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Epithermal muons from solid rare-gas moderators Morris, Gerald D 1990

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In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. 1 further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department The University of British Columbia Vancouver, Canada Date JIT V* ' DE-6 (2/88) Abstract Emission of positive muons (fi+) of low kinetic energy, with yield peaked at 5 to 20 electron volts and slowly decreasing to higher energy, has been observed by moder-ating a muon beam (p~28MeV/c, Ap/p~4%) with solid argon, krypton and xenon moderators. Emission at these energies is consistent with the muons escaping the tar-get material still carrying part of their initial kinetic energy. The underlying cause of this enhancement of low energy positive muons is not determined. Variation of charge-exchange, elastic or inelastic collision cross sections with energy, resulting in an increased charged fraction or greater mean free path for muons at these energies, are possible causes. ii Table of Contents Abstract ii List of Tables v List of Figures vi 1 Introduction 1 1.1 Muon Beams 1 1.2 uSR 3 1.3 Fundamental Physics With Slow Muonium Beams 6 1.4 Muonium Formation 7 1.4.1 Spur Model 8 1.4.2 Hot Atom Model 8 1.4.3 The Spur Model versus Hot Atom Model Debate 9 1.5 A New Approach 11 2 The Experimental Apparatus 13 2.1 Overview 13 2.2 Ultra High Vacuum Methods 15 2.3 Detectors 16 2.4 Target mounts 17 2.5 Accelerating Lens and High Voltage 18 3 Measurements 20 iii 3.1 Procedure . 20 3.2 The Time of Flight Data 25 4 Monte Carlo Simulation 32 4.1 Monte Carlo Procedure 32 4.2 Monte Carlo Results 35 4.3 Application of the Monte Carlo 39 5 Interpreting the Results 44 5.1 Comparison of Muon and Positron Energy Spectra 44 5.2 The Role of Charge Exchange Processes 45 6 Summary and Conclusions 50 Bibliography 51 iv List of Tables 3.1 Observed slow muon emission rates, normalized per 108 incident muons. 31 5.1 Electronic properties of rare gases 48 5.2 Diamagnetic, Mu and "lost" fractions in rare gases measured by //SR. . 49 v List of Figures 2.1 Schematic of the apparatus, viewed from above. Feedthroughs on the scattering chamber lid are omitted for clarity. 14 2.2 Cross section through the target and lens, along the lens axis. Surface muons are incident from the left 19 3.1 Data aquisition electronics 23 3.2 Histogram for reversed time-of-flight delay calibration 24 3.3 Time of flight histograms with LiF target 26 3.4 Time of flight histogram for copper target 27 3.5 Time of flight histogram for Si02 target 28 3.6 Time of flight histograms with solid argon target 29 3.7 Time of flight histograms with solid krypton target 30 3.8 Time of flight histogram with solid xenon target 31 4.1 Calculated time of flight as a function of emission energy with 10 V and 500 V target-lens biases 36 4.2 Acceptance as a function of muon kinetic energy with 500 V target-lens bias 37 4.3 Acceptance as a function of muon kinetic energy with 10 V target-lens bias 38 4.4 Measured Ar data and Monte Carlo generated time of flight histograms. 41 4.5 Muon energy spectrum with argon target 42 4.6 Muon energy spectrum with krypton target 43 vi Charge exchange collisions of / / + , Mu and M u " with neutral atoms of the medium vii Chapter 1 Introduction The positive muon (/x+) has proven itself to be a useful tool in experimental physics, either as a probe for studying chemical kinetics and materials or as the object of study itself in atomic and particle physics. Although the objectives of experiments in these fields are diverse, the interaction of positive muons with matter is important to each. There is, however, very little experimental data regarding the processes by which initially energetic positive muons passing through a medium end up in the states they do. A better understanding of how positive muons lose their kinetic energy to a medium, particularly at low energies ( <100 eV), and eventually thermalize would be of great use in experiments. It could also lead to the construction of sources of positive muons with very low kinetic energy, on the order of a few electron volts. Conversely, such a source of positive muons would enable the study of epithermal muons in media. This thesis describes the first experiments on such very low energy muons. 1.1 Muon Beams Muons are produced in quantity at the world's few intermediate energy accelerators (in particular the "meson factories" LAMPF, PSI and TRIUMF) from charged pions ( lifetime T„. = 26 ns ) according to the decays 7 T + fl+ + 7T" fl' + Un (1.1) 1 Chapter 1. Introduction 2 Muon beams were first (and sometimes still are) obtained from the in-flight decay of charged pions. Since the neutrino always has negative helicity (spin antiparallel to momentum) and the pion is spinless, conservation of linear and angular momenta requires that the muon also possess negative helicity in the pion's rest frame. In the lab frame, selection of a part of the muon beam's phase space by the beamline's acceptance and magnetic spectrometers produces a spin-polarized beam of muons with momentum up to a few hundred MeV/c. The polarization of this type of beam is typically 75 -95%. Pifer et al. [1] initiated the idea of producing an intense, highly polarized / / + beam from pions that had stopped at or just within the surface of the pion production target. The two-body final state of the pion's decay produces a longitudinally polarized 29.8 MeV/c (4.1 MeV) p,+. Since muons originating deeper in the pion production target are degraded by passage through the target, the overall momentum spectrum of those muons that escape depends to some extent on the distribution of stopped pions within the target. Typically the momentum spectrum of a "surface muon" beam (as it has come to be known) contains a peak at 28 - 29 MeV/c, sharply cut off at 29.8 MeV/c (the surface muon edge) with a tail to lower momenta. Since the secondary channel is usually operated with a small momentum bite (Ap/p of a few percent typically) a "sub-surface" muon beam can be obtained from the low energy tail. Negative pions are rapidly captured by nuclei of the target material so pT beams cannot be efficiently produced in a similar way. The surface muon beam has become the source of choice for many experiments due to its 100% spin polarization, short range (130 mg/cm2 in carbon) and consequent high stopping density in thin targets. Examples of such experiments are muon spin rotation (/iSR), measurement of the muonium (n+e~ or Mu) Lamb Shift and lepton conversion (e.g., p+e~ —> (J.~e+) searches. In each case the objective is to study the fate Chapter 1. Introduction 3 of the muon after it has been greatly decelerated or thermalized by passing through a medium. 1.2 pSR The muon spin rotation technique has become a well established tool for the study of matter. A complete discussion of pSR is not possible here and the unfamiliar reader may wish to consult the several reviews[2,3,4,5,6] which offer thorough discussions. The important properties of the muon that make pSK spectroscopy possible are: • The muon has an intrinsic spin angular momentum. © The muon has a magnetic moment. ^ = 3.1833452(10).[3] • The muon is produced 100% spin polarized in the pion's rest frame. • The muon decays via the weak interaction according to p+ ->e+ + u» + ue (1.2) with a lifetime rM = 2.19714(7) ps.{2] The positron is emitted preferentially along the muon spin polarization vector at such a high energy that it can penetrate several cm of material and is largely unaffected by normal magnetic fields. In a typical muon spin rotation experiment one injects transversely spin polarized muons from a surface muon beam into a gas, liquid or solid target located in a magnetic field. Each muon loses its kinetic energy within about a nanosecond in condensed media and stops (or thermalizes) within the medium. The processes by which this happens are not entirely understood and have been the topic of much debate. A muon, having stopped in the target, may be in one of several states. It may have picked up an electron from the medium to form the hydrogen-like atom muonium or Chapter 1. Introduction 4 become incorporated in a /^-substituted molecule or molecular ion. In some metals muons behave as screened positive charges. Muons in diamagnetic states (free muons and muons bound in molecules) precess in the local magnetic field at the muon Larmor frequency: u;M = 7„B (1.3) where 7 / 1 = - i e i - = 2TT x 13.55342 kHz G " 1 . (1.4) 2mMc Since the electrons of the stopping medium are unpolarized it is assumed that muo-nium is produced with p+ and e~ spins parallel and anti-parallel with equal probability. The |mM,me) = |§, — \) state, however, is not an eigenstate of the hyperfine interac-tion. In this state the muon polarization oscillates at the hyperfine frequency j£ = 4.46330235(52) GHz,[7] too high to be resolved by the timing electronics used in most //SR. experiments, so that the observed (average) polarization is due to only the \\,\) fraction. Larmor precession of muonium in weak transverse magnetic fields (WTF) is dominated by the much larger electron magnetic moment and exhibits the characteristic frequency WM = + wM) = 7A/B (1.5) where ~1M = 2TT x 1.394 MHz G" 1 . (1.6) The ensemble of muons or muonium atoms precess in phase until the muons decay or their polarization is de-phased by non-uniform magnetic fields in the medium. A detector that intersects some of the solid angle in the precession plane about the target Chapter 1. Introduction 5 detects some of the decay positrons. By timing the period between the arrival of each muon in the apparatus and the detection of its decay positron, a histogram is accumu-lated, one event at a time, that shows the exponential decay of the muon modulated by the precession signal which carries the information of interest. The histogram is fitted to a function of the form N(t) = Noe-"** (l + AMt) cos(w„t + ^)+ (1.7) ^RM(t) {(1 + S) cos [(WM - Sl)t + <f>M] + (1 - 6) cos [(WM + + <£M] } ) + b, where and AM are initial muon and muonium precession amplitudes (usually called asymmetry), R^(t) and .RM(') &ve * n e relaxation of the polarization and and <f>u are the initial phases. The parameters Cl = [^(1 + x2)* — 1] and 8 = x^, where x = (gefie — 9nHn)B/hwQ = B/Bo and Bo = 1585 G, are due to the Zeeman splitting of muonium precession and hyperfine oscillation frequencies. In weak magnetic fields where I < 1 these parameters can be neglected. Diamagnetic muon and muonium absolute fractions PD = A^/Ao and Pu = 2AM/AO are the asymmetries normalized to A0 = A M measured in CCI4 or aluminum. These materials produce the largest diamagnetic signals observed. It is the initial fractions PD a n d -PM and the relaxation functions R^(t) and Ru(t) that are the main interest in many /^SR experiments. Polarization is lost through several processes. Occupation of sites with different local magnetic fields results in de-phasing of the ensemble precession (analogous to T2 processes in NMR). Chemical reactions of fi+ or muonium with the stopping medium and spin-exchange (analogous to Tj processes) are directly reflected in the relaxation functions. Muonium is a unique light isotope of hydrogen that is particularly sensitive to tunneling and has been used in studies of chemical kinetics (see, for example, Reid et al.[8]). Chapter 1. Introduction 6 While the pSR technique is able to observe the evolution of muons and muonium from a few nanoseconds after thermalization until the last muon's decay, the thermal-ization process itself cannot be directly observed. Interpretation of pSK data is also hindered by the inability to distinguish free muons from muons bound in molecules. The diamagnetic muon and muonium fractions vary widely in different media and rarely ac-count for all the muons entering the target. The undetected fraction, }\ = 1 — Po — PM, is called the "lost" or "missing" fraction and is attributed to muons that are de-phased either during or immediately after thermalization. Variations of (and lack of varia-tions of) Po and PM with properties of the stopping medium have been used to make inferences about the thermalization process and are further discussed in a following section. 1.3 Fundamental Physics With Slow Muonium Beams Measurement of the differences between excited states of the hydrogen atom has long been used as a direct test of quantum electrodynamics. Since muonium is composed solely of leptons, the theoretical treatment of muonium is simpler than that of hydrogen. Measurements of the Lamb shift (2s - 2p energy difference) in muonium have used Stark mixing of the 2s and 2p states[9] and RF pumping of the 2s state[10,ll] to cause rapid quenching of 2s muonium in vacuum. The 2s muonium for these measurements was produced by passing a surface (or sub-surface) muon beam through thin metal foils. [12,13] The precision of these preliminary experiments was limited in part by low 2s muonium formation rates. The muon-electron interaction has also been studied with a view to lepton number conservation laws. A multiplicative law would allow the process p+e~ p~e+ (1.8) Chapter 1. Introduction 7 although such a law is not required by the standard model. Detection of this spon-taneous muonium to anti-muonium (Mu —• Mu) conversion would therefore be very interesting. Several experiments have set a progressively lower limit on the magni-tude of the coupling constant for this interaction. In these experiments thermal muo-nium was produced by a muon beam passed through gas,[14] SiC>2 powder targets in vacuum[15,16,17] or thin aluminum foils.[18] If present, p~e+ could then be identified by detection of characteristic muonic x-rays following absorption of the p~ by the Mu production gas,[14] or another target downstream,[16,18] or by radiochemical analysis of elements transmuted by p~ capture.[15] In these searches for (at best) an extremely rare event, it is ultimately the number of useful muonium atoms formed in vacuum that determines the final result. However, to quote Bowen, "There appears to be a need for more work to find a more efficient and convenient source of slowly moving muonium atoms in a vacuum, a source based on methods better than those that use thin, hot Pt foils, thin Au foils, thin layers of Si02 powder, or low momentum surface muon beams through degrader foils." [19] 1.4 Muonium Formation Since the use of muonium is central to these current areas of muon physics, it is not surprising that muon thermalization and muonium formation continue to be active topics also. Two models describing qualitatively the thermalization of initially energetic muons and formation of muonium were developed, based largely on the results of pSW experiments. Chapter 1. Introduction 8 1.4.1 Spur Model Following the first observation of muonium in degassed water by S^R[20] the diamag-netic fraction (free p,+ and //+-substituted molecules) was measured in the presence of electron scavengers. [21] The increase in PQ with [NOj] was attributed to an increased probability of thermal, unsolvated muons becoming hydrated and subsequently form-ing MuOH by fast proton transfer. The spur model proposed to explain this behavior pictures the p,+ losing energy near the end of the track by the creation of free electrons, ions and radicals in a radiation spur. [21,22] The thermal muon in the vicinity of the terminal spur can form muonium by recombination with a free electron. This model is borrowed from the analogous theory of positronium formation when positrons are injected into condensed media. [23] 1.4.2 Hot Atom Model The alternative hot atom model[4,5,6,24] proposes that the eventual distribution of muons among various states is determined by processes that occur while the //+ is still losing its initial kinetic energy. At high velocity (as from a surface muon beam) the muon should behave as any fast charged particle and undergo energy loss by Bethe-Bloch ionization of the medium; no significant amount of muonium should form until the kinetic energy has dropped to approximately 35 keV and the muon velocity is comparable to the orbital velocity of electrons of the medium. Then, in non-metals, charge exchange collisions become important as the muon undergoes a rapid series of ~100 electron pickup and stripping cycles, fi+ + A—> Mu + A + Mu + A ^ ^ + + A-|-e~. (1.9) Chapter 1. Introduction 9 At approximately 30 eV charge exchange is no longer dominant and most muons are expected to emerge from this stage as muonium in most insulators. Thermalization of muons and muonium atoms will continue by elastic and inelastic collisions and hot atom reactions which may, depending on the stopping medium, produce //'•'-substituted molecules and/or molecular ions. The final distribution of muon charge states will be determined by the reactions that p+ and Mu undergo in the last few steps. 1.4.3 The Spur Model versus Hot Atom Model Debate Since the spur model was first proposed for p+ thermalization in condensed media a number of arguments have been made against it. Walker[25] has pointed out that Ps (for which the spur model is believed to apply) and Mu should behave differently due to the small positron mass (me = ^)- One important consequence of the smaller mass is that the vacuum ionization potential of Ps is only 6.8 eV, compared to 13.6 eV for Mu (and hydrogen). Charge exchange collisions can occur only when energetically possible. Many materials such as hydrocarbons and water have ionization potentials intermediate between those of Ps and Mu, so charge exchange can proceed down to thermal energies for muons but not for positrons in these materials. Further, the linear energy transfer (LET) of positrons is believed to be much higher than that of muons, so it is more likely that positrons lose their last few hundred eV in a single large spur. The light positron should also be more easily and rapidly drawn to free electrons in the spur. Ito et a/. [26] measured in hydrocarbons and quartz subjected to electric fields up to 20 kV/cm and 60 kV/cm respectively. Electric fields have been shown to inhibit Ps formation,[27] presumably due to the electric field separating the positron from free electrons of the spur. The absence of any effect at all on Pn was concluded to Chapter 1. Introduction 10 be consistent with the hot atom model. If the spur mechanism were responsible for muonium formation, the spur size would have to be much smaller than that involved with positronium formation. Measurements of PD and PM in hydrocarbons, aqueous solutions of electron scav-engers, solvent mixtures and halogenated hydrocarbons have been made in efforts to find correlations with physical properties of the target material. The effects of electron scavengers in particular have been debated at length.[25,28,29,30,31,32] Very few clear trends emerge from the accumulated data. In water, saturated hydrocarbons and alcohols Pb is always about 0.5 - 0.7, despite widely varying dielectric constants and electron mobilities, which would be expected to influence PD if the spur mechanism were important. PD depends on the nature of bonding in the moderator molecules. It decreases with the presence of 7r bonding and electron derealization. Benzene is an example of this with PD = 0.17. Subtitution of H with halogens increases PD , as in the extreme case of CC1 4. Jean et al. [33] measured PD in mixtures of CCI4 and cyclohexane with benzene. (PD is 1.0, 0.69 and 0.15 in CC1 4, cyclohexane and benzene respectively). The almost linear dependence of PD on volume fraction, except at small fractions of CCI4, was argued to be qualitatively unlike that expected if the spur mechanism were at work. Further, it was concluded that in these mixtures hot atom reactions contributing to PD must proceed by direct (single step) reactions of muonium with a molecule, without competitive reactions or inhibition due to other molecules. The only consensus from the accumulated data is that the thermalization process is complex and depends on the detailed nature of the medium. Different properties and mechanisms are likely important to thermalization in different materials. Chapter 1. Introduction 11 1.5 A New Approach The purpose of this experiment is to contribute to the study of low energy muons on two fronts. First, to determine whether a few-eV muon beam can be produced by moderating an energetic (surface) muon beam with selected target materials. Second, to measure the energy spectra of muons emitted from these targets. Both of these objectives are guided by results from positron physics. Mills and Crane[34] recently found that positrons injected into ionic solids are emit-ted from the solid surface with kinetic energies peaked at 1 to 3 eV and maximum energy near the electronic band gap energy. They suggested that the positrons are Auger emitted when Ps, formed during thermalization, diffuses to the surface where the electron falls into an empty surface state. It has been argued that this is an unlikely mechanism because the Ps fraction in the bulk material should increase with the energy of the incident positrons since more free electrons are generated along the track.[35] A larger Ps fraction (formed by recombination - i.e., a spur mechanism) in the bulk material should lead to increased yields of Ps at the surface. This would result in more Ps and positrons being emitted, contrary to the result reported. An alternative "hot positron" mechanism suggested considers that diffusion lengths may change enormously when the positron kinetic energy drops below the threshold for positronium formation. Above this threshold energy is lost mainly by ionization of the medium. Once below this threshold positrons lack sufficient energy to form positronium and can lose energy only by elastic scattering. Consequently the much lower energy loss rate enables these positrons to reach the solid surface, where they are emitted directly, relatively easily. Gullikson and Mills[36] obtained high yields of positrons injected into neon, argon, krypton and xenon targets. Kinetic energies of the positrons emitted were less than the inelastic threshold. Measurements of positron and positronium re-emission yields \ Chapter 1. Introduction 12 with positrons injected at 0 - 30 eV identified the inelastic threshold as the threshold for positronium formation. These results are consistent with the "hot positron" model. They also noted that the yield far exceeded the positron fraction in argon and so cannot be explained by Ps diffusing to the solid surface. Small diamagnetic (but not necessarily free) muon fractions have been measured in solid argon, krypton and xenon to be 0.8±0.2%, 1.4±1.8% and 5.0±3.3% respectively. [37] Since the formation of muonium in rare gases appears to be strongly correlated with the relative ionization potentials of the medium and muonium, results similar to the positron experiments may be expected for muons in media with ionization potentials comparable with or greater than that of muonium. The present work seeks to determine whether slow positive muons can be efficiently extracted from various targets with high ionization potentials and to measure the energy spectra of non-thermal muons emitted into vacuum from the surfaces of these targets. i Chapter 2 The Experimental Apparatus 2.1 Overview The apparatus constructed for this experiment is shown schematically in Fig. 2.1. It is a high vacuum system incorporating a scattering chamber (containing a target) and a short tertiary beam line. Surface muons (nominally 28 Mev/c) enter the apparatus by passing through a beam counter assembly and a 0.025 cm stainless-steel vacuum window to reach the front of the target. Charged particles emerging from the back surface of the target are collected by an electrostatic lens and accelerated through a potential drop of 10 kV before entering a magnetic spectrometer. The magnetic spectrometer consisted of a 10 inch dipole followed by two 4 inch quadrupole electromagnets. The dipole magnet provided charge and momentum selec-tion while the quadrupole magnets focussed the 10 keV muons onto a microchannel plate detector. The principle measurement made was the time of flight of each muon from the beam counter to the microchannel plate detector. A pair of scintillator tele-scopes surrounding the channel plate at the end of the spectrometer, subtending a solid angle of about ^ sr (four sides of a cube), was used to detect positrons from muons decaying at the channel plate. Copper sheets approximately 0.3 cm thick inserted between the scintillators of each pair were found to reduce the background count rate. The target and electrostatic lens were oriented so that the lens axis was normal to the back of the target and the incident beam made an angle of 45° to the front 13 Chapter 2. The Experimental Apparatus 14 C H A N N E L P L A T E Q2 V. JJ VI J \ I > Qi Q L ' A D R U P O L E M A G N E T S 50 cm Figure 2.1: Schematic of the apparatus, viewed from above. Feedthroughs on the scattering chamber lid are omitted for clarity. Chapter 2. The Experimental Apparatus 15 surface of"the target. This geometry was chosen to reduce the number of energetic range-straggled muons entering the spectrometer. The apparatus is operated in two modes. By applying different voltages to the target and lens the electric field in the gap may be controlled. With the target held at +10.0 kV and the lens at +9.5 kV, the time of flight of low energy muons (< 50 eV) is nearly independent of initial kinetic energy. The resulting time of flight histogram establishes a timing standard used in the analysis of the data. With the front section of the lens set at 9.990 kV and the target biased +10 V with respect to the lens the time of flight varies significantly with initial energy. A retarding field may also be produced by setting the lens at 10 kV and biasing the target to a lower potential. In the future this technique may be useful in obtaining higher resolution muon emission spectra if higher event rates can be produced. 2.2 Ultra High Vacuum Methods In order to minimize the adsorption rate of residual vacuum gases on the target surfaces, it was desirable to operate the experiment in the best vacuum obtainable. The vacuum system built for this experiment was constructed of stainless steel and all but the most frequently broken seals were metal. Target and detector mounts and valves were sealed with Viton elastomer seals. Mounted below the scattering chamber were a gate valve, a radiant heat baffle, and a CTI Cryogenics Cryo-Torr 8 pump providing the main pumping. The baffle prevented radiant heat from hot targets from warming the uppermost condensing array in the pump. A Ti-Ball titanium sublimation pump, with a liquid nitrogen trap, was mounted on one of the 4 inch ports of the target chamber. This pump provided pumping for active gases (water and hydrogen) while cryogenic targets were being condensed. The I Chapter 2. The Experimental Apparatus 16 cryopump gate valve was closed during condensation of these targets. Vacuum diagnostics included UHV ionization gauges mounted in the top of the target chamber, in the heat baffle and near the Ti sublimation pump. A mass spec-trometer residual gas analyser positioned close to the target was used extensively in determining vacuum quality, for leak checking and for cryogenic target preparation. Prior to on-line runs the vacuum system was baked using heat tapes wrapped around the target chamber, ports and the spectrometer. The use of a number of Viton elas-tomer seals limited the maximum baking temperature to 150°C. A clean vacuum was considered to be a vacuum in which the residual gas analyser measured as much hydro-gen as water, and corresponds to a base pressure of 2 x 10 - 1 0 torr. During on-line runs however, the system was often opened and time constraints did not allow re-baking. Consequently, during runs the pressure was in the 5 x 10~9 to 5 x 10 - 8 torr range, the main contaminant being water. 2.3 Detectors The incident p+ beam counter assembly was mounted on a re-entrant port of the scattering chamber in order to locate the scintillator as close as possible to the target. It placed a 0.004 cm aluminized mylar mirror, 0.025 cm plastic scintillator (2.5 cm x 2.5 cm) and two 0.0013 cm aluminum foil heat reflectors in the beam path. Water cooling coils wrapped about the assembly and a thermocouple next to the scintillator provided additional protection from radiant heat during use of heated targets. Vacuum pumping for the assembly was provided by the muon channel. The incident beam counter was viewed by a single photomultiplier tube via the vacuum light guide and the mirror. A 4 cm diameter microchannel plate detector (MCP) (Galileo model 3040 chevron type) with a pulse amplifier was used to detect ~10 keV muons at the end of the Chapter 2. The Experimental Apparatus 17 spectrometer and surface muons when mounted on the scattering chamber opposite the beam counter. It is difficult to estimate the detection efficiency of the MCP for slow positive muons. Efficiencies for 2-50 keV electrons and protons are 10 - 60% and 60 -85% respectively.[38] Tests of the MCP used in this experiment, with electrons emitted from a filament, produced the broad pulse height distribution characteristic of these detectors. No measurement of detection efficiency was made. Assuming detection efficiencies of 90% and 50% for the beam counter and MCP respectively, the overall detection efficiency of the TOF spectrometer is about 38% for muons emitted from the target surface with zero kinetic energy. 2.4 Target mounts Two target mounts were used, one for solid materials and one for condensed gas targets. For the solid targets a mount was used that held three 5 cm x 5 cm targets. A screw drive on this target holder enabled each target to be positioned in the beam without venting the vacuum. Two coaxial heating elements silver-soldered into the edge of the stainless steel frame allowed heating of these targets to temperatures of 200 - 250°C in order to increase the ionic conductivity of lithium fluoride targets and to maintain clean emitting surfaces. Rare-gas targets were condensed onto a cold finger. The cold finger consisted of a stainless-steel cell with 0.025 cm thick windows of 2.2 cm x 3.5 cm area, spaced approximately 0.4 cm apart. Cooling was provided by cold helium gas continuously flowing through the cell, using an electrically isolated transfer line. Chapter 2. The Experimental Apparatus 18 2.5 Accelerating Lens and High Voltage The high voltages applied to the target and lens accelerated positively charged particles through a potential drop of 10 kV from the target to the exit aperture of the lens. The lens, shown in Fig. 2.2, was axially symmetric, with 5 cm diameter openings at each end, covered with a fine wire screen. A high voltage bias supply was used to set the potential of the lens front section. Low electric fields in the accelerating gap were produced with a variable battery powered supply to bias the target with respect to the lens. For 500 V-biased measurements, separate high voltage supplies were used to hold the target at 10.0 kV and lens at 9.5 kV. The middle section of the lens was always set at 4.5 kV, using another high voltage supply. The exit aperture of the lens was grounded. Chapter 2. The Experimental Apparatus 19 ACCELERATION GAP I SCREEN TARGET SCREEN 5 cm i i i i Figure 2.2: Cross section through the target and lens, along the lens axis. Surface muons are incident from the left. Chapter 3 Measurements 3.1 Procedure Data is presented in this chapter for targets of polycrystaline copper, single-crystal lithium fluoride (LiF), quartz, solid rare gases argon, krypton and xenon, and the bare stainless-steel cold finger. Runs on targets of polycrystaline copper and stainless steel were taken to provide a baseline since a band-gap dependent emission mechanism is not possible in metals. LiF and quartz have previously been studied with this apparatus,[39] but without the use of the 10 V-biased mode. From these previous measurements it was concluded that LiF and possibly quartz produced enhanced emission at less than 10 eV. But this conclusion is in doubt since the apparatus cannot resolve these low energies when run in the 500 V-biased mode. The LiF and quartz targets were 0.018 cm and 0.020 cm thick and cut with <100> and c-axes, respectively, normal to the target surfaces. Thicknesses of these solid materials were chosen and shimmed with 0.0015 cm aluminum foil so that the momentum required to just penetrate them was a few percent less than the surface muon edge at 29.8 MeV/c. Solid rare-gas targets were condensed onto the cold finger by manually operating a leak valve to obtain a pressure of 1.5 x 10 - 5 torr (measured with the residual gas analyzer) for 15 seconds. The temperature of the cold finger was not measured; the cold helium gas flow rate was simply maintained so that the condensed target did 20 Chapter 3. Measurements 21 not evaporate. Unfortunately, attempts to collect data with a neon target were not successful. In order to condense neon it was neccessary to flow liquid helium through the cold finger cell. With a mixture of boiling liquid and gas in the cell it was not possible to tune the incident beam momentum properly. Surface muons from the M20(B) or M13 channel at TRIUMF were incident on the targets at a rate of ~300,000/s with momentum width Ap/p ~ 4%. A positron telescope external to the scattering chamber was used to time the beam momentum to maximize the stopping density at the down-stream surface of each target. The thin beam counter provided very good discrimination between positrons and muons by pulse amplitude. In order to achieve high noise rejection with a low event rate the time of flight (TOF) of each muon arriving at the microchannel plate was measured with a reversed time of flight method. Fig. 3.1 shows the electronics schematically. In the reversed TOF method a clock is started on the microchannel plate signal and stopped on the delayed beam counter signal. The delay was about twice the TOF. A pile-up gate vetoed events in which a second muon triggered the beam counter within 1.5 ps, about twice the delay. Pile-up rejection reduced the useful beam rate to ~200,000/s. Resolution of the TOF clock was 1.25 ns. For the histograms shown, a good event did not require detection of the decay positron by the scintillators around the microchannel plate. Calibration of the beam counter delay period was obtained by relocating the mi-crochannel plate and preamplifier to the port opposite the beam counter on the scat-tering chamber. With targets removed from the beam path, the calculated time for surface muons (degraded by the beam counter and vacuum window) to traverse the chamber is 6 ns. Fig. 3.2 shows the delay calibration histogram. The true time of flight scale is calculated from the measured (reversed) time scale by the transformation *TOF = —<REV + d (3.1) Chapter 3. Measurements 22 where d = 755 ns is the delay period. This histogram also indicates that the timing resolution of the detectors and electronics is approximately 4 ns. Chapter 3. Measurements 23 TR2 TR, TL, d d d d TL, ^ d Good e+ Start2 Stop2 Startl Stopl CAMAC TDCs COMPUTER ® d PUG MB -750 ns V Figure 3.1: Data aquisition electronics. Chapter 3. Measurements 24 - 3 0 -20 -10 0 10 Time of f l ight, ns 20 30 Figure 3.2: Histogram for reversed time-of-flight delay calibration. Chapter 3. Measurements 25 3.2 T h e T i m e of Flight Da ta TOF histograms obtained for each target are shown in Figs. 3.3 - 3.8. In each, a uniform background (the average rate in the interval between 75 and 175 ns, typically 0.3 per 108 useful incident muons) has been subtracted and time scales transformed from reversed to true TOF. Errors shown are la derived from counting statistics. Runs taken with a target-lens bias of 500 V contain a peak at about 368 ns. TOF histograms for targets of LiF, quartz and Cu also contain tails to faster times. With a bias of 10 V the TOF histograms for Ar, Kr and Xe are broadened and shifted toward longer times compared to the 500 V data. Histograms for LiF and stainless steel are slightly broadened, however they remain peaked at about 370 ns. The kinetic energy of these muons is evidently sufficient to make the time of flight across the gap nearly independent of the target-lens bias. Clearly, the solid rare gases produce emission spectra with a larger low-energy component than LiF. Extracting an emission spectrum from the TOF data requires a relation between energy and TOF. However, due to the wide angular acceptance of the accelerating lens there is no direct relation between energy and time of flight. An estimate of the emission spectrum has been made using comparisons of the data with a Monte Carlo simulation, which is the topic of the following chapter. Slow muon production efficiencies calculated by integrating over the 500 V time of flight histograms (except for Xe and stainless steel, for which only 10 V runs were taken), normalized to the incident beam, are given in table 3.1. Errors indicated are la, from counting statistics. These observed rates do not consider detection efficiency, since the efficiency of the microchannel plate to ~10 keV positive muons is not well known, or acceptance of the spectrometer. It should also be noted that the cryogenic targets had a useful emitting surface area 1/3 that of the solid targets. Chapter 3. Measurements 26 .32 .34 .36 .38 'TOF' .40 .42 .44 .46 TOF' Figure 3.3: Time of flight histograms with LiF target. Chapter 3. Measurements u Q-t 2.0 1.5 -d - 1.0 -O 0.5 -* o.o -0.5 .32 .34 .36 .38 .40 .42 .44 .46 Figure 3.4: Time of flight histogram for copper target. Chapter 3. Measurements 28 o o 2.5 2.0 1.5 1.0 & 0.5 0.0 -0.5 1 1 l l l l T S i 0 2 Jjj 500 V [ H - llll ;„ I 1. ,<""1 ! 1 l l l l .32 .34 .36 .38 .40 .42 .44 .46 t T 0 F , pts Figure 3.5: Time of flight histogram for SiC"2 target. Chapter 3. Measurements 29 40 •30 o 220 0 1 1 T 1 1 i i Ar 500 V J J i i 1 i .32 .34 .36 .38 .40 .42 .44 .46 Figure 3.6: Time of flight histograms with solid argon target. Chapter 3. Measurements 30 8 \ 6 ci J 4 CO O q; 2 0 -2 25 1 1 1 1 1 1 Hi K r { Ti ft ll ( I X I _ • i i i i i i .32 .34 .36 .38 .40 .42 .44 .46 t T 0 F , fis + 20 6 i c co 2 io a) * 5 * 0 I l l l I I 1 K r h 500 V~ I i i ? I I I I .32 .34 .36 .38 .40 .42 .44 .46 t T 0 F , Ats Figure 3.7: Time of flight histograms with solid krypton target. Chapter 3. Measurements 31 Target Rate LiF 37±3 Quartz 23±1 Cu 18±1 Stainless Steel 29±2 Ar 175±4 Kr 93±3 Xe 35±2 Table 3.1: Observed slow muon emission rates, normalized per 10* incident muons. • i-H ID O 0 1 1 I I I I Xe I 10 V_ i i \ I " HWiV.ili.il.. -l l l l .32 .34 .36 .38 .40 .42 .44 .46 Figure 3.8: Time of flight histogram with solid xenon target. Chapter 4 Monte Carlo Simulation 4.1 Monte Carlo Procedure The beam transport properties of the electrostatic lens and magnetic spectrometer were studied by calculating the trajectories of muons from the target surface to the channelplate detector. This Monte Carlo calculation simulates only the path of muons from the target surface and does not consider in any way the processes involved in energy loss in the target material. The result of the calculation allows relations between initial energy, position, direction and time of flight as well as acceptance functions to be examined. The calculation was performed for target-lens biases of 500, 10, 0, -10, -20, -40 and -70 V, with negative potentials denoting retarding fields. The calculation proceeded through the apparatus as follows: first muons were generated at the target with desired distributions in coordinates and energy. Next, each trajectory was traced through the electric fields between the target and lens and inside the lens. A second program then calculated the coordinates and momentum components after each element of the magnetic spectrometer. Any muons that struck the walls of the vacuum system or electrostatic lens was considered lost. All calculations were performed on the TRIUMF VAX cluster using a number of programs. Most important were RELAX3D,[40] TRIWHEELRE[41] and REVMOC,[42] all of which are used at TRIUMF for cyclotron and beamline design. RELAX3D solves Poisson's equation over a volume for given boundary conditions. The program calculates the 32 Chapter 4. Monte Carlo Simulation 33 potentials at points on a grid by an iterative method that sets non-boundary points to the mean of nearest neighbors. TRIWHEELRE is a ray-tracing program that calculates step-wise the trajectories of particles subjected to electric fields. REVMOC calculates the coordinates in phase space of particles after each element of a beamline. This program does not trace particles in a step-by-step fashion along a trajectory. Instead the effect of each element of the beamline is calculated as a transformation of particle coordinates and momenta. This is much faster than would be possible by the step-wise method used by TRIWHEELRE. Additional short programs were written to prepare the output of each program for input to the next. Electric potentials inside the electrostatic lens and in the target-lens gap were cal-culated with RELAX3D. The input to the program consisted of FORTRAN subroutines that set fixed potentials on a grid that defined the shape of the stainless steel target and lens. Cylindrical symmetry of the lens was used to reduce computation time. Res-olution of the grid defining the lens was 0.1 cm. In the target-lens gap a rectangular grid with 0.05 cm spacing was used. For each calculation the program was allowed to run to convergence. The cumulative error over the lens grid was typically 10 - 7 V. The output of each field calculation was a list of the potentials of all grid points. Field calculations were performed for each bias voltage and target geometry. Using the program TRIWHEELRE random initial coordinates, energy and emission angles of particles at the target surface were generated. For all simulations the initial coordinates of the generated muons were uniformly distributed over a flat surface with dimensions as measured from the targets. This is a simplification in two ways. First, the cryogenic target surface (the downstream beam window) was not flat but bulged slightly making the edges of the target surface further from the lens than the central portion of the target. Second, the actual distribution over the target depends on the incident beam profile, which is not uniform. However, the central portion of the Chapter 4. Monte Carlo Simulation 34 cryogenic target, where the beam was most intense, was reasonably flat. Muons were generated with random initial energies uniformly distributed over a range, usually 0 to 500 eV. Random emission angles determining the initial direction of each muon were generated. Azimuthal angles were uniformly distributed over the interval 0 to 360°. The angle between the initial trajectory and the lens axis was generated with a cos2 9 dependence. Using the potential grid produced by RELAX3D to calculate electric field compo-nents, TRIWHEELRE traced the path of each muon from the target surface to the front surface of the electrostatic lens. The step size was the smaller of 0.025 cm or the dis-tance travelled in 0.25 ns. For each muon that reached the lens the coordinates, energy, velocity components and time of flight and the initial coordinates, emission angle and energy were written to a file. A program sorted out those muons with coordinates that did not fall within the 5.0 cm diameter entrance aperture of the lens. On a second pass through TRIWHEELRE the calculation continued through the lens interior, beginning with the intermediate results of the first stage. Again the electric field was calculated from the potentials provided by RELAX3D. The step size was increased to 0.5 ns or 0.05 cm in order to reduce computation time. The accuracy of the integration of the field along the path was found to depend on the step size. Ideally, the final energy of each muon should be exactly 10000 eV greater than its emission energy. The step size limits were set so that the width of the error distribution was approximately 2 eV. This was considered acceptable compared to the 10 keV minimum energy and corresponds to 0.01% error in velocity and time of flight. For each muon that was traced to the end of the lens the coordinates, momentum components, energy, time of flight and initial coordinates, energy and 6 were written to a file. Another short program was used to translate the output of TRIWHEELRE to a format readable by REVMOC. REVMOC was used to determine the final coordinates and time of Chapter 4. Monte Carlo Simulation 35 flight at the end of the spectrometer for each of the muons remaining after TRIWHEELRE. The physical dimensions of apertures, drift regions, magnets and magnet types and field strengths were defined in the input file for REVMOC. The exit aperture of the lens was the first element. The final output of the Monte Carlo procedure was a file constructed from the output of REVMOC and the output of TRIWHEELRE. For each particle accepted at the detector it contained initial and final coordinates, initial energy, initial 9 and time of flight from the target to the detector. 4.2 Monte Car lo Results Fig. 4.1 shows the time of flight, calculated by the Monte Carlo for 10 V and 500 V biases, plotted against initial kinetic energy. With a bias of 10 V the TOF of muons emitted with less than 100 eV varies about 70 ns, while at 500 V these muons all arrive at the microchannel plate in 360 - 365 ns, a range approximately equal to the spectrometer's resolution. Muons emitted along the lens axis fall on the lower edge of the scatterplot. At 10 V, muons emitted at angles to the axis have a longer TOF. Whether or not a given muon is accepted depends on its initial location, angle to the lens axis and energy. In each Monte Carlo calculation the initial energies were uniformly distributed by the program TRIWHEELRE. A histogram of the energies of the particles in each Monte Carlo output is then the acceptance as a function of energy, for a particular target-lens bias. Figs. 4.2 and 4.3 show the acceptance functions for 500 V and 10 V target-lens biases. Chapter 4. Monte Carlo Simulation 36 440 420 h -g.400 ° 3 8 0 a) a ^360 340 ^ i * - - . . , 10 V 500 V I 0 100 200 300 400 emiss ion energy, eV 500 Figure 4.1: Calculated time of flight as a function of emission energy with 10 V and 500 V target-lens biases. Chapter 4. Monte Carlo Simulation 37 0 100 200 300 400 500 Emiss ion energy, eV Figure 4.2: Acceptance as a function of muon kinetic energy with 500 V target-lens bias. Chapter 4. Monte Carlo Simulation 38 0.2 i i i i i I 0 100 200 300 400 500 Emission energy, eV Figure 4.3: Acceptance as a function of muon kinetic energy with 10 V target-lens bias. Chapter 4. Monte Carlo Simulation 39 4.3 Application of the Monte Carlo In principle it is a simple exercise to determine the kinetic energy of a particle given its time of flight over a known distance. For this experiment such a direct correspondence between time of flight and energy is obscured by the wide angular acceptance of the apparatus, required by the low rates involved. Time of flight depends on the trajectory taken through the accelerating electric field. Results of the Monte Carlo show that for few-eV muons the time of flight in the apparatus depends on the initial direction of emission. Two muons with the same energy, one along the lens axis and the other at an angle to the axis, will take different times to transit the apparatus. The axial path is shorter and results in the fastest time of flight for that energy. If the muons were emitted over a range of energies and angles, each will have at least the time of flight of a muon of the same energy along the axis. Timing resolution, limited by detector and electronics, also contributes to the width of the time of flight histograms. The resulting histograms are spread by both of these effects and preferentially to longer time. Muons emitted off-axis may, depending on initial position, direction and energy, strike the lens or walls of the spectrometer. In order to extract an emission spectrum from the data these effects must be taken into consideration. The Monte Carlo calculation samples a large number of trajectories, with muons emitted into a hemisphere from all points on the target surface. The output of the Monte Carlo calculation, performed for a particular target bias, is a file containing the initial coordinates, energy, emission angles and time of flight of each particle accepted at the detector. The effects of path length and time of flight differences and acceptances of the electrostatic lens and spectrometer are included in the Monte Carlo output. From this file a time of flight histogram may be made. Since the distribution of initial energies Chapter 4. Monte Carlo Simulation 40 assumed in the Monte Carlo determines the shape of this histogram, a real emission spectrum may be deduced by finding an initial energy distribution that results in the Monte Carlo reproducing the data. This could be done by repeating the whole Monte Carlo procedure with different distributions until the best fit is found, but this would be a tedious task. The same outcome can be obtained by simply weighting the muons in the Monte Carlo TOF histogram according to their initial energies, then finding those weights that fit the Monte Carlo result to the data. The relative weights are then the relative intensities, as a function of energy. The muon emission spectra from Ar and Kr targets were estimated by simulta-neously fitting both the 10 V and 500 V histograms generated by the Monte Carlo calculation to the time of flight data. In order to fit the 500 V data and Monte Carlo histograms, small shifts of the Monte Carlo time scale were allowed and the Monte Carlo histograms were also convoluted with a normal distribution of 7.5 ns width. Forcing the 500 V data and Monte Carlo histograms to be coincident in time greatly reduces the effect of a systematic error in calculated time of flight. In this way it is the difference in time of flight between the 500 V and 10 V data that determines the emission spectrum. That the Monte Carlo histograms needed to be "blurred" in order to fit the 500 V data suggests that there were variations in time of flight not considered by the Monte Carlo procedure. This is expected since timing resolution, jitter in detectors and electronics, curvature of the muon emitting surface and noise on the high-voltage power supplies all contribute to broadening the measured time of flight histograms. Fig. 4.4 shows the Monte Carlo result (solid lines) obtained assuming the muon emission spectrum shown in Fig. 4.5, superimposed on the measured time of flight data for Ar. The relative intensities in 18 energy bins from 0 to 500 eV were found by x 2 minimization, subject to the spectrum varying smoothly. The errors shown in Figs. 4.5 and 4.6 are under-estimates since they are uncorrelated errors. Chapter 4. Monte Carlo Simulation 41 50 I 1 1 1 1 1 r 320 340 360 380 400 420 440 460 t ime of flight, ns Figure 4.4: Measured Ar data and Monte Carlo generated time of flight histograms. These spectra should be considered qualitative estimates only, since it is possible to choose other emission spectra that fit as well. There is no doubt however that the peak at 5 - 20 eV and tail to higher energy, for both Ar and Kr, are generally correct. Chapter 4. Monte Carlo Simulation 42 co PS a) > •—I u 40 60 energy, eV 100 Figure 4.5: Muon energy spectrum with argon target. Chapter 4. Monte Carlo Simulation 43 1.0 h S °-a 0.6 > 'a) 0.4 h 0.2 0.0 0 20 40 60 energy, eV 80 100 Figure 4.6: Muon energy spectrum with krypton target. Chapter 5 Interpreting the Results 5.1 Comparison of Muon and Positron Energy Spectra The interesting feature in the muon energy spectra from targets of Ar and Kr is the presence of a broad peak in muon emission rate at 5 - 20 eV, with a tail to higher energy. Unfortunately the resolution obtained in these spectra is poor and it is not possible to correlate any features with obvious thresholds or mechanisms. These spectra are however qualitatively very different from those obtained by Gullikson and Mills[36] with positrons injected into solid rare gases. Positrons are efficiently emitted with energies up to the threshold for positronium formation, a few eV less than the band gap energy. In the gas phase the muonium formation threshold energy is simply where E\ is the ionization energy of the gas, so there is no threshold in most gases, with the exception of He, Ne, Ar and Kr (see Table 5.1). In the solid phase of the rare gases this becomes where Es, the band gap of the solid, replaces E\ and E\> is the muonium binding energy in the solid. Eb is probably a few eV less than 13.6 eV. There may also be a contribution to JE^th from the difference of bare muon and muonium work functions or affinities. Thus Eth = E{- 13.6eV, (5.1) Eth = Eg — E\, (5.2) 44 Chapter 5. Interpreting the Results 45 in solid Ar the minimum muonium formation threshold is estimated to be about 1 -5 eV. In order to escape the solid surface the muon would also have to overcome the work function. If this threshold were dominant in determining the probability of p,+ emission, one would expect a spectrum similar to that observed with positrons: muons emitted with energies up to a threshold. The high energy tail obtained is not consistent with a mechanism in which muons, having insufficient energy to form muonium, emerge from the target surface into vacuum. 5.2 The Role of Charge Exchange Processes We can at least speculate on the origin of the spectra observed. While a muon slows from about 30 keV down to about 30 eV the dominant energy loss process is expected to be charge exchange with the medium. During this period the muon will cycle between bare p+, neutral muonium or the negative muonium ion (Mu~ = p+e~e~). Six different single- and double-electron pickup and loss collisions with neutral atoms of the stopping medium are possible, as indicated in Fig. 5.1. The equilibrium fractions fx, fo and /_ i (XT/i = 1) of p,+, Mu and Mu~ can be obtained from the solution of where the subscripts of the cross sections {i, j, fc}={l,0,-l} indicate the charge of initial and final states. In terms of charge exchange cross sections these fractions are -T- = 0 = —fi(cTij + <7,fc) + fjCTji + fkCTki (5.3) fl = ( c - n O o i + < T _ n c r 0 _ i + c r _ 1 0 C T o i ) / a (5.4) fo = (cr_n<r10 + <T_ 1 0(T 1 0 + a_10<Ta_1)/a (5.5) f-i = ( o i r j C o - i + <7i-i0oi + o-\-\0~o-\)/a (5.6) Chapter 5. Interpreting the Results 46 Mu CT01=10 c r_ l 0 =5x10 ' a l 0 = 1 0 cr 0 _,=2x10 a_„=10 Mu Figure 5.1: Charge exchange collisions of p+, Mu and Mu with neutral atoms of the medium. Approximate cross sections (in cm2) at 1 keV in Ar are indicated. where a = 0_ii<7oi + 0 " - i i 0 o - i + 0"-io0oi + 0io0"-n + crio<7_i0 + O l - l < 7 - 1 0 + ClO^O-l + Ol-lOoi + (Tx.xCTo-l If the Mu" fraction is ignored (o"i_i and <70_i assumed negligible), Eqs. 5.4 and 5.5 reduce to fo = 0oi 010 + 0"oi 0"io (5.7) (5.8) 0"io + 001 In the pSK context this form is adequate since / _ i is very small and Mu" is indis-tinguishable from free p+. However in the present work the negative fraction should not be neglected since p+ yields are also very small and comparable to reported Mu" yields. [39,44] Chapter 5. Interpreting the Results 47 It is generally assumed that charge exchange cross sections depend on the velocity of the incident particle and not explicitly on its mass. In the absence of cross sections for fi+, Mu and Mu" in these target gases, cross sections for p + , H and H~ at the same velocity (Ep = -^E^ = 8.88£JM) provide the best available estimates. Unfortunately, velocity scaling of cross sections cannot be expected to be valid down to thermal energy. Also, known cross sections involving H~ do not extend below about 1 keV equivalent muon energy. Nonetheless, it has been noted frequently in the literature (see for example Ref. [5]) that fx calculated this way generally follows the thermal diamagnetic fraction observed by /JSR in gas targets. In Ar, Kr and Xe values of fi calculated this way are about 15%, 0.5% and 0.3% respectively as the kinetic energy approaches 1 keV or less. Yields of low energy muons reported here (see Table 3.1) also follow this trend. Unfortunately, since cross sections are not known at low energy (<100 eV), it is not possible to determine whether the peaks observed in muon energy spectra are a direct result of charge exchange processes increasing f\. Similarly calculated neutral fractions follow the overall trend of muonium fractions observed with MSR in these gases. [5] Often, though, the muonium fraction is less than the neutral fraction predicted by velocity-scaled proton cross sections. It has been suggested that hot atom reactions such as: Mu + He / i + + He + e~ - • Mu" + He+ Mu+He + e" (5.9) could account for the reduced thermal muonium fraction in He and Ne gas targets observed by MSR. [45] Such reactions likely occur at all energies but significantly in-fluence the charged and neutral fractions only after charge exchange cross sections are Chapter 5. Interpreting the Results 48 He Ne Ar Kr Xe E.- 24.58 21.56 15.75 13.99 12.13 E * - 21.42 14.16 11.60 9.28 10.98 7.96 2.15 0.39 -Table 5.1: Electronic properties of rare gases. E t are first ionization potentials (at low pressure), E g the band gaps of rare gas solids from Rossler.[47] Eth = E\ — 13.6 eV is the muonium formation threshold in low pressure gases. All energies in eV. diminished, at <50 eV. The small yield of muons at 5 - 20 eV observed here may be muons escaping between the end of charge exchange and the onset of efficient hot atom reactions at < 5 eV, or simply the product of such reactions. Recent calculations by Senba have indicated that processes other than charge ex-change are very important in determining the fate of a muon stopping in a helium target.[46] Inclusion of excitation, ionization and elastic processes in competition with charge exchange results in good agreement of both the thermal neutral fraction and the pressure dependence of residual polarization with experimental results. It therefore appears most likely that the p+ energy spectra observed in this work are determined by other processes in addition to charge exchange during the last few hundred eV of thermalization. Chapter 5. Interpreting the Results 49 PM, /M PL He g 1.2 - 3.1 atm 1 lOOil.O >90 O i l <2.0 <8 Ne g 1.2 atm 93±5 7i5 Ar g 1.0 - 2.9 atm 1 s 26±4 1.6il.0 0.8±0.2 74i4 97i30 91±9 3i29 8i9 Kr g 0.4 - 0.95 atm 1 s 0±5 6.5±0.1 1.4±1.8 100i5 57il0 lOOilO 36il0 OilO Xe g 0.4 - 0.65 atm 1 s 0±4 3.3±0.8 5.0±3.3 100i4 43i9 79i25 54il0 16i28 Table 5.2: Diamagnetic, Mu and "lost" fractions in rare gases measured by //SR. Rela-tive fractions in the gas phase from Fleming et al. [45]; condensed phases from Kiefl et al.[37] and Crane et al.[48]. Chapter 6 Summary and Conclusions Emission of low energy positive muons from solid rare gas targets into vacuum was studied with a time of flight spectrometer. Enhancement of the yield of low energy muons emitted into vacuum has been observed from targets of solid Ar, Kr and Xe. 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