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Positron annihilation in unsaturated, saturated and liquid argon with applied electric field Albrecht, Robert Stephen 1977

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POSITRON ANNIHILATION IN UNSATURATED, SATURATED AND LIQUID ARGON WITH APPLIED ELECTRIC FIELD by ROBERT STEPHEN ALBRECHT B.Sc, University of B r i t i s h Columbia, 1969 M.Sc, University of B r i t i s h Columbia, 1973 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES i n the Department of PHYSICS We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA November, 1977 © R o b ert Stephen Albrecht, 1977 In presenting th i s thes is in pa r t i a l fu l f i lment of the requirements for an advanced degree at the Univers i ty of B r i t i s h Columbia, I agree that the L ibrary sha l l make i t f ree l y ava i l ab le for reference and study. I further agree that permission for extensive copying of th i s thesis for scho lar ly purposes may be granted by the Head of my Department or by his representat ives. It is understood that copying or pub l i ca t ion of th is thes is fo r f inanc ia l gain sha l l not be allowed without my writ ten permission. Department of / ^ / / V/V C f The Univers i ty of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date ?/<h9-i i ABSTRACT The annihilation rate, X , of positrons i n unsaturated, saturated and l i q u i d argon has been measured over the temperature range of 100 K to 148 K with applied e l e c t r i c f i e l d s up to 40 V cm 1 amagat 1. The data was analysed i n terms of the c l a s s i c a l d i s t r i b u t i o n for s e l f -nucleating clusters which depends on the difference i n chemical potential, Ag, between the gas state and metastable l i q u i d state (at the same temperature and pressure). Results of th i s analysis suggest that the low temperature enhancement of the ann i h i l a t i o n rate arises from interactions of the positron with clusters of argon atoms, the size of the clusters being surprisingly small, of the order of 10 atoms or less. In addition, experimental evidence supporting the value of the pre-exponential factor determined by Kikuchi (1969) was also obtained. A growth model, consisting of a free and loc a l i z e d positron population was developed which accurately describes the saturation characteristic of the r e s u l t s , i n particular the t r a n s i t i o n i n the annihilation rate from a density proportional dependence to a rate, X , l i n e a r l y dependent on Ag. For the saturated gas, for example, where , -1 -1 * -1 Ag = 0, XQ/D = 40.0±0v6 ys amagat and X = 1.2 ns i i i TABLE OF CONTENTS page ABSTRACT i i LIST OF TABLES v i i LIST OF FIGURES v i i i ACKNOWLEDGEMENTS x CHAPTER ONE POSITRON ANNIHILATION IN DENSE AND LOW TEMPERATURE GASES 1 1.1 Introduction 1 1.2 Positron-atom Interactions 5 1.2.1 Free positron processes (Z f f Description) . . . 5 1.2.1.1 Direct A n n i h i l a t i o n 6 1.2.1.2 Experimental Results for 7 1.2.2 Modifications to the Z f f Description 7 1.2.2.1 "Single-Atom" E f f e c t s 7 1.2.2.1.1 Resonances . 7 1.2.2.1.2 Bound States 9 1.2.2.2 Many-Body E f f e c t s 10 1.2.2.2.1 Random Systems 10 1.2.2.2.2 Fluctuons 12 1.2.2.2.3 Clustered Systems 14 1.2.3 Positronium And Bubble Formation 14 1.3 Generalized Features of Positron Swarm Experiments ' . . 16 1.3.1 Slowing of Positrons i n A Gas 16 1.3.2 Positron V e l o c i t y D i s t r i b u t i o n . . . 18 1.3.3 E x i s t i n g Experimental S i t u a t i o n 21 1.4 Summary of Thesis Content 25 CHAPTER TWO NUCLEATION THEORY 27 2.1 Introduction . . 27 2.2 Energy Barrier to Nucleation 29 2.3 C l a s s i c a l Phase Integral 32 2.4 Structure of a Cluster 34 i v page 2.5 E f f e c t of Ions 35 2.6 Experimental Sit u a t i o n . 37 2.7 Positrons as a Probe for Clusters 40 CHAPTER THREE EXPERIMENTAL PROCEDURE AND RESULTS 43 3.1 Outline of Experimental Technique 43 3.1.1 Experimental Determination of Lifetime Spectrum 43 3.1.2 Analysis of a Low Temperature Spectrum . . . . 43 3.2 Experimental Method 52 3.2.1 Chamber Configuration 52 3.2.1.1 Chamber!Design 52 3.2.1.2 E l e c t r i c F i e l d Grid 58 3.2.1.3 Source Construction and Strength . . . . 60 3.2.1.4 Sample Gas P u r i f i c a t i o n 61 3.2.1.5 Attenuation and Scattering of Gamma-Rays 63 3.2.2 Parameter Control and Measurement • 63 3.2.2.1 Low Temperature 63 3.2.2.2 Pressure • 68 3.2.2.3 E l e c t r i c F i e l d 70 3.2.3 E l e c t r o n i c s 7 1 3.2.3.1 General Description • • • • 71 3.2.3.2 Pulse-Height Resolution 75 3.2.3.3 System L i n e a r i t y 75 '3.2.3.3.1 Integral L i n e a r i t y 75 3.2.3.3.2 D i f f e r e n t i a l L i n e a r i t y . . . . . 78 3.2.3.4 Time Resolution of the System . . . . . 80 3.2.3.5 Pulse Pileup and Background E f f e c t s - . 82 3.2.4 Data Reduction . ' 8 4 3.2.4.1 Ca l i b r a t i o n s 84 3.2.4.1.1 Temperature 84 3.2.4.1.2 Density . . . . . 85 3.2.4.2 Difference i n Gibbs Free Energy . . . . 85 3.2.4.3 Curve F i t t i n g Technique 89 3.2.4.4 Determination of Shoulder Width 91 V page 3.3 Presentation of Results 92 3.3.1 Summary of Data Runs 92 3.3.1.1 Random Background and El e c t r o n i c S t a b i l i t y • 93 3.3.1.2 Choice of Experimental Region 93 3.3.1.3 F i t t i n g the Ortho-Positronium Lifetime . . . 94 3.3.1.4 Goodness of F i t 96 3.3.1.5 Equipment F a i l u r e 96 3.3.1.6 Detection of a Saturated Gas or Liqu i d • • • 96 3.3.1.7 Approach to Equilibrium 97 3.3.2 Density Range Investigated 98 3.3.3 Ortho-Positronium A n n i h i l a t i o n Rate 3.3.4 Equilibrium Positron A n n i h i l a t i o n Rate 1 0 3 3.3.4.1 Saturated Gas and Liqu i d . . . . . 3.3.4.1.1 Density Dependence of X i r\-j 3.3.4.1.2 Temperature Dependence of A £  3.3.4.2 Unsaturated Gas 1 0 9 3.3.4.2.1 Density Dependence of A g 1 0 9 3.3.4.2.2, Temperature Dependence of ^ H 3 3.3.4.2.3 Density Dependence of Ag/D 1 1 3 3.3.4.2.4 Phenomenological Analysis . . . . . . 115 3.3.5 E l e c t r i c F i e l d Results f o r A /D 1 1 7 e 3.3.5.1 Saturated Gas 1 1 7 3.3.5.2 Unsaturated Gas • 1 1 7 3.3.6 Gibbs Free Energy Dependence 123 3.3.7 Temperature Dependence of A /D for 21 Amagat Gas • • 131 3.3.8 Time-Dependent A n n i h i l a t i o n Rate • • > 1 3 1 3.4 Error Analysis • . 3.4.1 Counting S t a t i s t i c s and Uncertainty i n A g and Ag/D • 1 3 5 3.4.1.1 El e c t r o n i c S t a b i l i t y 1 3 5 3.4.1.2 Temperature » 1 3 6 3.4.1.3 Gas Density X J / 3.4.1.4 E l e c t r i c F i e l d • 1 3 8 3.4.1.5 Difference i n Gibb's Free Energy 1 3 9 3.4.1.6 E f f e c t of F i n i t e Resolution • • • 1 3 9 3.4.2 Gas P u r i t y 1 4 0 VI page CHAPTER FOUR ANALYSIS OF RESULTS 1 4 5 4.1 Outline of Procedure -*-4^ 4.2 Inadequacy of Standard Analysis ^ 4^ 4.3 Cluster Model • 1 4 6 4.4 Growth Model • ' 1 5 5 1 4.4.1 Application to the Saturated Gas 4.4.2 Application to the Unsaturated Gas 4.4.2.1 Screening Region 1~'7 1 4.4.2.2 Dense Gas Region 4.4.23 100 K Gas 1 6 0 4.4.3 Dissociation Scheme . • • 1 ftl 4.4.4 Application to E l e c t r i c F i e l d Results 4.5 Temperature Dependence of Fluctuons • • . • • CHAPTER FIVE DISCUSSION 1 6 5 5.1 Clusters: Small or Large 1 6 5 5.2 Evidence for Clusters i n Unsaturated Argon 5.2.1 Previous Workers 1 6 6 5.2.2 Positron Annihilation 1 6 7 5.2.2.1; Self-Nucleating Clusters 1 6 8 5.2.2.2; Effect of Ions 1 6 8 5.2.2.3 Effect of Dimers 1 6 9 1 6Q 5.2.2.4 Phenomenological Theory x v y 5.2.2.5 . Pre-Exponential Factor and Surface Energy • • ^® 171 5.2.2.6 Screening Region 5.3 Growth Model 1 7 1 172 5.4 Localized Annihilation Rate 173 5.5 Diffusion Equation and Many-Body Considerations • • • • 5.6 Effect of an E l e c t r i c F i e l d 1 7 4 CHAPTER SIX CONCLUSIONS 1 7 6 •t -jc. 6.1 Summary of Results 6.2 Outline-of Possible Future Studies 1 7 7 BIBLIOGRAPHY . . • 1 7 9 APPENDIX A LIQUID NITROGEN FILLER SYSTEM 1 8 4 APPENDIX B DATA • • 1 8 7 v i i LIST OF TABLES TABLE I Ortho-Positronium Results . • • TABLE II Saturated Gas A n n i h i l a t i o n Rate TABLE III Zero F i e l d A n n i h i l a t i o n Rate • • TABLE IV Phenomenological Results f o r i TABLE V E l e c t r i c F i e l d A n n i h i l a t i o n Rate TABLE VI Cluster Model Results TABLE VII Surface Energy E f f e c t s TABLE VIII Pre-Exppnential Factor j' . . . . TABLE IX Results of Ag Dependence for X page 101 106 110 116 118 148 151 152 159 v i i i LIST OF FIGURES page Figure 1 Correspondence Plot f o r A l l Gas and Liquid Argon Data 2 4 Figure 2 Comparison of Low and Room Temperature Time Spectra - Linear Plot 44 Figure 3 Comparison of Low and Room Temperature Time Spectra - Log Plot 4 5 Figure 4 Growth Model Curves ^1 Figure 5 Growth Model F i t to Peak Region ^ 3 Figure 6 Experimental Chamber • • Figure 7 Gas and Coolant Handling System 57 Figure 8 E l e c t r i c F i e l d Electrode Assembly -^9 Figure 9 Gamma-Ray Attenuation - Nal Detector ^ Figure 10 Temperature Co n t r o l l e r 67 Figure 11 Temperature S t a b i l i z e r • • ^ 9 Figure 12 Photomultiplier C i r c u i t 7 2 Figure 13 E l e c t r o n i c Configuration f o r Timing Measurement . . . 73 Figure 14 Photomultiplier Pre-Amplifier C i r c u i t 7 4 Figure 15 Energy Windows and Discriminator Settings -NE 111 S c i n t i l l a t o r 7.6 Figure 16 Integral L i n e a r i t y of Time Measurement 7 7 Figure 17 D i f f e r e n t i a l L i n e a r i t y of Time Meausrement Figure 18 Thermocouple Voltage versus Temperature from Gas Pressure Measurement 8 ^ Figure 19 Temperature Dependent of 'Sp e c i f i c Volume' of Metastable Liquid Argon 8 8 Figure 20 Density Dependence of Free Energy Difference, Ag 9 0 99 Figure 21 Range of Gas Data • IX page Figure 22 Density Dependence of Ortho-Positronium A n n i h i l a t i o n Rate - Gas and L i q u i d 102. Figure 23 Density Dependence of A g - Saturated Gas 104 Figure 24 Density Dependence of A - Saturated Gas and Liqu i d . 6 105 Figure 25 Temperature Dependence of A g - Gas and Liq u i d . . . . 108 Figure 26 Density Dependence of A g - Gas 112 Figure 27 Density Dependence o f A /D - Gas 114 e 8 E l e c t r i c F i l  Dependence of A g/D - Saturated Gas . -1 Figure 29 Density Dependence of A g/D f o r E/D = 8.0 V cm -1 Figure 30 Density Dependence of A p/D for E/D = 15.0 V cm -1 Figure 32 Ag Dependence of A Figure 33 Ag Dependence of A /D Figure 38 F i t to Cluster Model - Ln ( A e/D- A^D) versus Ag 120 amagat ~ - Gas 1^1 amagat - Gas . . ,& 122 Figure 31 Density Dependence of A p for E/D - 15 V cm 125 amagat"^ - Gas . . ,& 124 e 126 u • • • • - - . e Figure 34 Ag Dependence of A for E/D =8.0 and 15.0 V cm--'- ama g a f l 129 Figure 35 Ag Dependence of A /D for E/E =8.0 and 15.0 V cm-!, amagat -! • 130 Figure 36 Temperature Dependence of A /D. S i m i l a r i t y of Saturated Gas and Unsaturated Gas (at 21 amagats) 132 Figure 37 Time Dependence of V e l o c i t y Averaged " D i r e c t " A n n i h i l a t i o n Rate • 133 149 Figure 39 Relative Size Dependence of Volume and Surface Energy of a C l a s s i c a l Cluster Figure 40 Control C i r c u i t f o r Liq u i d Nitrogen Level Sensor • • X ACKNOWLEDGEMENTS I wish to acknowledge the d i f f i c a l t task which my superviaor, Garth Jones, undertook when he ventured into a new f i e l d while giving his support to this l a s t positron venture of room 203. In spite of th i s c o n f l i c t his encouragement, electronic expertise and physicsl insight were invaluable i n bringing t h i s research to completion, Appreciation goes out to Mike Crooks for his helpful suggestions toward the design of the low temperature chamber and to A l Stevenson for his help i n putting together a workable temperature controller and l i q u i d nitrogen s e n s o r / f i l l e r system. Much thanks goes to those i n the machine shop who did an expert job i n the construction of the low temperature chamber. Thanks also to the l a t e Dick Haines, who kept high standards i n the student shop, and to Cy Sedger, the present Student Shop Technitian, for his useful suggestions and help. The help of Ron Deary, Storesman, i n obtaining a continuous supply of l i q u i d nitrogen i s also appreciated. My thanks i s extended to Frank Curzon for his quick translation of a Russian paper, months before the published translation was available. The friendship of Robert Orth, who introduced me to bubbles and clusters, i s much appreciated as were the many beers and meals we shared. Sim i l a r l y , the many discussions and dinners with M.J. w i l l not be forgotten, nor the many hockey games with B a l z a r i n i and Co.. I thank my committee for their suggestions toward the finished thesis and apologize that I did not get to pick their brains more often. x i A l o t must be said of my wife who suffered through many evenings alone i n order that I could do "my thing". This thesis i s dedicated to her. F i n a l l y I mention the encouragement of my parents which was very much appreciated over t h i s long haul. 1 CHAPTER ONE POSITRON ANNIHILATION IN DENSE AND LOW-TEMPERATURE GASES 1.1 Introduction For many years, positron annihilation i n gases has been used to gain insight into the interaction between positrons and single atoms. Although the unperturbed atomic potential i s a t t r a c t i v e for an electron and re-pulsive for a positron, the polarization of the atom by either charged p a r t i c l e leads to an overall a t t r a c t i v e potential for both cases at thermal energies. For th i s reason and because the positron i s d i s t i n -guishable from the atomic electrons, electron scattering theories can be c r i t i c a l l y tested. A comparison of recent electron and positron t o t a l cross-sections i s given by Massey (1976). Implicit i n such analyses i s the line a r dependence of the positron annihilation rate on density. This implies that the positron interacts with only one gas atom at a time (a free positron) and therefore that scattering theories with asymptotic plane waves are relevant. Normally these annihilation rates can be related to the positron interacting with a number of electrons of the order of the atomic number of the atom or the number of valence electrons (Cova and Zappa (1968). However, density proportional "enhanced rates", ten times the expected rate have been observed i n methane up to one tenth the c r i t i c a l density (Smith and Paul (1970), McNutt et a l . (1975). It i s now clear that at low temperatures and high densities positrons i n most noble gases (helium: Canter and Roelling (1970), Canter et a l . (1975); argon: Canter and R o e l l i g (1975)) do not s a t i s f y even this free 2 positron picture. In these gases, i n fact, at densities as low as one twentieth the c r i t i c a l density the annihilation rate approaches that of the l i q u i d . These effects do depend on the medium, however, as the l i f e t i m e of positrons i n neon gas (Canter and R o e l l i g (1974)) i s consistent with the free positron picture even for l i q u i d densities. Other gases, notably methane (at densities greater than one tenth the c r i t i c a l density)(McNutt et a l . (1974)) and ammonia (McNutt et a l (1974) ), also show effects similar to that seen i n helium and argon. However, the effect i n ammonia i s larger, and the annihilation rate approaches the l i q u i d value at densities 1/600 to 1/300 that of the l i q u i d . (That the p o l a r i z a b i l i t y and dipole moment of the atom plays a dominant role i n these phenomena i s suggested by the low p o l a r i z a b i l i t y of the neon atom and the fact that ammonia i s a polar molecule.) It i s thus apparent that two situations e x i s t . The f i r s t i s the presence of an enhanced annihilation rate which retains the proportionality to density. The second i s an enhanced annihilation rate that i s no longer proportional to density. This lack of proportionality to density i s very suggestive of positron " l o c a l i z a t i o n " . Possible explanations, to be discussed more f u l l y below, include: near-thermal resonances, positron-atom bound states, e l e c t r o s t r i c t i o n of gas atoms about a bound positron, many-body effects, and inherent structure of the host gas. Many-body effects seem to characterize the fate of positronium atoms in many of these substances, as we l l . The low annihilation rates for positronium i n the noble gases (Canter et a l . (1975), Canter and R o e l l i g (1975) ) and the narrow peak observed i n angular correlation experiments 3 (Briscoe et a l . (1968)) strongly suggest that the positronium atom forms a bubble i n the gas. Similarly interpreted are the reduced mobilities of electrons i n f l u i d argon (Schnyders et a l . (1966)). Thus, where bubble formation would require a repulsive potential between the charged p a r t i c l e and the gas atoms, that i s a positive scattering length, the p o s s i b i l i t y of the atoms clustering about the positron would require an attractive p o t e n t i a l , a negative scattering length. Both cases w i l l of course depend on the energetics of the medium, i e . the thermodynamics of the host gas. U n t i l quite recently, most positron-atom interactions were studied using "swarm" experiments. These experiments were characterized by the observation of the annihilation rate of positrons while the positron vel o c i t y d i s t r i b u t i o n relaxes to a thermal d i s t r i b u t i o n under i n t e r -actions of the positron with the gas atoms. Now, however, beam experi-ments are rapidly replacing the swarm experiments for determining positron scattering cross-sections i n gases i n the 1 to 20 eV range. Nevertheless, from the most recent swarm results i n the low temperature gases (Canter et a l . (1975), Canter and R o e l l i g (1975)) mentioned above, i t i s certain that the enhancement of the positron annihilation rate occurs for f i n i t e positron energies less than or near thermal energies. Thus beam experiments, because of the poor energy resolution ( t y p i c a l l y an energy width of .1/2 eV) currently available, cannot be used for such studies, and one has to r e l y on the averaged results of swarm experiments for information i n this energy regime. The work presented i n t h i s thesis was i n i t i a t e d i n order to both 4 assess the re p r o d u c i b i l i t y of the results of Canter and R o e l l i g (1974) as well as to extend the density range to higher values. Also, although the annihilation rates i n helium were known to approach the l i q u i d value even at moderate gas densities, i t was not clear that this was the case for argon.. In addition, a s p e c i f i c enhancement mechanism involving the independent presence of self-nucleating clusters of atoms i n the gas was considered. That clusters of gas atoms might provide an enhancement mechanism and possibly an e f f i c i e n t l o c a l i z a t i o n mechanism was f i r s t suggested to t h i s author by Orth (private communication - 1973). . The role of clusters i n the condensation of supersaturated gases i s well known and i t has been suspected (Frenkel (1936a,b; 1946), Band (1939)) that clusters are also of importance i n unsaturated vapors. Dimers have been observed i n unsaturated argon gas at 100 torr and 100 K to concentrations of 400 ppm (Leckenby and Robbins (1966)). The existence of such clusters w i l l c e r t a i n l y affect the l i f e t i m e of thermalized positrons i f only because of large energy losses through i n e l a s t i c c o l l i s i o n s . This effect w i l l manifest i t s e l f i n the temperature and density dependence of the positron l i f e t i m e s , r e f l e c t i n g the corres-ponding dependences i n the d i s t r i b u t i o n of clusters. This concept of considering clusters of atoms as e n t i t i e s i n the i r own r i g h t , fundamental to nucleation theory, i s discussed i n Chapter Two. For free positrons i n a swarm experiment, the l i f e t i m e i s given by the velo c i t y dependent annihilation rate averaged over the velo c i t y d i s t r i b u t i o n of the positron. If an e l e c t r i c f i e l d i s applied, then 5 the l i f e t i m e i s also related to the v e l o c i t y dependent momentum-transfer rate (Lee (1971)). If l o c a l i z e d states of the positron or cl u s t e r s of gas atoms exist then t h i s p i c t ure w i l l be a l t e r e d . This thesis deals experimentally with these aspects of positron a n n i h i l a t i o n i n argon gas, concentrating on the r o l e that c l u s t e r s may play, and observing the e f f e c t of an applied e l e c t r i c f i e l d on the enhanced a n n i h i l a t i o n rates. 1.2 Positron atom int e r a c t i o n s 1.2.1 Free positron processes (Z e££ Description) Positrons with energies greater than 500 eV. experience predominantly i o n i z i n g c o l l i s i o n s with gas atoms. Below 500 eV., down to the f i r s t i o n i z a t i o n l e v e l of the atom, e l a s t i c and i n e l a s t i c c o l l i s i o n s are of comparable importance. Near the i o n i z a t i o n l e v e l formation of positronium s t a r t s to become important and i t s formation i s important down to the threshold energy for positronium formation, E , re l a t e d to the i o n i z a t i o n s energy, E^, of the atom by the equation: E = E. - 6.8 eV. 1-1 s 1 Below t h i s threshold the positrons can only undergo e l a s t i c c o l l i s i o n s with the gas atoms, a n n i h i l a t i o n , and possibly formation of molecular complexes. The p o s s i b i l i t y of the formation of molecular complexes as well as positron-atom c l u s t e r s of a more general type i s the concern of t h i s t h e s i s . D e t a i l s of the above energy regions are discussed by Lee (1971). 6 1.2.1.1 Direct Annihilation The two photon annihilation rate of a positron and electron as calculated by Dirac (1931) i n a n o n - r e l a t i v i s t i c plane-wave approxi-mation i s : 2 v = 4iTr cN , 1-2 a o e 2 2 where r Q i s the c l a s s i c a l electron radius (e /mc ), c i s the ve l o c i t y of l i g h t and N g i s the number density of electrons. In order to take into account coulomb d i s t o r t i o n , i t i s conventional to represent the velocity-dependent, effective number of electrons per atom by the function: Z f^Cv), so that the spin averaged annihilation rate becomes v a(v) = * r o 2 c Z e f f ( v ) N r 1-3 where the spin averaging has removed the factor of 4 from 1-2, and i t i s assumed that the positron interacts with only one atom at a time i n a gas of number density N^. The measured;single-atom annihilation rate, X^(T), i s then related to v (v) and f(v,T), the positron v e l o c i t y d i s t r i b u t i o n at temperature T (Section 1.3.2), as follows: X 1(T) = / v a(v)f(v,T)v 2dv. 1 - 4 o A velo c i t y average of z e f f ( v ) c a n also be defined i n a similar manner and related to X^T) X (T) eff 2 .T irr cN, o 1 7 1.2.1.2 Experimental Results The direct annihilation rate (Section 1.2.1.1), A ^ , has been shown i n many experiments with argon to be proportional to density for temperatures from 135 K (densities less than 5 amagats) to 573 K (Lee (1971)) and densities up to 65 amagats at room temperature (Tao, (1970)). For t h i s temperature range, Lee (1971) obtained the following temperature dependence for A^/D (D = N^): A^D = 21.43 T - 0* 2 4 0 y s _ 1 amagat"1. 1-6 Lee (1971) also measured A^ for applied e l e c t r i c f i e l d s up to 35 V -1 -1* cm amagat (and obtained approximate functions for the momentum-transfer cross-section for e l a s t i c c o l l i s i o n s and the annihilation cross-section over the vel o c i t y range 0.012 eV to 0.057 eV). 1.2.2 Modifications to the Z Description ef f r 1.2.2.1 "Single-Atom" Effects 1.2.2.1.1 Resonances It i s quite easy to see how a resonance state would increase by increasing the time that the positron spends near the atom. Whether such a resonance enhancement exists however depends on whether a bound or v i r t u a l state (a state that would be bound with a s l i g h t l y stronger or wider potential) of the positron and gas atom, or cluster of atoms, exists. If a resonance state does exist then the calculation of the anni-h i l a t i o n rate i s i n i t s e l f very d i f f i c u l t . For, since t h i s rate depends _ _ __ * 1 amagat = 2.687 xlO molecules cm and i s the molecular density of 1 mole of ideal gas at STP. 8 on the s p a t i a l overlap of the positron and atomic electrons, t h e i r c o r r e l a t i o n at a l l times must be known. If the resonance i s narrow (less than 10 -5 eV) then the l i f e t i m e of the metastable state (6 x 10 -11 s) w i l l be long compared to the time between gas atom c o l l i s i o n s (10 -11 -1 s amagat at room temperature, where D i s the D density of the gas). Thus, c o l l i s i o n s of t h i s complex with a sin g l e gas atom w i l l modify the positron environment from that of the metastable state. For methane, applying a Breit-Wigner resonance to a v i b r a t i o n a l c o l l i s i o n Smith and Paul (1970) suggest that t h e i r experimental r e s u l t of = 140±1 might be explained by a sin g l e resonance at greater than 4.4 kT, which i s j u s t below the f i r s t v i b r a t i o n a l e x c i t a t i o n l e v e l of methane. Goldankski and Sayasov (1964), using a very crude model requiring only the existence of a bound or v i r t u a l state, also obtained a s a t i s f a c t o r y value for Z f^. However, Mao and Paul (1977) obtain an a n n i h i l a t i o n rate f or positrons i n methane decreasing l i n e a r l y with an applied e l e c t r i c f i e l d implying that such a resonance does not e x i s t . Note that since the positron i s s t i l l i n t e r a c t i n g with one atom at a time, the e f f e c t of such a resonance i s an enhancement of Z r j : ' err while s t i l l r e t a i n i n g the p r o p o r t i o n a l i t y to density of A^ as implied by 1-3. Thus, the large value of Z^^ for methane observed at low densities i s consistent with such a "resonance" p i c t u r e , whereas the n o n - l i n e a r i t y observed, at higher d e n s i t i e s (McNutt et a l . (197 )) implies the existence of some other mechanism, possibly the e f f e c t of a t h i r d body on the resonance. Nothing i s known about the p o s s i b i l i t y of positron-atom resonances involving s i n g l e atoms i n noble gases. 9 1.2.2.1.2 Bound State The formation of a bound state requires the l o s s of s i g n i f i c a n t k i n e t i c energy. Such a t r a n s i t i o n may be mediated by the d i r e c t emission of r a d i a t i o n or, more l i k e l y because of the low energies envolved, by t r a n s f e r r i n g the energy to a t h i r d body i n the v i c i n i t y . Therefore, the requirement for a second gas atom would give a quadratic density dependence to the a n n i h i l a t i o n rate i f a bound positron-atom state was a s i g n i f i c a n t contribution to the enhancement. For the noble gases a bound state seems highly u n l i k e l y on t h e o r e t i c a l grounds. The simplest known stable complex i s PsH, positronium hydride, (Navin and Schrader (1974)) whereas neither of the singly-charged structures e + - H (Aronson et a l . (1971) nor e + - He (Gertler et a l . (1968)) i s stable. Drachman et a l . (1976) by means of 3 a v a r i a t i o n a l c a l c u l a t i o n f i n d that helium i n a S state would support a bound state i f the positron were 6% l i g h t e r . It i s therefore only speculated that an improved c a l c u l a t i o n may prove that the positron i n t h i s state i s stable. Whether the much more complex argon atom supports a bound state w i l l , from a t h e o r e t i c a l viewpoint, remain unknown for some time. If c l u s t e r s of gas atoms ex i s t within the host gas, the problem of removing the excess k i n e t i c energy i n order to form a positron-bound state, i f such e x i s t s , i s simply achieved by either sharing t h i s k i n e t i c energy with a l l the atoms of the c l u s t e r or p a r t i a l l y breaking up the c l u s t e r . If bound states on single-atoms or small c l u s t e r s 10 atoms) do not e x i s t there s t i l l e x i s t s the p o s s i b i l i t y for bound states of 10 positrons on macroscopically-sized c l u s t e r s of atoms. This p o s s i b i l i t y w i l l be discussed i n Section 1.2.2.2.2. 1.2.2.2 Many-Body E f f e c t s Many-body e f f e c t s can r e s u l t from the overlap of the positron wave function with more than one atom at a time, or the presence of structure i n the gas, whether i t be density f l u c t u a t i o n s or actual c l u s t e r s . Either of these e f f e c t s w i l l manifest themselves at the higher densities and may i n fact r e s u l t i n actual l o c a l i z a t i o n of the positron. L o c a l i z a t i o n e f f e c t s are suggested because of the manner i n which the a n n i h i l a t i o n rate i n the low density gas approaches the rate i n the much denser l i q u i d . 1.2.2.2.1 Random Systems B a s i c a l l y the problem of describing a positron (or electron) i n a dense gas involves both a dynamical and a gas structure part. Unlike a c r y s t a l l a t t i c e containing randomly positioned impurity atoms, the structure of a dense gas i s completely random except for short range c o r r e l a t i o n s . The d i f f i c u l t i e s of solving the positron-dense gas problem become evident when the progress toward sol u t i o n of the l a t t i c e case i s con-sidered. The highly i d e a l i z e d models that must be applied i n these (electron) cases makes a useful s o l u t i o n of the t o t a l l y "amorphous" positron case seem quite remote ( E l l i o t et a l . (1974)). Nevertheless the continued generalization of the thermodynamic Green's function method from these basis systems to other non-basis systems, for example 11 electrons i n l i q u i d argon (Tekeno and Goda (1971), Takeno (1971)), gives this method potential for future application to the positron-dense gas sit u a t i o n . Such a treatment would require an analogy between the c r y s t a l system with impurities and the positron i n a dense gas. Such an analogy might be constructed by l e t t i n g the free positron be the "regular" known case and clusters of atoms the "impurities". I t i s not clear to t h i s author whether the Green's function theories reviewed by E l l i o t et a l . (1974) can be applied i n such a case. At any rate these random-system theories and the possible analogies with the positron case w i l l warrant further consideration by theoreticians for i t i s important to note that these theories predict both free and localized states i n a random system due solely to the random aspect of the system. Other idealized models pertaining to these l o c a l i z a t i o n effects have been considered by Anderson (1958) and Coopersmith (1971). (Kirkpatrick (1973)) Other analogies can be made to percolation theories^in which a c r i t i c a l concentration of some random system parameter i s introduced. This c r i t i c a l concentration separates system-states with and without localized states. However, a different analogy with the positron-dense gas case would have to be made. For example, whether a r e s i s t o r network from which resistors are randomly removed (leading to i n f i n i t e resistance for some c r i t i c a l number of resistors removed) can be used as a model for free positrons becoming lo c a l i z e d as the density i s increased i s at t h i s stage of speculative interest only. In t h i s analogy the free positron would be made to correspond to a conduction path i n the r e s i s t o r network and random addition of another scattering center to the host gas (that 12 i s increasing the density) to random removal of a r e s i s t o r . Returning to p r a c t i c a l solutions of the l o c a l i z a t i o n problem, two possible methods of considering the positron-gas system remain to be discussed. One of these r e l a t e s to the existence of l o c a l i z e d states and the other to the existence of s p e c i f i c structure i n the neutral system. These two are discussed i n the following sections. 1.2.2.2.2 Fluctuons Fluctuons are bound states of a l i g h t charged p a r t i c l e with a f l u c t u a t i o n of a p a r t i c l e system. The i n t e r a c t i o n between the charged p a r t i c l e and a single p a r t i c l e of the system i s considered to be a t t r a c t i v e but i n s u f f i c i e n t to form a bound state on a s i n g l e p a r t i c l e . Instead, the f l u c t u a t i o n , which i s assumed to be of macroscopic si z e and for which the free energy increases, i s s t a b i l i z e d by the presence of the positron which po l a r i z e s the atoms within the f l u c t u a t i o n (Krivoglaz (1974)). These l o c a l i z e d states i n gases can be considered as e i t h e r bubbles (Eggarter (1972)) or c l u s t e r s (Iakubov and Khrapak (1976), Khrapak and Yakubov (1976)) depending on the sign of the e l e c t r o n -atom, or positron-atom, s c a t t e r i n g length. Two methods of approaching the study of fluctuons can be considered. The f i r s t requires a high mobility of the gas atoms (Krivoglaz (1974)), and the second assumes a random f i e l d produced by the atoms i n which the electron (or positron) moves ( L i f s h i t z (1968)). It i s c l e a r that both of these are important at high d e n s i t i e s but the high mobility theory i s more s a t i s f a c t o r y at higher temperatures. Furthermore, the former theory can be related to the e l e c t r o s t r i c t i o n theories discussed by 13 R o e l l i g and Kelly (1967) and Canter et a l . (1975). Both L l f s h i t z (1970) and Krivoglaz (1974) present similar temperature dependences for the fluctuon radius for a lattice-gas model. However, the positron may not be sensitive to the fluctuon radius since the positron upon forming a fluctuon would annihilate with a rate approaching that of a positron i n a l i q u i d . In addition, a fluctuon state i s u n l i k e l y to exist for a large f r a c t i o n of the positrons l i f e t i m e due to the short l i f e t i m e of the positron and the fact that a barrier (decreasing with increasing temperature) exists to the formation of a fluctuon state ( L i f s h i t z (1970)). And f i n a l l y , the formation time of a fluctuon may d i f f e r greatly between gas and l i q u i d state medium due to the inherent differences i n the structure of these phases. For example, Wiegel (1973), from consideration of the p a r t i t i o n function of the Van der Waals gas, presents a picture of a pure thermodynamic phase as always containing a f i n i t e volume fr a c t i o n of the other phase. Thus, the l i q u i d phase i s pictured as a background of " l i q u i d " containing microscopic "gas" bubbles, inside which are l i q u i d drops, etc. (The gas picture i s just the opposite). The formation time of a fluctuon w i l l therefore vary, depending for example whether many atoms must be added to an already existing cluster or single-atom (for the gas phase), or a few bubbles excluded from a l i q u i d region (for the l i q u i d phase). In general, the calcutation of experimental quantities w i l l require consideration of both free and l o c a l i z e d states. Eggarter (1972) e f f e c t i v e l y applied such methods i n a semiclassical manner to adequately f i t the density 14 dependence of the mobility of electrons i n l i q u i d argon using two adjustable parameters. 1.2.2.2.3 Clustered systems Strickfaden and Sobrino (1970) have shown that for a Van der Waalsmetastable vapor, a c l u s t e r of 20 A° radius, containing about 350 atoms, would decay i n about 70 us. This time scale i s i n f i n i t e compared to the positron l i f e t i m e . However, for an unsaturated vapor, the p r o b a b i l i t y of formation of such a c l u s t e r by addition of atoms to smaller c l u s t e r s i s probably n e g l i g i b l e . It i s much more l i k e l y that dimers or other small polymers e x i s t i n the gas and that these either act as i n t e r a c t i o n centers for a random system according to Section 1.2.2.2.1 or as e f f i c i e n t l o c a l i z a t i o n centers for the fluctuon e f f e c t s mention i n the preceding section, 1.2.2.2.2. 1.2.3 Positronium and bubble formation The c h a r a c t e r i s t i c s of the r e l a t i v e l y long l i v e d ortho-positronium atom (oPs) are reviewed by Green and Lee (1964). It w i l l s u f f i c e here to state that for argon de n s i t i e s up to 65 amagats the a n n i h i l a t i o n rate i s given by A 0 = A + A D ' 1-7 2 o q where A = 7.2 us 1 i s the vacuum a n n i h i l a t i o n rate, and A = (.255±.009) o q ys 1 amagat 1 i s the quenching rate (Lee (1971)). Quenching i s c o l l i s i o n a l shortening of the l i f e t i m e . Molecules l i k e ^ and O2 which e f f i c i e n t l y "convert" ortho- to para-positronium can greatly increase t h i s rate. Thus 15 values of \^ greater than that given by the above r e l a t i o n would suggest the presence of noticeable amounts of impurities. If 2^ i s le s s than the generally accepted r e s u l t , then the implication i s that of positronium i n a reduced gas density, and suggests that positronium i s forming a bubble i n the high density gas or l i q u i d . The experimental r e s u l t s are indeed consistent with q u a l i t a t i v e predictions of such a model ( R o e l l i g and K e l l y (1967)). The existence of a bubble i s also implied from the presence of a narrow component i n the angular c o r r e l a t i o n measurements of positrons i n l i q u i d helium-4 (Briscoe et a l . (1968)). Both helium-4 (Canter et a l . (1975)) and helium-3 (Fishbein and Canter (1976)) show evidence of bubble formation i n the dense gas from reduced \^ values. In argon, no deviation of \^ from the above equation i s seen for gas densities up to 65 amagats (Tao (1970)). However, Paul's l i f e t i m e r e s u l t i n l i q u i d argon at 86.4 K i s 20% below that expected from an extrapolation of the normal l i n e a r dependence for argon gas and suggests the p o s s i b i l i t y of bubble formation i n l i q u i d argon. Also angular c o r r e l a t i o n measurements i n argon suggest that c a v i t i e s are formed at l i q u i d d e n s i t i e s (Varlashkin (1971)). For NH^, ^ i s l i n e a r up to and including l i q u i d d e n s i t i e s (McNutt et a l . (1974)). CH^ does show a s l i g h t n o n - l i n e a r i t y i n A2 i n the dense gas which i s also, attributed to bubble formation (McNutt et a l . (1975)). F i n a l l y , bubble formation i s also suggested for neon gas from the reduced values observed by Canter et a l . (1975). 16 1.3 Generalized features of positron swarm experiments Since the slowing of positrons i s discussed extensively by Lee (1971), only the basic features w i l l be reiterated here, with more emphasis placed on a discussion of the abnormalities i n the low-temperature time spectra. 1.3.1 Slowing of positrons i n a gas 22 The positrons emitted into a gas from beta decay of Na slow down rapidly v i a i n e l a s t i c c o l l i s i o n s from the i r i n i t i a l v e l o c i t y d i s t r i b u t i o n (maximum energy of 542 keV, peak energy of 170 keV) to 10 keV i n less than 0.7 ns (for 10 amagat of argon gas). From 10 keV to the l a s t i n e l a s t i c l e v e l i s estimated to take another 0.5 ns. The e l a s t i c slowing down of positrons to thermal energies takes a time comparable to the free anni-h i l a t i o n rate for room temperature gas (about 30 ns for 10 amagats). Thus the probability for annihilation p r i o r to thermalizing i s small, but s i g n i f i c a n t . Those events occuring during the e l a s t i c slowing down give r i s e to a shoulder i n the annihilation spectrum, which i s therefore evidence of a velocity-dependent annihilation rate for positrons i n the energy range 0.1 eV to 11 eV. Structure can be introduced into the shoulder region of the anni-h i l a t i o n spectra i f either the annihilation rate or momentum-transfer rate changes rapidly with decreasing energy. This i s more readily seen i n a plot of A(t), the velocity-averaged annihilation rate as a function of time, where the counts per channel, dN/dt, i n the annihilation spectra i s related to X(t) and the number of positrons remaining at time t , N(t): 17 ^ = -A(t)N(t). 1-8 Thus a large momentum-transfer cross-section occuring at some energy would rapidly remove the positrons at that energy to a s i g n i f i c a n t l y lower energy and so give the effect of an almost discontinuous increase i n the annihilation rate, since t h i s rate i s a decreasing function of energy. Of course, both rates could simultaneously contribute to the structure as i t i s quite possible that any mechanism causing one rate to increase would affect the other. From the low temperature results of helium and argon, the room-temperature picture i s indeed modified by the presence of a peak near the end of the shoulder. In addition the subsequent equilibrium anni-h i l a t i o n rate i s s i g n i f i c a n t l y greater than the rate for an equal density room temperature gas. In argon the peak i s at the shoulder terminus. The time at which the peak occurs i s inversely proportional to density i n exactly the same way as for the shoulder width i t s e l f . In helium the situation i s s l i g h t l y d i f f e r e n t . Here the room temperature shoulder edge i s s t i l l discernable from that of the narrower peak at the higher temperature end of the low temperature data. The position of th i s peak occurs e a r l i e r i n time as the temperature i s reduced from the higher temperatures eventually occuring at the shoulder terminus. This l a t t e r effect for helium i s very suggestive of inherent clusters i n the gas at low temperatures and high densities where the clusters act l i k e impurities with very low ly i n g ionization levels (approximately kT c > where i s the c r i t i c a l temperature). Therefore, i n the slowing down history of a positron, only the low energy region 18 i s affected due to the size of kT (Section 1.3.2). As the concentration c of clusters i s a decreasing function of temperature, the positron w i l l encounter more of these "impurities" at the lower temperatures and thus decrease the position of the shoulder terminus i n time. In general those positrons reaching thermal energies without anni-h i l a t i n g w i l l have a v e l o c i t y d i s t r i b u t i o n determined by the temperature, applied e l e c t r i c f i e l d ( i f any), v e l o c i t y dependence of the momentum-transfer rate (of the positrons with gas atoms and only i f an e l e c t r i c f i e l d i s present) and the annihilation rates, and the d i s t r i b u t i o n of any clusters. 1.3.2 Positron Velocity D i s t r i b u t i o n and Diffusion Equation Lee (1971) has summarised the d i f f e r e n t i a l equation which describes the positron v e l o c i t y d i s t r i b u t i o n for e l a s t i c scattering c o l l i s i o n s i n a monatomic gas. The stress here w i l l therefore be on i t s possible modification to take into account weakly bound "molecules". Massey et a l . (1972) have studied the behavior of small amounts of molecules added to rare gases with the idea that most of the annihilation w i l l take place with the atoms, whereas the v e l o c i t y and mean energy of the positrons w i l l be influenced by energy losses i n c o l l i s i o n s with the molecules. If i t i s assumed that only monomers and one cluster of size i i s present then the steady state d i s t r i b u t i o n function of positrons i n this system at temperature T and i n the presence of a uniform e l e c t r i c f i e l d E i s given by the solution of the following equation (Massey et a l . (1972)): 19 E 2 d , e_ df\ 2 d . 2„ . r ~ d T ( I ^ 7 dF> + 57 de" I B 1 \A 1 . 2kT d , 2„ . df. + d l ( £ I B I? + N cZ c(e) + / f (,r o 2C) [ N ^ Z ^ ^ - Z ^ ^ ( e ) ) + N c ( Z e f f , c - Z e f f s c ( £ ) 3 f =0 1-9 where k = Boltzmann constant and £ = positron energy. Here, subscript 1 refers to the monomer and c to the impurity molecule, i n t h i s case a single cluster of i atoms. For these two species: M = mass of species, N = number density of species, Q = momentum-transfer cross-section of species, m and Z en and Z ,.,.(•£) are defined i n Section 1.2.1.2. eft efr Also Q = Q . + Q N /N , A Tnl mc c 1 and Q - Q. + Q N/N.i. B ml mc c 1 Z (e) i s the contribution from i n e l a s t i c c o l l i s i o n s with the c l u s t e r , (e) = (e + e ) f ( e + e )Q,(e + e ) - ef(e)Q (e + e ) • c c t- c 1-10 + (e + e ) f ( e - £c)Q_(e - ec> - ef(e)Q_(e - £ c>, where i t i s assumed that there i s only one bound state. Q + are cross-sections i n which the positron gains or loses energy e^. The l a s t term of 1-9 expresses the effect of annihilations. c Z 20 The measured annihilation rate, A , i s then related to A, of ' e' 1 Section 1.2.1.2 (1-4) and Z^^ c^ e^ a s follows: oo A £ = A x + T r r o 2 c N c / Z e f f (v) f(v) f(v) v 2dv, 1-11 o ' A result of 1-10 i s that for l i g h t l y bound clusters ( i e . kT^ = .013 eV for argon, and .0004 eV for helium) Z c(e) i s negl i g i b l e except for thermal energies. Thus, the effect of clusters on the annihilation spectrum i n the v i c i n i t y of the shoulder i s neg l i g i b l e . However, i n a steady state, the peak of the positron v e l o c i t y d i s t r i b u t i o n function for argon (see Figure 5, Massey et a l . (1972)) i s above the rotati o n a l energy for a cluster. Thus unlike higher-lying molecular states, these cluster levels would contribute to the d i s t r i b u t i o n function and subsequently to Z ^ even at zero e l e c t r i c f i e l d s . For the addition of molecules to the rare gases ( s p e c i f i c a l l y CO and Massey et a l . note the following: 1) the mean energy of the thermalized positrons i s lowered, much more for CO than H^, due to the rotational excitation l e v e l of CO being 100 times larger than H^, causing a subsequent increase i n Z 2) As the concentration of molecules i s increased the equilibrium d i s t r i b u t i o n of positrons i s compressed into a narrow region at lower energies and Z ^ becomes nearly a constant equal to Ze£j(0). Also, when the applied e l e c t r i c f i e l d and/or momentum-transfer cross-section i s large or when the mass of the atom or rotati o n a l excitation cross-section i s small, the annihilation rate does not become constant u n t i l the concentration of molecules i s large. 3) For homonuclear diatomic molecules with no e l e c t r i c f i e l d applied the d i s t r i b u t i o n function becomes divergent unless the condition 21 N Q (O)kT m c - 2 M, Bo 1-12 N. 1 o i s s a t i s f i e d , where B i s the rotat i o n a l constant and q i s the quadrapole moment of the molecule. Note that this condition holds for use of the Born approximation for rotational excitation. F i n a l l y , as the concentration i s increased the molecules begin to contribute to the annihilation. 1.3.3 Existing experimental sit u a t i o n For argon gas, the only existing low temperature annihilation results are those by Canter and R o e l l i g (1975) for gas densities of 20, 30 and 40 amagats and temperatures as low as, but not on the co-existence l i n e . For 20 amagat gas i n a nearly saturated state, reached a value of 210 as compared to 26.6 for room temperature (Lee It i s interesting to compare the behaviour of these equilibrium annihilation rates, X^, with those of helium-4 by Canter et a l . (1975). As far as the dependence of X^ on temperature i s concerned, the q u a l i t -ative features are s i m i l a r , with X^ increasing rapidly with decreasing temperature and then f a l l i n g o f f . Differences arise between the two gases helium and argon i n the way i n which the results f a l l o ff. In helium, X G changes rapidly at the f a l l - o f f whereas i n argon the change i s more gradual. Also for a constant temperature (less than T C) but (1971)). 22 variable density of helium, X& becomes constant at approximately one seventh that appropriate to the l i q u i d density. For argon, lack of density variation for a constant temperature prevents such an i n -vestigation being made, although the near-saturated vapor X^ results for 30 and 40 amagats show quite s i m i l a r values. Fishbein and Canter (1976) have also measured X^ i n helium-3, thus enabling them to compare two gases with similar atomic structure, but different thermodynamic properties. They find that A g for helium-3 exhibits the same type of c r i t i c a l s e n s i t i v i t y to density and temperature at certain " t r a n s i t i o n " densities and temperatures as that observed i n helium-4. For example for helium-3 a t r a n s i t i o n to a constant X g i s observed at 160 amagats for 5.0 K gas and i n helium-4 at 96 amagats for the same temperature. In addition, the low c r i t i c a l density of helium-3 allowed investigations i n gas approaching and surpassing l i q u i d densities. This data indicated s t r i k i n g s i m i l a r i t i e s between X^ i n a gas and that i n a l i q u i d at 432 amagat. In addition above 400 amagats there i s indication that the enhancement of X^ above the free positron value i s reduced. I t i s not known whether the clustering vanishes or i s masked by the high free annihilation rate at these densities. These l i q u i d spectra also represent the f i r s t non-exponential l i f e t i m e spectra seen i n a l i q u i d . As mentioned i n the introduction, methane and ammonia both show s i m i l a r l y large Z^ ^^  values with associated low temperatures and high densities. Also of interest i s that the effect i n methane i s observable at the highest T/T^ r a t i o of a l l of the gases (1.53). A correspondence 23 plot of the regions of experimental data i s shown i n Figure 1 for a l l the gases with respect to the coexistence l i n e (C.L), (Note that the C.L. for helium i s narrower than the other gases). The range of the results for the rest of the present work (gas, T<Tc) are shown in-Figure 21. The present picture, however, i s not complete without a comparison of the r e l a t i v e value of the annihilation rates i n the gases to those i n the l i q u i d s at a corresponding temperature. For helium-4 and ammonia, the l i q u i d A g i s equal to that of the high density gas. However, the l i f e t i m e result of Spektor and Paul (1971) for l i q u i d argon at 1 atmosphere of 1.8 nsec i s not equal to 1.2 nsec given by Canter and R o e l l i g for 30 and 40 amagat gas. Note however, that the l i q u i d result i s taken at 86.4 K while that of the gas i s 118 K and 123 K respectively. For methane no l i q u i d result i s known. F i n a l l y one i s l e f t with the "anomalously" normal result for neon. Here A g appears to be proportional to density even for gases at l i q u i d densities. However,the near zero positron-neon scattering length of -0.6 a Q calculated by Montgomery and LaBahn (1970) and the fact that Z e f f m e a s u r e < * Canter and R o e l l i g (1975), i s less than the atomic number (the only noble gas exhibiting t h i s behaviour) suggests that the interaction potential i s small at thermal v e l o c i t i e s . Thus, the basic property necessary for enhancement of any kind i s absent (Section 1.2.2.2.2). Summarizing the positronium r e s u l t s , i t appears that a l l the noble gases and CH^ experience bubble formation either i n the dense gases (CH/t and helium-3 and -4), or i n the l i q u i d s (helium, argon and neon). 24 1.65-1 T/T ,c lineari 1.0H NH-0.5, CH^ — McNutt arid Summerour (1972) He-3 — Fishbein and Canter (1976) CH^ — (linear region) - Smith and Paul (1970) non-linear j i n °)• D-»-He-4 — Canter et a l . (1975) -Ar — Canter and Ro e l l i g (1975) Ne — Canter and R o e l l i g (1975) COEXISTENCE LINES Part of present work Remainder shown i n Figure 21 -NH3 — M c N u t t et a l . (]974) Spektor and Paul (1971) D A r (amagat) 100 200 300 400 500 600 700 , , 1 , , 1 r-0.0 1.0 D/D 2.0 Figure 1 Correspondence Plot for A l l Gas and Liquid Argon Data 25 For NH^ there i s a linear dependence of the positronium annihilation rate up to and including l i q u i d densities. As far as the application of d c e l e c t r i c f i e l d s to the low temperature gases, only one unpublished result for helium-4 i s known (Despande and R o e l l i g (1972)). Their results show a similar behaviour to that i n argon at higher temperatures (Lee (1971)), v i z . a s i m i l a r i t y between the effects of e l e c t r i c . f i e l d and temperature. Thermalization effects on positrons i n argon produced by molecular contaminants have been studied by Paul and Leung (1968). I t was concluded that thermalization of positrons with nitrogen molecules takes place by rotational excitation, but that the results for methane require further study. 1.4 Summary of Thesis Content Although the p o s s i b i l i t i e s for modifying the Z e f f description (Section 1.2.1 and 1.2.2) are t h e o r e t i c a l l y d i f f i c u l t , they do separate into two d i s t i n c t areas. That i s , those independent of the thermodynamics of the gas at suitably low temperatures (Section 1.2.2.1), and those that are always dependent on the thermodynamics, only less so at the low densities (Section 1.2.2.2). This thesis describes a research program i n which the role of the thermodynamics i n the enhancement of the equilibrium annihilation rate i s studied. Thus, the measurements have been made over an extensive range of states for argon gas at temperatures below the c r i t i c a l point and also i n states of the saturated gas and the l i q u i d . The experimental technique i s presented i n Section 3.1 and the results i n Section 3.3. 26 Evidence i s presented that small clusters of less than 10 atoms are dominating the positron interaction at the lower densities for the low temperature gas (Section 3.3.4.2.4 and 4.3). The success of a two-population model (Section 4.4) for the positrons i n the gas casts doubt on the v a l i d i t y of the interpretation of such results i n the current l i t e r a t u r e i n terms of hundred atom clusters. As i n any complete swarm experiment, results were also obtained for which the positrons were d i f f e r e n t i a l l y accelerated by an applied d c e l e c t r i c f i e l d . I t was found that the e l e c t r i c f i e l d was most  ineffective at accelerating the positrons i n the low temperature gas (Section 3.3.5 and 5.6), a result interpreted i n terms of the existence of large energy losses during c o l l i s i o n s of the positrons with the atomic clusters. 27 CHAPTER TWO NUCLEATION THEORY 2.1 Introduction It i s well known (Zettlemoyer (1969), Abraham (1974)) that the tra n s i t i o n of a supersaturated gas to a condensed state occurs by the random formation of small droplets of the new phase within that of the parent gas. At some c r i t i c a l supersaturation r a t i o these embrionic droplets subsequently and suddenly grow to macroscopic proportions. That a gas can exist i n a state of supersaturation i s explained by the increase i n energy of the system when the surface of a droplet i s formed (Section 2.2). This leads to the introduction of the Gibbs free energy of formation of a drop from the vapor as an activation energy or barrier to nucleation. Thus condensation depends on the formation of those droplets (containing 20 to 100 molecules) called c r i t i c a l n u c l e i , which are i n unstable equilibrium with the vapor, by addition of one atom or molecule at a time. In either unsaturated or saturated vapor, a c r i t i c a l size of nucleus does not exist. However, small clusters of atoms are expected to be present but at much smaller densities than for the supersaturated case. In addition, an individual cluster would be expected to exist for shorter periods of time and i t i s very l i k e l y that the clusters have a much more "open structure" of atoms. As far as positron annihilation i s concerned the increased length of time for which the atoms spend i n each others f i e l d s of force w i l l 28 affect the single atom picture of a positron i n a gas as discussed i n Section 1.2.2. H i s t o r i c a l l y , nucleation theory had i t s beginnings with the description of a l i q u i d drop having a chemical potential per molecule u^, and a surface tension, a, both equal to those of the bulk l i q u i d . The drop was considered a d i s t i n c t molecular species i n an id e a l gas mixture and thus to have a chemical potential of the form u ± = B±(T,V) + kT ln(n ±/N) 2-1 0 0 where n. i s the number of i - c l u s t e r s , N = I n . i s the number of molecules 1 1 1 i n the gas, T i s the temperature (K), V i s the volume, and k i s Boltzmann's constant. B^ i s the free energy of a drop composed of i single molecules. Thus the t r a d i t i o n was to put B. = i y T + 4ira(3v T/47r) 2 / 3 2-2 X Li Lt where v^ i s the s p e c i f i c volume of the l i q u i d . Then, using the standard condition for equilibrium among clusters, y i = i y l ' 2 - 3 and 2-1 for u^, the equilibrium d i s t r i b u t i o n was found to be n ± = N exp(-(B i - iy )/kT).. 2-4 Some controversy s t i l l surrounds the magnitude of the pre-exponential factor, N. The question i s whether this simple connection between B^ and y^ and a (2-2) i s relevant only for a droplet at rest. I f one assumes that the drop must be removed from the bulk l i q u i d to y i e l d the con-t r i b u t i o n B^ to the free energy, then characterizing the droplet with 29 t r a n s l a t i o n a l and rotational energy, results i n an increase of 12 17 the pre-exponential factor N by an additional factor of 10 to 10 (Lothe and Pound (1962)). This i s the essence of the " t r a n s l a t i o n a l -rotational paradox" (Reiss (1970)). This paradox has apparently been resolved by correct s t a t i s t i c a l mechanical treatment of the c l a s s i c a l phase in t e g r a l (Reiss (1970) Kikuchi (1971)), which i n effect analyses the entire assembly (Section 2.3). This transforms the problem into one of correctly defining a cluster so that numerical calculations of the p a r t i t i o n function for the defined cluster can be made. The result i s that for clusters large enough for application of 2-2 the pre-exponential factor i s of the order 3 6 of 10 to 10 . The more serious d i f f i c u l t y then becomes the application of 2-2 to clusters of less than 100 atoms. 2.2 Energy Barrier to Nucleation The exponential factor i n 2-4 with B^ defined by 2-2, can be con-sidered as a r i s i n g from the c l a s s i c a l Boltzmann factor: exp(-E f ( l )/kT), 2-5 where i s the minimum work required to form a droplet consisting of i atoms from the vapor. E^ can be related i n a simple manner to (B. - i u ) of 2-4 as follows (Landau and L i f s h i t z (1969), section 150; 1 JL Abraham (1974)). Considering only those i atoms involved i n the formation of the droplet with the rest forming a reservoir at temperature T q and pressure P , one can write that o 30 E , ( l ) = ( E - T S + P V + o-A), . _ f o o droplet - ( E - T S + P V ) . _ „ , o o 1 vapor atoms 2-6 or E £ = AE - T AS + P AV + aA, 2-7 f o o where E = internal energy, S = entropy, V = volume, P = pressure, A = surface area of droplet, and a = surface tension. The i-dependence i s not e x p l i c i t l y included. 2-6 (or 2-7) results from subtracting from the t o t a l increase i n the energy (characterizing the t r a n s i t i o n from a vapor only state to a vapor plus i atoms i n a droplet state) the heat added and work performed on the drop by the reservoir. Then the work remaining i s that required to actually form the drop, which without the presence of fluctuations i n the gas, a non-thermodynamic effect, requires some outside influence (otherwise the second law would be violated). The external influence i s required i n order to make both the i n i t i a l and f i n a l state equilibrium states to which thermodynamics can be applied. Since a fluctuation at constant temperature and pressure i s being considered, the Gibb's free energy G. = E. - T S. + P.V., j = L or g, 2-8 3 3 0 2 1 2 i s the relevant potential to employ. 31 However, because of the presence of the spherical surface, P J_i i s not equal to P (or P ) and i n fact the condition of mechanical g o ' equilibrium implies that PT = P + — 2-9 L g r where r i s the radius of the drop. Now E. = GT (P T) + ok - G (P ) + P VT - P TV T . 2-10 f L L g g o L L L But G (P.) = G (P ) + (P - P ) V_. 2-11 L L L g L o L and hence E. = G T(P ) - G (P ) + aA. 2-12 f LS %' g v g 7 F i n a l l y , since G(P) = i y ( P ) , 2-13 for a one component system, E = i [ y (P ) - y (P )] + aA. 2-14 Note that for a saturated gas (gas i n the presence of a plane l i q u i d surface), V V " V V ' 2- 1 5 and therefore that E f = aA. 2-16 However, i f a plane l i q u i d surface i s present then further condensation of additional gas takes place d i r e c t l y onto the surface of the l i q u i d without the need for nucleation. Nevertheless, within the body of the 32 saturated gas (above the l i q u i d surface) formation of clusters i s s t i l l governed by 2-4 with B^ (or E^) given by 2-16. 2.3 C l a s s i c a l Phase Integral The idea of considering the gas as made up of clusters of molecules was f i r s t conceived by Mayer (1937a,b;1938) (see also Zettlemoyer (1969)) and described by his cluster integral expansion of the free energy, F, of a gas. Thus, given F(T,V) i n terms of the c l a s s i c a l p a r t i t i o n function $ , exp(-BF) = A N$ N/N , X = (2^/gm) 3 / 2 2-17 and * N = /<*>./ e x p ( - e I ^ ( ? < 1 > , ? « > ) ) d ? a ) . . . d ? ( N ) , 2-18 (S = 1/kT) one can i n p r i n c i p l e calculate a l l thermodynamic quantities. Mayers main contribution involved analysing $^ i n terms of the various ways i n which molecules may associate under the influence of thei r mutual attractions, ->(i) ->(-j) <f> (r ,r ). Thus a molecule belongs to a cluster i f i t l i e s within the sphere of influence of at least one other molecule of the cluster. Soon after t h i s work Frenkel (1939), and Band (1939) observed that considerable s i m p l i f i c a t i o n results by considering interacting groups of molecules as systems to be discussed, as a whole. Thus, an imperfect gas i s to be thought of as made up of monomers, dimers and higher polymers which are a l l i n dissociative equilibrium. The assembly i s treated as a perfect gas mixture with the interaction potential involved i n the process of association and dissociation. Thus, whereas i n the usual theory of imperfect gases, c o l l i s i o n s higher than binary are assumed n e g l i g i b l e , the cluster theory allows for the presence of ternary c o l l i s i o n s and 33 subsequently the p o s s i b i l i t y of binary and higher clusters which because of their long l i f e t i m e compared to the time for a binary c o l l i s i o n w i l l y i e l d a large contribution to the gas description, depending of course on the concentration of clusters. This picture can be further s i m p l i f i e d by considering only the most compact of the various isomeric forms of each polymer. This assumption, the Zeldovitch 'single configuration assumption' i s essential for analytic calculations to be tractable. It i s at t h i s point that the c a p i l l a r i t y approximation (use of bulk thermodynamic quantities to describe the droplet) was made by the e a r l i e r writers. A more complete description;is obtained through evaluation of the phase int e g r a l 2-18. For an assembly of non-interacting clusters a p a r t i a l decomposition of 2-18 and use of 2-3 yields n i = q i ^ ( i V - j / k T ) • 2-19 where q^ i s the molecular p a r t i t i o n function for the i t h cluster. Calcu-l a t i o n of q^ i s the central problem of nucleation theory. q^ involves t r a n s l a t i o n a l motion of the cluster throughout the whole volume of the gas. .. However, the association 2-2 i s made for a "stationary" drop (but center of mass not necessarily stationary). Thus, the pre-exponential factor compensates for the two different points of view. Lothe-Pound (1962) made an " i n t u i t i v e " estimate of th i s factor and i n i t i a t e d extensive theoretical work on the v a l i d i t y of this estimate. It i s now clear that a more complete evaluation of q^, for which each approximation i n the calculation i s w i l l defined, yields a substantially smaller value for the pre-exponential factor. The main distinguishing 34 feature of the various calculations i s the manner i n which the stationary drop i s defined. Reiss (1970) considers a droplet free to move i n a suitably small sized container, whereas Kikuchi (1969) considers one molecule fixed and enumerates the various configurations of the other molecules (of the cluster) with respect to this fixed molecule. Both pictures y i e l d a pre-exponential factor of approxi-mately N,v /v T, where v and v T are the s p e c i f i c volume of the gas I g L ' g L and l i q u i d , respectively. In the future, the above pictures w i l l allow a more concentrated e f f o r t to be made using numerical methods to calculate q.. 2.4 Structure of a Cluster For low temperatures (less than 70 K for argon) clusters containing less than 100 atoms are expected to be s o l i d - l i k e , with a polytetrahedral structure, rather than l i q u i d - l i k e i n structure (Hoare and Pal (1975)). In this case the Zeldovitch assumption appears reasonable and permits the theoretical work to be content with calculating p a r t i t i o n functions for the configuration possessing a minimum free energy. In addition, the Tolman expression for the radius dependence of the surface tension (Tolman (1949)) yields a good representation of the surface free energy as obtained from molecular dynamic studies of clusters of 2 to 100 argon atoms for both s o l i d - l i k e ( 0 K) and l i q u i d - l i k e (40 K) structures (Briant and Burton (1975)). For higher temperatures (greater than the liquid-gas coexistence temperatures) the situ a t i o n i s e n t i r e l y different and at best a l l con-figurations w i l l have to be considered. Some idea of the differences i n _ Kikuchi (1971) made a minor but necessary improvement to Reiss's o r i g i n a l calculation. 35 "compactness" of the clusters can be inferred from an expression for the standard deviation, a^, of the radius of a cluster as derived by Lovett (see Zettlemeyer (1969)). Although this expression i s not v a l i d for clusters as small as ten atoms, i t does present a q u a l i t a t i v e picture of clusters at these temperatures. Thus for a cluster of 10 A radius, a, ^  4.5 A at 130K, a substantial v a r i a t i o n i n the s p a t i a l extent of the clusters of one size. Bernardes and Primakoff (1959) by solving the Schrb'dinger energy eigenvalue equation for two inert gas atoms described by a Lennard-Jones 12-6 interatomic potential found that up to 4 v i b r a t i o n a l states exist for argon at 300K (and ^ / n ^ i s about 3%). 2.5 Effect of Ions It i s well known that ions aid the formation of clusters i n supersaturated gases (Wilson (1951),Briant and Burton (1976)). However, even i n unsaturated gases ions induce cluster formation and i n fact add a d i s t r i b u t i o n of ion-induced clusters to the homogeneous d i s t r i b u t i o n (self-induced)(Section 2.1-2.3). Also the most probable "ion-cluster" i s not the ion i t s e l f . For example, for ^ 0 (in argon c a r r i e r gas) clusters of 4 to 6 atoms are the most l i k e l y clusters formed about a Lead ion for ^ 0 pressures of 1 to 20 torr (just less than the co-existent pressure) (Castleman, J r . and Tang (1972). The change of free energy due to adding atoms to an ion of radius r^ i n order to form a non-conducting cluster of radius r, d i e l e c t r i c constant e^ ,, i n a vapor of d i e l e c t r i c constant E c t involves adding a term 36 * 0"0(«g"'d) to (2-2)(modified to exclude the volume of the ion). In general the formation of an ion-induced cluster (less than 10 atoms) w i l l take place by many 3-body c o l l i s i o n s since the polarization potential i s of long range. Thus, the excess energy of the clustering atoms w i l l be removed i n stages. However, only clustering about a positron-atom bound state seems important because of the high thermal motion of a free positron r e l a t i v e to that of the atoms. Now, although information i s lacking on the formation times of ion-induced c l u s t e r s , i t seems that these times are longer than the addition times of one molecule to a large cluster (50 atoms) i n a saturated gas. This addition time, about 10 ns (McDonald (1963)), i s long compared to the l i f e t i m e (less than 1 ns) of a positron bound state. It seems then that the only ion-induced clusters possibly influencing the positron annihilation rate are those produced on the ions resulting from the slowing down of a previous positron. However, of these only those produced from the capture of the free electrons w i l l be important as the p o s i t i v e l y charged ions w i l l present a strong repulsive potential to the positrons and reduce rather than enhance the annihilation rate. Nevertheless, even i f 10 free electrons are formed during the slowing 9 of one positron, a 10 yCi source w i l l y i e l d only 4 x 10 ions/s i n a 3 10 cm volume of gas. The applied e l e c t r i c f i e l d , acting as a sweeping f i e l d , w i l l remove a charged p a r t i c l e i n 40 us (E/D = 5 V cm "'"amagat "*") (Bowe (I960)). Thus the number of negative ions present i n the active 37 volume at one time i s about 10 J and i s much less than lO"1"" dimers i n the same volume for 10 amagats gas at 100K. Even i f trimers 3 are less favourable than dimers by 10., the r e l a t i v e concentration of self-nucleating clusters to ion-induced clusters i s well below the impurity l e v e l (10 ppm) at which large polyatomic gases are expected to influence the positron-single atom annihilation rate (Lee (1969). It thus seems clear then that any clusters observed by positrons, at least i n low density gas w i l l be self-induced rather than ion-induced. In high density gases, and l i q u i d s , ion-induced clustering may be important and i n fact may be associated with the fluctuons discussed i n Section 1.2.2.2.2. 2.6 Experimental Situation Most experiments (up u n t i l t h i s time) have measured c r i t i c a l super-saturation ratios using expansion chambers, d i f f u s i o n i n 1-dimension or expansion nozzles and observed clusters which grew to macroscopic sizes (Zettlemoyer (1969), Abraham (1974)). A comparison i s then made to theory as follows. The equilibrium d i s t r i b u t i o n of clusters, d i s -cussed i n the preceding sections, i s calculated and assumed to relate i n Saturation r a t i o (saturated pressure/coexistent pressure) at which formation of c r i t i c a l nuclei becomes certain. 38 a simple manner to a steady state d i s t r i b u t i o n corresponding to some rate of formation of c r i t i c a l n u c l e i (with a suitable cutoff above the c r i t i c a l nucleus to sustain the steady state). Kinetic theory i s then applied to relate t h i s d i s t r i b u t i o n , the area of a c l u s t e r , and an empingment r a t i o (usually assumed to be unity) to the observed nucleation rate. Because of the c r i t i c a l nature of the onset of condensation this' comparison requires only an order of magnitude calculation of the cluster d i s t r i b u t i o n . In general, for these experiments i t i s found, using equation 2-4 for the calculation of the d i s t r i b u t i o n , that polar hydrogen-bonding l i q u i d s s a t i s f y the c l a s s i c a l theory whereas quasi-spherical or polar non-hydrogen bonding l i q u i d s do not. However, i t seems that a decrease i n the surface tension by 20% would bring these l a t t e r results into agreement with the c l a s s i c a l theory (Abraham (1974)). This would be i n agreement with the reduced surface tensions for smaller radius drops as calculated by Tolman (1949). It might be suggested here that the strongly hydrogen-bonding polar l i q u i d s would form a more compact cluster (and thus a more def i n i t e surface)and so f i t the assumptions behind the correspondence of equation 2-2. Such an explanation i s a b i t too simple, however, as i t i s inconsistent with the suggestion made by Reiss (1970) that s o l i d - l i k e clusters would require a larger correction to the pre-exponential factor, due to their, long range order, than would l i q u i d -l i k e clusters. 39 The emergence of new experimental techniques, such as that of employing a mass spectrometer (Foster et a l . (1969)) to measure cluster concentrations d i r e c t l y , w i l l make comparison between experiment and theory easier i n the future. Unfortunately the problems of specifying the exact experimental conditions and ve r i f y i n g an equilibrium state during nozzle-beam and o r i f i c e expansions s t i l l prevent a proper comparison with theory. In the case of unsaturated gases only dimer concentrations have been measured due to second order effects of ion-molecular reactions (important i n mass spectrometer experiments). The f i r s t evidence of dimer concentrations i n argon was obtained by Leckenby and Robbins (1966) using an ioni z i n g source and a mass spectrometer. They are quite convinced that the dimers observed do result from self-nucleating processes and that ion-molecular reactions are of second order, the dimers being ionized after forming. For argon at a pressure of 100 torr they f i n d , for example, 100 ppm at 300K and 400 ppm at 100K for the dimer concentrations. They also suspect that these dimers are mainly of a bound type, a large fraction of the metastable type having decayed i n the time taken for them to reach the spectrometer. These results do agree favourably with the results of the simple monatomic theory of Stogryn and Hirschfelder (1959) which i s concerned with the contributions of bound, metastable and free molecules to the second v i r i a l c o e f f i c i e n t . Evidence of dimers i n argon has also been indicated by an unresolved pure rotation band i n the Raman spectra from gaseous argon at a pressure of 2140 torr and at temperatures from 103K to 300K by Morgan and 40 Frommhold (1972). In t h i s case the behavior of the observed Stokes and anti-Stokes wings with respect to temperature and density i s representative of a biatomic process r e l a t i n g to bound pairs rather than c o l l i s i o n pairs. No estimate of dimer'concentrations was given. Milne and Green (1967) also measured dimer mole fractions i n a free j e t of argon. In t h i s case, however, the dimers were formed i n the supersaturated vapor resulting from the action of the j e t . However, by extrapolation of their results to zero o r i f i c e diameter (an apparently l i n e a r relation) they could approximate an effusion experiment. In this case they found, for example, 760 ppm dimers at an argon pressure of 1 atmos. and 3250 ppm at 5 atm., both at a temperature of 300K. Although these results agree with the results of Stogrn and Hirschfelder (1959), these experimental results have been multiplied by a factor of two to account for the effect of the ionization process discussed by Leckenby and Robbins (1966). Techniques based on the scattering of l i g h t have been used to measure cluster di s t r i b u t i o n s (Stein and Wegener (1967)) but are only sensitive to clusters larger than about 20 A i n diameter, and thus only useful i n practice for the study of supersaturated gases. 2.7 Positrons as a Probe for Clusters Evident from Section 2.6 i s the advantage (when probing for s e l f -nucleating clusters) of the f i n i t e l i f e t i m e of the positron compared to that of, say, an electron i n a mobility experiment. With allowance for possibly ion-induced clustering at high densities, the short l i f e t i m e 41 of the positron w i l l then allow probing of even the small clusters (2 to 3 atoms) i n the unsaturated gases which are presumably of an even more transitory nature than for supersaturated vapors below c r i t i c a l supersaturation. ' Since 100 ppm dimer concentrations exist i n argon gas at 300K,in whieh the positron s t i l l s a t i s f i e s a singl.e.~atom picture, the im-p l i c a t i o n i s that dimers are not important i n any enhancement mechanism. It i s conceivable though that the increase of these concentrations to 400 ppm at 100K i s enough of an increase of dimer energy loss mechanisms to s i g n i f i c a n t l y slow the positron. However, i t i s more l i k e l y that higher polymers, with their larger potential for energy loss (without completely breaking up the cluster) influcence the positrons motion to a much greater extent. In f a c t , i f the interaction of a cluster i s a rapidly increasing function of cluster si z e , then convoluting t h i s interaction with the rapidly decreasing d i s t r i b u t i o n of clusters (with size) presents a picture whereby the positron i s interacting with essentially only those clusters of 3 or 4 atoms. From the discussion by Kikuchi (1969), i t seems clear that the exponential dependence of 2-4 (with 2-2) w i l l describe the thermodynamic dependence of the cluster concentrations . Any discrepancy between the A Something not considered i n this present work i s the fact that the free energy for a l i q u i d cannot be measured for temperatures greater than the c r i t i c a l temperature, whether t h i s potential can be defined or not, or what ramifications i t s lack of d e f i n i t i o n has to the cluster theory i s not known. Although clusters of an extremely "open structure" have been considered for a l a t t i c e gas at temperatures greater than T (Reatto (1970)) the results are only applicable i n the immediate v i c i n i t y of the c r i t i c a l point. 42 inadequancy of the bulk surface tension to define the surface energy of a small cluster (less than 10 atoms) w i l l then be taken up by the 4 pre-exponential factor and w i l l be of the order of 10 . Since the probability of annihilation of a positron i n an i -cluster should be of the order of the annihilation rate with i atoms of the l i q u i d , an order of magnitude estimate of the pre-exponential factor can be obtained. Deviations of the measured annihilation rate from the exponential dependence on the bulk thermodynamic potential as the gas density i s increased w i l l represent possible ion-induced clustering (or electro-s t r i c t i o n ) resulting from the large energy loss to the positron by the clusters with possible formation of a bound state of the positron with a single atom or self-nucleated cluster, or simply shielding of the cluster from the positron by the higher density of gas. A t h i r d p o s s i b i l i t y i s the increase i n the cluster population to such an extent that the positron spends most of i t s time i n the v i c i n i t y of a cluster. The measured annihilation rate may then r e f l e c t the s t a b i l i t y of the clusters i n terms of the thermodynamic dependence. One may then hope to estimate the l i f e t i m e of the clusters. 43 CHAPTER THREE EXPERIMENTAL PROCEDURE AND RESULTS 3.1 Outline of Experimental Technique 3.1.1 Experimental Determination of Lifetime Spectrum The l i f e t i m e measurement involved measurement of the time i n t e r v a l between the detection of two photons, one associated with the " b i r t h " and the other with the "death" of a positron i n the gas. The b i r t h 22 signal was a 1.27 MeV photon emitted during de-excitation of Ne (Green and Lee (1964)), within 10 1 1 seconds of formation of th i s 22 excited state from the beta decay of Na which creates the positron (and associated neutrino). The death signal was a photon a r i s i n g from annihilation of the,positron, (either 0.511 MeV for two-photon anni-h i l a t i o n , or rather less i f the annihilation produced three photons). Events characterized by such a time i n t e r v a l signature were stored i n a NOVA computer. Each spectrum was taken at a fixed temperature, density and e l e c t r i c f i e l d . 3.1.2 Analysis of a Low Temperature Time Spectrum The difference between a room temperature and low temperature spectrum for the same gas density can be seen i n Figures 2 and 3. The most noticable difference i s the presence of a second peak for the low temperature spectrum, positioned at the end of the so-called shoulder found i n both spectra.: B a s i c a l l y , however, both spectra consist of four parts. The f i r s t , the negative time region, i s simply the random 44 1000 COUNTS 500 *: Prompt (25,000) Peak . - Room Temperature . - Low Temperature (114 K) v Shoulder >* . Random Background D = 25 amagats A - Intensity at channel 512. •'; Double Exponential ••••• / • •'• \'.< 512 0 Real Time CHANNEL Figure 2 Comparison of Low and Room Temperature Time Spectra - Linear Plot 45 Ln IdN [dt. Prompt > Peak Background and Ortho-Positronium Component Subtracted. + - Low Temperature Spectrum (114 K). Run Number 84. . - Room Temperature Spectrum (300 K). Run Number 77. Density = 25. amagats E D 0.0 V cm 1 amagat ^ Single Exponential e •Positive counts from background subtraction. CHANNEL Figure 3 Comparison of Low and Room Temperature Time Spectra - Log Plot 46 coincidence background displayed by e l e c t r o n i c a l l y delaying the death (stop) signal r e l a t i v e to the b i r t h (start) s i g n a l . This background i s assumed constant throughout the entire spectrum. The second region contains the very short l i v e d events at zero r e l a t i v e time which show up as a peak having a width equal to the experimental resolution of the system. This resolution i s t y p i c a l l y one to f i v e nanoseconds, depending on the energy resolution characteri-zing the sytem. The events i n t h i s "prompt peak" arise from positrons annihilating within the source, source holder, or the chamber walls and also from production of para-positronium i n the gas. Following the prompt peak i s the shoulder region consisting both of annihilation of free positrons not yet thermalized, characterized by a non-equilibrium vel o c i t y d i s t r i b u t i o n , and also ortho-positronium. Since the positrons, after t h e i r l a s t i n e l a s t i c c o l l i s i o n with an argon atom, are spread over a l l attainable energies, a small bump associated with those positrons finding themselves almost immediately i n the range of the enhancement at thermal energies (Section 1.3.1) i s expected to be present near zero time. Although t h i s small bump i s not easily seen i n the low temperature spectrum of Figure 3, i t s presence was c l e a r l y indicated by the work of Canter and Heyland (1974) who subtracted a room temperature spectrum from a low temperature spectrum at the same density. They then associated the d i s t i n c t separation of the peaks i n time to the presence of aRamsauer minimum i n the scattering cross-section. This minimum has a delaying action on the thermalization rate of the faster positrons. 47 The constancy of the shoulder-width density product i s further evidence that the free positron picture holds for positrons i n room temperature gas before the positrons are thermalized. This product as measured by Paul and Leung (1968) was t y p i c a l l y 340 to 360 nano-seconds-amagat. It i s very sensitive to impurities (Paul (1964), Orth (1966)), and a product less than the above values indicates impurity levels t y p i c a l l y greater than the 10 ppm l e v e l (Lee (1971)). In both spectra, the fourth region, that just after the shoulder, i s characterized by the sum of two purely exponential decays, one characterizing the annihilation of thermalized positrons and the other, that of the longer l i v e d thermalized positronium. For the low temperature spectrum, however, the rapidly increasing value of the annihilation rate of the thermalized positrons suggests the involvement of an additional mechanism, namely trapping of the thermalized positrons by clusters (Orth - private communication (1973)). This population of trapped positrons, designated " l o c a l " population would be assumed bound to "clusters" of gas atoms at least for times long compared to the posi-tron l i f e t i m e i n the l i q u i d . The "trapping rate", the rate of trapping of free positrons to the l o c a l i z e d d i s t r i b u t i o n , would then depend on the number density of such clusters (see Chapter Two). For the low-temperature spectra, i t w i l l be shown that the "pure" exponential following the peak can be expressed as the difference of two exponentials. However, i n the region of the second peak, the spread i n the thermalization times due to d i f f u s i o n results i n a s i g n i f i c a n t departure of the results from the predictions of t h i s 48 simple model. The peak i s decidedly wider than that obtained from the following simple considerations of two thermalized positron populations. Note that a complete derivation of the observed annihilation rate would include not only considerations of detector e f f i c i e n c i e s for both two and three photon annihilations as well as the presence of the ortho-positronium component, but also involve a solution of the appropriate d i f f u s i o n equation (1-9). In the following, i n order to point out the essential features of the process, we s h a l l assume that a l l positrons reach thermalization at the same instant. We s h a l l then consider the competing effects of the free positron anni-h i l a t i o n with single atoms versus trapping of the positrons i n clusters, followed by annihilation from the cluster i t s e l f . The ortho-positronium component though not considered here would simply add an exponential component to the result observed below with a decay rate lin e a r i n the densities. Consider a free population of positrons, N^, and l o c a l i z e d populations N^ (i=2 to °°) associated with clusters of i gas atoms. Let A^ be the vel o c i t y averaged free annihilation rate on single atoms ( i = l ) , A . the annihilation rate i n a l o c a l i z e d state, and A. ' c i 1 the rate of " l o c a l i z a t i o n " or trapping of the free positrons into the N.. population. The balance equations are then 49 — 1 = -XjN.. dt d 1 and 4^1 = -X .N. + X.N. i = 2,...» dt c i l l 1 where X^ = X^ + - j j ^ l These equations have the s o l u t i o n = N exp(-X.t) 1 o d NX. and N. = (exp(-X dt) - exp(-X c ±t) c i d The observed rate then follows: R e ( t ) . - N ^ + .E2N.Xc. N o [ ( x i + i l 2 T r ^ x T ) ) -pc-V) , cj • d cj d 3-1 3-2 3-3 Assuming for s i m p l i c i t y that one of the N. ( i . e N, ) dominates, t h i s can be s i m p l i f i e d by replacing X^ and A ^ by X c and X , res p e c t i v e l y , and disregarding a l l other N_^ . The rate equations 3-1 can then be diagramed as follows: 50 For t h i s case the observed rate i s * X X R (t) = N [( , + — ~ ) exp(-X t) a -v C - exp(-X t)] 3-4 " (x* -x d ) where now X, = X- + A . 3-5 d i e * Note that for X^ = X d R (t) = NX t exp(-X.t). 3-6 e o c d The time dependence of these rates for various values of X^, X , and X are displayed i n Figure 4, I t can be seen immediately that the rates q u a l i t a t i v e l y possess the experimental features of the low temperature l i f e t i m e spectra i n the form of a more pronounced peak with shorter 51 5 10 15 20 TIME (ns) Figure 4 Growth Model Curves 52 lif e t i m e s . However as seen i n Figure 5, where a f i t to a t y p i c a l low temperature spectrum has been made using the maximum li k e l i h o o d method (Orear (1958), Lee (1971)) and varying a l l the parameters of R e ( t ) , the experimental peak i s much too wide for th i s model. Since i t was not possible to incorporate the above model into a f u l l d i f f u s i o n treatment (1-9) only the exponential t a i l was f i t t e d according to the above model. A much more important result of t h i s f i t (using 3-4) i s that some care must be taken i n picking the single exponential region. For this reason and since the annihilation rate associated with this region w i l l no longer be equal to A^ (1-4), Section 1.2.1.1, the measured annihilation rate w i l l be denoted as the equilibrium anni- h i l a t i o n rate and written as A . This rate w i l l of course depend somewhat on the choice of the f i t t e d region (Section 3.3.1.2). The interpretation of A g w i l l now depend on l i m i t i n g cases of the above simple growth model (3-4), to be discussed i n the analysis. 3.2 Experimental Method 3.2.1 Chamber Configuration 3.2.1.1 Chamber Design A chamber capable of withstanding pressures greater than the c r i t i c a l pressure was essential i n order that argon gas i n i t s state of c r i t i c a l fluctuations could be studied. In turn t h i s would allow observations i n a l l the gas states i n close proximity to the coexistence l i n e (C.L.). In order to s a t i s f y this requirement, the minimum pressure requirement was 48.34 atm. (710 p s i ) . 53 X± = 0.34 ns 1 X =0.86 ns" 1 c * -1 X = 1.20 ns 1000-500" Run Number 2 j^. Density = 47 amagats 1 * * ' E n n „ -1 ,-1 j - , — = 0.0 V cm amagat • D T = 135 K \ COUNTS | \ A zero for f i t \ 100- " ' \ •*\ • \ T \ "* \ * 10-T * V • 7 \ + * \ Constant Background T \ • Subtracted \ * T \ t \ 1 150 290 CHANNEL Figure 5 Growth Model F i t to Peak Region 54 Because of the high pressures involved a stainless steel chamber was chosen over a copper one for i t s higher t e n s i l e strength at low temperatures. However, i n order to provide controlled temperature va r i a t i o n , i t s lower thermal conductivity required an isolated  chamber configuration i n order to obtain adequate thermal uniformity throughout the chamber, rather than, say, a more straight forward technique involving cooling with l i q u i d nitrogen vapor. A design problem with any high pressure system used for swarm experiments i s that of two c o n f l i c t i n g requirements for the r a d i a l dimen-sion. A small dimension f a c i l i t a t e s the pressure design, reduces y-ray attenuation, and allows for large detector s o l i d angles. However i n order to reduce the number of random coincidence background counts resulting from annihilation of positrons i n the walls of the chamber a large r a d i a l dimension i s advantageous. In addition, we desired enough room i n one of the chamber ends for a high voltage feedthrough (in our case, Ceramaseal 20kV, 900 psi) and four small thermocouple feedthroughs (Ceramaseal single mini-feedthrough cat. # 807B7844-3). The f i n a l choice of r a d i a l dimension, consistent with acceptably low levels of wall annihilations, limited the gas density to greater than 10 amagats; the C.L. limited the lowest gas temperature to 100 K. F i n a l l y but not of least importance, was the requirement of maintaining the purity of the gas over a s i g n i f i c a n t period of time. A technique i n which the chamber i s thermally isolated except for a small controllable conduction leak to a colder reservoir. 55 Since the effect of impurities on the annihilation spectrum i s well known (Lee (1971), Paul and Leung (1968)), s t r i c t precautions were followed as discussed below. Some general considerations involved use of only metal and teflon materials (in p a r t i c u l a r , elimination of butyl rubber 0-rings). Where possible, parts were welded into place to reduce the number of gaskets and welds were done i n an argon atmosphere to reduce microscopic cracks. The f i n a l chamber, pressure-designed according to Brownell and Young (1959), i s presented i n Figure 6 and the gas handling system i n Figure 7. The chamber volume i s about 750 ml. A 3/4" mini-valve (Varian # 951-5014) was o r i g i n a l l y placed i n the bottom of the chamber to f a c i l i t a t e evacuation, but the success of a much simpler flushing and baking technique to adequately obtain and hold a pure gas sample allowed this valve to be removed. The valve was removed for safety reasons as the valve was s p e c i f i c a l l y designed for vacuum work, and prolonged use at high pressures was not considered desirable. The 0-rings i n the chamber and vacuum jacket were teflon coated stainless steel (United A i r c r a f t Products #U-2420-02938-SEA and //U-2310-04500-NPD respectively). A l l parts including the feedthroughs were welded into place. The thermocouple wires, were epoxied into the hollow feedthroughs using Emerson "stycast" epoxy 2850 FT cured with catalyst # 11 (cured at 100 C for 2 hours and useable from 77 K to 200 C) The high temperature (200 C) was necessary to allow e f f i c i e n t baking of the chamber for outgassing of the inner chamber surfaces. 56 Radiation Cover (Cu.) Thermocouple Heat Sink, \ E [ (1/8" Dia. Cu.) Thermocouple Feedthrough (1 of 4) Aluminized Mylar Control Thermocouple Support Ring for E l e c t r i c F i e l d Assembly E l e c t r i c F i e l d Assembly, (Figure 8) Vacuum Jacket (77 K) Pumping Line ( 1-1/2" O.D.) Gas I n l e t , Support Tubes, and Heat Leaks , (1/4 O.D., 1 of 3) Cu. Heater Block (1 of 3 ) . 20 kV Feedthrough Gas Chamber (S.S.) (2.64" I.D.) (10 " inside Length) Radiation Shield (Cu.) Lower Thermocouple 'Vacuum Figure 6 Experimental Chamber 57 All-Metal Valve =<x>= Pressure Gauges Sample Chamber -Thermocouple . Feedthroughs H.V. Feedthrough Bursting Disc Ultra-High Purity 1  Argon Pumping Line Check Valve (1/3 atrnos.)! Level — Sensors Diffusion] Pump LN, Gas Chamber Glass Dewar Figure 7 Gas and Coolant Handling System 58 The chamber was designed for 950 p s i , but i n i t i a l l y , along with the mini-valve, was hydrostatically tested to 1000 p s i three times. The chamber was therefore considered safe for routine work up to a pressure of 600 p s i . After the mini-valve was removed, the chamber was subsequently used to 765 p s i for two runs, but was not hydro-s t a t i c a l l y retested. For personnel safety, the chamber and detection system was surrounded by a wall consisting of 1 f t . thick sand-box shielding up to waist high with walls of 3/4" plywood, with 3/4" of "gyproc" backing on the inner surface above. As a f i n a l precaution against over pressures a 900 p s i bursting disc sealed with teflon tape was placed on the %" gas f i l l i n g l i n e . This approach was chosen over the more convenient check-valve because the rubber 0-ring i n the valve would be a potential source of hydro-carbon contamination. In addition a -2 M-l/3 atmos. check-valve was placed i n the vacuum port of the vacuum shield i n case of a major gas leak from the pressure chamber into the vacuum region. 3.2.1.2 E l e c t r i c F i e l d Grid An e l e c t r i c f i e l d was produced within the chamber by a system of nine alternately biased thin wire meshes mounted within 2%" O.D. rings made from 3/16" dia. S.S. rod (to reduce the mesh edge f i e l d s to less than the breakdown value). A diagram of this assembly i s shown i n Figure 8. The mesh :contained 0.009" S.S. wire spaced at 20 to the s stainless steel inch, giving a 67%.opening. Two /\ sheets with similar ring-edges were used for the grid ends. In order to prevent the positrons from 59 Support Rods-(S.S., 2). Top (1/4" Teflon)-SECTION THROUGH MESH EDGE RINGS. Nylo* S c r e w s - ^ ^ F e ^ r u > ^ ^ ^ (Brass) Support Ring (Teflon) Mesh Edge Rings (S.S.) (3/16" Dia.) \P~0.020" Teflon Sheet, double Thickness. Glass Spacers (1.0 cm high) (FULL SIZE) 7777 Vent Holes Figure 8 E l e c t r i c F i e l d Electrode Assembly 60 entering the non-uniform f i e l d region outside of the meshed-rings and to accurately space the meshes, glass cylinders of 1.0 cm height, 4.4 cm I.D. and 0.2 cm thickness were used. The whole system was held together and insulated with a c i r c u l a r "blanket" of 0.020" teflon sheet screwed to 1/4" teflon end plates by nylon screws. The rings were alternately interconnected together by soft soldering a copper wire to brass hooks, hard soldered into the mesh rings. The high voltage end was secured with a nut to the H.V. feedthrough and the ground wire was l e f t long enough to make a spring f i t between the grid assembly and the chamber walls. The chamber and pumping port were therefore part of the return ground. 3.2.1.3 Source Construction and Strength For those measurements made prior to the introduction of the e l e c t r i c f i e l d assembly into the chamber the configuration of the 22 positron source was radioactive Na CI deposited onto a single sheet of 75 microinch n i c k e l f o i l . The f o i l was held i n a horizontal position at the chamber center by two 1/16" S.S. rods attached to the teflon ring, around the H.V. feedthrough (Figure 8). With the addition of the f i e l d assembly the source was deposited d i r e c t l y onto the mesh wires of the three center meshes. This l a t t e r technique was employed to prevent the l i q u i d from pooling i n the v i c i n i t y of the source when a saturated vapor was being investigated. An i n i t i a l source strength of 15 microcuries for both cases was si m i l a r to that used by Lee (1971). It was f e l t that this strength was necessary to obtain a satisfactory count rate from the i n e f f i c i e n t p l a s t i c s c i n t i l -l ators and thus to keep the counting time down to a reasonable length 61 for a temperature controlled experiment. The p o s s i b i l i t y of using a gold coating, a good backscatterer of positrons, on the mesh wires as a means of increasing the number of positrons annihilating i n the gas was contemplated but not pursued because the glass spacers contributed the greater percentage of wall annihilations, especially at the lower densities and these could not be coated for obvious reasons. 3.2.1.4 Sample Gas P u r i f i c a t i o n Previous workers (Lee (1971), Bird and Jones (1973)) used titanium f i l l e d p u r i f i e r s (at 650 C) operating either intermittently or continu-ously to compensate for outgassing from chamber surfaces and from 0-rings coated with high vacuum grease. However, as an i s o l a t i o n technique for temperature control was used i n t h i s work, th i s type of p u r i f i c a t i o n technique was not p r a c t i c a l because of the problem associated with such a non-constant heat load. I t was f e l t that adequate results would be achieved by use of an all-metal chamber (containing some tef l o n and epoxy) as a preventative method of p u r i f i c a t i o n . Thus the external gas f i t t i n g s to the chamber were either all-metal or teflon sealed. A l l the valves, except for the two closing off the sample chamber, were of the bellows type and the sample chamber was isolated from the main chamber by one of these. In addition, while f i l l i n g , a flushing technique to d i l u t e the chamber a i r to less than the 1 ppm l e v e l would remove the need to evacuate this a i r by means of a diff u s i o n pump system. Both high vacuum grease used on rubber 0-rings and back-streaming of pump o i l s are potential 62 sources of hydrocarbons. Only four flushings with u l t r a pure argon gas each from 400 ps i to 10 p s i (gauge) are necessary to reduce the a i r l e v e l to less than 1 ppm. Coupling t h i s flushing routine with baking of the chamber to just less than 200 C when f i l l e d with the high pressure argon gas was expected to e f f i c i e n t l y remove impurity molecules adhering to the chamber walls. In addition, the outgassing of the walls at the low temperatures employed during a run was expected to be much less than that from the room temperature case. Basically then, the p u r i f i c a t i o n technique consisted of four (or more) flushings with ultra-high purity argon while baking the chamber at approximately 200 C for a t o t a l of 24 hours. In addition, p a r t i a l replacement of the gas took place between runs due to the method of varying the density at constant temperature for a series of runs and complete replacement every three weeks. The observed con-stancy of the shoulder-width density product with time, i t s agreement with results for gas with known impurity levels of less than 10 ppm, and the results of a spectrographic analysis (Section 3.4.2), a l l serve to j u s t i f y the success of t h i s technique. F i n a l l y , i t i s of interest to note that when the stycast 2850 FT epoxy (catalyst # 11.) was cured at the required high temperature, i t l e f t a faint thin coating over a l l the inner chamber surfaces. This may have helped to reduce outgassing. Stewart (private communication -1976) reported to this author that an epoxy coating does exist that w i l l e f f e c t i v e l y seal up a l l small microscopic cracks which are potential 63 sources of impurities, but i t must be p r o f e s s i o n a l l y applied and i s somewhat c o s t l y . 3.2.1,5 Attenuation and Scattering of Gamma-Rays Because of the existence of the chamber walls and the r a d i a t i o n s h i e l d , one for pressure considerations the other for adequate thermal conductivity, considerable gamma-ray attenuation r e s u l t e d . The e f f e c t of t h i s attenuation through the three concentric c y l i n d e r s : chamber, r a d i a t i o n shield and r a d i a t i o n jacket, i s displayed i n Figure 9, where a 4" diam. x 3" Nal detector was used, An attempt to reduce the i attenuation involved machining two 2" x 2" square windows, i n l i n e with the detectors at 90°, into the r a d i a t i o n s h i e l d leaving a 1/32" " i thickness of copper. The improvement, however, was minimal. 3.2.2 Parameter Control and Measurement ! 3.2.2.1 Low Temperature j I I I s o l a t i o n of the chamber was attained by suspending i t i n a vacuum using three 10 cm long, 1/4" diameter S.S. pipes, of which one was used i as a gas supply l i n e . The vacuum container was held at 77 K by l i g u i d ! nitrogen i n a suuroiinding glass dewar. A r a d i a t i o n shield between the chamber and vacuum container wall consisted of a copper c y l i n d e r (water i pipe) having 1/8" thick walls, and with jl/4" thick top and bottom p l a t e s . j Only the top plate came i n contact with the chamber and was attached to i t by three copper, c y l i n d r i c a l heater blocks at the base of the suspension pipes (Figure 6). This copper plate and coupled to the chamber top and pipes by heater combination was thermally GE 70-31 varnish. A t o t a l of 64 8000 - Peak Normalized to 10,000 Nal(Tl) Detector 22 1 - Na i n A l . 22 2 - Na i n 9 i d e Gas Chamber. 6000" 22 3 - Na inside Chamber + Radiation Shield. COUNTS 22 4 - Na inside Chamber + Radiation Shield (with windows). 4000" \ \\w \\w - NE 111 P l a s t i c 2000" \ W \y { A \ \ \ \ \ X 0" 0 i i 0.511 1.27 ENERGY (MeV) Figure 9 :Gamma-Ray Attenuation - Nal(Tl) Detector 65 100 Q. of #32 insulated nichrome wire (4.686 ft/ft) was wound on the three c y l i n d r i c a l heater blocks as a heat source for compen-sating the heat loss along the three pipes and away from the radiation shield. I t was calculated that with the chamber at 200 K the heat flow to the 77 K reservoir was 600 mW (90 mW at 100 K). The outside of the radiation shield and gas chamber were both covered with two layers of aluminized mylar. I f an emissivity of 0.02 i s assumed for th i s covering then the radiation loss at 200 K to the 77 K vacuum jacket i s only 19 mW. The thermocouples were #30 copper-constantan wire obtained from Honeywell. I n i t i a l l y these were spot-welded to form the couple, but lat e r the use of a hydrogen torch produced a much better bead. Commercial units were not used because of s p a t i a l l i m i t a t i o n s on the chamber top and within the chamber. The control thermocouple was i n i t i a l l y placed around an isolated 1/8" copper rod near the top of the inner chamber space, but with the l a t e r introduction of the e l e c t r i c f i e l d assembly i t was epoxied to the top plate. A second thermocouple to check for thermal non-uniformity i n the gas between the top and bottom was hung i n thin teflon tubing near the chamber bottom and had the bead coated with epoxy for structural s t a b i l i t y and insulation. In addition immediately outside, the chamber the thermocouple wires were thermally anchored to 1/8" copper posts imbedded into the r a d i -ation shield p l a t e j using cigarette paper for insulation and GE 70-31 varnish as a glue. The wires were thermally insulated from the heater blocks and the vacuum chamber walls by layers of aluminized mylar. 66 It was found that the two sets of thermocouples systematically differed by only 1 uV (20 uV/K) at either room temperature, 77 K, or any controlled temperature between 100K and 160K. The i s o l a t i o n vacuum was held at 2 x 10 7 torr using a conven-t i o n a l d i f f u s i o n pump system, with a l i q u i d nitrogen cold trap situated above the dif f u s i o n pump. With the chamber at 150 K the l i q u i d nitrogen use was about 0.4 1/hour corresponding to a power loss of 18 watts. This loss includes an estimated 5 watts down the 1-1/2" diameter combination vacuum port and supporting tube and an unknown loss during the f i l l i n g of the glass dewar and from the sides and insulated top of this dewar when f u l l . 0.31/hour was also used i n the cold trap above the d i f f u s i o n pump. Both the trap and dewar were f i l l e d automatically (Appendix A). In order to speed up the cooling of the chamber between equilibrium positions, helium exchange gas was l e t into the vacuum region. However wrapping heater wires around the S.S. chamber for more rapid heating was not pursued as the heat loss along these wires during the periods of equilibrium would affect the chamber thermal uniformity to an unknown degree. The temperature controller consisted of four d i f f e r e n t i a l amplifiers and used the thermocouple epoxied to the inner chamber top as the sensor. As shown i n Figure 10, one of these amplifiers was used as a gain of 100 amplifier for the m i l l i v o l t thermocouple voltage and a second for a following active f i l t e r amplifier. A t h i r d amplifier used a 1.2 V zener-op-amp integrated c i r c u i t as a reference and the fourth compared the output of the active f i l t e r to this reference. This difference signal was then 67 +15 V .2.1k 1.2V, I LN113 r >9.53k 10k —VW-10k 10k (10 turn) -yfa Reference Voltage LM741 (0.25 - 0.50V) 2.87k Control Thermocouple 1M Chamber 10k I—O-A/W-100k -wv— 0.47 11 39k LM741 0.22 1M -L- (active f i l t e r * Reference 110V o-AC Chamber Heater 100 VW +10 V ° © V W - i — l ^ U O A l 200k 4.7 2N2646 MCR1304-4 47 Figure 10 Temperature Controller 68 passed to one side of a simple transistorized d i f f e r e n t i a l amplifier. The output drove a low frequency unijunction transistor (UJT) which controlled the a.c. l i n e current through a s i l i c o n controlled r e c t i f i e r (SCR). The SCR allowed for continuous variation of the heater current directed to the three s e r i a l heaters on the tubes suspending the experimental chamber i n i s o l a t i o n . Because of the temperature s e n s i t i v i t y of the input amplifier, i t was found necessary to put the entire controller into a temperature I I controlled enclosure, a small one foot square, 1/2 thick styrospan box (Figure 11). A 220 Q r e s i s t o r was controlled by a thermister sensor (part of a Simpson #389 room temperature measurement unit) which was part of a re s i s t o r bridge c i r c u i t with an d i f f e r e n t i a l amplifier as an error detector (Figure 11). The output of th i s amplifier drove one side of a d i f f e r e n t i a l transistor pair and associated UJT and SCR, similar to that co n t r o l l i n g the chamber heater (Figure 10). This temperature s t a b i l i z e r was also situated inside the styrofoam box. (Figure 11). The box temperature was kept at 30.8±0.2 C. By this means, an increase i n s t a b i l i t y of the chamber temperature from ±0.4K toiO.lK was obtained. 3.2.2.2 Pressure For most runs the chamber pressure was measured using two Ashcroft gauges. One read from 0 to 300 p s i (mirror backed) and the other from 0 to 1500 p s i . Both had a stated accuracy of 5/1000. The 300 p s i gauge was calibrated by Lee (1971) to an accuracy of better than 10/1000. A third :gauge manufactured by Marsh was used for a few low 69 680 -A/VV-Set Temperature 500 0 Thermister (from Simpson "Therm-O-meter" Model 389. -30 V To C i r c u i t Similar to That Following "C" of Figure 10 100 a Heater Cu Const 72 W Gas ;/ Chamber] • Cu Ice Bath Control Thermocouple Bottom Thermocouple to yV-meter. Cu Fan C O m CD o .O U S ^  S c CO rC o 4J uc_> CO Tl 32 C Thermistor 220ft Styrospan Box 3/4" wall Figure 11 Temperature S t a b i l i z e r 70 density runs (less than 15 amagat) and read from 0 to 150 p s i . (5/1000 mirror backed). This gauge was also calibrated by Lee (1971) to better than 10/1000 accuracy. In addition, a l l three gauges were found to agree to better than 10/1000. The gauges were kept at room temperature and isolated from the low temperatures by a metre of 1/4" S.S. tubing. Problems with gas leakage were of a temporary nature only. These consisted of leakage through the epoxy sealing the thermocouple wires into t h e i r respective feedthroughs, leakage through both metal 0-rings, and poor seating of one of the bellows valves. Re-epoxying, replacing the 0-rings, and careful closing of the faulty valve were the appropriate remedies. The residual gas loss was e n t i r e l y n e g l i g i b l e . 3.2.2.3 E l e c t r i c F i e l d The magnitude of the e l e c t r i c f i e l d was determined by the voltage applied by a Fluke 405B H.V. power supply (accurate to 1 V) and the 1.10±0.01 cm average mesh separation. For low density gases the glass separators (Figure; 8) prevented the positrons from being scattered into the non-uniform f i e l d at the mesh edges by transforming the resulting positrons into annihilation events i n a short l i f e t i m e component which i s part of the prompt peak (Section 3.1.2). For dense gases and l i q u i d s the range of positrons was much less than the radius of the glass-separators. Since the diameter of the glass rings was about 4.4 cm and the S.S. edge rings were outside the glass cylinders, the error i n the f i e l d due to edge effect was expected to be less than 1%. Thus the only si g n i f i c a n t inaccuracies i n the e l e c t r i c f i e l d were due to non-71 planar meshes, the f i n i t e thickness of the mesh wires and the spaces between the mesh wires. These sources of error are discussed i n Section 3.4.1.4. 3.2.3 Electronics 3.2.3.1 General Description The time-digitizing system employed a standard fast-slow coincidence c i r c u i t s i m i l a r to that used by Lee (1971). However, the Nal(Tl) crystals employed by Lee were replaced by the better time-resolution NE 111 u l t r a - f a s t p l a s t i c s c i n t i l l a t o r s (2" dia. by 3" i n length) with an NE 560 r e f l e c t coating and RCA 8575 photo-m u l t i p l i e r tube. The photomultiplier bias c i r c u i t i s i n Figure 12 and an overall block diagram of the timing system i s shown i n Figure 13. An Ortec design pre-amp (Figure 14) was used on the energy signal at the end of the photomultiplier to shape the pulses for the amplifiers (Ortec 471). The timing signal went to two discriminators placed i n series. The main timing discriminator was a constant-fraction Ortec 463 driven by the signal after popagation through a bridging input leading edge d i s -criminator (LRS Quad Disc 621-BLZ). The use of t h i s second discriminator was related to pileup problems discussed i n Section 3.2.3.5. The pileup rejectors (GP100/N), OR-gates (OR102/N), and Conuclear C126 time-to-amplitude converter (TAC) were a l l standard commercial fast NIM units. The coincidence f a c i l i t i e s provided i n the TAC were used for the slow coincidence. The TAC output was directed to a Northern S c i e n t i f i c 72 -H.V. ( ? " rfn 0.005 -777 150k 150k 68k 100k 68k 68k 68k 68k 68k 0.001 82k 120k A u x i l i a r y ; ; Power 0 T 10 (Unused)> ^ v v " 2.2qr 220k Timing Out (•-x i 470 P F 470 P F 470 pF 68k -T 470 pF 0.01 0.01 680k —VW— 0.01 120k =Jr0.01 10k < Cathode <D1 ~C D2 _£D3 -(_ D4 < M -Q D8 _£D9 D10 100 - A W — ( _ D l l "C D12 Anode 50 Figure 12 Photomultiplier C i r c u i t 73 NE 111 RCA 8575 RCA 8575 1.27 MeV SCA 0.51 MeV SCA UAJ Disc Disc. Disc. Start Pileup Rejector Rej ect TAC Stop Delay Disc. ^Amplitude Out ADC D i g i t a l NOVA Computer OR Pileup Rejector Figure 13 Electronic Configuration for Timing Measurement 74 100 +24 V o VvV-5.1k 750 -< IN — T lOOpfj^ 510 lOpf T I 2N3643 2N3638 2.2M rfn 10k £ 22 10k fn l k l k AT77 3.6k 100 -24 V o —vw I I 6.2k 5.1k 6.8 6.8 rfn 22 47 OUT —| | VW-2N3643 •750 "1 I 6.8 Units: R: ft C: uF Figure 14 Photomultiplier Pre-Amplifier C i r c u i t 75 NS 622 amplitude-to-digital converter (ADC) and then to a NOVA minicomputer i n which the events were stored using the NOVA data channel. 3.2.3.2 Pulse-Height Resolution The energy spectrum associated with detection of 0.511 MeV and 1.27 MeV photons by a NE 111 p l a s t i c s c i n t i l l a t o r i s shown i n Figure 15 along with the position of the appropriate energy windows. This spectrum i s compared to the spectrum from a Nal(Tl) detector, for which the photopeaks are c l e a r l y discernible, i n Figure 9. The lack of these photopeaks for a p l a s t i c s c i n t i l l a t o r c l e a r l y points out the need for discriminators with good timing resolution over a wide dynamic range, a characteristic of the constant-fraction type. 3.2.3.3 System Linearity 3.2.3.3.1 Integral Linearity The i n t e g r a l l i n e a r i t y was measured i n a manner similar to that described by Lee (1971) using a frequency generator and fixed delay. However i t was found that the fluctuation i n frequency at the higher frequencies did not permit a s u f f i c i e n t l y accurate c a l i b r a t i o n of the 50 nanosecond and 100 nanosecond ranges. For this reason the 200 nanosecond range was calibrated according to Lee (1971), and i s plotted in Figure 16, whereas the 50 and 100 ns ranges were calibrated using a * Data General. 76 Figure 15 Energy Windows and Discriminator Settings - Ne 111 S c i n t i l l a t o r 77 4.5-j 200 ns TAC Range J 4.0-1 / FREQUENCY (x 10~ 7s) / 3.5" f (0.331 + 0.0012) ns/channel 7 3.0-/ 2.9 / I 1 1 1 1 0 100 200 300 400 500 CHANNEL Figure 16 Integral Linearity of Time Measurement 78 fixed 30 nanosecond delay switched i n and out and comparison of this channel i n t e r v a l to that with the same delay on the 200 ns range. The following time calibrations resulted for the 50, 100, and 200 ns TAC ranges: 0.0834±0.0005, 0.177±0.001, and 0.331±0.001 (ns/channel) respectively. Note that the 100 and 200 ns ranges relate to conversion of the corresponding TAC ranges to 512 channels by the ADC, whereas the 50 ns range involved the conversion of the TAC 100 range to 1024 channels with 512 channels offset by the ADC. This setup for the 50 ns range was used to help eliminate some annoying o s c i l l a t i o n s i n the time ca l i b r a t i o n which characterized the beginning of th i s TAC range. These o s c i l l a t i o n s are discussed i n the next section. Thus for 512 stored channels the time ranges were 42.7, 90.6 and 169.0 ns. This range reduction from the expected 50, 100, and 200 (nanoseconds) resulted from the difference between the 5 V range of the TAC output and the 4 V range of the ADC input. 3.2.3.3.2 D i f f e r e n t i a l Linearity The setup for the d i f f e n t i a l l i n e a r i t y was also that used by Lee (1971). Basically t h i s consisted of random starts generated from nuclear radiation detected i n a s c i n t i l l a t i o n counter and periodic stops from a pulser.. The use of periodic stops permitted large counting rates without the attendant non-linearities of the form 1 - R T stop where R i s the stop side channel rate and T i s the time range stop 79 associated with random stops as discussed by Lee (1971). Except for a persistent 3 to 5 nanosecond o s c i l l a t i o n i n the time c a l i b r a t i o n of about 1% amplitude, the only non-linearity was a 3% positive slope i n the d i f f e r e n t i a l l i n e a r i t y which resulted from use of 93 fi cable between the TAC and ADC. It was more li n e a r with 50 Q cable but then the TAC output amplitude was twice the maximum amplitude converted by the ADC. The o s c i l l a t i o n s were reduced to this 1% l e v e l by careful choice of coaxial cables with securely f i t t e d end connectors. Although the o r i g i n of these o s c i l l a t i o n s remains uncertain, a close inspection of some of Lee's spectra (1971) reveal the presence of similar o s c i l l a t i o n s , which of course had negligible effect on his 500 ns range. The or i g i n of these o s c i l l a t i o n s were attributed to the combined effect of many small mis-terminations between 50 cables and the electronic equipment, and the non-perfect grounds provided by the mechanical aspects of the BNC and Lemo connectors. Interchange of equipment models involving both similar and different models, and temporary removal of the bridged-input discriminators, resulted i n minor frequency, phase and amplitude changes, but no sig n i f i c a n t amplitude improvement. In addition, although the o s c i l l a t i o n s persisted throughout a l l the time ranges, a few larger o s c i l l a t i o n s appeared at the beginning of the 50 ns range. Because of this and the non-linearity of the lower 25% of th i s range as quoted by the manufacturer,the top half of the 100 nanosecond TAC range was used instead by applying the d i g i t a l offset option of the ADC as mentioned i n the previous section. 80 The three d i f f e r e n t i a l l i n e a r i t i e s obtained by t h i s method and plotted i n Figure 17 were used d i r e c t l y i n the l i f e t i m e analysis of the data according to the range used for that run. Nevertheless, minor phase and amplitude changes i n the time c a l i b r a t i o n data did occur with time and some runs were rejected i f noticeable o s c i l l a t i o n s were present i n the region of the short l i f e t i m e . This was necessary as the short lifet i m e s being measured were of the order of the period of these o s c i l l a t i o n s . 3.2.3.4 Time Resolution of the System Generally a compromise must be made between timing resolution, coincident count rate and pileup rejection. Here this s i t u a t i o n was complicated by the 12-stage photomultiplier tubes used (14-stage tubes were used by Lee) and the detection of mainly Compton events by the p l a s t i c s c i n t i l l a t o r s (Figure 15). Since the shortlived "free" positron l i f e t i m e (vL ns) was the component of primary interest i n t h i s experiment, use of p l a s t i c s c i n t i l l a t o r s with good timing resolution was essential . This was obtained by f i r s t optimizing the value of the high voltage applied to the tubes (2120 V for both tubes) and setting the discriminators just above the noise. Although the timing resolution was more c r i t i c a l l y dependent on the high voltage setting than the discriminator setting, the 1.27 MeV timing discriminator could not be set higher than that shown i n Figure 15 (as desired for pileup considerations - Section 3.2.3.5) without compromising the timing The ortho-positronium l i f e t i m e i s more accurately and e f f i c i e n t l y measured using the better energy resolution and e f f i c i e n t Nal(Tl) detector. 81 20,000-J O j 60,000J COUNTS o- 1 60,000 200 ns Range 100 ns Range 50 ns Range (1024 Channels with 512 D i g i t a l Offset) 512 CHANNEL Figure 17 D i f f e r e n t i a l L i n e a r i t y of Time Measurement 82 resolution. F i n a l l y the side channels analysers should normally be set above the discriminator edges as i t i s near threshold that most time slew arises. As the cpnstant fraction discriminators did not exhibit t h i s c h a r a c t e r i s t i c , the 0.511 MeV side channel analyser was set below the 0.511 MeV discriminator setting to maximize the corresponding coincident count rate. With the discriminators and energy windows 22 set as shown i n Figure 15, the best resolution obtained for Na i n 60 aluminum was 0.95 nanoseconds full-width-half-maximum and for Co , 22 0.61 nanoseconds. The larger resolution for Na i s due to the f i n i t e l i f e t i m e of positrons i n aluminum. For a 15 uCi chamber source, the coincident count rate was 250 counts/min. 3.2.3.5 Pulse Pileup and Background Effect The various pileup effects are f u l l y discussed by Lee (1971). In summary, these result from high count rates causing baseline s h i f t s i n the ac coupling between the photomultiplier and discriminator ( i f applicable), increase i n the random coincidence background and non-uniform background i n the negative time portion of the spectra. As we employed dc coupling between the photomultiplier and discriminator the f i r s t problem does not arise. The random coincidence background depends on the source strength but i s constant over the entire spectrum 4 - 1 i f the side channel analyser 'stop' rate i s less than 10 s . The rate 3 - 1 of 5 x 10 s used i n t h i s work s a t i s f i e s this constraint. The t h i r d effect results from correlated negative time events and shows up as a shoulder (with l i f e t i m e components) i n the negative time region. Elimination of this effect i s most important when measuring 83 the long l i f e t i m e of ortho-positronium (^ 70 ns) for which a good estimate of the background (which we infe r from the negative time region of the spectrum) i s essential. As t h i s l i f e t i m e was not of prime importance i n t h i s work ("clustering" of positrons and bubble formation by positronium are independent) and the li f e t i m e s of the "free" positrons were short (1 to 15 ns) such negative time problems did not cause concern. I n i t i a l l y , however, for the low density runs (<20 amagats), the intensity of the negative region of the spectra was greater than the intensity at the most positive end of the time range. In t h i s case the u n r e a l i s t i c background estimate c r i t i c a l l y affected the convergence of the f i t t i n g program (Section 3.2.4.4). In order to reduce the back-ground estimate to a reasonable l e v e l , the following pileup considerations were taken. As negative time events originate from 0.511 MeV pulses sat i s f y i n g the 1.27 MeV discriminator and side channel analyser settings, the low setting of the 1.27 MeV discriminator, necessary to obtain adequate timing resolution (Section 3.2.3.4) contributes to this pileup. F o r t h i s reason the 1.27 MeV side channel analysers was set high as shown i n Figure 15. The pilup gates i n Figure 13 were then added to eliminate pileup of 0.511 MeV pulses through the 1.27 MeV side channel. In addition, as the deadtime of a constant fraction discriminator i s about 100 ns, a second set of discriminators (leading edge) with a bridging input was placed ahead of the timing (constant fraction) discriminators to reduce the (pileup) deadtime to 15 ns. By this means, the pileup was decreased from 1.6% to 0.3%. At the same time these second discriminators 84 were set lower than the timing discriminator settings i n order to test a l l timing pulses for pileup. Inclusion of t h i s pileup system, together with the second set of discriminators, did not compromise the timing resolution nor contribute to the o s c i l l a t i o n problem discussed i n Section (3.2.3.3.2). As seen i n Figure 2 (low temperature spectrum), the negative time events for the f i n a l system were always less intense than those i n the positive region, i n a manner consistent with expectation. 3.2.4 Data Reduction 3.2.4.1 Calibrations 3.2.4.1.1 Temperature The thermocouples were calibrated i n terms of the pressure-temperature r e l a t i o n for the coexistence of the gas and l i q u i d state of argon. This r e l a t i o n i s given i n tabular form for temperature increments of 1 K by Angus and Armstrong (1972). In Figure 18 the experimental relationship between the temperature (related to pressure) and the thermocouple voltage (for the hydrogen-beaded couple) follows approximately 15 microvolts below that given by the low temperature data fpr a copper-constantan thermocouple (NBS Mon. 124 (1972)). Results for the spot-welded thermocouple were i n even better agreement with these tables, but the overall s t a b i l i t y was not as good. Since the temperature range i n the figure covers the ranges of a l l three pressure gauges used, the continuity of the experimental points for the three gauges and agreement with the thermocouple tables provides additional v e r i f i c a t i o n of the c a l i b r a t i o n technique. 85 4950 H 4800 H > w § 4600 i-J o > w o u o 4400 H 4200 + — NBS Mon. 124 (1972) o — This Work ~1 108 110 176 p s i GAS PRESSURE 293 p s i 120 130 140 TEMPERATURE (K) Figure 18 Thermocouple Voltage versus Temperature from Gas Pressure Measurement 86 Assuming an accuracy of ±2 p s i i n the pressure (1% of 200 psi) and r e l a t i n g thermocouple voltage to temperature (NBS mon. 124 (1972)), the error i n the temperature as given by the slope i n Figure 18 i s ±0.2 K. The results of Canter and R o e l l i g (1975) give a maximum vari a t i o n i n the equilibrium annihilation rate with temperature of 0.032 ns/K. Therefore the above var i a t i o n of ±0.2 K would correspond to a variation i n X of ±0.006 nanoseconds. Since this i s smaller than e the experimental error i n A ( t y p i c a l l y ±0.04 ns) the temperature s t a b i l i t y of ±0.2 K was deemed adequate for the work reported here. 3.2.4.1.2 Density Using the temperature c a l i b r a t i o n from the saturation l i n e data and the measured pressure, the density of the unsaturated gas was determined using a 16 parameter f i t (published by Angus and Armstrong (1972)) to a l l available argon data. Assuming the errors for pressure and temperature given i n Section 3.2.4.1, th i s method propagated an error of less than 2% to the density determination. 3.2.4.2 Difference i n Gibbs Free Energy In the analysis of the results (Section 3.3.6) i t was found useful to relate the annihilation rate to the difference i n the Gibb's free energy of the gas and the metastable l i q u i d . This thermodynamic quantity i i s a measure of the s t a b i l i t y of the gas (or gas cluster) as discussed in Section 2.1. The free energy for a gas was calculated i n the usual fashion from: 87 where g = spec i f i c free energy (chemical p o t e n t i a l ) , h = sp e c i f i c enthalpy, s = sp e c i f i c entropy, T = temperature, and subscript g implies a gas state, h and s (each with a greatest maximum error of .25%) were obtained from the tabulation of: Angus and Armstrong (1972). Since the values for g T, the chemical potential for metastable l i q u i d (L) argon are not known, g^ was extrapolated downward from g^ for the stable l i q u i d at a presssure of 50 bar. The accuracy of th i s extrapolation was improved by using the temperature dependence of 3g /3p as calculated for the stable l i q u i d for pressures less than 1-4 50 amagats. It was then assumed that t h i s dependence could be applied to the metastable l i q u i d . The independence of 3g^/3p on the pressure (and therefore the density) was demonstrated by calculating: 3gT g T(50 bar,T) - g (p,T) W ^ - l - 50 - p 1 3 " 8 and p l o t t i n g the results (Figure 19). In th i s case, g L(P,T) for the stable l i q u i d states was calculated from the tables using a formula similar to 3-7. A v i s u a l f i t to these points was then used to estimate the temperature dependence of 9g^/3p and g^ for the metastable l i q u i d calculated as follows: § L(T,P) = g L(T,50 bar) + 3 § L/3p(T,P) (50 - P) 3-9 Thus, disregarding surface effects the increase i n the free energy upon the formation of a macroscopic metastable drop composed of i atoms 88 TEMPERATURE (K) Figure 19 Temperature Dependence of 'Specific Volume' of Metastable Liquid Argon 89 each of mass m, i s AG = i M ( g L - g ) = i M Ag. 3-10 Ag i s greater than zero since the vapor state i s the stable state i n an unsaturated gas and therefore g^ i s less than g^. For a saturated vapor g g = g L-Plots of Ag versus density, D, for various temperatures are given i n Figure 20. 3.2.4.3 Curve F i t t i n g Technique Following Lee (1971), we used the maximum li k e l i h o o d method (Orear (1958)) to f i t double exponential and constant background time spectrum. Basically t h i s method maximizes the j o i n t probability of obtaining the experimental values with respect to a set of variable parameters describing the phenomena. An i t e r a t i v e procedure was used to find the most probable parameter values, A^ (Orth et a l . (1968)) where the A^ are related to the mean count per channel, Y, as follows: Y = A 2A 1 exp(-A 2 t) + A^A3 exp(-A^ t) + A 5, 3-11 where t i s time, A^ the random background estimate, and following Lee, A-^  and A^ are the t o t a l counts associated with each exponential (rather than the inten s i t y, given by A^A^). As shown by Lee, the convergence of the f i t i s more rapid using t h i s set of parameters. In addition the f i t t i n g routine included integrating 3-11 over each channel width as given by the d i f f e r e n t i a l l i n e a r i t y (Section 3.2.3.3.2). Traditionally, the main d i f f i c u l t y i n applying t h i s i t e r a t i v e procedure was the lack of an e f f i c i e n t method for finding appropriate 90 DENSITY (AMAGATS) Figure 20 Density Dependence of Free Energy Difference, Ag 91 i n i t i a l estimates of the parameters. Orth and Smythe (private communication - (1974)) solved t h i s problem for the task of f i t t i n g sums of exponentials by developing a computer program to make i t e r a t i v e least square estimates of the lif e t i m e s and i n t e n s i t i e s from the logarithm of the data minus a background estimate. By this technique, i t was then necessary only to choose the spectrum region i n which each l i f e t i m e component was to be estimated. 3.2.4.4 Determination of the Shoulder Width As discussed i n Section 3.2.1.4, the shoulder "width" i s a useful monitor of the purity of the gas. In order to have an unambiguous measure of the shoulder width, the following technique was used. The average annihilation rate, A(t), as a function of time was computed according to the following formula = dN/dt(t) 3 _ 1 2 N ' - C O / dN/dt(t')dt' t where dN/dt(t) i s the number of counts per channel minus the ortho-positronium component and background, as derived from the maximum lik e l i h o o d f i t s to the exponential and background regions of the spectra. Then, following the example of Paul and Leung (1968) the following c r i t e r i a n was used to unambiquously f i x the position of the shoulder terminus: t : t was the time at which A(t) reached s s • 90% of the difference between i t s minimum value and i t s maximum, A . e In addition, a qua l i t a t i v e comparison was made between the shape of A(t) for various experimental conditions. 92 3.3 Presentation of Results 3.3.1 Summary of Data Runs A summary of the accepted data i s given i n Appendix B. Runs up to and including number 89 were made without the temperature regulated s t a b i l i z e r described i n Section 3.2.2.1 and so pertain to sample temperature variations of ±0.4 K. Runs 90 to 94 were taken while setting up the temperature " s t a b i l i z e r " (Section 3.2.2.1) and 95 to 180 with the controller s t a b i l i z e d , thus corresponding to temperatures d r i f t s of only ±0.1 K. At run 55 the spot-welded thermocouple was replaced with the arc-welded couple and the e l e c t r i c f i e l d grid assembly i n s t a l l e d . The f i r s t 16 runs were primarily for the purpose of consistency checks of the gas purity as well as for checking consistency with the results of Canter and Roelling (1975) (gas density 20 to 40 amagat). Runs 17 to 24 were with gas at densities higher than 40 amagats. As the temperature controller (Section 3.2.2.1) was being set up for these f i r s t 24 runs, only a portion were useful as data. These are reported i n the appendix. The l i q u i d state was then investigated i n runs 25 to 54 and the saturated vapor from 63 to 104. As with the f i r s t 24 runs some of the runs of the ' l i q u i d ' sequence were primarily of a diagnostic nature ( i . e . checking for the presence of a shoulder while changing temperature or waiting for equilibrium - Section 3.3.1.6) and are also not reported. From 63 onward a l l runs are recorded. Runs 106 to 180 consisted of a study of the density and e l e c t r i c f i e l d dependence of the equilibrium annihilation rate for the constant 93 temperatures of 100 K, 120 K, 130 K and 140 K. A few room temperature runs were interspersed among the t o t a l of 180 runs. C r i t e r i a r e l a t i n g to the a c c e p t a b i l i t y of the runs are presented i n the following sections. 3.3.1.1 Random Background and E l e c t r o n i c S t a b i l i t y As mentioned i n Section 3.2.3.5, the random (negative time) background was for some of the runs somewhat enhanced due to e l e c t r o n i c problems (Section 3.2.4.3). However, a r t i f i c a l l y f i x i n g the background at low or (reasonably) high values had no e f f e c t on the short 'free' positron l i f e t i m e (X "*") and so no X r e s u l t s were discarded for these r e e reasons. On the other hand, Lee (1971) noted that an anomalously short ortho-positronium l i f e t i m e was re l a t e d to the presence of such a back-ground problem. For t h i s and other reasons (Section 3.2.3.4) the ortho-positronium l i f e t i m e was allowed to vary only for the higher de n s i t i e s (Section 3.3.1.3). For Na(Tl) counters the p o s i t i o n of the photopeak can be monitored to detect photomultiplier gain changes which would increase such background problems (Lee (1971)). In the case of a p l a s t i c s c i n t i l l a t o r , however, t h i s photopeak i s not detected (Section 3.2.3.2) and such a useful monitor i s not possible. 3.3.1.2 Choice of the Exponential Region Because of the presence of the peak positioned at the r i g h t end of the shoulder of the time spectra, i t was found that an unbiased c r i t e r i a n for choosing the beginning channel for the exponential f i t was necessary. In p a r t i c u l a r , an a r b i t r a r y estimate of t h i s beginning channel based on a v i s u a l estimate of the log plot of the data, produced 94 a greater spread i n values for the a n n i h i l a t i o n rates versus density than the use of a formula r e l a t i n g the s t a r t of the f i t to the measured density. Therefore, a l l runs were analysed for the expon-e n t i a l components by s t a r t i n g the f i t at a distance 415/D ns amagats from the prompt peak. This density dependence related to the con-stancy of the shoulder-width density product (Section 3.2.1.4 and 3.2.4.4). The constant (415) was obtained from an examination of a low temperature spectrum (density = 20 amagats) of good s t a t i s t i c s . 2 This value maximized the x p r o b a b i l i t y for the f i t (Section 3.2.4.3), which decreased as the non-linear (on a log plot) region near the shoulder was approached. At the same time a compromise had to be made to the s t a t i s t i c a l p r e c i s i o n of the short l i f e t i m e which decreases ra p i d l y as the beginning of the f i t i s moved to higher channels. This f i n a l choice of ' s t a r t ' channel i s indicated i n Figure 37 (X(t) versus time, (Section 3.2.4.4)). For thosespectra having adequate s t a t i s t i c s i n the time region where A(t) = XQ (which excludes a l l of the low temperature runs) t h i s choice c l e a r l y lay i n the constant region of X(t). The adequacy of t h i s choice for the low 2 temperature (zero f i e l d ) spectra i s demonstrated by the good x proba-b i l i t i e s obtained for most runs during the en t i r e s e r i e s of runs. 3.3.1.3 F i t t i n g the Ortho-Postironium Lifetime R e l i a b l e estimates of the Ortho-positronium l i f e t i m e were impossible for densities l e s s than about 90 amagats because of the low i n t e n s i t y of t h i s component (Figure 2) and because i t s l i f e t i m e was long compared to the time scale (normally kept at 50 or 100 ns for measurement of the 95 enhanced 'free' positron l i f e t i m e ) . For the higher d e n s i t i e s , the reduction i n the shoulder width l e f t a region i n the negative time part of the time spectrum which was free of negative shoulder events (Section 3.2.3.5). A r e l i a b l e estimate of the random background could then be made i n t h i s region (where only a much lower i n t e n s i t y of 'negative' ortho-positronium events e x i s t s ) . Nevertheless, from time to time, such estimates of the background were affected by e l e c -t r o n i c i n s t a b i l i t i e s (Section 3.3.1.1). Although there i s some evidence for the uncertainties i n those r e s u l t s l y i n g on the high side of the data to be f r a c t i o n a l l y greater than those on the lower side, the evidence i s not conclusive. The lack of a well-defined c r i t e r i a for acceptance of the ortho-positronium r e s u l t s allows presentation of the r e s u l t s for q u a l i t a t i v e i n t e r e s t only. On the other hand, the shorter l i f e t i m e for the positron component was quite i n s e n s i t i v e to the value assumed for the almost f l a t ortho-positronium l i f e t i m e . Generally the Ortho-positronium l i f e t i m e was obtained from a f i t of the data only for gas de n s i t i e s greater than 88 amagats and for.the l i q u i d states. For the lower density gases the best density dependent r e s u l t given by Lee (1971) was used. This r e s u l t (1-7) i s given i n Section 1.2.3.; 96 3.3.1.4 Goodness of F i t 2 Generally x confidence l e v e l s of 5% to 95% were obtained for most runs (with the ortho-positronium l i f e t i m e fixed at the appro-p r i a t e value f or the lower d e n s i t i e s of gas (section 3.3.1.3)). Runs 2 having x - p r o b a b i l i t i e s lower than 0.5% were viewed with suspicion, and were generally found to be characterized by the o s c i l l a t i o n s discussed i n section 3.2.3.3.2. These runs, a t o t a l of 9, were discarded i n the subsequent analysis. 3.3.1.5 Equipment F a i l u r e Large temperature s h i f t s occurred i n a few runs due to f a i l u r e of the l i q u i d nitrogen f i l l e r system (Appendix C) or l o s s of vacuum around the chamber due to leaking metal 0-rings (Section 3.2.1.1). Five runs were discarded f or these reasons and two runs were l o s t because of computer problems (one hardware - one software). 3.3.1.6 Detecting a Saturated Gas or Liquid The presence of a saturated gas i n the chamber was indicated by the constancy of the pressure with addition of more gas to the chamber with 6 to 24 hours allowed for r e - e s t a b l i s h i n g an equilibrium state. The presence of a l i q u i d was c l e a r l y indicated by the disappearance of the shoulder i n the time spectra implying that the l i q u i d was covering 22 the source (Na i n Ni f o i l s i t u a t i o n ) and stopping most of the positrons i n the l i q u i d . For measurements of positrons i n l i q u i d argon, i t was necessary that a l l the positrons be stopped i n the l i q u i d as the shoulder, r e s u l t i n g from a n n i h i l a t i o n of any positrons i n the vapor 97 above the l i q u i d , would appear at the position i n the spectra where the equilibrium l i q u i d l i f e t i m e l i e s and would greatly affect t h i s l i f e t i m e . Therefore more l i q u i d was added even after the shoulder had apparently disappeared as noted by a v i s u a l inspection, to allow for minor fluctuations i n the l i q u i d surface l e v e l , as the range of positrons i n the l i q u i d i s less than a centimetre. In fact, one run (#33) was taken with the l i q u i d completely f i l l i n g the low temperature part of the chamber, indicated by an overpressure (pressure, P, greater than that, P •, given by the saturation l i n e data for the measured temperature) of 2.8. This run agreed with those with less l i q u i d at the same temperature. Twelve runs were used to monitor for the presence of a shoulder and were not analysed. 3.3.1.7 Approach to Equilibrium In general the time dependence of the state of the gas was monitored by taking periodic readings of the lower thermocouple and comparing i t to the setting of the upper one which was constant to, ±0.1 K. For checking the rate of attaining equilibrium, measurements were taken every 15 minutes and for checking the degree of equilibrium, every hour. Once the "rate" and "degree" were known readings of the upper thermocouple were taken 3 to 4 times a day and a pomparison of the lower and upper weekly. After changing temperatures, and waiting for agreement of the upper and lower thermocouple, the bottom "sense" thermocouple was monitored for up to 2 hours to check that the temperature had s t a b i l i z e d before a run was started. For rapid cooling with an exchange gas In the vacuum region, i t took about 1 hour for the top thermocouple to 98 approach the desired temperature from room temperature and then an additional 1 hour to remove the exchange gas and obtain an adequate vacuum. The lower thermocouple then took 2 to 6 hours to reach equilibrium. Heating was much slower and required monitoring the upper control thermocouple u n t i l the required temperature was reached. The controller was then set and a 6 to 24 hour wait required for the lower thermocouple to come to equilibrium with the top, depending on the amount of the temperature change. For changes i n temperature of 2 to 3 K (no exchange gas was required for cooling) only a 2 to 4 hour wait was required before starting a data run. During the i n i t i a l runs, after each temperature change, two runs were taken i n series with the same control settings, the f i r s t run just after equilibrium was expected and the second immediately following this run (each of 24 hours duration), when a stable state was assured. Generally these runs were i n agreement, as were those with gas i n a similar thermodynamic state, but with a temperature spread of ±0.4 K and ±0.1 K. 3.3.2 Density Range Investigated The range of the data for on and off the saturation l i n e i s displayed as dark bars i n Figure 21. The v e r t i c a l l i n e s at 20, 30, and 40 amagat are the constant, density states examined by Canter and R o e l l i g (1975). The higher, gas density of 130 amagat and the range of the l i q u i d states considered i n th i s thesis are displayed also as bars i n Figure 1. Also shown i n Figure 1 i s the l i q u i d state data of Spektor and Paul (1970) and a region delimiting the results of Canter and R o e l l i g . 99 TEMPERATURE (K) Figure 21 Range of Gas Data 100 3.3.3 Ortho-Positronium Annihilation Rate i Since the main interest i n thi,s thesis was the study of the equilibrium annihilation rate, the ortho-component was generally fixed at the best estimate of i t s value corresponding to the gas density involved. However, for a few high density gas runs and a l l the l i q u i d runs this component was s u f f i c i e n t l y well-defined that i t could be f i t t e d by the f i t t i n g algorithm. These results for the gas and l i q u i d are given i n Table I and plotted i n Figure 22. For densities less than 130 amagats, the results (generally) agree with the extrapolation of the l i n e a r density dependence observed by Lee (1971). This l i n e a r dependence implies that the positronium atom i s i n a free state. The single high result at 92 amagats i s probably due to electronic-related pileup problems (Section 3.3.1.3) rather than impurities (Section 3.2.1.4) as the uncertainty i n t h i s result i s considerably larger than that of the other gas results (taken l a t e r with the same chamber gas). In the l i q u i d results two phenomena may be apparent. For densities greater than about 600 amagats, the. results (including the result of Spektor and Paul (1971)) drop away from an extrapolation of the gas results i n a manner which suggests that the positronium atom has formed a bubble i n the l i q u i d . On the other hand, for densities less than 600 amagats the behavior i s probably better related to a picture whereby the positronium atom senses density fluctuations (and seeks out low density regions) i n the gas.. Such a mechanism was f i r s t suggested by Tseng et a l . (1977) to explain the results for O-Ps i n Xe gas near (within 10 K of) the 101 TABLE I Ortho-Positronium Annihilation Rate Run No. Density (amagats) Temp. (K) Ortho-positronium Annihilation Rate(us - 1) Confidence Level (%) State 18 87. 163.2 41.±6 35 gas 19 89. 154.0 38.1±2.2 69 gas 20 92. 143.5 30.8±1.5 42 gas 23 92. 142.6 34.5±2,0 42 gas 32 712. 106.5 147.±4. 22 l i q u i d 33 712. 107.0 . 147.±3. 0.8 l i q u i d 41 662. 117.8 149.±4. 4. l i q u i d 44 582. 132.7 130.±3. 0.7 l i q u i d 46 130. 130.0 36.±3. 0.8 gas 47 130 144.2 36.0.±3 0.8 gas 49 487. 144.1 91.8±2.2 15. l i q u i d 50 445. 147.9 76.811.9 40. l i q u i d Spektor and Paul 785 . 87.3 153.811.4 l i q u i d (1971) 102 DENSITY (amagats) Figure 22 Density Dependence of Ortho-Positronium A n n i h i l a t i o n Rate - Gas and Liq u i d 103 condensation (and c r i t i c a l ) temperature. However, since the results at 600 to 700 amagats may also be enhanced due to pileup problems (as the larger uncertainties might suggest)(Section 3.3.1.3) these results w i l l not be considered further. 3.3.A Equilibrium Positron Annihilation Rate 3.3.4.1 Saturated Gas and Liquid 3.3.4.1.1 Density Dependence of X g The density dependence of the annihilation rate for a positron i n saturated argon gas with no e l e c t r i c f i e l d applied i s shown i n Figure 23. Figure 24 i l l u s t r a t e s the continuity of the saturated gas and l i q u i d annihilation rate results corresponding to continuous d i s -placement along the saturation l i n e i n Figure 1. Table I I tabulates the saturation l i n e annihilation rates. It i s quite evident from Figure 23 and 24 that a "saturation" effect takes place i n the equilibrium annihilation rate of a positron i n a saturated gas near 30 amagat and continues to persist well past the c r i t i c a l density of 300 amagat. The l i q u i d results then asympptically approach a rate which i s proportional to density. However, the slope of this asymptotic l i n e i s much less than that found for argon gas obeying the single atom picture at room temperatures. Such a l i n e f i t t e d to the results of Lee (1971), Orth (1966), and Tao (1970) i s also shown i n Figure 24. The result of Lee i s X g = (6.9±0.7) us - 1amagat _ 1, 3-13 and i s derived from an extrapolation to 100 K of Lee's f i t (1-6) to the 104 X DENSITY (amagats) Figure 23 Density Dependence of X - Saturated Gas 105 DENSITY (amagats) Figure 24 Density Dependence of X - Saturated Gas and Liquid 106 TABLE I I Saturated Gas Results (Ag=0) Equilibrium Run Density Temp. Annihilation C.L. No. (amagats) (K) Rate(10 _ /ns- x) (%) 63 31.7 118.8 112.9 ± 5.4 65 25.3 114.9 66 21.3 112.0 69 42.2 124.0 119.2 ± 5.1 79 24.5 114.3 105.0 ± 12. 84 24.3 114.2 98.1 ±12. 88 23.5 113.7 97.5 ± 7.3 92 23.5 113.7 91.2 ± 4.3 94 17.2 108.6 65.8 ± 5.0 •96 17.3 108.7 73.2 ± 5.5 97 17.3 108.7 73.8 ± 5.5 99 21.0 111.8 76.3 ± 6.5 100 21.0 111.8 90.2 ± 12. 104 32.2 119.1 109.3 ±6.7 175 9.77 100.3 40.5 ± 5.1 LIQUID Results 32 706: 107.1 165±1 22 33 708 107.0 163±1 0.8 37 624 125.3 147±3 2 41 662 117.8 155±1 4 44 582 132.6 142±1 0.7 49 487 144.1 130±1 15 50 445 147.9 127±1 41 107 temperature dependence of his data which ranged from 135 K to 600 K. Returning to Figure 23 i t would appear that for a saturated gas at densities less than about 25 amagat, to be denoted as a "low density" gas, the equilibrium rate i s again proportional to density, but with a slope much larger than that experimentally observed for a single-atom gas. A least-squares f i t to the results of Figure 23 for densities less than 25 amagat yiel d s Xg/D = (40.0±0.6) us - 1amagat - 1. 3-14 Again Lee's single atom result i s shown. The rest of the s o l i d curve i s a f i t to the data using the growth model mentioned i n Section 3.1.2 and w i l l be discussed i n Section 4.4. This f i t to the gas region was plotted i n the following graphs when a normalization to the saturated gas results was required. 3.3.4.1.2 Temperature Dependence Figure 25 contains, besides some gas results which w i l l be referred to l a t e r , the l i q u i d results (broken l i n e ) and the f i t to the saturated gas data of Figure 24 plotted against temperature. By comparing the temperature dependence of the annihilation rate for the l i q u i d i n Figure 25 to the density dependence i n Figure 24 i t i s apparent that there i s less non-linearity i n the temperature dependence than i n the density dependence (see Section 4.5). If the experimental results are corrected for the systematic error caused by the f i n i t e resolution time of the system (Section 3.4.1.6), then the dashed l i n e above the l i q u i d results i n Figure 25 i s obtained. 108 1.8 J 1.51 1.2 J l.o H (ns 1) 0.5 J 0.0 . - Canter and Ro e l l i g (1975) • - Spektor and Paul (1971) • - This Work o - Interpolations from Figure 24 (Typical error i n gas data i s shown; error i n l i q u i d i s i n s i g n i f i c a n t . ) -3/5 T • dependence. -Liquid Data Corrected For Effect of F i n i t e Resolution. 55 ^90 \ 3 Q \40 amagats Triple Point C r i t i c a l Temperature "1 1 1 1 1 1 1 1 : I 80 90 100 110 120 130 140 150 160 170 180 190 TEMPERATURE (K) Figure 25 Temperature Dependence of A g - Gas and Liquid 109 Most apparent from t h i s figure i s the fact that the gas and l i q u i d results approach each other at the c r i t i c a l point. Below thi s point, gas results are approximately constant, implying that positrons may be captured on droplets with a density similar to that at T c > On the other hand, for the l i q u i d r e s u l t s , A g r i s e s as the temperature i s reduced because of increasing density. 3.3.4.2 Unsaturated Gas The data i s ordered according to temperature and increasing density i n Table I I I and plotted as a function of temperature i n Figure 25. 3.3.4.2.1 Density Dependence of X g The density dependence of the unsaturated gas data with zero e l e c t r i c f i e l d applied i s shown i n Figure 26. The ov e r a l l f i t to the saturated gas data of Figure 23 and 24 i s shown for reference. The densities for temperatures of 100 K, 120 K, 130 K, and 140 K are marked on the saturation l i n e (by c i r c l e s ) and used as reference points for extrapolation of the unsaturated data to the saturation l i n e . The l i m i t s for the single atom gas for.various temperatures are also shown (Tap (1969), Lee (1971)). At low densities (^10 amagats) the data appears asymptotic to the single-atom results of Tao and Lee. For densities less than about 30 amagat, the slopes of the data for constant temperature are approximately equal to that of the saturated gas data. At zero densities of course the annihilation rate must go to zero. As the density r i s e s above 30 amagat, the annihilation rate appears to be influenced by the same l o c a l i z a t i o n mechanism experienced by the 110 D Run Ag Density No. (J g ^ ) (amagat) 180 1.7 8.50 179 0.8 9.05 c r i t i c a l density 9.55 174 21.0 10.85 169 20.0 11.4 166 18.0 12.5 167 18.0 12.5 168 16.0 13.8 172 15.7 14.0 165 15.2 14.3 108 12.2 16.8 164 10.8 18.0 107 6.4 22.7 106 2.0 29.5 105 1.8 29.8 c r i t i c a l density 33.87 144 27.5 13.0 145 24.3 15.0 109 21.5 17.0 149 19.2 18.9 152 17.0 21.0 TABLE I I I ZERO FIELD DATA 100 K X • e Equilibrium Annihilation -2 -1 -1 -1 Rate(10 ns ) (us amagat ) 19.8 ± 1.3 23.4 + 1.5 25.4 ± 4.6 28.1 ± 5.1 120 K 11.2 + 0.8 10.3 + 0.7 14.0 + 1.8 12.2 + 1.6 16.1 + 1.1 12.9 + 0.9 16.5 + 0.7 13.1 + 0.6 27.2 + 2.8 19.8 + 2.2 27.4 + 2.0 19.5 + 1.4 35.2 + 2.1 24.6 + 1.5 47.4 + 1.8 28.2 + 1.2 55.8 + 6.9 21.1 + 3.8 76.0 + 4.0 33.4 + 1.8 101.0 + 7.0 34.2 + 2.3 116.0 + 10.0 33.9 + 3.2 130 K 10.7 + 0.7 8.24 + 0.5 16.4 + 0.5 10.9 + 0.3 22.7 + 0.7 13.5 + 0.5 32.2 + 1.6 17.1 + 0.9 45.3 + 2.1 21.5 + 1.0 I l l D Equilibrium X /D e (ys amagat ) Run No. Ag (J g"1) Density (amagat) Annihilation -2 -1 Rate(10 ns ) 153 16.0 22.0 48.1 ± 1.8 21.9 ± 0.8 110 15.1 23.0 60.6 ± 2.3 26.3 ± 1.1 155 14.0 24.3 61.4 ± 3.0 25.2 ± 1.2 110 10.0 29.9 89.2± 3.3 29.8 ± 1.2 112 5.0 40.0 103.6 ± 4.9 25.9 ± 1.3 113 0.8 53.9 121.8 ± 5.2 22.6 ± 1-0 c r i t i c a l density 58.34 140 K 162 29.1 17.9 18.0 ± 0.8 10.0 ± 0.4 163 26.0 20.2 22.7 ± 1.0 11.2 + 0.5 121 22.9 22.9 29.1 ± 1.1 12.8 ± 0.5 122 22.9 22.9 31.7 ± 1.4 13.8 ± 0.6 159 22.0 23.7 34.3 ± 1.1 14.5 ± 0.5 160 22.0 23.7 34.3 ± 1.0 14.5 ± 0.5 158 20.0 26.0 41.6 ± 1.8 * 16.0 ± 0.7 157 18.0 28.7 52.2 ± 2.2 18.2 ± 0.8 117 15.2 32.8 70.6 ± 3.8 21.5 ± 0.9 118 9.7 45.0 94.9 ± 3.5 21.0 ± 0.8 119 6.4 54.5 102. ±3.0 18.8 ± 0.6 120 6.4 54.5 102. ±3.0 18.8 ± 0.5 c r i t i c a l density 100.7 112 Figure 26 Density Dependence of \ - Gas 113 positron i n the saturated gas, but to a lesser extent. 3.3.4.2.2 Temperature Dependence of A g In Figure 25 are shown values of A g interpolated from the curves of Figure 23 for densities of 20, 30, and 40 amagat (points marked by large s o l i d c i r c l e s ) . Also shown are the results of Canter and R o e l l i g (1975) for the same densities. The few errors shown are similar for both sets of data. The only s i g n i f i c a n t disagreement i s i n the region near the saturation l i n e . This difference could e a s i l y result from s l i g h t l y different c r i t e r i a for choosing the exponential region (Section 3.3.1.2). This reaffirms the need for a f u l l d i f f u s i o n equation treatment (Section 1.3.2) so as to include the non-linear (on a log plot) region i n the f i t . A few higher density points of 55 and 88 amagat are shown i n Figure 25 with the corresponding temperature marked on the saturation l i n e . These generally follow the shape of the 30 and 40 amagat data near the saturation l i n e and also show evidence of following the temperature dependence of the l i q u i d (more so i f the corrected l i q u i d r e s u l t s , the dashed l i n e (Section 3.4.1.6) are considered). 3.3.4.2.3 Density Dependence of Ae/D U n t i l a positron becomes localized i n a gas i t i s expected to sample atoms or groups of atoms a cluster at a time. For t h i s "free" positron picture, Ae/D would be the quantity related to the interaction with the clusters. For t h i s reason a plot of Ae/D versus density i s presented i n Figure 27 for both the present results and those of Canter 114 DENSITY (amagats) Figure 27 Density Dependence of ^e/D - Gas 115 and Ro e l l i g . The saturation l i n e data (solid line)(from Figures 22 and 23) i s again included for comparison. Two observations can be made. The f i r s t i s that the peak i n each isotherm (for the saturation l i n e data, the point at which Ae/D starts to decrease) moves up i n density with increasing temperature. The second i s that the slopes of the data at densities less than these maxima, decrease with increasing temperature, being quite large for the 100 K data. A^D versus temperature i s presented l a t e r i n section 3.3.7. 3.3.4.2.4 Phenomenological Analysis of Data The v i r i a l expansion for an imperfect gas can be written as follows: P = D + B„(T)D2 + B 0(T)D 3 + ... 3-15 RT " ' " 2 V ^ " ' "3 where R i s the gas constant and B^(T) i s related to the interaction of i gas atoms. By analogy, i t i s l i k e l y a similar phenomenological expansion can be written for the annihilation rate of a positron with single, double and higher-order polymers. Thus, OO -1 L = A D + E A.(T) D 3-16 f S 1-2 1 . * or A- = A D + A..(T) D 1 3-17 f s 1 * where, for s i m p l i c i t y , i t i s assumed that one term dominates the sum i n 3-16. Now a simple application of 3-17 to the low density region of Figure 25, where deviations from the single atom picture f i r s t become 116 apparent should y i e l d a crude estimate of the value for i . A simple f i t of 3-17 to t h i s region yields the following values for i : TABLE IV PHENOMENOLOGICAL RESULTS FOR i * T (K) 120 130 148 Of interest i s the fact that the positron i s apparently interacting primarily with clusters s i g n i f i c a n t l y larger than dimers, the most probable cluster size. The value of i obtained i s , however, not very much larger than 2 and i s d e f i n i t e l y not of macroscopic size. These values of i w i l l be compared to those obtained from application of the cluster model using macroscopic thermodynamic gas data i n Section 4.3. The physical significance of A(T.) as given i n Table IV i s not readily apparent as i t contains information pertaining to both the di s t r i b u t i o n of clusters and the interaction of a positron with the cluster of size i . Since the d i s t r i b u t i o n of clusters i s expected to be a rapidly decreasing function of i while the interaction of a positron with an i - s i z e cluster i s expected to be an increasing function of i , the interpretation of A., i s expected to be d i f f i c u l t . i * A(T) _ ^ (us amagat ) 7.07 1.82 x 10 6.05 3.94 x 10' 3.81 9.8 x 10 117 3.3.5 E l e c t r i c F i e l d Results E l e c t r i c F i e l d data i s given i n Table V. 3.3.5.1 Saturated Gas In Figure 28, A£/D i s plotted as a function of E/D for the saturated gas, for both: a density of 40 amagats and densities up to 25 amagats. This was done i n order to compare the effect of an e l e c t r i c f i e l d on the positron i n a saturated gas for the two d i s t i n c t density regions ("low" and "high") observed i n the zero f i e l d case. Immediately obvious i s the s i m i l a r i t y of the E/D dependence of Ag/D for both regions of gas density for E/D values greater than about 10 volt-cm ^ amagat \ I t w i l l be shown (Section 4.4) that the elimination of the difference between the "low" and "high" density region can be simply explained i n terms of the Growth model (Section 3.1.2). 3.3.5.2 Unsaturated Gas In Figures 29 and 30 the results for X^/T) versus density are plotted for E/D values of 8.0 and 15.0 volt-cm "^-amagat ^ respectively. The constant values of \&/T> for D<10 amagats are the single-atom results of Lee (1971) (1-6, Section 1.2.1.2). In both graphs the position of the zero f i e l d data from Figure 27 i s given for reference and i n Figure 29, the 15 V cm "^amagat ^ data of Figure 30. The values of A /D for the saturated gas at th i s same f i e l d are also shown i n Figure 30. Figure 29 i s an intermediate case and as Figure 28 has shown, a temperature dependence was s t i l l expected for th i s low an E/D value. Figure 30, 118 TABLE V ELECTRIC FIELD DATA Run No. 177 178 98 95 90 83 87 91 82 93 81 85 86 80 70 71 74 72 D Density (amagat)_ 9.77 10.0 17.3 17.2 23.5 24.3 24.4 23.5 24.3 23.5 24.3 24.8 24.4 24.3 41.5 40.8 40.0 41.0 E/D (V cm"1 -1. amagat ) 14.5 15.0 4.99 7.98 3.00 3.00 4.66 5.00 5.69 8.00 11.5 14.7 20.5 43.2 10.95 17.8 23.2 41.8 Equilibrium Annihilation RateClO'^s" 1) 13.812.0 13.311.2 49.812.3 35.211.7 84.216.0 112.0117.0 82.716.8 77.213.6 90.917.6 56.412.7 36.212.3 29.711.4 21.611.1 11.210.7 69.412.7 41.111.0 33.210.9 21.210.6 X /D e (us amagat ) 14.012.0 13.311.2 28.811.3 20.511.0 35.612.5 46.217.0 33.912.8 32.511.7 37.413.2 23.811.2 14.911.0 12.010.6 8.8610.5 4.6110.27 16.710.7 10.010.2 8.2810.22 5.1710.15 170 173 147 148 154 14.0 17.7 15.0 17.0 22.0 15.0 15.0 8.00 8.00 8.00 120 K 14.910.9 21.411.1 130 K 12.810.6 18.811.0 35.811.2 10.710.6 12.110.6 8.4810.3 11.110.6 16.310.6 119 X e D E/D Equilibrium Run No. Density (amagat) (V cm amagat ^) Annihilation -2 -1 Rate(10 ns ) e (us "'"amagat 156 24.3 8.00 43.211.0 17.710.6 138 30.0 8.00 55.312.5 18.410.8 142 13.0 15.0 6.1910.4 4.7710.3 143 13.0 15.0 8.6010.7 6.6110.6 146 15.0 15.0 11.610.5 7.7010.3 141 17.0 15.0 14.610.6 8.6310.3 150 19.1 14.9 18.710.5 9.8410.3 151 19.1 7.96 25.710.8 13.510.4 140 20.0 15.0 21.210.7 10.610.4 137 30.0 15.0 34.811.4 11.610.5 139 40.0 15.0 47.711.9 140 K 11.910.4 161 23.7 8.00 29.410.9 12.410.4 136 30.0 8.00 49.312.2 16.510.7 133 39.9 8.00 65.813.2 16.510.8 128 71.0 7.89 94.813.0 13.310.4 129 71.3 7.85 94.813.0 13.310.4 123 , 22.9 15.0 20.010.8 8.9010.4 134 30.0 15.0 33.811.2 11.310.4 135 30.0 15.0 33.711.2 11.210.4 131 39.9 15.0 44.811.3 11.210.3 132 39.9 15.0 46.211.8 11.610.4 129 50.0 15.0 55.112.2 11.010.4 130 50.0 15.0 54.811.6 10.910.3 124 59. 15.1 60.211.6 10.010.3 125 60.0 15.0 60.611.4 10.110.3 126 71.3 14.7 69.511.8 9.7510.18 120 D (ys 1 |amagat-l)l o - 40 amagats • - < 25 amagats 40. \ 30. J K ft 20. V4 'I i o . H o«_. — I 40 10 20 30 — (V cm amagat ) Figure 28 E l e c t r i c F i e l d Dependence of X£/D - Saturated Gas 121 j 1 1 1 1 1 -0 10 20 30 40 50 DENSITY (amagats) Figure 29 Density Dependence of X /D for E/D = 8.0 V cm -1 e amagat - Gas 122 • - 120 K x - 130 K o - 140 K + - Saturated gas (ys 1 lamagat ^")| A - These points marked on Coexistence P l o t , Figure 21 for 120 K, 130 K, and 140 K. 20. 130 K 135 K. Lee (1971) 15 V cm ^ amagat ~~ "4 50 10 — r 20 30 40 DENSITY (amagats) Figure 30 Density Dependence of X /D for E/D = 15.0 V cm -1 e amagat - Gas -1 123 however, does show the expected independence of X^/D with temperature at l e a s t for intermediate de n s i t i e s (15 amagats<D<40 amagats, depending on temperature), but not for the lower d e n s i t i e s where a very d e f i n i t e temperature and density dependence of Ag/D s t i l l p e r s i s t s . Figure 31 shows how these isotherm data for 15 volt-cm ^-amagat ^ approach the saturation l i n e data asymptotically from both low and high d e n s i t i e s . Thus, for these intermediate d e n s i t i e s , the phenomenon seems to be independent of temperature and proportional to density i n a manner s i m i l a r to that of the z e r o - f i e l d , low-density, saturated-gas r e s u l t s . The dropping away at higher de n s i t i e s (D>40 amagats for 140 K gas) again r e f l e c t s the trend seen i n the z e r o - f i e l d saturated gas r e s u l t s (Figure 23) and w i l l be explained (Section 4.4) i n terms of the growth model (Section 3.1.2). The temperature and density dependence of Xg/D for the lower densities (with and without e l e c t r i c f i e l d ) i s very suggestive of the important of the 'distance' of the (T,D)-point from the saturation l i n e , as measured for example, by the Gibb's free energy. This dependence i s considered i n the next sections. 3.3.6 Gibb's Free Energy Dependence The dependence of A and A /D on Ag, the diff e r e n c e i n Gibb's free e e energy between the metastable l i q u i d and the stable gas state f or the same temperature and pressure (Section 3.2.4.2),are plotted i n figures 32 and 33, respectively. The saturation data (Figure 23) i s j u s t a sp e c i a l case for which Ag i s equal to zero. The values of Ag(T), corresponding to a density of 30 amagats (the density at which the 124 - 120 K - 130 K - 140 K - Saturated Gas DENSITY (amagats) Figure 31 Density Dependence of X for E/D = 15.0 V cm -1 e amagat - Gas -1 125 + - 100 K • - 120 K x - 130 K o - 140 K -^D >30 amagats f o r 120 K •I •D > 30 amagats f o r 130 K ^-D > 30 amagats f o r 140 K Solid l i n e s through data - growth and c l u s t e r model see text p. 128 Ag (J g" 1) Figure 32 Ag Dependence of X 126 (ys 1 _ 1 I amagat ) + -• -x -o -100 K 120 K 130 K 140 K D < 30 amagats for 120 K |D < 30 amagats for 130 K D < 30 amagats for 140 K 40. -k 30. A 20.4 10.-J 4 1 " . Numbered Points Correlate with Points of Figure 27 3 see text p. 128 4 | 1 uncertainty i n Ag Lu IT 20 3 3¥ Ag (J g"1) Figure 33 Ag Dependence of X /D 127 saturation data, Figure 23, turns over), are 2., 10., and 17. J g (for 120 K, 130 K, and 140 K, respectively) and are marked i n Figures 32 and 33. Thus i n Figure 32, the X& of the data for densities greater than 30 amagat seems to be approximately l i n e a r i n Ag, with a negative slope. A l l the lower density data then drop away from t h i s l i n e a r curve at the value of Ag appropriate to a density of about 30 amagats. Figure 33 shows that X^ /D also behaves i n a similar fashion for the low density data. Again a decreasing lin e a r dependence i s observed. However, because of the extended range of the low density data the line a r dependence on Ag, for Ag less than 15,0 J g ~~, appears more l i k e an exponential dependence on Ag, (Section 4.2.2)(for Ag greater than 15.0 J g "'"), and i n addition reveals a temperature dependence which becomes stronger as the temperature decreases. This deviation from a simple Ag "picture" becomes quite marked for the 100 K data which appears to be enhanced over a very narrow range of Ag or density (note Ae/D for saturated 100 K gas i s 40.0 us "^ amagat ^ (3-14)). This temperature dependence of the low-density, high-Ag data may be related to the surface tension of a cluster of atoms since the surface tension decreases with temperature and vanishes at the c r i t i c a l temperature of 150.7 K. That i s , since the surface tension vanishes for the c r i t i c a l temperature,A g(T c,Ag) would be expected to assume a l i m i t i n g value with respect to the lower temperatures. The measured ^ e ( T c ) appears to have this c h a r a c t e r i s t i c , and although Ag could not be calculated to any useful degree of precision for the low density gas at the c r i t i c a l 128 temperature because of the large uncertainty i n 8g^/3P and g-^ (P = 50 bar) (Section 3.2.4.2) near Tc» i t i s apparent from Figure 32 that the results do seem to approach a single l i n e a r curve, dashed i n Figure 32, as the temperature approaches Tc« In fact as i s evident i n Figure 32, for densities greater than 30 amagats, A i s approximately independent of e temperature except through the Ag dependence. Si m i l a r l y , as seen i n Figure 33, for densities less than 30 amagats, the Ag/D results are independent of temperature i n an analogous fashion. Of a l l the c h r a c t e r i s t i c s of the Ag-plots described above, the only one expected from the simple consideration of nucleation theory presented i n Section 2.1, was the exponential region which i s only just descernible i n the low-density high-Ag region of Figure 33. However, the existence of such a dependence does not of i t s e l f confirm the cluster model of the gas. I t i s quite conceivable that the presence of the positron exerts a profound effect on the gas structure. Nevertheless, the fact that t h i s exponential region occurs, but only for low density gas and that the surface term i s apparently of importance does suggest that the thermodynamics of c l u s t e r - l i k e structures does play an important role i n the behavior of the positron at low temperatures. Although Figure 34 exhibits an apparent l i n e a r i t y of A g with Ag, for E/D of 15 V cm ^amagat \ similar to the z e r o - f i e l d high-density re s u l t s , Figure 35 reveals the fact that for t h i s f i e l d value X^/D i s approximately independent of temperature. Thus the manner by which the Ag-transformation brings the data into alignment for the various tem-peratures, compared to a density p l o t , i s impressive (comparing Figure 27 and 33, and 30 and 35). 129 E D 8.0 15.0 Temperature (K) 120 130 140 C D A B (ns X) 1.01 0.8J 0.6-0.4-0.2J 0.0. (V cm ^ amagats ^) \ \ ^ j . O. r 10 ~5 20 30 Ag (J g ) Figure 34 Ag Dependence of A g for E/D = 8.0 and 15.0 V cm amagat -1 130 Ag Dependence of X /D for E/D =8.0 and 15.0 V cm -1 6 amagat 131 3.3.7 Temperature Dependence of Ae/D for 21 Amagat Gas Figure 36 i s given for speculative value only and displays the temperature dependence of Ae/D for the saturated gas and gas at 21±1 amagats. These data and their errors are interpolated from the guide curves of Figure 27. Although the proximity of the 21 amagat points (compared to the 19 and 23 amagats points) to the saturation l i n e i s intrigueing, this proximity i s probably circumstantial, and w i l l not be considered further. However, the s i m i l a r i t y of the shapes of the 10 amagat data and the 21 amagat data (with consideration for the fact that i n this work the temperature was generally held fixed at one of four values) does suggest that temperature may prove useful for interpreting the low density (D<30 amagats) data. 3.3.8 Time Dependent Annihilation Rate The time dependence of the velo c i t y averaged annihilation rate, A ( t ) , for low and room temperature gas at zero and high applied e l e c t r i c f i e l d s i s displayed i n Figure 37. Although these results are derived for gas at a density of 25 amagats, the results for gas at 40 amagats are similar except for a minor difference between the zero-field low-temperature results as noted i n the figure. As shown, the effect of a high e l e c t r i c f i e l d , 42 volt-cm ^ -amagat on a positron i n a room temperature gas i s to reduce A e ( t ) to a constant independent of the temperature and equal to the shoulder value, that value i n the zero f i e l d spectrum i n the neighbourhood of zero time, but beyond the influence of the prompt peak. This i s t r a d i t i o n a l l y 132 Points Interpolated from Figure 27. D (us ^ I amagat )| Saturation Temperature f o r : A - 10 amagats B - 19 amagats C - 21 amagats D - 23 amagats E - 40 amagats Saturation Line, G, from Figure 23. Figure 36 Temperature Dependence of Xe/D. S i m i l a r i t y of Saturated Gas and Unsaturated Gas at 21 amagats 133 4 2 . H X(t) - 1 dashed l i n e - 25 amagats ^-s o l i d l i n e - 40 amagats/ ^—X from l i f e t i m e e i i i a E D (ys lamagat """) 2 . 0 . 0 4 2 . Temperature (K) 114 300 -f-• (V cm ^ amagat ^) o Shoulder Edge |— F i t to X g — 0 Prompt Peak Jo0~ 415 TIME (ns amagats) Figure 37 Time Dependence of Velocity Averaged "Direct" Annihilation Rate 134 interpreted as r e s u l t i n g from the equilibrium positron v e l o c i t y d i s t r i b u t i o n being s h i f t e d to higher energies by the applied f i e l d , from whence the shoulder component a r i s e s and at which the anni-h i l a t i o n rate i s r e l a t i v e l y v e l o c i t y independent. However, for the low temperature gas with high f i e l d two i n t e r e s t i n g observations can be made: f i r s t , the shoulder width does not change from that of the low-temperature z e r o - f i e l d spectrum and secondly A g (with f i e l d ) approaches the room temperature.zero f i e l d r e s u l t . Since the shoulder-width i s a measure of the slowing down time, . the above two facts suggest the following: the mechanism responsible for the enhanced equilibrium a n n i h i l a t i o n rate i n the low temperature gas has l i t t l e e f f e c t on the positrons with energies higher than 1/40 eV, but i s so strong an i n t e r a c t i o n at lower energies that the high f i e l d , which i s capable of accelerating the positrons to tens of eV i n a room temperature gas, j u s t manages to accelerate the positron v e l o c i t y d i s t r i b u t i o n to approximately 1/40 eV. This strongly suggests that cl u s t e r s of gas atoms are present i n the argon gas independent of the presence of the positron and present a very large energy los s to thermal positrons. A high a n n i h i l a t i o n rate at low energies would then r e s u l t because of the near zero v e l o c i t y of the positron. L o c a l i z a t i o n would then take place only i f a bound, c l u s t e r state e x i s t s , which seems to occur for the higher density gas. 3.4 Error Analysis 3.4.1 Counting S t a t i s t i c s and Uncertainty i n X g and A£/D 135 Lee (1971) has shown that the l i k e l i h o o d function i s gaussian i n shape and therefore that the standard deviations derived i n the usual way are good estimates of the uncertainty i n X^ due to the counting s t a t i s t i c s . The contribution to the uncertainty i n X^/D from the uncertainty i n the gas density i s about 2% (Section 3.4.1.3). The combination of density and electronic s t a b i l i t y uncertainties (Section 3.4.1.1) contributes an uncertainty of 3% to X&/I). Since these effects only increased the uncertainty i n X^/T) a r i s i n g from the counting s t a t i s t i c s by a small amount (from 5% to about 5.8%) only the errors due to s t a t i s t i c s were plotted i n the graphs and used i n the various f i t s . 3.4.1.1 Electronic S t a b i l i t y The timing s h i f t of the prompt peak during one month of running was less than one channel (0.177 ns) and was well within the 1 nano-second resolution of the apparatus. As the d i f f e r e n t i a l l i n e a r i t y was used i n the maximum li k e l i h o o d f i t t i n g routine i t s only contribution to the uncertainty i n A g was due to structural changes r e l a t i n g to the 1% o s c i l l a t i o n observed i n t h i s l i n e a r i t y , Section 3.2.3.3.2. Contributions to uncertainty a r i s i n g from fluctuations i n the time c a l i b r a t i o n (see Section 3.2.3.3.2) were disregarded, as those runs for which the o s c i l l a t i o n s i n the d i f f e r -e n t i a l l i n e a r i t y contributed greater than 1% to the surrounding counting 2 l e v e l resulted i n poor x confidence l v e l s and therefore represented runs that were subsequently discarded. In any case the spectrum plots 136 were v i s u a l l y checked f o r any " u n r e a l i s t i c " o s c i l l a t i o n s . Therefore the main e l e c t r o n i c uncertainty originated, not from the d i f f e r e n t i a l l i n e a r i t y , but the i n t e g r a l l i n e a r i t y , Section 3.2.3.3.1, for which the uncertainty was about 0.6%. 3.4.1.2 Temperature The chamber temperature was measured with a copper-constantan thermocouple together with a DANA Multimeter 5000 with a r e s o l u t i o n of ±1 microvolt. One microvolt i s about 0.05 K. When using thermocouples for temperature measurement, errors can a r i s e due to temperature gradients along the sensing wires and temperature differences at connecting points. In t h i s experiment a 2 microvolt change was noticed when the heater was turned from o f f to f u l l on. However, as the heater was always p a r t l y on and the c a l i b r a t i o n of temperature, using pressure, was made under i d e n t i c a l conditions such e f f e c t s did not have to be considered. A second 2 microvolt change was noticed over some 24 hour periods. Since separate heating of i n d i v i d u a l chamber parts and thermocouple wires with a heat gun had no e f f e c t , i t was f e l t that t h i s was a r e s u l t of room temperature changes on the DANA multimeter for which a 2 microvolt/°C temperature c o e f f i c i e n t was quoted. From experience gained i n using t h i s system and from p e r i o d i c a l l y recording the temperature and pressure, which was a very s e n s i t i v e i n d i c a t o r of the average chamber temperature when a saturated gas was present, i t was f e l t that the combined systematic errors, of which some were random with time, contributed about 1.5 microvolts (±0.07 K). From recordings of temperature during many runs i t was found that the con-137 t r o l l e r d r i f t was about ±0.1 K. (The d r i f t was ±0.4 K prior to use of the temperature s t a b i l i z e r (Section 3.2.2.1)). A mixture of ice and water i n a mirrored glass dewar was used as the reference with the ice solution changed every evening. A measurement of this reference with a standard mercury thermometer gave a result of 0.0±0.1 C. I n i t i a l l y a warming of th i s reference by about 0.25 K was noticed over a 12 hour period as the ice melted. This was reduced to about 0.05 K by r a i s i n g the thermocouple reference junction (which consisted of the thermocouple wires epoxied into 1/8" glass tubes) to within 8 cm of the ice mixture surface. The ice mixture had a depth of about 24 cm. The above errors have a combined RMS sum of about ±0.17 K and combined with the ±0.2 K uncertainty resulting from the use of the pressure to calibrate the thermocouples resulted i n a t o t a l uncertainty i n the temperature of about ±0.3 K. 3.4.1.3 Gas Density As was mentioned i n Section 3.2.4.1, the gas density was calculated from a 16-parameter f i t made to the most r e l i a b l e experimental data found for the gas and l i q u i d regions. Although the older 2-parameter f i t (NBS C i r . 564 (1955)) used by Canter and Roe l l i g i s i n agreement with the 16-parameter f i t (Section 3.2.4.1) used here, the l a t t e r consists only of a v i r i a l expansion given i n increments of 10 K. In comparison, the 16-parameter f i t i s " s e l f - i n t e r p o l a t i n g " with an error at least less than that of the experimental data from which i t was derived. With a ±2 p s i uncertainty i n the pressure, Section 3.2.2.2, and the 138 ±0.3 K uncertainty i n the temperature given i n the previous section, an error i n density of ±0.6 amagat for 20 amagat gas at 140 K was derived numerically. Thus, the uncertainty i n the density i s about 3%. 3.4.1.4 E l e c t r i c F i e l d Since a c a l i b r a t e d p r e c i s i o n power supply (Fluke, model 405B) was used to generate the voltage for the e l e c t r i c f i e l d , the l a r g e s t uncertainty i n the e l e c t r i c f i e l d resulted from the u n c e r t a i n t i e s i n the mesh spacing, the mesh f l a t n e s s , and the g r i d edge-effect. The average mesh spacing was derived from the separation of the end plates (11.0±0.1 cm). The uncertainty i n t h i s average mesh spacing due to a ±0.1 uncertainty i n the thickness of the glass separators and the f i n i t e thickness of the mesh wires (twice the d i a . of 0.009") i s then about 2.2%. Since the glass spacers were positioned i n s i d e the glass rings (Figure 8) and r e s t r i c t e d the positrons to the inner regions, the edge e f f e c t i s expected to be n e g l i g i b l e . Because of the fineness of the mesh wires i t was not possible to keep a l l of the screens com-p l e t e l y f l a t . However, the deviation from f l a t n e s s was not more than 2% and was not expected to be affected by the high voltages applied to the screens because of the a l t e r n a t i n g arrangement of t h i s voltage. Some no n - l i n e a r i t y i n the e l e c t r i c f i e l d w i l l e x i s t i n the i n t e r s t i c e s which represent l e s s than 5% of the a v a i l a b l e volume. Combining the two errors of 2.2% and 2% to a 3% uncertainty i n the density y i e l d s a maximum uncertainty i n E/D of 4.2%, (for 95% of the a v a i l a b l e volume) which i s s i m i l a r to that obtained by Lee using a 139 r e s i s t o r - b i a s e d - r i n g g r i d . 3.4.1.5 Difference of Gibb's Free Energy For 120 K gas at a pressure of 7 bar, (17.5 amagats), and 140 K gas at a pressure of 15 bar, (33.7 amagats), the uncertainty i n Ag was calculated to be 1.3 j - g ^ and 0.90 j-g r e s p e c t i v e l y . These were derived using the equations given i n section 3.2.4.3 and assuming the following uncorrelated errors: 0.2% for h (140 K), g 0.25% for h (120 K), g 0.5% for h L(140 K), 1.0% for h L(120 K), 0.05% for s (120 and 140 K), g 0.15% for s L(120 K), 0.1% for s L(140 K), 0.14% for the temperature, Section 3.4.1.2. 3.4.1.6 E f f e c t of F i n i t e Resolution Because of the shortness of the l i f e t i m e i n the l i q u i d spectra and the fact that i t merges with the prompt peak at zero time, i t was suspected that some systematic error would r e s u l t . To gauge the magnitude of such an e f f e c t , pseudo-data c o n s i s t i n g of a gaussian peak with a width equal to the experimental r e s o l u t i o n representing the prompt peak and two exponentials of known l i f e t i m e were generated with a random "square root" error added. With the i n t e n s i t i e s , l i f e t i m e s and r e s o l u t i o n s i m i l a r to that of 140 an experimental spectrum, t h i s pseudo-data adequately represented the true data except for the presence of a s l i g h t secondary peak j u s t following the prompt peak. The small spacing of t h i s peak from the prompt peak depended on the r a t i o of the r e s o l u t i o n to short l i f e t i m e . By f i t t i n g t h i s data with the same algorithm used on the experi-mental spectra i t was found that the f i t t e d short l i f e t i m e was longer than the "true" l i f e t i m e by about 0.04 ns, t h i s d i f f e r e n c e being independent of the l i f e t i m e . The e f f e c t of subtracting 0.04 ns from the measured l i q u i d l i f e t i m e s increased the associated rates to those shown by the dashed l i n e i n f i g u r e 23. Note that the r e s u l t s of Specktor and Paul (1971) having approximately the same re s o l u t i o n as the system here would s u f f e r systematic error of s i m i l a r magnitude. The p o s s i b i l i t y of a s i m i l a r systematic error i n the shorter gas l i f e t i m e s i s not as obvious. Whereas the above error r e s u l t s from the e f f e c t of the r e s o l u t i o n on the "edge" of the l i q u i d spectra at zero r e l a t i v e time, the gas spectra does not have such an edge, but rather a f l a t area, the shoulder region, before i t s l i f e t i m e . It would seem, therefore, that such a systematic error a r i s i n g from the termination of the shoulder would be considerably smaller. 3.4.2 Gas P u r i t y The gas purity was monitored by computing the shoulder-width density product from the time dependence of the v e l o c i t y averaged 141 annihilation rate, Section 3.2.4.5. This product was found to be 340±10 nanosecond-amagat for room temperatures and 360±10 nanosecond-amagat for low temperatures spectra. These are i n agreement with the results of Lee (1971), and Tao (1970). The larger product for the low temperature spectra may be due to the fact either that the gas i s even cleaner at low temperatures or that i t takes a l i t t l e longer for the positrons to slow to lower temperatures or simply r e f l e c t s the effect of the larger X& at the low temperature on the determination of the shoulder width, which depends on X , Section 3.2.4.4. Matheson Gold Label U l t r a Pure, 99.999% pure argon gas was used to f i l l the chamber. The Matheson Company analysis of the gas was as follows: c o 2 ND °2 < 2 ppm H2 ND CO ND N2 < 5 ppm CH. 4 ND H20 ND The chamber was flushed over a 24 hour period while heating the chamber to 195 C. A sample chamber was used to obtain an analysis of the in-chamber gas after a long series of runs while the chamber was s t i l l at low temperatures. The analysis was performed by Gollob An a l y t i c a l Service Corporation of New Jersey and indicated impurity levels as. follows: 142 He < 4 ppm, minimum detection l e v e l , 4 ppm 4 ppm 4 ppm 10 ppm Hydrocarbons 19 ppm unknown, because of too small a sample. The impurity levels of a l l the gases with the exception of Nitrogen and the hydrocarbons, and possibly water vapor, were below the detectable l i m i t s . Water vapor was not expected to be a problem as i t would have been e f f i c i e n t l y flushed out by removal of the high the cold temperatures during a low temperature run. Lee (1971) found 30 ppm of Nitrogen i n his f i n a l analysis. These levels were achieved (in both cases) by heating the sample chamber prior to f i l l i n g . This heating of the sample chamber was found necessary as his previous analysis suggested an impurity l e v e l of 1000 ppm, which was thought to be a result of previous contamination of the sample chamber with nitrogen gas. The hydrocarbon impurity i s more than the "less than 0.5 ppm" found by Lee and i s probably a direct result of not using a p u r i f i e r i n t h i s system. This impurity possible results from soft solder f l u x used on the connections between the f i e l d rings or simply resulted from the large surface area of the doubly-rolled teflon cylinder, insulating pressure of argon gas present i n the chamber when heated for the 24 hours. Any residual would have been frozen onto the walls because of 143 the f i e l d rings from the inner chamber wa l l . Althought t h i s l e v e l of hydrocarbons i s s l i g h t l y above the 10 ppm l e v e l thought to be necessary from an order of magnitude consideration of the effect of impurity cross-sections on the slowing down time (Lee (1969), the agreement of the shoulder-width product with that of Lee (1971) suggests that t h i s l e v e l i s adequate. In addition the constancy of this product with time may be a result of the cold chamber surfaces keeping these sources on the walls. As mentioned i n Section 3.2.1.4 the epoxy used i n the inner part of the chamber was expected to lessen rather than increase the overall impurity l e v e l s . F i n a l l y to prevent any outgasing, which may have occurred during a room temperature run, from affecting the low temperature runs the chamber was always emptied of gas to 10 p s i and then cooled and flushed a few times before st a r t i n g a new series of low temperature runs. A l a s t check of the effects of the above impurity levels was the comparison of the room temperature A g observed here to that found by Lee (1971). The average of a l l the room temperature runs, s i x i n number, was 5.30±0.10 us ^ amagat ^ and agrees favorably with the value 5.35±0.12 found by Lee (1971). Impurities would be expected to increase these rates. In addition, the low temperature results were i n excellent agreement with the e a r l i e r measurements of Canter and Ro e l l i g (1975). The agreement of both the shoulder width and room temperature and low temperature annihilation rates with previous measurements attests to the l e v e l of impurity of the argon gas used i n t h i s experiment. The quenching rates for positronium, although of poor quality due to the 144 poor s t a t i s t i c s of t h i s component also suggest agreement with the results of Lee (1971) for the gas, and Spektor and Paul (1971) for the l i q u i d , but are not as conclusive because of s t a t i s t i c a l uncertainties. 145 CHAPTER FOUR. ANALYSIS OF RESULTS 4.1 Outline of Procedure Because of the lack of a theory for positrons interacting with clusters of atoms as opposed to interactions with isolated atoms, the analysis presented here w i l l of necessity be rather simple and attempt instead to relate a variety of diverse phenomena. In p a r t i c u l a r , the analysis w i l l center mainly on the apparent s i m p l i -c i t y supplied by the dependence of X g and X^/T) on the change of the Gibb's free energy, Ag, as i t relates to the existence of atomic clusters i n the gas. Results from t h i s analysis w i l l then be used i n the consideration of l i m i t i n g cases of the growth model i n order to interpret the "saturation" effects of X^ and ^/D. F i n a l l y Section 4.5 w i l l relates the size of possible bound states (of many atoms) to the temperature dependence of A g for the l i q u i d . 4.2 Inadequacy of Standard Analysis For the single atom case the application of a 'small' e l e c t r i c f i e l d i s equivalent, as far as the effect on the positron v e l o c i t y d i s t r i b u t i o n i s concerned, to an increase i n the gas temperature, the magnitude of t h i s increase dependent on both the small f i e l d change and the momentum-transfer cross-section (Orth and Jones (1966), Lee (1971)). In this manner an estimate of the momentum-transfer cross-section can be made. This type of analysis, however, depends on X^ being proportional to :density. For the low temperature situation 146 discussed here, t h i s i s not the case. In fact i t appears that the major a f f e c t on A G / D of a change of temperature (or density) i s a s i g n i f i c a n t change i n the state of the gas (as seen by the positron) and not 'heating' of the positron v e l o c i t y d i s t r i b u t i o n which must be assumed for t h i s analysis to be u s e f u l . On the other hand, the e l e c t r i c f i e l d dependence of A G / D does, at least q u a l i t a t i v e l y , agree with that for the single atom case - v i z , A £ / D decreases with E/D. In any case one i s l e f t with an incomplete a n a l y s i s : meaningful E/D behavior, but a temperature behavior which i s d i f f i c u l t to i n t e r p r e t without including more exact knowledge of the gas state. For t h i s reason the following analysis concentrates on the Ag dependence which i s a useful measure of the gas state (Chapter Two). 4.3 Cluster Model In Chapter Two i t was shown how the number of i - c l u s t e r s i s r e l a t e d , c l a s s i c a l l y , to a volume and surface term.(2-4 and 2-2): ,2/3 N. = N exp ,-iAg - a ( T ) i , 4 - 1 ^ kT kT 2/3 where a ( T ) i = A a,. s with A = 4TT3/(4TTD. ( T ) ) 2 / 3 i 2 / 3 4-2 S LJ and a = the surface tension. If the i n t e r a c t i o n of a positron with a s i n g l e atom or c l u s t e r of atoms i s independent of the presence of other atoms or c l u s t e r s then A \. + . L A . 4-3 e 1 1-2 x where A ^ i s the a n n i h i l a t i o n rate with single atoms (at density D), and A ^ i s the a n n i h i l a t i o n rate with c l u s t e r s of i atoms of density D^. 147 In the following, we assume for s i m p l i c i t y that the enhancement in the annihilation rate due to clusters can be accounted for i n terms of annihilation with an average cluster of size i . Thus EX. 'V/ X.. , with Y.J, — X.*/D.*. Moreover i f X./D i s given by 1-6 ^ X 1 * ' i * 1 1 * 1 (Section 1.2.1.2) as determined by Lee (1971) then e D X.* l _ Y D.* x Y ±*N i* EiN. x 4-4 If we make the assumption (valid for low densities) that EilsL -x-then (X /D - X /D) = y.*A(T,i*) exp(-Agi*(T)) e x x k T 4-5 where A(T,i ) = exp( - q ( T ) i * 2 / 3 ) kT 4-6 Deviations from this approximation would be expected at higher densities. Likewise, the ideal gas mixture assumption behind 4-1 wou also f a i l . In any case, for the low densities a value for i (T) can 148 now be obtained from a plot of ln(A /D) against Ag. Such a plot for Ag greater than 15 J g 1 i s shown i n Figure 38 with the results for i * and Y ±*A(T,i*) given i n Table VI. Table VI Cluster Model Results T Y ± A(T,i ) (K) i (ys amagats ) 100 8.86 ± 3.0 34 ± 2 120 7.49 ± 1.2 1560(+ 1800, - 800) 130 4.48 ± .21 236(+ 50, - 40) 140 3.20 ± .09 81 ± 7. The fact that the resulting i values are so small casts some doubt on the v a l i d i t y of using macroscopic thermodynamic data for calculating 149 Ag (J g ) Figure 38 F i t to.Cluster Model: Ln(A /D - A /D) versus Ag 150 Ag. Nevertheless, since small c l u s t e r s are expected and these i values are i n close agreement with those derived from a more general phenomenological model (Section 3.3.4.2.4), i t appears that the d e s c r i p t i o n of the phenomena i n terms of Ag i s at least q u a l i t a t i v e l y correct. According to the law of corresponding states, Guggenheim (1945) showed that the surface tension, a, i s quite well represented by a = a (1 - T/T ) 1 1 / 9 4-7 o c with a Q = 36.31 dyne-cm for argon assuming that the surface area, A g, i s related to the temperature-dependent l i q u i d density by 4-2. Using the l i q u i d d e n s i t i e s given by Angus and Armstrong (1972) the values for A g (and consequently A ( T , i )) were determined. Then with the A(T,i ) given by Table VI, values of were r e a d i l y calculated and are given i n Table VII: Table VII Surface Energy E f f e c t s L i q u i d Surface T i density, D_ tension, a kT A a Y. L 1 3 (K) (amagats) (dyne cm"1) (10 1 4 e r g s ) (10 1 4 e r g s ) (ys 1amagats 100 8.86 736. 9.58 1.38 27.1 1.1 x 10 120 7.49 651. 5.19 1.66 14.3 8.6 x 10 6 130 4.48 598. 3.21 1.79 6.63 9.6 x 10 3 140 3.20 529. 1.43 1.93 2.56 310. 152 Now the "abnormally" low annihilation rates for the 100K data r e l a t i v e to the 120K, 130K and 140K data as shown i n Figure 38 i s simply explained i n terms of the greatly increased surface tension at these lower temperatures (see Table V I I I ) . These lower annihilation rates, t y p i f i e d by the low value of y^A-UOOK, i ), are brought into perspective when Y^*> the annihilation rate, per cluster i s considered. For now Y^*» unlike Y^*A, does increase with the number of atoms per cluster for a l l temperatures considered. The magnitude of Y-*> however, i s of some concern as y , the X -Li annihilation rate per atom of l i q u i d i s (from Figure 24) only 2.3 us ^ -1 * amagat . A comparison of Y t to y*/± (the annihilation rate per atom of a cluster) i s given i n Table VIII. Table VIII Pre-Exponential Factor T Y.*/i X Y i / i YL (K) (ys ^ amagat "*") 100 1.2 x 10 9 5 x 10 8 120 1.1 x 10 6 5 x 10 5 130 3 2.1 x 10 9 x 102 140 9.7 x 10 1 4 x 10 1 However, as discussed i n Sections 2.1 and 2.3, 4-1 should contain an 4 additional pre-exponential factor whose magnitude i s of the order of 10 . It i s then expected that y . * / i y T should be of t h i s magnitude. 153 In fact l o g ( y . * / i Y T) from Table VIII i s 5±3.5, consistent with the 1 Li value of 4 referred to. Although this v a r i a t i o n of 3.5 due to the systematic temperature dependence evident i n Table VIII could be increased by an additional factor of two to account for the uncertainty in the surface tension and y as applied to small drops, the results do L i argue against exponents as large as 17 which have been suggested (albeit generally from " i n t u i t i v e " considerations)(Section 2.1 and 2.3). On the other hand the observed temperature dependence of the pre-exponential factor i s i n agreement with the idea that the pre-exponential factor i s expected to increase as the cluster becomes more s o l i d - l i k e i n structure (Reiss (1970)). The r e l a t i v e importance of the surface energy with respect to the volume energy, iAg, i s e x p l i c i t l y shown i n figure 39. Here A^a of 4-1 i s plotted against i with the values of i (T) from Table VII marked. The volume energy, iAgEAG, has also been plotted for different values of AG. The larger surface energy r e l a t i v e to the volume energy for 100K ( i = 8.86) i s quite s t r i k i n g . For Ag<15 J g ^ the exponential dependence gives way to a linea r dependence of ^ £/D on Ag. This may imply i n terms of the cluster model that the number of clusters has increased to the point that the positron i s interacting with more than one cluster at a time. However, i t i s also equally possible that either the ideal gas mixture assumption or the approximation of EiN^ by EN^ (on pg. 147) has f a i l e d . This value of CLUSTER SIZE, i Figure 39 Relative Size Dependence of the Volume and Surface Energy of a C l a s s i c a l Liquid Drop 155 Ag (15 J g """) w i l l be termed the screening l i m i t and Ag<15 J g ^ the screening region, including only those states for which A^ /D i s lin e a r i n Ag. The designation of t h i s region with a screening t i t l e does not infer that screening i s the only mechanism describing the phenomena but instead i s to be considered more as a useful label for those states for which A g i s no longer exponential on Ag but s t i l l proportional to density (and now a lin e a r function of Ag). This region w i l l be discussed more f u l l y i n terms of many body theories i n the next chapter. 4.4 Growth Model As shown i n Section 3.1.2 the general shape of the time spectrum at low temperature (peak plus shortened A ) i s consistent with a positron popu-l a t i o n indergoing a t r a n s i t i o n from having a low annihilation rate to one with a much higher annihilation rate. In this section, we attempt to develop a model which i s consistent with the observations over the whole density (or Ag) range, a model which incorporates the cluster model discussed i n Section 4.3. The f i r s t region w i l l consist of the very low densities where the single atom annihilation rate i s relevant (less than about 5 amagats, as observed by Lee at 135K). The next region, the low density region (densities 5 to about 25 amagats, see figure 26), i s regarded as the region where clusters start to contribute s i g n i f i c a n t l y to the single atom rate. This i s the same region for which the cluster model of 156 Section 4.3 was applied, v i z . the region of which Ag i s greater than 15 J g \ In terms of a two-population model y^* n°w becomes a capture rate into the clusters, with A , the annihilation rate i n the cluster, so rapid i n practice that capture implies immediate annihilation. In this case, since the X term i n 3-4 i s negligible i n the f i t t e d region (Section 3.3.1.2): A % X, = A, + A , 4-8 e d 1 c which i s similar to equation 4-3. The dense region (densities i n the range 25 to 300 amagats) i s then characterized by: A N, » 4-9 c 1 1 1 or A » A, 4-10 c 1 and a l l positrons are trapped before annihilating. In t h i s case A « A* 4-11 e where the observable annihilation rate i s that of the denser (cluster) medium. 4.4.1 Application to the Saturated Gas The growth model i s applied by calculation of R ( t ) , (3-4) of Section 3.1.2, with A^, A^ and A given values appropriate to the state of the gas. The slope i n the exponential region (linear on a log plot) i s then the desired growth model A (denoted A ), found 157 i n a manner completely analogous to the determination of the experimental A G from a time spectrum (Section 3.3.1.2). For the saturated gas region X^/D (4-8) i s set equal to A e/D * -1 (3-14) and A to 1.22 ns , the constant value of A ^ for saturated gas at densities greater than 40 amagats (Figure 23). The resulting dependence of X on the density (the s o l i d l i n e i n Figure 23) i s seen to adequately describe the saturated gas results very nicely. An a t t r a c t i v e feature of this growth model i s the very natural way i n which the switching of the value of A i s accommodated from a X ° e e measuring A ^ (4-8), the 'free' rate, to measuring A (4-11), the 'localized' rate. 4.4.2 Application to the Unsaturated Gas 4.4.2.1 Screening Region As discussed at the end of Section 4.3, when A = A , + A ('free' ' e 1 c picture), deviations from the cluster model exist of the nature of a screening effect as the number of clusters increases. That A G i s proportional to density for the saturated gas before the switch to A * takes place (Section 4.4.1) and that the saturated gas si t u a t i o n i s just a special case (Ag=0) of the unsaturated gas results suggests that the cluster density i n the saturated gas even for the lowest saturated gas densities i s high enough for screening to be present. In fact from the discussion of the results of Section 3.3.6, i n particular comparison of Figures 32 and 33 for the screening region at the densities of 30 amagats, i t i s clear that a separation into 158 'free' and 'localized' rates takes place at a density of about 30 amagats for unsaturated gases as we l l . However, for the isothermal (experimental) states for which D<30 amagats, a switching takes place not between a free and lo c a l i z e d rate picture Section 4.4.1 but between a cluster- and screened-positron-picture (Section 4.3). The application of the growth model i n t h i s case means not that a trapping rate i s measured for high cluster densities, but that a screening effect i s turned on when the cluster density becomes large. Thus, for D<30 amagats one chooses A* = (X - AJ'Ag)D/30 4-12 a o d * 1 where A^ = A^ for the screening region. A value of A^ can now be derived by applying the growth model to the results for which D<30 amagats and for which a screening region exists (the 100 K and 120 K data), setting A = ^ c»^ 0 = 1.22 ns (Section 4.4.1) and l e t t i n g A^ be given by the cluster results (Table VI). In addition, however, i t was necessary to vary y.jA(T,i ) of Section Table VI. This procedure was followed for the 120 K data and the f i t i s shown i n Figure 32. This f i t i s not good near Ag = 0 as this i s part of the dense gas region to be discussed next. The f i t to the 100 K data w i l l be discussed l a t e r . The 130 K and 140 K data which do not contain a screening region but only a cluster and dense gas region (because Ag<15 J g ^  for D>30 amagats) are discussed i n the next sections. 4.4.2.2 Dense Gas Region By analogy to 4-12 and as suggested by the r e s u l t s , the 159 annihilation rate i n the dense (cluster) medium was chosen as follows A * = Xq - A*Ag 4-13 The results were then f i t t e d with the growth and cluster models i n a manner ent i r e l y analogous to the application to the region with D<30 amagats for which y^ A(T,i ) and A ^ (but independent of temperature) were allowed to vary and A q was set equal to 1.22 ns ^ (Section 4.4.1). These results are shown i n Table IX and the f i t i n Figure 32. Again the f i t i s excellent. Table IX Results of Ag Dependence for A T \* A , A . A(T,i ) d 1 x (K) (g J - 1 ns" 1) (ns-^-amagats-1) 100 0.0276 34.3 120 0.0276 2050. 130 0.0294 319. 140 0.0294 130. 160 4.4.2.3 100 K Data The 100 K data was now f i t t e d by choosing X"^  = X^  (120 K) and d d extrapolating to 100 K the f r a c t i o n a l difference of A(T,i ) from Tables VII and IX for 120 K, 130 K and 140 K. This predicted f i t i s shown i n Figure 32. The abnormality of the 100 K data, enhanced only for Ag<5 J g \ i s now c l e a r l y seen as resulting from the reduction i n the number of clusters caused by the greatly increased surface tension at the lower temperatures. 4.4.4 Dissociation Scheme A model which takes into account more d i r e c t l y the p o s s i b i l i t y of the positron returning to the free population i s that with an escape channel added to the radioactive decay model of Sections 3.2 and 4.4.2. If this i s done then 3-1 become dN1 - r - i = -(X. + X.)Nn + X N. 4-14 dt 1 i 1 esc x dN : * _ i = -(A + X )N. + X.N. dt esc l l 1 where X g g c i s the required escape rate. These have the solutions: No i , at $t. N. = -7—rr (e -e ) i (a-3) N and = f ° ((a+X*)e a t-(g+X*)e 3 t) 1 (a-B) 4-15 161 where 2a = + x. + A*+X ) )2g\ 1 i esc \ + f ±(A..+X.+X*+X ) 2 4-16 I - I 1 l esc - 4(X1+X.)X ] 1 1 1 Again the observed rate i s R (t) = X*N. + X,N, 4-17 e i l l As i n Sections 4.4.1, X was obtained by f i t t i n g a single exponential e * -1 to 4-17. In t h i s case X was set equal X (1.22 ns )(Section 4.4.1) * and the cluster model used with A varied. I t x^ as found, however, that because of the coupling between y^ A and ^ e s c J i t was not possible to obtain a Ag dependence for X which was physically reasonable. If esc ^esc ^ S t 0 r e P r e s e n t a probability for d i s s o c i a t i o n , then i t should increase monotonically with increasing Ag. Instead i t was found that the expected rapid increase was followed by a slow decrease after about 2 J g \ Changing y^A affected the value of ^ e s c at i t s maximum, but did not affect i t s o v e r a l l behavior. Thus, neither parameter was uniquely determined. The relevance of t h i s extended decay scheme over that for which X was given by 4-13 i s therefore not demonstrated. 4.4.4 Application to the E l e c t r i c F i e l d Results The e l e c t r i c f i e l d results can now be simply interpreted i n terms of the zero f i e l d results (Sections 4.4.1 and 4.4.2). In p a r t i c u l a r the cluster and screening regions are c l e a r l y discernable with the 162 following observation for the screening region. The range over which the positron obeys a screening picture has been expanded at the expense of the l o c a l i z a t i o n picture but not at the expense of the * cluster picture: X i s now being measured at densities greater than 45 or 50 amagats, whereas the screening l i m i t i s s t i l l at a Ag value of 15 J g ~~. In addition the annihilation rate i n the screening region loses i t s Ag dependence implying that the e l e c t r i c f i e l d prevents the formation of a clustered bound state since t h i s state should be dependent on the energetics of the gas. 4.5 Temperature Dependence of Fluctuon Size * From the behavior of X represented by 4.-13 i t i s apparent that some type of bound state i s present i n the gas and that the annihilation rate i n t h i s state i s dependent on Ag i n a simple manner (Section 4.4.2.1). Figure 23 and 24, however, display an obvious difference between the annihilation rate i n the saturated gas and the l i q u i d : A g i s independent of temperature i n the former and strongly temperature dependent i n the l a t t e r . In fact consideration of the l i q u i d rate i n the v i c i n i t y of the c r i t i c a l point reveals that (AX D /(D-D )) i s only 0.1 ns 1 whereas e c c (AX T /(T-T )) ^ 1.5 ns" 1, a much stronger temperature dependence. The e c c dependence of X on only Ag for the gas and the contrasting strong temperature dependence revealed above for the l i q u i d could be interpreted to mean that fluctuons are forming only i n the l i q u i d (due possibly to the higher densities of the l i q u i d ) . 163 If fluctuons are formed i n the l i q u i d , the annihilation rate would depend on the volume of the fluctuon since the polarization potential depends on the number of atoms. As an example(for s i m p l i c i t y ) we assume the case where the annihilation rate i s simply proportional to the volume of the fluctuon. Both L i f s h i t z (1970) and Krivoglaz (1974) showed that the radius of a fluctuon i n a l a t t i c e gas model at suitable low temperatures and high densities, i s related to the temperature as follows: R « ( T / T 3 ) ~ 1 / 5 for T « T 3 4-18 where T 3 i s the temperature above which the atoms of the gas act as free particules i n the e l e c t r i c f i e l d of the l i g h t e r charged p a r t i c l e . Here T 3 would be about 200 K, the temperature below which deviations from the free positron picture are evident. Then * , 3 / 5 A = k(T 3/T) . 4-19 Letting t h i s function pass through the resolution-corrected l i q u i d data of Figure 25 at 110 K (k=1.20 n s " 1 ) , the dotted curve i n Figure 23 results. The r i s e of t h i s curve above the l i q u i d data at the lower temperatures (higher densities) i s expected to occur as the volume dependence (and therefore temperature dependence) vanishes as the positron wave-function " f i l l s " the fluctuon. The behavior of the annihilation rate (the temperature dependence of the size of the fluctuon) i s not known and the poor f i t to these higher temperatures i s not surprising since 4-18 i s not v a l i d at these temperatures. F i n a l l y , although the temperature of A G for the lower temperatures 164 does f i t a "volume" interpretation, t h i s agreement may be s t r i c t l y coincidental. For as noted by both L i f s h i t z (1970) and Krivoglaz (1974), the formation time of a fluctuon i s quite slow (no estimate given) due to the existence of a thermodunamic potential barrier to the formation of the fluctuon. Thus, even i n a l i q u i d the positron l i f e t i m e may be too short for complete formation of a stable fluctuon and therefore the predicted temperature dependence invalidated. 165 CHAPTER FIVE DISCUSSION 5.1 Clusters: Small or Large For gas temperatures greater than 200 K, the equilibrium rate, A g (denoted A^ for single atom annihilations i n t h i s case), i s proportional to density and decreases with increasing temperature or increasing E/D due to the ve l o c i t y dependence of the single-atom annihilation rate. However, for low temperature gaseous argon at a constant density the temperature dependence of X^ deviates substantially from that expected from the ve l o c i t y dependence of the single-atom annihilation rate and the temperature dependence of X g becomes as strong or stronger than the density dependence. It has been suggested that t h i s rapid increase of the annihilation rate with temperature and density can be associated with formation of clusters of hundreds of atoms about the positron i n the gas (Nieminen (1977). However, this picture i s incomplete for the following reasons: a l o c a l i z a t i o n mechanism i s lacking and the cluster formation times are as long or longer than the positron l i f e t i m e . In fact the present results and subsequent analysis suggests that much smaller clusters (<10 atoms) exist i n the gas and dominate the enhancement at least for the lower densities. Basically both pictures can be considered i n terms of the free energy to form a droplet (with or without the aid of the positron): AF = AE + AV + E o + Ag i * 5.1 166 where AE and AV are the k i n e t i c and potential energy changes for the positron on forming a droplet from i t s free state, E g the surface * energy of the droplet and Ag i the volume energy due to the difference i n chemical potential between the gas (stable) and l i q u i d (metastable). AV i s calculated using a pseudo potential related to the scattering length (Coopersmith (1965)). For self-nucleation: AE=AV=0 and the positron neither assis t s nor i s bound i n the cluster. The results of Chapter Two are then applicable. The positron induced models set AF=0 and calculate the droplet size i . The work presented here calculates the droplet size from the Ag-dependence i t s e l f . The evidence that small clusters are present i s discussed i n the next section and the consequences of the growth model to the application of the positron-induced theories i s discussed i n the subsequent sections. 5.2 Evidence for Clusters i n Unsaturated Argon I t should be appreciated that most experiments r e l a t i n g to clusters i n gases involve supersaturated gases and the assumption of an equilibrium concentration for calculation of the supersaturation pressure for which a fog would be v i s i b l e . Although i n p r i n c i p l e t h i s could be extrapolated to a non-saturated gas, the need for only an order of magnitude precision for the cluster d i s t r i b u t i o n (for these fog experiments) makes th i s method for studying unsaturated gases highly imprecise and suspect. 5.2.1 Previous Workers Evidence for clusters i n unsaturated single atom gases i s of both a 167 direct and indirect nature. Most evidence to date either exhibits only the existence of dimers or actually measures thei r concentration (Section 2.6). Ion-molecular reactions, important i n the more direct mass spectrometer experiments, obscure the higher polymer concentrations. The direct evidence suggests that at least 100 ppm dimer concentrations exist i n low density (100 torr) argon gas at 300 K and 400 ppm at 100 K. As some of the metastable dimers have time to decay i n these experiments these values are possibly lower l i m i t s . 5.2.2 From Positron Annihilation Although Canter and Ro e l l i g (1975) f i r s t observed enhanced anni-h i l a t i o n rates i n argon at low temperatures and high densities, this enhancement was attributed to e l e c t r o s t r i c t i o n of the gas atoms about the positron. These considerations w i l l be discussed i n Section 5.3. The aim of the present work was to consider the extent to which s e l f -nucleating clusters can contribute to th i s enhancement. The c l a s s i c a l d i s t r i b u t i o n of clusters of gas atoms and bulk properties of the metastable l i q u i d were employed throughout. Although the relevance of such a model for clusters of such small size as those obtained can be questioned, i t remains the simplest and only p r a c t i c a l model available at the present time. At the same time, the thermodynamic pot e n t i a l , Ag, which i s the measure of cluster s t a b i l i t y , introduces (on i t s own) substantial s i m p l i -f i c a t i o n into the results and isolates a temperature dependence i n X^/T) which r e f l e c t s the temperature dependence of the l i q u i d surface tension. Again, however, the use of macroscopic surface tensions for small droplets of small curvature i s of a qua l i t a t i v e basis only. 168 5.2.2.1 Self-Nucleating Clusters Application of self-nucleating cluster theory to positron anni-h i l a t i o n has a few major drawbacks. I t involves averaging the positron-cluster interaction over a d i s t r i b u t i o n of clusters for which the pre-exponential factor, surface tension for small clusters (100 atoms) and positron-cluster interactions are generally unknown. These reasons increase the uncertainty of the cluster sizes resulting from such an application. Nevertheless, the actual presence of an exponential dependence i n the results and the resulting prediction, on f i t t i n g the cluster model to this dependence, that small clusters (10 atoms) are important for describing the low temperature enhancement i s a major step i n understanding this low temperature phenomena. Although these values are surprising (for self-nucleating clusters) the positron-cluster interaction no doubt p r e f e r e n t i a l l y senses the larger clusters which, i n any case, w i l l s t i l l be much smaller than those clusters sizes (50-100 atoms) required to i n i t i a t e nucleation i n a supersaturated gas and those sizes predicted by e l e c t r o s t r i c t i o n theories. Only with better calculations of the cluster d i s t r i b u t i o n for unsaturated gases and some knowledge of the positron-small cluster interaction w i l l these cluster sizes of 3 to 8 atoms be substantiated or refuted. General con-sideration of the absolute cluster concentrations i s not feasible due to the large uncertainty i n both the annihilation rate per cluster and the pre-exponential factor (to be discussed i n Section 5.2.2.4). 5.2.2.2 Effect of Ions The effect of ions was discussed thoroughly i n Section 2.5. From 169 t h i s discussion i t appears that ions produced by the positron upon slowing down are too few i n number to y i e l d s i g n i f i c a n t cluster den-s i t i e s . (It i s also assumed that at least for the lower densities -v i z . X £ exponential i n Ag - that the positron i n a free state i s too energetic to cause s i g n i f i c a n t clustering). This does not exclude the p o s s i b i l i t y that a positron bound to a self-nucleating cluster or one substantially slowed by interaction with such clusters can i n i t i a t e e l e c t r o s t r i c t i o n . 5.2.2.3 Effects of Dimers As mentioned i n Section 2.6 dimers are known to exist i n room temperature argon to concentrations of at least 100 ppm and i n 100 K gas to 400 ppm. Since no enhancement (X s t i l l proportional to density) i s evident for the room temperature gas i t i s not suspected that the increase of dimer concentrations by a factor of four w i l l cause the strong temperature and density dependence of X& at the lower temperatures. Instead i t i s much more l i k e l y that trimers are becoming s i g n i f i c a n t at the lower temperatures and that the interaction of a positron with a trimer (or higher polymer) results i n not only a more si g n i f i c a n t energy loss but i n the p o s s i b i l i t y of the positron either finding i t s e l f with near-zero v e l o c i t y near a dimer (part of a trimer after c o l l i s i o n with a positron) or heating up of the trimer. Thus the capture of a positron i s more probable i n a cluster containing a number of atoms than with a single dimer and so the presence of dimers may be de-emphasized. 5.2.2.4 Phenomenological Theory The phenomenological model based on a v i r i a l - t y p e expansion i n the 170 density Section 3.3.4.2.4) hasa two-fold purpose. F i r s t , i t showed that the size of clusters which influence the positron l i f e t i m e i s of a magnitude expected on the basis of other evidence. Secondly, since the sizes predicted by t h i s simple model are consistent with those resulting from use of the more sp e c i f i c cluster model the v a l i d i t y of the cluster model i s greatly enhanced. The surface tension dependence of the cluster model can then be used to explain the large discrepancies of the data for the lower temperatures, Section 3.3.6. This point i s discussed further i n Section 5.2.2.5. 5.2.2.5 Pre-Exponential Factor and Surface Energy From calculations of the cluster d i s t r i b u t i o n from phase int e g r a l considerations, i t became clear that the c l a s s i c a l cluster d i s t r i b u t i o n (2-4 with 2-2) must be modified to include a pre-exponential factor 4 (besides the single atom density factor) of the order of 10 . The manner of handling the surface energy i n terms of the phase int e g r a l expansion i s presented by Kikuchi (1969). However, the d i f f i c u l t y of actually calculating the surface energy contribution allows only the bulk surface tension to be used. The difference i s then included i n the pre-exponential factor, possibly increasing i t by a factor of ten. In this case, at least for the moment, the use of the cluster model i s mostly of a qua l i t a t i v e nature. Nevertheless the temperature dependence of the surface tension i s apparent i n the results which i s not the case for the e l e c t r o s t r i c t i o n theories where the size of clusters i s necessarily large (vLOO atoms) and the surface energy small compared to the volume energy. With the assumption that the annihilation 171 rate per cluster i s equal (at least to an order of magnitude) to that of an equivalent number of l i q u i d atoms i t was possible to derive a pre-exponential factor possibly correct to a factor of 100. Thus 1 8 the calculated pre-exponential factor of 4 x 10 to 5 x 10 i s within a 3 6 factor of 100 of the theoretical estimates of 3 x 10 to 10 (reported by Kikuchi (1969). In addition the temperature dependence i s q u a l i t a -t i v e l y correct as discussed by Reiss (1970), v i z . an increasing pre-exponential factor for decreasing temperature. Thus, the factor of lO 1 ^ , i n t u i t i v e l y suggested by Lothe and Pound (1968) i s not acceptable. However, i t i s also possible that the pre-exponential factor i s nearly 8 constant (^ 10 ). In th i s case the annihilation rate per cluster would rapidly increase with cluster s i z e . This also i s not impossible. 5.2.2.6 Screening Region The results for D<30 amagats show evidence of deviations of X&/J) from an exponential (and temperature) dependence to a line a r dependence on Ag (and not e x p l i c i t l y temperature dependent) for Ag<15 J g \ This has been interpreted i n the analysis as resulting from the increase of the cluster density to such an extent that screening of the positron by additional clusters i s taking place. Again i t should be stressed that t h i s i s only one possible extension of the cluster model to high cluster concentrations. This region w i l l be discussed i n terms of many-body theories i n Section 5.5. 5.3 Growth Model The growth model q u a l i t a t i v e l y generates the low-temperature peak and equilibrium component (A ). However, this i s not proof that s e l f -172 nucleating clusters are causing the enhancement. Instead the value of thi s model l i e s i n i t s description of the switching behavior of.X from measuring a 'trapping' rate to measuring a 'localized' rate. The application of this model to both the saturated and unsaturated results i s very successful and sheds some l i g h t on part of the peculiar behavior exhibited at the low temperatures. In part i c u l a r i t suggests that the saturation assumed by some authors to be the onset of positron induced clusters i s nothing more than a chara c t e r i s t i c of the time spectra analysis, representing the fact that the slowest rate of those characteri-zing the phenomena i s measured. Considered i n th i s l i g h t the choice of selecting the 'saturation' point as r e f l e c t i n g l o c a l i z a t i o n or electro-s t r i c t i o n i s inappropriate. More importantly, the prediction that hundreds of atoms are involved i n the l o c a l i z a t i o n i s incorrect. What i s needed i s a complete d i f f u s i o n analysis which considers the information i i n the low temperature peak. 5.4 Localized Annihilation Rate The growth model suggests that X g i s measuring the loc a l i z e d annihilation rate, X , for D>30 amagats. Of some relevance i s the fact that the value of X i s less than that of the annihilation rate for either the positronium atom (spin averaged, 2.0 ns "*") or the positronium ion (3.0 ns - - 1) . This would be consistent with the positron being i n a free ' l i q u i d ' state ( i n the cluster) rather than i n a positron-argon bound state for which the positronium ion annihilation rate i s expected to be a lower l i m i t ( F e r r e l l (1956)). This could be s i g n i f i c a n t i n terms of 173 denying the existence of t h i s positron-argon bound system. The existence of the X component, however, i s important for i n i t i a t i n g e l e c t r o s t r i c t i o n of additional gas atoms about the positron plus cluster. However i t i s not clear whether the decreasing dependence of X on Ag r e f l e c t s the Ag dependence of the e l e c t r o s t r i c t i v e - c l u s t e r environment or the s t a b i l i t y of the self-nucleating clusters. 5.5 Diffusion Equation and Many-Body Considerations The need for a complete d i f f u s i o n analysis involving both the vel o c i t y and time dependence of the positron, inportant for a complete description of the low temperature phenomena, points out the value of the cluster model for the lower gas densities (small cluster concentrations) where the impurity-equation (1-9) can be used. In this case the low temperature peak information may be exploited. On the other hand the cluster theory i t s e l f needs to be checked for i t s v a l i d i t y and more general many-body considerations (Monte Carlo, molecular dynamics) need to be applied. Here the Green's function method mentioned i n Section 1.2.2.2.1 may prove useful. In general the theoretical calculations become quite involved as soon as the gas structure becomes important. In addition, the degree to which this basic structure i s influenced by the positron i s unknown. However, the application of the cluster theory i n an improved form for the description of small clusters (10 atoms) and consideration of the positron-cluster interaction may y i e l d t h i s information. 174 5.6 Effect of an -Electric F i e l d Two observations can be made concerning the effect of an e l e c t r i c f i e l d on a positron i n argon gas. The f i r s t i s that a shoulder s t i l l persists i n the time spectra, as noted i n Figure 37, even for values of E/D which remove the shoulder for the room temperature gas. This implies that a very large energy loss mechanism i s acting i n the low-temperature, high-density gas, a mechanism that only affects thermal  positrons since the times to thermalize remain inversely proportional to density, with the same proportionality factor as i s found for the room temperature gas. Both the thermodynamic dependence (predicting small clusters, Section 4.3) and the low energy aspect of the phenomena are strong evidence for the existence of clusters i n the gas independent of the presence of the positron. The strongly v e l o c i t y dependent trapping rate then required to s a t i s f y the fact that t h i s trapping behavior depends mainly on the low energy positrons i s then not unreasonable, i n i t s e l f . One could argue, on the other hand, however, that the fact that epithermal positrons are not trapped i s evidence that the clusters develop about the positron. In t h i s case i t would appear; that the size of clusters, 3 to 8 atoms, i s incompatible with what would be expected (^  100 atoms per cluster) from fluctuon theory (especially i n view of the lack of experimental evidence for a single-atom bound state) or what other authors predict for the size of the clusters on the basis of simple positron-induced cluster theories. The second observation i s the loss of the Ag dependence i n X^ /D for small Ag and moderate e l e c t r i c f i e l d s less than about 15 V cm ^ 175 amagat X (in the screening region) while generally preserving the -1 -1 overall Ag-dependence: v i z . exponential region for Ag>15 V cm amagat and linear region for Ag<15 V cm ~~ amagat ^. The implied picture i s that of a positron being accelerated i n an unchanged gas environment. Thus the main effect of the applied f i e l d i s to increase the v e l o c i t y of the positron between c o l l i s i o n s with the result that the positron i s taken out of the v e l o c i t y range for which trapping i s energetically possible or for which a large annihilation rate exists due to the presence of the clusters. The persistence of a s l i g h t l y enhanced annihilation rate i s then more compatible with self-nucleating clusters than a v e l o c i t y -dependent, positron-induced clustering model. 176 CHAPTER SIX CONCLUSIONS 6 .1 Summary of Results The equilibrium annihilation rate of positrons i n unsaturated, saturated and l i q u i d argon has been measured with and without an applied e l e c t r i c f i e l d and considered as a function of the difference in chemical potential between the gas state and metastable l i q u i d (at the same temperature and pressure), Application of a simple growth model consisting of a free and loc a l i z e d population of positrons implies that the abnormal behavior at the higher densities - that i s the turnover of the annihilation rate with density - occurs simply because the measure-ments determine the slowest rate associated with the enhancement mechanism. The low density results were considered i n terms of a density dependent phenomenological model and a thermodynamic cluster model involving s e l f - nucleating clusters. Both models are consistent with the picture of a positron interacting with small clusters (clusters of less than 20 atoms) at a rate which i s dominated by capture into these clusters. In addition i t has been shown that the thermodynamic predictions of the cluster f r e -quency i s consistent with the experimental results. The effect i n the low temperature gas of an e l e c t r i c f i e l d of a magnitude adequate to remove the shoulder i n the room temperature spectra i s only that of transforming a low temperature spectrum to one quite similar to the zero- f i e l d room temperature spectra. The shoulder persists and the equilibrium annihilation rate, A , i s near that for the room 177 temperature gas. This implies that a large energy loss mechanism characterizes those positrons with thermal energies much less than room temperature. In addition the coarse dependence of A^ on the chemical potential also persists i n the e l e c t r i c f i e l d data, the main difference being the expected overall reduction i n the anni-h i l a t i o n rate due to the increased v e l o c i t y of the positron. Although the dependence on the chemical potential for these f i e l d results are not as well-defined as for zero f i e l d , the implication i s that the gas structure i s unaffected by the positron's vel o c i t y - that i s , the clusters are self-induced rather than ion-induced (or positron-induced). 6.2 Outline of Possible Future Studies For positrons i n low temperature argon, further experiments should await more extensive theoretical calculations i n which small clusters are considered as part of the single atom population. However, confirmation of the usefulness of thermodynamic potentials could be obtained by comparing the behavior of the annihilation rate as a function of the chemical potential for other noble gases, i n particular helium, and non-polar gases l i k e CH^. For higher densities, the switching of the annihilation rate from a localized rate to a capture rate with application of an e l e c t r i c f i e l d to the positrons may make further measurements with an e l e c t r i c f i e l d d i f f i c u l t to interpret at these densities. A more complete consideration of the low temperature peak to include d i f f u s i o n effects i s then warranted. In general, the region around the c r i t i c a l point appears to be un-178 Interesting i n terms of the positron l i f e t i m e . However, the Ortho-positronium l i f e t i m e i s quite sensitive to density fluctuations near the c r i t i c a l point i n helium and should be for other gases as w e l l . Of some interest i s the question of whether the reduced annihilation rate per atom for near l i q u i d and l i q u i d densities i s strongly temperature dependent. This would help to interpret the localized rates measured at high densities which could become of a non-localized nature at some high temperature and density. However, the high pressures (3000 psi) required at the higher temperatures makes these experiments d i f f i c u l t . The strongly enhanced annihilation rate for CH^ . (even for high temperatures, T/Tc = 1.5) which i s proportional to density for the lower densities i s similar to the behavior for saturated argon. Measure-ments of annihilation rates i n CH^ at higher temperatures would allow the d i s t i n c t i o n to be made between resonance and clusters as the enhancement mechanism for these gas regions. In general, the effect of ion-induced clusters has not been s a t i s f a c -t o r i l y answered. 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University Press, Cambridge. Zettlemoyer, A.C. (1969). "Nucleation". Marcel Dekker, Inc., New York. 184 APPENDIX A LIQUID NITROGEN FILLER SYSTEM The e l e c t r o n i c part of the l i q u i d nitrogen f i l l e r system i s shown i n figure 40. The system i s set up by varying the 2.5 K pot. to set 20 uA on the 100 uA monitor when the sense diode i s i n l i q u i d nitrogen and the 200 Q. pot. to set 80 uA when the diode i s out of the l i q u i d . Each dewar to be f i l l e d required both an upper and lower l e v e l sense diode as shown i n Figure 7. As the f i l l e r was to ac t i v a t e only when the lower diode followed the upper diode out of the evaporating l i q u i d , relays were used to handle t h i s l o g i c and also to switch the 24 V to the appropriate valve. This l o g i c , shown i n the i n s e r t of Figure 40 included the means to independently c o n t r o l the f i l l i n g of both the chamber dewar and a cold trap positioned over the top of the d i f f u s i o n pump which sustained the high vacuum around the gas chamber. Relays C and D i n the i n s e r t of fig u r e 40 were necessary to con t r o l t h i s independent operation and themselves were controled by the appropriate upper and lower relays (LI and Ul or L2 and U2 of the i n s e r t ) . NCl and NC2 activated the relays f o r the NC valves i n the two branches of the f i l l i n g l i n e s . Relay F simultaneously activated both the gas i n l e t valve (NC) from the regulated high pressure nitrogen gas cylinder and the vent value (NO). An i n i t i a l problem with the system was experienced due to cooling of the diodes when f i l l i n g . This was solved by placing the diodes inside 3/4" tu f n o l tubes and covering the top ends with cotton wool. 185 rfn 3.9k-5.6k' 2.7k 1.5k 560 2N706 22 :560 1.5k -VW-2N2219 2N2219 22 100 uA monitor 15k L2, U2' -24 V •10 +12 V o-680 I — W W -1N5227 3.6 V '4.7k RELAY LOGIC _I_ U 1 J L , T - L l T - C C I Dl D —-L2 - j -0.01 1.8k Hww—' 100 - W W 1 2N555 — — o -12 V 1N5232 200 • A W -l k 2.5k Level Sensor Refer to Figure 7 and This Figure for Relay Locations. 1N100 rftr Figure 40 Control C i r c u i t for L i q u i d Nitrogen Level Sensor 186 In addition some v a r i a b i l i t y of the relay c i r c u i t s (top of fi g u r e 40) was experienced because some of the 1/4 A fuses were found to have resistences of 1 to 2 ft's, which i s s i g n i f i c a n t compared to the 10 Q, load. In general, once the system was set up i t operated without problems. However, i t was found that the use of a power bin with a fan was h e l p f u l i n controling the heat produced by the power tran-s i s t o r s . Thus, e f f i c i e n t removal of heat from these t r a n s i s t o r s allowed for better s t a b i l i t y of the 20 uA and 80 uA settings mentioned above and resulted i n better response times to f i l l i n g . 187 APPENDIX B DATA The following data was obtained. Where runs were discarded or not analysed, the run number and reason (Section 3.3.1) i s given. The error on A /D i s not given for the l i q u i d and high density gas states (90 amagats). e D E/D A n n i h i l a t i o n X /D e Conf idei Run Temp. Density (V cm"1) Rate (ys 1 Level No. (K) (amagats) -1 amagat ( 1 0 0 _ 1 ns" 1) "IN (amagat ) (%) 10 130.8 23.2 0.0 54.4(2.8) 23.5(1.2) 17 11 119. 20. 0.0 81. (15.) 41. (7. ) 6 12 160.4 22.5 0.0 17.3(0.3) 7.67(0.13) 62 18 163.2 86.5 0.0 79.4(1.5) 0.918 35 19 154.0 88.5 0.0 98.3(1.2) 1.11 69 20 143.5 91.5 0.0 113.1(1.5) 1.24 42 23 142.6 92.0 0.0 116.0(1.5) 1.26 42 25 Shoulder i n l i q u i d run 26 I n s u f f i c i e n t l i q u i d nitrogen 29 chi-squared bad 30 shoulder i n l i q u i d nitrogen run 32 107.1 706. 0.0 165. (1.) 0.234 22 33 107.0 708. 0.0 163. (1.) 0.230 0. 34 chi-s< quared bad 37 125.3 624. 0.0 147. (3.) 0.36 2 .8 188 E/D X /D e Run T D (V cm"1 X& ( y s _ 1 C.L. No. (K) (amagats) amagat" 1) (100 ns 1 ) amagat 1 (%) 39 d i f f e r e n t i a l l i n e a r i t y 40 added to run 41 41 117.8 662. 0.0 155.(1.) 0.234 4 44 132.6 582. 0.0 142.(1.) 0.244 0.7 45 chi-squared bad 46 144.2 129.8 0.0 123.(3.) 0.948 0.8 47 144.2 130. 0.0 124.(3.) 0.954 0.8 48 added to run 49 49 144.1 487. 0.0 130.(1.) 0.267 15 50 147.9 445. 0.0 127.(1.) 0.285 41 54 shoulder i n l i q u i d run. 55 296.6 26.1 0.0 13.5(0.6) 5.17(0.18) 63 56 295.4 26.1 0.0 15.8(1.4) 6.06(0.5) 52 57 d i f f e r e n t i a l l i n e a r i t y 58 d i f f e r e n t i a l l i n e a r i t y 59 d i f f e r e n t i a l l i n e a r i t y 60 295.4 31.7 0.0 16.8(0.6) 5.29(0.19) 75 61 296.1 31.9 0.0 16.9(0.4) 5.30(0.13) 49 62 296.9 31.9 31.3 12.0(0.3) 3.76(0.09) 67 63 118.8 31.7 0.0 113.0(5.3) 35.6 (1.6) 26 189 E/D X /D e Run T D (V cm"1 X e (us 1 C.L, No. (K) (amagats) -1 amagat ) (100 ns" 1) -1 amagat (%) 64 116.9 28.4 0.0 104.0(8.2) 36.6(2.8) 75 65 114.9 25.3 0.0 74.9(2.6) 29.6(1.0) 25 66 112.0 21.3 0.0 78.2(3.7) 30.9(1.5) 34 67 No run number 68 large temperature v a r i a t i o n 69 124.0 42.2 0.0 119.2(5.0) 28.3(1.1) 85 70 123.7 41.5 11.0 69.4(2.7) 16.7(0.7) 26 71 123.5 40.8 17.8 41.1(1.0) 10.0(0.2) 74 72 123.5 40.8 41.8 21.2(0.6) 5.17(0.15) 0. 73 data l o s t i n computer 74 123.4 40.1 23.2 33.2(0.9) 8.28(0.22) 18 75 295.5 40.7 42.0 13.6(0.3) 3.34(0.07) 18 76 295.5 40.7 0.0 19.9(0.4) 4.89(0.09) 7 77 296.0 25.2 0.0 12.7(0.6) 5.06(0.22) 95 78 296.0 25.2 41.7 8.06(0.2) 3.20(0.08) 71 79 114.3 24.5 0.0 105.0(12.3) 42.9(4.9) 94 80 114.2 24.3 43.2 11.2(0.6) 4.61(0.26) 54 81 114.2 24.3 11.5 36.2(2.3) 14.9(0.9) 11 82 114.2 24.3 5.69 90.9(7.6) 37.4(3.1) 97 83 114.2 24.3 3.00 112.1(17) 46.2 (6.8) 26 84 114.2 24.3 0.0 98.1(12) 40.4(6.8) 85 6 190 E/D X /D Run T D (V cm 1 X& (us 1 C.L. No. (K) (amagats) amagat 1 ) (100 ns 1 ) amagat 1 (%) 85 114.6 24.8 14.7 29.7(1.4) 12.0 (0.5) 30 86 114.3 24.4 20.5 21.6(1.1) 8.86(0.4) 70 87 114.3 24.4 4.66 82.7 (6.8) 33.9 (2.8) 35 88 113.7 23.5 0.0 97.5(7.1) 41.5(3.1) 93 89 large temperature v a r i a t i o n 90 113.7 23.5 3.00 84.2(6.0) 35.6 (2.5) 12 91 113.7 23.5 5.00 77.2(3.6) 32.5 (1.7) 53 92 113.7 23.5 0.0 91.2(4.1) 38.5(1.8) 0.05 93 113.7 23.5 8.00 56.4(2.7) 23.8 (1.2) 0.0 94 108.6 17.2 0.0 65.8(5.0) 38.3(3.0) 0.0 95 108.6 17.2 7.89 35.2(1.7) 20.5(1.0) 30 96 108.7 17.3 0.0 73.2(5.4) 42.3(3.1) 73 97 108.7 17.3 0.0 73.8(5.6) 42.7 (3.3) 94 98 108.7 17.3 4.99 49.8(2.3) 28.8(1.3) 46 99 111.8 21.0 0.0 76.3(6.5) 36.4(3.2) 18 100 111.8 21.0 0.0 90.2(12) 43.0(4.2) 0.0 101 Run 99 + Run 100 102 Power supply problems (computer interface) 103 Vacuum leaking 104 119.1 32.2 0.0 109.(9) 33.9(2.0) 95 105 119.8 29.8 0.0 116.(10) 39.2(3.2) 25 E/D X /D e Run T D (V cm"1 X e (ys 1 C.L. No. (K) (amagats) -1. amagat ) (100 ns" 1) -1 amagat (%) 106 120.1 29.5 0.0 101. (7) 34.2 (2.3) 66 107 120.2 22.7 0.0 76.0(4.0) 33.4 (1.8) 10 108 120.2 16.8 0.0 47.4(1.8) 28.2 (1.2) 51 109 130.1 17.0 0.0 22.9(0.7) 13.5(0.5) 94 110 130.1 23.0 0.0 60.6(2.3) 26.3 16 111 130.1 29.9 0.0 89.2 (3.3) 29.8 (1.2) 29 112 130.1 40.0 0.0 103.6(4.9) 25.9 (1.3) 23 113 130.1 53.9 0.0 121.8(5.2) 22.6(1.0) 60 114 130.0 58.4 0.0 112. (4.5) 19.2(0.8) 20 115 139.1 95.7 0.0 122.8(3.8) 12.9(0.4) 3 116 139.3 . 96.5 0.0 125.7(4.5) 13.1(0.5) 8 117 140.1 32.8 0.0 70.6 (3.8) 21.5(0.9) 59 118 140.1 45.0 0.0 94.9 (3.5) 21.0(0.8) 0.7 119 140.1 54.5 0.0 102. (3.4) 18.8(0.6) 0.3 120 140.1 54.5 0.0 102. (3.0) 18.8(0.5) 42 121 140.1 22.9 0.0 29.1 (1.1) 12.8(0.5) 6 122 140.1 22.9 0.0 31.7 (1.4) 13.8(0.6) 31 123 140.1 22.9 15.0 20.3 (0.8) 8.90(0.4) 92 124 140.1 59.8 15.1 60.2 (1.6) 10.0(0.3) 74 125 140.1 60.0 15.0 60.6 (1.4) 10.1(0.3) 0.0 126 140.0 71.3 14.7 69.5 (1.8) 9.75(0.2) 0.5 127 140.0 71.3 7.85 94.8 (3.0) 13.3(0.4) 19 128 140.1 71.0 7.89 94.8 (3.0) 13.2(0.4) 51 192 X /D Run T D (V cm 1 X e (vs 1 C.L. No. (K) (amagats) amagat "*") (100 ns" 1) -1 amagat (%) 129 140.1 50.0 15.0 55.1(2.2) 11.0(0.4) 45 130 140.1 50.0 15.0 54.8(1.6) 10.9(0.3) 93 131 140.1 39.9 15.0 44.8(1.3) 11.2(0.3) 11 132 140.1 39.9 15.0 46.2(1.7) 11.6(0.4) 76 133 130.1 39.9 8.0 65.8(3.1) 16.5(0.8) 0. 134 140.1 30.0 15.0 33.8(1.2) 11.3(0.4) 65 135 140.1 30.0 15.0 .33.7(1.2) 11.2(0.4) 94 136 140.1 30.0 8.0 49.3(2.2) 16.5(0.7) 78 137 130.1 30.0 15.0 34.8(1.4) 11.6(0.5) 18 138 130.1 30.0 8.0 55.3(2.5) 18.4(0.8) 18 139 130.1 40.0 15.0 47.7(1.9) 11.9(0.4) 32 140 130.1 20.0 15.0 21.2(0.7) 10.6(0.4) 17 141 130.1 17.0 15.0 14.6(0.6) 8.63(0.3) 23 142 130.1 13.0 15.0 6.19(0.4) 4.77(0.3) 4 143 130.1 13.0 15.0 8.60(0.7) 6.61(0.6) 26 144 130.1 13.0 0.0 10.7(0.7) 8.24(0.5) 32 145 130.1 15.0 0.0 16.4(0.5) 10.9(0.3) 33 146 130.1 15.0 15.0 11.6(0.5) 7.70(0.3) 29 147 130.1 15.0 8.0 12.8(0.6) 8.48(0.3) 38 148 129.8 17.0 8.0 18.8(1.0) 11.1(0.6) 77 149 129.8 18.9 0.0 32.2(1.6) 17.1(0.9) 14 150 129.8 19.1 14.9 18.7(0.5) 9.84(0.3) 21 193 X /D Run T D (V cm -1 X e (ps 1 C.L. No. (K) (amagats) -1 N amagat ) (100 ns" 1) amagat 1 (%) 151 129.8 19.1 7.96 25.7(0.8) 13.5(0.4) 4 152 130.0 21.0 0.0 45.3(2.1) 21.5(1.0) 23 153 130.0 22.0 0.0 48.1(1-8) 21.9(0.8) 57 154 130.0 22.0 8.0 35.8(1-2) 16.3(0.6) 92 155 130.0 24.3 0.0 61.4(3.0) 25.2(1.2) 19 156 130.0 24.3 8.0 43.2(1.0) 17.7(0.6) 93 157 140.0 28.7 0.0 52.2(2.2) 18.2(0.8) 46 158 140.1 26.0 0.0 41.6(1.8) 16.0(0.7) 42 159 140.0 23.7 0.0 34.3(1.1) 14.5(0.5) 10 160 140.0 23.7 0.0 34.3(1.0) 14.5(0.5) 41 161 140.0 23.7 8.0 29.4(0.9) 12.4(0.4) 78 162 140.0 17.9 0.0 18.0(0.8) 10.0(0.4) 31 163 140.0 20.2 0.0 22.7(1.0) 11.2(0.5) 1 164 120.1 18.0 0.0 55.8(6.9) 31.1(3.8) 16 165 120.0 14.3 0.0 35.2(2.1) 24.6(1.5) 49 166 120.0 12.5 0.0 16.1(1.1) 12.9(0.9) 74 167 120.0 12.5 0.0 16.5(0.7) 13.1 (0.6) 42 168 120.0 13.8 0.0 27.2 (2.8) 19.8 (2.2) 0.04 169 120.0 11.4 0.0 14.0(1.8) 12.2(1.6) 70 170 120.0 14.0 15.0 14.9(0.9) 10.7(0.6) 12 171 d i f f e r e n t i a l l i n e a r i t y 172 120.0 14.0 0.0 27.4(2.0) 19.5(1.4) 97 194 X /D e Run T D (V cm 1 X& (ys 1 C.L. No. (K) (amagats) amagat 1 ) (100 ns 1 ) amagat 1 (%) 173 120.0 17.7 15.0 21.4(1.D 12.1(0.6) 38 174 120.0 10.9 0.0 11.2(0.8) 10.3(0.7) 0 175 100.3 9.77 0.0 40.5(5.1) 41.5 (5.2) 74 176 large temperature v a r i a t i o n s 177 100.3 9.7 14.8 13.8(2.0) 14.0(2.0) 11 178 100.3 10.0 15.0 13.3(1.2) 13.3(1.2) 99 179 100.3 9.05 0.0 25.4(4.6) 28.1(5.1) 52 180 100.3 8.50 0.0 19.8(1.3) 23.4(1.5) 3 

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