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Electric field and temperature dependence of positron annihilation in argon gas Lee, Gregory Frank 1971

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ELECTRIC FIELD AND TEMPERATURE DEPENDENCE OF POSITRON ANNIHILATION IN ARGON GAS by GREGORY FRANK LEE B.Sc, University of British Columbia, 1966 M.Sc, University of British Columbia, 1969 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of PHYSICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA December, 1971 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia , I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study . I f u r t h e r agree t h a t pe rmiss ion f o r e x t e n s i v e copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department o r by h i s r e p r e s e n t a t i v e s . It i s understood that copying or p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l ga in s h a l l not be a l lowed without my w r i t t e n p e r m i s s i o n . Department of • PHYSICS The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada Date MARCH 8 , 1972 i i ABSTRACT The annihilation rate of positrons in argon has been measured over the temperature range of 135 K to 573 K vith applied d.c. electric fields up to 35 V cm ^ amagat ^. Analysis of this data vas carried out assuming that the annihilation rate and the momentum-transfer rate can be approx— imated by a function of the form Av over this velocity range. Results of this analysis indicate a strong velocity dependence in both the anni-hilation rate and the momentum-transfer rate. For velocities in atomic 2 —1 units (e /K) corresponding to positron wave number from .03 to .065 a Q the annihilation rate was found to be adequately represented by i^(v) = (1.195 x 10^)v"*^^ see -* amagat The momentum-transfer rate was similarly determined as ^ ( v ) = (2.11 x 10^)v sec ^ amagat ^. The errors in these results are velocity dependent. The largest error is 16$. This velocity dependence at low energies is not reproduced satisfac-to r i l y by any current theoretical models. A modified effective range parameterization of the momentum-transfer cross—section gives a scattering length of A Q = - 4.4 - .5 a Q for positron-argon collisions. No evidence of a velocity dependence in the orthopositronium quenching rate at room temperature was detected. The orthopositronium quenching rate vas measured at .255 - .009 -fsec * amagat \ in good agreement vith previous results. i i i TABLE OF CONTENTS page ABSTRACT i i LIST OF TABLES v i i LIST OF FIGURES v i i i ACKNOWLEDGMENTS x CHAPTER ONE POSITRON ANNIHILATION IN THE NOBLE GASES 1 1.1 Introduction 1 1.2 Positron-Atom Interactions 2 1.2.1 Elastic Scattering Existing Theoretical Situation Existing Experimental Situation 1.2.2 Non-Elastic Scattering Inelastic Scattering Direct Annihilation Positronium Formation Positron-Atom Complexes 1.3 Generalized Features of Positron Swarm Experiments . . . 10 1.3.1 Slowing of Positrons 1.3.2 Positron Velocity Distribution Interpretation of Time Spectrum Diffusion Equation 1.3.3 Existing Experimental Situation 1.4 Summary of Thesis Content 16 CHAPTER TWO EXPERIMENTAL PROCEDURE AND RESULTS 17 2.1 Outline of Experimental Techniques 17 2.1.1 Experimental Determination of Lifetime Spectrum 2.1.2 Analysis of Time Spectrum 2.2 Experimental Method 21 2.2.1 Chamber Configuration iv Chamber Design Electric Field Grid Source Construction and Strength Purifier Design and Operation 2.2.2 Parameter Control and Measurement High Temperatures Low Temperatures Pressure Electric Field 2.2.3 Electronics General Description System Linearity Integral Linearity Differential Linearity Time Resolution of System Pulse Pile-up and Effects on the Random Background 2.2.4 Analysis of Results Density Corrections Curve Fitting Techniques Presentation of Results 2.3.1 Acceptability of Data Random Background and Electronic Stability Convergence of Fitting Procedure Goodness-of-Fit Equipment Failure 2.3.2 Orthopositronium Quenching Rate Temperature Dependence 2.3.3 Direct Annihilation Rate Temperature Results Electric Field Results Error Analysis 2.4.1 Counting Statistics and Uncertainty in ^/D Electronic Stability V Temperature Gas Density Electric Field 2.4.2 Gas Purity CHAPTER THREE ANALYSIS OF RESULTS AND EXTRACTION OF CROSS-SECTIONS 68 3.1 Outline of Procedure 68 3.2 Calculation of Velocity Dependent Annihilation Rate. . . 69 3.2.1 Zero Field Thermal Positron Velocity Distribution 3.2.2 Determination of Velocity Dependent Annihilation Rate 3.3 Calculation of Velocity Dependent Momentum-Transfer Cross-Section 70 3.3.1 Thermalized Positron Velocity Distribution with Electric Field 3.3.2 Determination of Momentum-Transfer Cross-Section Comparison of Electric Field and Temperature Results Method of Determination Evaluation of Velocity Dependence 3.4 Velocity Dependence of Orthopositronium Quenching Rate . 81 CHAPTER FOUR DISCUSSION OF RESULTS 82 4.1 Annihilation Rate Dependence on Density 82 4.1.1 Theoretical Discussion 4.1.2 Experimental Evidence 4.2 Comparison to Published Theoretical Results 84 4.3 Modified Effective Range Theory 85 4.3.1 Description 4.3.2 Effect of Bound State 4.3.3 Calculation of Parameters from Results 4.4 Positron-Argon Complexes 88 4.5 Numerical Solution of Diffusion Equation 90 4.5.1 Technique of Solution vi 4.5.2 Use in Consistency Tests Application to Annihilation Rate Application to Momentum-Transfer Cross-Section CHAPTER FIVE CONCLUSIONS 96 5.1 Summary of Results 96 5.2 Outline of Possible Future Studies 96 BIBLIOGRAPHT 100 APPENDIX A ELECTRONIC PILE-UP EFFECTS 103 APPENDIX B ON THE FITTING OF DOUBLE EXPONENTIAL LIFETIME SPECTRA 109 APPENDIX C DENSITY CORRECTED DATA I l l APPENDIX D CALCULATION OF DERIVATIVES 117 APPENDIX E CALCULATION OF ANNIHILATION RATE FROM TEMPERATURE RESULTS 123 v i i LIST OP TABLES page TABLE I Temperature Dependence of Orthopositronium Quenching Rate 48 TABLE II Error Matrix Resulting from Fit to Zero Field Temperature Results 52 TABLE III Results of Fitting Electric Field Data 52 TABLE IV Error Matrix Resulting from Fit to Combined Electric Field and Temperature Results 74 TABLE V Values of Effective Range Parameters 88 v i i i LIST OF FIGURES Figure 1 Typical Time Spectrum page 18 Figure 2 Experimental Chamber 22 Figure 3 Chamber Configuration - External Features 24 Figure 4 High Voltage Electrical Feedthroughs 25 Figure 5 Electric Field Assembly 27 Figure 6 Effect of Purifier 29 Figure 7 Low Temperature Chamber Configuration 32 Figure 8 Photomultiplier Circuit 35 Figure 9 Electronic Configuration 36 Figure 10 Electronic Configuration for Linearity Measurements 37 Figure 11 Integral Linearity 39 Figure 12 Temperature Dependence of Orthopositronium Quenching Rate 49 Figure 13 Temperature Dependence of Direct Annihilation Rate at Zero Electric Field 51 Figure 14 Likelihood Function - hft) versus Temperature 54 Figure 15 Electric Field Dependence of Direct Annihilation Rate at T = 135 K 55 Figure 16 Electric Field Dependence of Direct Annihilation Rate at T = 209 K 56 Figure 17 Electric Field Dependence of Direct Annihilation Rate at T = 298 K 57 Figure 18 Electric Field Dependence of Direct Annihilation Rate at T = 365 K 58 Figure 19 Electric Field Dependence of Direct Annihilation Rate at T = 464 K 59 ix Figure 20 Electric Field Dependence of Direct Annihilation Rate at T = 573 K 60 Figure 21 Likelihood Function - Area Fitting of Double Exponential 62 Figure 22 Velocity Dependence of Annihilation Rate 71 2 Figure 23 Temperature Dependence of a(^/D)/d(E/D) 75 Figure 24 Likelihood Contour - d(;>/D)/d(E/D)2 76 Figure 25 Likelihood Contour - A/D 78 Figure 26 Velocity Dependence of Momentum-Transfer Cross-Section 79 Figure 27 Likelihood Function - d(/VD)/d(E/D)2 80 Figure 28 ?^ /D versus D for Several Temperatures 83 Figure 29 Effective Range Fits to Experimentally Obtained Momentum-Transfer Cross-Section 89 Figure 30 Temperature Dependence of Direct Annihilation Rate -Boltzmann Solution 92 Figure 31 Electric Field Dependence of Direct Annihilation Rate -Boltzmann Solution 94 Figure 32 Schematic Diagram of Gated and Ungated Time Spectrum 106 X ACKNOWLEDGMENTS « I wish to express my sincere appreciation to Dr. Garth Jones for his help over the past five years. He displayed keen interest in the progress of this work and was always readily available for consultation. Dr. Jones' ability and willingness to communicate his extensive knowledge physical insight, and exacting standards made the completion of this project immeasurably easier. At the completion of a Ph.D thesis I would be remiss i f I did not thank my parents for their encouragement over the many years leading to this degree. Most importantly, I wish to thank my wife, Fiona. Words are not sufficient to describe the importance of her contribution and I dedicate this thesis to her. 1 CHAPTER ONE POSITRON ANNIHILATION IN THE NOBLE GASES 1.1 Introduction Despite the fact that the positive particle predicted by Dirac's relativistic quantum theory of the electron, the positron, was discovered some 39 years ago (Anderson (1932)) the detailed nature of its interaction with gas atoms is s t i l l not well understood. One of the main reasons is that the experimental data concerning such interactions has been by necessity ambiguous. This ambiguity arises because, until mono-energetic positron beams are developed, the only experimental information available is obtained from swarm experiments using the decay of a nucleus, 22 usually Na , as a positron source. The experimental results so obtained are necessarily the averaged value over the positron velocity distribution and therefore the velocity dependence of either the annihilation rate or the momentum-transfer rate, the quantities which can be predicted by theory, is not easily determined. The theoretical understanding of positron-atom interactions is difficult because of the many-body nature of the problem. It might be expected, at first glance, that those theoretical models which have explained electron-atom scattering should also work for positrons, but experimental data has shown this to be untrue for many electron atoms such as argon (Orth and Jones (1969b), Montgomery and LaBahn (1970)). Helium, because of its simpler structure, is a more tractable problem and the results of theory and experiment are much closer (Lee et al. (1969)) but s t i l l not completely in agreement. This disagreement for 2 the case of positrons arises because the annihilation rate depends upon the positron-electron vavefunction overlap and therefore severely tests the validity of the approximate wavefunctions chosen. For many years the study of positron-atom interactions was concerned principally with the study of positronium formation. The suggestion of a bound positron-electron system was made by Mohorovicic (1934) and named by Ruark (1945). Deutsch (1951) established its existence while investigating the time dependence of the annihilation of positrons in gases. This technique of obtaining the time distribution of annihilation events has been the standard method used to investigate the positron-atom interaction (Tao et al. (1964), Paul (1964), Falk and Jones (1964)). The lifetime of thermalized free positrons in a noble gas at a particular temperature can be related, as will be demonstrated, to the velocity dependent annihilation rate. If a d.c. electric field is applied a dependence of the lifetime on the momentum-transfer rate is obtained. The experimental investigation of the lifetime of free positrons as a function of electric field and temperature and the determination of momentum-transfer and annihilation cross-sections from this data forms the basis of this thesis. 1.2 Positron-Atom Interactions 1.2.1 Elastic Scattering A positron with an energy below that of the first excited state of the atoms of the host gas can undergo elastic collisions, annihilation, positronium formation, or formation of a molecular-like complex with the host atoms i f such exist. From the practical point of view the determination of momentum-transfer cross-sections for elastic collisions 3 and the annihilation rate are of importance since these parameters, integrated over the positron velocity distribution, determine the measured lifetime of the free positron when subjected to electric fields. 1 . 2 . 1 . 1 Existing Theoretical Situation In gases most of the theoretical effort has been directed toward" the calculation of elastic scattering cross-sections and annihilation rates in atomic hydrogen, helium, and argon. Such calculations are of interest because of the comparisons that may be made with electron-atom collisions. The most obvious difference is that the positron is a dis-tinguishable particle from the atomic electrons whereas for electron-atom scattering the indistinguishability must be taken into account. The average interaction of the undisturbed atom, the static atomic field, is repulsive for the positron and attractive for the electron. The polarization effects, on the other hand, are attractive in both cases. Hence these two potentials cancel in the case of positron-atom collisions and add for electron-atom collisions. It is therefore possible to obtain new information about the applicability of atomic collision theory by applying i t to positrons. In general a theoretical model with simplifying assumptions must be used because of the complexity of the many-body problem. The calculation of the annihilation rate is a particularly good test of the validity of the approximated wavefunctions of a particular model since this parameter is a direct measure of the positron-electron wavefunction overlap. Of the noble gases helium is the one which offers the most hope of an "exact" solution. With the exception of atomic hydrogen, which 4 is experimentally impractical, helium has received the most careful study. A review article by Fraser (1968) outlines the various approx-imations used in the calculations. Of the current theoretical models the results of Drachman (1966, 1968) appear to be most consistent with experiment (Lee et al. (1969)). Theoretical calculations in argon are very dependent upon the approximations employed. Early calculations took the polarization effect into account using the Holtzmark polarization potential in a manner similar to that used successfully in electron-argon scattering (Orth and Jones (1966), Massey (1967)). Such calculations failed, however, to yield results consistent with experimental annihilation rates, presumably because of their neglect of positron-electron correla-tion effects within the atom. • More recent calculations have used a polarized orbital method which approximates the distortion interaction with a dipole approximation (Montgomery and LaBahn (1970)) and which has been used successfully for electron scattering from the noble gases (LaBahn and Callaway (1964), Thompson (1966)). This method was also unsuccessful in reproducing the experimental results. The theoretical model which best fits existing experimental data is that of Hewson et al. (to be published) which takes into account the enhancement of the electron density at the positron position. The theoretical—experimental agreement is good for room temperature electric field results but does not agree vith the experimental deter-mination of temperature dependence. One can in principle parameterize the positron-atom momentum-transfer cross-section at low energy using the modified effective range theory 5 (E.R.T.) of O'Malley et a l . (1962). This theory describes the scattering of a charged particle by a neutral polarizable system which thus contains an attractive l / r ^ polarization potential explicitly. For the case of single channel elastic scattering the modified E.R.T. yields a momentum-transfer cross—section that contains a term linear in k (the positron wave number in units a Q *) and higher order terms, the lowest of which 2 is proportional to k ln(k). For the case of a bound positron-atom system the normal effective range theory does not require modification (O'Malley et a l . (1961)) and the cross-section expansion retains terms only in k^. The modified E.R.T. has been used for the case of electron-atom scattering to extrapolate total and momentum—transfer cross—sections' to zero energy with only moderate success. The measured values of the momentum-transfer cross-sections from diffusion experiments are incon-sistent with the cross-sections obtained by extrapolating the total cross-section to zero energy (O'Malley (1963), Golden (1966)). Several suggestions have been offered to explain such inconsistencies including a pressure effect on the long range polarization (O'Malley (1963)) or the difficulty of extracting diffusion cross-sections from swarm experiments (Golden (1966)). To date the modified effective range theory has not been applied to the case of positrons in the noble gases since the velocity dependent annihilation rate has not been available and consequently comparison with experiment impossible. Existing Experimental Situation Experimental estimates of the momentum-transfer cross-section 6 for positrons in gases have been obtained in two vays each of which is applicable to a different velocity range. By measurement of positron-ium formation at high electric fields the momentum-transfer cross-section in the neighbourhood of the positronium formation threshold of the host gas can be obtained (Teutsch and Hughes (1956)). The r e l i a b i l i t y of published values (Teutsch and Hughes (1956)) based upon this method is now questioned since the original measurements in helium (Marder et a l . (1956)) have been found to be in error (Lee et a l . (1969), Leung and Paul (1969), Albrecht and Jones (private communication - 1971)) and there is also disagreement in the rate of positronium formation increase in argon (Orth (1966)). A second method of obtaining an estimate for the momentum-transfer cross-section vas described by Orth and Jones (1969b). By assuming a constant momentum-transfer rate and using the results of both temp-erature and electric f i e l d measurements they obtained a rough estimate for the momentum-transfer cross-section in argon at thermal velocities. 1.2.2 Non-Elastic Processes Inelastic Scattering Positrons vith energy greater than 500 eV almost exclusively undergo inelastic scattering (ionizing collisions) and consequential rapid energy loss. Betveen this energy and the f i r s t excited state of the gas atom the probability of inelastic and elastic collisions is comparable. Belov the f i r s t excited state elastic collisions are, excluding positronium formation, the only type possible (Green and Lee (1964)). Direct Annihilation 7 During any collision with a gas atom the positron may annihilate with an atomic electron. Such processes are also therefore non-elastic events. The collision of a positron and an electron occurs in the 3S (triplet) spin state three quarters of the time and in the IS (singlet) state one quarter of the time. Dirac (1931) calculated the annihilation cross-section for annihilation from the singlet state. In the non-relativistic limit this "plane wave" approximation for the decay into two photons is (Y - 4irr ^ — s ~ o v where r is classical electron radius o v is relative velocity. The spin-averaged two-photon annihilation rate for a gas of atomic number Z is then A = Zirr ^ c/atom. Since the Dirac rate is calculated for plane waves the actual rate, taking into account Coulomb distortion, is usually significantly larger. It is convenient to express the true annihilation rate in the same form as that of the Dirac rate, with the Coulomb enhancement included in a Z term. Thus e i i X = Z „ .TTr c/atom f's eff o where ^eff ^ eP e n^ s o n ^ n e positron velocity. The triplet state is prevented by selection rules from decaying via two photons. The probability for two-photon decay compared to three-photon decay is 1115:1 (Ore and Powell (1949)). Hence the ratio of the cross-sections for two and three photon decays (spin averaged) is 372:1. Thus for'free positron annihilation three-photon decays are negligible. 8 Positronium Formation An alternate non-elastic process is the formation of positronium. The energetics of positronium formation limits the energy range in which i t can be produced. As the binding energy of positronium is 6.78 eV, the minimum energy required by a positron in order to remove an electron from an atom with ionization energy E. is 6 , 7 ion E x u = E. - 6.8 eV. thr ion For energies above the ionization energy of the atom, , inelastic collisions will predominate because of phase space considerations. For energies between E. and the first excited state, E , excitation b ion exc collisions compete strongly with positronium formation and therefore most positronium formation occurs between E,, and E (Green and Lee r thr exc (1964)). This region is known as the "Ore gap" (Ore and Powell (1949)). Positronium in its ground state occurs in one of two spin config-urations; parapositronium, in which the spins of the positron and electron couple to give total spin equal to zero, and orthopositronium in which total spin is one. Formation of positronium in its ground state appears to follow the statistical weighting expected; the ratio of orthopositron-ium: parapositronium is 3:1. Formation in an excited state is unlikely (Deutsch (1953)) and has not been observed (Brock and Streib (1958), Bennett et al. (1961), Duff and Heymann (1963)). Since, as mentioned previously, free orthopositronium can only annihilate by three photons its mean lifetime is 1115 times larger than that of parapositronium. Ore and Powell (1949) calculated the values -7 -10 to be 1.39 x 10 and 1.25 x 10 seconds respectively. The most recent measurement of the free orthopositronium lifetime is that of Beers and 9 Hughes (1968) in which they obtained (1.37 - .01) x 10 seconds. In addition to the free three-photon decay, orthopositronium can decay via two-photons if i t is quenched by interaction with a gas atom. There are several quenching mechanisms possible and these are well summarized by Fraser (1968). The most important of these in the noble gases is "pick-off quenching". The positron of ortho-positronium may, during a collision, annihilate i f i t finds itself at the location of an atomic electron with which i t can form a singlet state. This effect depends both on the density of scatterers and also the energy of the collision since the degree of penetration of the orthopositronium and the atom determines the annihilation prob-ability. Positron-Atom Complexes Measurement of the annihilation rates in hydrocarbon gases and diatomic molecules (Paul and Saint-Pierre (1963), Osmon (1965)) have yielded values of UP "to several hundred times the Dirac value. The interpretation of results of this magnitude in terms of free positron annihilation is unreasonable. A tentative explanation is that a positron-molecule complex is formed which greatly enhances the probability of annihilation (Paul and Saint-Pierre (1963), Green and Tao (1963), Massey (1967)) or that a resonant state exists (Paul and Smith (1970)). The lifetime of such a system would be of the same order of parapositronium, that is 10 ^  seconds. One could expect such complexes, i f formed, to be formed at energies below the parapositronium formation threshold since at any higher energies positronium formation would likely dominate. 10 1.3 Generalized Features of Positron Svarm Experiments 1.3.1 Slowing of Positrons From the discussions of Section 1.2 i t is evident that a positron emitted into a gas with several hundred keV of energy will very rapidly lose energy by inelastic collisions and therefore the probability of annihilating in this time is small. The energy of emission from a 22 Na source, the positron source used for the work described in this thesis, is up to 542 keV with a maximum in the modified Fermi distribution at about 170 keV. Falk (1965) estimated the time taken for a positron to reach 10 keV in argon at 10 amagats as less than 0.7 nsec. From 10 keV to the last inelastic level is less well defined. Orth (1966) estimated 0.5 nsec as an upper limit under similar conditions. The elastic slowing down of positrons to thermal energies takes a longer time and i s comparable to the free annihilation lifetime (Tao et a l . (1963)). Thus a significant probability for annihilation prior to attaining thermal equilibrium results. These events in contrast to the annihilations occurring during the rapid inelastic energy loss are experimentally detectable and in fact give rise to a shoulder in the annihilation lifetime spectrum as f i r s t observed in argon in 1964 (Tao et a l . (1964), Paul (1964), Falk and Jones (1964)). The existence of the shoulder is evidence of the velocity dependence of the direct annihilation rate for positrons in the energy range 0.1 to 11 eV. During the time in which the positron is in the energy range of the "Ore gap" positronium can, of course, be formed. The positronium formation threshold in argon is 8.9 eV, only 2.7 eV below the last inelastic level, and therefore a l l positronium is formed within a short time, certainly 11 well within the shoulder width. (Tao et a l . (1963) estimated the formation time to be approximately 1.5 nsec.) The decay of parapositronium then would be experimentally indistinguishable from prompt annihilations in the source and walls for an experimental resolution of several nano-seconds. The orthopositronium decay on the other hand will obviously contribute to the annihilation time spectrum (Section Once the positrons have slowed to energies below the positronium formation threshold they can conceivably form positron-atom complexes i f such exist. The lifetime of such complexes is such that they annihilate well within the experimental resolution time by two—photon decay. This depopulation of the positron distribution would therefore be equal to the complex formation rate which would be linearly density dependent. These events would therefore be indistinguishable from direct annihilations. Those positrons which reach thermal equilibrium without annihilating or forming positronium or a positron—atom complex have a velocity distribu-tion characterized by the temperature of the host gas, the magnitude of an applied electric f i e l d , and the velocity dependences of the momentum-transfer and annihilation rates. It is the annihilations from this distribution that comprise the measured direct annihilation rate in positron swarm experiments. 1.3.2 Positron Velocity Distribution Interpretation of Time Spectrum The time spectrum which one attains in a positron annihilation experiment is built up of individually measured events. It i s , however, convenient to think of a l l the positrons as having been emitted at one time and existing simultaneously in the positron velocity distribution 12 from which they annihilation. It is customary to assume that these two approaches are equivalent. In the f i r s t case one can define the prob-ability at any time, t, of a positron having a velocity between v and v + dv. In the second case one can define the fraction of positrons that, at time t, have a velocity between v and v + dv. Properly normalized these definitions are equivalent. The one difference which arises is that in describing the positron velocity distribution positron-positron interactions can be neglected. For a weak source this is an obvious assumption in the f i r s t case but only holds in the second description i f the density of positrons is very much less than the density of the host gas (Lorentz Gas). Diffusion Equation For elastic scattering collisions the differential equation which describes the positron velocity distribution at time t has been shown to be (Falk (1965)) bf(v,t) 1-1 where f(v,t) probability density in velocity space a = — m acceleration of positron due to electric field e positron charge E applied electric f i e l d m mass of positron momentum-transfer rate for positron-atom collisions 13 ")) (v) = velocity dependent annihilation rate ^,(v) = positronium formation rate ^ M M = mass of scattering atoms T = temperature of host gas in K k = Boltzmann's constant. The velocity distribution of positrons is then proportional to v 2f(v,t) and the velocity averaged annihilation rate at time t, / ^ ( t ) , CO i s * i v e n b y jv(v)v 2f(v,t)dv Mt) = ° *„ . Jv 2f(v,t)dv o The time independent or steady state equation has been derived from 1-1 by Orth (1966). The resulting equation i s , neglecting positron-ium formation, ( * A y . ( v ) k T \ A W v f 0 [n^J + = J o ( ia ( v ,> - ^ 2 f ( v ) d v . . This equation can then be solved for f(v) either numerically (Orth (1966), Montgomery and LaBahn (1970)) or analytically for special cases. The velocity averaged, time independent, annihilation rate is given as before by 1-2. For the case of a velocity independent annihilation rate \\J (v) = }\ and the right hand side of 1-3 vanishes. In this case 1-3 can be solved by integration / v \ 1-3 a 2 , kT -dv 1-4 \ ° U V d < v ' ) 2 + mj / 14 This result is also approximately correct i f the integral on the right hand side of 1-3 is much smaller than the terms on the left hand side in which case the effect of removing positrons from the distribution does not appreciably affect the shape of the distribution. Numerical solutions of 1-3 for low electric fields support this approximation and this will be discussed further in Section 4.5. This approximate solution will be used extensively throughout this work. If the applied electric field is zero i t is readily seen from 1-4 that the positron velocity distribution, and hence the average annihilation rate as given by 1-2, does not depend on the momentum-transfer cross-section but only on temperature. That is For the case of a constant momentum-transfer cross-section, that is no dependence of V, on velocity, 1-4 can be solved again to get 1-5 This equation can therefore be solved for V ( v ) i f a functional dependence of A on T can be obtained. 1-6 which can be written as where 1-7 15 is the effective temperature of the steady state Boltzmann distribution. Thus for constant momentum-transfer cross-sections the effect of an applied d.c. electric f i e l d is to raise the effective temperature of the positron velocity distribution. The exact solution of 1-3 that does not neglect the effect of anni-hilations must be obtained by numerical integration. For such an inte-gration the values of ^ ( v ) and V ( v ) , assuming no positronium formation, must be supplied either by theory or experiment. The techniques involved in such numerical solutions are further described in Section 4.5. It is because the experimentally determined value, ^ , cannot be obtained from X(v) and V(v) without solving 1-3 and subsequently 1-2 Q a that the comparison between theory and experiment is d i f f i c u l t . In order to obtain estimates of i/(v) and V(v) from experimental data d a i t is convenient to neglect the effect of annihilations on the shape of the positron velocity distribution. This assumption is made exten-sively throughout this work. The validity of this approximation wi l l be tested in Section 4.5. 1.3.3 Existing Experimental Situation Many swarm experiments in the noble gases have been performed in the last seven or eight years but until the temperature dependence of the annihilation rate in argon vas measured by Miller et a l . (1968) there vas no method of extracting an unambiguous o r ^**v) from the results. Early vorkers could only test the prediction of theoretical models and perhaps give an averaged annihilation rate over the therraalized positron velocity distribution. Although d.c. electric fields could change this distribution, unless the ^(v) vere knovn the resulting 16 shape of the distribution was unknown. The temperature measurements of Miller et a l . (1968) used the approximate solution 1-5 to obtain the functional dependence of V(v). Following this result Orth and Jones (1969b) obtained an estimate of the momentum-transfer cross-section in argon at thermal velocities by using the approximation inherent in 1-7. They assumed that a small electric f i e l d had the equivalent effect of a small temperature increase and by comparing temperature and electric f i e l d measurements obtained a value f o r l ^ ( v ) . With the exception of the value near E^^, inferred by Marder et a l . (1956) from measurements of positronium formation enhancement (Section, no other exper-imental values of the momentum-transfer cross-section are available. 1.4 Summary of Thesis Content In terms of the current experimental tests of theory (Section 1.3.3) i t is possible for two widely differing theoretical models to both f i t the experimental data. This could occur because the measured annihilation lifetime is averaged over the velocity distribution and therefore in principle two differing models could give two different values of which result in the same averaged annihilation rate. This thesis describes a research program in which the combined electric f i e l d and temperature dependence of the annihilation of positrons in argon was measured. Based upon these results, the velocity dependences of the momentum-transfer cross-section and annihilation cross-section in the region of thermal velocities are separately determined. The determination of these cross-sections provides criteria upon which the correctness of any theoretical model may be judged. 17 CHAPTER TWO EXPERIMENTAL PROCEDURE AND RESULTS 2.1 Outline of Experimental Techniques 2.1.1 Experimental Determination of Lifetime Spectrum The experimental determination of the lifetime of positrons is generally performed by measuring the time interval between a suitable "birth" signal, indicating the generation of a positron, and a "death" signal, indicating annihilation of the positron. The beta decay of 22 22 -11 Na to the f i r s t excited state of Ne is followed within 10 seconds by a 1.28 MeV nuclear gamma ray. This signal is used to indicate the birth of a positron. The annihilation, either by two or three photon decay, is signaled by the detection of the characteristic radiation. These time intervals are then stored as events in a composite time s p e G t r u m . Each spectrum is taken at a fixed temperature and applied electric f i e l d . 2.1.2 Analysis of Time Spectrum A typical time spectrum is shown in Figure 1. The characteristics of such a spectrum can be divided into four areas. The f i r s t of these is the negative time region characterized by the presence of a random coincidence background. Operation with a portion of the time spectrum in the negative time region is obtolned. by electronically delaying the stop signal so that random coincidence events, in which the start signal is preceded by a stop signal, are recorded. The random coincidence background which is the sole component in this region occurs at a constant level throughout the rest of the spectrum as well. 18 10,000 . Prompt Peak RUN NUMBER 39 DENSITY = 9.10 atnagats — = 0 V/cm-amagat D 1,000 • ". , Shoulder • « • i « • • « * 4 • \ Double Exponential 100 \ 1 • t a 1 • • • o o « » Random . .• • ' Background * • • 10 • • 100 200 300 400 CHANNEL Figure 1 Typical Time Spectrum 19 The second region is the "prompt peak" which occurs at zero relative time. This peak corresponds to the detection of annihilations of the positrons within the source, the source holder, or chamber walls and also from parapositronium decay. The width of the prompt peak, defining the instrumental resolution for the experiment, was typically 5 nanoseconds. The third region is the "shoulder" area which, as mentioned earlier (Section 1.3.1), is due mainly to annihilations of free positrons before they reach thermal equilibrium in the gas and partly to ortho-positronium decay. The typical density-width product of the shoulder region in argon is 350 nsec amagats, where the width is defined as the time between the prompt peak and the estimated beginning of the double exponential decay. This feature of the spectrum is very sensitive to impurities (Paul (1964), Orth (1966)) and can therefore be used as an indication of gas purity. The fourth region is that characterized by annihilation of free positrons in thermal equilibrium with the host gas and by the decay of thermalized orthopositroniumai The removal of free positrons from the equilibrium distribution is described by 2-1 where > , is the direct annihilation rate d ^ is the positronium formation rate ^ c is the molecular complex formation rate i f such exists and N(t) is the total thermalized positron population integrated over the velocity distribution 20 solving + + ^ H N(t) = N(0)e a r C 2-2 or the observed rate for decay from the free positron distribution i s / \f x v -OV, + \ + A )t E f ( t ) . e 2 ( > d + ^ t ^ ) H ( o ) . a f • where £ is the detection efficiency for two-photon decay, taking into account both detector efficiency and solid angle, and i t is assumed that a l l parapositronium and any molecular complexes formed annihilate immediately. (Parapositronium lifetime is 1.25 x 10 seconds and can therefore be considered as depopulating the distribution directly.) For the low electric fields such as employed in this experiment no positronium formation occurs, so for this experiment + X)t Rf = ^ 2 ( \ + \)N(0)e C . 2-3 The orthopositronium population is given by dN (t) — 2 - — = - (V + h )N (t) + \ /Y,N(t) 2-4 dt o q o 4 r where }\ is the free orthopositronium decay rate }\ is the orthopositronium quenching rate. Again, for low electric fields, no positronium formation occurs from the thermalized positron distribution and therefore 2-4 can be written dN (t) -~-=- + K)N (t) dt o q o solving -(A + A )t N (t) = N (0)e ° q . 2-5 o o The observed rate from orthopositronium decay is therefore R (T) = GAN(t) + , N (t) 2-6 o 2 q o 3 o o where €^ is the detection efficiency for three-photon decays. 21 Combining 2-3 and 2-6 the observed annihilation spectrum from the thermalized components is R(t) = R.(t) + R (t) i o - \ t -At = V + V 2 " 7 where I. = '€ (Pi + />\ )N(0) 1 2 d c I. = (e* + U ) N ( O ) 2 2 q 3 o o A. = X + * 1 d c A- = >\ + A 2 o q It is assumed that for the low electric fields encountered in this experiment no positronium formation occurs from the free positron thermalized distribution, that i s , = 0. It is evident from 2-7 that i f molecular complexes are formed they would be experimentally indistinguishable from direct annihilations. For the purposes of this thesis the short-lived component obtalnted_from a two-component analysis of the equilibrium part of the spectrum has been designated, as the "direct rate" although i t would contain any molecular complex channels i f they exist. 2.2 Experimental Method C 2.2.1 Chamber Configuration Chamber Design The chamber used for this work was basically that used by Miller (1968) and is illustrated in Figure 2. Modifications to Miller's system were i . inclusion of a titanium gas purifier i i . provision for eleven high voltage feedthroughs in order to be able to externally bias the electric f i e l d grid 22 To pur i f ier O-Ring Grove 7" 6 g- Nominal O.D. I" 2 T 1" 1 Bolt Holes 17" 32 ' 2 4 i " To pur i f ier To accessories -3"-12 evenly spaced / n 0.18" 0.28" 3" q 4 Material : Stainless Steel Figure 2 Experimental Chamber 23 (Section i i i . an iron-constantan thermocouple. The external fixtures such as sample bottles, pressure gauges, and so forth were attached as shown in Figure 3. The pressure gauges were connected to the chamber by stainless steel tubing a meter long to ensure that the gauge readings would be unaffected by temperature changes. The chamber and purifier were hydraulically pressure tested to 500 p.s.i.g. at room temperature, sufficient to ensure its capability of withstanding 250 p.s.i.g. at 350° C. The high voltage and thermocouple feedthroughs were Ceramaseal type 800A0211-1 purchased from Ceramaseal Incorporated, New lork. In order to facilitate replacement of defective units these were held in place mechanically as shown in Figure 4 rather than being welded. Solder, of course, could not be used because of the high temperatures to which the chamber was raised. The limiting feature of these feed-throughs was their rather low breakdown voltage which although rated at 3.1 kV was in practice somewhat less. This restricted the size of the electric f i e l d that could be attained (Section The chamber was baked at 650 K under vacuum (estimated at less than 0.01 mm Hg) for forty-eight hours before f i l l i n g with 99.999^ ultra pure argon gas (Section 2.4.6). A l l pumping was done either by absorption pumping (roughing out of chamber) or by a conventional diffusion pump system. In the latter system a cold trap was employed in the vacuum line to reduce the backstreaming and contamination of the chamber by mechanical or diffusion pump o i l . Electric Field Grid 24 High V o l t a g e f o r F i e l d Rings H . V . i n 0 To chamber t e r m i n a l s " X " Sample B o t t l e Thermocouple Water "out M 0-110 v; Water i n • P u r i f i e r Heat ing C o i l P r e s s u r e Gauges Pumping L i n e JLL T0I F i l l i n g L i n e High V o l t a g e Termina ls 0-110 V Chamber Heat ing C o i l Figure 3 Chamber Configuration - External Features 25 ess P l u g Ceramaseal 800AD211-1 Ceramic Terminal Figure 4 High Voltage Electrical Feedthroughs 26 The large temperature variation to which the chamber was subjected in the experiment required that the electric f i e l d grid be externally biased, unlike previous f i e l d configurations (Falk (1965), Orth (1966)) in which the resistor chain was inside the chamber. The f i e l d rings j were 5T inches in outer diameter and were made of polished copper -g- inch 3 in cross-section. The spacers between rings consisted of •g- inch ceramic terminals, giving a center-to-center spacing of j inch on the rings. Ceramic spacers were chosen because of their good electrical insulating properties over a wide temperature range and their chemical stability at high temperatures (unlike Teflon which had been previously used). Figure 5 shows the electric f i e l d assembly. The end plates were -g- inch brass and the source ring consisted of a fine wire mesh similar to that of Falk (1965). As no part of the f i e l d assembly was internally grounded, the voltage gradient could be applied from end-to—end or, as in previous experiments (Falk (1965), Orth (1966)), from the center to each end. In order to attain a wider range of voltage gradients and also to check the consistency of operation both of these configurations were used (Section Source Construction and Strength The source, placed in the center of the electric f i e l d grid, was 22 composed of radioactive NaCl salt deposited on a single layer of 30 -<(inch nickel f o i l . Its strength, 15 -^Ci, was larger than that previously used (Falk (1965), Orth (1966), Miller (1968), Lee (1969)) but the disadvantages of increased random background were fel t to be outweighed by the advantages of a shorter counting time while doing a temperature controlled experiment. Nickel f o i l was used rather than 27 Figure 5 Electric Field Assembly 28 aluminum f o i l since i t was found that AlCl^ was formed when the radioactive salt was deposited, this compound subliming at 178° C. The result was a large hole where the source was originally situated. The somewhat higher atomic weight of the nickel f o i l increases slightly the fraction of positrons that annihilate in the source and consequently increases the random background and prompt peak. These effects were considered to be satisfactorily small. Purifier Design and Operation The use of the closed loop titanium purifier is illustrated in Figure 3. Previous work (Lee (1969)) on positron annihilation in helium had demonstrated the effectiveness of such a purifier for removing non-noble gases. The necessity of a thermocouple to measure the titanium temperature was also demonstrated since gas at high density is an efficient coolant. Before operating the purifier i t was f i r s t heated in vacuum to 450° C. Titanium absorbs significant amounts of hydrogen gas at room temperature and this is re-emitted at temperatures above 400 C. (Stout and Gibbons (1955)). No other gases are re-emitted once absorbed. In the steady state operation of the purifier the temp-erature was held constant at 650° C. Tests with nitrogen impurity added to the argon indicated that impurities were removed within twenty-four hours after turning on the purifier. Figure 6 compares a spectrum taken with 2fo nitrogen added to one taken after running the purifier for forty-eight hours. It can be seen that the increase in shoulder width, to the acceptable level of 350 nsec amagats, is evidence of the effectiveness of the purifier. As standard procedure during the experiment the purifier 29 85 . 1 • • RUN 51 6 .03 amagats CCOUNTS) ' ': i i 5 0 .0 V/cm-amagat 1.27 nsec /channel Shoulder w i d t h = 230 nsec amagats C D /—\ _ i • • • • • 100 150 CHANNEL 200 250 17 1 • • • RUN 57 5 .38 amagats 0 .0 V/cm-amagat Zn *• 1.27 nsec /channel t — \ 70 Shoulder w i d t h = o o •* . . . . . . . ..1 360 nsec amagats - *• _1 •... 50 100 CHANNEL 150 200 Figure 6 Effect of Purifier Top spectrum taken vith added N2» Bottom spectrum taken after purification. 30 vas turned on for a minimum of forty-eight hours whenever new gas was added to the chamber or whenever the temperature of the chamber was increased. 2.2.2 Parameter Control and Measurement High Temperatures The chamber was heated with a coil of #22 nichrome wire wrapped around the central area and covered with asbestos. The whole chamber was then wrapped in aluminum f o i l . The a.e. heating current was drawn, via a variac, from the regulated voltage lines to reduce fluctuations. It was found that once a steady state was attained the subsequent temperature variation was less than 5° C. depending mainly upon the changes in room temperature. Since this degree of stability was tolerable no further control was used. The temperature was measured using an iron-constantan thermocouple and a Hewlett-Packard Model 425A micro-voltmeter. The reference point for the thermocouple was an ice bath at 0° C. Tests of the temperature measurement system at liquid nitrogen temperature and at the boiling point of water indicated that the error in the system was dominated by the uncertainty in reading the voltmeter. Depending upon the temperature, and therefore the voltage range, this uncertainty varied from less than 1° C. at temperatures near 0° C. to 5° C. at temperatures above 190° C. The output of the voltmeter vas connected to a chart recorder to facilitate continual monitoring of the temperature fluctuations during a run. Low Temperatures The method used, to attain low temperatures involved a simple control 31 system designed to cool the chamber by adding a small amount of liquid nitrogen as required, followed by slow warming back to the temperature at which the liquid nitrogen control system was activated. The low temperature configuration is shown in Figure 7. The chamber was placed in a styrospan container lined with a layer of aluminized mylar. Stainless steel tubes were inserted into the container through which liquid nitrogen was forced. The liquid nitrogen flow was controlled by a voltage sensor circuit (Figure 7) attached to the thermocouple micro-voltmeter output. Although the sensitivity of this control system was less than 2° C. the intrinsic hysteresis of the pressurized liquid nitrogen system was such that the actual temperature variation was between 5° C. and 10° C. For safety reasons compressed nitrogen rather than compressed air was used to pressurize the liquid nitrogen system. The same thermocouple and micro-voltmeter were used for temperature measurement as for the high temperature case. Pressure The pressure of the gas contained within the chamber was measured using two test gauges. One, manufactured by Marsh, read from 0 to 150 p.s.i.g. (2$ mirror backed) and the other, manufactured by Ashexoft, read, from 0 to 300 p.s.i.g. (2fo mirror backed). Calibrations obtained using a dead weight tester (Civil Engineering Department, U.B.C.) enabled pressure to be obtained to an error of less than Vfo. In order to be able to neglect temperature effects on the gauges themselves connections to the chamber were made via one meter stainless steel tubing (-g- inch inner diameter) and the gauges were kept at room temperature. The pressure variation over a run because of lost gas was negligible 32 +24° R l +24V - i S o l e n o i d V a l v e SI S o l e n o i d 3 V a l v e S2 <&S2 2.2K { j! R l (see SN524A 0-1 v "2 gas si L i q . N 2 Thermal I n s u l a t i o n -V o l t m e t e r Thermocouple Input To c h a r t r e c o r d e r A l u m i n i z e d M y l a r S tyrospan LT w • I • . • • • « w ^ , M ; K - l t i » 7 ; Three l i q u i d N 2 l i n e s i n t o c o n t a i n e r Figure 7 Low Temperature Chamber Configuration 33 at high temperatures but was measurable at room temperature and below. The cause of such leaks is uncertain but was probably due to the large number of electrical feedthroughs and* the effect of thermal expansion. During such runs the pressure difference from start to finish, normalized for slight temperature variations, was always less than 3$. The pressure used to calculate the gas density was, of course, the average pressure over the run. Electric Field The electric f i e l d grid (Section was externally biased by a resistive chain consisting of ten 130 Kfrlfo wire-wound resistors. The voltage was supplied by a Northeast Scientific Corporation regulated high voltage supply capable of 5 kV at 10 mA. Previous workers had used only a line regulated supply (Falk (1965), Orth (1966), Lee (1969)) since they required much higher voltages. The voltage was measured using an Avometer (2% f u l l scale accuracy) in parallel with the resistor chain. The calibration of this meter with a Fluke 853A null-meter indicated the validity of this quoted accuracy. The uncertainty in the high voltage due to this measurement was therefore the 2fo error associated with the meter. Two configurations of the electric f i e l d were used in this experiment. In the f i r s t , the one used for most of the work, the highest voltage was put on the top plate of the electric f i e l d grid and the bottom plate was attached to ground. The source was then at an intermediate potential. Previous workers in this laboratory (Falk (1965), Orth (1966), Lee (1966)) had connected the source to high voltage and kept both end plates at ground. This latter configuration is capable of 34 yielding a higher electric field gradient since the distance is halved, high fields being of interest to the previous workers. For the low fields required in this experiment the f i r s t configuration was sufficient, simpler, and presumably yielded a field gradient of better uniformity. As a consistency check, however, the second configuration was used for some runs at room temperatures and higher fields. 2.2.3 Electronics General Description The data acquisition system employed a standard fast-slow coin-cidence circuit simi lar to that used by Orth (1966), Miller (1968), and Lee (1969). Modifications to the previous system (Lee (1969)) included the following. i . The RCA 7046 photomultipliers were replaced by Philips 58AVP photomultipliers. The biasing circuit for these is illustrated in Figure 8. i i . The home-made discriminators, pile-up-rejectors, and time-to-amplitude converters were replaced by standard fast commercial NIM units. The slow coincidence circuit remained the same as that used previously. Figure 9 outlines the electronic configuration used. System Linearity Integral Linearity The integral linearity was measured by the method of Taylor (1968) using a frequency generator and fixed delay. The complete system is shown in Figure 10. For a fixed delay, d, the time between two conseq-35 H.V, Slow o u t ' 50 250K 10M 10M 50K 50K 50K 50 K 50K 50K 50K 50K 50K 50K i 68pf 50K 500K JL .01 y 220pf lOpf Timing out 500K (J) — .01 =t 680pf • W v ' 500K CD " .01 " .002 cathode 91 g2 g3 s i s2 s3 s4 s5 s6 s7 s8 s9 slO acc s l l s l 2 s l 3 s l4 anode Two I neon V b u l b s I in series Figure 8 Photomultiplier C i r c u i t 36 P h i l i p s 58 AVP A = anode D = dynode P h i l i p s 58 AVP 0.51 SCA TAC P u l s e He ight A n a l y s e r C o i n . In P i l e u p R e j e c t o r 1.28 SCA P i l e u p Re jec tor Or A n t i c o i n . C o i n . C o i n . Figure 9 Electronic Configuration 37 V a r i a b l e O s c i l l a t o r +-• >-~i_r D i s c . Frequency Counter Delay I Stop S t a r t TAC P u l s e Height A n a l y s e r INTEGRAL LINEARITY Photo-m u l t i p l i e r D i s c . / 600 osec Gate [Generator! P u l s e r And S t a r t TAC Stop P u l s e Height A n a l y s e r DIFFERENTIAL LINEARITY Figure 10 Electronic Configuration for Linearity Measurements 38 utive peaks obtained on the pulse-height-analyser i s , as a function °ff~y- v=(*H-(H A plot of l / f versus channel number is given in Figure 11. The straight line fitted to the data corresponds to an integral linearity of the system of 1.27 - .01 nsec/channel. Differential Linearity The differential linearity was measured using the method of Falk et a l . (1965). The electronic configuration is shown in Figure 10. The start pulses are generated from nuclear radiations detected by a scintillation counter and are thus random in nature. The stop pulses, however, are produced by the pulser. For a completely linear system, the counts per channel accumulated are equal within statistics. Non-linearities to a level of %fo were determined by accumulating f i f t y thousand counts in each channel. The relative channel widths so determined were used in the data analysis. The differential linearity was essentially counstant (within %fo) over the region from channels 5 to 399 on the 400 channel pulse-height-analyser. Time Resolution of System The timing resolution of the electronic system, as measured by 22 the width of the positron lifetime spectrum characterizing Na in aluminum, was found to decrease, as might be expected, both for lower discriminator settings and for higher applied voltage to the photomulti-pliers. The procedure used to define these settings was as follows. 39 1/FREQUENCY (nsec) 650 550 450 A 1.27 nsec / channel 350 250 -150 .. i 1 100 200 300 400 CHANNEL F i g u r e 11 I n t e g r a l L i n e a r i t y P l o t o f f r e q u e n c y - 1 v e r s u s c h a n n e l number. 40 The discriminator setting was required to be above that corresponding to triggering by thermal noise in order to maintain a low random back-ground. Since the discriminators had only a small range of threshold settings (100 mV to 500 mV) this limited the photomultiplier H.T. to about 2400 volts. The best resolution obtained in this manner was 22 less than 4 nsec full-width-half-maximum for Na in aluminum. During the experimental runs, resolution was about 5 nsec since a background problem required that the discriminator level be set significantly higher than the noise (Section Pulse Pile-up and Effects on the Random Background The effect of a high count rate manifested itself in several ways. The f i r s t effect was that of time—slewing due to pile-up causing a baseline shift in the a.e. coupling between the photomultiplier and discriminator. This effect was also observed by Falk (1965) and i t was demonstrated that the use of a pile-up gate would eliminate the problem. The other major effect of high count rates is an increase in the random coincidence background. This effect arises for one of the following reasons. The f i r s t is the straightforward one of detection of uncorrelated events which happen to satisfy the energy window criteria and occur within the time interval spanned by the system. The probability of such random events can be shown to be constant in time for low count rates (less than 10^/sec) from the slow coincidence stop single channel analysers (Appendix A). For this experiment the count rates were low enough to satisfy this condition. The second random background effect is more troublesome since 41 i t leads to a non-uniform random coincidence time spectrum in which the random background on the negative time region of the spectrum is affected to a greater extent than on the positive time side. This effect is discussed in detail in Appendix A. Even i f such an effect causes a difference of only a few percent between the negative and positive time backgrounds, i t manifests it s e l f as an apparent increase in the orthopositronium quenching rate when the time spectrum is analysed assuming a constant background equal in both time regions. Use of a pile-up-rejector reduces this second effect only i f the pile-up-rejector has a deadtime that is short compared to the mean period between counts from the single-channel analyser. Miller (1968) observed anomalously low orthopositronium lifetimes which he interpreted as large quenching rates. Since he used a strong source and pile-up-rejectors whose deadtime was l-*<sec (Falk (1965)) the explanation for these results could easily be that just discussed. Miller's results = .39^ sec amagat ^) were similar to those obtained in this work before this problem was detected. For the much weaker sources used by Orth (1966) and Lee (1969) this effect was not observed. The inherent limitations of the operation of the pile-up-rejector illustrated in Figure 9 is the output pulse width of the discriminator. Pile—ups occuring within this time cannot be detected by the pile-up-rejector. For the updating discriminators used, the width is determined by the length of the input pulse from the photomultipliers. This pulse, originally in excess of 300 nsec, was shortened by differentiating the photomultiplier anode pulse through a 10 pf capacitor in series with the 50 JLload. In this way the length of the pulse at threshold 42 vas shortened to 150 nsec and consequently the limitations on the pile-up-rejector were reduced to 150 nsec. In addition, the count rate was reduced by raising the discriminator level slightly. This reduced the number of longer output pulses (and thus longer deadtimes) which are caused by the updating of the discriminator when a second pulse exceeds the triggering threshold within 150 nsec. The use of an a.e. coupled system could lead to time-slewing problems due to baseline shifts, as earlier mentioned. Falk (1965) demonstrated that the use of pile-up-rejectors removes this problem although he observed a prompt peak broadening at very high count rates even with the pile-up-rejectors in operation. At similar count rates no significant 22 broadening in the prompt peak of Na in aluminum was: observed in this experiment with the pile-up-rejectors in use. The effect noticed by Falk could easily be due to the rather long deadtime (l^sec) of his pile-up-rejectors. 2.2.4 Analysis of Results Density Corrections The density of argon differs from that calculated using the perfect gas law by an amount that depends on the temperature and the pressure. A complete tabulation (N.B.S. Circular 564 (1955)) of such corrections was used to determine the appropriate densities in this work. For the results presented here the perfect gas density was corrected by as much as 5$ and the resulting density was estimated to be accurate to within Vfo. Curve Fitting Techniques 4 3 Throughout this thesis a l l curves were fitted using maximum like-lihood techniques (Orear (1959)). Since the applicability of this method to Poisson and Gaussian statistics has been described by Orth et a l . (1968) only a brief summary will be given here. For Poisson statistics the probability of an experimental value N is Y Ne" Y P ~ N! where Y is the mean value. For Gaussian statistics 2 i 2 V t f J where (f^ is the variance associated with the experimental value N. The joint probability, or likelihood, of m specific values being attained is T „ = n V 1=1 If Y depends upon some undetermined parameters, a^, then those * values of a. which maximize L are the best values a. . l l Generally one maximizes the logarithm of L, V = InL, by solving the system of equations = 0 i = l,p 2-8 a ai where p is the number of parameters, a^, to be fitted. The solution of 2-8 is easily obtained i f I is a linear function of the a^. Otherwise iterative methods may be required. For the case where L is approximated by a multi-dimensional gaussian 44 in the neighbourhood of the maximum, the variances associated with the a. are 1 (a.. - a.*)(a - a *) = ( H _ 1 ) i j 2-9 where _ ^ V If a physical quantity X is a function of the a^, that i s , X = X(a.,a.,...a ) 1 2 m then the best value for X, X , and the uncertainty in X, AX, are given by X* = X(a.*) v 1 J 1 J V 2-10 The iterative method used to solve 2-8 was that of Orth et a l . (1968) in which the partial derivatives in 2-8 are Taylor expanded about the i n i t i a l (or best) f i t and the resulting simultaneous equations are solved for the f i r s t order corrections. The application of this program to the f i t t i n g of a function con-sisting of the sum of two exponentials plus a constant random background has been described by Orth et al . (1968). Essentially the same procedure was used in this work. The one difference was that for each exponential, instead of f i t t i n g to the intensity at zero time and the lifetime, the f i t was made to the area under the curve ( I T product) and the lifetime. This resulted in somewhat faster convergence due to the reduced degree of correlation between these variables (Appendix B). The f i t t i n g of electric field and temperature results also used the same basic maximum likelihood technique. A l l physical quantities determined from these results were calculated according to 2-10. 45 2.3 Presentation of Results 2.3.1 Acceptability of Data Appendix C contains a summary of the accepted data. There were a total of 160 runs taken of which 34 were discarded for various reasons. Acceptance was based on the following criteria. Random Background and Electronic Stability The i n i t i a l forty spectra indicated an anomalously short ortho-positronium lifetime. This was attributed to the background problem outlined in Section Once the corrective measures described in Section were taken, however, the orthopositronium quenching rate deduced from the data was found to be consistent with expectations. As the discriminator level was set near noise, this condition was very susceptible to small changes in the gain of the photomultiplier and' thus was dependent upon the stability of the H.T. supply. Occasionally such gain shifts did occur resulting in the generation of non-uniform random backgrounds. The positions of the 0.51 MeV and 1.28 MeV energy peaks were periodically monitored. When a shift in these indicated that the photomultiplier gain had varied, the data from the immediately preceding runs was neglected i f the orthopositronium quenching rate was statistically high compared to the rest of the experimental values for that temperature. Such shifts occured five times. As a result four single runs and one group of twelve runs were compromised. The electronics was readjusted before continuing. Convergence of Pitting Procedure No runs were accepted unless the spectrum-fitting procedure con-46 verged while varying a l l five parameters. Convergence could normally be attained in di f f i c u l t cases by f i r s t holding one or more parameters constant and so obtain better estimates for the remaining parameters before allowing simultaneous fi t t i n g to a l l five parameters. The start-ing channel for the f i t was normally chosen to be that point, no less than 5 channels away from the end of the shoulder, at which both con-vergence and an acceptable chi-squared (Section were obtained. If the starting position was too close to the shoulder the chi-squared goodness-of-fit was reduced even though convergence was attained. For starting positions further away from the shoulder the chi-squared values were comparable but the errors in the parameters increased. Comparison of the values of the parameters obtained from f i t s starting further away indicated that, within statistics, the choice of starting channel had no effect. For d i f f i c u l t cases the choice of starting position had l i t t l e effect on the rate of convergence. A total of thirteen runs were rejected on this basis and a l l of these were at low densities where convergence is known to be di f f i c u l t to attain (Orth (1966)). For these low densities even fixing the long lifetime did not result in good enough statistics to make this approach worthwhile. Goodness-of-Fit The chi-squared probability was calculated for a l l spectrum f i t s both for the complete data as well as after combining channels to get better statistics (Orth (1966)). Individual channel counts were normally sufficiently large that these two values were consistent. The criteria for acceptability of a run was that either of the chi-squared probabilities 47 described be greater than 0.1. Only two runs were rejected on this basis. The chi-squared was found to be essentially independent of the starting position provided this position was out of the region of the shoulder. Equipment Failure Two runs were rejected due to electrical and mechanical failure and one was rejected because of a temperature variation that was un-acceptable. 2.3.2 Orthopositronium Quenching Rate Temperature Dependence The statistical errors on the orthopositronium lifetimes were poor since the primary interest was the direct annihilation rate and hence the lifetime spectra were taken in a direct enhanced mode (Orth (1966)). That i s , the stop signal single channel analyser was centered about the 0.51 MeV peak. In order to obtain any significant information about the orthopositronium quenching rate, }\ , the theoretical value for the free orthopositronium rate of 7.2 x 10 sec 1 was assumed and the ortho-positronium lifetime, was fitted to the functional relationship A. = 7.2 + A D ^ s e c - 1 2-2 q for a l l values of a^ a particular temperature. No attempt was made to find a dependence upon electric field since no effect was expected and previous work (Orth (1966)) had failed to observe such an effect. The results of these f i t s are shown in Table I and Figure 12. From these i t can be seen that there is no statistical evidence for a temp-erature dependence in the orthopositronium quenching rate. The averaged 48 TABLE I Temperature Dependence of Orthopositronium Quenching Rate Temperature (K) Average Quenching Rate (fysec ^ amagat ^) Number of Point 573 .27 + .06 17 465 .18 + .14 9 367 .22 + .04 13 298 .255 + .009 48 209 .23 + .02 14 135 .24 + .08 8 298 .23 + .03 39 298 .23 + .02 22 49 Figure 12 Temperature Dependence of Orthopositronium Quenching Rate 50 results for the quenching rates obtained below room temperature were = .23 - .02 -fsec 1 amagat * and for temperatures above room temperature ^ = .23 - .03 isec 1 amagat \ The results for room temperature, ft = 0.255 - .009 "fsec - 1 amagat-1 are in excellent agreement with the recently reported results of Tao (1970) of ?\ = 0.255 - .015 Hsec"1 amagat-1. For the room temperature results there were enough data points to get a good f i t without constraining the free orthopositronium lifetime. Thus >\„ = /) + }\ D "fsec - 1 2-12 Z o q and the resulting f i t was ?\2 = (7.35 - .4) + (.235 - .05)D -Ysec-1 which is again in good agreement with previous results (Tao (1970), Beers and Hughes (1968)). 2.3.3 Direct Annihilation Hate Temperature Results The temperature dependence of the zero field direct annihilation rate is shown in Figure 13. The function fitted to these results was of the form > mb -1 .-1 o n ^ = aT -Ysec amagat 2-13 and the resulting best values obtained were a = 21.43 - 2.33 b = - 0.24 - .02. The error matrix obtained from this fit t i n g procedure is shown in Table II. This matrix is necessary in order to calculate results 51 200 300 400 500 600 TEMPERATURE (K) Figure 13 Temperature Dependence of Direct Annihilation Rate at Zero Electric Field 52 TABLE II Error Matrix Resulting from F i t to Zero Field Temperature Results a. _b a 5.44 -.0455 b -.0455 .000381 TABLE III Results of Fitting Electric Field Data Temp. (K) a o a x x 10 2 b 0 b x 10 2 b2 X io6 135 6.8 - .2 -.45 + .17 6.7 ± .2 -.46 + .16 1.8 + 1.2 209 5.80 i .08 -.40 + .05 5.80 - .08 -.52 + .10 4.5 + 2.0 298 5.35 - .03 -.28 + .05 5.35 i .03 -.27 + .03 1.0 + 0.2 367 5.32 - .16 -.23 + .05 5.33 - .16 -.27 + .08 1.0 + 0.8 465 5.10 i .18 -.21 + .06 5.10 - .20 -.24 + .13 0.8 + 1.7 573 4.89 - .11 -.20 + .09 4.90 - .11 -.23 + .11 1.0 + 1.2 5 3 using 2-10. The errors quoted, and the error matrix presented, are based upon the assumption that the likelihood function resulting from the f i t to 2-13 is a two dimensional gaussian in the region of the best values (Orear (1958)). That this is the case is demonstrated in Figure 14 where the relative likelihood is plotted for points near the best values obtained. The current results are slightly different from those of Miller (1968) in that the dependence upon temperature is somewhat less. This is attributable to the fact that this earlier work (Miller et a l . (1968)) did not take into account the density correction necessary at low temp-eratures (Section If in fact the current results are fitted using values of ^/D uncorrected for this density change the agreement with Miller is good. Electric Field Results The electric f i e l d results at various temperatures are shown in Figures 15 through 20. At each temperature the data was fitted to the power series expansions — = &Q + a^ ^  — J -*rsec amagat 2-14(a) "* D = b0 + bl (I) 2+ b2 (f)4 ^eC_1 aJnagat_1 2"14(b) depending upon the magnitude of the electric fields included in the data. A power series expansion was chosen because knowledge of the 2 derivative ^(tyD)/a(E/D) at zero f i e l d i s required in order to obtain values for the scattering cross-sections (Section 3.3). The results of this f i t t i n g procedure are summarized in Table III. It is seen that for low electric fields (E/D < 25 v cm 1 amagat 1 for T 54 i i • ' i a* - 6 a* - & a* a* + £ a * + <5 PARAMETER Figure 14 Dependence of Likelihood Function on Parameters Involved in F i t of tyl) versus Temperature One parameter was varied while the other was held at its best value. 55 10 20 30 40 — (V c m " 1 amagat" 1 ) Figure 15 Electric Field Dependence of Direct Annihilation Rate at T = 135 K 56 10 20 30 40 E D ^ (V c m " 1 a m a g a t - 1 ) Figure 16 Electric Field Dependence of Direct Annihilation Rate at T = 209 K 57 T = 298 K T. 5 A Or th (1966) ^ c u r r e n t work source a t mid p o t e n t i a l c u r r e n t work source a t h igh v o l t a g e 5.35 - .0028 / - \ 2 ( - = 5.35 - .0027 D I 2 + (1 .0 x l O " 6 ) I V 10 20 30 40 — (V c m - 1 a m a g a t - 1 ) Figure 17 Electric Field Dependence of Direct Annihilation Rate at T s 298 K 58 10 20 30 40 — (V c m " 1 amagat" 1 ) Figure 18 Electric Field Dependence of Direct Annihilation Rate at T = 365 K 59 T = 464 K 6L — i 1 1 1 10 20 30 40 — (V c m - 1 amagat" D Figure 19 Electric Field Dependence of Direct Annihilation Rate at T = 464 K 60 10 20 30 40 E -1 -1 — (V cm amagat ) Figure 20 Electric Field Dependence of Direct Annihilation Rate at T = 573 K 61 > 300 K and E/D < 20 V cm amagat for T < 300 K) the values of ^ so determined agree within statistics with the value of b^ obtained by f i t t i n g over a larger range of electric fields (E/D < 40 V cm 1 amagat-1). The difference in the electric f i e l d limits as a function of temperature arises since one expects, on theoretical grounds, that the higher order terms in the power series expansion have a greater effect at lower temperatures (Appendix D). It would thus appear that for low fields the expansion containing only the f i r s t term in (E/D) (Equation 2-14(a)) adequately represents the data. Comparison with previous results is shown in Figure 17 where the results of Orth (1966) are also plotted. In this figure the higher electric f i e l d results were, as indicated in the figure, taken with the central or source plane at high voltage and the end plates at ground (Section The results do not appear to depend upon the electric f i e l d configuration used and agreement with the results of Orth is excellent. 2.4 Error Analysis 2.4.1 Counting Statistics and Uncertainty in y^*D Standard deviations of the fitted parameters due to counting statis-tics were determined from the maximum likelihood f i t t i n g procedure (Equation 2-9) with the assumption that the likelihood function is gaussian in shape near the best values. This assumption was substantiated by Orth (1966) fitting similar data using an intensity-lifetime function. The validity of this assumption for the case of f i t t i n g an area-lifetime function (Section is demonstrated in Figure 21. The likelihood space is in fact more gaussian—like for the latter case, possibly 62 1 • i i i i a* - 26 a* - cr a* a* + a* + 26 PARAMETER Figure 21 Dependence of the Likelihood Function on Parameters Involved in Area Fitting of Double Exponential Each parameter was varied vith the other three held fixed at their best values. 63 reflecting the reduced degree of correlation between the parameters. A l l errors displayed in Figures 15 to 20 characterize such statistical f i t s to the data. Additional uncertainties in arising from electronic instability (Section and uncertainties in gas density (Section are approximately 3.2$, obtained by combining the RMS uncertainties in these parameters. As this uncertainty is less than that contributed by counting statistics (about 5$), the overall uncertainty obtained by combining these is comparable, that i s , less than 6fo. For this reason the uncertainties due to density and electronic stability were not considered when fit t i n g the electric f i e l d and temperature results of Section 2.3.3. In addition, the values of E/D and T were assumed to be well defined while the f i t s to 2-13 and 2-14 were carried out. The errors in E/D (Section and T (Section are small and are equivalent, for the dependencies obtained in 2-13 and 2-14, to a change in A/D of less than 1%. Electronic Stability As mentioned in Section electronic instability was responsible for the rejection of certain runs. During an acceptable run the effects of electronic instability on the timing were almost certainly less than the inherent resolution of the apparatus (5 nsec). No shift in the prompt peak by even as much as 1 channel (1.27 nsec) was noted during a run. Uncertainties in timing due to the integral and differential linearity are of the order of Yfa. The absolute error in the integral linearity 64 (Section is less than 0.8$ and the uncertainty in the d i f f -erential linearity is about 0.4$ (Section Since the d i f f -erential linearity is already a correction terra, the total uncertainty in timing due to linearity measurements is less than 1$. Temperature The temperature of the chamber was measured with an iron-constantan thermocouple and H.P. 425A micro-voltmeter (Section The uncertainty in temperature thus varied between 1° and 5° C. depending upon the temperature range. The major temperature uncertainty over a run was the inaccuracy due to temperature fluctuations, rather than measuring inaccuracy, and these fluctuations were within a maximum of 10° C. Thus the maximum error in temperature due to measurement was about 1.5$, whereas the total error in temperature over a run was a,bout 5$. Gas Density The gas density was determined from measurements of pressure and temperature by using the perfect gas law together with published correc-tions for the non-ideal nature of argon (Section The uncer-tainty in the gas density, obtained by combining the errors in pressure (Section, temperature (Section, and the correction, is therefore of the order of 2$. The major uncertainty, however, vas due to gas leakage during a run and amounted to a maximum of 3$ for any one run. As an upper limit on the error in gas density this value of 3$ is used. Electric Field 65 The spatial uniformity of the electric field within the chamber for a grid structure similar to the one used in this experiment was investigated by Falk (1965). He found that the non-uniformities were confined to a region immediately surrounding the field rings encompassing about 8$ of the volume. The uniformity of the field for the current geometry is assumed to be similar. The reproducibility of the E/D lifetime results over a range of densities is evidence of the unimpor-tance of this small non-uniformity since the number of positrons reaching the neighbourhood of the rings is pressure dependent. The accuracy of the electric field gradient depends in turn on the measurement accuracy of both the applied voltage and the spatial distances. For the grid assembly used, the distance between end-plates (5 inches) was known to y|- inch, or 1.3$. As the applied voltage was measured to 2$ accuracy (Section, the combined uncertainty is of the order of 2.3$. There is also some possible non-uniformity in the f i e l d arising from variations in the values of the bias resistors. The resistors, which are external to the chamber and at room temperature, are accurate to 1$. This effect and the non-uniformity introduced by variations in ring spacing are a perturbation on the average f i e l d as calculated above. For the relatively slight dependence of the positron lifetime with electric f i e l d , an upper estimate for the effect on the measured lifetime of the non-uniformity due to bias resistors and ring spacing is 1$. The combined error in the electric field is therefore less than 3$. The error in E/D obtained by combining the error in E and D is then 4.2$. 66 2.4.2 Gas Purity The gas purity was monitored by constant checking of the "shoulder width"-density product which has been found to depend strongly on the presence of impurities (Falk (1965), Paul (1964)). This product was found to be about 350 nsec amagats. The gas used to f i l l the chamber was Matheson Gold Label Ultra Pure, 99.999$ pure. The Matheson Company analysis of the gas was C0 2 < 1 ppm 0 2 < 5 ppm H 2 < 1 ppm CO < 1 ppm N2 < 5 ppm CH4 < 2 ppm H20 < 4 ppm. The chamber was outgassed under vacuum (0.01 mm Hg) for forty-eight hours at 650° K before f i l l i n g and the purifier (Section was used after every f i l l i n g and after every addition of gas since i t was f e l t that significant contamination could arise from the valves and f i l l i n g lines which were exposed to atmosphere between usage. Tests of the purifier (Section showed i t s effectiveness in this respect. In order to obtain a more quantitative estimate of the purity of the gas contained in the chamber during a series of runs, a sample bottle was attached to the chamber and used to obtain a gas sample for analysis. An analysis performed by Gollob Analytical Service Corporation of New Jersey (recommended by Matheson Company) indicated the impurity levels as 67 30 ppm 0 2 < 4 ppm ^\ C0 2 < 4 ppm H 2 < 4 ppm detectable limits He < 4 ppm Hydrocarbons < 0.5 ppm H20 < 20 ppm.^ / The impurity level of a l l gases, with the exception of nitrogen, was belov the detection limit. The small, but detectable amount of nitrogen gas, is di f f i c u l t to explain but could be due to outgassing of the sample bottle or chamber walls since nitrogen was added to the chamber for a test of the purifier at an earlier time. In support of this hypothesis, an earlier sample analysis indicated greater than 1000 ppm nitrogen yet the shoulder width was as good. This earlier sample had been obtained without f i r s t baking out the sample bottle and the large impurity level indicated is attributed to outgassing from i t s walls. That the results agree well with previous room temperature results (Orth (1966), Tao (1969)), particularly the agreement in the ortho-positronium quenching rate, corroborates the results of the quantita-tive analysis. 68 CHAPTER THREE ANALYSIS OP RESULTS AND EXTRACTION OF CROSS-SECTIONS 3.1 Outline of Procedure From the discussion of Section 1.3 i t is evident that the positron lifetime in argon depends upon the temperature of the gas, the electric f i e l d applied, and the velocity dependence of the momentum-transfer and annihilation cross-sections. The equation relating the measured annihilation rate, ?\(T), and the velocity dependent annihilation rate, V (v), (Equati on 1-5J is CD V(v)v 2f(v)dv A ( T ) = ^ - ^ . 3-B v 2f(v)dv Given V (v), the relationship (Equation 1-4) f(v) = Cexp X O \ 3 m A ^ mj tlv' 3-in principle outlines a method for obtaining the momentum-transfer rate, V^(v). The momentum-transfer cross-section can be parameterized and the f(v) of 3-2 written in terms of these parameters. This f(v), used in 3-1, in principle allows the experimental results, A(T), to be expressed in terms of these same parameters. In practice, however, an analytic solution to 3-1 and. 3-2 is not possible for non-zero electric fields. A more practical technique involves consideration of the quantity d(^yD)/d(E/D) at E = 0. An analytic solution of the resulting equations 69 is possible for certain parameterizations of and an equation relating cM?y*D)/d(E/D) at E = 0 with these parameters can be obtained. By f i t t i n g the set of experimental values at different temperatures, T, with this expression, the best values of these parameters and thus ^(v) can be obtained. This technique was the one adapted for analysis .of results in this thesis. 3.2 Calculation of Velocity Dependent Annihilation Rate 3.2.1 Zero Field Thermal Positron Velocity Distribution The zero f i e l d thermal positron velocity distribution, with the approximation that the velocity dependence of the annihilation rate does not appreciably affect the shape of the distribution, is given 2 by v f(v) where f(v) is given in 3-2 with E = 0. Thus \ 2kT J f(v) = C e x p l - ^ ] 3 -3 2 and v f(v) is recognized as the Maxwell-BoItzmann distribution at temp-erature T. 3 . 2 . 2 Determination of Velocity Dependent Annihilation Rate For a positron velocity distribution of the form 3 -3 the experiment-ally determined annihilation rate at temperature T is related to V(v) by U ., . 2 "2kT , \ y (vjv e dv ^(T) = ° a , 3 . 3 -4 ,2 If in 3—4 the experimental values of the annihilation rate are fitted by the functional form of 2-13, that is by — = aT *fsec amagat , 3 -5 70 then the solution of 3-4 for V(v) is readily obtained analytically Or (Appendix E) and is given by \ i / x a \Tfr / m \ b 2b -1 -1 V (v) = Y " : — v Hsec amagat 2r(|+b)i2k/ 3-6 where a, b are the best f i t values to 3-5 m = mass of positron k = Boltzmann1s constant. Hence from the experimental data y(v) can be determined over EL the velocity range encountered in the experiment. The result is model-independent but does depend on the assumptions that the annihilation rate does not affect the positron velocity distribution (Section and that the data can be adequately represented by the form of 2-13 over the velocity range involved. Using the results of Section for a and b, i A v ) = 1.195v *^^^Vsec - 1 amagat-1. 3-7 £L The error in V(v) is determined using 2-10 and is therefore velocity dependent. The result (Equation 3-7) with i t s associated errors (Section is displayed in Figure 22 where for comparison various theoretical results are also displayed. It is seen that the curves of Falk et a l . (1965) and Montgomery and LaBahn (3p-d) (1970) most closely reproduce the experimentally derived velocity dependence. The curve of Falk et a l . (1965) was not, however, the result of a theoretical calculation but was derived from a semi-empirical f i t of lifetime spectra. 3.3 Calculation of Velocity Dependent Momentum-Transfer Cross-Section 3.3.1 Thermalized Positron Velocity Distribution with Electric Field Again, the approximation is made that the annihilations do not affect 71 Figure 22 Velocity Dependence of Annihilation Rate 72 the shape of the positron velocity distribution. Hence the steady state 2 distribution is of the form v f with f given by 3-2. The validity of this approximation is checked in Section 3.3.2 Determination of Momentum-Transfer Cross-Section Comparison of Electric Field and Temperature Results Previous workers (Orth and Jones (1969b)) derived a value for the 2 momentum-transfer cross-section (C£ = 39tfa ) at v = 0.045 atomic units d o 2 (e /fi) by comparing the relative effects of temperature and electric f i e l d at room temperature. An obvious extension of this procedure involves comparison of electric f i e l d effects at various temperatures. Method of Determination The momentum-transfer cross-section calculation of Orth and Jones (1969b) was based upon the assumption that the momentum-transfer rate was essentially constant over the positron velocity distribution. This restriction is unnecessarily stringent. It is only necessary for the method described here to require that the momentum-transfer rate, ^j( v)> be adequately represented, over the velocity range of interest, by the B —1 —1 parameterization ^ j ( v ) = ^ v s e c amagat 3-8 The analysis technique used involves consideration of the effect of electric fields in the limit of zero f i e l d and thus the rate of change of the theoretical annihilation rate (Equation 3-1) with respect to the square of the electric f i e l d at E = 0 is calculated. That i s , d(§)2 ^ / jV(v)v 2f(v)dv E = o \ fa(v)dv / E = 0 73 where f(v) i s given by 3-2. This expression can be analytically evaluated using 3-7 and 3-8, yielding the result (Appendix D) Mi o i i n — 2e S /2kT 2 E = 0 3«k 2T 27TA 2(l - B) \ m "(^ rd—|)-r(f-!)r(!--)) « where A, B are the parameters defined by 3-8 S, R are the parameters from V(v) = Sv ^  of 3-7 EL e = charge of positron T = temperature (K) k = Boltzmann's constant H = reduced mass = — M m = mass of positron M = mass of host gas atom. This function (Equation 3-9) when fitted to the experimental results of Section yields the best values and errors of the parameters A and B and thus the magnitude and velocity dependence of V ( v ) . The momentum-transfer cross-section, cX(v), is thus given d a by x V,(v) 6^ - V where N is the number density of atoms. Evaluation of Velocity Dependence 2 The experimental values of <M/VD)/d(E/D) at E = 0 used in the f i t of 3-9 are the a^ listed in Table III. A plot of these values 3-10 74 as a function of temperature together with the corresponding maximum likelihood best f i t is given in Figure 23. The best f i t parameters to 3-9 are A = (2.11 - 1.43) x 10 1 1 3-11 B = - .49 - .24 2 for v of 3-9 in atomic units (e /ft). The rather large errors on these parameters are a direct result of the very strong correlation between them. The likelihood contour plot shown in Figure 24 gives visual evidence of the degree of this correlation. It is evident that although the range of uncertainty in either A or B is large the combined error applicable to (5^ is much less. The error matrix characterizing this f i t t i n g procedure is given in Table IV. TABLE TV Error Matrix Resulting from F i t to Equation 3-9 A B A 2.032 .3344 B .3344 .0553 The momentum-transfer cross-section given by 3-10 can now be written tf ( y ) = Av -rfa 2 3 - 1 2 d (5.171 x 1 0 U ) v 0 where v is in atomic units and A, B are given in 3-11. The uncertainty in ^ ( v ) is velocity dependent since the errors 75 Figure 23 Temperature Dependence of d(A/D)/8(E/D) 76 Figure 24 Likelihood Contour of Parameters from Fitting <B(VD)/d(E/D)2 as a Function of Temperature 77 are calculated using 2-10. In addition to these statistical errors resulting from the f i t to 3-9 there is additional uncertainty from the errors in S and R used in 3-9. These errors reflect the uncertainty in the f i t t i n g procedure to the temperature results of Section Again the errors in a and b of 2-13 are strongly correlated and the degree of correlation is demonstrated in Figure 25. Since the extremities in the allowable range for the parameters a and b are at points X and Y, the function 3-9 vas fitted for A and B using the values of S and R corresponding to these points. The uncertainty in C^(v) vas then taken as the largest of the deviations between the best value, (5^  (v), and the values obtained from the calculation of (5, (v) and (5, (v) and d d the corresponding range of uncertainty in each of these results. The best value, (5^  (v), and the resulting uncertainty is shown in Figure 26. The errors as calculated from 2-10 depend upon the assumption that the likelihood space is gaussian in the region of i t s maximum. The validity of this assumption was checked for the f i t to 3-5 and is demonstrated in Figure 27. It can be seen that the likelihood function closely approximates a gaussian for the parameter B but is somewhat skew for the parameter A. In order to check the effect this might have on the uncertainty in C^(v) the values of ^ ("v) at points A^ and were calculated. These points enclose an area of roughly 68$ of the skew curve and therefore are analogous to the definition of error for the gaussian case. The values B^ and B^ used to find C^(v) were those which maximized the likelihood at A^ and It is evident that the results, shown in Figure 26 and labelled 1 and 2, axe within the 78 - . 2 2 2 \ - . 2 2 8 \ V L i k e l i h o o d R e l a t i v e to \v\ Best Value = .4 - . 2 3 6 L i k e l i h o o d R e l a t i v e to w \ Best Value = .8 - . 2 4 4 \ - . 2 5 2 - . 2 6 0 I I I ! 18.63 20.50 22.36 24.23 26.09 a Figure 25 Likelihood Contour of Parameters from Fitting /^D as a Function of Temperature 79 Figure 26 Velocity Dependence of Momentum-Transfer Cross-Section Experimental result is darkened area. 80 a* - 16 a* a* + 2c5 a* + 4c5 PARAMETER Figure 27 Dependence of the Likelihood Function on Parameters 2 Involved in F i t of 5(VD)/c)(E/D) versus Temperature One parameter was varied while the other was held at its best value. 81 errors as quoted. From this i t can be assumed that the effects of deviation from the gaussian shapes of the likelihood function for parameter A can be neglected. For comparison, the results of published theoretical estimates are also displayed in Figure 26. 3.4 Velocity Dependence of Orthopositronium Quenching Rate The orthopositronium velocity distribution is presumed to be that of a Maxwell-BoItzmann distribution at temperature T. No change in the distribution occurs when electric fields are applied since ortho-positronium is electrically neutral. Since no change in the quenching rate is observed as the temperature is varied, the quenching rate over the velocity region corresponding to the temperature extremes is constant, at least within the errors shown in Figure 12. 82 CHAPTER FOUR DISCUSSION OF RESULTS 4.1 Annihilation Rate Dependence on Density 4.1.1 Theoretical Discussion The assumption that has been made throughout this thesis i s that the positron interacts with only one atom at a time and that the anni-hilation and scattering rates are therefore linearly dependent upon density. Thus the experimental annihilation rate,A(T)» is A = AjD. If such was not the case, for example at very high densities or low temperatures, then clearly A/D is no longer density independent and the analysis of Chapters Two and Three is invalid. 4.1.2 Experimental Evidence Near room temperature there is good experimental evidence for an annihilation rate linearly dependent upon density up to 50 amagats (Tao (1970)). Orth (1966) showed some evidence of non-linearity at densities greater than 20 amagats but these probably arose from system-atic errors in the analysis techniques. A non-linearity in the positron annihilation rate in argon at high density (greater than 20 amagats) and low temperature (less than 180 K) has been reported (Canter and. Roellig (1971)). The extent of this non-linearity is shown in Figure 28 where A /D versus D for various temperatures is plotted. The values for densities greater than 20 amagats are those quoted by Canter and Roellig and the lower density values are results from Miller (1968) and the present work. The extent 83 L e e , M i l l e r * — | — » Canter and R o e l l i g i i i p 10 20 30 40 DENSITY [amagats) Figure 28 ^/D versus D for Several Temperatures 84 of the non-linearities for densities less than 6 amagats at 135 K and less than 15 amagats at 160 K is negligible. As these were the upper limits for the densities employed for this work the assumption of linearity is assumed to be valid. 4.2 Comparison to Published Theoretical Results Generally the theoretical results do not reproduce the large velocity dependences observed (Figures 22 and 26). The semi-empirical f i t to lifetime spectra of Falk (1965) and the (3p-d) curve of Montgomery and LaBahn are the two that most closely reproduce the velocity depend-ences, although not the magnitudes, of V(v) and lA(v). £L CL The Montgomery and LaBahn (3p-d) result is derived from a polarized orbital calculation (Temkin (1957), Temkin and Lamkin (1961)) which approximates the distortion interaction by a dipole polarization potential. The unperturbed wavefunctions were taken to be the Hartree-Fock wave-functions of argon. The correction which accounts for the distortion was calculated using perturbation theory to f i r s t order. The label (3p-d) is a designation of the perturbed orbital components considered. For argon the (3p-d) component calculated in this fashion was found to give too high a value for the theoretical polarizability so was normalized to give the experimental polarizability and labeled (3p-d) Norm. Although these two results, (3p-d) and (3p—d) Norm, were found to give better agreement with experiment than other components, (3p-s) and (3s-p), this agreement is s t i l l poor (Figure 31). A semi-phenomenological model of Hewson et al . (1971) gives reasonable agreement with electric f i e l d results at room temperature (Figure 31) but does not reproduce the temperature dependence (Figure 30). In 85 this work the effect of the positron on a Thomas-Fermi model of the atom is calculated. Increased electron density or enhancement, effective charge, and effective mass are calculated and used to predict the exper-imental dependencies. The ability of a theoretical model to give good agreement with electric f i e l d effects at room temperature but to f a i l completely to reproduce the correct velocity dependences of the individual cross-sections is well demonstrated by this example. Other results (Orth and Jones (1969a)) used a Hartree-Fock potential and an empirical polarization potential, a combination which had been used in f i t t i n g the electron-argon case (Holtzmark (1929), Kivel (1959)). The velocity dependence arising from this method is too small. This is probably due to neglect of positron-electron correlations near or inside the atom. 4.3 Modified Effective Range Theory 4.3.1 Description Effective range theory describes the scattering and momentum-transfer cross-sections in terms of two parameters, the scattering length, A, and the effective range, r^. This approximation is valid for low energies. It is derived by expanding the phase-shifts, d , about a selected value Li of the wavenumber, k, corresponding to a low energy and then using these approximations in the standard expressions for the cross-sections. The derivation of effective range theory is discussed by Mott and Massey (1965). Standard effective range theory requires the asymptotic value of the potential to decrease faster than l / r (Mott and Massey (1965)). / 4 This is not the case for a l / r polarization potential. O'Malley, 86 Spruch, and Rosenburg (1961) have developed a modified effective range theory that takes into account the l/r* dependence explicitly. For the case of scattering of a charged particle by a neutral polarizable system (O'Malley (1963)) the result is A 2 + -r^-A2c*k2ln(ka ) + -r^-Acxk + Ck2 3a o 5a o o + ... 4-1 where (5, is in units of TTa d o A is in units of a where A is the zeroth order o scattering length k is in units of a ^ o 3 c* is in units of a^ where c< is the polarizability C depends upon both the f i r s t order scattering length and the zeroth order effective range (r ) o A and C are parameters to be determined from experiment. This expression differs from that of standard effective range theory 2 in that terms in k and k Ink are present. This formula, 4—1, is based upon the assumption that there is no permanent dipole distortion of the target atom, low incident energy (small k), and only single channel elastic scattering. It does not hold for bound state formation (Section 4.1.2). This formula is valid for values of k such that the appropriate terms in the phase shift expansions are small. In particular (O'Malley (1963)) 2 ^ << 1 or k « .29 a Q" for argon. 4-2 4.3.2 Effect of Bound State The existence of a weakly bound state requires significant modification 87 to 4-1. In fact, the usual effective range formula is applicable for a bound state system (O'Malley et a l . (1961)). Taking the expansion about the energy of the bound state rather than zero energy, the approx-imate solution for low energies is (5d = 4(A'2 + C'k2 + ...) 4-3 2 where (T, is in TTa d o A' and C* are again determined by experiment. 2 2 For a bound state of energy, Eg, such that t = - 2mEjj/n , the coefficients A' and C for the momentum-transfer cross-section are both dependent upon Y and the zeroth and f i r s t order (L = 0, L = l) scattering lengths and effective ranges in such a manner that a value for ti, the scattering length, or the effective range cannot be deduced from knowledge only of the A' and C values (O'Malley et al. (1962)). It is interesting that a, similax situation characterizes the modified theory of 4-1. The coefficient A is the zeroth order scattering length, but C depends on the zeroth order effective range and the f i r s t order scattering length, A^. Therefore the effective range cannot be obtained from knowledge of A and C. Only for the total scattering cross-section 2 expansions (rather than momentum-transfer cross—sections) is the k coefficient unambiguously related to the zeroth order effective range (O'Malley et a l . (1961)). 4.3.3 Calculation of Parameters from Results Values for the coefficients in 4-1 and 4-3 were obtained by a maximum likelihood f i t to the momentum-transfer cross-section of Figure 26. The results of these f i t s and the chi-squared goodness-of-fit are shown in Table V. The best chi-squared was attained from the fitt i n g 88 of 4—1, the modified effective range theory. However, as Figure 29 indicates, there is l i t t l e difference in the two curves for C^( v) i n the region of the experimental values. TABLE V Values of Effective Range Parameters A (a ) o C <ao4) Chi-Squared Probability Modified Effective Range Standard Effective Range -4.38 - .16 181 i 217 -3.89 - .15 -2292 ± 348 .59 .35 The standard deviations in Table V are statistical. The actual uncertainty is larger since the assumed shape ( i ^ = Av ) is different from the shape predicted by the modified E.R.T. From consideration of the cross-section at v = 0 an estimated value for the zeroth order scattering length is A = - 4.4 - .5 a Q. This is significantly larger in magnitude than the corresponding result for electrons, A^ = - 1.70 a Q (O'Malley (1963)). 4.4 Positron-Argon Complexes If a bound state, as previously mentioned in the discussion of effective range theory, exists i t would significantly enhance the positron annihilation rate at low velocities. Canter and Roellig (1971) postulate the existence of a positron-argon complex, which is stable at low temper-atures and high densities, in order to explain the very high annihilation 89 Figure 29 Effective Range Fits to Experimentally Obtained Momentum-Transfer Cross-Section 90 rates obtained. It is not clear, however, i f such effects could not also be due to interactions of the positron with more than one atom. Resonance scattering (Paul and Smith (1970)) could also account for an increased annihilation rate. However, such a resonance would be expected to be above thermal energies and therefore should have the effect of increasing the annihilation rate at higher temperatures and electric fields. These effects are not observed. 4.5 Numerical Solution of Diffusion Equation 4.5.1 Technique of Solution The numerical methods employed to solve the diffusion equation (Equation l - l ) were those developed by Orth (1966). This iterative method assumes the zeroth order solution to be that of Equation 1-4, that is f Q ( v ) = C e x p ^_ f _vj ^ j ^ ^ 8, ^ kX U ^ v 1 ) 2 m The zeroth order solution for A is then given as in 1-2 A0 = ^ H ( v ) v 2 f o d v > 4 - 5 This expression, using the appropriate W (v) and V (v), was eval-d a uated using eight-point gaussian integration. The i n i t i a l values obtained, fQ( v) and ^Q> were then used in the right hand side of Equation 1-3 which was solved for the next approximation, f^(v). Hence where ^ V / J W g(v) = - / ( V ( v ) - ^ 0 ( V ) ) v ' 2 f 0 ( v ' ) d v ' . a + kl \ ° 3 ^ ( v ) 2 + m 91 The constant % is required since, in general, the limit v 0 of f (v)/fg(v) is not unity. This constant was evaluated by requiring the integral of f^,(v) and ^Q^) to be normalized. In a similar manner any approximation f. (v) can be evaluated in terms of f.(v) and )\. is calculated, as indicated for zero order, by 4—5. Numerical solution of 4-6 vas carried out using Simpson's rule integral techniques. In most cases the iterative method described converged vithin four itera -r tions. 4.5.2 Use in Consistency Tests The numerical solution of the diffusion equation does not involve any approximations about the shape of the distribution nor does i t neglect the effect of annihilations on this shape. Throughout the analysis of Chapters Tvo and Three the explicit assumption vas made that the effect of the annihilations vas negligible. If such is indeed the case, numerical solution of the diffusion equation using the derived values of V (v) and \J (v) should reproduce the experimental lifetime dependences. On the other hand, an inconsistency at this point vould indicate that the region of applicability of the assumptions implicit in the analytic treatment vas exceeded. Application to Annihilation Rate The temperature dependence of the annihilation rate as determined from the numerical solution of the diffusion equation using V(v) from 3-7 is shovn in Figure 30. The agreement between this result and the best f i t to the temperature results as given by 2-13 is excellent. This agreement confirms the validity of the assumption that the effect 92 200 300 400 500 600 TEMPERATURE (K) Figure 30 Temperature Dependence of Direct Annihilation Rate as Determined from Boltzmann Solution Experimental values are shown. 93 of annihilations on the shape of the velocity distribution is negligible for the case of zero electric f i e l d . Application to Momentum-Transfer Cross-Section The room temperature electric f i e l d dependence of the positron annihilation rate as calculated by the diffusion equation is given in Figure 31. The dependence was calculated for both the i^(v) of 3-8 and 3-11 and the modified effective range expansion for ^ ( v ) given by 4-2 and Table V. For the modified effective range theory a cut-off was imposed to prevent from going negative for k ) .10 a Q \ This value (illustrated in Figure 29) was chosen to be 0^  = lOTTa^ since this gave best agreement with the experimental results. The excellent agreement with experiment for values of E/D less than 30 volts/cm-amagat is confirmation that the approximation used to obtain the M(v) of 3-11 is valid. At these low fields the f i t to d experiment using the modified effective range results is not as good, probably because of the a r t i f i c i a l cut-off employed. Further confirmation of the adequacy of the analysis carried out 2 in Chapter Three is obtained by comparing the values of d(^/D)/c5(E/D) calculated from experiment with those deduced from solution of the diffusion equations. The value so obtained was in excellent agreement with the experimental value of - .0028 -Vsec = - .0029 4sec -1 .r-2 2 V cm -1 V - 2 cm2 (Table III). It is therefore apparent that the values for X1 (v) and M(v) a 94 7 -1 10 20 30 40 - (V cm amagat ) D Figure 31 Electric Field Dependence of Direct Annihilation Rate at T = 298 K as Determined from Boltzmann Solution Experimental values are shown. 95 obtained by the approximate analytic analysis of the experimental data do indeed yield agreement with the experimental results when an "exact" numerical solution of the relevant diffusion equation is carried out. 96 CHAPTER FIVE CONCLUSIONS 5.1 Summary of Results The velocity dependences of the annihilation rate and the momentum-transfer cross-section for free positrons in argon have been measured 2 over the velocity range of 0.03 to 0.065 atomic units (e /n) by comparing the effect on annihilation rates of electric fields and temperature changes. In terms of the mean velocity of a distribution at thermal equilibrium, this range corresponds to a temperature variation between 135 K and 573 K. The velocity dependences of both the annihilation rate and momentum-transfer cross-section were found to be larger than that predicted by most theoretical models. This strong dependence at low velocities prob-ably reflects strong positron-electron correlation effects during c o l l i -sions between the positron and the argon atom. Further theoretical study of the effects of such correlations is now required. No velocity dependence in the orthopositronium quenching rate was observed (within 8fo) over the same temperature range. 5.2 Outline of Possible Future Studies The experimental determination of the electric f i e l d dependence of the free positron lifetime at higher electric fields at different temperatures will contribute l i t t l e in the way of additional knowledge of these cross-sections because of the difficulty of analysing such results (Section 3.1). Furthermore the available results at higher fields (Orth and Jones (1969)) are sufficient to assess the validity 97 at higher velocities of velocity dependences obtained from any theoretical model which reproduces the experimental velocity dependences derived in thi3 work. Repeating the current experiment with greater accuracy would not significantly reduce the errors in the momentum—transfer cross-section since much of this error is attributed to the assumed parameterization (which cannot completely reproduce the shape predicted by effective range theory). A less restrictive from of the cross-sections could in principle be employed but analytic solution of the pertinent equation (Section 3.1) is not possible by the techniques described. The use of numerical methods to determine the maximum likelihood f i t to the data is also possible in principle but the amount of computer time required is excessive (approximately three months CPU time for the amount of data in this thesis). Extension of the outlined technique to other noble gases would be of use only for those heavier noble gases which exhibit a significant velocity dependence in the annihilation rate as evidenced by the existence of a shoulder region (Falk and Jones (1964)). For neon and helium the shoulder is not as pronounced (Lee et a l . (1969), Paul and Leung (1969), Roellig and Canter (private communication - 1971)). Therefore the velocity dependence of the annihilation rate, and consequently the electric field and temperature dependences of the positron lifetime, is small. Experimentally this makes both the measurement and analysis, in terms of velocity dependent cross-sections, extremely d i f f i c u l t i f not impossible. More promising is the concept of "doping" a host gas with minor 9 8 amounts of known impurities. Variation in the amount of positronium formation in a pure noble gas, an effect which can easily be measured as a function of electric f i e l d (Orth (1966), Marder et a l . (1956)), depends primarily upon the momentum-transfer cross-sections of the gas atoms for positron energies near that of the positronium formation threshold (Teutsch and Hughes (1956)). For the case of impurity doping, the positron velocity distribution is determined primarily by the char-acteristics of the host gas. If the impurity has a low positronium formation threshold an increase in positronium formation should be observed at this threshold. Current work is being carried out using this technique (Albrecht and Jones (private communication 1971)). There is also the possibility of analysing the non-thermalized portions of the positron annihilation lifetime spectra in order to obtain some estimate of the scattering and annihilation rates at higher velocities. Paul and Leung (1968) have made a start at this type of analysis for helium but again the difficulty of two unknowns V(v) and V,(v) makes analysis d i f f i c u l t , a Development of mono—energetic positron beams would allow the deter-mination of higher velocity cross-sections by direct measurement. Such beams are currently under development (Jaduszliwer et a l . (1971)). The non-linear density dependence of positron annihilation in argon at low temperatures is also of some interest. The cause of this effect is not understood (Canter and Roellig (1971)). Measurements are needed over a wider range of densities in order to determine the exact pressure and temperature dependence of the non-linear behavior. Experimental problems will arise, however, i f densities less than four 99 amagats are required. The analysis of the decay spectrum into its two lifetime components becomes di f f i c u l t at such a low density because the two lifetimes become comparable in magnitude. In summary, the swarm type of experiment for positrons in argon has probably reached its limit as a source of useful information and effort should now be directed to other gases and to the determination of the momentum-transfer and annihilation cross-sections at higher velocities. 100 BIBLIOGRAPHY Albrecht, R.S. and Jones, G. (1971). private communication. Anderson, C.D. (1932). Phys. Rev. 41, 405. Beers, R.H. and Hughes, V.W. (1968). Bull. Am. Phys. Soc. ^ 3, 633. Bennett, W.R., Thomas, W., Hughes, V.W. and Wu, C.S. (1961). Bull. Am. Phys. Soc. 6, 49. Brock, R.L. and Streib, J.P. (1958). Phys. Rev. 109, 399. Canter, K.F. and Roellig, L.O. (1971). to be published. Deutsch, M. (1951). Phys. Rev. 83, 866. Deutsch, M. (1953). in "Progress in Nuclear Physics" (O.R. Frisch, ed.), Vol. 3, pp. 131-158. Pergamon Press, Oxford. Dirac, P.A.M. (1931). Proc. Roy. Soc. A126, 360. Drachman, R.J. (1966a). Phys. Rev. 144, 25. Drachman, R.J. (1966b). Phys. Rev. 150, 10. Drachman, R.J. (1968). Phys. Rev. 173, 190. Duff, B.G. and Heymann, F.F. (1963). Proc. Roy. Soc. (London) A272, 363. Falk, V.R. (1965). Ph.D. Thesis, University of British Columbia. Falk, W.R. and Jones, G. (1964). Can. J. Phys. 42, 1751. Falk, V.R., Jones, G. and Orth, P.H.R. (1965). N. I. M. 33, 345. Falk, W.R., Orth, P.H.R. and Jones, G. (1965). Phys. Rev. Letters 14, 447. Fraser, P.A. (1968). in "Advances in Atomic and Molecular Physics" (D.R. Bates and I. Estermann, ed.), Vol. 4, pp. 63-107. Academic Press, New York. Golden, D.E. (1966). Phys. Rev. 151, 48. Green, J.H. and Lee, J. (1964). "Positronium Chemistry". Academic Press, New York. Green, J.H. and Tao, S.J. (1963). J. Chem. Phys. 39, 3160. Hewson, P.W., Jones, G. and Vogt, E.W. (1971). to be published. 101 Holtzmark, J. (1929). Z. Physik 55_, 437. Jaduszliver, B., Keever, W.C. and Paul, D.A.L. (1971). in "Positron Annihilation, 2nd International Conference". Queen's University, Kingston, Canada, unpublished. Kivel, B. (1959). Phys. Rev. 116, 4. LaBahn, R.W. and Callaway, J. (1964). Phys. Rev. 135A, 1539. Lee, G.P. (1969). M.Sc. Thesis, University of British Columbia. Lee, G.F., Orth, P.H.R. and Jones, G. (1969). Phys. Letters 28A, 674. Leung, C.T. and Paul, D.A.L. (1969). J. Phys. B2, 1278. Marder, S., Hughes, V.W., Wu, C.S. and Bennett, W. (1956). Phys. Rev. 103, 1258. Massey, H.S.W. (1967). in "Positron Annihilation" (A.T. Stewart and L.O. Roellig, ed.), pp. 113-125. Academic Press, New York. Miller, D.B. (1968). M.A.Sc. Thesis, University of British Columbia. Miller, D.B., Orth, P.H.R. and Jones, G. (1968). Phys. Letters 27A, 649. Mohorovicic, S. (1934). Astron. Nachr. 253, 94. Montgomery, R.H. and LaBahn, R.W. (1970). Can. J. Phys. 48, 1288. Mott, N.F. and Massey, H.S.W. (1965). "The Theory of Atomic Collisions" (3rd edition). Oxford Univ. Press, London and New York. National Bureau of Standards Circular 564 (1955). O'Malley, T.M., Spruch, L. and Rosenberg, L. (1961). J. Math. Phys. 2, 491. O'Malley, T.M., Rosenberg, L. and Spruch, L. (1962). Phys. Rev. 125, 1300. O'Malley, T.M. (1963). Phys. Rev. 130, 1020. Ore, A. and Powell, J.L. (1949). Phys. Rev. 75, 1696. Orear, J. (1958). "Notes on Statistics for Physicists", UCRL-8417 Report. Berkeley, California. Orth, P.H.R. (1966). Ph.D. Thesis, University of British Columbia. Orth, P.H.R., Falk, W.R. and Jones, G. (1968). N. I. M. 65, 301. Orth, P.H.R. and Jones, G. (1966). in "Positron Annihilation" (A.T. Stewart and L.O. Roellig, ed.), pp. 401-407. Academic Press, New York. 102 Orth, P.H.R. and Jones, G. (1969a). Phys. Rev. 183, 7. Orth, P.H.R. and Jones, G. (1969b). Phys. Rev. 183, 16. (There is a numerical error in this paper. The momentum-transfer cross-section obtained should be 39TTa02.) Osmon, P.E. (1965). Phys. Rev. 138 B, 216. Paul, D.A.L. (1964). Proc. Phys. Soc. (London) 84, 563. Paul, D.A.L. and Leung, C.T. (1968). Can. J. Phys. 46, 2779. Paul, D.A.L. and Smith, P.M. (1970). Can. J. Phys. 48, 2984. Paul, D.A.L. and Saint-Pierre, L. (1963). Phys. Rev. Letters 11_, 493. Roellig, L.O. and Canter, K.F. (1971). private communication. Ruark, E. (1945). Phys. Rev. 68, 278. Stout, V.L. and Gibbon, M.D. (1955). J. A. Phys. 26, 1488. Tao, S.J. (1970). Phys. Rev. A l , 1257. Tao, S.J., Bell, J. and Green, J.H. (1964). Proc. Phys. Soc. 83, 453. Tao, S.J., Green, J.H. and Celitans, G.T. (1963). Proc. Phys. Soc. 81, 1091. Taylor, H.E. (1968). N. I . M. 68, 160. Temkin, A. (1957). Phys. Rev. 107, 1004. Temkin, A. and Lamkin, J.C. (1961). Phys. Rev. 121, 788. Thompson, D.G. (1966). Proc. Roy. Soc. A294, 160. Teutsch, V.B. and Hughes, V.V. (1956). Phys. Rev. 103, 1266. 103 APPENDIX A ELECTRONIC PILE-UP EFFECTS High count rates have three possible effects which can influence the measured lifetime spectra (Section i . There may be time-slewing due to baseline shifts in a.e. coupled parts of the system, i i . There will be an increase in the relative size of the random coincidence background, i i i . There may be a non-linearity introduced in the random background because of negative time events which are correlated. A.l Time-Slewing In the electronic system used (Section 2.2.3) there was a.e. coupling between the photomultiplier and the discriminator. Time-slewing occurs i f an accepted pulse triggers the discriminator while sitting on the overshoot arising from the differentiation of a previous pulse. Falk (1965) demonstrated the effectiveness of pile-up gates in removing this effect. The lack of any observable broadening of the prompt peak (the f i r s t evidence of such effects) in this experiment is attributed to the fact that pile-up gates were employed. A.2 Random Background - Uncorrelated Coincidences The random background due to uncorrelated coincidences i s , strictly speaking, non-constant with an exponential time structure. However, for low count rates and short time scales the deviation from a constant 104 is negligible. The probability of n counts per second is defined as P(n) and the probability of n counts in time t as P(n,t). If there are R counts per second on average, the probability P(n) is given by the Poisson distribution _/ s Rne ^ p ( n ) = —aT-Similarly, i f there are Rt counts in t on average n -Rt n l — R r l " f Hence P(l,dt) = Rdte . The probability of a single count in dt when there has been no count in previous t is P = P(0,t) x P(l,dt) _ -Rt p, + -Rdt P = e Rdte _g For small dt (dt corresponds to channel width a 10 sec) e _ R d t « (1 - Rdt). Therefore P a Re dt where terms in dt only are kept. Hence given a start pulse the probability of the stop pulse occurring in time t to t + dt is -R , t P = R . e s t ° P dt stop where R , is count rate from stop side of discriminator. stop For R , t small this becomes stop P = R dt(l - R , t + ...). stop stop The random coincidence rate is then R , ,R , dt(l - R , t). start stop stop 105 The magnitude of this effect for the current experiment can be calculated from the observed 0.51 MeV count rate of 7500 counts/sec. The fractional effect on a 500 nsec time scale was therefore R , t / l stop or 7500 x 500 x IO"9 = ^ x 1 Q-5 = ^ This degree of non-constant background is small and can be neglected. A.3 Random Background - Correlated Coincidences The effect of correlated coincidences can be most easily seen by considering the idealized diagram of Figure 32. Since the discrim-inator outputs are indistinguishable regarding their origin (0.51 MeV or 1.28 MeV pulses) and since the zero time is delayed, the time-to-amplitude converter output is symmetric with respect to time as indicated. The events selected for analysis corresponding to t > 0 are chosen by use of the slow coincidence channel, a system which is energy sensitive. A probl em arises, however, i f the deadtime of the pile—up gates is long. Consider a negative time event consisting of detection of the correlated pulses in the fast channel. For this case the slow channel is not driven since the two channels are inverted as far as energy is concerned. For long deadtimes in the pile-up gate i t is possible for this event to be followed by a pair of pulses which satisfy the energy conditions in the slow channel and thereby actuate the slow coincidence circuit. This second pair may or may not be correlated. However, only the time interval between the f i r s t correlated events will be recorded for this situation. If the pile-up gate has had time to recover then a pile-up pulse is generated and such negative time events are vetoed. The length of time during which a pile-up pulse 106 Ungated Number of Counts t< 0 t = 0 t 70 Gated P i l e - u p (exaggerated) No p i l e - u p J . ——b> Number o f Counts t < o t = 0 t > 0 Figure 32 Schematic Diagram of Gated and Ungated Time Spectrum 107 is generated must, of course, "be greater than the slow coincidence resolving time. These unwanted events occur only on the negative time side, resulting in an apparent asymmetric random coincidence background about t = 0. Since the magnitude of the random coincidence background for the t > 0 region is inferred from i t s magnitude on the t < 0 region this effect can lead to serious errors. As a consequence of this problem the pulse width and hence the deadtime of the pile-up gates was shortened by differentiating the photomultiplier output (Section This problem could also arise when encountering excessive count rates such as would occur for discriminator settings which were in the noise region. Because the discriminators were of an updating type their output pulse width and hence the deadtime of the pile-up gates would increase. The magnitude of such an effect can be estimated by considering the original electronic configuration in which no consideration was given to the effect of such pulse pile-ups in the slow channel. The effective deadtime was then the slow coincidence resolving time of 8 -<<sec. The count rates from the single-channel analysers were of 3 the order of 5 x 10 from the 0.51 MeV channel. The probability of getting a second energy-correct event in both channels is P = P, 0.51 x P. 1.28 or P = N, 0.51 dt x N. 1.28 ,dt where dt is the effective deadtime. Hence for this case P * 3 x 10 .-4 Since the time-to-amplitude converter ungated output rate was 1 0 8 1 7 0 sec 1 and about half of this corresponds to negative time events the count rate for "slow pile-up" events was „ „ ( 1 7 0 \ - 1 R = P VT7 S E C or R - . 0 3 sec" 1. The random background rate is given by (Section A.2) KRB = N 0 . 5 1 N 1 . 2 8 ( 5 ° ° N S E C ) or RJJQ = 2 . 5 sec 1 . Therefore R^g on negative time was approximately l/sec. Hence the effect of the "slow pile-up" situation on the random background was R „ » 0 3 _ \& 1 This yfo effect was calculated assuming that the negative time events are uniformly distributed. In fact, since they correspond in shape to an ungated time spectrum the region near t = 0 is emphasized. Since the estimate of background is made within 25 channels of t = 0 (channels 20 to 6 0 for peak at 8 5 ) the effect is somewhat larger. Direct experimental verification of this shape is di f f i c u l t since statis-t i c a l significance would require about 1 0 0 0 counts/channel in the back-ground. This small increase in the measured random background did manifest it s e l f , however, in causing significant errors in the determination of the long lifetime component in the annihilation spectrum. 109 APPENDIX B NUCLEAR INSTRUMENTS AND METHODS 91 (i 971) 665-666; © NORTH- HOLLAND PUBLISHING CO. ON THE FITTING OF DOUBLE EXPONENTIAL LIFETIME SPECTRA USING A MAXIMUM LIKELIHOOD TECHNIQUE G. F. LEE and G. JONES Physics Department, University of British Columbia, Vancouver, B.C., Canada Received 5 October 1970 A modification of a previously presented iterative method for fitting double exponential lifetime spectra is presented. A different choice of parameters is shown to reduce the number of iterations required. Orth et al.1) described an iterative maximum likeli-hood technique for fitting lifetime spectra consisting of two exponentials with constant background. Thus *k yk = [/, exp( - t\x0 +12 exp( - t/x2) + B]dt is the theoretical yk (per channel), /j and I2 are the intensities, tj and x2 are the lifetimes of the two com-ponents respectively and B is the constant background per unit channel. The integration is performed over the kth channel. In this letter, a modification of this technique is described which reduces the number of iterations needed for convergence and in addition gives the standard deviation of the intensity-lifetime (Ix) prod-uct, a quantity equal to the total number of events within one of the exponentials. Determination of the standard deviation of this quantity normally requires manipulation of the re-sulting error matrix. In particular the standard devia-tion A(Ix) associated with Ix is given by 2) 4 / T ) R M S = [ / 2 ( / / - 1 ) t t + 2/T(//-])/L + T2(//-1)//]*, (1) where H~l is the error matrix 1 , 2). Note that for uncorrected parameters (H~1)ij = 0 for i 7*= j. In this case (1) reduces to the well known special case J(/T)RMS = [ / W + * W J * > (2) since (//-")„ = {Aaf. Since / and x are correlated it is not sufficient to simply combine the errors in / and T as in (2) but the full expression (1) must be evaluated. In the modification described here, the theoretical yk (per channel) is of the form where Ax and A2 are the areas under each exponential (the intensity lifetime product). Otherwise the same technique as that of Orth et al.1) was used. The correlation between A and x was found to be rather less than that between / and x. This is responsible for the reduction in the number of iterations required. The degree of correlation between two parameters can be easily seen on a contour plot of the likelihood function as shown in fig. 1. This plot is determined by varying /t and xt (or At and T j ) with I2(ov A2), x2 and B set at their optimum values. The value of the likeli-hood function relative to its peak value is shown. For uncorrected parameters the contours would be ellip-tical with minor and major axes parallel to the co-ordinate axes. It can be seen that in neither case are the parameters uncorrected, but the degree of correla-tion is less between Al and Tx then between /, and T,. That this results in faster convergence is demonstrated in table 1. The errors are calculated from the error matrix 1) and convergence is defined to occur when the change in all parameters is less than 0.1% between iterations. W, the logarithmic likelihood function is a maximum at convergence. The number of iterations and hence the amount of computing necessary is re-L1KELIH00D CONTOURS (PEAK - 1) 0.25 VA=J K ^ i / T i ) e x p ( - f / T , ) + (A2lx2)cxp(-tlx2) + B]dl, AREA OR INTENSITY Fig. ]. Maximum likelihood contour plots for A \ and 1/n (solid line) and h and 1/TI (dashed line). The remaining parameters are fixed at their optimum values. 665 110 666 G. F. L E E A N D G. J O N E S TABLE 1 Iteration Analysis of spectrum fitting intensities number h h T 2 B W t> 852 259 17.9 40.7 6.85 -867.9 1 926 ± 25 220 + 25 17.7 + 0.5 43.6 + 1.7 6.89 ± 0.29 -861.3 2 943 ± 32 202 ± 27 18.0 ± 0.6 45.2 ± 2.7 6.85 ± 0.29 -860.0 3 967 ± 24 174 ± 28 18.5 ± 0.5 47.6 ± 2.5 6.77 + 0.29 -859.6 4 970 ± 32 170 + 35 18.5 + 0.6 48.3 + 3.7 6.75 ± 0.30 -859.6 5 976 + 30 164 ± 33 18.6 ± 0.6 49.0 ± 3.7 6.73 + 0.29 ^859.6 6 977 ± 32 163 ± 35 18.7 ± 0.7 49.1 + 3.9 6.73 ± 0.29 -859.61 7 977 ± 32 163 ± 35 18.7 ± 0.7 49.1 ± 3.9 6.73 ± 0.30 -859.6j c Analysis of spectrum fitting areas Iteration /4i(x 102) At T 2 B W 0 153 105 17.9 40.7 6.85 -867.9 1 180 ± 7.0 81 ± 6.8 18.3 + 0.5 47.2 ± 2.0 6.67 ± 0.28 -862.5 2 179 ± 11.5 83 ± 11.1 18.5 ± 0.7 48.0 ± 3.7 6.75 + 0.30 -859.6 3 182 + 11.2 80 + 10.8 18.6 ± 0.6 49.0 + 3.8 6.73 + 0.30 -859.6 4 182 ± 11.4 80 ± 10.9 18.7 ± 0.7 49.1 ± 3.9 6.73 ± 0.30 -859.61 5 182 ± 11.4 80 ± 10.9 18.7 ± 0.7 49.1 ± 3.9 6.73 ± 0.30 -859.6/ a Iteration number 0 refers to initial estimates. duced by the modification described by about one third. That the correlation between / and T in fits to lifetime spectra influence the standard deviation in the resulting estimate of the total number of events (IT) to a con-siderable degree is readily apparent in the example illustrated in table 1. In this case, the value obtained for the standard deviation in Ax was 11.4 x 102. This is the same as the value obtained using (1) and the likelihood fit described by Orth et al. 1). It differs markedly from the value (2 x 102) for the standard deviation obtained if the parameters IT and are treated as if uncorrelated, and is about 30% larger than the value (8.9 x 102) obtained if the correlated values of ATt and Azt as determined by Orth et al.1) are used in (2), rather than the correct expression (1). References !) P. H. R. Orth, W. R. Falk and G. Jones, Nucl. Instr. and Meth. 65 (1968) 301. 2) J. Orear, Notes on statistics for physicists, UCRL-8417 (1958). I l l APPENDIX C DENSITY CORRECTED DATA The following data was obtained. Where runs were disregarded or not analysed, the run number and the reason (Section 2.3.1) is given. ^1 ^2 E/D Direct Orthopositronium Run Temp. Density (V cm"1 Annihilation Annihilation No. (K) (amagats) amagat 1) Rate (-<rsec 1) Rate (><rsec 1) 63 296.16 9.94 0.0 51.6 + 1.4 9.00 - 0.37 64 296.16 9.68 0.0 53.1 + 1.9 9.93 i 0.43 65 296.16 9.25 0.0 49.9 + 0.7 9.43 i 0.17 66 295.96 8.91 0.0 48.9 + 1.3 9.71 - 0.32 67 296.16 8.78 9.0 46.1 + 2.1 9.70 i 0.50 68 296.66 8.65 4.6 45.4 + 1.8 9.70 - 0.44 69 297.66 8.51 2.3 44.7 + 1.8 9.04 i 0.38 70 298.56 8.44 0.0 45.4 + 1.8 9.44 - 0.47 71 298.46 8.36 7.1 44.5 + 1.7 9.49 i 0.46 72 297.16 8.15 12.0 42.7 + 1.8 9.28 i 0.48 73 297.16 7.93 2.5 41.1 + 1.2 8.99 i 0.36 74 296.86 7.84 1.3 42.8 + 1.2 9.60 - 0.38 75 297.26 7.75 8.1 38.9 + 1.4 9.24 - 0.44 76 297.46 7.64 0.0 42.0 + 1.8 9.31 i 0.44 77 297.16 7.49 3.0 40.7 + 0.9 9.18 - 0.25 78 296.36 7.34 1.1 38.8 + 1.3 9.26 i 0.42 79 295.76 7.13 2.9 38.2 + 1.3 9.15 - 0.47 80 295.16 6.76 0.0 37.8 + 0.9 9.78 - 0.32 81 295.46 6.20 2.6 32.5 + 0.5 8.76 - 0.27 82 295.56 5.76 0.0 30.7 + 1.2 8.57 - 0.59 83 background 84 468.16 5.00 7.9 24.8 + 1.9 7.81 i 1.58 85 466.16 5.01 15.9 23.8 + 1.2 7.40 - 1.72 112 ^1 ^2 E/D Direct Orthopositronium Run Temp. Density (V cm 1 Annihilation Annihilation No. (amagats) amagat ) Rate (^sec *) Rate (^sec 1) 86 464.16 5.01 3.9 24.4 - 1.5 8.85 - 2.47 87 464.16 5.01 11.8 23.7 - 3.9 9.67 - 3.48 88 463.16 4.99 19.7 21.7 - 2.5 7.07 - 2.84 89 463.66 4.97 0.0 25.2 i 1.9 7.79 - 1.65 90 462.16 4.97 23.8 19.1 - 1.3 5.04 - 2.70 91 459.16 4.97 31.7 19.6 ± 4.1 10.21 i 4.20 92 chi-squared. bad 93 464.16 4.95 0.0 27.0 i 3.2 9.33 - 1.38 94 574.66 4.68 0.0 25.4 i 2.0 9.41 i 1.26 95 579.16 4.64 8.5 24.6 i 3.1 10.31 - 1.35 96 579.16 4.63 17.0 23.8 - 3.1 9.56 - 1.38 97 568.16 4.66 25.4 19.2 i 2.5 8.56 - 1.58 98 570.66 4.63 4.2 24.5 - 3.7 9.61 - 1.81 99 570.16 4.63 0.0 25.2 i 5.4 9.88 - 1.12 100 571.16 4.60 21.4 21.1 ± 3.0 9.38 i 1.73 101 573.16 4.58 12.9 21.5 - 2.1 8.26 i 1.50 102 569.16 4.60 0.0 26.3 - 3.8 10.97 - 1.20 103 567.16 4.59 0.0 21.2 i 3.9 8.18 - 1.44 104 367.16 5.01 0.0 26.0 - 2.2 6.90 - 1.43 105 364.66 5.03 31.2 18.2 - 1.3 4.02 - 1.27 106 did not converge 107 368.16 4.97 7.9 23.6 i 1.7 5.67 - 1.26 108 368.16 4.98 15.8 20.8 i 1.5 4.78 i 1.92 109 background 110 364.66 5.00 19.7 20.5 - 1.5 6.57 - 1.40 111 did not converge 112 electronics failure 113 366.16 5.00 23.6 20.7 - 2.0 8.12 - 1.52 114 367.16 4.97 26.1 20.9 - 1.2 8.91 - 1.48 115 368.16 4.96 3.9 25.9 - 3.2 9.22 - 1.34 113 E/D Run Temp. Density (V cm-1 No. (K) (amagats) amagat ^) 116 367.16 4.92 0.0 117 294.16 5.16 0.0 118 did not converge 119 294.16 5.10 23.2 120 chi-squared bad 121 291.91 4.91 16.0 122 294.66 4.68 0.0 123 295.66 4.58 0.0 124 295.66 4.55 0.0 125 large temperature variation 126 211.66 3.52 0.0 127 did not converge 128 211.16 3.45 11.4 129 did not converge 130 did not converge 131 did not converge 132 did not converge 133 did not converge 134 did not converge 135 did not converge 136 293.66 10.05 0.0 137 213.16 9.96 0.0 138 206.16 9.88 0.0 139 209.66 9.32 0.0 140 background 141 background 142 background 143 background 144 background 145 background / h ^ 2 Direct Orthopositronium Annihilation Annihilation Rate (/<sec ) Rate (fsec ) 27.8 + 1.5 9.47 + 0.73 26.5 + 2.7 8.31 + 1.39 19.0 + 1.5 6.19 + 2.50 22.1 + 2.1 8.19 + 1.48 25.8 + 4.3 8.79 0.67 25.2 + 2.3 8.92 + 1.41 27.2 + 3.3 10.05 + 1.79 19.4 + 2.9 6.19 + 6.13 17.4 + 1.5 9.54 + 4.89 53.2 + 1.3 11.07 + 1.04 57.4 + 1.9 9.69 + 0.40 57.5 + 2.9 9.83 + 0.51 53.0 + 2.6 9.06 + 0.54 114 M >\2 E/D Direct Orthopositronium Run Temp. Density (V cm 1 Annihilation Annihilation No. (K) (amagats) amagat 1) Rate (-^ sec "*") Rate (<<sec "*") 146 background 147 background 148 background 149 background 150 background 151 background 152 background 153 136.16 5.86 33.6 23.8 + 1.3 5.95 + 1.40 154 136.16 5.62 0.0 39.9 + 2.4 9.09 + 0.83 155 140.16 5.17 15.2 29.9 + 1.6 8.39 + 0.99 156 138.16 4.98 7.9 31.9 + 1.6 8.29 + 1.18 157 137.16 4.80 0.0 34.9 + 1.8 9.22 + 0.87 158 139.16 4.52 26.1 19.8 + 2.9 8.78 + 1.80 159 132.16 4.52 34.8 16.9 + 1.3 5.74 + 2.77 160 135.16 4.24 0.0 25.6 + 1.8 7.68 + 1.58 161 166.16 3.83 0.0 22.6 + 1.7 7.33 + 1.68 162 did not converge 163 188.16 3.56 0.0 22.5 + 3.3 8.14 + 2.20 164 did not converge 165 212.16 14.77 0.0 86.3 + 1.9 11.07 + 0.39 166 159.66 14.23 0.0 99.7 + 3.3 11.25 + 0.33 167 180.16 13.01 0.0 84.4 + 1.3 11.14 + 0.41 168 229.16 10.64 0.0 61.3 + 2.4 9.58 + 0.46 169 250.66 9.46 0.0 51.1 + 1.4 9.09 + 0.40 170 240.16 9.25 0.0 51.4 + 1.1 10.09 + 0.32 171 208.16 8.97 0.0 51.7 + 1.9 10.02 + 0.49 172 209.16 8.77 9.0 45.5 + 1.8 9.12 + 0.60 173 209.16 8.70 22.6 37.4 + 1.4 8.34 + 0.74 174 209.66 8.70 18.1 39.7 + 1.4 8.33 + 0.65 175 212.16 8.60 13.7 43.7 + 1.5 8.67 + 0.55 115 Run Temp. Density No. (amagats) 176 209.66 8.72 177 207.16 8.80 178 206.16 8.82 179 208.66 8.77 180 equipment failure 181 299.16 7.13 182 background 183 421.16 6.36 184 472.16 6.24 185 background 186 586.66 5.91 187 298.16 6.90 188 298.16 6.82 189 298.16 6.79 190 298.16 6.76 191 298.16 6.73 192 298.16 6.65 193 297.66 6.61 194 298.16 6.67 195 298.16 6.52 196 373.66 6.08 197 373.66 6.08 198 369.66 6.04 199 373.16 6.02 200 298.16 6.33 201 298.16 6.28 202 298.16 6.25 203 298.16 6.21 204 298.16 6.15 205 298.16 6.10 Al E/D Direct (V cm Annihilation amagat ^) Rate (<<sec "*") 4.5 50.3 + 1.7 0.0 53.4 + 1.8 15.6 42.8 + 1.4 20.2 39.2 + 1.4 0.0 38.6 + 1.5 0.0 31.8 + 3.6 0.0 34.4 + 1.9 0.0 32.3 + 2.5 0.0 36.7 + 1.7 5.8 38.0 + 2.9 23.2 28.2 + 1.1 17.5 29.9 + 1.2 11.7 32.7 + 1.3 14.8 31.9 + 1.2 20.8 29.1 + 1.4 0.0 33.6 + 2.0 0.0 34.9 + 1.1 19.5 27.9 + 0.8 13.0 30.2 + 1.3 6.5 33.1 + 2.3 9.8 31.2 + 1.8 24.9 26.2 + 1.3 50.1 20.5 + 2.1 63.1 17.3 + 1.9 38.0 20.7 + 1.1 57.6 18.0 + 2.1 0.0 33.7 + 1.8 Orthopositronium Annihilation 9.37 + 0.48 9.85 + 0.46 8.16 + 0.46 8.82 + 0.85 9.43 + 0.42 9.34 + 1.10 9.36 + 0.61 9.35 + 0.55 9.45 + 0.61 9.00 + 0.50 7.88 + 0.61 8.31 + 0.59 9.47 + 0.62 8.61 + 0.48 8.76 + 0.72 8.08 + 0.62 8.94 + 0.38 8.51 + 0.41 8.49 + 0.65 9.47 + 0.65 8.90 + 0.56 8.21 + 0.84 8.40 + 1.27 7.72 + 2.28 6.98 + 1.19 8.77 + 1.78 8.91 + 0.57 116 ^1 ?\2 E/D Direct Orthopositronium Run Temp. Density (V cm"1 Annihilation Annihilation No. (K) (amagats) amagat 1) Rate (<<sec 1) Rate (<<sec 1) 206 298.16 6.09 32.3 22.6 - 1.1 7.94 i 0.95 207 298.16 6.09 34.9 21.4 i 1.2 7.36 - 1.10 208 298.16 6.07 0.0 30.9 - 1.3 8.55 - 0.63 209 298.16 6.02 29.4 22.3 - 1.2 7.78 - 1.08 210 298.16 5.97 27.7 23.7 - 1.3 7.88 - 0.99 211 298.16 5.95 30.5 20.7 - 1.3 7.50 - 1.52 212 331.66 5.67 0.0 30.9 - 1.4 8.78 - 0.59 213 571.16 5.10 0.0 26.1 i 1.3 7.95 - 0.82 214 571.16 5.08 31.0 17.9 - 1.1 5.45 - 2.41 215 572.66 5.06 15.6 21.0 i 1.1 6.59 - 1.38 216 572.66 5.06 7.8 23.4 i 1.4 8.20 i 1.45 217 571.66 5.07 0.0 24.0 - 1.2 7.47 - 1.00 218 572.66 5.05 5.0 23.5 - 1.3 8.02 i 1.07 219 571.16 5.06 6.4 24.0 - 1.2 7.90 i 0.97 220 521.16 5.14 0.0 26.8 - 1.8 9.33 - 0.99 221 430.16 5.32 0.0 27.2 - 1.4 7.70 i 0.96 222 410.16 5.38 0.0 29.7 - 2.7 10.89 - 1.36 117 APPENDIX D CALCULATION OF DERIVATIVES D.l Determination of d (>/D)/a(E/D)2 at E = 0 The total positron annihilation rate is from 1-2 C O 2. (y'(v)v 2f(v)dv 1 J e a A i - i . - i • — sec amagat v f(v)dv where V* 1(v) is in sec \ a Therefore m at (M 1 where ,° E and \) (v) is density independent annihilation rate in units of sec 1 amagat 2 Taking C)(VD)/£, into integral * C O 0 0 o o D' 0 f v 2fdv fv v 2-r^fdv - f V v 2fdv fv 2-^rfdv hi f dv Using the approximation (Section V f v'dv' f (v) = Cexp / - ? 2 e "E kT m 23/t^'(v) 2 m where y '(v) is in sec \ 118 Therefore where £ § = f g ( v ) v'dv' 2 2 e E kT , m 3^ <V, JJ m Q where Hence V^( v) is density independent momentum-transfer rate in units of sec 1 amagat \ CD Cp CO v 2fdv f V &v 2fgdv CO OP - j v / f d v j , 2 v fgdv Specifically g(v) = - dv» im23-vK2 m, d dv'. For the special case E = 0 g(v) E = 0 m^M4 m 2 3 ^ 2 mi v' 1 2 V 2 d If then Then h(v) - h(A) g(v) * d rdx E = 0 3>Kk2T2 h(v) - h(0) CO CO j v 2 f d v j y v f(h(v) - h(0))dv a J r l 2 3«kV I '* CP oo \ / °° \ 2 - J ^ a v 2 f d v j V 2 f ( h ( v ) - h(0 ) ) d v V ^ [ v 2 f d v J • 119 Since h(0) is independent of v CO CO 2 e o 2„2 V v 2 f hdv - V T 2 f dv v 2 f hdv cjc' 3<<k T For the special case y^(v) = Av^ sec 1 amagat 1 h(v) - h(0) = f - ^ d V = - i j v 1 - 2 3 o d A 0 2-2B 2A (1 - B) 0 Therefore 2-2B h(v) 2A (1 - B) valid for B ^  1. For B = 1 If one uses and h(v) = Mdv' = -knv. JLJ V' )T \) (v) = Sv ^  sec 1 amagat 1 (Section 3.2.2) Cexp 2 mv "2kT | E = 0 then a l l the integrals (from above) that must be calculated are of the type °o 2 2 Using this integral result and simplifying (for B ^  l) For B = Si2 1 i j | d S 2 2e S /2kT| 2 E = 0 3*k 2T 27TA 2(l - B) I m E = 0 3^k 2T 2 JIT A ( f r ( i - B - f ) - r f l - f ) r ( t - B ) ) . 120 where ^ ( x ) = . T ^ l In fact, since the limit as B 1 of the expression derived from = Av 2 (B ^  1) is identical to the second expression, f i t t i n g 3(^ /D)/6£ at E = 0 using the f i r s t expression only does not discriminate against the value B = 1. D.2 Derivatives for Maximum Likelihood Program 2 In order to use the expression obtained for 3(^/D)/^£. at E = 0 in a maximum likelihood f i t t i n g program the f i r s t and second partial derivatives with respect to the variables A and B must be determined. Such calculations are straightforward but clumsy. For completeness the results are included in this section. 121 x - f f i r g - B - f ) -r(M)r(!-B) *r(i-i)r (f-B )r(f-B) -r(i-f)(r(f-») ( Y ' ( I - B ) + f ( f - B ) ) ) and 4\*) 1 d fix ) dx Hx) = digamma function trigamma function Y 2(x) = ?(x)T(x). D.3 Calculation of Higher Order Terms If tyb is expanded as a power series in (E/D) , that is (!)2 + c ( ! ) 4 + -^ = A + B then and a(f) = B + 2C E = 0 ( ! ) = B E = 0 m m = 2C. E = 0 Therefore an estimate of the temperature dependence of the higher order term can be obtained by calculating <Hi)2Uf)V E = 0 This laborious task will not be described in detail here. However, 122 the result of such a calculation shows that this derivative and hence the value of C is related to T by C c x 1 2 + 2B + f T Z The effect of the higher order term is thus expected to increase at lower temperatures and therefore the range of values of E/T) over which the approximation tyb = A + B(E/D) is valid is reduced. 123 APPENDIX E CALCULATION OF ANNIHILATION RATE FROM TEMPERATURE RESULTS From Section *(T) 1 D " D co ( v 2 y '(v) l K a fdv -1 ,-1 •ssec amagat v 2fdv where \P (v) is in fsec -1 For the case of E = 0, f can be approximated by 2 mv ~2kT Hence f = Cexp Jv 2V a(v) A(T) 4 D 2 mv ~2kT. e dv Wsec 1 amagat 1 mv 2 2kT, v e dv where Therefore If then V ( v ) is density independent in units of <«sec - l amagats -1 co mv D " C 2.,, \ 2kT, \v V(v)e dv J a D KjaT k\2 [TT T2 2 aT (Section 2 mv 2.,, , 2kT, v V(v)e dv a where K, = -124 Making the substitutions Therefore s Using the Laplace transform of and Therefore or 


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