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The measurement of transition probabilities of atomic neon Robinson, Alexander Maguire 1966

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THE MEASUREMENT OF TRANSITION PROBABILITIES OF ATOMIC NEON Alexander Maguire Robinson B.Ao Sc., U n i v e r s i t y o f B r i t i s h Columbia, 1961 M. Sc., U n i v e r s i t y of B r i t i s h Columbia, I963 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF Ph. D. i n the Department of P h y s i c s We accept t h i s t h e s i s as conforming t o the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA September, I966 In presenting t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y avail able f o r reference and study. I furt h e r agree that permission, f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s representatives. I t i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n permission. The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8 , Canada THE UNIVERSITY OF BRITISH. COLUMBIA FACULTY OF GRADUATE STUDIES PROGRAMME OF THE FINAL ORAL EXAMINATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY ALEXANDER MAGUIRE ROBINSON B.A.Sc, The U n i v e r s i t y of B r i t i s h Columbia, 1961 M.Sc, The U n i v e r s i t y of B r i t i s h Columbia, 1963 WEDNESDAY, SEPTEMBER lk, 1966 at 3 :30 P.M. IN ROOM 301, HENNINGS BUILDING COMMITTEE IN CHARGE Chairman: B. A. Dunell External Examiner: R. W. N i c h o l l s Professor of Physics York U n i v e r s i t y Research Supervisor: R. A. Nodwell of A. M. Crooker F. L„ Curzon A. J. Barnard T. J. Ulrych W. F. Slawson R. A. Nodwell THE MEASUREMENT OF TRANSITION PROBABILITIES OF ATOMIC NEON ABSTRACT The transmission of neon l i n e r a d i a t i o n through the p o s i t i v e column of a neon dc glow discharge has been measured. Six lengths of the column were used and a graphical comparison of the t h e o r e t i c a l and experimental transmissions were made. This permitted a determination of the absorption c o e f f i c i e n t of the gas ? f o r the case of Doppler~broadened sp e c t r a l l i n e s , „- The r e l a t i v e t r a n s i t i o n p r o b a b i l i t i e s f o r t r a n s i t i o n s with the same lower l e v e l were obtained from the values of the absorption c o e f f i c i e n t s . Radial v a r i a t i o n of the density of absorbing atoms and the presence of isotopes i n the column were taken i n t o account. The r e l a t i v e i n t e n s i t i e s of several p a i r s of spectral, l i n e s emitted by neon gas excited by a pulsed e l e c t r o n beam have been measured. The neon was at a low pressure (.1 mm Hg) and excited f o r a short time (200 nsec) to supress se l f ~ a b s o r p t i o n of the emitted r a d i a t i o n . The r e l a t i v e t r a n s i t i o n p r o b a b i l i t i e s f o r l i n e s with the same upper l e v e l were determined from the i n t e n s i t y measurements, A weighted averaging technique was used to connect the r e l a t i v e t r a n s i t i o n p r o b a b i l i t i e s of the absorption • and emission measurements and a complete set of r e l a t i v e t r a n s i t i o n p r o b a b i l i t i e s was obtained. The p r o b a b i l i t i e s were placed on an absolute scale using the r e s u l t s of a l i f e t i m e measurement r e c e n t l y made by van Andel, GRADUATE STUDIES F i e l d of Study: Physics Molecular Spectroscopy Magnetohydr odynaiaic s Special R e l a t i v i t y Plasma Physics Related Studies: Electron Dynamics F. W. Dalby F. L. Curzon P. R. Smy P. R a s t a l l L. G. de Sobrir;o G. B. Walker PUBLICATION The Measurement of Line Absorption of Excited Gases, R„A„ Nodwell and A. M. Robinson, Proceedings of the Seventh International Conference on Phenomena i n Ionized Gases s Beograd 9 19&5 ( i - n press). i i ABSTRACT The t r a n s m i s s i o n of neon l i n e r a d i a t i o n through the p o s i t i v e column of a neon dc glow d i s c h a r g e has been measured. S i x l e n g t h s of the column were used and a g r a p h i c a l comparison of the t h e o r e t i c a l and experimental t r a n s m i s s i o n s were made. T h i s p e r m i t t e d a d e t e r m i n a t i o n of the a b s o r p t i o n c o e f f i c i e n t of the gas, f o r the case o f Doppler-broadened s p e c t r a l l i n e s . The r e l a t i v e t r a n s i t i o n p r o b a b i l i t i e s f o r t r a n s i t i o n s w i t h the same lower l e v e l were obtained from the v a l u e s of the a b s o r p t i o n c o e f f i c i e n t s . R a d i a l v a r i a t i o n o f the d e n s i t y of abs o r b i n g atoms and the presence of i s o t o p e s i n t h e column were taken i n t o account. The r e l a t i v e i n t e n s i t i e s of s e v e r a l p a i r s of s p e c t r a l l i n e s emitted by neon gas e x c i t e d by a p u l s e d e l e c t r o n beam have been measured. The neon was at a low p r e s s u r e (.1 mm Hg) and e x c i t e d f o r a short time (200 nsec) t o supress, s e l f - a b s o r p - t i o n of the emitted r a d i a t i o n . The r e l a t i v e t r a n s i t i o n p r o b a b i l i - t i e s f o r l i n e s w i t h the same upper l e v e l were determined from the i n t e n s i t y measurements. A weighted a v e r a g i n g technique was used t o connect the r e l a t i v e t r a n s i t i o n p r o b a b i l i t i e s of the a b s o r p t i o n and emission measurements and a complete set of r e l a t i v e t r a n s i t i o n p r o b a b i l i t i e s was o b t a i n e d . The p r o b a b i l i t i e s were p l a c e d on an a b s o l u t e s c a l e u s i n g the r e s u l t s of a l i f e t i m e measurement r e c e n t l y made by van Andel i i i TABLE OF CONTENTS Abstract, o o o o o o o o o o c o o u o o o o © o o o o o o o o © o ooo o o oo o o o o © o © o © © i i Table of Contents O O O O O 0 O O ' o o o o o o o o ' o ' o o ooo oo O O O O 9 0 0 0 0 oo ooo i i i Index of Tables.. o o o o o © o o © o o o o o o © o o o o oo o o o o o ooo © o* © © o « © © . vi Index of Figures. v l i i o o o o o o o o o o o o o o o o o . ooo, © oooooooo «o. ooooooo Acknowledgements 0 © © © O O O O O O O O O O O O O O O O O O, O O 9 0 0 ooooo ooooooo X CHAPTER 1 Xltl'PHOIDTJC'P XOM o o o o o o o o o o o o o o o o o o o o o o o o o o 1 CHAPTER 2 1?!E£EiOSYo O O O O O O O O O O O O O O O O O o o o © © © • • oeoooo 7 2 o 1 A . ' b . S O I ^ t l 0 H s o oo o o o o ooooooooo oooo ooooooo 7 JjYXifOGO.S « « « O • oooo ooooooooo o o o o ooooo 7 2.1.2 NOin."** XJXX i f O Gf l lS tt « « « 0 0 o © o o o • o, © o oooooooo 9; 2*1*3 Hyp erf Ine Structure© © © © © © © © © ©. © • © o ©«© • 13 2.1.4 A n a l y s i s of Data©©©o©• «•«•.••©-••©©o*©«• 16 2.2 S i l l !LS S 3. OX1 oooooooooo O O O O © » © O O O © • O O O O O O O O 1.8 CHAPTER 3 ABSORPTION EXPERIMENT.. . . . . . . . . . . . . . . . 21 3.1 Apparatus o e «o o o o oo oooo oo oo e* o o o o 0 0 , 0 0 0 © 21 3.1.1 Background Light Source and Modulator. 23 3.1.2 Absorpt ion Tubes and Apertures.. . . . . . ' . 24 3.1.3 Monochromator..... . . . . . « . . <>..0» 0 . . . . . . 27 3.1 .^ Phot omul t ip l 1 er a . . . . . . . . . . . . 0 . . . . . . 0 . . . 28 3.1.5 Amplifier and Phase-Sensitive Detector 29 3,2 Experimental M e t h o d 6 . . . . . . . . . . . . . . . . . . 33 3.2.1 Circular A p e r t u r e s . . . .v. . . . . . . . . . . . . . . 33 3 e 2 © 2 Annul1«e . . . 0 0 . . . 0 . . . . . . . . . p . o e . . . . . . . . 36 CHAPTER 4 EMISSION EXPERIMENT...... . . . . . . . . 37 Measurement of Relative Intensities... 37 4 .2 Preliminary Intensity Measurements.... 3? ^.3 Final Intensity Measurements.......... ^3 ^.3.1 Apparatus © 0 0 . 0 . « . « . . . «. . . . . . . . 0 0 . • 44 i v TABLE OF CONTENTS (cont'd) CHAPTER k (cont'd) .1 Sampling Technique...............*\. kk .2 Monochromator.. « . . . . o . . . . . . . « • > » • . . . . fh5 .3 Spectral C a l i b r a t i o n , . . . . . . . . . . . . . . . 5̂ *K3.2 Experimental Method................. 7̂ CHAPTER 5 RESULTS AND DISCUSSION.............. 52 3.1 Neon S p e c t r u m . . . . . . . . s o . . . . . . . . . . . . . 52 5„2 Absorption-Circular A p e r t u r e s . . . . . $ k 5.2.1 Cross-Sectional of Transmission..... 5̂ -3 © 2«2 Curv© FAttin££©© © o © © © • © * • © © • « © © « • © © © © 5^ ,5.2.3 ' Relative O s c i l l a t o r Strengths....... 58 5*3 A b s o r p t i o n — A n n u l i o . . . . . . . . . . o . o . « o . . 64- 5.̂ - Discussion of Results and Errors of Absorption Method............... 68 5J^.1 F i c t i t i o u s Line-Shapes.............. 68 5.^.2 Discussion of Errors................ 71 .1 Stimulated Emission................. 72 .2 Transmission Accuracy............... 72 .3 Curve F i t t i n g and Slopes............ 73 ,k Optical Alignment............. • • 73 .5 End-Window Reflections....»......... 7^ .6 Uniform Discharge Conditions........ 75 .7 End-Effects of Absorption Column.... 76 . 8 Estlmated Errors............... ...... 79 3 b 5 EmiSS 1 On . . . . s o o . o . o o o o . . . . . . . o o . o . . . 81 5*5*1 Relative T r a n s i t i o n P r o b a b i l i t i e s . . . 81 5.5.2- Discussion of Errors................ 82 il Intensity-Ratio Error............... 82 .2 Emisslvity and Temperature Error.... 82 .3 Pile—Up Error.............. »••««.... 8̂ .k Radiation Trapping.................. 85 V CHAPTER 5 (cont/d) 5.6 5 o 6 .1 5e6 02 5 . 6 . 3 CHAPTER 6 APPENDIX 1 APPENDIX 2 APPENDIX 3 APPENDIX 4 APPENDIX 5 APPENDIX 6 REFERENCES Absolute Transitions Probabilities "Multi-Path" Method of Relative Transition P r o b a b i l i t i e s . . . . . Lifetime M e a s u r e m e n t . . . . . « . . 0 . . . . . . Comparison and Discussion of Absolute P r o b a b i l i t i e s . . . . . . . . . CONCLUSIONS... O 0 0 O « 0 0 O Q 0 0 0 0 0 0 0 0 0 0 0 0 . i o o o o * o o « « « o • o o o o o 88 89 94 97 99 101 ISOTOPE CORRECTION EVALUATION OF TRANSMISSION EQUATION 107 SELF-ABSORPTION CORRECTION FACTOR.. 109 RELATIVE INTENSITY MEASUREMENT OF.. SPECTRAL L I N E S . . . . 0 . . . • • • • . . « « 112 CURVE-FITTING OF NON-DOPPLER SPECTRAL DISTRIBUTION TO DOPPLER DISTRIBUTION 117 END CORRECTION A T T E M P T S . . . . . . . . . . . . 124- o « o t > Q © o « o » e o e o © © © © • • • * « « o © o o o Q O O © » v e o o e v o o o o 130 vi INDEX OF TABLES TABLE PAGE 1. Circular Aperture Determination of f r e l : s 2 Lines, c . . . . . . . . . . . . . . . . . 59 2. Circular Aperture Determination of k _ . CC • S /s Line S e o . o o . o . e o . o . . . . . o « . o . . o o . ' 6 0 O * C . > 3. Circular Aperture Determination of ^rel*^3 4̂ ^l^®^• • • • • • • • • • . . . . . . . . o . 61 ke Circular Aperture Determination of • cc• S^QJTICL ^ L i 2 i . © s o « o * • • » « • « • • • • • • « • « 62 5. Annulus Determination of fre-^: s^,s^, and ŝ Lines 65 6. Annulus Determination of k Q , a t s^jS^, and ŝ Lines. 66 7. Estimated Errors of Relative Oscillator S t r e n g t h s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 • 8. Relative Transition Probabilities From Intensity Measurements... 83 9, Estimated Errors of Transition Probabilities (Emission and Absorption). 92 10. Relative and Absolute Neon Transition Probabilities and Oscillator Strengths. 93 11. Comparison of Absolute Transition Probabili- t ies . . . . . . . . . . . . . . . . . . 95 12. Comparison of Relative Transition Probabili- ties. 96 13. Comparison of Upper Level Lifetimes 98 v i i TABLES (cont'd) A3. Self-Absorption Correction Factors For a One and Two Isotope G a s . 0 o o . . . « o . . . . . . . . . . I l l A5.1 Self-Absorbed Doppler Line-Shape Fit . . . 120 A5.2 Calculation of k- ( r e l ) for self-Absorbed Doppler Line-Shapes 121 A5.3 Voigt Line-Shape Fit . 120 A6.1 End-Correction Attempt #1 . . . . . . . . . . . . . . . . . . . . 128 A6.2 End-Correction Attempt #2. 128 v i i i INDEX OP FIGURES FIGURE PAGE 1. P l o t of F^ as a F u n c t i o n of k i . , a.... 10 o o * 2. Geometry of Absorbing Column................. 12, 3. P l o t of (1/1 + a ) ( F 0 ( k o i , a) + a F 0 ( b k 0 , a ) ) as a Fu n c t i o n of k 4. Schematic of A b s o r p t i o n Apparatus.. 22 5. A b s o r p t i o n Tubes and Annulus................. 25 6. P h o t o m u l t i p l l e r and A m p l i f i e r C i r c u i t . . . 30 7. P h a s e - S e n s i t i v e D e t e c t o r C i r c u i t 31 8. Polynomial F i t of R 2T. 3^ 9. R e l a t i v e I n t e n s i t y of M.6096,6678. .......... 42 10. P a r t i a l Term Diagram o f Neon f o r I n t e n s i t y Measurements ... kQ 11. R e l a t i v e Response of I n t e n s i t y - M e a s u r i n g C i r c u i t 50 12. Neon Term Diagram? F i r s t and Second E x c i t e d S t a t e C o n f i g u r a t i o n s . . . . 53 13. T r a n s m i s s i o n as a Fu n c t i o n of R,............. 55 14. Curve F i t t i n g . . o . . . . . . . . o 57 15. ^ r e l a s a ^ u n c ^ l ° n °^ R . • • . « • o . • • • « 63 16. Photograph of D i s c h a r g e . . . . . . . . . . . . . . . . . . o . . . 77 17. E n d - E f f e c t s o f Disc h a r g e . . . . . . . . . . . . . . 78 18. T r a n s i t i o n Array f o r " M u l t i - P a t h " Method..... 90 A l . Isotope S p l i t t i n g E r r o r . . . . . . . . . 105 Ix FIGURES (cont'd) A*r„ Pass-Band of Monochromator„« 0...... • • . . . . . . 1 1 3 A6.1 Geometry of Ends of Absorbing Column...... 0.. 125. A 6 . 2 P a r t i a l l y Blocked-Off Annulus................ 1 2 6 ACKNOWLEDGEMENTS I would like to sincerely thank Dr. R. A. Nodwell for his helpful guidance and assistance to me in this work, and to other members of the plasma physics group for their h e l p f u l d i s c u s s i o n s and suggestions. Thanks is also given to the members of the physics workshop for assistance in con- struction of equipment. I am indebted to my committee members Drs. P. L. Curzon, A. J . Barnard, and A. M. Crooker for suggestions regarding my research and the writing of this thesis. Special acknowledgement is given to Mr. H. W. H. van Andel for making available the apparatus used for measuring intensities and his assistance given in performing them. CHAPTER 1 INTRODUCTION It was In 1 9 1 6 and 1 9 1 7 that Einstein / ! / published his classic papers In which he Introduced transition probabilities Since that time much effort has been expended in the theoretical and experimental determination of these probabilities or the associated oscillator strengths because of their Importance in many fields of basic scientific research. For example, transition probabilities are essential in astrophysics. Here, research is restricted to analysis of radiation emitted and absorbed by stellar bodies. Measurement of spectral intensities can in principle determine such variables as electron density and temperature, number density of atoms and degree of ionization, and the abundance, of the elements in the universe but these determinations depend explicitly- on the transition probabilities,, In the field of plasma physics a similar situation exists. Of course the scientist is not con- fined to mere observation of electromagnetic radiation and can to some degree control his experimental environment. Neverthe- less transition probabilities are needed for virtually a l l spectroscopic diagnoses of plasmas. The recent interest in lasers has produced a spate of research In excitation and de- excltatlon processes In connection with excitation decay rates and population inversions in solid and gaseous materials. A knowledge of probabilities is imperative for both experimental and theoretical studies of these processes. Although quantitative values of transition probabili- -2- ties are essential in these f ie lds of research, the actual number of values known accurately Is extremely small. The need for additional and more accurate values is apparent and has led to marked Increase in work in this direction (for a recent bib- liography of transit ion probabi l i t ies , see Glennon and Wlese / 2 / ) . Theoretically, the transit ion probabil ity can be c a l - culated from the eigenfunctions of the atom, and the eigenfunc- tions may be obtained from the potential acting on the electron(s) involved in the radiative transi t ion. The simplest atomic system is the hydrogen-like atom whose eigenfunctions may be found exactly because the electron moves in a simple Coulomb f i e l d . Thus for these atoms, transit ion probabil i t ies may be computed, as has recently been done by Karzas and Latter /3/ and Herdon and Hughes /4/. However, for more complicated atoms, exact analytic solution of Schroedinger's equation is not possible and thus theoretical calculations involve some degree of approxi- mation. The most widely used approximation method is due to Bates and Damgaard /5/ which uses asymptotic solutions of a Coulomb potential . Because these approximations are somewhat doubtful there is some uncertainty in the computed values, and hence i t is desireable to determine the values experimentally. The experimental techniques f a l l into two general classes. The f i r s t class measures the rate of decay of atom populations d irect ly . This method can provide accurate deter- minations under appropriate experimental conditions but i t is usually d i f f i c u l t to know the rate at which the upper level is being populated. In addition, i f the upper state decays to more than one lower level individual probabilities cannot be determined by lifetime measurements alone. For most elements, this means absolute transition probabilities can be obtained from lifetime measurements for a few transitions only. The second general technique is less direct and measures parameters related to the transitions probabilities. For example, measure- ments of emission and absorption of radiation can lead to estimates of transition probabilities. To obtain absolute values the absolute population densities of the two levels must be known. Values of population densities of sufficient accuracy are not usually available. However measurements involving relative emission and absorption can be made accurately, so that relative transition probabilities can be acquired.The lifetime methods usually yield at least one (absolute .transition probability so when combined withthe more numerous relative values, accurate absolute probabilities can be obtained. In the experiment reported in this,thesis the rela- tive transition probabilities have been measured for most of the transitions between the 2p%s and 2p̂ 3p configurations of Ne I. This gas was chosen for several reasons. Neon appears in the hotter stars and in some cases is as abundant as oxygen /6/ , making it an interesting subject for astrophysical research and it is often used for high-temperature shock tube work in plasma research so that a knowledge of neon transition proba- bi l i t ies is important. From an experimental point of view, neon is rich in spectral lines in the visible region. Finally, there are some significant discrepancies in previous evaluations of neon transition probabilities and a new determination is highly desireable. Transitions between the above mentioned configurations of neon have been the subject of several experimental investi- gations In the past four decades. One of the f irst was performed by Ladenburg, et a l . / ? / who measured the relative transition probabilities from the anomalous dispersion of the gas by the Roschdestwensky "hook method" /8/ 8 This technique was recently repeated by Pery-Thorne and Chamberlain /9/. Doherty /10/ determined the absolute transition probabilities from absolute intensity measurements using a shock tube. Frledrichs / l l / also measured absolute intensities, but with a stabilized arc as a source. More recently, relative values for these lines have been determined by Irwin /12/ using line- reversal and relative intensity methods. The method of determining the time dependence of populations of the excited states was first attempted on some of these neon levels by Griffiths /13/ who estimated the lifetimes by measuring the phase lag between periodic excitation and emission. Other direct measurements of lifetimes has been performed by Bennett / l V , et a l . and KLose A 5 A l n which the decay of the radiative intensity is measured by sampling. There is considerable disagreement amongst the values of the transition probabilities obtained by these workers. The presence of self-absorption in some of the experiments and other limitations on the accuracy of the measurements prompted the undertaking of the experiment described in this thesis. -5- The experiment reported here consists of two separate parts, an absorption experiment which relates probabilities of transitions having a common lower level, and an emission or intensity experiment which relates probabilities of transitions with a common upper level. Combining the results of these two experiments allows one complete set of relative values to be obtained. The absorption experiment employs a modification and extension of a method due to Bamberger / l 6 / , who measured; the transmission of a plasma to intensity-modulated hydrogen radia- tion to determine the density of excited atoms as a function of time. In the experiment described in this thesis, the trans- mission of the positive column of a neon dc glow discharge is measured. Use of six absorption lengths enables a graphical comparison of the theoretical and experimental transmissions to be made to determine the absorption coefficient of the gas, assuming Doppler-broadened spectral lines. The values of the absorption coefficients of appropriate spectral transitions allows calculation of relative transition probabilities. Radial variations of the density of excited atoms is observed and compensated for by varying the diameter of the incident beam or by using an annular cross-sectional beam. The presence of isotopes is also taken into account. The emission experiment measures relative intensities of several pairs of neon spectral lines by a method not believed to have been used previously. The radiation is emitted by a neon source excited by a pulsed electron beam and the intensity -6- Is determined by counting the photons with a photo-electric counting device. The pressure of the gas is low (.1 mm Hg) to eliminate self-absorption. The apparatus was developed by van Andel / l ? / for lifetime measurements but was readily adapted for this experiment. The values of the transition probabilities obtained by this method are estimated to be the most accurate to date and should prove to be valuable for future spectroscopic diagnosis of neon. The theory of the experiment is developed in Chapter 2. Starting with the general laws of total absorption of a spectral line in a uniform gas, the specific case of Doppler broadened lines is considered. Non-uniformity in a radial direction of a cyllndrlcally symmetric system is next considered, and two methods overcoming this difficulty are discussed. The equations necessary to account for the presence of isotopes in the source and absorber are given. Finally emission theory is developed and discussed. Chapter 3 describes the experimental apparatus and procedure of the absorption experiment while Chapter 4 does the same for the emission experiment. The results of the work are given in Chapter 5 and discussed; attempts are made to account for broadening other than pure Doppler in the absorption experiment and errors in absorption and emission are investigated. The chapter is concluded with a presentation of the absolute transition probabilities based on a lifetime measurement by van Andel / l ? / . The conclusions are contained in Chapter 6. -7- CHAPTER 2 THEORY The theory developed In this chapter indicates how relative transition probabilities may be determined from a measurement of the absorption and emission of radiation by a gas. 2 8 1 Absorption The absorption coefficient integrated over a spectral line is proportional to the transition probability (see for example, Garbuny / 1 8 / ) . The theory given in this section calculates the transmission of Doppler-broadened spectral radiation through a gas and relates the absorption coefficient and hence the transition probability to the transmission measurements. 2.1.1 Uniform Gas When a plane-parallel beam of monochromatic radiation at frequency "0 passes through a uniform gas of length ^ , the intensity of the beam is decreased because of absorption. The fraction of the energy transmitted per second, or the trans- mission T, is given by e v ' where k ( V ) is called the absorption coefficient of the gas. If the incident beam consists of a single spectral line of small but finite width, integration over the line yields the transmission, thus: -8- £Jkwl ' T = A-__1_J ...(2.1) where 1(9 ) Is the intensity of the incident beam at frequency\> The absorption coefficient is a parameter of the absorbing gas and depends on the transition probability. In principle the absorption coefficient may be determined from a measurement of the transmission of a beam of light of known spectral distribution,, The transition probability may then be determined from the absorption coefficient. In this experiment conditions are chosen such that both the source and the absorber have predominantly Doppler- broadened lines. For a source and an absorber having tempera- tures T_ and T respectively, and a spectral line of centre- s a f r e q u e n c y 9 the emission intensity and the absorption coefficient are given by (see for example, Mitchell and Zemansky /19/) where O = ' . 6 M > S A v K = • — / — = Doppler width of the -9- spectral line of the source, and a = VTs/Ta .= ratio of the Doppler widths of the source and the absorber. When stimulated emission is negligible, as it is in this experiment, the value of the absorption coefficient at the centre of the spectral line, kQ , Is given by a - / " ^ N t j - - K 3j Ni An , 2 2 ) where N^ = the number density of atoms in the lower state of the transition, and 'f^j and Aj^ are the oscillator strength and the transition probability respectively, for the absorption transition 1—*J (i s lower level, J • upper level). Equation (2.1) then becomes. Fig. 1 is a plot of F Q as a function of kQi? for several values of a. 2.1.2 Non-Uniform Gas A modification of equation (2.1) Is necessary when  - l i - the density or the temperature of the absorbing gas is not constant perpendicular to the axis of the beam (but uniform along the axis) because the absorption coefficient wil l vary with radial position and an integration over the transverse area is necessary. Equation (2,1) then becomes For an absorbing column with axial symmetry and an annular cross-sectional beam (with inner and outer radii R i and R respectively) centred on the axis of the column, T = T = (2.5) where r is the radial distance from the axis. Pig. 2 shows the -12- geometry of the system. INCI DENT RADIATION Pig. 2 Geometry of Absorbing Column If R Q - ^ is small enough so that k("v> ,r) is nearly constant over the annulus, then the form of T becomes identical with equation (2.1). If R is treated as a variable so that T becomes a o function of RQ, an expression depending only on the parameters 2 2 at r = RQ may be obtained by differentiating (R Q - R^T with 2 respect to RQ; d(Rl) ^ I M C K > - -13- For Doppler broadened lines this beomes For a circular beam (R̂  = 0) equation (2.6) becomes finally where R has been written instead of RQ for simplicity. This enables the absorption coefficient to be related to the trans- mission as the beam diameter is varied. 2.1.3 Hyperfine Structure So far only spectral lines having a simple Doppler profile have been considered. Many lines possess a structure, which can lead to error i f ignored. The fine structure of neon is large enough to separate easily with a spectrograph,;, but the hyperfine structure of the spectral lines is not and must be accounted for in the analysis. In neon, the hyperfine struc- ture of the lines is due solely to the mass difference of the isotopes, since the nuclear angular momentum of the predominant -14- Isotopes Is zero /20/. Naturally occurlng neon Is composed approximately of 91% atomic mass 20, 9% mass 22, and „3# mass 21. The quantity 21 of Ne is so small that the effect of its presence wi l l be negligible in this experiment, making neon essentially a two- isotope gas. For a two-isotope gas, the total profile of a spectral line is the sum of the profiles of two components. Each individual profile is centred on a slightly different frequency and if the splitting is large enough the total line will have two peaks or even two separate components. In neon the isotope of mass 22 only contributes about 10$ of the emiss- ion energy but in absorption the effect of this isotope can be predominant. To see qualitatively the effect of this Isotope in absorption consider a neon source and absorber whose spectral lines have two isotope components with profiles completely or almost completely non-overlapping. Because of the non-overlapp- 20 ing condition, the Ne component in the absorber wil l absorb 20 only the Ne component emitted by the source, with a similar 22 situation existing for the Ne portion of the spectral l ine. The energy entering the absorber wi l l be predominantly from the 20 Ne isotope. However this isotope has a correspondingly large absorption coefficient and the transmission varies exponentially with absorption coefficient, not linearly. That is , the gas is more transparent to the weaker component of the radiation. With strong absorption the energy transmitted by the absorber can be almost completely due to the less -15- abundant Isotope Ne . Gf course If the line-width of the components are. larger than the isotope separation, the two components blend into one line and the isotope effect is not so significant. A quantitative analysis has been made relevant to the conditions in this experiment. Only the results are given here and the detailed calculations and discussion reserved for Appendix 1. The main postulate of the analysis is that the spectral profile may be represented by two separated Doppler curves. Experimental conditions are such that this is a good approximation, as shown in the appendix. For a uniform gas the transmission is shown to be where F Q Is defined in equation (2.3) and a = K VM/M* « ,094 b = K V M V M = aM'/M « .104 K = relative abundance of the two isotopes « .099 M = mass of greatest-occurring isotope = 20 atomic mass units M* = mass of least-occurring isotope = 22 atomic mass units The numerical values given pertain to neon. The two terms on the right hand side of equation (2.8) represent contributions to the transmission from the two isotopes. The relationship between this equation and the qualitative discussion of the previous paragraphs may be seen with reference to Fig. 1. F Q is close to 1 for small k Q i but -16- decreases rapidly with increasing k Q £ . a and b are approxi- mately equal and small i f one of the isotopes is not very abundant. Therefore for small absorpt ion ( i . e . , k Q l small) the f irst term dominates, but at large k Q i , the argument in the second term is s t i l l small because of the factor b and the second term dominates. Equation (2.8) is plotted in Fig. 3. For comparison F 0(k QJ2 , a) is plotted for several values of a (dotted line). The calculations of these functions is discussed in Appendix 2, The equation corresponding to equation (2.7) is = ^ [ F 0 ( 4 ( R ) ^ ( R ) ) - f - a F 0 ( b 4 ( R ) i M ( R ) ) J - ~ { 2 ' 9 ) 2.1.4 Analysis of Data If k and a can be determined from measurement of T, o the relative probabilities (or oscillator strengths), tempera- tures and populations can be calculated as can be see from consideration of equation (2.2). For measurements made on a single spectral line the relative temperature and number density of the lower state may be determined as a function of radial position; FIG. 3. Plot of ( I /I + a)(FB(k0l,a) +aF0(bk0l,a)) as a function of k0l and a TJR') L<*(R). •18- N,-CR) = MB) Its) H(R') A(R') ~^R'> ...(2.10) where R» is some arbitrary reference radius. But more important, i f two spectral lines having a common lower level are compared at the same radius, is the same for both and « • « ( 2 e H ) At a . /\ K 9 , 2.2 Emission Einstein / l / showed that the power spontaneously emitted by a group of atoms is proportional to A^h-p where -19- Aj^ is the spontaneous transition probability and "9 is the frequency of the emitted radiation. For Nj atoms per unit volume in the upper state j emitting in a layer of thickness ! the energy flux due to spontaneous emission is proportional to NjAjjhtfi . , , ( 2 . 1 3 ) Thus relative transition probabilities may be calculated from a determination of the relative intensities of lines with a common upper level since Nj is equal for both transitions and hence If the density of atoms in the lower state is non- zero, there is a finite probability of emitted photons being reabsorbed with a consequent decrease of output intensity. This radiation imprisonment is largest for resonance line since most of the atoms are in the ground state. Ladenberg and Levy /7/ derived for a one-isotope gas a correction factor to be applied to the measured intensity for Doppler lines. They showed that the intensity of a finite layer of the luminous gas is equal to the intensity of a layer of unit thickness in the absence of self-absorption multiplied by the factor J^S , where 20 S = 2 Co+Oi/rvTT (2.15) When two isotopes are present,-equation (2.15) must be modified to i where a and b are defined in equation (2;8) and - f l is the frequency separation between the isotoplc components, in units of 60 The derivation of equation (2.16) as well as a table of numerical values is given'in Appendix 3• with -21- CHAPTER 3 ABSORPTION EXPERIMENT 3.1 Apparatus The detecting apparatus used in these studies is based on a method described by Hamberger /16A A brief description of the apparatus is given with the help of Fig. 4 which Illustrates schematically the experimental arrangement. Radiation from a neon-filled Geissler tube passes through a pinhole and is intensity-modulated at 210 cps by a rotating disk chopper. It is then collimated by a lens;.and passed through a cylindrical neon-filled absorption tube. The absorption tube has four internal electrodes so that six different discharge lengths may be obtained. The radiation is then focused on the entrance s l i t of a monochromator set to pass the spectral line under investigation. The cross-sectional area of the transmitted beam is controlled by apertures placed in the collimated beam at both ends of the absorption tube. The intensity of the light is measured at the exit s l i t of the monochromator by a photomultiplier and a narrow-band amplifier centred at 210 cps. The resultant signal is then fed into a phase-sensitive detector which receives a reference signal of the same frequency. The detector only accepts signals coherent with the reference signal and eliminates unwanted signals due to emission of radiation from the absorption tube and noise in the photomultiplier and amplifier. The dc signal from the detector is measured with a chart recorder and is proportional C H O P P E P I N H O L E / LENS- SOURCE P H O T O - DETECTOR' G R A T I N G MONOCHROM- A T O R 4 - E L E C T R O D E A B S O R P T I O N T M B E - PHOTOMULT1PLIER- REFERENCE SIGNAL DC RECORDER PHASE-SENSITIVE DETECTOR AMPLIFIER i to to 1 F \ 6 . 4- S C H E M A T I C OF A B S O R P T I O N A P P A R A T U S -23- to the Intensity of the modulated beam fall ing on the photo- multiplier. Thus the apparatus provides a means of separating the background light from the absorption tube and furnishes several absorbing lengths so that it' is possible to calculate kQ and a using equation (2„9) The apparatus wi l l now be discussed in detail. 3.1.1 Background Light Source and Modulator The source used was a neon-filled Geissler tube 10 cm long and approximately 2 mm inner diameter. The tube was operated from a 600v dc power supply and current regulator. The current regulator was necessary to stabilize the intensity of the emitted radiation. The direct current through the tube could be varied upwards from 2 ma and was operated at a maximum of 9 ma. The tube did not heat up and remained essentailly at room temperature. The modulator was a seven-slotted circular disk placed between the Geissler tube and the collimatlng lens. It was mounted on the shaft of a Bodine Electric Co. 1/25 hp, 30 cps synchronous motor, chopping the beam at a frequency of 210 cps. The modulating frequency of 210 cps was chosen to avoid harmonics of 60 cps. It was found that best overall performance was attained when the motor was driven by a 60 cps crystal oscillator and power amplifier, rather than using power from the mains. The frequency of both the oscillator-amplifier source and the mains voltage was measured with a digital counter. The variation of the frequency had a standard deviation of 9 x 10 % for the former source and 12 x 10 % for the latter. This larger -24- deviation resulted in a 5% variation of the output signal measured on the chart recorder. This was because the frequency of the modulated light did not remain at the centre-frequency of the pass-band of the amplifier. The variation of the output signal with the oscillator-amplifier source was less than \%. 3.1.2 Absorption Tubes and Apertures The absorbing neon for this experiment was the plasma of a positive column dc discharge contained in a cylindrical absorption tube. Two absorption tubes were used for this experiment and are Illustrated in Fig. 5. Both were constructed with glass tubing of 5 ° cm length, one with 25 mm inside diameter and the other 13 mm. Optically flat windows were mounted on each end. Four side-arms containing electrodes were spaced along the length of the tube so that absorption lengths of 10, 15, 20, 25, 35, and 45 cm could be obtained. The electrodes were partially hollowed aluminum cylinders joined to a tungsten wire sealed through the glass. The dimensions of the electrodes are given in Fig. 5« The glow discharge was maintained by applying 600v dc across the pair of electrodes corresponding to the discharge length desired. For both the absorption and emission tubes, the discharge was Initiated with a Tesla co l l . Before f i l l ing , the tubes were baked at 400°C for - 9 several hours and then evacuated to a pressure of 10 ' mm Hg on a high-vacuum system built by van Andel / 2 1 / before being 13mm ~« 1 0 cm — » 1 5 cm 20 cm 1/2 25 mm FIG. 5. ABSORPTION TUBES AND ANNULUS -26- f i l l e d w i t h r e s e a r c h grade A i r c o neon t o a p r e s s u r e of 2,mm Hg and s e a l e d o f f . Centred on each end of the narrow a b s o r p t i o n tube were aluminum h o l d e r s i n t o which c o u l d s l i d e s h o r t s t r i p s of aluminum w i t h c i r c u l a r h o l e s i n them. By u s i n g holes of v a r y - i n g diameter, a beam of c i r c u l a r c r o s s - s e c t i o n o f known diameter passed down the a x i s o f the a b s o r p t i o n tube. Mounted a t each end of the l a r g e r diameter tube was a b r a s s sheet from which was cut an annulus. T h i s a l l o w e d a beam of 2.2 cm out e r diameter and 2.14 cm i n n e r diameter t o pass through the tube, (see F i g . 5) The dimensions of the annulus was governed by experimental o p e r a t i n g c o n d i t i o n s . The width of the annulus had to be s m a l l f o r c o n d i t i o n s t o be approximately constant i n the an n u l a r beam. T h i s meant the diameter c o u l d not be too s m a l l , otherwise the c r o s s - s e c t i o n a l area of the annulus and hence the i n t e n s i t y of the l i g h t from the background source would be s m a l l and the s i g n a l - t o - n o i s e r a t i o low. As w e l l , the l e s s h i g h l y a b s o r b i n g gas w e l l away from the a x i s of the a b s o r p t i o n tube was of i n t e r e s t and l e d to the above c h o i c e of an n u l a r dimensions. Because the diameter of the annulus was l a r g e r than t h a t of the narrow a b s o r p t i o n tube, the a n n u l ! were on l y used w i t h the l a r g e a b s o r p t i o n tube. It i s obvious t h a t the o p t i c a l alignment f o r t h i s experiment i s c r i t i c a l and s p e c i a l c a r e i s necessa r y . For t h i s reason a P r e c i s i o n T o o l and Instrument Co. 2-meter o p t i c a l bench w i t h x, y, and z v e r n i e r motion saddles was used i n c o n j u n c t i o n w i t h a S p e c t r a - P h y s i c s He-Ne l a s e r . The l a s e r was -27- in i t ia l ly lined up along the axis of the monochromator and defin- ed the optic axis for alignment of tubes, leases, annuli\, etc. Correct alignment of each component was estimated to be within . 25 mm. 3.1.3 Monoohromator The monochromator used for the absorption experiment was built in this laboratory. It Incorporated an f/15 3-meter spherical mirror, 10 cm diameter, and a 3-meter concave grat- ing, 10 cm diameter. The arrangement was in a standard, Wads - worth mount with the exit s l i t mountedon ways perpendicular to the centre of the grating. The spectrum was observed In the first order and gave a reciprocal dispersion of an almost cons- tant 22.4 AVW over the total wavelength region observed (5800A - 7300A). The maximum signal-to-noise resulted from maximum' width of entrance and exit s l i t s . Ideally the exit s l i t would be set exactly at the exit. plane position corresponding to the spectral line under investigation. The s l i ts (assuming unit magnification) could then be opened to a width equal to the distance between the centre of this spectral line and the centre of the nearest strong adjacent spectral line and the two spectral lines would then be contiguous. However because the exit s l i t cannot be positioned exactly there is danger of some radiation from the.wrong, line passing through the exit s l i t . To safeguard against this the width of the entrance s l i t 28« was made smaller than the exit s l i t by a factor of about 2/3. Care must be taken in deciding which weak adjacent lines produce effects small enough to ignore and so can be encompassed by the exit s l i t ; by the same reasoning as given in Chapter 2, a line which is very weak in emission can also be weak in absorption and with strong absorption of the main line, a large proportion of the energy transmitted may be due to the weak line. For this reason the spectrum from the Geissler tube was examined with a high resolving power 3.5- meter Ebert mount grating spectrograph and the approximate relative intensities of a l l weak lines near the main lines were measured. The criterion was set that the weak lines must have intensity less than 1% of the main line. For most lines the exit s l i t width was 1 mm or wider, although some had widths as small as 350 microns. 3.1.4 Photomultlplier Affixed directly behind the exit s l i t in a light- tight brass container was a Philips 150 CVP end-on photo- multiplier. The circuit is shown In Fig. 6. Because the spectral response of the photomultlplier peaked in the near Infra-red, the dark current was found to be excessively large at room temperature. By surrounding the brass container with crushed dry ice, the noise was reduced by a factor of approxi- mately 50, giving an acceptable signal-to-noise ratio of approxi- mately 5 for the less intense lines. Depending on the humidity, -29- a small stream of air blowing on the edges of the exit s l i t was sometimes necessary to keep ice from forming on the edges, 3.1»5 Amplifier and Phase-Sensitive Detector The signal from the photomultiplier was fed into a four-stage amplifier with a twin-tee feed-back circuit /22/ from stage 4 to stage 2, The resulting amplifier had a one- half amplitude bandwidth of 13 cps centred at 210 cps and a gain at this frequency of approximately 32,000. The output signal was tapped off. a potentiometer and reduced in amplitude so the resulting signal into the phase-sensitive detector was never larger than ,.25v peak-to-peak to ensure linearity. The circuit diagram is shown in Pig. 6. The phase-sensitive detector is the same as that described by Schuster /23/ and the circuit is drawn in Pig. 7. Essentially i t operates by having a sine-wave reference voltage turning one triode on and the other off on each half cycle; a signal with the same frequency is fed through the cathodes and alters the anode currents, but with opposite polarities in each anode circuit . Upon rectification, the resulting dc voltage depends sinusoldally on the phase difference between the reference voltage and the signal. The phase difference was set to 0° or 180° for a maximum output. The dc output was then monitored on a Heathkit model EUW - 20A chart recorder. The RC time constant in the rectifying unit indicated in Pig. 7 was chosen to obtain a response time appropriate to the signal- — I ooo V oc <\if.< 41t< 7k t - 3 o o v D C 1/p - ° 5 > f > 2 ! O K I2AX7 I — " W I 3«>ff TWIN-TEE olS>f 12.AX7 ~ 7 ~7. 150 C V P P H O T O M U L T I P L I E R AMPLIFIER ^ I O O K i I Z A U 7 A ZS/<f Fig. 6. Photomultiplier and Amplifier Circuit + 3<=»ov P^- /V\r 50 K 5W r t AU7A _ 33* 33* vwjgsRp—'W j - 1 - XM 41H 4-1 M '/lW R C N E T W O R K 1 r— 1 \M 1 1 • 1 1 1 1 1 V* 1 1 v 1 4--7H _ J Sot. 5V IOTURW I CWVRT I RECORDER. F i g . 7 Phase-Sensitive Detector Circuit -32- to-noise ratio at the output of the amplifier. Thus the detector acts essentially as a frequency-mixer which f i l ters out a l l the resulting signals except the difference-frequency signal, which in this case happens to be a dc signal. The reference voltage was generated by a 6v tungsten lamp Illuminating a Philips OCP-71 photo-transister. The intensity was strong enough to saturate the photo-transister making the output voltage Independent of variation of lamp illumination. The two components were mounted as a unit, but with the rotating disk intervening and modulating the light, as indicated in Pig. 4. The resulting square-wave signal was sent through a low-pass f i l ter and re-shaped into a 1.4v rms sine-wave before entering the phase-sensitive detector. The phase difference between the reference signal and the signal from the amplifier could be adjusted for maximum detector out- put by mechanically rotating the lamp-photo-transister unit around the axis of the disk. One-seventh of a revolution was necessary for a phase-shift of 3^0°. The linearity of the intensity - measuring circuit (photomultlplier, amplifier, phase-sensitive detector, and chart recorder) was measured using a 6-step neutral density f i l t er which had transmissions varying from .06 to 1. For maximum rms voltages Into the amplifier and phase-sensitive detector of 4 mv and 90 mv respectively, the system was linear to within 2%. The entire system, including the intensity of radia- tion from the Geissler tube, was stable to within 3% over periods of 5 minutes or more. -33- 3.2 Experimental Method The procedure followed in determining absorption coefficients and oscillator strengths is outlined in the following paragraphs. 3.2.1 Circular Apertures The circular apertures were used in conjunction with the 13 mm inner diameter absorption tube and the transmission of the gas determined for the six lengths at each diameter of aperture. The diameters were varied from 3 mm to 13 mm In steps of 1 mm. For a particular discharge length the transmission is determined as the ratio of the intensity of the modulated light from the source when the gas in the absorption tube is excited to the intensity when the gas is not excited. Prom this data 2 of T as a function of R, R T can be plotted as a function of R2 from which is to be calculated the slope d(R2T)/d(R2) at some value of R. For this purpose, an IBM 7040 digital compu- ter was employed to process the data and determine a least- squares f i t of a second-order polynomial to the experimental points. The slope of this curve determined d(R T)/d(R ). A second order polynomial was chosen as it fitted the points fairly well and was simple; an example is shown in Fig. 8 of the experimental points and the calculated polynomial.  -35- This procedure is repeated for a l l six discharge 2 2 lengths and a plot at a particular R of d(R T)/d(R ) against log ( 1 ) prepared. This differs from a plot of equation ( 2 , 9 ) as a function of log (kQ I ), as shown in Pig, 3» "by the vadditive constant log (kQ). By shifting the two plots along their a b s c i s s a e the theoretical curve may be found which best f its the experimental points. The curve found in this manner determines a, and the origin shift determines k Q . Thus the problem of determining the two unknowns from equation ( 2 . 9 ) is solved by using more than one length; six lengths allows a quick averaging graphical solution. In this manner k̂  and a were measured for each of the o four series of lines; a series consists of those lines with a common lower level. The excitation current in the absorption tube had values varying from 2 to 12 ma during the course of these trials while the current in the source was kept at a constant 7 noa. Some of the lines were extremely highly absorbing with transmissions as low as 2%, For these lines the accuracy of the measurement of T was increased by the use of an optical attenuator. This consisted of Kodak Wratten gelatin "neutral density" f i l ters which were used to attenuate the background signal when no absorption was taking place. The sensitivity of the chart recorder was then changed unti l f u l l deflection was achieved. When absorption occurred upon switching on the excitation current in the absorption tube, the f i l ters were removed increasing the otherwise small deflection of the -36- recorder due to the transmitted light. The f i l ters were not truly neutral and the transmission had to be measured at each wavelength for which they were used, 3.2,2 Annull Using the circular apertures, the transmission of some of the highly absorbing spectral lines was less than 2%„ Most of the absorption occurs near the axis of the dis- charge tube where the atom density is higher (see section 5.2.1), For such lines, the annulus method is useful because the region near- the axis is avoided. Conversely, more accurate measure- ments may be performed on weakly absorbing lines by using the circular apertures. As well, use of the annul! affords a means of confirming the values of the transition probabilities measured with the circular apertures. The larger diameter absorption tube was used with an annulus at each end and determinations of k̂  made. Over a o number of measurements the current in the absorption tube was varied from 1 to 10 ma, and the current in the source, from 5 to 9 ma. For a given set of conditions the transmission was measured 3 times for each wavelength and an average transmission calculated. The procedure followed to analyse the data using the annull was simpler than with the circular apertures; a plot of T against log (!.) was compared with the theoretical curves, eliminating the necessity of measuring slopes from the experimental T's. - 3 7 - CHAPTER 4 EMISSION EXPERIMENT 4.1 Measurement of R e l a t i v e I n t e n s i t i e s In measuring i n t e n s i t i e s the e r r o r due to s e l f - a b s o r p t i o n may be eliminated e i t h e r by c o r r e c t i n g f o r s e l f - a b s o r p t i o n when the v a l u e of the a b s o r p t i o n c o e f f i c i e n t i s known (see f o r example, Irwin /12/), or by l o w e r i n g the gas d e n s i t y so the s e l f - a b s o r p t i o n i s n e g l i g i b l e . Once s e l f - a b s o r p t i o n has been e l i m i n a t e d or allowed f o r , the v a r i a t i o n i n response of the d e t e c t i n g apparatus w i t h wavelength must be taken i n t o account. " N o n - f l a t " response r e q u i r e s a c a l i b r a t i o n a g a i n s t a standard lamp of known s p e c t r a l v a r i a t i o n . In t h i s case a tungsten s t r i p f i l a m e n t lamp was used. The output from the p h o t o m u l t l p l i e r can be w r i t t e n 1 U ) « G(X) S(\) T U ) P U ) where G i s a f a c t o r which accounts f o r the geometry of the system, S r e p r e s e n t s the s e n s i t i v i t y of the d e t e c t o r , T the t r a n s m i s s i o n of the monochromator, and P the power emitted by the source. I f two d i f f e r e n t sources are compared, the r a t i o of t h e i r power outputs a t a p a r t i c u l a r wavelength can be determined by measuring the r a t i o of the photPmultiplier'out,puts without knowledge of S or T. I f the r a t i o of the P's at two d i f f e r e n t wavelengths of one source i s r e q u i r e d i t may be measured i f the P r a t i o f o r the o t h e r source i s known. Absolute c a l i b r a t i o n of a l i g h t - d e t e c t o r and d e t e r - -38- mination of spectral output of a continuous light source has been described by Christensen and Ames /24/, Their theory relevant to intensity determinations is given in Appendix 4, with modification for unequal monochromator entrance and exit s l i t s . The results of the analysis show that i f the relative response of the detector at two wavelengths to the line radiation is designated (X-pJig) and the corresponding ratio for a continuous source is R (X^jXg) then Rjt lA.^O = I A ) ^ c ( ^ K ( x Q DfoQ ...(4.1) where I« and I are the intensities of the line and continuous I. c sources, respectively, D is the dispersion of the monochromator, and K is a correction factor defined in the appendix which accounts for spectral variation of the continuous source, photomultiplier, and the monochromator transmission over the pass-band of the monochromator. In this experiment, K differs . from one by an amount less than the estimated experimental error. The intensity Ig includes the effect of self-absorption, i f present, and must be corrected later. For a tungsten lamp I can be written c where £ is the emissivity of tungsten, t is the transmission of the glass envelope surrounding the tungsten strip, T. Is -39- the temperature of the tungsten strip, and J is the blackbody spectral intensity. V changes very l i t t l e over the range of wavelengths considered here and wil l be treated as a constant. Values of £ have been measured for tungsten by de Vos /25/ and Larrabee /26/ for different X and T t . Although the values given in these two references differ by a few percent the ratios £ (X^Tj. ) / £ (X 2 »T t ) agreed to within .5^, and in fact were at most about 2% from the value of unity over the wave- length range 5800A' - 7000A\ Thus equation (4.1) becomes Rlf*o>Q _ Ijfo.) H(h^Jt) T(^Tt) D(%) ...(4.2) from which I g (X-^J/l^ (X2) may be calculated. 4.2 Preliminary Intensity Measurements Initially it was thought that the Intensity measure- ments could be made using the absorption tube as an emitting source, with corrections for self-absorption being determined from the measured values of k Q . Apart from the fact that the optical arrangement required for these measurements was extremely cr i t i ca l with respect to alignment, i t turned out that the uncertainty in the value of kQ rendered the results unsatis- factory (see section 5.4.1). -40- The alternate method of using a discharge tube with no self-absorption was then attempted but it was found that self-absorption could not be eliminated. This preliminary experiment is described here, for although no "true" relative intensities resulted, i t pointed out the difficulties involved In making Intensity measurements. Two neon sources were used. One was a discharge tube similar to the narrow absorption tube but with windows mounted perpendicular to the axis so that the discharge could be viewed transversely. This gave a discharge thickness of about 1 cm. The other source was the Geissler tube used as a background source in the absorption experiment. The source was mounted behind the modulating chopper and imaged on the s l i t of the spectrograph. Beyond the source was a General Electric "Sun Gun" photographic tungsten lamp for calibrating the. spectral response of the system. Only one pair of spectral lines was investigated thoroughly. It was chosen because of the low absorption of each line, and because both transitions started from the same upper level (see equation (2.14)). The lines were X.6678 and X6096. First the side-on tube was used and the centre of the discharge investigated. As the discharge current was reduced from 5 ma to 1 ma in steps of 1 ma the relative response to the two spectral lines was measured at each step. If self- absorption was not present, the relative response should be independent of current. The relative response of the equipment -41- to the continuous radiation source was measured at these two wavelengths after each pair of line measurements, A change in this ratio would indicate a variation somewhere in the apparatus. No significant change was observed. Two other procedures were attempted. The discharge tube operated at a current of 3 ma was moved perpendicular to the optic axis after each measurement so that succeeding measure- ments "saw" regions of the discharge further from the centre and nearer the cylindrical edge. Towards the outer regions, the discharge is less dense as well as being of smaller thickness. Because of this, the extreme edge of the discharge was examined where the discharge thickness is minimum as the current was again reduced in steps. The Geissler tube was investigated in a similar manner except that only the centre of the tube was examined. Also the discharge extinguished at low currents and the Intensities were measured only down to a current of 2 ma. The reproducibility of a l l the measurements was extremely good. A variation between similar measurements never exceeded 2% and in most cases was less than 1%. It was observed in a l l the above tests that the apparent relative intensities varied monotonically as the current or position of the discharge was changed. The total variation was approximately 10$. Pig. 9 shows a plot of the relative intensity as a function of current and discharge position. The observed variation could have been due to the ^ 1 RELATIVE. INTENSITY 1} fj) • — P r > < o> - m * - z 1 2. H </> en § H -< 0 o — - y N - 0 '" 0 » - 0\ GN N l CD I t I o TI H c o» •m 0 M z c n ;» o 3 > i X I c p i a . n i o z. o m 1(6678) e f f e c t of s e l f - a b s o r p t i o n o r s t i m u l a t e d emission, Irwin /12/ showed t h a t s t i m u l a t e d emission i s n e g l i g i b l e f o r c u r r e n t s up to 100 ma u s i n g a d i s c h a r g e tube s i m i l a r t o the s i d e - o n tube, Ladenburg /7 / came to the same c o n c l u s i o n about a d i s c h a r g e tube comparable to the G e i s s l e r tube used here. Thus the observed e f f e c t must be due to s e l f - a b s o r p t i o n . I t i s a l s o to be noted t h a t the v a r i a t i o n of r e l a t i v e i n t e n s i t y i s i n the d i r e c t i o n c o n s i s t e n t w i t h X.6096 b e i n g the more h i g h l y a b s o r b i n g of the two l i n e s . A decrease o f the s e l f - a b s o r p t i o n would i n c r e a s e the i n t e n s i t y of \6096 by a l a r g e r p r o p o r t i o n than f o r X.6678 (see the t a b l e of s e l f - a b s o r p t i o n f a c t o r s In Appendix 3). The i n a b l i l i t y to e l i m i n a t e s e l f - a b s o r p t i o n i n the sources prompted the use of a d i f f e r e n t type of source. The d e t a i l s o f the ensuing experiment a re g i v e n below. 4.3 F i n a l I n t e n s i t y Measurements The procedure f i n a l l y adopted t o ensure the absence of s e l f - a b s o r p t i o n was to measure the emi s s i o n from low d e n s i t y neon which had only been i n the e x c i t e d s t a t e f o r a sh o r t time. The low d e n s i t y combined w i t h the short e x c i t a t i o n time ensured t h a t the e x c i t e d s t a t e p o p u l a t i o n d e n s i t i e s was s m a l l . Thus the p r o b a b i l i t y of a b s o r p t i o n of r a d i a t i o n by atoms i n an e x c i t e d s t a t e was s m a l l . Because the d e n s i t y i s low, the emission i n t e n s i t y i s a l s o low and the problem c o n s i s t s of c o u n t i n g i n d i v i d u a l photons. -44- The average photon f l u x i s equal to the i n t e n s i t y . The c o u n t i n g r e q u i r e s somewhat s o p h i s t i c a t e d apparatus and techniques which are d e s c r i b e d below. 4.3.1 Apparatus T h i s p a r t i c u l a r apparatus and the g e n e r a l techniques have been developed f o r l i f e t i m e measurements by van Andel /17/ who helped perform t h i s experiment. As the apparatus i s des- c r i b e d by him i n d e t a i l , only a c u r s o r y d e s c r i p t i o n w i l l be g i v e n here. .1 Sampling Technique An e l e c t r o n gun s i t u a t e d w i t h i n a bakable, u l t r a - h i g h vacuum system was used to e x c i t e the neon gas which was at a p r e s s u r e of .1 mm Hg. The gun was p u l s e d 485 times per second w i t h a p u l s e d u r a t i o n of 200 nsec, and r a d i a t i o n from the e m i t t i n g neon was observed w i t h a g r a t i n g monochromator and a p h o t o m u l t i p l i e r . The r e s u l t a n t s i g n a l from the c o n t i n u o u s l y m o n i t o r i n g RCA 7265 p h o t o m u l t i p l i e r was sampled 20 nsec a f t e r the c e s s a t i o n of each p u l s e w i t h a sample width of 0.3 nsec. The sampling was performed w i t h a T e k t r o n i x 66l o s c i l l o s c o p e , which puts out a v o l t a g e p r o p o r t i o n a l t o the height of the sampled p h o t o m u l t i p l i e r s i g n a l and was f o l l o w e d by an a m p l i f i e r , d i s c r i m i n a t o r and d i g i t a l counter. The purpose of the d i s c r i m i - n a t o r was t o i s o l a t e the counter from p u l s e s below a c e r t a i n -45- v o l t a g e ; i n t h i s manner p u l s e s o r i g i n a t i n g from n o i s e i n the sampling c i r c u i t was e l i m i n a t e d . Counts were recorded over a p e r i o d of te n seconds and y i e l d e d the number of samples which contained a s i g n a l . T h i s s i g n a l i n d i c a t e d the a r r i v a l of one or more p h o t o - e l e c t r o n p u l s e s at time of sampling. .2 Monochromator The monochromator was b u i l t by van Andel /17/ f o r h i s experiment, and i s a 42" f o c a l l e n g t h Ebert mount w i t h a 6" by 8" concave g r a t i n g having 300 lines/mm. The r e c i p r o c a l d i s p e r s i o n i s approximately 4 i n the f i f t h o r d e r and the t h e o r e t i c a l r e s o l v i n g power i s 300,000. A p p r o p r i a t e Corning f i l t e r s were used t o i s o l a t e the f i f t h o r d e r and gave an e f f e c - t i v e pass band from 5500 % t o 7500 A. To guard a g a i n s t e r r o r s i n p o s i t i o n i n g the s p e c t r a l l i n e image i n the e x i t p l a ne of the monochromater, the e x i t s l i t width was 1.75 mm and the entrance s l i t w i d t h 1.25 mm f o r a l l measurements. The unequal widths ensured t h a t the image of the entrance s l i t was always encompassed by the e x i t s l i t . .3 S p e c t r a l C a l i b r a t i o n The s p e c t r a l response of the system was measured u s i n g a G. E. 6v-9A tungsten s t r i p f i l a m e n t lamp p l a c e d approximately a t the p o s i t i o n occupied by the e l e c t r o n gun w i t h the f i l a m e n t focused on the monochromater s l i t . The c u r r e n t through the lamp was 5 amps and the temperature o f the tungsten .46. was c a l c u l a t e d by measuring the b r i g h t n e s s temperature w i t h a Hartmann and Braun f i l a m e n t pyrometer and then c a l c u l a t i n g the t r u e temperature by the method g i v e n by Rutgers and de Vos /27/. The pyrometer i n t u r n had been c a l i b r a t e d at two tempera- t u r e s , 1800°K and 1340°K, w i t h a G. E. tungsten r i b b o n f i l a m e n t standard lamp, type T-24 86-P-50. The pyrometer was found to agree w i t h i n one per cent of the c a l i b r a t i o n temperatures. U n f o r t u n a t e l y the standard lamp I t s e l f was too l a r g e t o be used d i r e c t l y to measure the response without s e r i o u s m o d i f i c a t i o n of the equipment. I n i t i a l l y the standard source was the G. E„ "Sun Gun" used i n the p r e l i m i n a r y i n t e n s i t y experiment d e s c r i b e d i n s e c t i o n 4.2. Upon measuring the i n t e n s i t y d i s t r i b u t i o n of the Sun Gun over the range 58O0X to 6800°v i n s t e p s , i t was found t h a t a dip o f approximately i n the d i s t r i b u t i o n o c c u r r e d a t 6000°i w i t h a width of 100 °v. The o p e r a t i n g v o l t a g e o f the lamp f o r t h i s measurement was onl y 20v dc as opposed t o the r a t e d 120v dc which had been used i n the p r e l i m i n a r y measurements. I t i s not known whether t h i s d i p i s present at the h i g h e r v o l t a g e as no f u r t h e r i n v e s t i g a t i o n s were made. The cause of t h i s d i p i s a l s o u n c e r t a i n ; p o s s i b l y i t i s caused by the g a s ( i o d i n e ) c o n t a i n e d i n the envelope s u r r o u n d i n g the tungsten f i l a m e n t f o r purposes o f c l e a n s i n g . The s t r i p f i l a m e n t lamp f i n a l l y used showed no such d i p . 4.3.2 Experimental Method The t r a n s i t i o n p r o b a b i l i t i e s c a l c u l a t e d by a b s o r p t i o n are d i v i d e d i n t o f o u r groups c o r r e s p o n d i n g to the f o u r lower l e v e l s on which a l l the s p e c t r a l l i n e s t e r m i n a t e . To r e l a t e the p r o b a b i l i t i e s between any two groups the r e l a t i v e I n t e n s i t y of two common u p p e r - l e v e l l i n e s must be measured, one i n each group (see equation (2.14)). The si m p l e s t way t o r e l a t e a l l f o u r groups i s t o choose one l i n e from each group w i t h each l i n e having the same upper l e v e l . In t h i s experiment the c h o i c e of l i n e s was governed by d e s i r e a b i l l t y of a h i g h s i g n a l - t o - n o i s e r a t i o f o r the l i n e s measured as w e l l as the n e c e s s i t y of choosing from a p a r t i c u l a r group a l i n e t h a t had been measured i n a b s o r p t i o n . For these reasons i t was not p o s s i b l e to choose one upper l e v e l common t o the f o u r groups and i n s t e a d t h r e e s e p a r a t e p a i r s o f l i n e s were chosen t o l i n k the f o u r groups t o g e t h e r . They were X.X.6096, 6678 ( s ^ and s 2 ) ; 6163, 6599 ( s 3 and s 2 ) ; 6334, 6506 ( s ^ and s ^ ) . (The l e v e l d e s i g n a t i o n s are i n the Paschen n o t a t i o n (see s e c t i o n 5.1). Three other l i n e s from the p^ l e v e l were measured to g i v e two independent s e l f - c o n s i s t a n c y checks; they were \\6217, 6383, and 6533. F i g . 10 shows a p a r t i a l term diagram i n d i c a t i n g the t r a n s i t i o n s . The i n t e n s i t i e s o f the t h r e e p a i r s o f l i n e s were measured one p a i r at a time; the monochromator was set on the f i r s t l i n e , and th r e e t e n second counts were r e c o r d e d b e f o r e p r o c e e d i n g on t o the next l i n e of the p a i r . T h i s was repeated -48- •+4 -f<5 2 cc N •C IN X < « IT < < < S) < F i g . 10 P a r t i a l Term Diagram of Neon; I n t e n s i t y - Measurement T r a n s i t i o n s t e n times and the r a t i o of sums of the counts c a l c u l a t e d as the measured r e l a t i v e i n t e n s i t y , t o be c o r r e c t e d f o r instrument response t o get the t r u e r e l a t i v e I n t e n s i t y . T h i s procedure was repeated twice more f o r each p a i r ; s i m i l a r measurements were made on the p^ l i n e s . E r r o r counts due to s t r a y room l i g h t , p h o t o m u l t l p l i e r n o i s e , and continuum l i g h t from the glowing cathode of the e l e c t r o n gun had to be s u b t r a c t e d from the neon counts by o b s e r v i n g the counts at a time 5 microseconds a f t e r the e x c i t a - t i o n p u l s e , when the neon e x c i t a t i o n had decayed. For c a l i b r a t i n g the s p e c t r a l response of the system, an analogous procedure i s f o l l o w e d . The s i g n a l due to the tungsten - 4 9 - lamp was sampled and counted over a t e n second p e r i o d at each of the wavelengths f o r which neon measurements were made. Stray l i g h t and n o i s e counts were made by b l o c k i n g o f f the lamp. F i g . 11 shows the response of the system t o the r a d i a t i o n from the tungsten lamp. R^X.^jJig) m a y b e computed from t h i s curve. One of the b a s i c s y s t e m a t i c e r r o r s of c o u n t i n g methods i s p i l e - u p e r r o r ; w i t h t h i s p a r t i c u l a r c o u n t i n g apparatus t h e r e can occur e s s e n t i a l l y two types of p i l e - u p e r r o r and they w i l l now be d e s c r i b e d . P i l e - u p i s the o v e r l a p p i n g of two p u l s e s . Consider what happens i f the d i s c r i m i n a t o r l e v e l i s set h i g h and the s i g n a l i s sampled at a time when two p u l s e s o v e r l a p at the anode of t?ie p h o t o m u l t i p l i e r . I f i n d i v i d u a l l y these p u l s e s have a height g r e a t e r than one-half but l e s s than one times the d l s c r i m i ' n a t o r l e v e l no count i s r e g i s t e r e d f o r separate a r r i v a l . Added t o g e t h e r they w i l l r e g i s t e r a count of one, whereas t h e r e should be no count; i f the i n t e n s i t y i s assumed p r o p o r t i o n a l to the count, the apparent i n t e n s i t y i s g r e a t e r than the t r u e i n t e n s i t y . On the o t h e r hand, i f the d i s c r i m i n a t o r i s set low, the a r r i v a l of the two o v e r l a p p i n g p u l s e s w i l l a g a i n r e g i s t e r as one count, a l t h o u g h a r r i v i n g s e p a r a t e l y they should g i v e a count of two, so the apparent i n t e n s i t y i s l e s s than the t r u e i n t e n s i t y . I f the l e v e l i s set low enough, the number of p u l s e s of height l e s s than the d i s c r i m i n a t o r l e v e l i s s m a l l , and the former p i l e - u p case occurs i n f r e q u e n t l y . In bo t h cases the e r r o r i n c r e a s e s as the apparent count i n c r e a s e s . Thus f o r a l l measurements the d i s c r i m i n a t o r l e v e l was 5 co- F i g u r e 11 R e l a t i v e Response of I n t e n s i t y - Measuring C i r c u i t -51- kept low and the count s m a l l compared t o the number of samples (4850 p e r 10 seconds). The n e u t r a l d e n s i t y f i l t e r s from the a b s o r p t i o n experiment were used on the e s p e c i a l l y " b r i g h t " l i n e s t o keep the count below 250 per 10 seconds. The presence of p i l e - u p at h i g h counts was apparent when the t r a n s m i s s i o n of one of the f i l t e r s was measured on the apparatus. When the u n f i l t e r e d count was approximately 250 per 10 seconds the t r a n s m i s s i o n agreed to w i t h i n 1% of the v a l u e as measured on the a b s o r p t i o n apparatus. On the o t h e r hand an upward d e v i a t i o n of more than 5% was observed when the u n f i l t e r e d count was approximately 700 per 10 seconds. The estimated accuracy of the t r a n s m i s s i o n as measured w i t h the a b s o r p t i o n apparatus i s about 2% so at 250 counts per 10 seconds the c o u n t i n g apparatus i s l i n e a r . -52- CHAPTER 5 RESULTS AND DISCUSSION For d i s c u s s i o n of the r e s u l t s , i t i s a p p r o p r i a t e to b e g i n w i t h a short d i s c u s s i o n of the neon term scheme and s p e c t r a . F o l l o w i n g t h i s , the r e s u l t s are p r e s e n t e d i n two s e c t i o n s , one p e r t a i n i n g e x p l i c i t l y to the a b s o r p t i o n experiment and the o t h e r to the emission. The f i n a l s e c t i o n of t h i s chapter concludes w i t h a combination of the two p r e v i o u s s e c t i o n s and p r e s e n t s f i n a l r e s u l t s . A g e n e r a l d i s c u s s i o n i s g i v e n i n each s e c t i o n on the s i g n i f i c a n c e of the r e s u l t s and. the v a l i d i t y of the t h e o r y . E r r o r s and e s t i m a t i o n of the accuracy of the r e s u l t s are a l s o d i s c u s s e d . 5.1 Neon Spectrum T h i r t y s p e c t r a l l i n e s emitted by neon i n the v i s i b l e and i n f r a - r e d . r e g i o n r e s u l t from t r a n s i t i o n s between the f i r s t and second e x c i t e d s t a t e s o f the atom, 2p-*3s and 2 p^3p. F i g . 12 shows the allowed t r a n s i t i o n s w i t h the c o r r e s p o n d i n g wave- l e n g t h s g i v e n i n Angstroms; the f u l l l i n e s r e p r e s e n t t r a n s i t i o n s on which measurements were taken i n t h i s experiment. The d e s i g n a t i o n of the l e v e l s i s g i v e n i n both the LS /28/ and the Paschen / 2 9 / n o t a t i o n ; the l a t t e r n o t a t i o n w i l l g e n e r a l l y be used here. -53- L S N o t a t i o n N o t a t i o n Grooncl Stare F i g u r e 12 Neon Term Diagram; F i r s t and Second E x c i t e d S t a t e C o n f i g u r a t i o n 5.2 A b s o r p t i o n - C i r c u l a r Apertures As was mentioned i n s e c t i o n 3.2.2, the c i r c u l a r a p e r t u r e method was more a p p l i c a b l e e x p e r i m e n t a l l y f o r l i n e s showing s m a l l a b s o r p t i o n . The s 2 l i n e s were g e n e r a l l y l e s s a b s o r b i n g than the other l i n e s by a f a c t o r of t h r e e or more and were measured o n l y by t h i s method. Measurements were a l s o made on some of the o t h e r groups of l i n e s . 5.2.1 C r o s s - S e c t i o n a l V a r i a t i o n of T r a n s m i s s i o n If b o t h N 4 and T are constant over the c r o s s - s e c t i o n l a of the a b s o r b i n g column, eq u a t i o n (2.3) shows t h a t the t r a n s - m i s s i o n T should not vary as R, the r a d i u s of the a p e r t u r e s , i s v a r i e d . P i g . 13 shows a t y p i c a l p l o t of the t r a n s m i s s i o n of the a b s o r b i n g gas as a f u n c t i o n of R, f o r the v a r i o u s d i s c h a r g e l e n g t h s . I t can be seen t h a t the shape of these curves are s i m i l a r f o r each l e n g t h , d e c r e a s i n g m o n o t o n i c a l l y towards the c e n t r e of the tube w i t h a c o r r e s p o n d i n g tendency to l e v e l o f f . The curves f o r the other s p e c t r a l l i n e s behaved i n the same manner i n d i c a t i n g that c o n d i t i o n s are d e f i n i t e l y not constant over the c r o s s - s e c t i o n of the a b s o r b i n g column. The t r a n s - m i s s i o n should i n c r e a s e w i t h R i f decreases. Whether i t i n c r e a s e s w i t h d e c r e a s i n g T depends on the v a l u e of k_ and a but roughly speaking an i n c r e a s e occurs i f k Q £ i s " l a r g e " and a decrease i f k Q i ! i s " s m a l l " . There i s reason to b e l i e v e how- ever, t h a t the observed i n c r e a s e i n t r a n s m i s s i o n i s caused 14 tl z o co oo Z CO | - K > J ,06H I Z 3 6 7 F I G 13 T R A N S M I S S I O N A S A F U N C T I O N O F R wholly by a decrease i n (see discussion i n section 5.^.1). 5.2.2 Curve P i t t i n g 2 2 For the process of f i t t i n g the s i x values of d(R T)/d(R ) corresponding to the s i x absorption lengths to the th e o r e t i c a l curves (see section 3.2.1 and equation (2.9)), the slopes were calculated as a function of R, as R was varied from the minimum to the maximum diameter of aperture used, i n steps of .5 mm. Generally some portion of the t h e o r e t i c a l curves could be found which f i t t e d the s i x points well although i n some cases the s i x points showed scatter at large R (5.5 to 6.5 mm) and the f i t was not as good. F i g . 14 shows an example of curve f i t t i n g where the curves of F i g . 3 are p a r t i a l l y redrawn and the experimentally determined slopes of \6l64 f o r R = 4 mm, I = 10 ma are superimposed, a), b), and c) show f i t t i n g s to the curves corresponding to a = .5, .7, and .9 respectively, with the best f i t being f o r a = .7 giving k Q « .245 cm - 1. Generally the uncer- t a i n t i e s i n estimating a and k Q from the t h e o r e t i c a l curves were no greater than about 10% and 5% respectively. It can be seen from F i g . 3 that the curves are c l o s e l y spaced f o r small values of k Q! , and f o r large k Q^ at values of a ~ 1. In these regions a l e s s precise value of a i s determined. Curve f i t t i n g was also attempted using the data of T versus log( I ) at a p a r t i c u l a r aperture radius. It was found that a reasonably good f i t could be made but the values of a and k Q were much d i f f e r e n t from the values determined using  -58- d(H T)/d(R ). The v a l u e s of f ^ c a l c u l a t e d t h i s way were e f f e c t i v e l y the same. 5.2.3 R e l a t i v e O s c i l l a t o r Strengths The r e l a t i v e o s c i l l a t o r s t r e n g t h s f r e l can be computed from equation (2.11). In t h i s s e c t i o n these q u a n t i t i e s are c a l c u l a t e d and are converted i n s e c t i o n 5»6 to t r a n s i t i o n p r o b a b i l i t i e s . Tables 1 - 4 l i s t the v a l u e s of f r e l » k Q» and a f o r the S g , s^, and s^ l i n e s as a f u n c t i o n of R f o r v a r i o u s I a . The a b s o r p t i o n by the s^ l i n e s was g e n e r a l l y g r e a t e r than 90% and the experimental p o i n t s d i d not f i t the t h e o r e t i c a l curves w e l l so these q u a n t i t i e s c o u l d not be determined by the c i r c u l a r a p e r t u r e method. P i g . 15 shows some of the data f o r f r e l f r o m T a D l e 1 i n g r a p h i c a l form. The e r r o r bars r e p r e s e n t maximum p o s s i b l e e r r o r s due to the f i t t i n g to the t h e o r e t i c a l c urves; In most cases the best f i t o c c u r r e d at the same value of a f o r l i n e s w i t h the same lower l e v e l and under the same o p e r a t i n g c o n d i t i o n s . Sometimes i t was d i f f i c u l t t o t e l l which curve f i t t e d b e s t ; i n t h i s case the average v a l u e s of k Q and a were taken from the curves f o r which the f i t was not unreason- a b l e . I t was found t h a t even c a l c u l a t i n g f p e l u s i n g v a l u e s of k Q determined from curves f o r which t h e f i t was d e f i n i t e l y poor, the d e v i a t i o n of f r e ^ from the b e s t - f i t curves was not more than 10$ and o f t e n much l e s s . The d e v i a t i o n of a between the b e s t - f i t and p o o r - f i t curves was as h i g h as 30$ and i n d i c a t e s a slow -59- TABLE 1 CIRCULAR APERTURE DETERMINATION OF f r e l : s 2 L I N E S » Is=7ma R V ^ i ) (ma) (mm) 5852\ 6599 66?8 6717 6929 6 2.0 1.00 1.32 2.23 1.34 2.02 2.5 1.30 2.22 1.31 1.99 3.0 1.29 2.22 1.31 1.97 3.5 1.28 2.18 1.27 1.87 4.0 - 1.26 2.13 1.23 1.82 4.5 1.19 2.06 1.17 1.71 5.0 1.10 1.92 1.10 1.51 Average 1.25 2.14 1.25 1.85 10 1.5 1.00 1.10 2.15 1.22 1.72 2.0 1.10 2.17 1.24 1.72 2.5 1.09 2.17 1.22 1.68 3.0 1.13 2.19 1.20 1.72 3.5 1.10 2.11 1.19 1.69 4.0 1.10 2.04 1.18 1.65 4.5 1.16 2.10 1.14 1.67 5.0 1.14 1.97 1.60 Average 1.12 2.11 1.20 1.68 10 2.0 1.00 1.19 2.06 1.21 1.76 2.5 1.18 2.04 1.18 1.74 3.0 1,18 2.04 1.19 1.74 1.18 2.05 1.18 1.74 4.0 1.18 2.05 1.17 1.74 4.5 1.17 2.05 1.18 1.74 Average 1.18 2.05 1.18 1.74 12 2.0 1.00 1.2? 2.02 1.21 1.78 2.5 1.24 2.04 1.20 1.76 3.0 1.24 2.01 1.20 1.75 3.5 1.23 2.02 1.19 1.74 4.0 1.23 2.03 1.19 1.73 4.5 1.22 2.04 1.18 1.73 5.0 1.2.1 2.05 1.17 1.76 5.5 1.20 2.05 1.16 1.74 6.0 1.18 2.06 1.13 1.73 Average 1.22 2.04 1.18 1.75 T o t a l Average 1.19 2.08 1.20 1.75 -60- TABLE 2 CIRCULAR APERTURE DETERMINATION OP k Q, a : s 2 LINES, I = 7 ma Xa 5852 6599 6678 6717 6929 (ma) R(mmrv k 0(cm _ 1) a k 0(cm _ 1) a k 0(cm _ 1) a k o(cm _ 1) a k^cm - 1) a 6 2.0 .020 0.9 .032 1.0 .0495 0.85 .031 0.9 .0525 1.05 2.5 .024 1.3 .031 1.1 .050 1.0 .032 1.1 .053 1.2 3.0 .025 1.5 .034 1.4 .050 1.15 .033 1.3 .0525 1.35 .0230 1.55 .0350 I .65 .0505 1.35 .0335 1.55 .0495 1.5 .0235 I .85 . 033 1.85 .051 1.65 .034 1.9 .04? 1.7 k.5 .0198 1.85 .0293 2.0 .0465 1.85 .0275 1.9 .0415 1.9 5.0 .0175 2.0 .0277 2.05 .038 2.0 .021 1.9 .033 2.0 10 1.5 .052 I .05 .067 1.1 .113 0.9 .071 1.0 .103 1.0 2.0 .050 1.1 .064 1.15 .114 lvO .071 1.1 .098 1.05 2.5 .052 1.25 .064 1.25 .113 1.1 .070 1.2 .098 1.15 3.0 .050 1.35 .066 1.4 .114 1.25 .072 1.4 .089 1.3 ?'5 .050 1.5 .064 1.55 .116 1.45 .074 1.6 .096 1.^5 4.0 .050 1.7 .064 1.75 .111 1.6 .069 1.8 .091 1.6 .046 1.9 .062 1.95 .108 1.9 .068 2.1 .O87 1.8 5.0 .040 2.0 .053 2.1 .095 2.05 .088 2.2 10 2.0 .043 1.15 .050 0.95 .0865 0.95 .054 1.0 .073 0.9 2.5 .042 1.2 .053 1.15 .0855 1.05 .054 1.15 .078 1.1 3.0 .046 1 .5 .049 •1.2 .084 1.2 .0555 1.35 .0775 1.35 .044 1.65 .053 1.5 .084 1.4 .054 1.5 .076 1.4 4.0 . 0405 1.75 .0513 1.7 .081 1.55 .053 1.75 .074 1.6 k.5 .0385 2.0 .0503 2.0 .075 1.75 .0523 2.0 .0715 1.85 5.0 .0305 2.0 .062 2.05 5.5 .0235 2.05 12 2.0 .046 0.95 .0645 0.95 .093 0.8 .053 0.7 .085 0.8 2.5 .047 1.1 .0635 1.05 .100 1.0 .060 1.0 .0865 0.95 3.0 .045 1.2 .O63 1.2 .0975 1.15 .0595 1.15 .0895 1.15 3.5 .046 1.4 .063 1.4 .095 1.3 .0595 1.35 .088 1.3 4.0 .0435 1.55 .061 1.6 .096 1.5 .059 1.55 .083 1.5 .0405 1.75 .060 1.9 .091 1.7 .053 1.75 .079 1.7 5.0 .037 2.0 .052 2.1 .0795 I .85 .0485 2.05 .071 1.9 5.5 .0295 2.1 .044 2.4 .064 2.0 .035 2.1 .055 2.0 6.0 .0238 2.5 .025 2.2 .045 2.0 .0225 2.05 .039 2.05 -61- TABLE 3 CIRCULAR APERTURE DETERMINATION OF f r e l ! s 3 a n d s 4 L I N E S » ! s ~ 7ma Xa ^ \ X ( ? ) LINES (ma) R(mnO x\ 6164 6266 6533 10 4,0 1.22 1.73 1.00 4.5 1.24 1.71 5.0 1.20 1.85 5.5 1.14 1.96 6.0 1.91 • • 6.5 2.08 Average 1.20 1.87 > \ \ ( A ) LINES (ma) R ( m m ) v \ 6074 6074 6096 6383 6507 7245 2 2.5 .714 1.00 1.16 1.69 .596 3.0 • 733 .745 1.16 1.65 .567 3.5 .681 .744 1.16 1.71 .542 4.0 .675 .696 1.15 1.68 .497 4.5 .653 .678 1.18 1.70 .487 5.0 .652 .644 1.17 1.74 .443 5.5 .638 .626 1.16 1.69 .430 6.0 .585 1.16 1.67 .391 6.5 .546 1.20 1.69 Average .678 .658 1.17 1.69 .494 10 2.0 1.00 1.28 2.5 1.25 i 1 3.0 1.18 1 3.5 .74 1.20 1.55 .47 ; 4.0 .72 1.16 1.60 .46 1 4.5 .72 1.16 1.59 .44 5.0 .66 1.16 1.58 .43 1 5.5 .66 1.12 1.59 .42 6.0 .64 1.12 1.56 .40 6.5 .65 1.11 1.60 .40 Average .68 1.17 1.58 .43 T o t a l Average .673 1.17 1.64 .465 j TABLE 4 CIRCULAR APERTURE DETERMINATIONS OF k Q , a : and LINES I = ? ma s a (ma) M B ) R(mm) 6164 ^ ( c m - 1 ) LINES 6266 k o ( c m - 1 ) a 6533 k 0 ( c m _ 1 ) a k o ( c m - 1 ) a lc (cm o -1, K ( e n - 1 ) o 10 4.0 4.5 5.0 5.5 6.0 6.5 0.245 0.280 0.290 0.210 0.7 1.0 1.2 1.2 0.700 0.575 0.390 0.350 0.275 0.235 l . l 1.15 1.1 1.15 1.15 1.15 0.270 0.240 0.220 P.195 0.150 0.116 0.9 l . o 1.1 1.2 1.15 1.15 s u LINES "724T 2.0 2.5 3-0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 6074 6074 .0935 0.103 0.114 0.114 .0.106 0.111 0.093 0.075 0.45 0.75 0.95 1.15 1.25 1.5 1.55 1.6 0.110 0.110 0.098 0.098 0.083 0.06I 0.046 0.9 I.05 1.15 1.35 1.45 1.4 1.45 6096 0.140 0.143 0.151 0.151 0.137 0.121 0.106 0.084 0.0615 0.7 0.9 1.1 1.2 1.3 1.4 1.5 1.5 1.45 6333 O.I63 .0.135 0.182 0.193 0.170 0.170 0.148 ^ 0.133 0.098 O.O79. 0.5 0.3 0.95 1.1 1.15 1.3 1.4 1.5 1.45 1.5 6507 0.232 0.260 0.250 0.230 0.212 0.200 0.165 0.135 0.103 0.6 0.9 1.0 1.1 1.2 1.3 1.35 1.4 1.35 0.093 0.102 0.112 0.114 0.106 0.096 O.O76 0.055 0.039 10 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 0.280 0.280 0.290 0.270 0.260 0.230 0.185 0.150 0.120 0.5 0.7 0.3 1.0 1.1 1.2 1.2 1.2 1.2 0.500 0.740 0.720 0.570 0.450 O.36O 0.300 0.245 0.205 0.135 0.65 1.0 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.15 0.710 0.500 0.520 0.645 0.500 0.440 0.430 0.315 0.230 0.230 0.7 0.65 0.85 1.05 1.05 1.1 1.2 1.15 1.2 1.2 1.00 0.75 0.55 0.455 0.375 0.315 1.2 1.2 1.15 1.15 1.15 1.15 0.240 0.250 0.260 0.245 0.220 0.205 0.165 0.130 0.103 f 'rel 2.o - j 18- 1.6 1-2'- 7-o - .8- 6- 6506A I | t J ^ 6 074^ ~* 1 . I ...a . I- t 3 4 5 6 7 P i g . 15 f , as a F u n c t i o n of R -64. dependence of f p e ^ w i t h a. I t can be seen upon comparing T a b l e s 2 and 4 t h a t the v a l u e s of a are s i g n i f i c a n t l y h i g h e r f o r the s 2 l i n e s than the o t h e r s e r i e s . The p r e s s u r e i n the source was 5 mm Hg which i s s l i g h t l y h i g h e r than i n the absorber. Lang /30/ has shown t h a t p r e s s u r e broadening of the S g l i n e s i s much more p r e v a l e n t than the o t h e r l i n e s ; i t i s c o n c e i v a b l e that i f t h e r e i s p r e s s u r e broadening of the s 2 l i n e s i n the source the e f f e c t i v e a measured c o u l d be h i g h e r than f o r other l i n e s . The s i g n i f i c a n c e of the v a l u e s of a obtained w i l l be d i s c u s s e d i n s e c t i o n 5.^» 5»3 A b s o r p t i o n - Annul! The l a r g e r a b s o r p t i o n tube was used w i t h the annulus c o n f i g u r a t i o n . T h i s allowed the t r a n s m i s s i o n of the more s t r o n g l y a b s o r b i n g l i n e s i n the l e s s dense r e g i o n s away from the a x i s of the tube to be a c c u r a t e l y measured. The a b s o r p t i o n of the S g l i n e s was consequently too s m a l l to measure a c c u r a t e l y and f i t to the k Q - a curves. Tables 5 and 6 l i s t the v a l u e s of f r e ^ » k Q , and a f o r measurements made on the s^, s^, and s^ l i n e s f o r v a r i o u s v a l u e s of I and I„. I t can be seen that t h e r e seems to be a s no d e f i n i t e s y s t e m a t i c v a r i a t i o n of f r e l w i t h I or I g . The remarks made p r e v i o u s l y c o n c e r n i n g the f i t t i n g o f the p o i n t s t o the curves ( i n the c i r c u l a r a p e r t u r e method) are g e n e r a l l y a p p l i c a b l e here a l s o . I t was observed t h a t i n some t r i a l s , the e x p e r i m e n t a l v a l u e of T was l e s s than the t h e o r e t i c a l v a l u e 65 TABLE 5 ANNULUS DETERMINATION OP f r e l : s 2» s 4 a n d s 5 L I N E S \ X ( A ) f r e ] LINES (ma) I s ( m a ) \ 6164 6266 6533 8 7.6 1.19 1.82 1.00 10 7 1.21 1.88 10 7 1.14 l s 8 4 10 7 1.18 1.83 X a \ x ( X ) f r e l ! s 4 LINES (ma) I f i ( m a > \ 6074 6096 6305 6383 6507 7245 2 7 1.00 1.44 .401 4 7 1.44 .402 6 7 .592 .852 1.41 .423 8 7 .600 .860 .271 1.41 .448 8 7.6 .265 .440 10 7 .611 .869 .266 1.43 10 7.6 ,251 J a \ MA) f r e l ! 85 LINES (ma) I s ( m a > ^ 5882 5945 6143 6217 6334 6402 7032 1 7.6 .347 .592 1.43 .244 1.00 2.79 .988 1 8 2.71 2 7 1.38 2.60 1.01 2 9.2 1.40 2 9.2 1.40 2 6 1.42 2 7.0 2.52 2 6.0 2.59 2 5.0 2.60 TABLE.6 ANNULUS DETERMINATION OF k Q , o l : S g , s 4 , a n d s g LINES s. LINES o 6164 6266 6533 I a I s k 0 ( c m _ 1 ) a \Ccnf1) a k 0 ( c m - 1 ) a k ( c m - 1 ) a k ( c m 1 ) a k ( c m - " ' ' ) cr a k ( c m 1 ) a ( m a ) ( m a ) 8 7.6 .0187 .45 .0293 .5 .0168 .5 10 7 .0164 .65 .0261 .65 .0144 .65 10 7 .0176 .7 .0289 .65 .0163 .65 10 7 .0177 .55 .0276 .5 . 0156 .5 .8. LINES 4 6074 6096 6305 6383 6507 6245 2 7 .0133 1.45 .0195 1.45 .0060 1.4 4 7 .0196 1.3 .0288 1.3 .0095 1.4 6 7 .0118 .85 .0157 .7 . 0199 .75 .0296 .8 .0099 .8 8 7 .0135 .5 .0190 .4 .0064 .5 .0244 .5 .0425 .85 .0129 .6 8 7.6 .0090 1.0 .0343 1.0 10 7 .0161 .65 . 0216 .55 .0073 .7 '. 0277 .7 .0518 1.05 .0135 .6 10 7.6 .0096 1.0 .0386 1.0 s.-LINES 0 5882 5945 6143 6217 6334 6402 7032 1 7.6 .0205 1.35 .0356 1.35 .0807 1.35 .0153 1.35 .0638 1.35 .163 1.3 .077 1.45 1 8 .0625 1.4 .174 1.35 2 7 .0790 1.15 .059 1.15 .165 1.2 .072 1.25 2 9.2 • .102 1.35 .0755 1.35 2 9.2 .101 1.3 .0734 1.3 2 6 .104 1.3 .0755 1.3 2 7.0 .0700 1.25 .193 1.3 2 6.0 .0726 1.25 .190 1.25 2 5.0 .0735 1.25 .194 1.25 -67- when i = 10 cm. Since the other experimental p o i n t s agreed w e l l w i t h the t h e o r e t i c a l p r e d i c t i o n s the d i s c r e p a n c y may be due t o the e f f e c t i n the d i s c h a r g e as i t "rounds the c o r n e r " i n the tube (see s e c t i o n 5.4.2). When the above d i s c r e p a n c y o c c u r r e d , was measured from the curves w i t h and without i n c l u s i o n of the 10 cm p o i n t ; the v a l u e of f r e ^ c a l c u l a t e d by these two methods was s u b s t a n t i a l l y the same. Again i n c a l - c u l a t i n g f r e i from curves which d e f i n i t e l y were not the best f i t , v a r i a t i o n s of 5% a t most were observed. I t i s apparent a l s o , i n comparing l i n e s common to Tables 1 and 5» that t h e r e i s e s s e n t i a l l y no d i f f e r e n c e i n the v a l u e s of f r e l o b t a i n e d by the annulus and the c i r c u l a r a p e r t u r e method. I f t he experimental p o i n t s were f i t t e d t o the t h e o r e t i - c a l curves c a l c u l a t e d f o r the presence o f o n l y a s i n g l e i s o t o p e , p o r t i o n s of the curves c o u l d be found which f i t t e d adequately. The v a l u e s of a were approximately the same, w h i l e k Q was lower. Q u a l i t a t i v e l y i t can be seen t h a t t h i s should be so, s i n c e the one-isotope curves tend t o zero more q u i c k l y w i t h i n c r e a s i n g k Q i . ; f i t t i n g the p o i n t s t o a curve of the same a "pushes" the experimental T versus l o g ( i ) p l o t t o the l e f t , i . e. s m a l l e r k . The f r a c t i o n a l decrease of k^ between the o o one-isotope and two-isotope f i t s depends on the a b s o r p t i o n of the s p e c t r a l l i n e and i s l a r g e r the hi g h e r the a b s o r p t i o n . The v a l u e of f p e l depends on the k Q v a l u e s from two s p e c t r a l l i n e s . Thus f r e l f o r the one-isotope f i t w i l l be g r e a t e r or l e s s than the v a l u e computed from the two-Isotope curves depending on whether the a b s o r p t i o n of the r e f e r e n c e l i n e i s 6 8 . g r e a t e r or l e s s than the o t h e r l i n e . T h i s observed d i f f e r e n c e i n f* ^ w a s n 0 g r e a t e r than approximately 5% and was l e s s f o r low k . o 5.4 D i s c u s s i o n of R e s u l t s and E r r o r s of A b s o r p t i o n Method 5.4.1 F i c t i t i o u s Line-Shapes One of the t h i n g s to be n o t i c e d from Tables 2 and 4 Is the v a r i a t i o n of the v a l u e of a over the c r o s s - s e c t i o n of the a b s o r p t i o n tube. Under the c o n d i t i o n s of t h i s experiment, the r e l a t i v e i n c r e a s e of a from R = 1.5 mm t o R = 6.5 mm was g e n e r a l l y about two, going from approximately 1 to 2 f o r the S g l i n e s and approximately .6 to 1.3 f o r the and s^ l i n e s ; a p p a r e n t l y the temperature changes by a f a c t o r of about f o u r i n t h i s d i s t a n c e . Bamberger /31/ has measured the Doppler widths of s p e c t r a l l i n e s emitted by a h y d r o g e n - h e l i u m - f i l l e d G e i s s l e r tube and concluded the temperature of the e m i t t i n g gas c o r - responded to the w a l l temperature of the tube. Assuming, t h e r e f o r e t h a t both the source and the o u t e r r e g i o n s of the absorber a r e a t room temperature, a f o u r - f o l d Increase i n temperature i n the absorber would y i e l d a temperature of 1200°K at the c e n t r e . T h i s seems h a r d l y p o s s i b l e . Irwin /12/ d i d not observe t h i s i n h i s l i n e - w i d t h measurements on a s i m i l a r a b s o r p t i o n tube. Ecker and Z o l l e r /32/ have c o n s i d e r e d t h e o r e t i c a l l y the temperature v a r i a t i o n s a c r o s s a c y l i n d r i c a l p o s i t i v e column -69- and f o r the column used i n t h i s experiment, a c u r r e n t of about 4 amps would be r e q u i r e d to get such a v a r i a t i o n ; they p r e d i c t a constant temperature, equal t o the w a l l temperature, w i t h the o p e r a t i n g c o n d i t i o n s used here. G r a n t i n g t h a t both the absorber and e m i t t e r should have temperatures c l o s e to room temperature, the v a l u e s of a should then be approximately u n i t y , c o n t r a r y t o what was found i n f i t t i n g the data t o the t h e o r - e t i c a l curves.. The reason f o r these r e s u l t s i s p r o b a b l y t h a t the s p e c t r a l l i n e s do not have a pure Doppler p r o f i l e . Lang /33/ has measured t h e s p e c t r a l d i s t r i b u t i o n o f the neon X5852 l i n e as a f u n c t i o n of p r e s s u r e and showed t h a t a t p r e s s u r e s o f the or d e r of 1 mm Hg, Doppler broadening i s the dominant p r o c e s s . I t i s expected, however, that t h e r e w i l l be some d e v i a t i o n , p a r t i c u l a r l y at the edges of the s p e c t r a l l i n e , due to v a r i o u s combinations of n a t u r a l broadening, and c o l l i s i o n and Stark broadening, as w e l l as some s e l f - a b s o r p t i o n of r a d i a t i o n from the source. Indeed, s e l f - a b s o r p t i o n seems t o be p r e s e n t , as mentioned i n s e c t i o n 4.2. S l i g h t d e v i a t i o n s from a pure Doppler d i s t r i b u t i o n i n the wings of the l i n e can be s i g n i f i c a n t when a b s o r p t i o n i s l a r g e ; most of the c e n t r e of the l i n e i s "absorbed out" and a l a r g e p a r t o f the t o t a l amount of the l i g h t t r a n s m i t t e d comes from the non-Doppler wings. (The i s o t o p e e f f e c t d i s c u s s e d In s e c t i o n 2.1.3 i s a comparable p r o c e s s . ) A s o l u t i o n of t h i s problem woultt be to measure the s p e c t r a l d i s t r i b u t i o n by the standard methods such as a h i g h - r e s o l u t i o n s p e c t r o g r a p h o r Pabry-Perot i n t e r f e r o m e t e r . In -70- a c t u a l p r a c t i c e i t would be a time-consuming and t e d i o u s pro- cedure t h a t would have t o be undertaken on each l i n e and f o r each operating c o n d i t i o n . A l s o i t i s d o u b t f u l whether, w i t h the equipment at hand, the accuracy of the measurements would be s u f f i c i e n t l y h i g h to a l l o w more a c c u r a t e a n a l y s i s than has been done here. Lang a p p a r e n t l y has an accuracy of o n l y about 20% i n the wings of the l i n e . A-reasonable e x p l a n a t i o n of the good f i t of the exper- imental p o i n t s to the Doppler t h e o r e t i c a l curves c o u l d be t h a t t h e r e i s non-Doppler broadening pr o c e s s e s p r e v a i l i n g which approximately match a t r a n s m i s s i o n curve f o r pure Doppler broaden- i n g having the parameters k Q and a c o r r e s p o n d i n g to those c i t e d i n Tables 2, 4, and 6. That t h i s c o u l d happen i s seen by c o n s i d e r i n g two s p e c i f i c examples of non-Doppler l i n e - s h a p e s s a s e l f - a b s o r b e d . Doppler d i s t r i b u t i o n , and a V o i g t ( i . e. Doppler p l u s n a t u r a l ) d i s t r i b u t i o n . Numerical c a l c u l a t i o n s f o r a s i n g l e - i s o t o p e gas have been performed f o r these two d i s t r i b u t i o n s and t h e r e s u l t s a r e g i v e n i n Appendix 5» I t i s shown i n the appendix t h a t V o i g t and s e l f - a b s o r b e d Doppler p r o f i l e s can be found which f i t the t r a n s m i s s i o n curves of a pure Doppler p r o f i l e . Because of the u n c e r t a i n t y i n the a c t u a l l i n e - s h a p e s and parameters p e r t a i n i n g to. the source and the absorber, no q u a n t i t a t i v e c o n c l u s i o n s can be drawn from these c a l c u l a t i o n s . Q u a l i t a t i v e l y ; they show the p o s s i b i l i t y of e q u i v a l e n c e between non-Doppler l i n e - s h a p e s and " f i c t i t i o u s " Doppler l i n e - s h a p e s . A more d e t a i l e d c a l c u l a t i o n p e r t a i n i n g to a s e l f - a b s o r b e d Doppler -71- source was undertaken In the appendix/ The r e s u l t s i n d i c a t e t h a t f r e-|_ o b t a i n e d from f i t t i n g t r a n s m i s s i o n s u s i n g a s e l f - absorbed source i s not very dependent on d e v i a t i o n s from a pure Doppler l i n e - s h a p e , a l t h o u g h the k Q and a f o r the two p r o f i l e s d i f f e r markedly. T h i s i s i n accordance w i t h o b s e r v a t i o n , s i n c e In the experiment the va l u e of f p e ^ was g e n e r a l l y independent of absorber and source c u r r e n t , and method of measurement. T h i s i s the main j u s t i f i c a t i o n i n the acceptance of the v a l u e s of f p e l w h i l e r e j e c t i n g the v a l u e s of k Q and a; w i t h v a r i a t i o n of c o n d i t i o n s and consequent v a r i a t i o n of the " f i c t i t i o u s " v a l u e s of k Q and a, f r e l remained constant and g e n e r a l l y independent of these v a r i a t i o n s . The r e s u l t s of the i n t e n s i t y measurements d i s c u s s e d i n s e c t i o n 4.2 show t h a t most l i n e s emitted by the source have some s e l f - a b s o r p t i o n so that the l i n e s a r e d e f i n i t e l y not pure Doppler. The e f f e c t of the d e v i a t i o n s from a non- Doppler d i s t r i b u t i o n i s s m a l l e r the l e s s the a b s o r p t i o n so that more weight should be p l a c e d on f p e ^ f o r l i n e s which are l e a s t a b s o r b i n g . Because of the u n c e r t a i n t y i n the v a l u e of k Q , v a l u e s of the r e l a t i v e d e n s i t i e s of the atoms i n the lower s t a t e s as a f u n c t i o n of R c a l c u l a t e d from e q u a t i o n (2.10) are i n doubt and computations have not been c a r r i e d out. 5.4.2 D i s c u s s i o n of E r r o r s The e s t i m a t i o n of e r r o r s i s u s u a l l y d i f f i c u l t and the s i t u a t i o n here i s no e x c e p t i o n 0 The most l i k e l y and most -72- important causes of e r r o r i n t h i s experiment are d i s c u s s e d below, and e s t i m a t i o n s of the magnitude of the e r r o r s are made. .1 S t i m u l a t e d Emission It has been assumed t h a t s t i m u l a t e d emission i n the a b s o r p t i o n tubes i s n e g l i g i b l e . I f i t i s not, k Q (see equation (2.2) must be c o r r e c t e d by m u l t i p l y i n g by the f a c t o r (see f o r example, M i t c h e l l and Zemansky /19/0 Q j ^ = 1 - NjS^/Nj^gj. and Nj a r e the atom d e n s i t i e s of the lower and upper l e v e l s , r e s p e c t i v e l y , and g^ and gj the c o r r e s p o n d i n g s t a t i s t i c a l weights. Irwin /12/ has measured the r a t i o N^/^ i n a d i s c h a r g e tube of dimensions s i m i l a r t o t h a t used here and c o n t a i n i n g neon at the same p r e s s u r e (2 mm Hg). At I = 15 ma, he found t h e l a r g e s t c o r r e c t i o n f o r s t i m u l a t e d emission was f o r the X.5852 l i n e which amounted to = .996. T h i s c o r r e c t i o n i s n e g l i g i b l e compared to the accuracy of the measurement of k Q . As mentioned i n s e c t i o n 4.2, Ladenburg's /7 / r e s u l t s i n d i c a t e the absence of s t i m u l a t e d emission i n the G e i s s l e r tube. .2 T r a n s m i s s i o n Accuracy Throughout the course of t h i s experiment the l i n e a r i t y of the i n t e n s i t y - m e a s u r i n g c i r c u i t was checked p e r i o d i c a l l y by v a r y i n g the l i g h t i n t e n s i t y i n t o the monochromator u s i n g n e u t r a l d e n s i t y f i l t e r s . The output was never found to d e v i a t e from l i n e a r i t y by more than 2%a The c h a r t r e c o r d e r c o u l d u s u a l l y be -73- read to b e t t e r than 2% and t r a n s m i s s i o n s were measured t h r e e times f o r each d i s c h a r g e l e n g t h and the average taken f o r subsequent a n a l y s i s . E r r o r s here should be random and l e s s than about 2%. .3 Curve P i t t i n g and Slopes 2 2 In f i t t i n g the experimental p o i n t s of T or d(R T)/d(R ) to the t h e o r e t i c a l curves, i t was found t h a t the u n c e r t a i n t y of the f i t t i n g was u s u a l l y l e s s than 5% of k Q so t h a t maximum u n c e r t a i n t y of f p e l would be l e s s than 10% f o r a s i n g l e c a l c u l a - t i o n of f r e ^ . Over s e v e r a l measurements the standard d e v i a t i o n of f r e i s hould be no g r e a t e r than 5%. The e r r o r i n v o l v e d i n 2 2 computing d(R T)/d(R ) i n the c i r c u l a r a p e r t u r e method i s hard to estimate and depends oh the q u a l i t y of the approximation to a second order polynomial assuming a c c u r a t e experimental p o i n t s . The apparent v a r i a t i o n of f r e ] _ at l a r g e R f o r some wavelengths (see Table 1) and the poor f i t t i n g i s no doubt due i n p a r t t o c a l c u l a t i n g the s l o p e s by the p o l y n o m i a l method. .4 O p t i c a l Alignment When measurements were repeated on a s e r i e s of spec- t r a l l i n e s a f t e r a short p e r i o d of time such as the o r d e r of a day, without changing the alignment of the a b s o r p t i o n tube, source, or the o p t i c s , i t was found t h a t the v a l u e s of k Q would g e n e r a l l y agree w i t h i n 5%, whereas i f r e p e t i t i o n took p l a c e a f t e r a p e r i o d of a few weeks or the equipment was d i s t u r b e d and r e a l i g n e d , t h e r e would be a l a r g e r d e v i a t i o n between the v a l u e s of k Q . I t can be seen, however, t h a t the v a l u e s of f r e 1 _ c o r r e s - ponding t o these cases do not d e v i a t e i n n e a r l y as d r a s t i c a manner, s y s t e m a t i c a l l y or otherwise, i n d i c a t i n g an Independence to alignment. Ther e f o r e the v a r i a t i o n of f r e ^ due t o alignment should be g e n e r a l l y of a s t a t i s t i c a l nature and the e r r o r i s d i m i n i s h e d by i n c r e a s i n g the number of measurements, r e a l i g n i n g the o p t i c s between each measurement. .5 End-Window R e f l e c t i o n s F r i s h and Bochkova / 3 V measured a b s o r p t i o n c o e f f i c i e n t s of a gaseous d i s c h a r g e by p l a c i n g the d i s c h a r g e tube between two p a r a l l e l m i r r o r s and measuring the l i g h t output a f t e r the emitted r a d i a t i o n had undergone m u l t i p l e r e f l e c t i o n s through the a b s o r b i n g gas. Conceivably i n t h i s experiment the r a d i a t i o n from the source c o u l d be r e f l e c t e d back and f o r t h between the end-windows undergoing a b s o r p t i o n w i t h each pass and i n t r o d u c e an e r r o r i n the a n a l y s i s as g i v e n p r e v i o u s l y . I f the windows are c o n s i d e r e d as s i n g l y - r e f l e c t i n g p l a n e s u r f a c e s of r e f l e c t i v i t y r , and are p o s i t i o n e d p e r p e n d i - c u l a r to the a x i s of the tube, a procedure s i m i l a r to t h a t g i v e n i n the above r e f e r e n c e shows that the t r a n s m i s s i o n i s g i v e n by T . S „ y ^ « j j . . . . 5 . D -75- For g l a s s , r = .04 and the f a c t o r ( l - r 2 ) / ( l - r 2 e ~ 2 k ^ ^ ) thus approaches .998 f o r h i g h a b s o r p t i o n c l o s e to the c e n t r e of the l i n e and i s e f f e c t i v e l y u n i t y elsewhere. Even i f the windows of the a c t u a l a b s o r p t i o n tube were p a r a l l e l and p e r p e n d i c u l a r to the a x i s , which i s probably not the case, the e r r o r due to m u l t i p l e r e f l e c t i o n s i s n e g l i g i b l e . .6 Uniform Discharge C o n d i t i o n s The c o n f i g u r a t i o n of the a b s o r p t i o n tube was chosen w i t h the e l e c t r o d e s l o c a t e d away from the main column (see F i g . 5 ) . T h i s ensured t h a t the v a r i o u s glow and dark space r e g i o n s a s s o c i a t e d w i t h the anode and cathode would not p e n e t r a t e i n t o the a b s o r b i n g column, and o n l y the p o s i t i v e column r e g i o n would be a s s o c i a t e d w i t h the a b s o r b i n g p r o c e s s . Uniform c o n d i t i o n s a l o n g the l e n g t h of t h e a b s o r p t i o n tube have been assumed; t h i s assumption depends on the c o n d i - t i o n s o c c u r r i n g i n the p o s i t i v e column. In a low-pressure glow discharge t h i s p o s i t i v e column i s g e n e r a l l y c o n s i d e r e d as a r e g i o n of constant and equal c o n c e n t r a t i o n s of p o s i t i v e ions and e l e c t r o n s and i s c h a r a c t e r i z e d by a constant low v o l t a g e - g r a d i e n t . T h i s s i t u a t i o n i s notalways so, however, and under c e r t a i n c o n d i t i o n s a l t e r n a t e b r i g h t and dark s t r i a t i o n s appear which may be s t a t i o n a r y o r move a l o n g the column /35/» G e n e r a l l y the motion of these running s t r i a t i o n s i s so r a p i d t h a t the column appears uniform. Donahue and Dieke /36Y contend t h a t these s t r i a t i o n s are o n l y e x c e p t i o n a l l y absent and p l a y 76 an e s s e n t i a l p a r t i n the mechanism of the glow d i s c h a r g e . Krebs /37/ found, u s i n g neon i n a s i m i l a r tube t o that used here, that the s t r l a t i o n s o n l y disappeared above a c u r r e n t of 2 amps. Rudimentary examination of the p o s i t i v e column w i t h a p h o t o c e l l confirmed the e x i s t e n c e o f running s t r i a t i o n s i n t h i s a b s o r p t i o n tube. A r e l a t i v e l y l a r g e s l i t (approximately 2mm) i n f r o n t of the p h o t o c e l l l i m i t e d the r e s o l u t i o n but y i e l d e d a p e r i o d of o s c i l l a t i o n of approximately 1.8 msec and an i n t e n s i t y - modulation o f no l e s s than 25$. Because of the r e l a t i v e l y short p e r i o d of the s t r i a t i o n s compared t o the a b s o r p t i o n time («10 seconds or g r e a t e r ) , l o c a l v a r i a t i o n s of c o n d i t i o n s i n the p o s i t i v e column would be averaged out i n any measurements. One oth e r v a r i a t i o n of c o n d i t i o n s i n the p o s i t i v e column was observed when a photograph of the tube was taken; when the d i s c h a r g e had an unused e l e c t r o d e s i t u a t e d between the two c o n d u c t i n g e l e c t r o d e s , a s l i g h t d i f f u s i o n o f the d i s c h a r g e i n t o the unused side-arm took p l a c e . Although not n o t i c e a b l e by eye, t h i s m a n i f e s t e d i t s e l f by a s l i g h t decrease of i n t e n s i t y above the side-arm when observed w i t h a camera. F i g , 16 shows t h i s I n t e n s i t y decrease. The e f f e c t of t h i s would be t o y i e l d a s h o r t e r e f f e c t i v e a b s o r p t i o n l e n g t h , a l t h o u g h i n the a n a l y s i s , i t s presence was i g n o r e d . .7 E n d - E f f e c t s of A b s o r p t i o n Column So f a r no mention has been made co n c e r n i n g t h e e r r o r Introduced by the rounding of the d i s c h a r g e at the ends. -77- (see F i g . 16). I t i s evident from F i g . 17 that Incident rays Arrows i n d i c a t e p o s i t i o n of side-arms, (opaque segments near ends of d i s c h a r g e are clamps). F i g . 16 Photograph of Discharge such as CD are more s t r o n g l y absorbed .in the end r e g i o n s than the d i a m e t r i c a l l y o p p o s i t e rays such a AB. T h i s i s due t o the more dense end r e g i o n t h a t CD passes through and the l o n g e r a b s o r b i n g l e n g t h . By assuming an a b s o r p t i o n l e n g t h equal to the d i s t a n c e between the c e n t r e s of the e l e c t r o d e s , the e r r o r s f o r the two rays shown i n the diagram are i n o p p o s i t e d i r e c t i o n s and tend to c a n c e l each o t h e r . However u s i n g t h i s a b s o r p t i o n l e n g t h may l e a d to an e r r o r f o r the t o t a l e f f e c t of a l l the rays p a s s i n g through the a b s o r p t i o n tube. For the annular i n c i d e n t beam, two attempts have been made to c a l c u l a t e the e f f e c t the ends have on the t r a n s m i s s i o n . The d e t a i l s of these c a l c u l a t i o n s are given i n Appendix 6. I t i s assumed that the d i s c h a r g e bends through 90° In a c i r c u l a r p a t h at each end, and as a f i r s t approximation t h a t k and a are constant i n the end r e g i o n . With these s i m p l i f y i n g assumptions the r e s u l t s of the F \ G . 17 E N D - E F F E C T S O F D I S C H A R G E calculations show that no correction for the ends is necessary. Consequently no correction was applied. The basic fault in this analysis is that kQ is not constant in the end zone. A better approximation i f the pro- f i le of k across the diameter of the tube were known would be o to assume this profile is preserved in "going around the corner" and perform a triple numerical Integration over the frequency, length, and angular position on the annulus to calculate the error. Because the form and magnitude of kQ is not well known and because of the complexity of the calculations, no attempts were made in this direction. It is likely that for the less highly absorbing lines the end-effect error is small and of course is highest for the 10 cm discharge length. As mentioned previously the 10 cm point sometimes did not seem to f i t very well in the curve- fitt ing, although whether this point was included or not made l i t t l e difference in the value of f r e l determined. As a result of the analysis, no corrections for end- effects for the method of circular apertures was applied either. .8 Estimated Errors The averaged values of the relative oscillator strength and a numerical estimate of the errors are given in Table 7. The values of the errors given represent the ratio of the standard deviation of f p e l to the value of f r e l , in percent. These hopefully account for contributions to random errors from such 80- TABLE 7 f r e l AND ESTIMATED ERRORS (The number i n parentheses f o l l o w i n g the e r r o r i s the number of measurements) \d) f r e l Estimated E r r o r s2 Lines 5852 6599 6678 671? 6929 1.00 1.19 2.08 1.20 1.75 Reference 5.5 ,̂(4) 2.3^(4) 2.6#(4) 4.0^(4) s 3 Lines 6163 6266 6533 1.18 1.85 1.00 2.3^(5) 1.5^(5) Reference s 4 Lines 6074 6096 6304 6383 6506 7245 .589 .861 .263 1-00 1.42 .416 2.9#(6) 0.9^(5) 3.3% (*> Reference 1.6^(7) 6.1^(7) s 5 Lines 5882 5945 6143 6217 633^ 6402 7032 .347 .592 1.41 .244 1.00 2.64 1.00 10 %(1) 10 %{1) 1.4^(5) 10 %(1) Reference 3.7%(6) 7.5^(2) causes as alignment, curve f i t t i n g , t r a n s m i s s i o n accuracy e t c . and are made from a p u r e l y s t a t i s t i c a l approach. The standard d e v i a t i o n a r e p r e s e n t i n g the e r r o r i s the best estimate t o the standard d e v i a t i o n of an i n f i n i t e number of measurements o f f p e l . The standard d e v i a t i o n s c a l c u l a t e d from the n measurements 2 of f r e l a c t u a l l y made i s r e l a t e d t o a by the r e l a t i o n a = (r)/(ri-i))s*~ (see f o r example, Topping /38/). The v a l u e of n f o r each s p e c t r a l l i n e i s i n d i c a t e d i n b r a c k e t s f o l l o w i n g the e r r o r . For l i n e s which were measured on l y once or twice the v a l u e s of 10% and 7*5% r e s p e c t i v e l y were a s s i g n e d t o the e r r o r . For most l i n e s s y s t e m a t i c e r r o r s a re b e l i e v e d t o be s m a l l and are not Included i n t h i s a n a l y s i s . Non-Doppler l i n e - shapes and incomplete i s o t o p e s e p a r a t i o n (see Appendix .1) c o u l d c o n t r i b u t e a s i g n i f i c a n t s y s t e m a t i c e r r o r i f a b s o r p t i o n i s h i g h . By f a r the most h i g h l y a b s o r b i n g l i n e i s \6402 so tha t the e r r o r estimate g i v e n here may be o p t i m i s t i c . Futher d i s c u s s i o n i s g i v e n i n s e c t i o n 5«6. 5.5 Emission 5.5-1 R e l a t i v e T r a n s i t i o n P r o b a b i l i t i e s For the c a l c u l a t i o n of r e l a t i v e t r a n s i t i o n p r o b a b i l i - t i e s equations (2.14) and (4.2) y i e l d A ( H VvjA) = JARtfo^Jfa.Jt) . . . ( 5 . 2 ) 82 where . Table 8 g i v e s the v a l u e s of the f a c t o r s i n e q u a t i o n (5.2) f o r each l i n e p a i r and the f i n a l two columns l i s t s A r e l and the estimated e r r o r . Only two of the l a s t t h r e e p a i r s of l i n e s are independent so ^^(6383/6533) i s merely c a l c u l a t e d from the two v a l u e s immediately p r e c e e d i n g I t . 5.5.2 - D i s c u s s i o n of E r r o r s The e r r o r a n a l y s i s i s more amenable t o numerical e s t i m a t i o n here than i n the a b s o r p t i o n experiment; the main sources of e r r o r are d i s c u s s e d below and estimates g i v e n . .1 I n t e n s i t y - R a t i o E r r o r The e r r o r s i n v o l v e d i n measurelng R^ (X.1SX.2) a n d R c ^ l , J l 2 ^ f o r e a c n W a v e l e n S t h P a i r were estimated from the f r a c t i o n a l standard e r r o r of the t h r e e s e t s of measurements. T h i s amounted from between 1% and 3% f o r R| , and k% to 5$ f o r .R . f o r XA.6217, 6383, and 6533 was d e r i v e d from the c c curve of F i g . 11 r a t h e r than by d i r e c t measurement and the e r r o r was estimated at 6% t o 9$. .2 E m l s s i v i t y and Temperature E r r o r The e r r o r s "in £.U,T. ) as g i v e n by Larrabee /26/ amd de Vos /25/ TABLE 8 RELATIVE TRANSITION PROBABILITIES FROM INTENSITY MEASUREDSITT3 X 1 ! X 2 x n / x 2 •->. \ % ( X 1 » X 2 } R;u l f x 2 ) J ( X 1 , T t ) A r e l ( X 1 , X 2 ) J ( X 2 , T t ) 6096/6678 .913 1.02 17.5 10.8 .535 .802 k.5% 6164/6599 .934 1.01 6.47 6.49 .626 .591 4.1$ 6334/6506 .974 1.00 1.63 2.32 .334 .572 5.6% 6217/6383 .974 1.00 .542 1.83 x .832 .241 6.2% 6217/6533 .952 1.01 3.37 3.95 .711 .668 8.0% 6383/6533 2.77 8.9% 84- are both approximately a l t h o u g h t h e i r v a l u e s d i f f e r by about 2.5$, i t i s estimated here t h a t the e r r o r i n the r a t i o £i\j_,Tt)/£ (^2Tt^ i s n o S r e a t e r t h a n l%e The b r i g h t n e s s temperature of the tungsten lamp measured w i t h the pyrometer was found t o be 1768°K"143'K where the u n c e r t a i n t y f a c t o r i s the standard d e v i a t i o n of 10 measurements. The t r u e temperature T t was c a l c u l a t e d t o be 1900°K + l 6 ° K where the d i f f e r e n c e between u s i n g Larrabee's o r de Vos*s v a l u e s of £ amounted to only 4° K. From t h i s temperature and i t s u n c e r t a i n t y , the u n c e r t a i n t y i n J U-pTj.)/j(X 2»T t) was l e s s than 1% f o r a l l wavelengths used. .3 P i l e - U p E r r o r The second type of p i l e - u p e r r o r d i s c u s s e d i n s e c t i o n 4.3.2 can be roughly estimated In the f o l l o w i n g way. n i s the average number of p h o t o - e l e c t r o n p u l s e s a r r i v i n g at the anode of the p h o t o m u l t i p l i e r p e r sampling p e r i o d , and hence i s p r o p o r t i o n a l to the I n t e n s i t y . Gadsden /39/ has shown t h a t the time d i s t r i b u t i o n of the a r r i v a l of p u l s e s i s c l o s e l y P o i s s o n , so t h a t the p r o b a b i l i t y of no p u l s e s a r r i v i n g d u r i n g the sampling p e r i o d i s e " n . The p r o a b i l i t y of one o r more a r r i v i n g i s l - e " n . The apparatus used here counts how manysamples c o n t a i n one o r more p u l s e s . I f t h e r e a r e C counts f o r the normal 4850 samples, then i t f o l l o w s from above t h a t l - e ~ n = C/4850 a.o(5.3) T h i s equation may be s o l v e d e x a c t l y f o r n but f o r a low count (and t h e r e f o r e low n) equation (5*3) becomes approximtely -85- n(l-n / 2 ) = C/4850 and f o r the C = 250 the e r r o r i n assuming n = C/4850 i s a p p r o x i - mately 2%, The more equal the counts are f o r each wavelength p a i r , the l e s s i s the e r r o r of the r a t i o s of the counts; by u s i n g the o p t i c a l f i l t e r s t o do t h i s and t o keep the count below 250, the p i l e - u p e r r o r was expected t o have been kept below 1$. An o b j e c t i o n may be r a i s e d t h a t a P o i s s o n d i s t r i b u - t i o n i s i n v a l i d f o r t h i s a n a l y s i s because the sampling width (.3 nsec) i s s m a l l e r than an i n d i v i d u a l p h o t o - e l e c t r o n p u l s e w i d t h (approximately 5 n s e c ) . However i f the w i d t h of the p u l s e s at the v o l t a g e l e v e l set by the d i s c r i m i n a t o r has an average v a l u e ft, on average a count i s r e g i s t e r e d whenever the c e n t r e of a p u l s e f a l l s w i t h i n °Q/2 of e i t h e r edge of the sampling gate. T h i s g i v e s an e f f e c t i v e sampling width of ft + w, where w i s the w i d t h of the sampling gate, and the P o i s s o n d i s t r i b u t i o n i s v a l i d . .4 R a d i a t i o n Trapping One f a c t which has been t a c i t l y assumed i n measuring the e mission i n t e n s i t i e s i s the absence of s e l f - a b s o r p t i o n of the l i n e s . T h i s demands t h a t the number d e n s i t y of atoms of the lower l e v e l o f the t r a n s i t i o n be s m a l l . C o n s i d e r a t i o n of the e x c i t a t i o n and d e - e x c i t a t i o n p r o c e s s e s o c c u r r i n g i n the gas a f t e r s w i t c h i n g on the e l e c t r o n 86- beam can g i v e an upper bound on the number d e n s i t i e s of the lower l e v e l s of the t r a n s i t i o n s i n the f o l l o w i n g way. The important processes which populate the f o u r l e v e l s o f the 2p^3s c o n f i g u r a t i o n a re e x c i t a t i o n of atoms i n the ground s t a t e by c o l l i s i o n w i t h e l e c t r o n s i n the e l e c t r o n beam and decay from upper l e v e l s . The processes which de-populate the l e v e l s a re spontaneous t r a n s i t i o n s to the ground s t a t e w i t h accompanying em i s s i o n of r a d i a t i o n , d i f f u s i o n t o w a l l s of the c o n t a i n e r and subsequent d e - e x c i t a t i o n , and a b s o r p t i o n of r a d i a t i o n of approp- r i a t e frequency. As w e l l , because the energy d i f f e r e n c e s between the f o u r l e v e l s a r e s m a l l , c o l l i s i o n s w i t h n e u t r a l atoms can b r i n g about t r a n s i t i o n s between these e x c i t e d s t a t e s . T h i s r e d i s t r i b u t e s the p o p u l a t i o n s among the f o u r l e v e l s but does not change the t o t a l p o p u l a t i o n of the c o n f i g u r a t i o n . The upper bound on the number d e n s i t y can be obtained by c o n s i d e r i n g o n l y the p o p u l a t i n g p r o c e s s e s mentioned above and d i s r e g a r d i n g the d e - p o p u l a t i n g p r o c e s s e s . Imprisonment of the resonance r a d i a t i o n need not be c o n s i d e r e d as a p o p u l a t i o n process s i n c e every atom r a i s e d t o an e x c i t e d energy s t a t e by s e l f - a b s o r p t i o n r e q u i r e s the t r a n s i t i o n of at l e a s t one atom i n the r e v e r s e d i r e c t i o n , and t h a t atom w i l l have a l r e a d y been "counted". Por p o p u l a t i o n by e l e c t r o n c o l l i s i o n the t o t a l number d e n s i t y of the c o n f i g u r a t i o n i s governed by the f o l l o w i n g equation — T = N e N 0 Q T ...(5.4) d t 4 where NL = E N , Is the number d e n s i t y o f a l l f o u r l e v e l s , Q„ i s 8 ? the t o t a l c r o s s - s e c t i o n f o r e x c i t a t i o n t o any of the l e v e l s In the c o n f i g u r a t i o n , N Q i s the number d e n s i t y of atoms i n the ground s t a t e , and N Is the f l u x of e x c i t a t i o n e l c t r o n s per u n i t area In the e l e c t r o n beam. The o t h e r p o p u l a t i n g process i s c a s c a d i n g from h i g h e r l e v e l s and i n neon the only d i r e c t c a s c a d i n g i s from the 2p-^3p c o n f i g u r a t i o n . The c r o s s - s e c t i o n s f o r each of the ten l e v e l s i n t h i s c o n f i g u r a t i o n has been measured by F r i s h and Revald /40/ -19 2 and are a l l approximately 10 7 cm . I f a l l these atoms cascade i n t o the 2p^3s c o n f i g u r a t i o n the e f f e c t i v e c r o s s - s e c t i o n f o r p o p u l a t i n g the lower c o n f i g u r a t i o n s by t h i s method i s a p p r o x i - mately 10" cm . According to Revald / 4 l / the c o n f i g u r a t i o n -17 2 c r o s s - s e c t i o n f o r d i r e c t e x c i t a t i o n i s approximately 10 'cm so t h a t c a s c a d i n g c o n t r i b u t e s an e f f e c t which i s one order of magnitude s m a l l e r . D e t a i l e d v a l u e s of c r o s s - s e c t i o n s f o r the h i g h e r c o n f i g u r a t i o n s are not known, but i t i s probable that the t o t a l e f f e c t of c a s c a d i n g f o r the e x c i t a t i o n time used here can a t best make a c o n t r i b u t i o n which i s comparable t o t h a t from d i r e c t e x c i t a t i o n . Thus f o r rough c a l c u l a t i o n s , e q u a t i o n (5.4) i s v a l i d f o r e s t i m a t i n g p o p u l a t i o n d e n s i t i e s . From t h i s e q u a t i o n i t f o l l o w s N T « ^e^o^T* 8 3 1 ( 1 ^ d e r t h e experimental 17 -2 -1 c o n d i t i o n s used here, the approximate v a l u e s N » 10 'cm sec , N 0 « 101^cm"-^, Q T » 10~ 1^cm 2, and the e x c i t a t i o n time t = 200 nsec y i e l d s N,p(t)£ 10 cm~^ f o r the maximum number d e n s i t y of the lower c o n f i g u r a t i o n . f o r each l e v e l w i l l of course be s m a l l e r . For Doppler l i n e s , the e f f e c t of s e l f - a b s o r p t i o n i s 88 g i v e n by the f a c t o r S of equations (2.15) or (2.16). Rough c a l c u l a t i o n s g i v e k Q from equation (2.2) as and as ̂  f o r the e l e c t r o n gun i s approximately 1 cm, k '^-^ 10" Reference to the t a b l e i n Appendix 2 shows t h a t f o r t h i s v a l u e of k P the s e l f - a b s o r p t i o n i s n e g l i g i b l e . 5.6 Absolute T r a n s i t i o n P r o b a b i l i t i e s As mentioned b e f o r e , by combining the r e l a t i v e t r a n s - i t i o n p r o b a b i l i t i e s between l i n e s having the same lower l e v e l ( a b s o r p t i o n ) w i t h those having the same upper l e v e l (emission) a complete set of r e l a t i v e t r a n s i t i o n p r o b a b i l i t i e s may be obt a i n e d . Here, only t h r e e r e l a t i v e emission p r o b a b i l i t i e s a re needed to t i e the f o u r l o w e r - l e v e l groups of l i n e s t o g e t h e r . There were however, f i v e independent measurements made i n emiss i o n thus a l l o w i n g a s e l f - c o n s i s t e n c y check. I t was found that the r e s u l t s were c o n s i s t e n t w i t h i n the estimated e x p e r i - mental e r r o r and so a weighted a v e r a g i n g method was employed to a r r i v e at the f i n a l r e l a t i v e t r a n s i t i o n p r o b a b i l i t i e s (see s e c t i o n 5.6.1). The f i n a l s et of r e l a t i v e p r o b a b i l i t i e s are pl a c e d on the a b s o l u t e s c a l e u s i n g one l i f e t i m e measurement r e c e n t l y made i n t h i s l a b o r a t o r y and the a b s o l u t e v a l u e s a re di s c u s s e d and compared w i t h those of ot h e r workers In s e c t i o n s 5.6.2 and 5.6.3. - 8 9 5 . 6.1 " M u l t i - P a t h " Method :of R e l a t i v e T r a n s i t i o n P r o b a b i l i t i e s With f i v e independent r e l a t i v e i n t e n s i t y measurements i t i s p o s s i b l e to o b t a i n up to f i v e independent ways of c a l c u - l a t i n g A y e ^ f o r one p a r t i c u l a r l i n e , w i t h X.5852 chosen as the r e f e r e n c e l i n e . The d i f f e r e n t " r o u t e s " are b e s t seen when the s p e c t r a l l i n e s i n v e s t i g a t e d are l a y e d out i n a t r a n s i t i o n a r r a y w i t h the lower l e v e l s l a b e l i n g the rows, and the upper l e v e l s l a b e l i n g the columns. P i g . 18 shows t h i s a r r a y w i t h the wavelengths marked a t the i n t e r s e c t i o n of the a p p r o p r i a t e rows and columns. The v e r t i c a l arrows connect the l i n e - p a i r s i n v o l v e d i n the emission measurement and hence I n d i c a t e the the I n t e r c o n n e c t i o n between the rows. To u t i l i z e completely the i n f o r m a t i o n gained from the I n t e n s i t y measurements, the f o l l o w i n g method was adopted to c a l c u l a t e the e n t i r e set of t r a n s i t i o n p r o b a b i l i t i e s , r e l a t i v e to X.5852. S t a r t i n g at the wavelength i n q u e s t i o n , one proceeds to \5852 by moving h o r i z o n t a l l y a long the rows and v e r t i c a l l y a l o n g the arrows of F i g . 18 by as many independent psths as p o s s i b l e . Since the t r a n s i t i o n p r o b a b i l i t i e s i n a p a r t i c u l a r row are c a l c u l a t e d r e l a t i v e t o an a r b i t r a r i l y chosen s p e c t r a l l i n e i n the row, a p a t h which jumps to more than two squares without l e a v i n g the row f i r s t i s not independent. A l s o no p a t h may pass through the same square more than once a l t h o u g h i t may "Jump" over a p r e v i o u s l y used square. The number i n parentheses f o l l o w i n g the wavelengths i n F i g . 18 i n d i c a t e s the number of Independent paths f o r t h a t wavelength. The i n d i v i d u a l sz L I M E 6539(3) t 6678(3) 67/7(5) 6266(3) 6 533&) - 63CV&) 6383(4 72+S@> £382(4) *9*5(+) 62/7(4) 6*02(4) 7£XS2(4) FIG.T8 TRANSITION ARRAY FOR MULTI-PATH METHOD - 9 1 r e l a t i v e t r a n s i t i o n p r o b a b i l i t i e s of each step i n a p a t h are used to c a l c u l a t e the p r o b a b i l i t y r e l a t i v e to X.5852. A weighted average i s then c a l c u l a t e d from these independent v a l u e s of the t r a n s i t i o n p r o b a b i l i t i e s , , The weight g i v e n t o a v a l u e obtained along a p a r t i c u l a r p a t h i s d e r i v e d from the estimated e r r o r s of the t r a n s i t i o n p r o b a b i l i t i e s of which the path i s composed. Table 9 l i s t s these e r r o r s , where the number at the i n t e r s e c t i o n of a row and column i n d i c a t e s the standard d e v i a t i o n of the r e l a t i v e t r a n s i t i o n p r o b a b i l i t y f o r the wavelengths at the head of the column and row. These d e v i a t i o n s have been ob t a i n e d from the a b s o r p t i o n and emission e r r o r a n a l y s i s and o n l y those which are r e q u i r e d here have been c a l c u l a t e d . An example of one independent, c a l c u l a t i o n f o r the X5882 l i n e i s g i v e n here f o r the p a t h 5 8 8 2 — * - 6 3 3 4 — 6 5 0 6 — » - 6096—*^6678—»-5852. The standard d e v i a t i o n aj_ f o r each step i s 10% 5*6%, 1.2%, ka5%, and 2 . 3 $ r e s p e c t i v e l y , as g i v e n i n Table 9 . The product of the r e l a t i v e p r o b a b i l i t i e s d e r i v e d from Tables 1, 3 , 5 , and 8 y i e l d A r e l (5882/5852) = .144 w h i l e s^ = (|pĵ ) g i v e s a standard d e v i a t i o n of 12.6%, The w e i g h t i n g f a c t o r t o be g i v e n the A ^ ^ a s s o c i a t e d 2 ^ w i t h a p a r t i c u l a r path j i s Wj = 1/s ^ where s j i s the s t a n d a r d d e v i a t i o n a s s o c i a t e d w i t h the path. The weighted average i s thus computed u s i n g the formula (see f o r example, Topping / 3 8 / ) . The standard d e v i a t i o n of the 5852 5882 5945 6074 6096 6143 6163 6217 6266 6304 6334 6383 6402 6506 6533 6599 6678 6717 6929 7032 7245 5852 5.5% 2.3% 2.6% 4.0% 5882 10% 10% 5945 10% 10% 6074 2.5% 2.9% 4.1% 6096 2.5% 7.5% 0.9% 1.2% 4.5% 7.6% . 6143 10% 1.4% 6163 2.3% 2.3% 4.5% 6217 10% 10% 10% 10% 6.2% 10% 8.0% 10% 6266 2.3% 1.5% 6304 7.5% 3.3% 7.5% 6334 10% 10% • 1.4% 10% 3.7% 5.6% 7.5% 6383 2.9% 0.9% 6.2% 3.3% 1.6% 8.9% 6.1% 6402 10% 3.7% 6506 4.1% 1.2% 7.5% 5.6% 1.6% 5.7% 6533 2.3% 8.0% 1.5% 8.9% 6599. 5.5% 4.1% 4.2% 2.2% 6678 2.3% 4.5% 1.8% 3.7% 6717 2.6% 4.2% 1.8% 6929 4.0% 2.2% 3.7% 7032 10% 7.5% 7245 7.6% 6.1% 5.7% TABLE 9 ESTIMATED ERRORS OF TRANSITION PROBABILITIES (EMISSION AND ABSORPTION) -93- TABLE 10 RELATIVE AND ABSOLUTE NEON TRANSITION PROBABILITIES AND OSCILLATOR STRENGTHS ^ e l xlO"' fabs ERROR % 5852 . 10.0 6.58 .113 1.6 5882 1.44 .948 .0295 13.0 59̂ 5 1.44 .948 .0503 13.0 6074 8.54 5.62 .104 10.7 6096 2.50 1.64 .152 7.0 6143 3.17 2.09 .118 11.3 6163 1.90 1.25 .214 7.6 6217 .948 .624 . 0216 13.0 6266 2.87 1.89 .334 7.5 6304 .691 .455 ; . 0449 12.4 633^ 2.11 1.39 .0832 11.2 6383 4.33 2.85 .17^ 8.4 6402 3.91 2.57 .221 11.7 6506 3.68 2.42 .256 9.1 6533 1.43 .941 .181 8.0 6599 3.17 2.09 .136 6.0 6678 3.20 2.11 .235 3.9 6717 3.01 1.98 .132* 5.3 6929 2.50 1.64 .197 7.3 7032 2.88 1.90 .0845 12.6 7245 1.39 .915 .0720 12.5 94- values of A^-^ i s used as an e s t i m a t i o n of the e r r o r of A^^CA.) and i s computed by the formula The f i n a l r e l a t i v e t r a n s i t i o n p r o b a b i l i t i e s and e r r o r s determined i n t h i s manner are l i s t e d i n Tab l e 10. The determina- t i o n of the a b s o l u t e p r o b a b i l i t i e s a l s o l i s t e d i n the t a b l e i s d i s c u s s e d i n the next s e c t i o n . 5.6.2 L i f e t i m e Measurement The l i f e t i m e of l e v e l 1 i s connected to the spontaneous t r a n s i t i o n p r o b a b i l i t i e s through the r e l a t i o n X. = I / 2 A i k •••(5-7) For the l e v e l of neon t h e r e a re onl y two allowed downward t r a n s i t i o n s , X.5852 and X.5400. Recent measurements by van Andel /17/ g i v e s the v a l u e o f 15.2 + .25 nsec. f o r the l i f e - time of t h i s l e v e l . The i n t e n s i t y of X5400 was measured t o be -2 l e s s than 10 of the I n t e n s i t y of X.5852 so t h a t n e g l e c t i n g the * c o n t r i b u t i o n of the X5400 t r a n s i t i o n t o the l i f e t i m e would amount t o an e r r o r of l e s s than \%„ Consequently the v a l u e of A a b s(5852) was taken to be (15.2 X 10'9f- (6.58 ± .11) X 107 s e c . " 1 . From t h i s a b s o l u t e v a l u e , the complete s et of a b s o l u t e v a l u e s may be computed and are shown i n Table 10. A l s o i n c l u d e d are the p o s s i b l e e r r o r s a s s o c i a t e d w i t h them and the o s c i l l a t o r s t r e n g t h s . -95- TABLE 11 COMPARISON OF ABSOLUTE TRANSITION PROBABILITIES (XIO" 7) s e c " 1 x ( i ) THIS IRWIN DOHERTY FRIEDRICHS LADENBURG EXP'T /12/ /10/ A l / /7 / 5852 6.58 6.8 3.51 6.0 12.8 5882 •948 1.15 .69 .96 2.12 59̂ 5 .948 1.05 .61 .93 1.70 6074 5.62 4.8 3.0 5.90 7.8 6096 1.64 1.6 .89 2.4 6143 2.09 2.75 1.34 3.25 6163 1.25 1.5 .78 1.5 2.44 6217 .624 .61 .35 .77 1.46 6266 1.89 2.2 1.20 3.84 6304 .455 .38 .27 .4 .93 633^ 1.39 1.4 .84 2.07 6383 2.85 2.65 1.5 4.17 6402 2.57 5.3 2.7 5.78 6506 2.42 2.22 1.3 6533 .941 1.2 .7 2.14 6599 2.09 2.35 1.2 4.32 6678 2.11 2.35 1.1 4,10 6717 1.98 2.4 1.1 4.17 6929 1.64 1.8 .8 3.56 7032 1.90 2.0 1.06 3.24 7245 .915 .72 .42 1.80 TABLE 12 COMPARISON OF RELATIVE TRANSITION PROBABILITIES \ THIS EXP'T IRWIN /12/ DOHERTY /10/(30*) FRIEDRICHS /ll/(20-30^) . LADENBURG /7/(20-30*) 5852 10.0 ( 1.6*) 10.0 UP*) 10.0 10.0 10.0 5882 1.44 (13.0#) 1.69 (15*) 1.97 1.60 1.66 59̂ 5 1.44 (13.0*) 1.54 (15%) 1.74 1.55 1.33 6074 8.54 (10.7*) . 7.06 (10%) 8.55 9.85 6.09 6096 2.50 ( 7.0*) 2.35 (10%) 2.54 I.87 614-3 3.17 (11.3*) 4.04 (20%) 3.82 2.54 6163 1.90 .( 7.6%) 2.20 (15%) 2.22 2.50 1.91 6217 .943 (13.0*) .897 (15%) .997 1.29 1.14 6266 2.87 ( 7.5%) 3.24 (15%) 3.42 3.00 6304 .691 (12.4*) .559 (10%) .769 .668 .726 6334 . 2.11 (11.2*) 2.06 (15%) 2.39 1.62 6383 4.33 ( 3.4*) 3.68 (10%) 4.27 3.26 6402 3.91 (11.7%) 7.79 (25%) 7.98 4.51 6596 3.63 ( 9.1*) 3.26 (10%) 3.70. 2.66 6533 1.43 ( 8.0*) 1.76 (15%) 1.99 I.67 6599 3.17 ( 6.0*) 3.45 (10%) 3.42 3.37 . 6678 3.20 ( 3.9*0 3.45 (10*0 3.13 3.20 6717 3.01 ( 5.3*) 3.53 (10*). 3.13 3.26 .6929 2.50 ( 7.3*) 2.65 (10*) 2.28 3.78 7032 2.88 (12.6*) 3.21 (15*) 3.02 2.53 7245 1.39 (12.5*) 1.13 (10*) 1.20 l . 4 l -97- 5.6.3 Comparison and D i s c u s s i o n of Absolute P r o b a b i l i t i e s For comparison w i t h the r e s u l t s of o t h e r workers /7/, /10/, / l l / , /12/ by d i f f e r e n t methods, T a b l e s 11 and 12 l i s t b o t h the a b s o l u t e and r e l a t i v e t r a n s i t i o n p r o b a b i l i t i e s measured here w i t h those from the r e f e r e n c e s c i t e d above. The e r r o r s as g i v e n by the authors are a l s o i n c l u d e d . I t can be seen that Ladenburg's a b s o l u t e v a l u e s are g e n e r a l l y h i g h e r whereas Doherty's are lower than those of the p r e s e n t work. F r i e d r i c h s shows g e n e r a l agreement as does Irwin, who.claims the b e s t accuracy to date. However on the r e l a t i v e s c a l e Ladenburg and Doherty agree w i t h the present work w i t h i n experimental e r r o r . T h i s I n d i c a t e s t h a t the d i f f e r e n c e s may be a t t r i b u t e d t o the f i x i n g of the a b s o l u t e s c a l e . Ladenburg p l a c e s h i s r e l a t i v e v a l u e s on an a b s o l u t e b a s i s by a d o p t i n g the v a l u e f = .5 f o r \6k02, a f t e r computing the bounds of .85 and .21 by the f-sum r u l e . The u n c e r t a i n t y of t h i s v a l u e can e x p l a i n the d i s c r e p a n c i e s , w h i l e from van Andel's measurement of the l i f e t i m e of \5852, Doherty's a b s o l u t e v a l u e s must be c o n s i d e r e d to be In e r r o r . Because of i t s extremely h i g h a b s o r p t i o n , X.6402 seems to be one of the most d i f f i c u l t l i n e s to measure and t h e r e i s a l a r g e v a r i a n c e between the v a l u e measured here and those of the o t h e r s . I r w i n c i t e s evidence t h a t h i s v a l u e i s somewhat hig h but n e v e r t h e l e s s t h i s l a r g e d i f f e r e n c e seems to i n d i c a t e the presence of systematic e r r o r s so that the e r r o r estimate g i v e n i n the present work i s too o p t i m i s t i c f o r t h i s l i n e . - 9 8- The l i f e t i m e s of the 2p^3p l e v e l s may be computed from a b s o l u t e t r a n s i t i o n p r o b a b i l i t i e s by means of equation (5-7)• T h i s has been done and i s co n t a i n e d i n Table 13 and i n c l u d e s s i m i l a r data computed by Irwin /12/ and Klose /15/. S e v e r a l of the l i f e t i m e s computed from t h i s work do not c o n t a i n a l l the p r o b a b i l i t i e s , but the ones m i s s i n g a re small and c o n t r i b u t e an e r r o r l e s s than 15$. Again agreement i s good except f o r the p^ l e v e l , f o r which the l i f e t i m e i s due on l y to the \6402 t r a n s i t i o n . TABLE 13 COMPARISON OP UPPER-LEVEL LIFETIMES (X10 9) s e c . LEVEL THIS EXP'T IRWIN /12/ KLOSE /15/ p l 15.2* 14 .7* 14 .7 P2 23.6* 18.1 16 .3 P3 17.7* 20 .7* 23 P4 21.3 20.0 22 p5 25.8** 20 .1* 18 .9 P6 23.5 20 .3 22 p 7 23.0* 23.2* 20 .3 *8 26.5* 25.4 24.3 p 9 37.7 19 22.5 PlO 35.3** * M i s s i n g 1 t r a n s i t i o n p r o b a b i l i t y ** M i s s i n g 2 t r a n s i t i o n p r o b a b i l i t i e s -99- CHAPTER 6 CONCLUSIONS R e l a t i v e t r a n s i t i o n p r o b a b i l i t i e s f o r t r a n s i t i o n s between the e x c i t e d s t a t e c o n f i g u r a t i o n s 2p^3s and 2p^3p of Ne I have been s u c c e s s f u l l y measured by a new method. Measure- ments were made on both a v a r i a b l e l e n g t h c y l i n d r i c a l dc glow d i s c h a r g e absorber and a p u l s e d e l e c t r o n gun e m i t t e r t o o b t a i n a complete set of r e l a t i v e p r o b a b i l i t i e s . The t r a n s m i s s i o n of the glow d i s c h a r g e absorber v a r i e d over the c r o s s - s e c t i o n of the column; t h i s i s a t t r i b u t e d mainly to a v a r i a t i o n of the e x c i t e d atom d e n s i t y . The t r a n s - m i s s i o n was compared w i t h that which would be obtained from Doppler-broadened s p e c t r a l l i n e s and a g e n e r a l l y good comparison was found. The v a l u e s of the parameter a were anomolous however and subsequent numerical c a l c u l a t i o n s showed the p o s s i b i l i t y of equ i v a l e n c e between non-Doppler and " f i c t i t i o u s " Doppler l i n e - shapes. The r e l a t i v e t r a n s i t i o n p r o b a b i l i t i e s computed from t h i s method are v a l i d a l t h o u g h the accurac y i s l e s s w i t h h i g h l y a b s o r b i n g l i n e s due to bot h non-Doppler l i n e - s h a p e s and a p p r o x i - mations used t o account f o r the presence of i s o t o p e s . E x p e r i - mental evidence e x i s t s f o r the presence of s e l f - a b s o r p t i o n i n the G e i s s l e r tube source. Use o f a p u l s e d e l e c t r o n gun e m i t t e r has been found f e a s i b l e f o r measurement of r e l a t i v e l i n e i n t e n s i t i e s . The advantage i s t h a t s e l f - a b s o r p t i o n i n the e m i t t i n g neon i s n e g l i g i b l e . F i n a l l y a l i f e t i m e measurement by van Andel / ! ? / -100- allowed a b s o l u t e t r a n s i t i o n p r o b a b i l i t i e s t o be c a l c u l a t e d as w e l l as l i f e t i m e s of the upper l e v e l s of the t r a n s i t i o n s . -101- APPENDIX 1. ISOTOPE CORRECTION Equation (2,9) was g i v e n i n s e c t i o n 2.1.3 f o r the case of a gas c o n t a i n i n g two i s o t o p e s . The d e r i v a t i o n o f t h i s equation i s c o n t a i n e d i n t h i s appendix. Consider the case of a gas having two i s o t o p e s o c c u r r i n g i n the r a t i o K ( K - l ) . I f both the i s o t o p e s have Doppler-shaped s p e c t r a l l i n e s then i n the n o t a t i o n of Chapter 2, w i t h primed and unprimed q u a n t i t i e s r e f e r r i n g t o the two isot o p e s ; 1., T- w i t h cj- in / - C O eX and ZU = I > " ^ , X(*)=X* w i t h CO ->— _ {.(O - LX) y *) CA - ~r, ? ^ o — — ^ r7~ where <S i s the c e n t r e frequency s e p a r a t i o n of the s p e c t r a l l i n e s of the two i s o t o p e s . The energy i n t o the absorber p e r u n i t time i s thus -102- and the energy t r a n s m i t t e d per u n i t time i s E =\ f ( IW + XVv)U dWA . . . U i . i ) I f the r a t i o 161 / A S ) ^ i s l a r g e enough, the two components may be c o n s i d e r e d as separate and equation ( A l . l ) becomes so that the t r a n s m i s s i o n i s ( [ Pd̂ v) + X'<r*0)dv>]dA •'A- 'o D i f f e r e n t i a t i o n w i t h r e s p e c t t o the ar e a f o r a x i a l l y symmetric geometry y i e l d s d(R^T) = Sj*™* + . . . ( A i . 2 ) The f o l l o w i n g assumptions are now made: 1. There i s ho p r e f e r e n t i a l thermal h e a t i n g between the Isotopes, so t h a t T = T', T„ = T i . and t h e r e f o r e a = a'. r ' . a a ' s s' 2. The p o p u l a t i o n s of co r r e s p o n d i n g l e v e l s of the Isotopes are i n the r a t i o K and the t r a n s i t i o n p r o b a b i l i t i e s f o r corr e s p o n d i n g t r a n s i t i o n s are equal, o r l e s s s t r i n g e n t l y , -103- Prom these assumptions, and the f a c t t h a t "k+'fr - \ + Ot\o~s) ̂  | , i t can be seen t h a t 4 = ^ / f f . ^ b = / H ^ L l f .. (Al . 3 ) and s i n c e I Q 5 ^ . - K ... (A1.4) so t h a t e q u a t i o n (Al . 2 ) becomes: = 7 T ^ [ F o < 4 J U ) +aFft(bJU.*)] ...(AI.5) which i s i d e n t i c a l w i t h equation ( 2 . 9 ) . A few comments about the assumptions are appro- p r i a t e here. In t h i s experiment the times of o b s e r v a t i o n and measurement a r e of the o r d e r of minutes. A rough c a l c u l a t i o n of the c o l l i s i o n frequency between the i s o t o p e s g i v e s a v a l u e of 10̂  c o l l i s i o n s per second. With the energy t r a n s f e r f r a c t i o n -104- b e i n g .5 f o r e l a s t i c c o l l i s i o n s , ample time occurs f o r e q u i l i - brium t o be e s t a b l i s h e d between the Isotopes. Half of assumption 2. has been e x p e r i m e n t a l l y v e r i f i e d by Conner and B l o n d i /42/, as w e l l as Lang /33/, who observed f o r neon the r e l a t i o n g i v e n by equ a t i o n (Al . 4 ) . It seems reasonable t h a t t h i s should h o l d f o r a l l l e v e l s and p a r t i c u l a r l y the lower l e v e l 1, J u s t i f y i n g the assumption. The e r r o r s i n t r o d u c e d i n c o n s i d e r i n g the two Doppler l i n e s completely separated can be estimated by c o n s i d e r i n g the r a t i o of the Doppler width t o the s p l i t t i n g . Nagaoka and Mishima /kj/ have measured the i s o t o p e s p l i t t i n g of neon, and found i t t o be between .02 and .035 A*. Irwi n /12/ measured the Doppler widths of the neon l i n e s u s i n g a r a d i a t i o n source s i m i l a r t o t h a t used here, and found the widths t o correspond t o room temperature, o r approximately .017A. T h i s g i v e s an i s o t o p e s e p a r a t i o n t o Doppler l i n e - w i d t h r a t i o g r e a t e r than 1 to a p p r o x i - mately 2. Knowing t h i s r a t i o , e q uation ( A l . l ) may be i n t e g r a t e d n u m e r i c a l l y as a f u n c t i o n of k Q i and a, and T (or d ( R 2 R ) / d ( R 2 ) ) c a l c u l a t e d e x a c t l y . T h i s has been done u s i n g an IBM 7Q40 d i g i t a l computer and the e r r o r between equations ( A l . l ) and (Al . 5 ) i s shown g r a p h i c a l l y i n F i g . A l as a f u n c t i o n of k Q i . . For those l i n e s t h a t a re h i g h l y a b s o r b i n g , the v a l u e of k Q i s no doubt i n e r r o r due to t h i s cause, p a r t i c u l a r l y f o r h i g h absorber c u r r e n t s and near the a x i s of the tube where a b s o r p t i o n i s g r e a t e r . T h i s ' c o u l d be one of the reasons f o r the poor f i t t o the curves t h a t was observed f o r some s t r o n g l i n e s (see remarks  -106- i n s e c t i o n 5.2.3). G e n e r a l l y most l i n e s had a kQl of l e s s t h a n 3 and an XL of 1.5 or g r e a t e r making the e r r o r i n assuming separated curves t o be l e s s than 5*. The e r r o r i n d e t e r m i n i n g f r e ] _ can be c o n s i d e r a b l y l e s s than t h i s however, s i n c e f r e l depends on the r a t i o of two k 0 » s (see equ a t i o n (2.11)) For two l i n e s having comparable a b s o r p t i o n and v a l u e s of XL , the e r r o r i n the va l u e s of k Q w i l l tend i n the same d i r e c t i o n and make the e r r o r i n f r e ^ l e s s . -107- APPENDIX 2. EVALUATION OF TRANSMISSION EQUATION The method of computing the r i g h t - h a n d s i d e of equat i o n (2.8) i s o u t l i n e d i n t h i s appendix. Equation (2.8) can be ev a l u a t e d e i t h e r by numer i c a l i n t e g r a t i o n of equation (Al . 5 ) or by use of the i n f i n i t e s e r i e s form of equations (2.8) and (2 . 3 ) : w i t h x = kQjH , a = 0.094, and b =0.104. For s m a l l x the s e r i e s converges r a p i d l y and the number of terms necessary i n order t o eva l u a t e i t i s s m a l l . For l a r g e x the o p p o s i t e i s t r u e and rec o u r s e t o n u m e r i c a l i n t e g r a t i o n i s b e s t . The problem i n c a l c u l a t i n g the s e r i e s f o r l a r g e x i s t h a t the terms a re not sm a l l u n t i l n > x and e v a l u a t i o n c o n s i s t s of computing the d i f f e r e n c e between p a i r s of l a r g e numbers. With a f i n i t e "word l e n g t h " such as i s present i n a computer, t r u n c a t i o n and round-off of the terms becomes c r i t i c a l . Using an IBM 7040 d i g i t a l computer w i t h double p r e c i s i o n , i t was found t h a t f o r the T's encountered, i n t h i s experiment, the val u e s o f x necessary were low enough t o use t h i s s e r i e s . How- ever f o r the l a r g e r x used i t probably would have been q u i c k e r t o c a l l on numer i c a l i n t e g r a t i o n . The FORTRAN IV program i s g i v e n below. The f i r s t -108- sum i n e q u a t i o n (A2.1) i s c a l l e d the major s e r i e s and the second sum the minor s e r i e s . The procedure i n c a l c u l a t i n g these sums f o r a g i v e n x i s to e v a l u a t e s u c c e s s i v e terms u n t i l t h e i r a b s o l u t e v a l u e i s l e s s than 10 , a f t e r which computation ceases. For speed of computation and m i n i m i z a t i o n of round-off e r r o r , terms are e v a l u a t e d from p r e c e e d i n g terms, thus: It was found t h a t the major s e r i e s becomes u n s t a b l e above x = 35, a t which v a l u e the number of terms computed was approximately 100. A spot-check of the v a l u e s was c a r r i e d out X u s i n g a Simpsons Rule n u m e r i c a l I n t e g r a t i o n and was found to agree to t h r e e s i g n i f i c a n t f i g u r e s . I B F T C SUM DOUBLE P R E C I S I O N T»TERM»RAT I 0 » R » A 2 * A I »X I » R 1 » X F , D S Q R T . X 1 » X 2 » T 2 » T 3 — PTWE-N'S-rO N RATI 0"( 5 0 0 ) , R ( 5 0 0 ) " R E A D ( 5 » 1 ) X I , X F , D E L X , A » A F , A D » M 1 1 F O R M A T ( 6 F 8 . 4 , 1 4 ) ~C X I =X ( I N I T I AL ) >XF = X ( F I N A L ) »DELX= I INCREMENT IN X, D I T T O FOR A ? AF » AD WR T A L P H A C M1=.MAX. NO. OF TERMS IN S E R I E S • DO 12 I1=1»NN ^ W R I T E ( 6 » 4 ) • : 4 F O R M A T ( 1 H 1 > 7 X J 3 H S U M > 6 X » 2 H X , 6 X » 5 H A L P H A ) A I = A**2 R ( 1 ) - 1. D0 + A2 : — R A T I O t 1 ) = 1 . D 0 / D S Q R T ( R { 1 ) ) M2 = M1 DO 6 I - 2 > M 2 : A I = 1 R ( I ) =R( I - l ) +A2 T> RAT 1 0 ( I ) - DSQRT ( R ( I -1 ) /R ( I ) ) / A I : — C C C A L C U L A T I O N OF MAJOR S E R I E S X 1 = X I 7 T = 1 . D 0 N--i TERM--RAT 1 0 ( 1 ) * X 1 3 I F ( A B S ( T E R M ) . L T . . 0 0 0 1 ) GO TO 1 7 5 T-T + TERM I F C N . G T . M 2 ) GO TO 1 8 9 N = N + 1 R l - R A T I 0 ( P s l ) " X l • TERM=-TERM * R 1 GO TO 3 C — C C A L C U L A T I O N OF MINOR S E R I E S C i r7 -5 X 2 = 0 . 1 0 4 X X 1 T 2 = 1 . D 0 N = l T E R M - R A T I O ( 1 ) « X 2 3 0 I F ( A B S ( T E R M ) . L T . . 0 0 0 1 ) GO TO 1 7 0 T 2 = T 2 + T E R M i r ( N . G T . M 2 ) GO TO 1 0 0 N = N+1 R1 = RA TI 0 ( N ) * X 2 170 —T~E~R M — T E R M " RT: GO TO 30 T3=(T+.094*T2)/1.094 i ! 180 183 GO TO 18 WR I T E ( 6 »18 3) FORMAT(IX»41HNUMSER OF TERMS I N MI NOR SERIES TOO LARGE) i 189 184 GO TO 12 WR I T E ( 6 J 1 8 4 ) FORMAT(IX»41HNUMBER OF TERMS I N MAJOR SERIES TOO i LARGE) 18 2 GO TO i 2 WRITE(6 > 2) T3?X1?A FORMAT(5X>3F8.3 - i r i A i . O t i A M 1 U 1 U X1=X1+DELX GO TO 7 1 u i h l A . b t . A I - ; G U 1 U 1<L A=A+AD GO TO 99 TT. S I OP END -109- APPENDIX 3. SELF-ABSORPTION CORRECTION FACTOR S e l f - a b s o r p t i o n reduces the I n t e n s i t y e m i t t e d by a gaseous source. The c o r r e c t i o n f a c t o r S t o be a p p l i e d t o the measured i n t e n s i t y f o r Doppler-broadened l i n e s i s d e r i v e d i n the f o l l o w i n g paragraphs (see s e c t i o n 2 . 2 ) . By c o n s i d e r i n g the equation of t r a n s f e r of r a d i a t i o n i t can be shown th a t the t o t a l l i n e i n t e n s i t y emitted by an o p t i c a l l y t h i c k l a y e r of l e n g t h $ can be re p r e s e n t e d by (see f o r example, Irwin / 1 2 / ) 1 = B J O - - ^ 9 ^ ) ^ . . . ( A 3 . 1 ) where, i f s t i m u l a t e d e mission i s n e g l i g i b l e B = S u b s t i t u t i o n of the simple Doppler p r o f i l e f o r k( S> ) r e s u l t s i n the c o r r e c t i o n f a c t o r g i v e n by eq u a t i o n (2.15). However when two i s o t o p e s a re p r e s e n t , k( ~i> ) has the form g i v e n i n Appendix 1. , -co'* -110- n-o 1 * L S u b s t i t u t i n g f o r k Q from equation (2.2) and B from above i n t o e quation (A3 .1) , and changing the i n t e g r a t i o n v a r i a b l e from ~P to co then y i e l d s e q u a t i o n (2 .16). A t a b l e of numerical v a l u e s of S f o r s e v e r a l v a l u e s of the i s o t o p e s e p a r a t i o n are g i v e n i n Table A3. The val u e of S i s d e f i n e d such t h a t S = 1.0 f o r no s e l f - a b s o r p t i o n when onl y one i s o t o p e i s p r e s e n t ; f o r two Isotopes i n the r a t i o 9.9:100, S = 1.099 when the o p t i c a l t h i c k n e s s i s zero. As w e l l , the c o r r e c t i o n f a c t o r f o r a s i n g l e i s o t o p e has been c a l c u l a t e d from equation (2.15) and has been i n c l u d e d . It agrees w i t h the v a l u e s g i v e n by Ladenburg and Levy /7 / f o r the va l u e s of k Q $ which are common to both. The computer program used t o eva l u a t e e q u a t i o n (2.16) i s g i v e n below. To c a l c u l a t e the f a c t o r n u m e r i c a l i n t e g r a t i o n was r e s o r t e d t o . The method of i n t e g r a - t i o n chosen was Simpson's Rule because of i t s . s i m p l i c i t y , as w e l l as the f a c t t h a t the accuracy needed was not h i g h — only t o t h r e e s i g n i f i c a n t f i g u r e s . A f t e r computation of sSn , -111- the s e r i e s was ev a l u a t e d i n a manner e n t i r e l y analogous t th a t used i n the program g i v e n i n Appendix 2. The s i n g l e i s o t o p e s e l f - a b s o r p t i o n f a c t o r was computed by p u t t i n g K (and t h e r e f o r e a = b = 0) i n the program. TABLE A3 SELF-ABSORPTION CORRECTION FACTOR FOR A ONE AND TWO ISOTOPE GAS S TWO ISOTOPES) s n-=1.0 XI =1.5 n =2.0 (ONE ISOTOPE) 0.0 • 1.099 1.099 1.099 1.000 .62 1.091 1.091 1.092 .993 .04 1.083 1.084 1.084 .986 .06 1.075 1.077 1.077 .979 .08 1.068 1.069 1.070 .972 .1 1.060 1.062 1.063 .966 .2 1.023 1.02? 1.030 .933 '? .988 .994 .998 .902 .4 .956 .962 .967 .873 .5 .924 .933 .938 .845 .6 .895 .904 .911 .818 .7 .867 .877 .884 .793 .8 .840 .852 .860 .769 .9 .915 .827 .836 .7^7 1.0 .791 .804 .813 .725 1.2 .7^6 .761 .771 .685 1.4 .705 .721 .733 .648 1.6 .668 .685 .698 .615 1.8 .635 .653 .666 .584 2.0 .604 .622 .636 ' .556 2.4 .550 .569 .584 .507 2.8 .504 .524 .540 .466 3.2 .466 .485 .502 .430 3.6 .432 .451 .469 .399 4.0 .403 .422 .440 . 372 SIBFTC ISOABS REAL OMEGA »OMEGAF DOUBLE PRECISION SUM»TERM>RAT I 0»A I»DSQRT»S DIMENSION RATIO(100) »S( 100) READ(5»10) X,XF>DX,OMEGA,DOMEGA »OMEGAF 10 FORMAT(6F8 . 1 ) I F ( X F . G T . 3 5 . )STOP • I F ( X F . L E . 3 5 . )N = 120 I F ( X F . L T . 2 0 . ) N = 6 0 • I F U F . L T . 1 0 . )N = 35 I F ( X F . L T . 5 . 0 ) N = 2 0 I F ( X F . L T . 1 . 3)N=10 N l = 20 N2=40 N3 = 20 B = 2. 5 A = 2 . 5 RT=SQRT(1.1) HK=.099 HK1 = RT#H,< XI =X 23 WR 1 I E ( 6 »11 ) 11 FORMAT ( 1 HI >7X? 1HS » 1IX » 4-HK0#L » 3X »8HNO.TERMS,5X »5HOMEGA) C • c COMPUTE S ( J ) • c J l = l 7 DO 50 J = J1 >N c c INTEGRATION FROM - I N F . TO -B BN1=N1 H = l . / ( B N 1 * B ) SUME = 0 . SUMO=0. L = l K = l N l 1 = N1-1 DO 100 1 = 1 , N 11 A I = I Y=AI*H 103 Y = 1 . / Y Y2=Y*Y B R AK = ( E X P ( - Y 2 ) + H K 1 * E X P ( - 1 . 1 * ( (Y +OMEGA)**2) ) ) F1=Y2*( (BR AK)*#J) I F ( L . E Q . 2 ) GO TO 101 IF(K.EO.l)SUMO=SUMO+Fl - IF(K.EQ.O)SUME=SUME+F1• I F ( I . E Q . N l 1 )G0 TO 102 GO TO 100 102 Y = l ,/B L = 2 GO TO 103 101 FF1=F1 100 K=l-K SUM1 =( 1 . 33 333*SUMO+.66667*SUME + .33 3 3 3 * F F l ) * H c c INTEGRATION FROM -B TO A BN2=N2 H=(A+B)/BN2 L= 1 K = 0 SUME=0. SUMO=0. DO 200 I=1»N2 AI = I-1 W = -B + AI-«-H 204 VI2 =W-«-W BR AK =(EXP(-W2)+HK1*EXP(-1.1*((W-OMEGA)**2))) F2=BRAK*#J I F ( I . E Q . 1 ) G O TO 20 1 I F ( L . E Q . 2 ) G O TO 202 I F ( K . E O . l ) S U M O = SUMO + F 2 I F ( K . E Q « 0)SUME = SUME + F2 I F ( I.EO.N2 )GO TO 203 GO TO 200 201 FI2=F2 GO TO 200 203 W = A L = 2 GO TO 204 202 FF2=F2 200 K = 1-K SUM2=( 1 .33 333*SUMO+.66667*SUME+.33 3 3 3 * ( F F 2 + FI 2) )*H c INTEGRATION FROM A TO INF. BN3=N3 H=1./(A*BN3) SUME=0. SUMO=0. L = l K=l N31=N3-1 DO 300 .1 = 1 >N31 A I = I ; Y=AI*H 1 Y = l . / Y 303 Y2=Y*Y , ' 1 BRAK= ( EXP (--Y2 ) +HK1*EXP (-1 . 1* ( ( Y-OMEGA ) **2 ) ) ) F 3 = Y 2 * ( ( B R A K ) * * J ) ! 1 F ( L . E Q . 2 ) G O 10 301 1 I F ( K . E O . l ) S U M O = SUMO+F3 IF(K.EQ.0)SUME=SUME+F3 j 1 h I I.EQ.N3 1 )GO 10 302 { GO TO 300 1 302 Y = A ] L = 2 I GO TO 303 301 FF3=F3 ; 300 K=l-K SUM3 = ( 1 . 3333 3*SUM0+.66667*SUME+. 333 33*FF3 ) * H 50 S(J)=SUM1+SUM2+SUM3 i C C S ( J ) NOW COMPUTED C 18 J2=J1+1 DO 55 I=J2»N A l = I 55 R A T I O ( I ) = ( S ( I ) / S ( I - l ) ) / A l , I F ( J 1 . GE • 2)G0 TO 8 TERM = -S (,2 ) *X/2 . SUM=S(1) 8 J3=J2+1 DO 2 I=J3>N I F ( A B b ( ItKM) .LI . l . D - 4 ) G 0 TO 3 SUM=SUM+TERM 2 TERM=-TERM*RATIO(I)*X 3 I F ( I - N ) 5 » 6 » 6 ' j 6 J1=N+1 i N=N+10 GO TO 7 i 5 SUM=SUM/1.77245 | WRITE(6 » 4 )SUM »X » I »OME GA 4 FORMAT(5X»F7.4»5X,F5.3,5X,I 3,5X,F4.1) 1 X-X+DX I F ( X - X F ) 16 » 16 »17 16 J l = l 1 GO TO 18 j 17 X = XI I OMEGA=OMEGA+DOMEGA IF(OMEGA-OMEGAF)23 >23?19 ! 19 STOP ; END -112- APPENDIX 4. RELATIVE INTENSITY MEASUREMENT OP SPECTRAL LINES The theory developed i n t h i s appendix d e r i v e s e q u a t i o n (4.1) f o r the q u a n t i t a t i v e measurement of r e l a t i v e I n t e n s i t i e s o f s p e c t r a l l i n e s (see s e c t i o n 4.1). Consider the case i n which the i l l u m i n a t i o n covers the s l i t of a monochromator and i s of un i f o r m i n t e n s i t y . The t r a n s m i s s i o n pass-band i s then t r i a n g u l a r o r t r a p e z i o d a l , depending on the r e l a t i v e s i z e s of e x i t and entrance s l i t . I f the monochromator i s set on wavelength xM and the w i d t h of the e x i t s l i t i s denoted by Wxt, then i t encompasses a wavelength r e g i o n A \^ g i v e n by where D(.XM) i s the l i n e a r d i s p e r s i o n . The corresp o n d i n g r e g i o n which the image of the entrance s l i t c o v e r s , A x n t , i s g i v e n by where M(x) i s the l a t e r a l m a g n i f i c a t i o n of the instrument i n the e x i t p l a n e and i s the widt h of the entrance s l i t . I t i s assumed here t h a t ^ A ^ N ^ S O the t r a n s m i s s i o n of an i d e a l monochromator i s t r a p e z o i d a l of w i d t h ^ X x t • T h i s i s shown i n F i g . A4,.a. t(x,Xjj) i s the t r a n s m i s s i o n of the monochromator and reaches a maximum, T ( X M ) i n the r e g i o n . _ A * z t - A K t ^ X ^ >\ + ^ Z 2. The d e t e c t o r output c u r r e n t due to r a d i a t i o n i n a s m a l l band of wavelengths dX, c e n t r e d at X i s -113- 6i = J T l t ^ A M ) S(*)1M6?\ ...(A4.1) where <TL i s the s o l i d angle subtended at the source by the monochromator and input l e n s , S(\) i s the s e n s i t i v i t y of the d e t e c t o r , and I(X) i s the i n t e n s i t y p e r u n i t s o l i d angle per u n i t wavelength i n t e r v a l o f the source. ••tew,) > M z. t / \. ; 1 1 P i g . A4,I Pass-Band of Monochromator I n t e g r a t i n g over a l l wavelengths y i e l d s the t o t a l response, thus: L = j df X i t S X d A ...(A4.2) 2. -114- I. For a s p e c t r a l l i n e of w i d t h < S/N«A>\ 2 : t and wavelength X where the s u b s c r i p t i i n d i c a t e s l i n e r a d i a t i o n . T h e r e f o r e f o r two l i n e s X-^, x 2 of the same l i n e - w i d t h UK) _ p / >. v N Tfro S(*,)T£(?\ .) For a continuous source, the v a r i a t i o n of I and S over A X ^ must be taken i n t o account. I n t r o d u c i n g a new v a r i a b l e x' = ( X - X M ) / A ' X ' ^ a n d w r i t i n g a T a y l o r s e r i e s expansion ~2 f o r I and S about x' = 0 Now X'(o) ^fC(0-36o)J/| and i f o n l y the f i r s t two terms i n the expansion are r e t a i n e d , then w i t h s - X 0 ) / x ^ o ) , <r = S6)/3tf>) -115- and the t r a n s m i s s i o n i s g i v e n by (see P i g . A^.b) v \ - ex. J 7 where l - o- a = » - — n t * * * * Equation (A^.2) then becomes w i t h K M = [ ^ [ » + i ( 6 - O d o - - 0 - ' o > - 2 : * U - 0 ^ - 0 ] 5 and a, 81 , and T are f u n c t i o n s of X. T h e r e f o r e i f two wavelengths X-p X 2 a r e compared, but A A ^ ' ] = where D i s the l i n e a r d i s p e r s i o n . The r a t i o -116- i s then i d e n t i c a l w i t h equation (4.1). -117- APPENDIX 5. CURVE-PITTING OP NON-DOPPLER SPECTRAL DISTRIBUTION TO DOPPLER SPECTRAL DISTRIBUTION Attempts are made here t o equate the t r a n s m i s s i o n s of pure Doppler-broadened s p e c t r a l l i n e s t o two types of non- Doppler s p e c t r a l l i n e s , as d i s c u s s e d i n s e c t i o n 5.4.1. 1. Self-Absorbed Doppler D i s t r i b u t i o n The i n t e n s i t y d i s t r i b u t i o n of a gaseous source of i n wh e (see Appendix 3 ) . l e n g t h J(_ n ich s e l f - a b s o r p t i o n takes p l a c e i s g i v e n by 6 1(9} - B d - ^ ^ 4 ) ...(45.1) which, f o r k (9 ) giv e n by a Doppler d i s t r i b u t i o n and k Q j l &~ 1 reduces to the form g i v e n i n Chapter 2. For a uniform, mono-isotopic e m i t t e r and absorber gas w i t h a pure Doppler a b s o r p t i o n c o e f f i c i e n t , the t r a n s - m i s s i o n T i s a c t u a l l y g i v e n by T - \ (i - E*P c -JkB(^h)) E x P( - A . (SO A ) d v> -118- = 2 2 (->) y " ^ O-o Ante (Tv -r1) \ m\ J/0-+I + ^ d i 1 - ...(A5.2) where y = k „ i _ and x = k ,0 , and the s u b s c r i p t s e and a r e f e r OG 6 OS. Q. i to e m i t t e r and absorber, r e s p e c t i v e l y . The q u e s t i o n i s r a i s e d , can t h i s be equated t o a Doppler t r a n s m i s s i o n w i t h no s e l f - a b s o r p t i o n ? i . e. f o r the s i x a b s o r p t i o n l e n g t h s l a ,, i s t h e r e some va l u e of x* and a* such t h a t a i T s ^ 5 ^ ° 0 =TVz<'<*0^#=//--> -? ...(A5.3) where x, = k ,/ . and x' = k» „ , and i oa a^ oa a^' T ' C Z \ = ^^K^ ...(A5.4) as g i v e n i n equation (2.3). An attempt has been made t o s o l v e equation (A5.3) n u m e r i c a l l y i n the IBM 7040 d i g i t a l computer; v a l u e s of k» r t and a* determined e x p e r i m e n t a l l y (see s e c t i o n s 3.2.1 and 3.2.2) were used t o compute T* (xj^, a ' ) . T s ( y » xi» a ) was ev a l u a t e d as k was i n c r e a s e d i n ste p s u s i n g assumed oa , • -119- values of the parameters y and a unti l the root-mean-square deviation between T* and T s for the six absorption lengths was a minimum. Some examples of T g and T' for two wavelengths are shown in Table A5..1. It can be seen that a f i t with a low rms error is possible; the error is better than some of the experi- mental f its to the theoretical curves in determining k ' o a and a'. Another point to be noticed is that k Q a substantially differs from k£ & when a 4 a* indicating a large uncertainty in approximating the 'fictitious* k» to k . Further compu- Ocx Oct tations were carried out on the two lines in Table A5 .1; since f r e l is proportional to k Q a ( re l ) , this latter ratio was computed as k Q e i e was varied, for a = .9 and 1.0 , and compared: with the measured value of .0285/.0193 = 1.48. The results of these computations are shown in Table A5 .2 . The value of l&^JL^ for X.6383 was increased in steps of .01. The corresponding value for X6505 was assigned by assuming k f r e l ) = k' (rel). Exact values of k _ i ^ for the Geissler tube are not oe e known although from measurements of kQ made by Irwin /12/ k ^ i ' could be of the order of .3 for some of the strongly oe e , absorbing lines; as Well, in the present calculations the determined value of k is uncertain to 5 units in the last oa figure. Both these facts do not allow any quantitative conclu- sions to be made from the above tables, but the variation of k Q a (rel) appears to be much less than that of "absolute" k Q a as a function of k„ i . oe e TABLE A5.1 SELF-ABSORBED DOPPLER LINE-SHAPE FIT oa (cm - 1) a* k i oe e ABSORPTION LENGTH (CM). k o a , -1 (cm a RMS ERROR 10 15 20 25 35 45 6506 .0285 1.3 .05 T i .844 .778 .719 .666 .575 .501 .0230 1.0 .003 T s .845 .780 .722 .668 .575 .497 6383 .0193 1.3 .05 T» T s .891 .886 .842 .838 .797 .792 .755 .750. .679 .673 .614 .605 .0157 1.0 .006 TABLE A5.-3 VOIGT.LINE-SHAPE FIT k» oa cc1 a ABSORPTION LENGTH (CM) k o a a RMS (cm" 1) 10 15 20 25 35 45 (cm - 1) ERROR 6506 .0285 1.3 .01 .844 .778 .719 .666 .575 .501 . 0238 1.0 .004 T v .854 • 783 .723 .681 .575 .497 •121- TABLE A5.2 CALCULATION OF k ' ( r e l ) FOR SELF- oa ABSORBED DOPPLER LINE-SHAPES k ^ c n T 1 ) a' k o 2 / k o l x 2 ( i ) 6383 6506 .0193 .0285 1.3 1.3 y 1 •1.48 .6383 6506' ! a ( k o^e k 0 (cm _ 1 ) V c n f 1 ) V k o l .9 .01 . O I 5 2 .0148 .0220 1.45 •9 .02 .0152 .0295 .0220 1.45 .9 .03 .0150 . .0443 .0218 1.45 .9 .04 .0149 .0591 .0216 1.45 1.0 .01 .0161 . 0140 .0234 1.45 1.0 .02 .0160 .0295 .0232 1.45 1.0 .03 .0159 .0443 .0230 1.45 1.0 .04 .0159 .0591 . 0230 1.45 1.0 .05 .0157 .0738 .0228 1.45 1.0 .06 .0156 .0886 .0226 1.45 1.0 .07 .0154 .1034 .0224 1.45 1.0 .08 . 0154 .1181 .0222 1.44 1.0 .09 .0153 .1329 .0220 1.44 -122- The computer program used t o e v a l u a t e equation (A5.3) and compose Tables A5.1 and A5.2 i s g i v e n below. T.» i s computed u s i n g the FUNCTION TFALS subprogram, w h i l e T_ i s computed from the FUNCTION TTRUE subprogram as k Q & i s v a r i e d f o r some a u n t i l T* = T g f o r minimum rms e r r o r . 2. Voigt D i s t r i b u t i o n When the l i n e - b r o a d e n i n g processes are a combination of Doppler and n a t u r a l broadening, the s p e c t r a l l i n e shape i s g i v e n by the V o i g t p r o f i l e (see f o r example, M i t c h e l l and Zemansky /19/) AM = Jjo. rv*v/ -(A5-5) where Cu - J&iSL and i s the n a t u r a l h a l f - w i d t h of the s p e c t r a l l i n e , and the other parameters are as d e f i n e d p r e v i o u s l y . I t should be p o i n t e d out t h a t t h i s d i s t r i b u t i o n i s perhaps a more " r e a l " d i s t r i b u t i o n , as most i n d i v i d u a l broadening pr o c e s s e s are Gaussian or L o r e n t z i a n . For the case of n e g l i g i b l e s e l f - a b s o r p t i o n i n the source and a l i n e - s h a p e g i v e n by e q u a t i o n (A5«5)» the c o r r e s - ponding t r a n s m i s s i o n becomes, a g a i n f o r u n i f o r m and mono- -123 i s o t o p l c e m i t t e r and absorber T^C-X^OL) ..-.(A5.-6) where' VOIGT( OJ , a) i s d e f i n e d from equation (A5.5) as k(-i>)/k Q and x i s d e f i n e d as before„ Again, can t h i s be equated to a pure Doppler t r a n s - m i s s i o n f o r some va l u e s of k'-and a*? 1. e. can o T h i s equation has been evaluated i n a manner e q u i v a l e n t to equation (A5»3) f o r k u s i n g an assumed v a l u e of a and a, OQ, to see whether a s o l u t i o n does e x i s t ; T a b l e A5.3 shows the r e s u l t s of t h i s computation. I t i s seen t h a t an approximate s o l u t i o n i s p o s s i b l e f o r a k and a much d i f f e r e n t than k* and : oa o a'5 e x c e s s i v e computing time as w e l l as the g a i n i n g of informa- t i o n of l i t t l e v a lue p r e c l u d e d f u r t h e r computational i n v e s t i g a - tion,, The program used to e v a l u a t e e q u a t i o n (A5.7) i s a l s o g i v e n below and i s s i m i l a r i n form t o the program In the p r e v i o u s s e c t i o n . FUNCTION TPKA e v a l u a t e s T*, but by n u m e r i c a l i n t e g r a t i o n r a t h e r than by the i n f i n i t e s e r i e s , and FUNCTION TKAA c a l c u l a t e s T . The VOIGT i n t e g r a l Is e v a l u a t e d i n FUNCTION VOIGT from a program w r i t t e n by Young / 4 V . ) V O I 6 T c ' t O J G - ) d c o IBFTC FALST1 REAL KP0,L,KP0L,K0LE,K0,KOI,K0L,K0F,K0MIN DIMENSION L<6) »TP(-6) »T(6) ,E(6) 1 6 DATA L / 1 0 . > 15 . , 20 ., 25 . ,35. ,45 . jOl R EAD ( 5,1) KP0,ALFAP • . 'TT c K P C A L F A P = F I C T I T I O U S VALUES OF K AND ALPHA Zl 1 FORMAT(2F6.3) K0=KP0/5.5 A READC 5,1)KOLE,ALFA c KOLE =K*L FOR EMITTING SOURCE, ALFA = K0F =10.*K0 OTRUE'? ALPHA KOI= KO DK = KO DO 7 1=1,6 KPOL = KPO*L( I ) 7 T P ( I ) = T F A L S ( K P O L , A L F A P ) 77 WRITE(6 ,5)KOLE,ALFA FORMAT(1X//1X,7H(K0L)E=,F7.4,3X,5HALFA=,F5.2) J= 1 E2 = 0. DO 10 1=1,6 K 0 L = K 0 * L ( I ) T ( I ) = T T R U E ( K O L E , K P L , A L F A ) E < I ) =T( I )-TP( I ) 10 E 2 = E 2 + E ( I ) * E ( I ) ERMS = SQRT(E2/6. ) WRITE(6 ,8 ) (T( I ) , I = 1 ,6) , KO,ERMS j 6 FORMAT(1X///1X,22HMEASURED TRANSMISSIONS , 5X,6F9.3) | WRI TE ( 6 ,6 ) ( TP ( I ) , 1 = 1 ,6 ) j 8 FORMAT( IX,22HSELF-ABS TRANSMISSIONS,5X,6F9.3/1X,3HK0=,F9.4,5X,5HER \ 1MS= ,3X , F6 . 3 ) i 12 I F ( J . EQ. 1 ) GO TO 15 I Li IF(ERMS.GE.EMIN) GO TO 11 )I0 EM I N= ERMS K0MIN=K0 b GO TO 11 15 EM IN = ERMS KOMIN = K0 5 J = 2 11 K0=K0+DK IF ( KO.LE.KOFjGO TO 9 'J I F ( KOMIN.EQ.KOF ) GO TO 20 j " I F ( K O M I N . E Q . K O I ) G 0 TO 20 N U - K U r l l IN U N 1 K 0 I = K 0 K 0 . F = K O M I N + D K j D K = D K / 5 • ! I F ( D K . L E . 1 . E - 4 ) S T 0 P G O T O 77 2 0 W R I T E ( 6 J 2 1 ) 2 1 F O R M A T ( 1 X / 1 X , 2 2 H E N D - P T . M I N I M U M , B E W A R E ) S T O P E N D I F U N C T I O N T F A L S t X l . A L F A ) DOUBLE P R E C I S I O N T , TERM>RAT I 0>R,A2>A I »RI >DSQRT D I M E N S I O N R A T I O ('100) »R(100) I F ( X l . G T . 1 0 . ) M l = 1 0 0 I F ( X 1 . L E . 1 0 . )M1 = 35 I F ( X 1 . L E . 6 . 5 ) M 1 = 2 5 I F ( X 1 . L E . 4 . 5 ) M 1 = 2 0 I F ( X 1 . L E . 3 . ) M l = 15 I F ( X 1 . L E . 1 . ) M 1 = 1 0 I F ( X 1 . L E . . 2 ) M1 = 5 . A2 = ALFA-"-ALFA R ( 1 ) = 1 . D 0 + A 2 R A T I O ( 1 ) = 1 . D 0 / D S Q R T ( R ( 1 ) ) DO 6 I=2>M1 A I = I R( I ) =R( I - l ) +A2 6 R A T I O ( I ) = D S Q R T ( R ( I -1 ) / R ( I ) ) / A I T = l . D O N = l T E R M = - R A T I O ( 1 ) X1 R l = R A T I O ( 1 ) * X 1 3 I F ( A B S ( T E R M ) . L T . . 0 0 0 1 ) G O TO 17 T = T-f TERM N=N+1 R 1 =R AT I 0 ( N ) -x-Xl TERM = - T E R M » R 1 GO TO 3 1 1 1 17 T F A L S = T i I RETURN END • 1 i i i ! I F U N C T I O N T T R U E ( X ? Y ? A L F A ) ! R E A L NUM I F ( X . G T . 1 .5)MMAX=2 5 1 I F ( X . L E . 1 . 5 ) M M A X = 1 0 ! I F ( Y . G T . 1 0 . >MAX=100 I F ( Y . L E . 1 0 . )MAX = 35 NMAX=50 9 DEN0M=SUM1(X,MMAX) 8 SUM = SUM 2 < Y i A L F A ? 0 ? MAX) TERM=-SUM2 ( Y , A L F A , 1 , MAX ) -*X /2 . DO 2 N = 2 » N M A X I F ( A B S ( T E R M ) . L T . 1 . E - 4 ) G 0 TO 3 AN=N+1 SUM=SUM+TERM 2' TERM=-TERM*SUM2 ( Y , A L F A >N,MAX)*X/AN 3 NUM=SUM TTRUE=NUM/DENOM RETURN END F U N C T I O N S U M K X . M M A X ) 12 D I M E N S I O N R ( 5 0 ) . R A T I O ( 5 0 ) J i R ( 1 ) = 2 • |:o R A T I O ! 1 ) = ( 1 . / R ( 1 ) ) * * 1 . 5 9 DO 1 I =2»MMAX 8 A l = I |7 R( I ) =R( I - l ) +1. 3 1 R A T I 0 ( I ) = ( R ( I -1 ) / R ( I ) )-"--"-1. 5 / A I S SUM1=1. |4 T E R M = - R A T I O ( 1 ) * X 3 DO 2 I=2»MMAX I F ( A B S ( T E R M ) . L T . 1 . Z - 4 ) R1TURN SUM1=SUM1+TERM 2 T E R M = - T E R M * R A T 1 0 ( I ) * X END F U N C T I O N SUM2(X,ALFA>M*MAX) DOUBLE P R E C I S I O N TERM*RAT I 0 > R » A 2 » D S Q R T » A I>AM,DABS D I M E N S I O N R( 100) »RAT 10( 1 0 0 ) A 2 =AL F A * A L FA A M = M R ( 1 ) = A M + 1 . D 0 + A 2 RAT 10( 1 )=1.D0/DSQRT(R( 1 ) ) DO 1 I=2»MAX AI=I R ( I ) = R ( I - l ) + A 2 1 R A T I O ( I ) = D S G R T ( R ( I -1 )7 R ( I ) ) / A I 5 SUM2=1.D0 T ER: i = -RAT I 0 ( 1 ) * X - DO 2 1=2*MAX I F ( D A B S ( T E R M ) . L T . 1 . D - 4 ) G O TO 3 SUM2=SUM2+TERM 2 T E R M = - T E R M * R A T I O ( I ) * X 3 RETURN END 12 J i 110 S I d F T C F A L S T 6 I R E A L K P O , L , K P O L , K O L E , K O , K O I , K O L , K O F , K O M I N i D I M F N S I O N L ( 6 ) , T P ( 6 ) , T ( 6 ) , F ( 6 ) D A T A L / 1 0 . , 1 5 . , 2 0 . , 2 5 . , 3 5 . , 4 5 . 1 I R E A D ( 5 » 1 ) !<P0 , A L F A P i C K P O , A L F A P = F I C T I T I O U S V A L U E S O F K A N D A L P H A l 1 F O R M A T ( 2 F 6 . 3 ) R E A D ( 5 , 1 ) A L F A , A ! C A L F A = @ T R U E @ A L P H A , A = ( N A T U R A L T O D O P P L E R L I N E - W I D T H R A T 1 0 ) - - - S O R T ( A L O G ( 2 ) ) K 0 = K P 0 / 5 . 5 K 0 F = 1 0 . * K 0 i K 0 I = K 0 • i D K = K 0 D O 7 1 = 1 , 6 1 K P 0 L = K P 0 * L ( I ) i 7 T P ( I ) = T P K A ( K P O L , A L F A P ) 7 7 W R I T E ( 6 » 3 ) A , A L F A i 3 F O R M A T ( 1 X / / 1 X , 1 7 H L I N E W I D T H R A T I 0 = , F 6 . 3 , 3 X , 5 H A L F A = , F 5 . 2 ) J = l i 9 E 2 = 0 . D O 1 0 1 = 1 , 6 i K 0 L = K 0 * L ( I ) T ( I ) = T K A A ( K 0 L , A L F A , A ) I E ( I ) = T ( I ) - T P ( I ) 1 0 E 2 = E 2 + E ( I ) * E ( I ) E R M S = S Q R T ( E 2 / 6 . ) W R I T E ( 6 , 6 ) ( T P ( I ) , I = 1 , 6 ) 6 F O R M A T ( 1 X / / / 1 X , 2 2 H M E A S U R E D T R A N S M I S S I O N S , 5 W R I T E ( 6 , 8 ) ( T ( I ) , 1 = 1 , 6 ) , K O » E R M S X , 6 C ° . 3 ) 8 F O R M A T ( I X , 2 2 H D O P P L E R T R A N S M I S S I O N S , 5 X , 6 F 9 . 3 / l X , 3 H K 0 = , F 9 . 4 , 5 X , 5 H E R 1 M S = , 3 X , F 6 . 3 ) I F ( U . E Q . 1 ) G O T O 1 5 I F ( E R M S . G E . E M I N ) G O T O 1 1 E M I N = E R M S K 0 M I N = K 0 G O T O 1 1 1 5 E M I N = E R M S K O M I N = K 0 J = 2 1 1 K 0 = K 0 + D K I F ( K 0 . L E . K 0 F ) G O T O 9 I F ( K O M I N . E Q . K O F ) G O TO 2 0 I ? I F ( K O M I N . E Q . K 0 I ) G O TO 2 0 £ K O = K O M I N - D K 17 0 K O I = K 0 s K 0 F = K 0 M I N + D K 9 D K = D K / 5 . L % I F ( D K . L E . 1 . E - 4 ) S T 0 P 8 G O T O 7 7 6 2 0 W R I T E ( 6 , 2 1 ) 0l# 2 1 F O R M A T ( 1 X / 1 X » 2 2 H E N D - P T . M I N I M U M » 3 E W A R F . ) 11 S T O P • ' E N D F U N C T I O N T P K A ( X , Y ) N = 1 0 A l = 2 . A N = N H = A 1 / A N S U M E = 0 . S U M O = 0 . L = l K = 0 D O 2 I = 1 » N A I = I - 1 W = A I * H W A = W * Y W 2 = W * W W A 2 = W A * W A E = E X P ( - W A 2 E 2 = E X P ( - W 2 ) F = E 2 * E X P ( - X * E ) T I F ( I . E Q . 1 ) G O TO 4 I F ( L . E Q . 2 ) G O TO 5 12 I F ( K . E Q . O ) G O TO 7 4 11 S U M O = S U M O + F | i o G O T O 7 5 • 9 7 4 S U M E = S U M E + F • 8 7 5 I F ( I . E Q . N ) G O T O |7 G O T O 2 • 6 4 F I = F I 12 11 9 8 | 7 G O T O 2 ! 6 W = A 1 1 = 2 ' • 1 I i G O T O 3 1 5 F F = F . - 2 K = l - < S U M 1 = ( 1 . 3 3 3 3 3 * S U M O + . 6 6 6 6 7 * S U M E + . 3 3 3 3 3 * ( F I + F F ) ) * H H = 1 . / (. A 1 * A N ) S U M E = 0 . S U M O = 0 . L = l K = l • N 1 = N - 1 D O 1 0 I = 1 » N 1 A I = I W = A I * H i W = l . / W I 1 3 - W 2 = W * W 1 W A = W * Y W A 2 = W A * W A 1 E = E X P ( - W A 2 ) ' E 2 = E X P ( - W 2 ) F = E 2 * E X P ( - X * E ) * W 2 I F ( 1 . F Q . 2 ) G O T O 1 1 I F ( K . - E Q . O ) G O T O ' 7 6 S U M O = S U M O + F . •' G O T O 7 7 i 7 6 S U M E = S U M E + F 1 j 7 7 I F ( I . E Q . N l ) GO T O 1 2 G O T O 1 0 1 i 1 2 W = A I 1 = 2 ' • •• G O T O 1 3 1 1 . F F = F 1 0 K = l - K S U M 2 = ( 1 . 3 3 3 3 3 * S U M O + . 6 6 6 6 7 * S U M E + . 3 3 3 3 3 * F F ) * H S U M = S U M 1 + S U M 2 ' • • T P K A = 2 . * S U M / 1 . 7 7 2 4 5 R E T U R N E N D • • • F U N C T I O N T K A A ( X » Y > Z ) i I i N = 1 0 I A l = 2 . j A A = Y * Z A N = N I 7 H = A l / A N S U M E = 0 . i i S U M O = 0 . L = l K = 0 D O 2 I = 1 » N i A I = I - l • W = A I * H 3 W A = W * Y W 2 = W * W W A 2 = W A * W A i V = V O I G T ( W A » A A ) V V = V O I G T ( W » Z ) F = V V * E X P ( - X * V ) i I F ( I . E Q . 1 ) G O TO 4 - I F ( L . E Q . 2 ) G O TO 5 I F ( K . E Q . 0 ) G O TO 7 4 S U M O = S U M O + F G O T O 7 5 7 4 S U M E = S U M E + F 7 5 I F ( I . E Q . N ) G O T O G O T O 2 6 4 F I = F G O T O 2 6 W = A 1 L = 2 G O T O 3 5 F F = F 2 K = l - K S U M 1 = ( 1 . 3 3 3 3 3 - - S U M O + . 6 6 6 6 7 - - - S U M E + . 3 3 3 3 3 - " - ( F I ' + F F ) ) * H H = l o / ( A 1 * A N ) S U M E = 0 . S U M O = 0 . L = l K = l • N 1 = N - 1 D O 1 0 1 = 1 , N l W = A I * H | 9 W = 1 . / W \ L 1 3 W2 = W * W , ; \ _ J 8 W A = W * Y ' ! 6 W A 2 = W A * W A V = V O I G T ( W A , A A ) : ; [»l V V = V O I G T ( W , Z ) JZI- F = V V * E X P ( - X * V ) * W 2 I F . ( L . E Q . 2 1 G Q T O 1 1 I F ( K . E Q . O ) G O T O 7 6 S U M O = S U M O + F G O T O 7 7 7 6 S U M E = S U M E + F 7 7 I F ( I . E Q . N 1 ) G O T O 1 2 G O T O 1 0 1 2 W = A I L = 2 G O T O 1 3 1 1 F F = F 1 0 K=l-K S U M 2 = ( 1 . 3 3 3 3 3 » S U M O + . 6 6 6 6 7 * S U M E + . 3 3 3 3 3 * F F ) * H S U M = S U M 1 + S U M 2 T K A A = S U M / . 8 8 6 2 • R E T U R N E N D i y. F U N C T I O N V Q I G T ( X I N , Y I N ) T D I M E N S I O N A ( 4 2 ) » H H ( 1 0 ) » X X ( 1 0 ) | v D I M E N S I O N R A ( 3 2 ) » C A ( 3 2 ) » . R 5 ( 3 2 ) , C a ( 3 2 ) , 8 ( 4 4 ) , A K ( 5 ) . , A M ( 5 ) , D Y ( 4 ) '2 D A T A H H ( 1 ) , H H ( 2 ) , X X ( 1 ) , X X ( 2 ) , A / 0 . 8 0 4 9 1 4 0 9 E - 0 0 , 0 . 8 1 3 1 2 3 3 5 E - 0 1 , 0 . 5 7 4 _±L 1 6 4 7 6 2 , 1 . 6 5 0 6 8 0 1 2 , 0 . , 0 . 1 9 9 9 9 9 9 9 E 0 » 0 . , - 0 . 1 8 4 E 0 , 0 . , 0 . 1 5 5 8 3 9 9 9 , 0 . , - 0 |10 2 . 1 2 1 6 6 4 , 0 . , 0 . 8 7 7 0 8 1 5 9 E - 1 , 0 . , - 0 . 5 8 5 1 4 1 2 4 E - 1 , 0 . , 0 . 3 6 2 1 5 7 3 E - l , 0 . , - 0 . 9 3 2 0 8 4 9 7 6 5 E - 1 , 0 . , 0 . 1 1 1 9 6 0 H E - 1 , 0 . , - 0 . 5 6 2 3 1 8 9 6 E - 2 , 0 . , 0 . 2 6 4 8 7 6 3 4 E - 2 > 0 . 8 4 , - 0 . 1 1 7 3 2 6 7 E - 2 , 0 . , 0 . 4 8 9 9 5 1 9 9 E - 3 , 0 . , - 0 . 1 9 3 3 6 3 0 S F - 3 , 0 . , 0 . 7 2 2 8 7 7 4 5 E - 4 |7 5 , 0 . , - 0 . 2 5 6 5 5 5 1 2 E - 4 , 0 . , 0 . 8 6 6 2 0 7 3 6 E - 5 , 0 . , - 0 . 2 7 8 7 6 3 7 9 E - 5 , 0 . , 0 . 8 5 6 6 8 7 3 6 6 6 E - 6 , 0 . , - 0 . 2 5 1 8 4 3 3 7 E - 6 , 0 . , 0 . 7 0 9 3 6 0 2 2 F - 7 5 4 0 1 X = X I N £ » Y = Y I N - t'i X 2 = X * X - s Y 2 = Y * Y 9 » I F ( X - 7 . ) 2 0 0 , 2 0 1 , 2 0 1 L 2 0 0 I F ( Y - 1 . ) 2 0 2 , 2 0 2 , 2 0 3 8 2 0 3 R A ( 1 ) = 0 . 6 - C A ( 1 ) = 0 . 01 R B ( 1 ) = 1 . 11 C B ( 1 ) = 0 . Zl » R A ( 2 ) = X t C A ( 2 ) = Y 1 R B ( 2 ) = . 5 - X 2 + Y 2 C B ( 2 ) = - 2 . * X # Y C B 1 = C B ( 2 ) C A 1 = 0 . U V 1 = 0 . D O 2 5 0 J = 2 , 3 1 J M I N U S = J - l J P L U S = J + l F L O A T J = J M I N U S R B 1 = 2 . - : ; - F L O A T J + R B ( 2.) R A l = - F L O A T J * ( 2 . - » - F L O A T J - l o ) / 2 . R A ( J P L U S ) = R B 1 * R A ( J ) - C B 1 * C A ( J ) + R A 1 * R A ( J M I N U S ) - C A 1 * C A ( J M I N U S ) C A ( J P L U S ) = R B 1 * C A ( J ) + C 3 1 * R A ( J ) + R A 1 * C A ( J M I N U S ) + C A 1 * R A ( J M I N U S ) R B ( J P L U S ) = R B 1 * R B ( J ) - C 3 1 * C S ( J ) + R A 1 * R S ( J M I N U S ) - C A 1 * C 3 ( J M I N U S ) C B ( J P L U S ) = R 3 1 * C ° ( J ) + C 3 1 * R 3 ( J ) + R A 1 * C ? ( J M I N U S ) + C A 1 * R B ( J M I N U S ) U V = ( C A ( J P L U S ) * R 3 ( J P L U S ) - R A ( J P L U S ) * C B ( J P L U S ) ) / ( P B ( J P L U S ) * R 3 ( J P L U S ) + i 1 C B ( J P L U S ) * C B ( J P L U S ) ) I I F ( A . B S ( U V - U V l ) - l . E - 7 ) 2 5 1 , 2 5 0 * 2 5 0 j 2 5 0 U V 1 = U V 2 5 1 V O I G T = U V / l . 7 7 2 4 5 4 R E T U R N 2 0 2 I F ( X - 2 . ) 3 0 1 , 3 0 1 * 3 0 2 3 0 1 A R N T = 1 . 12 M A X = 1 2 . + 5 . * X 2 K M A X = M A X - 1 (10 D O 3 0 3 K = 0 , K M A X 9 A J = M A X - K 8 3 0 3 A R N T = A R N T # ( - 2 . * X 2 ) / ( 2 . * A J + 1 . ) + 1 . I 7 U = - 2 . * X - " - A R N T jS 6 G O T O 3 0 4 j i s " Ii 4 |« 3 . ; J 3 0 2 I F ( X - 4 . 5 ) 3 0 5 , 3 0 6 , 3 0 6 3 0 5 B ( 4 3 ) = 0 . B ( 4 4 ) = 0 . J = 4 2 1 D O 3 0 7 K = l , 4 2 1 3 ( J ) = o 4 * X * E ( J +1 ) - 8 ( J + 2 ) + M J ) 1 3 0 7 J = J - 1 ! U = B ( 3 ) - B ( 1 ) G O T O 3 0 4 3 0 6 A R N T = 1 . M A X = 2 . + 4 0 . / X . | A M A X = M A X . : D O 3 0 8 K = l , M A X I A R N T = A R N T * ( 2 . * A M A X - i . ) / ( 2 . * X 2 ) +1 . i 3 0 8 A M A X = A M A X - 1 . 1 U = - A R N T / X ; 3 0 4 V = 1 . 7 7 2 4 5 4 * E X P ( - X 2 ) \ H = . 0 2 ; J M = Y / H 1 I F ( J M ) 3 1 0 , 3 1 1 , 3 1 0 .1 3 1 1 H = Y . i 3 1 0 z = o . ' ! L=O • • • ! D Y ( 1 ) = 0 . "I 3 1 2 D Y ( 2 ) - - - ; / 2 . ' D Y ( 3 ) = D Y ( 2 ) • ! D Y ( 4 ) = H i 3 1 8 A K ( 1 ) = 0 . ' ! A M ( 1 ) = 0 . i D O 3 1 3 J = i , 4 Y Y = Z + D Y ( J ) 1 U U = U + . 5 * A K ( J ) : V V = V + • 5 * A M ( J ) 1 ' A K ( J + l ) = 2 . - ( Y Y - x - U U + X - V V ) * H • , A M ( J + l ) = - 2 . * ( l . + X * U U - Y Y * V V ) * H I F ( J - 3 ) 3 1 3 , 3 1 4 , 3 1 3 3 1 4 A K ( 4 ) =2 . - » - A K ( 4 ) A M ( 4 ) = 2 . * A M ( 4 ) 3 1 3 C O N T I N U E Z = Z + H L = L + 1 U = U + . 1 6 6 6 6 6 7-"- ( A K ( 2 ) + 2 . * A K ( 3 ) + A K ( 4 ) + A K ( 5 ) ) 3 1 5 V = V + . 1 6 6 6 6 6 7 * ( A M ( 2 ) + A M ( 3 I F ( J M ) 3 1 5 , 3 2 0 * 3 1 5 I F ( L - J M ) 3 1 3 , 3 1 7 * 2 2 0 ) + A M ( 3 ) + A M ( 4 ) + A M ( 5 ) ) 3 1 7 A J M = J M H = Y - A J M * H G O T O 3 1 2 3 2 0 2 0 1 V O I G T = V / 1 . 7 7 2 4 5 4 R E T U R N F 1 = 0 . 3 3 0 D O 3 3 0 J = l , 2 F l = F l + H r i ( J ) / ( Y 2 + ( X - X X ( J ) 1 ) ) -» ( X - X X ( J ) ) ) + H H ( J ) / ( Y 2 + ( X + X X ( J ) ) * ( X + X X ( J ) ) V O I G T = Y * F l / 3 . 1 4 1 5 9 2 7 R E T U R N E N D -124- APPENDIX 6. END CORRECTION ATTEMPTS An i n v e s t i g a t i o n i s undertaken here i n t o the e f f e c t s the ends of the absorbing column have on the t r a n s m i s s i o n (see s e c t i o n 5o4 .2.7). The geometry of the ends of the discharge are drawn i n P i g . A6.1. The c y l i n d r i c a l discharge has an outer r a d i u s R and bends c i r c u l a r l y at the ends through 90° w i t h an inner and outer r a d i u s of curvature D and D + 2R r e s p e c t i v e l y . I t i s assumed k A and n are constant i n the ends of the discharge. o • Consider a s i n g l e ray of l i g h t which leaves the s t r a i g h t p o r t i o n of the discharge at p o s i t i o n r,© i n the c r o s s - s e c t i o n of the annulus of rad i u s r ( r = 1.07 cm). I t emerges from the curved p o r t i o n of the discharge at p o i n t B a f t e r t r a v e r s i n g a d i s t a n c e <̂ t( & ) on plane A. <?C may be computed from the geometry on the end as, a f u n c t i o n of 9 . The t o t a l d i s t a n c e t r a v e l l e d i s JLQ + 2oC JtQ being the l e n g t h of the s t r a i g h t p o r t i o n of the discharge* Let £,(.9) be d i v i d e d i n t o two segments, z and A z( 6 5 ) , where z i s independent of 9 . Therefore f o r each p a r a l l e l ray of l i g h t the i n t e n s i t y out of the discharge i s where L = Jt^ + 2z, and the t r a n s m i s s i o n i s then o 7 T = ..(A6.1)  -126- I f the bottom of the annulus ( i , e. the s i d e toward the e l e c t r o d e s ) i s s y m m e t r i c a l l y b l o c k e d o f f at an a n g l e &M. (see F i g . A6.2) and the u s u a l assumption of Doppler l i n e - s h a p e s h o l d s , equation (A6.1) becomes, -? \ ' ' T = ...(A6.2) where Z \ r ^ = — J £ A ^ y " c l © and cm O H ) n t f Q L ) F i g . A6.2 Annulus w i t h P o r t i o n Blocked Off -127' That Pm tends to zero rapidly as m increases can be seen for k Q « l , where the presence of the term k^ ensures the rapid decrease; on the other hand the same tendency is present as kQ increases because although the k̂  factor is then not so small, the Integral (or sum) Pm is . At any rate it seems advantageous to try to choose conditions so that the first correction term (T^ is small or zero by making £"~z as small as possible. The f irst case which was tried was to choose L equal to the distance between the centres of the electrodes, i . e. z = D + R and calculate ^ on the computer such that £ ^ - 0 . The results of this calculation are shown in Table A6.1 using some experimentally determined values of kQ and a for the case in which the end-effects should be the greatest, which corresponds to L = 10 cm. It was found that A J<0 for O == B ̂ TT which means the effective absorption length for the first order correction is less than the distance between the electrodes for any O , although | A/^| is a minimum at & =TTwhere A£ has the value -.08 cm. Also included in this table are the succeeding terms (Tm) of equation (A6.2) up to m = 4. It can be seen that even for the highest value of kQ , the correction necessary is not greater than .003. For absorption lengths greater than 10 cm the correction terms rapidly decrease. Table 6 shows that kQ attains this highest value only for X.6402, Because the value of kQ is uncertain due to its "fictitious "quality (see section 5 .^.1), as well as the unjustified but simplifying assumption of constant conditions in the ends, it is apparent 128- \ TABLE A6.1 END-CORRECT I ON ATTEMPT #1: 0-1T ,L • 10 cm cm . ^e= .©an on" 1 9 3 CAW-1 "Tim "Ton l -.16 .019 - . 0 0 3 .03 - . 0 0 5 .04, -.006 2 2.26 .0005 .001 .001 .002 .004 ,009 . 3 «•»16 «10"' 5 « - 1 0 " 6 .0001 .0006 1 — 4 «»10 k . 8 2 « 1 0 ~ 7 wIO"7 «10" 6 « 1 0 ~ 6 wlO"4 TABLE A6.2 END-CORRECTION ATTEMPT #2. = 1.99 cm, R + D = 2.03 cm, L =* 10 cm m ( - 2 ) m ^ 7 m ! Tm kQ=i.0217 kQ=.082 kQ=.193 1 0 0 0 0 2 2.26 .001 - .002 .00Q 3 - . 3 5 « - 1 0 ' 6 « - 1 0 " 5 -4 «-10 * k .63 « 1 0 ~ 7 « 1 0 " 6 « 1 0 ~ 4 -129- that these corrections cannot be quantitatively applied. It is likely however, that those spectral lines Table 6 indicates as having a small value of kQ do not need end-effect corrections; this cannot be said of lines like \6402 quite so definitely. The second correction attempt sought an effective absorption length for a fixed value of 0 , which in this case was chosen as TT . Again T^ was to be as small as possible so a value of z was looked for such that A z = 0. For this equality to occur, z has the value 1.99 cm. D + R for the absorp- tion tube (see Fig. A6.1) is 2.03 cm so the effective absorp- tion length is essentially equal to the distance between the electrodes. This of course follows from the first correction attempt i f S^TT a n d the general comments given concerning that attempt are generally applicable here also. The first four terms of the correction series are given in Table A6.2. The three computer programs used to compute F m and A ^ ™ are given below; the first two were used in the first end-correction attempt, and the f irst and the third in the second attempt. The only remarks necessary concerning these programs is that the presence of two isotopes was taken into account, and that numerical integrations were done using Simpson*s Rule. 1 SIBFTC NDC0R1 | 1 FORMAT(314) 2 76 FORMAT(2F10.3) FORMAT(IX > F6.4> 2X » F6.2 * 2X >F4.0 » 2X D0U3LE PRECISION T ? T ER M ? F 6 o 4 9 2 X , 3 14) C DIMENSION RATI0(100)»R(100) s E K(5C R E A D ( 5 J 1 ) N » M1 J M 2 N=NO. OF CARDS 5M1=MAX.N0. OF TERMS, ) s A ( 5 0 ) M 2 = 0 R D E R OF CORRECTION C 25 R E A D ( 5 9 2 ) ( E K ( I ) , A ( I) 9 I = 1 ,N) EK( I )= K (0) 9 A( I ) =ALPHA DO 70 I = 1 9 N EL=10. J l = l A 2 = A ( I ) * * 2 1 AM2=M2 RR=1.+AM2*A2 RA=1./SORT(RR) R(1)=RR+A2 RAT 10( 1)=5QRT(RR/R ( 1 ) ) . DO 6 K=2,Ml 6 AK = K R ( K ) = R ( K - l ) + A 2 RAT 10 ( K ) = SGRT ( R ( !<- 1 ) /R ( X ) ) / A X C 624 MAJOR SERIES EKM2 = EK ( I ) *-"-M2 T = E K M 2 -»-R A EKL = EK ( I ) - " -EL E K L 1 = - E K L * R A T I 0 ( 1 ) TERM=T-x-Ei<Ll 28 M = 2 I F ( ABS ( TERM ). LT . .0.001') GO TO 17 T=T+TERM I F ( M . G E . M l ) GO TO 17 E<L2=EKL*RATI0(M) TERM =-TERM-*Ei<L2 1 7 M = M+1 GO TO 28 M 3 = M C MINOR S E R I E S E K 2 = ( . 104 - - - E X ( I ) )>#y.2 T 1 = E K M 2 R A EKL= . 104-x-EK ( I E K L 1 = - E K L * R A T I 0 ( 1 ) T E R M = T 1 * E K L 1 : 1 2 8 M = 2 I F ( A B S ( T E R M ) . L T . . 0 0 0 1 ) GO T O T 1 = T 1 + T E R M 1 1 7 i i ' 1 i I F ( M . 6 E . M 1 ) GO T O 1 1 7 1 E K L 2 = E K L - " - R A T l O ( M ) ! T E R M = - T E R M E K L 2 1 1 1 7 M = M + 1 G O T O 1 2 8 T = ( T + T l ) / l . 0 9 4 I 5 0 W R I T E ( 6 , 7 5 ) El< ( I ) , A ( I ) , Z L , T , M 3 G O T O ( 5 0 , 5 0 , 5 0 , 5 3 , 5 3 , 7 0 ) , J 1 E L = E L + 5 • ! 5 3 J 1 = J 1 + 1 G O T O 6 2 4 E L = E L + 1 0 . 7 0 J 1 = J 1 + 1 G O T O ' 6 2 4 C O N T I N ' J E i 6 0 G O T O ( 6 0 , 6 1 , 6 2 , 6 3 ) , M 2 M 2 = 2 G O T O 2 5 i i ! 6 1 6 2 M 2 = 3 G O T O 2 5 M 2 = 4 ! i 6 3 G O T O 2 5 S T O P E N D i i i ! $ I B F T C C N D C O R 2 T O C O M P U T E M O M E N T S C F D E L T A i L F O R E N D C O R R E C T I O N S O F A B S T U B E I C 1 R E A D ( 5 , 1 ) N N = 0 RD E R O F M O M E N T F O R M A T ( 1 2 ) 1 ! c T H E T A = 3 . 1 4 1 5 9 M = 5 0 M = D I V I S I O N O F I N T E G R A T I O N I i M T E R V A L ; R = 1 . 2 7 ! £ 1 s ' 9 D = . 7 6 i L * R R = 1 . 0 7 : 8 R D = R + D ', 6 R 2 = R * R Oil R R 2 = R R * R R ! U A M = M 1 z i D O 8 J = l , 5 0 H = T H E T A / A M SU|ViE = 0 . SUMO= 0 . K = 0 D O 2 I = 1,M A I = I - 1 X = A I * H 6 R M = R D + S 3 R T ( R 2 - R R 2 * ( S I N ( X ) * * 2 ) ) E L = S Q R T ( R M * R M - ( R D + R R * C O S ( X ) ) * * 2 ) D E L = E L - R D F = D E L * * N I F ( I . E Q . 1 ) GO T O 3 I F ( L . E Q . 2 ) G O T O 4 I F ( K . E Q . l ) S U M O = S U M O + F I F ( K . ' E Q . O ) S U M E = S U M E + F I F ( I . E Q . M ) G O T O 5 G O T O 2 3 F I = F G O T O 2 I 5 X = T H E - T A L=_2 ? G O T O 6 Vr 4 F F = F 12 2 K = l - K J i S U M = 1 . 3 3 3 3 * S U M O + . 6 6 6 6 7 * S U M E + . 3 3 3 3 3 * ( F I + F F ) II 0 S U M = 2 . * S U M * H / T H E T A 9 A N G L E = T H E T A * 1 8 0 . 7 3 . 1 4 1 5 9 3 W R I T E < 6 > . 7 ) S U M > A N G L E 17 7 F O R M A T ( I X , E 1 4 . 7 » 4 X , F 6 . 2 ) 6 8 T H E T A = T ; - I ' E T A - 3 . 1 4 1 5 9 / 5 0 . 5 S T O P »4 c N D 61 M M ID 0 _ 0 ) 0 > c: c: II II ro i — • • C) 1 0 C >.») 1.0 :t co o + U l CJN 3 ON <IN -.1 m + 0 o 1 0 '. J ; o A "11 O r~ II "Tl O n I—' II IV> 1 "Tl —1 A o 0* "n + "Tl CD T| O — II o t-j rv) u i O • 1—1 t — i 1—1 i—i "Tl Tl Tl Tl Tl A A r~ - H • O • e M rn rn rn rn £ 3 o O a o 0 o o o e , o 1—' rv) I—1 cn Co Co Q CD o cz C O O - i rn 6 - i —1 o n II O Co Co Ul c c -p- 10 . - ;n o + + T) T I rn II m r~ o i~ ll m M co r n o M rn A 3 * r~ —i * I ~ 2 N D rv> 0 " - A ) x II n > x i >— + co o A ; XI ru i A 3 A J I V ) I I 1 C/l 2 L. CRA IN L I M I T E D O A O II O ro — o 4> A ) A 3 (NJ A 3 II A3 rv> II * A 3 A 3 XI A3 A ) A 3 O a A 3 II II II o A 3 (-". -J + • ON o o A 3 2 II II -J U l » o ro - J w _ n —I n A 3 m > 2 o o — n Ul o A 3 N W If) ^1 M u i en s j oo to m c o 2 —I o o t—( M Tl "Tl II O Ni 73 1 o o > — 1 m — « X ~ .• N* rn i—• — 4 > e o ~J o N. -f> — 1 X o V9 NO o © R. l_. C R A I N U I M L T E O ^ o i « o N © i n ^ -130- REFERENCES / ! / A. Einstein, Verhandl. Deut. Phys. Ges. 18, 318 (1916). A. Einstein, Physik. Z. 18, 121 (1917). /2/ B. M. Glenhon and W. L. Wiese, N.B.S. 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