to the quantization axis w i l l tend to mix the magnetic states, and i f t h i s magnetic f i e l d has a component i n the d i r e c t i o n of observation, the p o l a r i z a t i o n i s decreased. This depolari-zation i s known as the Hanle effect and i s described by M i t c h e l l and Zemansky. I f the directions of observation and of the magnetic f i e l d are the same, the reduction i n p o l a r i z a t i o n i s given by where H i s the magnetic f i e l d and X i s the radiative l i f e t i m e of the excited state. For the l i n e s studied i n t h i s thesis we have (26) P "Po I -r- ( a-9 H ) x where H i s measured i n gauss. I I 26 I f we want the depolarization to be less than 1%, say, we must reduce any transverse magnetic f i e l d to less than 3.5 x 10 gauss. •2.7 P o l a r i z a t i o n as a Function of Angle In the theory of the p o l a r i z a t i o n of l i g h t due to electron impact on atoms, we assumed that the observer looks from a d i r e c t i o n at r i g h t angles to the electron beam. But in practice, the observer looks at a cone of l i g h t rays. In thi s •section we f i n d how the p o l a r i z a t i o n varies with the angle of observation.-three mutually perpendicular dipole antennae. The z axis antenna radiates with i n t e n s i t y I" i n a di r e c t i o n perpendicular to i t s axis, and the x and y antennae each radiate with i n t e n s i t y I x i n directions perpendicular to their axes. The rad i a t i o n i n t e n s i t y from a rad i a t i n g 2 dipole varies as s i n 0, where 0 i s measured from the dipole axis . perpendicular to the electron beam, the p o l a r i z a t i o n we see i s -2 We consider the l i g h t r a d i a t i o n to originate from I f we look at the l i g h t at an angle d from the I " 4- Is-- ( I 1 1 - M s i n 1 " ^ II 2 7 ( 2 7 ) I - P ( o ) ^ I f oc i s small, this becomes The t o t a l e f f e c t on the observed p o l a r i z a t i o n i s found by-aver aging over the directions of observation. 2 . 8 Intensity as a Function of Angle Not only the p o l a r i z a t i o n , but also the i n t e n s i t y of l i g h t varies with the angle of observation. If 9 i s the angle from the z axis, the i n t e n s i t y i s given by (We have used the same model as i n the previous section.) The t o t a l i n t e n s i t y i s The importance of this r e l a t i o n s h i p i s that i n order to measure c o l l i s i o n cross sections by measuring l i g h t i n t e n s i t i e s i n a d i r e c t i o n perpendicular to the electron beam, one must know the p o l a r i z a t i o n of the l i g h t . 1(9) - I "+ l x - ( I 1 1 - ! ^ ) c o s a G-1(G) - 1 (%)( l - Pcos* &) ( 2 8 ) ( 2 9 ) I I 2 8 2 . 9 Effect of Dispersion of Electron Beam on Po l a r i z a t i o n I f the axis of p o l a r i z a t i o n i s rotated by angle Q, the new p o l a r i z a t i o n i s P' = I " c o s ^ 6 + I1- sln^e- I u s l h v & - I x c o ^ 9-I " +• I x I 4- X = P (i - a s \ n % ^ (30) Por small angles, t h i s becomes P ' = P ( i - a ^ ) . The t o t a l e f f e c t on the observed p o l a r i z a t i o n i s obtained by averaging over the v e l o c i t y directions i n the elec6ron beam. I I 29 References and Footnotes for Chapter I I 1. I. C. Pe r c i v a l and M. J . Seaton, P h i l . Trans. Ser. A 251, 113 (1958). 2. E. U. Condon and G. H. Shortley, The Theory of Atomic Spectra, (Cambridge University Press, London, 1963) 3. H.S.W. Massey and B. L. Moiseiwitch, Proc. Roy. Soc. (London) Ser. A 258, i h j (i960). 4. A.C.G. M i t c h e l l and M, ¥. Zemansky, Resonance Radiation and Excited Atoms (Cambridge University Press, London, 1961). CHAPTER I I I EXPERIMENTAL DETAILS 3.1 Vacuum System The vacuum system i s shown in Figure 4. The various parts are demountable, being joined by nuts and bolts and sealed with neoprene "0" rings. I t i s a conventional system, for the most part, and i t i s possible to obtain pressures as -7 low as 2 x 10 torr with I t . The main pump i s an o i l d i f f u s i o n pump (Balzers, D i f f . 170) with a speed of 90 litres/second with the b a f f l e i n place. The d i f f u s i o n pump i s backed by a mechandial pump (Welch 1402) with a pumping speed of 100 litres/minute. In series with the d i f f u s i o n pump i s a water cooled b a f f l e which reduces o i l backstreaming, and a l i q u i d nitrogen trap. The trap was f i t t e d with a copper c o l l a r that was kept cold by thermal conduction through a copper rod from the inner part of the trap. The purpose of this c o l l a r was to discourage o i l creep along the warm outside walls of the trap. When low energy electron beams are being used i t i s important to keep o i l out of the vacuum system, because o i l can be deposited as an ins u l a t i n g layer which w i l l charge up to large potentials. When the vacuum pumps were operating, the trap was always kept cold. A f i l l i n g of l i q u i d nitrogen Cold Trap Cold Collar Liquid Nitrogen Z3 Water Cooled Baffle 1 Diffusion Pump Valve i Collision Region , Helium Leak H i n p e - - Q — Valve Needle Valve Charcoal Trap, Mechanical Pump 10 cm. F i g . 4. Vacuum Sys tem. The components are drawn t o s c a l e , b u t t h e vacuum c o n n e c t i o n s are s c h e m a t i c o n l y . I l l 32 lasted for about 16 hours. Normally the trap was f i l l e d twice d a i l y . The next component upstream from the l i q u i d nitrogen trap i s a plate valve. I t was useful during the preliminary stages of the experimental work when the vacuum chamber had to be opened often. I t was not used once the experiment was working. However, the poxts i n the valve housing were used. One pasrt was used for rough pumping, and another was used for e l e c t r i c a l wires. The e l e c t r i c a l connections into the vacuum were made through glass insulated feed-through components that were soldered to a brass plate. The vacuum chamber was made from a piene of i n d u s t r i a l pyrex pipe, with 4 inches nominal inside diameter. I t was bolted to a brass flange and sealed with an "0" r i n g . The chamber could be supported from below and removed whenever i t was necessary to change anything inside. The electron gun mount slides into the vacuum chamber and the end w a l l of the mount divides the chamber into two sections. One section contains the electron gun, and the other i s the c o l l i s i o n region. (See Figure 5) The hole connecting the two sections has a con-ductance of 0.6 l i t r e s / s e c . for helium. The pumping speed for helium at the po s i t i o n of the vacuum chamber i s estimated to have been 60 l i t r e s / s e c . Hence when helium was admitted to the c o l l i s i o n region, the pressure r a t i o between ELECTRON GUN TUNGSTEN 1OTE5H ELECTRO COLLECTOR PYREX VACUUM CHAMBER WALL POLAROID CYLINDRICAL T E N S 5 C M INTEFER EN€E~ FILTER P H O T O M U L T I P L I E R Eig. 5 . Arrangement of Apparatus I l l 34 the c o l l i s i o n r e g i o n and the s e c t i o n c o n t a i n i n g the e l e c t r o n gun was approximately 100/1. Because the pumping speed out of the c o l l i s i o n r e g i o n was s m a l l , t h a t p a r t of the vacuum chamber had to be baked i n order to achieve low p r e s s u r e s . Baking a t 120° C f o r a day or two was s u f f i c i e n t to allow the _7 p r e s s u r e 'to go down to 5 x 10 t o r r , as measured by an i o n i z a t i o n gauge. I f the baking was continued l o n g e r , or i f the pumps were l e f t r u n ning f o r a number of weeks, the pre s s u r e would go down to 2 x 10 t o r r . Pressures were measured with a B a y a r d - A l p e r t type i o n i z a t i o n gauge (Veeco RG -75P)• This gauge was c a l i b r a t e d a g a i n s t a McLeod gauge when the p o l a r i z a t i o n experiments were f i n i s h e d . The c a l i b r a t i o n curve i s shown i n F i g u r e 6. 3.2 Helium Source Tank helium was used. I t i s s a i d by the s u p p l i e r ( L i q u i d A i r Co.) to be 99-5$ pure. T h i s was e v i d e n t l y not good enough f o r t h i s experiment, s i n c e near the e x c i t a t i o n t h r e s h o l d , the im p u r i t y l i g h t was at l e a s t as inten s e as the helium l i g h t . The helium was t h e r e f o r e p u r i f i e d by p a s s i n g i t through a l i q u i d n i t r o g e n c o o l e d c h a r c o a l t r a p . T h i s proved to be an e f f e c t i v e enough procedure that the im p u r i t y l e v e l was very s m a l l , and was probably determined by the c l e a n l i n e s s of the vacuum system. 35 Fig« 6« Ion Gauge Calibration for Helium. The ion gauge readings correspond to a factory calibration for dry nitrogen. I l l 36 The helium from the tank was admitted by means of a needle valve through the charcoal trap into a reservoir region with a volume of roughly one l i t r e where i t was l e f t at a constant pressure of up to 0.5 atmospheres during a run of several hours. The helium passed from the reservoir into the c o l l i s i o n region through a very fine glass c a p i l l i a r y leak, made by drawing out a piece of c a p i l l i a r y tubing i n a flame. A pressure of 0.5 atmospheres i n the reservoir gave r i s e to a pressure of about 4 m i l l i t o r r i n the c o l l i s i o n region. 3.3 Electron Gun - Design and Operation The electron gun i s shown i n Eigure J. I t was made according to a design by Simpson and Kuyatt."*" In t h i s experiment the requirement i s for an electron gun that w i l l produce a stable, w e l l defined electron beam at low energy, and with as high a current as possible. At low energies, the current density i s l i m i t e d by the dispersive effect of space charge. An example of the magnitude of th i s e f f e c t which i s relevant to the electron gun that was used i s the following. A 25 v o l t , 10/cA, electron beam with a diameter of 2 mm. w i l l , i n a length of 4 cm. be dispersed to an extent that the electrons at the edge of the beam w i l l be moving at an angle of 0.05 radians from the beam axis. (The derivation of th i s quantity i s given i n Appendix IIIA.) I l l 38 I t w i l l be seen l a t e r that the observed behaviour of the electron beam at i t s best i s consistent with this picture. Because of the space charge e f f e c t , there i s a d e f i n i t e upper l i m i t to the amount of current that can be forced through a 2 given space at a given energy by e l e c t r o s t a t i c focusing. The electron beam can be confined magnetically, but the author considers a magnetic f i e l d objectionable, because i n a p o l a r i -zation experiment i t i s the d i r e c t i o n of the electron's v e l o c i t y that i s important rather than the electron's p o s i t i o n . A magnetic f i e l d confines the po s i t i o n of the electrons, but allows the electrons to move i n a s p i r a l motion with a v e l o c i t y component perpendicular to the electron beam. The design of the electron gun used here i s intended to put the maximum possible current into a beam of given dimensions. I t does t h i s by f i r s t accelerating the electrons to 10 times th e i r f i n a l energy i n order to draw s u f f i c i e n t current from the outside, and then decelerating them and projecting them into the electron beam at the proper angles. Voltages were applied to the plates of the electron gun as indicated i n the fi g u r e . The gr i d potentiometer Had l e f t at a constant setting such that the p o t e n t i a l difference between the cathode and gr i d was 8$ of the cathode p o t e n t i a l . The anode p o t e n t i a l was adjusted as necessary to be 10 times the magnitude of the cathode p o t e n t i a l . ( I t r e a l l y should have been 9 times, but the change made very l i t t l e difference I l l 39 i n the beam shape.) To the cathode was applied a square wave pot e n t i a l at a frequency of 500 Hz. The l i g h t from the electron beam was chopped i n t h i s way, as w i l l be explained l a t e r . The p o t e n t i a l applied to the cathode determines the electron energy since the c o l l i s i o n chamber i s at ground p o t e n t i a l . At 25 v o l t s , the electron gun delivered from 5 to 10/sA, depending on the condition of the cathode. The electron beam, which i n the presence of helium could be seen with dark adapted eyes, was 2 mm. i n diameter where i t emerged from the gun, and spread to a diameter of 4 mm. at the electron c o l l e c t o r which was 4 cm. away. After the gun had been i n service for several weeks, the beam would spread to 1 cm. at the c o l l e c t o r . Removing the gun and cleaning i t with trichloroethylene vapour restored the o r i g i n a l beam shape. 3.4 Electron Gun - Construction The electron gun was made from a k i t supplied by Nuclide General Corporation. I t consists of stainless s t e e l plates and ceramic spacers connected by ceramic rods. Before t h i s k i t was obtained, the author spent a large amount of time i n f r u i t l e s s attempts to make an electron gun. These attempts w i l l not be described here, but a few comments may be i n order. The main point i s that the number of materials that can be used at elevated temperatures i n a vacuum i s severely I l l 4o r e s t r i c t e d . This i s p a r t i c u l a r l y true of ins u l a t i n g materials. Ceramics and glass are about the only materials that can be used. As for metals, brass cannot be used, because of the high vapour pressure of zinc. Ordinary e l e c t r i c a l soft solder should not be used, even below i t s melting point, because when i t becomes warm, i t sprays the vacuum chamber with metallic vapour. A ceramic material that promised to be very us e f u l i s Sauereisen Insa-Lute cement. When i t i s dry, i t forms a hard, strong, ceramic-like body. Unfortunately, ^after o being heated to only 120 C i n a vacuum for a few days i t dries out, and crumbles to powder under the s l i g h t e s t stress. The chief d i f f i c u l t y i n making electron guns, then, is the i,ack of a suitable i n s u l a t i n g material thaft can be worked e a s i l y and p r e c i s e l y . This i s important, because the metal parts must be aligned p r e c i s e l y by i n s u l a t i n g pieces that are small and d i f f i c u l t to make. A suitable material that has recently become available Is Boron N i t r i d e , available from The Carborundum Co., Latrobe, Pa., U.S.A. There appear to be three possible approaches to making electron guns for experimental work. One i s to make precise i n s u l a t i n g parts. This i s what has been done at the factory i n the electron gun k i t . As an al t e r n a t i v e , 3 a suggestion by Krotkov was to use synthetic sapphire b a l l s as pre c i s i o n i n s u l a t i n g spacers. A second approach I l l 41 i s to a l i g n the metal parts pr e c i s e l y on some sort of j i g , and f i x them i n place with a p l a s t i c i n s u l a t i n g material. This i s what i s done i n the commercial manufacture of electron guns, i n which the p l a s t i c material i s glass. But this, requires special equipment. A t h i r d approach i s to use ready-made electron guns, from t e l e v i s i o n tubes for instance. This i s a good approach, of course, only i f one can f i n d a gun 4 suitable for the experiment. Heddle and Keesing used a low energy electron gun from a magnetron tube, which, however required a magnetic f i e l d to operate. The k i t solves the problem of making the electron gun, but one s t i l l has to decide what kind of cathode to use and how to attach i t . To st a r t with, the electron gun used i n t h i s experiment requires a planar cathode, and for the best energy res o l u t i o n , and least thermal l i g h t , the cathode should operate at a low temperature. These c r i t e r i a suggest oxide cathodes. These were t r i e d . The disadvantage i s that they are short l i v e d i n practice. They poison e a s i l y , and should be kept under continuous high vacuum. They can be taken out of the vacuum, put back, and reactivated i f care i s taken to keep them dry, but their emission i s substa n t i a l l y reduced, and the procedure can be repeated at most two or three times. They also tend to make the gun a b i t d i r t y , since the binder evaporates during a c t i v a t i o n and set t l e s on other parts of the gun. I l l 42 Dispenser cathodes were f i n a l l y used. They run at a o n o . somewhat higher temperature, (1100 C as opposed to oOO C.) but they recover a f t e r p o i s o n i n g , and they survive being taken to atmospheric pressure quite w e l l . They can even be l e f t under rough vacuum f o r a while without r e q u i r i n g r e a c t i v a t i o n . Only three dispenser cathodes were used i n a l l . The cathodes were mounted to a s t a i n l e s s s t e e l p l a t e by means of three tungsten wires 0.01" i n diameter and 1 cm. long. The wires were spot welded w i t h tantalum f o i l being used as f l u x . (The manufacturers recommend that only r e f r a c t o r y metals be used at the cathode temperature to avoid p o i s o n i n g the cathode.) Shorter, t h i c k e r wires were t r i e d , but i t was found t h a t the greater heat l o s s n e c e s s i t a t e d heater temperatures high enough to burn out heater f i l a m e n t s a f t e r a few hours of operation. With the arrangement j u s t described, however, heaters o u t l a s t e d the cathodes. The heater was spot welded (with tantalum f o i l ) to tungsten wires which were supported by a gla s s s t r u c t u r e . (See Figure 7.) The e l e c t r i c a l leads were bare n i c k e l wires spot welded to the s t a i n l e s s s t e e l p l a t e s of the e l e c t r o n gun. These wires were spot welded at the other end to p l u g connectors made from one inch f i n i s h i n g n a i l s cemented i n t o pieces of ceramic tubing. On the other side of the I l l 43 connectors, within the valve housing which was always cool, enamelled copper wire was used. o The cathodes were activated at 1150 C brightness o temperature and operated at approximately 1000 C brightness temperature. In order to f i n d the relationship between temperature and heater current, the cathode was heated i n a vacuum, without the remainder of the electron gun, and i t s temperature was measured with an o p t i c a l pyrometer. 3 . 5 Electron Gun Mount This i s shown i n Figure 8. The electron gun rests on ceramic rods. The rest of the mount i s made of copper. The end of the electron gun i s attached with screws to the wa l l separating the c o l l i s i o n region from the rest of the vacuum chamber. The whole mount was made to slid e i n and out of the vacuum chamber, and to make a f a i r l y .close f i t with i t . The idea was to make the hole that the electrons passed through the only important passageway for helium between the two regions of the vacuum chamber. The f i t between the electron gun and the vacuum chamber was close enough that baking at a o temperature a great deal higher than 120 C would have been r i s k y because of the expansion of the copper. The whole assembly of electron gun and mount, except for the cathode, was cleaned with trichloroethylene vapour, 45 as shown i n Figure 9, just before putting i t under vacuum. 3 . 6 C o l l i s i o n Chamber Everything i n the c o l l i s i o n chamber except o p t i c a l parts was coated with c o l l o i d a l graphite to reduce surface 5 • 6 charging-^ and r e f l e c t i o n s of l i g h t and electrons . A gr i d of fin e tungsten mesh 2 cm. i n diameter concentric with the electron beam was used to shie l d the electron beam from surface charges on the viewing.window. That such surface charging was very important was shown by the following observation when the shield was not present. I f one looked at the electron beam (with helium i n the c o l l i s i o n region), what was seen depended on the energy of the electrons. Up to 24 v o l t s , nothing was seen. Then as the energy was increased, the beam would become v i s i b l e at both ends, and lengthen from each end towards the middle and unite when the electron energy was approximately 30 v o l t s . (See Figure 1 0 . ) D C l 24- v o l t s • = = = = > c = Z ] c=:zi:::__:iZZZ) 3 0 v o i t s F i g . 1 0 . Evidence of Potential Minimum. 46 t i g * 9. "Vapour Degreasing" Method of Cleaning Vaouum Parts. I l l 47 This effect i s interpreted as being due to a po t e n t i a l gradient along the beam. The po t e n t i a l was 6 volts lower at the centre of the c o l l i s i o n region than at the ends. In order to remedy this,' an e l e c t r o s t a t i c shield made of wires spaced at 3 nmu intervals was placed around the beam. However i t was not e n t i r e l y e f f e c t i v e . The tungsten mesh f i n a l l y used has a wire spacing of 0 . 8 mm. and i t eliminated the e f f e c t . The e l e c t r o s t a t i c p o t e n t i a l calculations necessary to determine how fine a mesh i s needed are given i n Appendix IIIB. The electron c o l l e c t o r i s a copper cup f i l l e d with f i n s i n order to reduce electron r e f l e c t i o n s . The vacuum chamber, except for the viewing window, was surrounded by two layers of magnetic shielding (Conetic AA, Perfection Mica Co.). The transverse magnetic f i e l d at the po s i t i o n of the electron beam was measured to be 0 . 0 1 gauss. The long i t u d i n a l f i e l d was of the same order of magnitude. 3 . 7 Optics Because the l i g h t i n t e n s i t y i n th i s experiment i s small, i t i s important to c o l l e c t as much of the l i g h t as possible, within certain l i m i t a t i o n s . One l i m i t a t i o n i s the p o l a r i z a t i o n of l i g h t changes with the angle of observation (equation 2 7 ). Therefore we cannot c o l l e c t I l l 48 l i g h t over a large angle oc . However we may c o l l e c t l i g h t from as large an angle 0 as we l i k e . Another l i m i t a t i o n i s imposed by the size of the l i g h t detector, which i s e f f e c t i v e l y about 1 cm. i n diameter i n t h i s experiment. The o p t i c a l system used i s shown i n Figure 11. The l i g h t from the excited helium Is focused on the photocathode, i n the plane of the electron beam by a c y l i n d r i c a l lens, and i n the perpendicular plane by an e l l i p t i c a l aluminum mirror. The f o c a l r a t i o of the mirror i s approximately f/0.5 and that of the lens f/5. Thus l i g h t was collected from a s o l i d angle of approximately 4$ of 4rr , .and from an area of approximately 1 cm. (along the beam) by 2 mm. The image size of this part of the electron beam i s 1 cm. x 1 cm. Thus i n one plane, l i g h t i s collected from a large angle and small object, and i n the other plane l i g h t i s collected from a small object and a large angle. The mirror was cut from a s o l i d piece of aluminum with a m i l l i n g machine with i t s head t i l t e d at the proper angle to form the desired e l l i p s e . I t was polished with "Brasso" and " S i l v o " . I t was found that s o l i d aluminum i s not an id e a l material for making mirrors because i t i s somewhat porous. I t should be noted that the in t e n s i t y of l i g h t received from an o p t i c a l system l i k e the one just described i s quite sensitive to changes i n the size and shape of the SIDE VIEW 49 Top View Fig. 11. Focusing Properties of Optical System.c i l l 50 electron beam. Therefore e x c i t a t i o n curves obtained with i t cannot be r e l i e d upon to be accurate. However, t o t a l i n t e n s i t y changes should not affect the accuracy of the p o l a r i z a t i o n measurements. The spectral l i n e s were isolat e d with interference f i l t e r s , and analysed for p o l a r i z a t i o n by a sheet of polaroid o which turned 90 at automatically timed i n t e r v a l s . Type HNP'B polaroid was used for the 3889A l i n e and type HR for the 1 0 , 8 2 9 A l i n e . Transmission curves for the interference f i l t e r s are shown i n Figure 12 and Figure 1 3 . 3 . 8 Photomultipliers For measurements on the 3889 A l i n e , an E.M.I. 6256s photomultiplier was used. I t had a dark cureent of about 130 counts/sec. at room temperature and about 5 counts/sec. at 260°K. I t was usually not necessary to cool i t . The 7 6256s was operated at 1500 v o l t s and had a gain of 2 . 5 x 10 . For measurements•on the 10,829A l i n e , a P h i l l i p s CVP 150 photomultiplier selected for r e l a t i v e l y high •infrared response was used. I t s quantum e f f i c i e n c y at 10 ,o29A was approximately o x 10 . I t was cooled to 135 K where i t had a dark current of 10 counts/sec. The CVP 150 5 was operated at 1420 volts and had a gain of 6 . 5 x 10 . 3 2 0 0 3 4 0 0 3600 3800 400d 4200 Wavelength A Fig. 1 2 . Optical Transmission of Interference Filter 52 o I l l 53 3 . 9 Photo-multiplier Cooling The housing msed to cool the CVP 150 photomultiplier i s shown i n Figure 14 . Liquid nitrogen i s boiled o f f , and the r e s u l t i n g cold, dry nitrogen i s passed around the photo-m u l t i p l i e r . A thermister attached to the photomultiplier was used to measure i t s temperature. The cooling device was supplied by Spex, but had to be modified somewhat. A brass sleeve was added to improve the cold nitrogen flow and to help provide e l e c t r o s t a t i c shielding. Insulation was added on the inside. F i n a l l y , i n order to prevent the polaroid turner from getting cold i t was necessary to warm i t by passing hot water through a c o i l i n thermal contact with i t ( c o i l not shown i n diagram). The device f i n a l l y operated s a t i s f a c t o r a l l y down to 135°K, as measured by the thermister. 3 . 1 0 Signal Processing A block diagram of the electronics i s shown i n Figure 15. The functions of the various components are described i n the following sections. Electron gun Control. The electron gun control supplies the various potentials to the electron gun and measures the various currents. In order to separate the l i g h t due to the electron beam from stray l i g h t , the Optical Filter Quartz Evacuated Cell Polaroid' rs-Tr Foam Plastic ((k\l Bakelite Aluminum Wire Spacer O T T T T i ( ( ; i ( i (/i i i Photomultiplier Resistor Chain o » / / / / ATF^ o ° o o o o o q i .i i i i i i i i i i t i i i i i b5 5 & • o o v c PM Cables Gas Outlet Heat Exchanger 10 cm. F i g . 14. Photomultiplier Cooling. The photomultiplier was wrapped i n aluminum f o i l whioh was connected at cathode potential through the wire spaoer. Pre -Amplifier Lock - in Amplifier Photomultiplier F — H Polaroid Turner E l e c t r o n Gun Electron Gun Control Analog to Digital Converter Switch Control Scaler Scaler Fig. 15. Electronics? Block Diagram I l l 56 electron beam was chopped. The usual way to do t h i s i s simply to turn the beam on and off at some frequency, but i n t h i s experiment i t was done i n a s l i g h t l y more subtle way. The beam was always on, but the energy given to the electrons alternated between the working values, and a value one or two volts below the e x c i t a t i o n threshold. This was done to minimize the amount of background l i g h t coherent with the signal when measurements were made close to threshold. The chopping was done, then, by applying a square wave p o t e n t i a l to the cathode. The c i r c u i t used to do t h i s i s shown i n Figure 16 . Also shown i s the way i n which the applied p o t e n t i a l was measured. I t was compared to the p o t e n t i a l along a precision (0.1$ l i n e a r i t y ) potentiometer, which was i n turn calibrated with a mercury battery as standard. The method of chopping made a special c i r c u i t necessary for measuring the electron beam current. I t was necessary to measure the current during the "on" part of the cycle. This c i r c u i t i s also shown i n Figure 17 . The electron gun control-also supplies the l o c k - i n amplifier with a reference signal derived from the same o s c i l l a t o r that drives the cathode supply. Preamplifier. The photomultiplier output was loaded with a 100K r e s i s t o r and fed into a broadband preamplifier. 5 7 - V . - V a 10 Turn Precision P o t e n t i o m e t e r rig. 16. Eleotron Gun Cathode Supply„ Potentials -Vo and -V* are supplied alternately to the cathode at a frequency of 500 Hz. Vo and V*. are independ-ently adjustable from 0 to 50 volts. The greater of Vi and V a is measured with the peak rectifier and potentiometer circuits. 58 Fig. 17. Mioroammeter for Electron Beam. 1$ acouraoy resistors are used. Peak ourrent is measured. I l l 59 _5 This arrangement had a time constant of 10 seconds. The preamplifier (Micronoise, Denro Labs) has a gain variable between 10 and 5 0 . Lock-in amplifier . The l o c k - i n amplifier consists of a tuned amplifier and a phase sensitive detector. The phase sensitive detector responds only to that part of the signal that i s coherent with, and i n phase with the reference s i g n a l . The l o c k - i n amplifier used i n this experiment i s a model JB - 4 , Princeton Applied Research. I t has a gain of 9000 and a l i n e a r i t y of 1%. The output i s a D.C. d i f f e r e n t i a l voltage which i s 5 volts at f u l l scale. In t h i s experiment, i t was used with a one second time constant.. Analog to d i g i t a l converter. I t was found that longer intergration times were required than were con-venient to provide with RC c i r c u i t s , so d i g i t a l averaging was used. The anagog to d i g i t a l converter i s 'a device constructed from operational amplifiers that puts out pulses at a rate p r e c i s e l y proportional to the p o t e n t i a l difference applied to i t s input. (Linearity i s 3 parts i n 10 .) Five v o l t s gives r i s e to 100 pulses per second. Thus the number of pulses registered over a given time i n t e r v a l i s a measure of the average output of the l o c k - i n amplifier. I l l 60 Switch control. The switch control determines both the p o s i t i o n of the polaroid and the flow of signal pulses. The sequence of operations, repeated every 20 seconds, i s as follows. To sta r t with, the pulses are flowing to one scaler. Then, at a signal from the timer, the pulses are switched o f f , and the polaroid turns 90°• Six seconds l a t e r (the time required for transient signals to die away) the pulses are switched on again, t h i s time going to the other scaler. In th i s way, a r b i t r a r i l y long integration times can be achieved, and slow d r i f t s i n the t o t a l l i g h t i n t e n s i t y do not affect the r e s u l t . In practice, integration times up to 20 minutes were used. The switch control was made up of multivibrator c i r c u i t s with long time constants, and electro-mechanical relays. The timer was a free running multivibrator that completed a cycle every time the polaroid turned. Thus there was no p o s s i b i l i t y of a bias i n the lengths of time given to the " p a r a l l e l 1 ! s i g n a l and the "perpendicular" s i g n a l . When a constant voltage (a dry c e l l ) was placed across the input terminals of the analog to d i g i t a l converter, the scaler readings were found to be equal to within a few parts i n 10 . 3 . 1 1 Polaroid Turner This i s shown i n Figure l 8 . I t i s turned by a small wheel with a f r i c t i o n drive. The small wheel i s 61 Fig. 18. Device Used to Rotate Polaroid. I l l driven by a f l e x i b l e cable that i s rotated by a reversible e l e c t r i c motor. The motor i s turned on only during the time that the polaroid i s moving. I l l 63 References and Footnotes for Chapter I I I 1. J . A. Simpson and C. E. Kuyatt, Rev. S c i . In s t r . 34, 265 ( 1 9 6 3 ) . 2. J . R. Pierce, Theory and Design of Electron Beams (Van Nostrand, New York, 1 9 5 4 ) . 3 . R. Krotkov, Private Communication. 4 . D.W.O. Heddle and R.G.W. Keesing, Proc. Roy. Soc. (London) Ser. A 25_8, 124 (1967) . 5 . E. Lindholm, Rev. S c i . I n s t r . 3 1 , 210 ( i 9 6 0 ) . 6 . P. Marmet and L. Kerwin, Can. J . Phys. 3 8 , 7 8 7 , i 9 6 0 . CHAPTER IV EXPERIMENTAL RESULTS 4.1 'Data The p o l a r i z a t i o n of l i g h t as a function of applied p o t e n t i a l i s shown i n Figures 23 to 33• Each of the eleven figures represents data taken on one day. The error bars represent r.m.s. s t a t i s t i c a l errors only. These were determined from estimates of photelectron pulse rates. No corrections have been applied to the data. The v e r t i c a l dashed l i n e s at the bottom of some of the graphs indicate "off" voltages used. (See section 3.10) The t h e o r e t i c a l curve i n Figure 28 i s derived from values given by Massey and Moiseiwitch"*" as the r e s u l t of a distorted wave cal c u l a t i o n '(see section 2 . 5 ) . The curves drawn through the p o l a r i z a t i o n data near threshold are taken from the threshold p o l a r i z a t i o n model which w i l l be discussed l a t e r , and have been f i t t e d to the data. The "excitation" data are simply plots of I" - l x divided by the electron beam current. They are plotted i n order to give some idea of the l i g h t i n t e n s i t y , and because they are of some i n t r i n s i c i n t e r e s t . The exc i t a t i o n scale i s a r b i t r a r y and i s not shown on the graphs. Among the i — i — i — r 24 22 20 18 16 14 12 3 3P-2 3S 3889 A -3 2 x 1 0 t o r r 1 yd A 0 10} o + J 8| m N 161 o Q. 41 2 0 -2 Polarization f Excitation •-J - L 24 Fig. 23. 28 32 36 40 lied Potential (volts) 44 66 3 6 -34-32-30 — 28-26-24-22 20 § 18 _ g 16 c 10 8 6 4 2 0 © • o J L 33P--2*S 3 889 A 4x10 torr Polarization { E x c i t a t i o n • J" *1-1 ^A 24 28 J 1 l I 1 32 F i g . 24. 36 40 44 Potential (volts) 48 20 18 16 14-12 5? 10 g 8 h - H 03 M 6 h S-ro O 4 h 2 h 0 - 2 k Figo 25, 23 24 25 3 3P-2 SS 3889 A -3 2x10 t o r r Polarization, f • Til + T4. Excitation • •* A 26 (volts) 27 14 12 10 5s 8 c .2 6 +-> n» Z 4 OJ £ 2 0 t I 1 l 1 | 1" j T "j- J 0 3 3 P - 2 3 S 3889 A — - 3 4 x10 t o r r o 0 0. • —' ' \> \ \ e — 0 e • e \> X 0 O o ° o o — _ Polarization 4 — Excitation ° i e 1 1 i I , I . I t Fig. 26. 24 25 26 27 Applied Potential (volts) 28 24 22 20 18 16 14 o o - , 2 C o 10 nj N Z 8 O °- 6 4 2 0 I I- I I I • 1 1 I I 1 1 • 2 3P-2 3S • 10.829A 0 0 0 a o o ~ 0 o 0 • O o 0 • • • e • — • • Polarization $ -Excitation 8 ^ A « • • • a "~* * I ! . • i i _ i ! 1 1 1"" ! I I I I 1 1 I 20 24 28 32 36 40 44 Fig. 87. Applied Potential (volts) c o m o CL 24 22 20 18 16 14 12 10 8 6 4 2 T — i r ! <y» L- 8 J L 2 3P-2 3S 10,829 A O o o Polarization 4 Excitation • Theoretical P o l a r i z a t i o n I I I F i g . 28, 24 28 32 36 40 44 Applied Potential (volts) 48 T 2*P- 2*S 10 829 A 20 18 16 14 • 0 12! o i o | o N r 8| o £ 61 4 2 0 22 Polarization > Excitation • J i ± J i • MA J Fig. 29. 23 24 Applied Potential 25 (volts) 24 -22 20 18 16 14 12 10 8 :':e\ A 2 0| 2 SP- 2*S 10,829 A* Polarization Excitation • 21 Fig. 30, 22 23 24 25 Applied Potential (volts) 26 27 71 24 22 20 18 16 14 12 10 8 6 4 2 0 2 SP-2 3S 10,8 29 A Polarization $ E x c i t a t i o n • >tA 20 Fig. 31. i 1 21 22 23 24 Applied Potential (volts) 25 26 24 22 20 18 .tcr 14 12 I 1 0 1 H3 Q i M O O 6| 2 o 20 Fig. 32. 7 » i 1 2 3P- 2 3S 10,829 A Polarization 4 -Excitation 1" • P-MA i 21 22 23 24 Applied Potential (volts) 25 26 24 22 20 18 16 14 12 10 •s\ 6 4 2 ' 0| L 21 Fig.33. i 7ff 2 JP-2 , SS 0 10,829A Polarization { Excitation • 22 23 24 . 25 Applied Potential (volts) 26 27 I V 76 graphs that show the whole energy range only Figures 24 and 28 show the correct e x c i t a t i o n curve. For the other graphs, the apparent electron beam current i s an average of the "of f " and "on" currents. The e x c i t a t i o n data are expected to be less accurate than the p o l a r i z a t i o n data. This i s because slow d r i f t s i n signal i n t e n s i t y , and variations i n the size of the electron beam affect the i n t e n s i t y data but not the po l a r i z a t i o n data. The ex c i t a t i o n data are not considered to be accurate enough to j u s t i f y p o l a r i z a t i o n corrections i n order to obtain r e l a t i v e cross sections. The helium pressure and electron beam current are indicated on the graphs of the 3889A l i n e p o l a r i z a t i o n . For the 10,829A data the experimental conditions are a l l _ -3 approximately as follows: 4 x 10 t o r r , 7/^ A near threshold and higher currents at higher energies. The v a r i a t i o n of po l a r i z a t i o n with pressure i s shown i n Figure 3^. 4.2 Energy Scale The "applied p o t e n t i a l " i s the po t e n t i a l difference applied between the cathode and the scattering chamber, which was at ground p o t e n t i a l . The thresholds for ex c i t a t i o n are taken to be the values obtained by extra-polating the steepest parts of the i n t e n s i t y curves. The differences between these apparent thresholds and the T X* 10,829 A 5- — . X? 3 8 8 9 A o I I 1 1 " 2 3 4 HELIUM PRESSURECmiHitdrr) Fig. 34. Polarization as a Function of Pressure -si IV 78 3 3 3 spectroscopic values are, for the 2^P, 3 Pj 3 S l i n e s respectively 1 .95 v o l t s , 1.8 v o l t s , 1 . 9 v o l t s , or 1 .9*0.1 v o l t s . (Some of the curves do not show th i s agreement. These were taken before the voltage applied to the cathode was properly calibrated.) The threshold energies obtained from f i t t i n g curves to the p o l a r i z a t i o n data tend to be a b i t higher than the ones just given, but not by more than about 0 . 2 v o l t s . 4 . 3 Experimental Sources of Error P o l a r i z a t i o n due to o p t i c a l elements. There were no elements i n the o p t i c a l system capable of p o l a r i z i n g p a r a x i a l l i g h t rays, but for the highly convergent l i g h t that was used, the mirror, the curved glass w a l l sBf the vacuum chamber, the c y l i n d r i c a l lens, and the interference f i l t e r can a l l effect the p o l a r i z a t i o n . The contribution to the p o l a r i z a t i o n due to the glass w a l l and the lens can be calculated, and are -0.5% and - 0 . 4 $ respectively. (See Appendix IVA) The effects of the mirror and f i l t e r can be larger and were found by varying the aperture at the p o s i t i o n of the lens. o In the case of the 3889A l i n e , the p o l a r i z a t i o n as a function of aperture varied over a range of 2 or 3% (depending on the electron energy) i n a way that i s d i f f i c u l t to interpret. (See Pigure 3 5 . ) At electron energies 8 7 6 5 c o o ° N o o CL 11 • 10 11 10 I I _ •'Spherical lens at f/10 I I 8 0 0 o 8 0 — 0 0 o o 3889 A Mirror o ° ° 0 o — 3889 A Lens 0 0 _ O 0 ° o 10 829 A Mirror o O 0 0 10 829 A ~ Lens I I 79 2 3 Aperture (cm.) Fig. 35. Polarization as a Function of Optical Aperture. i v 8o s u f f i c i e n t l y low that the 3889A l i n e was the only one within the bandpass of the f i l t e r that could be excited, i t was found that the p o l a r i z a t i o n at f / 1 0 agreed with the f u l l aperture value, while the value obtained with a spherical lens at f / 1 0 was 1% higher. The 3889A p o l a r i z a t i o n must be considered uncertain to at least ±% on t h i s account. In the case of the 1 0 , 8 2 9 A l i n e , these problems were o evidently absent, 'since the observed p o l a r i z a t i o n was independent of aperture to better than Yfo (see Figure 35) • As a further check on the p o l a r i z i n g effect of the 3 3 ° optics, the p o l a r i z a t i o n of the 3 S-2 P (7065A) l i n e was measured. This l i n e originates from an upper S state, and should have zero polarizations The results are shown i n Figure 3 6 . The observed p o l a r i z a t i o n i s about -O.hfo. The r i s e i n p o l a r i z a t i o n s t a r t i n g at 25 volts occurs at the threshold of the 3 1D - 2 1P (6678A) l i n e , for which the transmission of the interference f i l t e r i s 0 . 1 times that of the 7065A l i n e . V a r i a t i o n of p o l a r i z a t i o n with angle. The cone of l i g h t c ollected from the excited helium extends to 0 . 1 radians on either side of the d i r e c t i o n perpendicular to the electron beam. For the polarizations that were EXCITATION 12 10 8 2 2 < O 0. -21 i 5 POLARIZATION 6 * 4 < o X UJ 24 25 1 26 27 AR P LI ED ROT ENTI AL (volts) Fig. 36. Polarization and Excitation Curves for the 33S-23P (7065 A) Line 28 co IV 82 observed, t h i s angular effect reduces the p o l a r i z a t i o n by 0.1% (for P - 20%) at most, which i s e n t i r e l y n e g l i g i b l e . (Refer to section 2.7.) Dispersion of electron beam. When the electron beam was at i t s worst, the half angle of the electron cone was 0 . 1 3 radians. This results i n a p o l a r i z a t i o n of 20% being reduced by 0.2%, which again i s n e g l i g i b l e . (Refer to section 2.9.) Coherent background. The coherent background consists of any unwanted signal that i s coherent with the chopping- of the electron beam. This signal was too small to measure d i r e c t l y . In order to estimate the effect of the coherent background s i g n a l , i t s magnitude was changed by a factor of perhaps 2 or 3 . This was done by changing the "off" p o t e n t i a l applied to the electron beam by 1 or 2 volts when the "on" po t e n t i a l was near threshold. At the lowest 0 signal l e v e l s , the p o l a r i z a t i o n of the 1 0 , 8 2 9 A l i n e seemed to change by an amount comparable with the s t a t i s t i c a l error. This test was not done for the 3889A l i n e , although d i f f e r e n t "off" potentials were used for d i f f e r e n t runs. For the dif f e r e n t runs, the plateau of p o l a r i z a t i o n near threshold i s reproducible, but the way i n which the p o l a r i z a t i o n f a l l s IV 83 off below threshold i s not. I t i s assumed that t h i s f a l l i n p o t e n t i a l below threshold i s due to the background sign a l . Incoherent background. In observations of the 3889A l i n e , the incoherent background consisted of dark current and unmodulated l i g h t from impurity gases. The background -7 l i g h t at 5 x 10 torr was 50 counts/sec. ( i . e . 50 photo-electrons/sec.). In observations of the 10,829A l i n e , the dominant background was l i g h t from the hot cathode which amounted to 800 to 1000 counts/sec. These known background levels together with shot noise were used to determine the size of the error bars. For comparison, the weakest signal i n the infrared p o l a r i z a t i o n curve i s equivalent to 40 counts/sec. Unwanted o p t i c a l wavelengths. The interference f i l t e r s have bandpasses wide enough to transmit l i g h t from other helium l i n e s . (See Figures 12 and 13-) Near threshold, however, none of these l i n e s are excited. 3 8 8 9 A F i l t e r : The width of the transmission curve i s 140A. Several unwanted helium l i n e s are transmitted by t h i s f i l t e r , which have thresholds of 0.7 volts or more above the 3 8 8 9 A threshold. The r e l a t i v e i n t e n s i t i e s of these l i n e s transmitted by the f i l t e r were measured with a IV 8 4 monochrometer. The f r a c t i o n of the l i g h t that was due to these l i n e s was found to be approximately 10$ at 30 volts and 20$ at 45 v o l t s . The increase i s due almost e n t i r e l y to the 4 1 D - 2 1 P (3964A) l i n e . 1 0 , 8 2 9 A F i l t e r : The width of the transmission curve at half maximum i s 125A. The only i n t e r f e r i n g l i n e s are weak ones among states' of high p r i n c i p a l quantum number. They begin at 3 volts above the 2^p threshold and together contribute perhaps 1$ of the l i g h t . Cascading. Some of the l i g h t r e s u l t s from cascading from higher l e v e l s . Again, t h i s does not occur near threshold. 2 Cascading i n helium has been discussed by Gabriel and Heddle. Electronics. The amplifiers and other components together are lin e a r to within about 1 $ . IV 85 References and Footnotes for Chapter IV 1. H.S.W. Massey (London) Ser. 2. A. H. Gahriel (London) Ser. and B. L. Moiseiwi A 2 5 8 , 147 ( I 9 6 0 ) . and D.W.O. Heddle, A 2 5 8 , 124 ( I 9 6 0 ) . ch, Proc. Roy. Soc. Proc. Roy. Soc. CHAPTER V DISCUSSION OF RESULTS AND CONCLUSIONS 5 . 1 P o l a r i z a t i o n Structure In the 2 3p_2 3S p o l a r i z a t i o n curve (Figures 2 8 , 2 9 , 3 0 ) , there appears a bump at 1 .2 volts above threshold. The bump is quite reproducible and the uncertainty'in i t s peak r e l a t i v e to the apparent threshold i s perhaps - 0 . 2 v o l t s . I t i s tempting to i d e n t i f y t h i s structure with the resonance i n forward i n e l a s t i c electron scattering at 1 .6 above threshold observed by Chamberlain."'" However, because of the 0.'l_4 v o l t discrepancy i n energy and the fact that the structure does not appear as an increase i n the. p a r a l l e l e x c i t a t i o n function, the relationship between the bump and the resonance is unclear. An experiment with better energy resolution would help. 5 . 2 Threshold P o l a r i z a t i o n 3 3 In the case of the 2-T-2 S l i n e p a r t i c u l a r l y , the structure of the p o l a r i z a t i o n curve" near threshold i s buried under the broad energy d i s t r i b u t i o n of the electron beam. What follows i s an attempt to estimate the threshold p o l a r i -zation by using a model of the electron energy d i s t r i b u t i o n and the p o l a r i z a t i o n and i n t e n s i t y curves. We assume that: V 87 (i) The i n t e n s i t y I has the form I=kx where x i s electron energy i n electron volts r e l a t i v e to the threshold energy. ( i i ) The true p o l a r i z a t i o n P near threshold i s given by P = P o (1-yx) where P o i s threshold p o l a r i z a t i o n . ( i i i ) The energy d i s t r i b u t i o n of the electron beam having mean energy x 0 i s of the form exp ,-lx-Xol/cr. Assumption ( i i i ) i s correct at least for the high energy edge of the energy d i s t r i b u t i o n , since the apparent e x c i t a t i o n curve below threshold i s exponential. The r e s u l t i n g apparent p o l a r i z a t i o n Pa(x) i s then given by P a ( x ) / P o = 1-2^0- • x * 0 P ^ ( x ) / P c = l-2y<r - 4jcrR(x/o-) x ^ 0 where R(t) =• (l-t+^t 2-exp. (-t))/(exp. (-t)+2t) The derivation of t h i s expression, and a graph of Po.(x) are given i n Appendix VA. There are two adjustable parameters cr and y . cr was determined by f i t t i n g exponential curves to the e x c i t a t i o n functions below threshold, y was determined by f i t t i n g -R(x/o~) curves to the p o l a r i z a t i o n curves near threshold. . The result s are shown i n Table I I I . The threshold polarizations obtained i n t h i s way are 32 ^6$ for the 2~)P12-:>S l i n e and 3 3 1 5 ± 3 $ for the 3 P-2 S l i n e . The error l i m i t s are somewhat V 88 a r b i t r a r y , since they depend on the model, the accuracy of which i s d i f f i c u l t to judge. However, i t can be said that the data are consistent with the predicted threshold p o l a r i -zation for the 23p_2 3S l i n e but not for the 3 3P - 2 3S l i n e . TABLE I I I THRESHOLD POLARIZATION AND OTHER PARAMETERS POUND BY CURVE FITTING Transition M ° ) (*) o~ (volts) Hvolts) P o ( * ) . 3 3 3 P - 2 JS 1 0 . 0 0 . 2 0 0 . 8 0 1 5 . 0 1 1 . 4 0 . 1 6 O.78 1 5 . 4 average 0 . 7 9 1 5 . 2 3 3 2 JP - 2 JS 2 0 . 8 0 . 1 7 1.13 3 3 . 5 1 9 . 0 0 . 2 0 0 . 5 2 4 * 2 1 . 0 0 . 1 5 1 .07 3 2 . 0 2 0 . 3 0 . 1 5 1 .13 3 1 . 0 average 1.11 - 31-8 t h e o r e t i c a l 0 . 2 7 * * 3 6 . 6 *Not included i n the average **Massey and Moiseiwitch 5 . 3 E x c i t a t i o n Curves I t was mentioned before that the measured e x c i t a t i o n functions cannot be r e l i e d upon to be accurate because the V 89 o p t i c a l system i s sensitive to changes i n the shape of the o electron beam. Nevertheless, the 3889A e x c i t a t i o n curve seems to exhibit more or less the same features as those 2 3 4 measured by other workers. ' ' I t • i s expected, then, 0 that the same i s true of the 10,829A e x c i t a t i o n curve. In p a r t i c u l a r , . i t appears to f l a t t e n out s l i g h t l y at about one vo l t above threshold i n the same way as the cross section curve estimated by Holt and Krotkov. 5.4 Conclusions This thesis has reported the f i r s t measurement of 3 3 the p o l a r i z a t i o n due to electron impact of the 2 P-2 S multiplet i n helium. The shape of the p o l a r i z a t i o n curve i s similar to that predicted by the distorted wave cal c u l a t i o n of Massey and Moiseiwitch, but the observed p o l a r i z a t i o n d i f f e r s from the the o r e t i c a l one i n d e t a i l and 3 3 i n magnitude. Some structure i n the 2 P-2JS p o l a r i z a t i o n curve provides a l i k e l y connection between p o l a r i z a t i o n measurements and electron scattering measurements. This thesis has also reported a measurement of tone 3 3 p o l a r i z a t i o n of the 3 P-2 S multiplet with better than usual s t a t i s t i c a l accuracy near threshold. In answer to the question of whether the p o l a r i z a t i o n approaches the t h e o r e t i c a l value at threshold, i t can be 3 3 said that the p o l a r i z a t i o n of the 2 P-2 S multiplet does, v 90 within the uncertainties of the experiment, and that the 3 3 p o l a r i z a t i o n of the 3 P-2 S multiplet does not, at least not on an energy scale comparable with the energy resolution of the electron beam. 5.5 Suggestion for Further Work The importance of this type of experiment l i e s i n finding p o l a r i z a t i o n values near threshold, and the obvious l i m i t a t i o n of the work reported i n t h i s thesis i s the lack of electron energy resolution. However, the energy resolution obtained i n the present work i s close to the l i m i t imposed by the cathode temperature. I suggest therefore, that the next step for anyone wishing to pursue t h i s l i n e of investigation would be to b u i l d an electron energy selector of the type currently becoming popular i n electron scattering experiments.^ The use of such a device e n t a i l s a severe reduction i n electron beam current, and therefore i n l i g h t i n t e n s i t y . However, i n the experiment of th i s thesis there was more l i g h t than necessary at the wave-length 3889A, and i t should be possible to trade some of thi s for better energy resolution. V 91 References and Footnotes for Chapter V 1. G. E. Chamberlain, Phys. Rev. 155, 46 (1967). 2. C. Smit, H.G.M. Heidman, J . A. Smit, Physica 2 9 , 245 (1963). 3. A. H. Gabriel and D.W.O. Heddle, Proc. Roy. Soc. (London) Ser. A 258, 124 ( i 9 6 0 ) . 4. I. P. Zapesochnyi and 0 . B. Shpenik, Soviet Phys. JETP (English Transl.) _23, 592 (1966) . 5. H. K. Holt and R. Krotkov, Phys. Rev. 144, 82 (1966). 6. J . A. Simpson, Rev. S c i . Instr. 35, 1698 ( 1 9 6 4 ) . 92 APPENDIX I I I A Properties of an Electron Beam In t h i s section, we calculate the effect of the space charge within the electron beam on two properties of the electron beam that are important i n p o l a r i z a t i o n studies. These properties are the energy d i s t r i b u t i o n of the electrons and the r a d i a l dispersion of the electron beam. (i) P o t e n t i a l D i s t r i b u t i o n i n the Cross Section  of an Electron Beam We assume the electron beam to be a uniform c i r c u l a r cylinder, and we assume the current to be evenly di s t r i b u t e d within that cylinder. We use c y l i n d r i c a l coordinates ( z , r , 0 ) . The r a d i a l e l e c t r i c f i e l d within the electron beam i s given by where R i s the radius of the electron beam. In terms of the electron current I , we have where Ve i s the energy of the electrons. Hence and the* p o t e n t i a l r e l a t i v e to the centre of the electron beam i s N u m e r i c a l l y , t h i s i s U[r) = l. 5 a x 10 ^ — where I i s measured in/<A and V i s measured i n v o l t s . To give an example r e l e v a n t to the experiment d e s c r i b e d i n t h i s t h e s i s , we put V=25 v o l t s , I - 1 0 j u A . We f i n d t h a t the p o t e n t i a l v a r i a t i o n from the centre to the edge of the beam _ o i s 3 x 1 0 v o l t s . T h i s i s e n t i r e l y n e g l i g i b l e because the energy v a r i a t i o n of the e l e c t r o n s i s much gr e a t e r than t h i s f o r other reasons. ( i i ) D i s p e r s i o n of E l e c t r o n Beam due to Space Charge As we have j u s t seen, the r a d i a l e l e c t r i c f i e l d a t the edge of the beam i s giv e n by I 7 3 V V-m 3. TT £ 0R The r a d i a l a c c e l e r a t i o n i s then g i v e n by a - E ( R ) - ^ 1 &TT i0 JJy R For s m a l l d i s p e r s i o n s , the r a d i a l v e l o c i t y becomes r ^ <q--rrt0V ft where -£ i s the l e n g t h of the beam and ./v i s the l o n g i -t u d i n a l v e l o c i t y . Then the angle of d i s p e r s i o n ( s t i l l assuming the d i s p e r s i o n i s small) i s giv e n by r\ _ jy>__ r i 1 J- _ For our example o f V=25 v o l t s , I-10/{A, we have 6 = 1.2 2 X ! 0 " 3 4 -Then i f the beam i s 4 cm. l o n g and 0.1 cm. i n r a d i u s , we have 0 - 0 , 0 5 The i n c r e a s e i n r a d i u s of the beam i s g i v e n by When the e l e c t r o n beam was on i t s b e s t b e h a v i o u r , t h i s i s a p p r o x i m a t e l y the i n c r e a s e i n r a d i u s t h a t was a c t u a l l y o b s e r v e d . 95 APPENDIX I I I B The Pote n t i a l i n a Region E l e c t r o s t a t i c a l l y Shielded by Grids We are interested i n knowing how closely we must space the grids i n order to shield the electron beam from stray e l e c t r i c f i e l d s . We use a simple two dimensional model of the s i t u a t i o n i n which an array of i n f i n i t e p a r a l l e l wire separated by distance a i s used to shield a space from a uniform e l e c t r i c f i e l d . -I- - f ± + ' E o a F i g . 19. E l e c t r o s t a t i c Shield The assumption i s made that the radius r e of the g r i d wires i s small compared to a. A l l of the g r i d wires are grounded and have p o t e n t i a l V = 0. The average surface charge i n the plane containing the grids i s f.a e0, so the charge per unit length on the wires i s 96 Consider the e l e c t r i c f i e l d beneath one of the wires (along the dotted l i n e ) . The f i e l d due to that wire i s r£ ^  = S^l. = - £ ° _Qr.. &7l t o V" a TT v% And the corresponding p o t e n t i a l i s 1 a T l ro n = i rv=a. n-3 F i g . 20. E l e c t r o s t a t i c Shield. The f i e l d due to the remaining wires i s En = ? : - ~ 4 4 — 1 — .^in ©-r >i = i air € o v + n ^ ^— -•2- j . y-,-> n % The p o t e n t i a l i s - ^ 5 IT vi -1 r i - o . " V . = Ayv TT f i - t- - - - - -9 7 ( r 0 i s small and i s taken as zero) The f i e l d due to the .upper plate (the plane charge d i s t r i b u t i o n somewhere above the grids) i s x and the p o t e n t i a l i s Then the resultant p o t e n t i a l along the dotted l i n e i s V = v, + V, + V, - ^ \JU -- +• -#sn-T\ - TLT_ 1 3, TV [ r0 ^ (X Introduce the dimensionless variable x r/o. Then V -where 3. TT (" + n = i X 4- iUx, X + x6rv TT — TT X + X J L t J_ where N Using this method of f i n d i n g Iwe arrive at the values of V shown i n Figure 21. We see that we are e s s e n t i a l l y i n a f i e l d free region beyond one gri d spacing from the gr i d . At r>a, then, we have air I r a J or, with regard to the diameter of the wire, 2v0, we have 5TT JyYV 3. rQ . 1 5 98 V 1 9 ? oL 1 1 !.0 2.0 x-F i g . 2 1 . P o t e n t i a l Inside Shield. The p r a c t i c a l r e s u l t s of these calculations are the following. (a) In order to be free of periodic variations i n po t e n t i a l due to the f i e l d that leaks through the spaces between the grids, we must make the gr i d spacing smaller than the distance between the gr i d and the electron beam. (b) Provided we abide by (a), the p o t e n t i a l i n the shielded region i s approximately A V = ^ y r ' , since the factor \JLYL ay^rc — I.15 i s of order unity i n p r a c t i c a l cases. In the present experiment, we are concerned with f i e l d s of the order of 6 volts i n 2 cm., or 0 . 3 volts per mm. The gr i d spacing f i n a l l y used i s 0 . 8 mm. so A V * * — x 0 ' - = 0.04 v o l t s . The fact that the shi e l d that 5 TT was used i s a mesh rather than an array of wires i n only one d i r e c t i o n should improve the shielding somewhat. We see then, that the shielding used was just about what was needed, and shielding that was very much poorer would have made a s i g n i f i c a n t contribution to the electron energy spread. APPENDIX I I I C Signal Processing Theory Transient Signals i n the Output of the Lock-in Amplifier The output signal of the phase sensitive detector i s smoothed by a double section f i l t e r which i s shown schematically below. -— Voi + o-oe F o l l o w e r A M / V -C /\/\/VV-p. e c F i g . 22. RC F i l t e r of Phase Sensitive Detector. For p o l a r i z a t i o n experiments, a time constant of RC = 1 sec. was used. The response of such a f i l t e r to a step function input voltage of the form e(in) = e 0 (a constant) for time t< 0, e(in) - 0 for t >0 i s given by eo \ RC I f we put RC = 1 sec., then i f we wait for 6 s e c , e/e0 = 0.018. The average signal during the next 14 seconds (during which time we are observing another signal) i s given by e/e0 (ave.) - 0.002. That i s , the transient 101 s i g n a l causes an e r r o r i n the next s i g n a l , (which i s of a s i z e comparable to ec) of about 0.2$, which i s s m a l l as compared, with other e r r o r s . S t a t i s t i c a l E r r o r s Suppose i n an o b s e r v a t i o n of l i g h t i n t e n s i t y we count N s i g n a l p u l s e s and N b background p u l s e s . The r.m.s. s t a t i s t i c a l e r r o r i n the (total number of counts i s ./N+Nb. The f r a c t i o n a l e r r o r i s then +• N b J \ + N b . N j N N Next we f i n d the r.m.s. s t a t i s t i c a l e r r o r i n the p o l a r i z a t i o n . The f a c t o r V±.+ N b/N i s independent of the d i r e c t i o n of the o b s e r v a t i o n , and'we r e f e r to i t as f . The p o l a r i z a t i o n i s g i v e n by P = M" - -f - (N^ t f /N1) N " + N X The p o l a r i z a t i o n i s assumed to be s m a l l enough t h a t we can ignore the e r r o r i n the denominator. Thus we have P - N " - N-1- t f YN U + N X N " - N X ± -f N" + N x N"*^ /iVHNfz T h i s i s the formula t h a t was used to c a l c u l a t e the r.m.s. e r r o r s r e p r e s e n t e d by the e r r o r b a r s . The numbers N, and N^ r e f e r co the number of p h o t o e l e c t r o n s . The c o r r e s -pondency between' the magnitude of the l o c k - i n amplifier-output s i g n a l and the c o u n t i n g r a t e was found by doing a run i n which the s i g n a l was measured both by the u s u a l method and by p u l s e c o u n t i n g . 102 Lower Limit on Useful Signal Strength The analog to d i g i t a l converter responds to p o s i t i v e signals only. This fact places a lower l i m i t on the signal strength that can be used, because a s u f f i c i e n t l y weak signal w i l l be noisy enough that the phase sensitive detector output w i l l be negative for part of the time. A lower l i m i t on the signal strength implies an upper l i m i t on useful integration times. We now make an estimate of t h i s upper l i m i t under the assumption that we require r.m.s. errors of 1% or l e s s . The output voltage x of the phase sensitive detector varies with time, and presumably the amount of time spent at each value of x i s given by a gaussion d i s t r i b u t i o n where x 0 i s the average signal l e v e l . The signal w i l l be negative 1% of the time i f we choose x Q = 2.3cr . Now the ef f e c t i v e integration time of the double section f i l t e r i s approximately 2 seconds. I f we integrate the output signal from t h i s f i l t e r over time T, the s t a t i s t i c a l uncertainty i s reduced by a factor J2/T . I f we now i n s i s t that t h i s uncertainty be 1$, we require that 72/T cr - 0.01 x 0 = 0.01 x 2.3 cr from which we f i n d that T = 4000 seconds. In practice, integration times of about 2000 seconds.were the longest used. 103 APPENDIX IV A P o l a r i z a t i o n of Light due to Optical Elements P o l a r i z a t i o n of Light due to Passage through a Glass Surface Assume that a ray of l i g h t has an angle of incidence to the normal of a glass surface of 0 , and an angle of r e f r a c t i o n of 0'. The amplitudes of the e l e c t r i c f i e l d s of the transmitted l i g h t are given 'by E P s I n" ( ) " c o s\ i -$') EJL = 3 sin 0 ' cos 0 (Jenkins and White, Fundamentals of Optics) where the primes refer to refracted l i g h t and where Ep'^and Es('^ are the e l e c t r i c f i e l d components respectively p a r a l l e l to, and perpendicular to the plane of incidence. We assume the incident l i g h t to be unpolarized; i . e . Ep = E$ . Then the p o l a r i z a t i o n with respect to the plane of incidence of the refracted l i g h t i s given by E p z + El 2 - ^ i n ^ ( t f - 0') I t i s now a straightforward matter to calculate the effect on p o l a r i z a t i o n of glass o p t i c a l elements, es p e c i a l l y i f the angles 0 and 01 are small. I t i s necessary to average over a l l the angles 0 contained i n the cone of l i g h t . 104 P o l a r i z a t i o n of Light due to Reflection  from a Metal Surface The s i t u a t i o n here i s somewhat more d i f f i c u l t than i n the l a s t section. The p o l a r i z a t i o n depends on the re f r a c t i v e index and conductivity of the metal i n a complicated way. However, for a very good r e f l e c t o r the effects cannot he large. An aluminum surface r e f l e c t s about 90$ of blue l i g h t and i s somewhat better i n the infrared. In general, l i g h t polarized perpendicular to the plane of incidence i s re f l e c t e d p r e f e r e n t i a l l y . This means that i n the p o l a r i -zation experiment, any p o l a r i z a t i o n due to the mirror i s negative. : 105 APPENDIX V A Pol a r i z a t i o n Model Consider f i r s t the in t e n s i t y of l i g h t as a function of the nominal electron energy. We assume that the ex c i t a t i o n cross section has the form -P (IL) = kx (x.> ox) - 0 (x^ 0) and that the energy d i s t r i b u t i o n of the electron i s - IT'- ^ a (iL'-x) = -— £ ^ a v ' n o -where x i s the nominal electron energy r e l a t i v e to the exc i t a t i o n energy, and k and cr are constants. Pig. 37. Intensity Model. Then the i n t e n s i t y of l i g h t i s given by Upon performing the integration, we obtain I (x) - ^ - e + k x ( x * O) IX I 2 ' 106 Thus, below threshold, the i n t e n s i t y function i s exponential i n energy, and s u f f i c i e n t l y far above threshold the i n t e n s i t y follows the e x c i t a t i o n cross section correctly. In order to f i n d apparent p o l a r i z a t i o n functions, we must know how to add po l a r i z a t i o n s . That i s , i f i n a source of l i g h t there are two components with i n t e n s i t i e s I ( and I ^ , and polarizations P, and P z (referred to the same a x i s ) , the resultant p o l a r i z a t i o n i s P _ Pi I I + Iz I . + U • This may be shown e a s i l y from the definition' of p o l a r i z a t i o n . I t follows then, that i f the p o l a r i z a t i o n function i s P(x), the apparent p o l a r i z a t i o n function P 0 L(x) i s given by $ U')q Cx'-X) dx' The denominator i s simply I ( x ) , which we have just calculated, In order to calculate the numerator, we assume the p o l a r i -zation function to be P ( x ) = Po ( 1 - X t.) X * 0 The f i n a l r e s u l t i s given by Po. U)/pc = i 1-0 P o . ( x ) / p 0 = R ( x A ) X ^ O where 107 One apparent feature of this r e s u l t i s that the apparent p o l a r i z a t i o n i s constant below threshold. This feature i s common to a l l such p o l a r i z a t i o n models i n which an exponential electron energy d i s t r i b u t i o n i s assumed. The reason for this i s clear i f i t i s r e c a l l e d that the shape of the function exp.(x'-x), considered as a function of x', i s independent of the value of x. Another feature of this p a r t i c u l a r model i s that the apparent p o l a r i z a t i o n Po_(x) approaches the "true" p o l a r i -zation P(x) for large x, but more slowly than might be supposed. Por x >> cr, we have P*. W « Pn (i - * x - a Graphs of the various functions are shown i n Figure 3 8 . A convenient feature of the function Po_(x) i s the way i n which the dependence on tf and the dependence on a- are separated i n the product 4!fcr R(x/cr ) . This means that only one 4tfc> R(x/cr ) curve has to be calculated; the others are obtained by multiplying the coordinates by appropriate factors. The parameters cr and H are found from the data i n the following way; Exponential curves are made up on transparent sheets, and f i t t e d to the in t e n s i t y data i n order to determine cr . Then 4ftcr R(x/cr) curves are made 108 x (Electron Volts) F i g . 38. Mathematical Model of P o l a r i z a t i o n 109 up, for several values of cr^f, and. f i t t e d to the p o l a r i z a t i o n data i n order to determine f . I t was found necessary to make up only two sets of 4 fcr R (x/cr ) curves, one for cr = 0 . 1 5 v o l t s , and one for o~ = 0 . 2 0 v o l t s . 

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