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A µSR study of spin relaxation of small molecules in the gas phase Pan, James Jun 1995

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A / i S R S T U D Y O F S P I N R E L A X A T I O N O F S M A L L M O L E C U L E S I N T H E G A S P H A S E By JAMES JUN PAN B. Eng. (Engineering Physics) Qinghua University, 1983 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF T H E REQUIREMENTS FOR T H E DEGREE OF DOCTOR OF PHILOSOPHY i n T H E FACULTY OF GRADUATE STUDIES Department of Chemistry We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y Q F BRITISH C O L U M B I A December 1995 © James Jun Pan, 1995 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. I 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 of The University of British Columbia Vancouver, Canada Date 2. DE-6 (2/88) Abstract Current understanding of the underlying mechanisms of muonium and Mu-radical spin relaxation in the gas phase is presented. Models and formulae describing the effects of the three contributing processes, spin exchange, chemical reaction and collisional relaxation, on the muon spin polarization are developed and employed to extract reaction rate constants, cross sections and other kinetic parameters in reactions of M u with atoms and small molecules; notably M u + Cs, M u + N O , M u + N 2 O and M u + C O . The experimental data obtained are consistent with theory and the models so introduced. The radical relaxation rates obtained for larger molecules ( M U C 2 H 4 and M U C 4 H 8 ) are well described by the phenomenological model presented, which serves as a useful bridge linking the observed relaxation rates and the physical, chemical and magnetic properties of the reactants and offers invaluable insight into the underlying mechanisms causing muon spin relaxation. The ratios of thermal spin-flip cross sections (<TH/<TMu) in electron spin exchange interactions (Mu + Cs and M u + NO) are found to be about 3, consistent with previous experimental measurements for Mu + N O and M u -I- O 2 systems. Reaction rates for M u + N O , Mu + N 2 O and M u + C O were measured over a wide range of pressures (1 to 60 atm, up to 272 atm for CO) and in one case ( N 2 0 ) over a range of temperatures (300-600 K ) . Dramatic kinetic isotope effects are observed in all of these systems: a small "inverse" effect (^Mu/^H=0.23) in the Mu + N O reaction and large "normal" effects (&MUAH > 100) for the M u + N 2 0 and M u + C O reactions at room temperature. These kinetic isotope effects can be qualitatively understood within the trends indicated by reaction rate theory and the experimental results of the analogous H(D) reactions, but a quantitative comparison with theory must await specific calculations of the M u reaction rates. This thesis data represents the first observation of large tunneling effects at room temperature in H-isotope addition reactions in the gas phase. i i Contents Abstract ii List of Tables vii List of Figures viii Acknowledgements x 1 Introduction 1 1.1 Positive Muons, Muonium Atoms and Muonium-Substituted Radicals 1 1.2 Muon Spin Relaxation and Rotation: the /zSR Technique 3 1.2.1 Slowing Down Process and M u Formation 5 1.2.2 Time Evolution of the Muon Spin State 7 1.3 Interaction of Muonium with Gases . 12 1.3.1 Spin Exchange 12 1.3.2 Chemical Reaction 14 1.3.3 Free Radicals and Collisional Relaxation 16 1.4 The Importance of Isotope Effects 19 2 Experimental and fiSK Techniques 22 2.1 Muon Production and Transport 22 2.1.1 The T R I U M F Accelerator 22 2.1.2 Surface Muons: M15 and M20 Beamlines 24 2.1.3 Backward Muons: M9B Beamline 27 2.2 fiSR Setup 27 2.2.1 Longitudinal Field (LF) 28 2.2.2 Transverse Field (TF) 29 2.2.3 fiSR Spectrometer 30 2.3 Data Acquisition and Analysis 34 2.4 Target Vessels 39 2.4.1 High Pressure Targets 39 2.4.2 High Temperature Target 41 ii i 2.4.3 Target for Radical Studies 41 2.4.4 Cs Target 43 2.5 Reagent Gases 44 3 Spin Dynamics of Muonium and Muonium-like systems 50 3.1 Time Evolution of the Muon Spin in a Diamagnetic Environment 51 3.2 Polarization Function of the Muon Spin in Muonium 52 3.2.1 Time Evolution of the Muonium Spin State in Vacuum 52 3.2.2 Polarization Function of Muonium in Matter 58 3.3 Polarization Function of the Muon Spin in Mu-Radical 64 3.4 Competition among Depolarizing Processes 69 3.4.1 Spin Exchange and Chemical Reaction with Diamagnetic Final State 70 3.4.2 Chemical Reaction with Mu-Radical Final State 73 4 Theoretical Background 75 4.1 Chemical Reaction 75 4.1.1 Reaction Kinetics 75 4.1.2 fiSR and Reaction Mechanisms 92 4.2 Spin Exchange 101 4.2.1 Potential Energy, Phase Shift and Cross Section 101 4.2.2 Isotope Effects 105 4.2.3 fiSR Relaxation Rate 108 4.2.4 Slow Spin Exchange 112 4.2.5 Fast Spin Exchange 114 4.3 Collisional Relaxation (Spin Relaxation of Free Radicals) 115 4.3.1 A Phenomenological Model 115 4.3.2 T i Spin Relaxation 122 4.3.3 T 2 Spin Relaxation 123 5 Results and Their Interpretation 125 5.1 Mu + Cs: Spin Exchange 125 5.1.1 Results 127 5.1.2 Comparison of Cross Sections 132 5.1.3 Summary 137 5.2 Spin Relaxation of Free Radicals 137 5.2.1 Results 138 5.2.2 Interpretation of the Parameters 148 iv 5.2.3 Summary 157 5.3 Mu + N 2 O : Addition/Decomposition and Collisional Relaxation 158 5.3.1 Results 160 5.3.2 Reaction Pathways of H(Mu,D) + N 2 0 168 5.3.3 Comparison of Reaction Rate Constants 174 5.3.4 Summary 182 5.4 M u + N O : Addit ion and Spin Exchange 183 5.4.1 Results 187 5.4.2 Combination Reaction 188 5.4.3 Spin Exchange 200 5.4.4 Summary 201 5.5 M u + C O : Addit ion, Collisional Relaxation and "Spin F l ip " 202 5.5.1 Experimental Results to Date 203 5.5.2 Interpretation 208 5.5.3 Summary and Future Experiments 219 6 Summary and Conclusions 220 6.1 Spin Exchange Interaction 221 6.2 Chemical Reactions 222 6.3 Collisional Relaxation of Mu-radicals 223 6.4 Future . 224 Bibliography 226 Appendices 246 A Theoretical Details 246 A . l Spin Dynamics, a Boltzmann Equation Approach 246 A . 1.1 Weak Transverse Field 248 A.1.2 Intermediate Transverse Field . 249 A.1.3 Longitudinal Field 249 A.2 Decay Problems, Rate of Reaction 250 A.2.1 Simple Consecutive Decay 250 A.2.2 Simultaneous Decay 251 A.2.3 Opposing Decay in the First Step 251 A.2.4 A Complex Case 252 A.3 Competing Processes 253 v B Relaxation Rate Data Tables C List of Acronyms and Symbols List of Tables 1.1 Properties of positive muon and muonium 2 2.1 Cs vapor pressure test results 49 5.1 Spin-flip cross sections for Mu + Cs 130 5.2 Relaxation rates (Ti) for M u C 2 H 4 139 5.3 Relaxation parameters of free radicals 143 5.4 Pressures dependence of T i and r c 145 5.5 T F relaxation rates of M u C 2 H 4 146 5.6 Moments of inertia 151 5.7 M u + N 2 0 results in transverse field 161 5.8 Rate constants of the Mu + N 2 0 reaction 163 5.9 M u + N 2 0 , pressure dependence 166 5.10 Mu(H,D) + N 2 0 , activation energy 179 5.11 Mu + N O , relaxation due to addition reaction 191 5.12 Bimolecular chemical reaction rate constants for M u + N O 193 5.13 Spin-flip rate constants for M u + N O at 298 K 201 B. l Tabulated Results 257 C. l List of Acronyms 273 C.2 Symbols 274 C.2 List of Symbols 275 vii List of Figures 1.1 Counter arrangement 8 1.2 fiSR histogram in L F 9 1.3 T F / /SR histogram for 800 torr neon 10 1.4 Typical /zSR asymmetry plots 11 2.1 The Triumf Cyclotron 23 2.2 The M15 secondary beamline 25 2.3 / /SR spectrometer: Gas Cart 32 2.4 fxSR spectrometer: O M N I ' 33 2.5 nSR spectrometer: HEL IOS 35 2.6 Logic diagram for time-differential / /SR data acquisition 38 2.7 High pressure target vessel 40 2.8 High temperature target vessel 42 2.9 Cs target and gas handling 44 3.1 Breit-Rabi diagram of muonium 54 3.2 Effect of spin exchange on /zSR signal 59 3.3 Typical / iSR signal in an intermediate transverse field 61 3.4 Effect of chemical reaction on [iSR signal 63 3.5 Breit-Rabi diagram for three-spin-system 67 3.6 High field Breit-Rabi Diagram for M u C H 2 C H 2 69 3.7 "3/4 effect" of electron spin exchange in 0 2 72 4.1 Perspective view of a potential-energy surface 77 4.2 Profile of a potentialrenergy surface 78 4.3 Pressure dependence of recombination reaction rate constant 84 4.4 Energy diagram of R R K ( M ) theory 86 4.5 Energy diagram of S A C M theory 89 4.6 Typical spin-exchange potentials 103 4.7 H(Mu) + 0 2 partial-wave spin-flip cross sections 107 4.8 H 4- Cs spin exchange cross sections 109 5.1 Relaxation rate of M u + Cs 130 5.2 Thermal spin-flip cross sections of M u + Cs 131 5.3 Representative Relaxation Rates of M u C 2 H 4 Radical 140 5.4 M u C 2 H 4 relaxation rates 141 vi i i 5.5 Mu 1 3 C 2 H 4 relaxation rates in L F 141 5.6 Representative relaxation rates of M11C2D4 radical 142 5.7 T 2 relaxation rates of M11C2H4 radical 147 5.8 M u + N 2 0 relaxation rates at 303 K 163 5.9 M u + N 2 O relaxation rates at various temperatures 164 5.10 M u + N 2 0 relaxation rates at low pressures 164 5.11 M u + N 2 0 pressure dependence at 300 K 165 5.12 M u + N 2 0 relaxation rates in L F at 303 K 167 5.13 M u + N 2 0 L F relaxation rates at 38 atm 168 5.14 Energy diagram for M u + N 2 O 169 5.15 Arrhenius plot for Mu + N 2 0 176 5.16 Addit ion rate constant for the M u + N 2 O reaction 180 5.17 Addit ion and decomposition contributions to Mu + N2O 181 5.18 M u + N O relaxation rate in L F at different total pressures 189 5.19 M u + N O relaxation rates in L F at 40 atm 189 5.20 M u + N O chemical reaction rates 192 5.21 M u + N O bimolecular rate constant 193 5.22 M u + C O relaxation signal in T F 204 5.23 M u + C O relaxation rates in T F 205 5.24 Ratio of M u + C O relaxation rates in weak and intermediate T F 205 5.25 Relaxation signal in L F for M u + C O 206 5.26 Relaxation rates in L F with pure C O 207 5.27 Relaxation rates in L F with mixture of C O and N 2 207 5.28 Effective reaction rate constants of the M u 4- C O reaction 215 ix Acknowledgements It is a pleasure to thank my supervisor, Professor Donald G . Fleming, for introducing me to this topic, for his guidance throughout the acquisition and analysis of the data and for his patience during the writing of this thesis. I would also like to thank Professors Robert Snider of U B C and Paul Percival of the S F U Chemistry Department for useful discussions on muon spin dynamics related topics. I have benefitted immensely from interactions with my fellow students and research associates and with the staff of T R I U M F in general. Special thanks are extended to the former and current members of the / /SR Gas Chemistry group, which include Drs. Masayoshi Senba, Donald J . Arseneau, James R. Kempton, A l ic ia C . Gonzalez, Susan Baer and Mee Shelley as well as Ms. Alexandra Tempelmann and Mr. Rodney Snooks. I am very grateful to members of the / iSR technical support group: Dr. Syd Kreitzman, Mr . Keith Hoyle, Mr. Curtis Ballard and Mr. Mel Good for their technical support and humorous remarks and for the most enjoyable moment I had at T R I U M F (with them as teammates, I had the pleasure of winning two T R I U M F Volleyball Championships). My family, in particular my partner in life, Clara, and my lovely son, Adam, is the source of my energy. I thank them dearly for their love, encouragement and understanding from the very start to the very end. No word can express my gratitude to my parents, Peiyin Y u and Chunsheng Pan, who brought me into this world and gave me the opportunity to explore the world and to do what I want. If not for them, none of this would have been possible. I thank them by dedicating this thesis to them. x Chapter 1 Introduction 1.1 Positive Muons, Muonium Atoms and Muonium-Substituted Radicals Muons are unstable elementary particles which were first observed in cosmic rays [1,2] and can be produced artificially with particle accelerators [3], such as the cyclotron at T R I U M F , where this research was carried out. Muons have two possible charge states, fi~ and fi+. The negative muon (fx-) is often described as a "heavy electron" by elementary particle physicists. However, from a chemist's point of view, the behavior of a positive muon in matter resembles that of a light proton far more than that of its closer relative, the positron. The / i + mass is 0.113 times that of the proton but 207 times that of the positron; its magnetic moment is 3.18 times that of the proton but only 0.484% of the positron's. Like the proton , a muon has a spin of one half; but unlike the proton, the muon is unstable, decaying weakly with a mean life of 2.2 fis. The most useful attribute of the muon stems from violation of the parity invariance principle both as it is formed, resulting in 100% spin polarization, and when it decays. This wil l be discussed in detail in the next section. This thesis is only concerned with positive muons and unless stated otherwise, "muon" wil l be used to mean "positive muon" hereafter. Although muons do not feel the strong force and do not form nuclear matter as protons and neutrons do, a positive muon does form an atomic bound state with an electron just as a proton does in hydrogen. The formation of this bound state was proposed by Friedmann and Telegdi in 1957 as the probable cause of the muon spin depolarization in nuclear emulsions and they named it 1 Table 1.1: Some Properties of the Positive Muon and Muonium Atom. Compared with Proton, Electron and Hydrogen Atom. Property Symbol Positive M u o n Value Charge e +1 Spin J 1/2 Mass 105.6595 MeV 0.112610 mp 206.769 me Magnetic moment Pp. 4.4905 x 10~ 2 3 e r g G " 1 3.18333 nP 0.004836 ne Mean Free lifetime 2.19713 us Gyromagnetic ratio > 13.5544 k H z G - 1 3.18333 7 P 0.004836 7 e Muonium Value Mass mMu 0.113978 amu 0.113093 m H Reduced mass 0.995187 me 0.995729 un Ionization potential I.P.Mu 13.533 eV 0.9952 I .P .H Bohr radius <*Mu 0.5315 A 1.0044 an Hyperflne frequency 4.4633 GHz 3.1423 v$ 2 "muonium" (Mu=/i+e~) [4,5]. Three years later, it was experimentally observed by Hughes et al [6]. Since a muon is much heavier than an electron, it remains an almost stationary nucleus, giving muonium a reduced mass 0.996 as great as hydrogen's. Consequently, its Bohr radius, ionization potential, and other properties are nearly identical to those of H, as shown in Table 1.1. Unlike the positronium atom, muonium can truly be regarded as an isotope of hydrogen; and it is an ultralight isotope. Muonium is only 1/9 as massive as hydrogen or 1/27 as heavy as tr i t ium! This great difference is commonly exploited by / iSR to investigate the mass dependence of H-atom reactions and other physical processes involving hydrogen. Of particular importance for this thesis is the reactivity of muonium in gases and its associated scattering cross sections, chemical reaction rate constants and other kinetic properties compared to the analogous ones of the hydrogen atom. When Mu adds to a double bond, as in Eq. (1.1), a free radical (here Mu-ethyl) is produced. The first experimental evidence of the existence of a muonium-substituted free radical was found in liquids in 1978 [7]. In a free radical, the muon is removed from the center of the unpaired electron density by at least one atom. The muon-electron hyperfine coupling is, therefore, much weaker than in Mu . These molecules resemble their hydrogen analogues so closely that their reaction kinetics can be expected to be nearly identical, which allows use of the "muonated" version to learn about the chemical and collisional behaviour of the "protonated" version even when information on the latter is unavailable by other methods. This is especially appealing for free radical studies in the gas phase, where, unlike in the condensed phase, there is almost no E S R and reaction data available except for studies of a few diatomics and linear triatomics [8,9]. 1.2 Muon Spin Relaxation and Rotation: the //SR Technique The acronym " / iSR" stands for Muon Spin .Rotation, Relaxation pr Resonance, or even just simply Research. It was coined in 1974 to draw attention to the analogy with N M R or E S R , the range of (1.1) 3 whose application is well known. In fact, any study of the interactions of the muon spin using its asymmetric decay property could go under the banner of fiSR. As early as in 1944, muons were used as probes of magnetism in matter [10], but the history o f / i S R did not begin until the eventful days of January, 1957, when three manuscripts [4,11,12] arrived at the editorial office of The Physical Review almost side by side, confirming the violation of parity (V) symmetry in the weak interaction, the very property that makes most of /zSR possible and which was suggested one year earlier by Lee and Yang [13,14] to explain anomalies in kaon decay experiments [15,16]. Short of winning a Nobel prize, 1 Garwin, Lederman and Weinrich left their mark in history by pointing the way to the muon spin rotation technique and suggesting as well that 7>-nonconservation in n —• fi —> e decay might furnish a sensitive general-purpose probe of matter [11]. It was also in one of these three papers [4] that the concept of the muonium atom was introduced with its distinguishing behavior in an external magnetic field. Since then fj.SK has blossomed into a standard magnetic tool with applications not only in subatomic physics where it was first developed, but also in chemistry, in such fields such as kinetics, radicals and molecular structure, and in condensed matter physics. In fact, these latter subjects are the primary areas of application of / iSR today [17,18]. Muons can be produced in a variety of high-energy processes and elementary particle decays, but low energy muons with high intensities are available only from the ordinary two-body weak decay of charged pions: 7 T + - > / i + + Z^ (1.2) from which the muon emerges (in the rest frame of the pion) with a momentum of 29.79 M e V / c or a kinetic energy of 4.119 MeV. The lifetime of a free charged pion is 26.03 ns. Because the (massless) neutrino is only produced with negative helicity (spin antiparallel to momentum), the simultaneous conservation of linear and angular momentum forces the fi+ also to have negative helicity in the rest frame of the spin 0 pion. Thus muons emitted from pion decay at rest are also 100% spin polarized, opposite to the direction of their momenta. There is no other known spin 'Dr. Wu won a Nobel prize for the work described in one of these papers [ 1 2 ] . 4 probe in matter with such a high degree of initial polarization. The unstable / / + a lso decays via the weak interaction, according to the parity violating process: emitting a positron preferentially along the muon spin axis [19]. Under normal conditions, this muon decay is unaffected by its chemical and magnetic environment since it is mediated by the weak nuclear force. Consequently, the relatively simple process of monitoring the direction of the decay positrons is a direct measure of the orientation of the muon spin. This 'P-nonconservation in the weak interaction is what allows us to read out the information encoded in the evolution of an initially polarized muon spin ensemble. The information is delivered to the experimenter in the form of rather high energy (up to 52.8 MeV) positrons, which readily penetrate target vessels, shielding and the detectors used to establish the time and direction of the muon decay. This is one of the greatest advantages of / /SR as a magnetic technique: whereas N M R and E S R rely upon a thermal equilibrium spin polarization, usually achieved at low temperatures in strong magnetic fields, / /SR begins with a perfectly polarized probe, regardless of conditions in the medium to be studied. It can also be applied in any target system of interest in any, even zero, magnetic field. 1.2.1 Slowing Down Process and M u Formation In a / /SR experiment, muons wil l be stopped in some medium, be it solid, l iquid or, as of interest in this thesis, gas. The process of muon slowing down and of M u formation as well as their consequent thermalization in matter is, in general, a rather complex phenomenon and there is stil l no consensus as to the mechanisms of these processes in condensed media. It is generally accepted, however, that in the gas phase the muon slows down via three different energy/time regimes where one (sometimes more) type of collision dominates in each regime [20-26]. When a muon enters the target, it has a kinetic energy (a few MeV) far in excess of those of chemical interest and loses most of this energy in Bethe-Bloch ionization of the medium, which takes about 10-30 ns at pressures ~ 1 atm in the gas phase. In this regime, the muon polarization is unaffected. A t some energy (of the (1.3) 5 order of 50 keV) which is determined by the competition between the electron capture process and moderation effects, where the muon velocity is comparable to those of the bound electrons in the medium, a series of about 80-100 charge exchange cycles with the moderator ensues + M # M u + M + ) as the muon thermalizes further to about 10 eV. At this point cyclic charge exchange ceases and the muon emerges either as M u or as a bare muon, depending on the mass and ionization potential (I.P.) of the moderator. In this charge-exchange regime, the hyperfine interaction between muon and electron spins plays a crucial role in the depolarization of the muon. Since electrons of the medium are not polarized, muonium can be produced in two different spin states: (i) ortho-muonium (o-Mu) in which the muon and electron spins are parallel at the time of muonium formation and (ii) para-muonium (p-Mu, sometimes referred to, incorrectly, as "singlet" muonium) in which the spins are antiparallel. The muon spin polarization during cyclic charge exchange is not affected by the formation of o-Mu [23], since this remains an eigenstate of the hyperfine interaction. However, p-Mu is not an eigenstate of this Hamiltonian and consequently oscillates between the triplet (total spin one) and singlet (total spin zero) states with the muonium hyperfine frequency 4463 Mhz. The formation of p T Mu for a time comparable with the hyperfine mixing time (0.035 ns) will effectively depolarize the muon spin [20,24,27], a result which will be most apparent at low pressures, where the time between collisions is long. On the other hand, at higher pressures, where this time is much shorter than hyperfine mixing time, this charge-exchange depolarization is effectively quenched. It is noted that the total time spent in this region is only ~ 0.1 ns at normal gas pressures. After the last charge exchange collision, the muon or M u are thermalized by elastic or inelastic scattering processes as well as reactive collisions down to kT (~ 0.025 eV at room temperature) energies. During this last stage, some "hot" atom (or ion) reactions are likely to occur, and after the thermalization processes, a muon ends up in one of three possible environments: 1. in a paramagnetic M u state (for example, with fractions 85% in N 2 and 75% in Ar. ) ; 2. in a diamagnetic state, such as a muon molecular ion [28-30] or as part of a saturated 6 molecule [31,32], and possibly also as a free muon (which is only likely in a metal); 3. in a paramagnetic radical state [33-36]. Since the initially 100% muon polarization (P) is shared by all three states and there maybe a "lost" (or missing) fraction of polarization, the total polarization is: P = PD + PM + PR + PL, (1.4) where PD, PM and PR are the polarizations retained in Diamagnetic, M ion ium and Radical environments, and PL is the total Lost polarization during the slowing down process. In most gases, PL approaches zero at pressures of order of 1 atm [20,21,37], though it can again increase at much higher pressures, an indication of possible "spur" processes [38], and there are notable exceptions [30,37,39]. It is the ultimate fate of those muons thermalized as M u or as Mu-substituted radicals that is of prime concern in this thesis, which provides the basis of the subsequent discussion. 1.2.2 Time Evolution of the Muon Spin State For a perfectly polarized ensemble of muons, the spatial anisotropy of positron emission is given by N(0)/N = \ + A cos0 (1.5) after averaging over the decay positron energy (0.0-52.8 MeV) and where 9 is the angle between the muon spin and the direction of the detected positron. The technique of fiSR relies on detecting these decay positrons and tabulating the events in a histogram of positron counts vs. time. In this time-differential (TD) method, the muon that created each decay positron must be unambiguously identified, necessitating that only one muon be in the target at a t ime 2 . This requirement is ensured by the electronic logic used in the experiment's data acquisition system (see Section 2.3 and Fig. 2.6). If all positrons were detected with equal efficiency, and the muon beam was 100% polarized, then the "asymmetry" A = 1/3 [40]. In practice, higher energy positrons are detected more easily while low energy ones may not even get out of the target, the beam is somewhat less 2 T h e phrase "ensemble of muons" is used with reference to time rather than space. 7 Figure 1.1: Schematic of the general arrangements for the counters, muon spin and magnetic field in transverse-field (TF) and longitudinal field (LF) / /SR. T M labels the Thin Muon counter. B, L, F, R are labels for the Back, Left, Forward and Right positron counters. The arrows under the column labelled "muon spin" indicate the muon spin direction while the arrows under the column "field" show the direction of the applied magnetic field. In practice, in the experiments described in this thesis, the gas target, positioned in the center of the counter array, is much larger than indicated in the figure. See more details in Chapter 2. than 100% polarized, the solid angle covered by the positron detector is large and variable depending on the actual experimental setup, and the polarization wil l inevitably decrease with time (relaxation) so A is always treated as an empirical factor, AQ, multiplied by a relaxation function R(t), which describes the loss of polarization over time. Moreover, 6 may also be time-dependent (precession). F ig. 1.1 shows a schematic diagram of a / /SR experimental setup. In the simple case where there is no coherent muon precession, such as when the magnetic field is zero or it is aligned with the muon spin, 6 is constant, and the corresponding longitudinal field (LF) relaxation function is denoted Rz(t), in analogy with T\ relaxation in N M R . Thus, the histogram of positron decays is described by N(t) = N0e-^T" [1 + A0 R,(t)cos0] + b (1.6) where No is a normalization, 6 is a time-independent background due to random events, is the mean muon lifetime (2.2 /zs), and t is the time the muon spent in the target before decaying. The subscript z identifies the relaxation as longitudinal or as T\ relaxation in N M R parlance. In local 8 0 2 4 6 8 Time in microseconds Figure 1.2: /zSR histogram for 247 torr N O in 20 atm N 2 at 7.65 k G L F . The points represent the number of positrons counted within each time bin, (80 ns wide for the plot, but only 2.5 ns in the raw data) and the curve is the fit to Eq . 1.6. The error bars are due to Poisson counting statistics alone: a- = VW, but they are smaller than the squares so are not visible. field environment in solids, Rz (t) is often Gaussian but in the gas phase, of interest in this thesis, it is invariably exponential. F ig. 1.2 shows a typical muon decay histogram in longitudinal field for muons stopped in a mixture of N O and N 2 . Many experiments for this thesis were performed with a magnetic field applied perpendicular to the initial muon spin direction. In this transverse field (TF) environment, the muon precesses with a specific Larmpr frequency characteristic of its environment until it decays. The resulting variation of 9 with time is seen as oscillations in the muon decay histogram, a typical example of which is shown in Fig. 1.3 for fx + stopped in Ne. Such a simple spectrum is described by the equation N(t) = W o e - ' / T " [1 + A0Rx(t) cos(wr. + cj>)] + b (1.7) where u is the appropriate Larmor frequency3 and <f> is the initial phase angle between the muon spin and the direction of the detector; uit + <f> is 9 in equation 1.5. The relaxation functions, 3For muons in diamagnetic environments wn/B = 85.165 k s - 1 G - 1 ; for paramagnetic muonium at low fields o^iu/B = 8.7336 ^ s - 1 G _ 1 ; for radicals the manifold of precession frequencies at low fields could be replaced by a mean-field approximation, single frequency U / R , which is often of the order of 10 ns~1G~1, an order of magnitude larger than WMU and is not detected in low TF experiments [41]. 9 20000 -•J 15000 a. » 10000 c 3 O o 5000 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Time / /us (20 ns per bin) 7.0 8.0 Figure 1.3: / /SR histogram for / / + stopping in 800 torr neon at room temperature, with T F applied. The points represent the number of positrons counted within each time bin, (20 ns wide for the plot, but only 2.5 ns in the raw data) and the curve is the fit to Eq. 1.7. The error bars are due to Poisson counting statistics alone: a = V~N, but they are smaller than the squares so are not visible. Taken from D. J. Arseneau's P h . D thesis [28]. describing transverse field dephasing in analogy with T 2 relaxation in N M R , are here denoted Rx{t). In the gas phase, lineshapes are generally simple "Lorentzian" corresponding to exponential collisional relaxation. The Eqs. (1.6) and (1.7) can then be expressed in one form: JV(<) = ^ o e - ' / T " [ l + i 4 ( < ) ] + 6 I (1.8) for a single environment. The relaxing term constitutes the signal of interest: the decay asymmetry A(t), often referred to as the / /SR "signal" and denoted S(t) in the / /SR literature. When muons exist in more than one environment (this is usually the case), more than one term is required in A(t) to describe the muon decay (see below). It is easy to see from Eq. 1.8 that A(t) must be time dependent. Any time-independent term in A(t) wil l be buried in No and cannot be measured by / /SR. Two representative signals or "asymmetry plots" in L F (top) and T F (bottom) are shown in Fig. 1.4, and such plots wil l be used for illustration through the remainder of this thesis. 10 0.04 0.00 0) E E > N CO < -0.04 -0.08 -0.12 0.35 Time ( fj. s) -0.35 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Time / (is 8 HMIIIMIJWM! M 7.0 8.0 Figure 1.4: The top plot is the corresponding longitudinal field asymmetry plot for F ig . 1.2, after removal of normalization, decay, and background. The solid line is a fit of the data to Eq. 3.34, giving the relaxation rate \L= 0.6/ZS - 1 . The seemingly random scatter at later times is simply a reflection of the 2.2 /zs lifetime of the positive muon. The bottom plot is the corresponding transverse field asymmetry plot for F ig 1.3. The very slow signal relaxation of A = 0.03 ps'1 is due to magnetic field inhomogeneity [42,43]. 11 1.3 Interaction of Muonium with Gases Once a muonium atom is formed and thermalized in a gas, there are a number of ways the spin polarization, retained during the slowing down process,4 can be lost. First of al l , M u could react with other molecules to form a new muonium-containing molecule, placing the muon in a different environment. Or it could encounter another paramagnetic atom/molecule and undergo an electron spin exchange interaction. Although electron spin exchange itself, like charge exchange, will not directly alter the spin direction of the muon in muonium, the consequent hyperfine coupling, as in p-Mu, wil l . A similar process is seen in the collisional relaxation of a muonium-substituted free radical. Even if a Mu-radical collides with an inert diamagnetic molecule (hence no chemical or electron spin exchange interaction), the collision can stil l induce some local magnetic field fluctuation which causes spin relaxation. By observing the relaxation of the muon spin, one may then be able to extract kinetic parameters for these interactions. Absolute rate constants and cross sections thus obtained are important to such diverse fields such as combustion, chemical kinetics, scattering theory, optical pumping, and others. However, not every interaction will result in a loss of polarization and the relation between the rate of muon spin relaxation and the rate of interaction is not always a simple linear one and may depend on other conditions (e.g., orientation and strength of the applied magnetic field). Complicating the situation as well is the fact that all these different interactions may co-exist in one system and the task of differentiating them is not always straight forward. Part of this thesis work is to develop a comprehensive approach towards these problems in addition to providing new reaction rate constants and cross sections. 1.3.1 Spin Exchange First recognized by Bates [44] and Purcell and Field [45] as the dominant mechanism for the establishment of the spin temperature of H atoms in the upper atmosphere and in outer space, electron spin exchange is one of the most fundamental of quantum processes. It is not only 4Typically, 50% of the initial muon polarization is lost upon muonium formation. See page 55. 12 amenable to theoretical calculations in that it can be viewed as a quasi-elastic scattering process, but also important in many branches of chemistry and physics such as optical pumping, masers, plasmas and astrophysics. Electron spin-exchange has been studied experimentally in many H(D,Mu)-atom systems in the gas phase, such as H-H(D) [46-51], H - 0 2 and H - N O [47,52,53], M u - 0 2 , M u - N O and Mu-Cr-complex [54-60], H (D ,Mu) -A [61-65] as well as A - A [66-76], utilizing techniques such as masers, E S R , microwave absorption, optical pumping [76-80] and, of interest here, (J.SR. ("A" represents an alkali-metal atom.) Spin exchange of atomic H(D) with alkali metals has been employed to produce polarized hydrogen, deuterium and other atoms [61,63,64,81-86], which have many applications including polarized nuclear targets [81], cyclotron ion sources [63,64,85] and possibly spin-polarized fusion [61]. Accurate knowledge of electron spin-exchange cross sections for these and related systems is therefore valuable to both theorists and experimentalists alike. Since Mu has an unpaired electron, it can undergo electron spin exchange with other species with unpaired-electrons, causing muon spin relaxation through the (isotropic) muon-electron hyperfine interaction during which the muon spin direction is changed. The first study of muonium spin exchange in gases was the work of Mobley et al. [87] carried out at high moderator pressures, necessitated at that time by the use of high momentum conventional muon beams. These workers obtained spin exchange cross sections for M u + 0 2 and M u + N O in argon gas at 40 atm by measuring the n+ polarization in a strong longitudinal magnetic field. At such moderator pressures and high fields, the rates of termolecular chemical reactions between Mu and 0 2 or N O can be expected to be comparable to the spin exchange rates, an effect ignored by Mobley et al.. This, coupled with the generally poor quality of the low statistics data, prompted a second investigation by Fleming and Miku la and coworkers [58-60] with "surface" muon beam and low pressure (1 and 2.5 atm) moderators in weak transverse magnetic fields. The latter study also measured temperature dependences of the rates. But again the chemical reaction contribution was argued to be insignificant without checking experimentally, though this time it was based on a more credible argument. A more complete survey was done 13 almost 10 years later [55,56] to search for possible sources of the poor agreement among experimental data of Mu(H) + 0 2 and Mu(H) + N O reactions [47,53,88,89]. These seeming inconsistencies were actually due to an incorrect interpretation of these data which was not corrected until recently [56,90,91]. A better understanding of the underlying mechanisms for muon spin relaxation in gases, different experimental techniques to differentiate contributions from different mechanisms and, very importantly, correctly relating the observed muon spin relaxation rate to some well defined physical quantity (e.g., reaction cross section) are all essential in the measurement and interpretation of these reaction rate constants. Aspects of this nature will be discussed in this thesis. In addition, due to experimental limitations, some of the earlier measurements, e.g., Mu+NO, had large uncertainties and more accurate measurements were made during the course of this thesis research for this system. Since 1988, Senba has developed a formalism to describe the spin dynamics of muons in the gas phase using a time-ordered approach and has published several papers concerning the phenomena of charge and spin exchange in different situations [23,24,91-99]. Turner and co-workers [90] also treated the same problem using the Boltzmann equation with a broader range of applications. These papers have formed the basis of our current understanding of spin exchange in pSR and will be discussed in due course. In this thesis, the effect of electron spin exchange on the muon spin polarization in both longitudinal and transverse magnetic field will be treated and measurements of spin exchange cross sections for Mu+Cs and new data for Mu+NO wil l be reported. 1.3.2 Chemical Reaction The birth of chemical kinetics could be taken to have occurred in 1850, when the German chemist Ludwig Ferdinand Wilhelmy studied the rate of inversion of sucrose [100]. Chemical kinetics deals with the rates of chemical reactions and with how rates depend on experimental conditions such as concentration and temperature. Such studies are important in providing essential evidence as to the mechanisms of chemical processes. Valuable evidence about the mechanism also is provided by non-kinetic investigations, such as the detection of reaction intermediates and equilibrium isotope 14 exchange studies, but knowledge of a mechanism can be satisfactory only after a careful kinetic investigation has been carried out and it is very important to have accurate individual elementary reaction rate constants. A l l of the measurements in this thesis were made on systems that were in thermal equilibrium, which means that energy is distributed among the molecules according to the Boltzmann distribution. Despite the growing sophistication of molecular beam experiments and measurements of state-to-state reaction cross sections [101-104], the study of bulk chemical kinetics, as manifest by thermal reaction rate constants, remains important. First, rate constants provide some measure, albeit indirect, of absolute cross sections, rarely reported in beam experiments. Second, only thermal rate constants test the nature of the potential energy surface near the reaction threshold, thus usually providing the best determination of both the position and height of the potential barrier. The measurement of kinetic isotope effects (KIE) is of vital importance in this regard and more discussion on this topic wil l be given in the next section. Muonium chemistry can be broadly defined as the study of the structure, dynamics and transformations (reactions) of molecules containing positive muons. This thesis is primarily concerned with the study of reaction rates and chemical dynamics. Chemical reaction breaks (alters) the hyperfine coupling between the muonium electron and the muon and places the muon in a different magnetic state, which consequently affects the time evolution the spin direction of the muon and hence the polarization of the ensemble. Muonium formation was proposed, as mentioned in section 1.1, as the cause for spin depolarization in nuclear emulsion [4,5], and it was also offered very early on as an explanation for the observation that the "residual muon polarization" is not the same in all condensed media [105]. The first extensive experimental study of thermal M u reactions was by Brewer et al [106] who studied the residual polarization of muons (resulting from M u reactions) in liquids, by applying a modified form of the muonium mechanism of Ivanter and Smilga [107,108] to the measurement of bimolecular rate constants of simple M u reactions. These experiments grew out of an experiment to determine the muon's magnetic moment precisely [109] which required small corrections due to 15 chemical effects. Brewer et al. also found it necessary to extend the model to include epithermal reactions of M u as well as reactions in which transient muonium-substituted radicals are formed. The study of muonium chemistry in liquids is complicated though by several processes such as solvolysis and "spur" reactions [110,111]. From the viewpoint of understanding elementary chemical rate processes, the gas phase provides an environment which is much more theoretically tractable than the liquid phase. The aforementioned work of Mobley et al [87,112,113] also pioneered gas phase Mu chemistry studies, but at much higher pressures such that moderator effects could not, in retrospect, be ignored. Fleming and Garner et al carried out the first reliable measurements of M u reaction rates in low pressure gases [114-116]. This necessitated the development of low energy ("surface") muon beams (see Chapter 2). In the two decades thereafter, the rate constants for many reaction systems at low to intermediate pressures in the gas phase have been reported [22,26,31,32,55-59,114,117-125], the most recent being M u + C H 4 [125,126]. The chemical reactions studied in this thesis, Mu+NO, M U + N 2 O , and M u + C O are part of this systematic investigation. 1.3.3 Free Radicals and Collisional Relaxation Many chemical reactions, important in the environment and in industry, involve transient free radicals as reactive intermediates. The elucidation of the structures of free radical species, of their reaction mechanisms and rates is a fundamental importance, not only to theoretical chemistry but also to technological applications of radical reactions, such as combustion, production of plastics and rubber, oxidation of hydrocarbons, stabilization of polymers, etc. Numerous reactions in cosmic space on one hand and in living cells on the other also proceed by radical mechanisms. Radical reactions have thus significant impact on many processes in the chemical industry and energetics and on the clarification of chemical changes in both living and nonliving nature. Nevertheless, some fundamental aspects of free radical chemistry, particularly radical-radical reactions are still not well understood. Being entities with unpaired electrons radicals are, as would be expected, generally 16 extremely reactive and often only capable of transient existence. Thus it is no wonder that, although there had been no lack of interest to make radicals and isolate them, no attempts were successful until 1900 when Moses Gomberg generated for the first time his famous triphenylmethyl radical, and produced experimental evidence to substantiate his claim[127,128]. Some twenty-eight years elapsed before the simple methyl and ethyl radicals were prepared. In the past three decades a wealth of information on radical properties and reactions has been accumulated mostly by use of the powerful methods of optical spectroscopy and E S R [8,9,129-134]. At the same time, one of the major areas of progress in muon chemistry in the late 1970's was the discovery of muonium-containing free radicals, in the / /SR spectrum of unsaturated liquid hydrocarbons at relatively high magnetic fields [7]. It was almost another decade, however, before similar radicals, notably the M U C H 2 C H 2 radical, were observed in the gas phase [33]. This was an important result, since it is generally not feasible to do E S R studies on polyatomic molecules in the gas phase due to experimental difficulties of separating too many resonance lines which are broadened as well by collisions, specifically, by the spin-rotation (SR) interaction. It appears that there are essentially no gas-phase E S R studies of (transient) polyatomic free radicals, other than that of Ref. [9], many years ago. In comparison with E S R , the / iSR technique offers a considerable simplification, for studies of both hyperfine spectroscopy and spin relaxation of gas-phase free radicals. In most eases, in transverse magnetic fields of order only ~ 2kG, the multitude of transitions between the many nuclei and unpaired electron spins, characteristic of E S R spectra in general [8,132], and the ethyl radical in particular [135-137], collapse to a single pair of lines at well resolved frequencies for a given radical environment, revealing the (isotropic) p,+-e~ hyperfine coupling [35,133,138-143]. Although this well known " E N D O R effect" was, of course, first realized and exploited in E S R spectroscopy in condensed phases [8,79,144,145], it has received little attention in the gas phase, certainly for transient radicals [132]. As mentioned, in 1986, a muonium radical was first observed in the gas phase at SIN (now PSI) by Roduner and Garner [33], who studied the M U C H 2 C H 2 radical at relatively high pressures 17 (25-50 atm) and in low (w 0.1 T) magnetic fields. Their conclusion was that the Spin-Rotation interaction likely dominated the /zSR linewidths, but there was really insufficient data to draw any firm conclusions as to the detailed nature of the operative relaxation mechanism(s). There are two broad motivations for the study of spin relaxation of muonium substituted radicals in the gas phase. The first is simply to compare the data so obtained with the basic predictions of (magnetic resonance) spin relaxation theory in gases [79,145-151], particularly in view of the fact that E S R studies for the same kinds of radicals are really not feasible. In contrast to N M R studies, where most work of this nature has been done [134,147,152-156], the presence of the unpaired electron in a radical allows one to probe fluctuations in local fields on a faster time scale and over a much broader range of spectral densities, consistent with the muon lifetime of 2.2 / i s - 1 . In principle different features of the intermolecular potential (eg., induced dipole) are accessible via measurements of the (electron) relaxation times. Also, very recently, Turner and Snider [157] have treated the spin relaxation of the M U C H 2 C H 2 radical in a first-principles calculation from a correct Hamiltonian, with which the results of the model developed herein can be compared. The second, which is part of the primary motivation in this work, relates to studies of reactive processes of muonium-substituted free radicals in gases, including the study of radical-radical reactions, which would be virtually impossible to do by ESR. In order to extract information on rate constants for chemical and/or spin-exchange reactions, it is first necessary to understand the intrinsic nature of the mechanisms contributing to spin relaxation of the Mu-radical itself as mentioned above. A representative paper on the M u C H 2 C H 2 + O2 reaction is in press [158]. The spin relaxation rates in a variety of Mu-substituted radicals ( M u C H 2 C H 2 , M u C H 2 C ( C H 3 ) 2 , and MuCO) were measured and these rates and their interpretation wil l be discussed in this thesis. Some hyperfine coupling constant will also be reported, though this aspect of "hyperfine spectroscopy" is not the principle focus of the present thesis. It can be noted though, again from a theoretical point of view, that the measurement of hyperfine couplings in the gas 18 phase, free from solvent or matrix interaction, is an important contribution to the field. 1.4 The Importance of Isotope Effects The study of isotope effects on the rates and products of chemical processes has long been useful in increasing chemists' understanding of the detailed nature of chemical reactions. For example, by substituting deuterium (D) for hydrogen (H) in a simple bimolecular abstraction reaction, information about the contribution of quantum mechanical tunneling to the total reaction rate becomes available. In reactions which may occur either by simple abstraction or even by a more complicated mechanism, such as complex formation followed by dissociation, isotopic substitution may help discriminate between reaction pathways. The first definitive experimental demonstration of an isotope effect in chemical kinetics dates back to the discovery in 1932 by Washburn and Urey [159] that deuterium is enriched in the liquid phase in the electrolysis of water. This discovery came at a time when the transition state theory of reaction rates was in a stage of active development and was immediately explained by the zero-point energy (ZPE) difference between isotopic molecules [160]. The subsequent world-wide production of laboratory quantities of concentrated deuterium by the electrolytic method provided material for new experiments on kinetic hydrogen isotope effects in a multitudinous array of reactions. Many of these had an important bearing on the development of transition state theory (e.g., H + H 2 ) [160,161]. In the succeeding years there has been a considerable interaction between experimental and theoretical progress on kinetic isotope effects and their application to the elucidation of reaction mechanisms. See for example Refs. [26,162-178]. The isotopes of hydrogen have been used extensively in the study of kinetic isotope effects [51,162,165,170,171,174,177-188], including as well the study of hot tr i t ium chemistry [189]. A theoretical discussion from first principles of kinetic isotope effects requires accurate potential energy surfaces (PES) for the calculation of energies and frequencies. Being the simplest molecules, systems involving hydrogen and its isotopes are often the only systems that can be treated theoretically with high accuracy. Moreover, for the isotopes of hydrogen both 19 tunneling transmission coefficients and Z P E effects can be expected to be especially important in establishing kinetic isotope effects. Several such calculations have been carried out on both ab initio and semi-empirical potential energy surfaces [164,168,170,174,190-197]. The extremely light muon mass in the Mu atom renders it a particularly sensitive probe of K IE 's . The light mass of the muon (though still ca. 200 times the mass of an electron) does raises the question of the validity of the Born-Oppenheimer approximation in the atomic and molecular interactions of muonated species. It is well known that this approximation begins to break down as the mass difference between the nucleus and the electrons becomes small, and the comparisons in Table 1.1 suggest that this could be of some concern in the case of M u . However, theoretical calculations of one-electron problems, comparing for example, H D + and H M u + [198], as well as calculation of non-adiabatic effects in K + H spin exchange [199], indicate that the Born-Oppenheimer approximation remains valid for muon interactions. The main advantage in the study of M u reaction kinetics over that of traditional hydrogen isotopes lies in the remarkable range and magnitude of possible isotope effects it affords. Wi th in the Born-Oppenheimer approximation, isotopic species share a common P E S ; any differences that arise in their respective reaction rates depend on mass effects only. The true surface must be able to account for the behavior of all isotopes, no matter how light or how heavy. Wi th the inclusion of Mu , the available mass ratio range of hydrogen isotopes is increased from 3 to 27! This unprecedented mass range therefore provides a uniquely sensitive probe of reaction dynamics and of the underlying potential energy surface, particularly near threshold. Perhaps the most propitious result of the low M u atom mass is its greatly enhanced predilection for quantum tunneling of M u relative to H, enabling observation of tunneling effects at easily accessible temperatures (e.g.> 100 K ) , indeed even at room temperature. This has facilitated experimental observation of pronounced tunneling regimes in some exothermic reactions, notably Mu + F2, where the experimental activation energy approaches zero near 100K [123], indicative of Wigner threshold tunneling [194,200], the first measurement of its kind in gases. In the present work, the M u + N 2 0 reaction reveals a tunneling enhancement of a factor of 120 compared to H + 20 N2O, even at room temperature. Another important advantage inherent in the study of M u lies in the experimental technique. As mentioned earlier, Muonium atoms are easy to form and M u events are individually monitored, thereby eliminating the self interactions that often plague H-atom experiments [185,201-204]. The experimentally obtained Mu rate constants may therefore be more accurate than those of their heavier atom counterparts and can, in principle, be used to predict H atom reaction rates, providing an accurate potential energy surface is available. Although this application lies in the future for most reactions, as calculation methods become faster and more accurate, the study of M u reactivity may well be used more in this predictive capacity. 21 Chapter 2 Experimental and / iSR Techniques 2.1 Muon Production and Transport 2.1.1 The T R I U M F Accelerator The T R I U M F cyclotron and experimental facilities [205] are shown in Fig 2.1. The accelerator is a six-sector isochronous cyclotron accelerating H~ ions to 183-520 MeV at a maximum unpolarized current of 150 fiA at 520 MeV. A l l of the experiments described in this thesis were conducted with 100-140 /iA at 500 MeV. The proton beam has a 100% "macroscopic" duty cycle and a microscopic time structure consisting (normally) of a 5 ns burst every 43 ns. A proton beam, extracted from the cyclotron by stripping both electrons from the H~ ions, passes down beamline-1 (BL-1A) in the "meson hal l" and strikes two pion production targets, first 1AT1 and then 1AT2. Three secondary beamlines simultaneously extract pions or muons produced at each target. The first production target (1AT1) is thin (typically 1 cm carbon) and provides a small source for the three high quality channels: namely M13, a low energy pion and surface muon channel at 135°; M i l , a high energy, good resolution pion channel at 215°; and M15, a dedicated surface muon channel. The second target (1AT2) is thick (10 cm Be) and provides the source for three high intensity but generally poorer optics channels: M9, a low energy pion and backward muon channel; M20, a general purpose muon channel now dedicated to surface muons; and M8, a high flux pion channel for medical treatments. The experiments described in this thesis were mostly performed on either M20 or M15 with a few exceptions, which were conducted on M9. 22 Figure 2.1: The layout of the Triumf Cyclotron, Beamlines and facilities. 23 2.1.2 Surface Muons: M15 and M20 Beamlines For most experiments presented in this thesis, a "surface" | i + beam was used [205-208], in which muons are produced by the decay of pions at rest on the surface of the production target. Surface muons form an essentially morioenergetic 4.1 MeV beam with almost 100% longitudinal spin polarization, a natural consequence of pion decay at rest. This beam is collected, focused, and momentum-selected by a series of magnets and other devices which form the secondary beamline; in the case of these experiments, principally the M20B or M15 channels [205,208]. F ig . 2.2 shows the layout of the M15 channel which was specially designed for surface muon beams. The M20 secondary beamline originates at the 1AT2 pion production target, which is generally a strip of Be 1 c m 2 x l 0 cm long (in the beam direction). The channel views the target at 55°. It was originally used as a general purpose channel, capable of producing forward, backward or surface muons. There is an alternating gradient decay segment of 10 quadrupole magnets for focusing either surface muons or those from pions which decay in flight. The beamline is divided into two legs, A and B. Leg A was used primarily for backward decay muons with a typical momentum of 90 M e V / c , but this leg has now been decommisioned. Only leg B was used in these experiments; it was always operated in the surface muon mode and was equipped with a spin rotator (see below). M15 views the 1AT1 thin target at 150° in the vertical direction. It was designed specifically for surface muons; the first two quadruples are permanent samarium cobalt magnets, tuned to 28 M e V / c . This limits the momentum range of the channel, which can be tuned in the range of approximately 20 - 30 M e V / c , maintaining reasonable fluxes. There are three sets of slits along the channel to control the muon rate, beam profile and momentum acceptance. In this case, two spin rotators (see below), which each rotate the beam a maximum of 45°, are separated by a quadrupole triplet which allows the vertical momentum dispersion caused by the first rotator to be cancelled by the second, making the system achromatic in the vertical plane. The lack of such a feature in the M20 separator causes a loss of approximately 50% in the the muon rate when 24 Figure 2.2: The M15 beamline at T R J U M F . This beamline is specially designed for (low-energy) surface muons, and uses a train of magnetic quadrupoles (un-labeled devices) and dipoles (B) to deliver muons to the experiment. It rises above the muon production target 1AT1, and climbs to ground level, as it needs no heavy shielding. Its length eliminates pion contamination, and the dual separators/spin-rotators eliminate positron contamination. The spin rotator is split, with a triplet of quadrupole magnets in the middle, to reduce beam dispersion at full spin rotation. 25 running in spin rotated mode. A considerably smaller loss is encountered in M15. A special feature of both these two channels is the presence of the spin rotator(s), also called a "dc-separator" or Wien filter [209]: essentially a large vacuum box containing crossed magnetic and electric fields, both perpendicular to the beamline axis, which act as a particle velocity selector (whereas the dipole magnet "benders" act as momentum selectors). The crossed magnetic and electric fields in the separator are adjusted such that they exert equal and opposite forces on the muons. Only particles with a velocity given by (where E and B are the electric and magnetic fields, respectively) will pass through the separator undeflected. It also removes positron contamination from the beam as positrons have much higher velocities than muons of the same momentum. Positron contamination in the beam is reduced from approximately 10 positrons per muon (in M20) to approximately 1% through the use of this spin rotator. The separator can also be used to rotate the muon spins (since the spins precess in the applied field when in the separator, through an angle proportional to the applied magnetic field strength), allowing the injection of the muon beam into large transverse fields. For large fields (greater than about 200 G) one must inject the beam along the field lines, otherwise the beam wil l be bent and hit the walls of the target vessel, or even miss the target altogether. This implies that for a transverse field (TF) geometry, one must have the muon polarization perpendicular to the muon momentum, rather than parallel to it as created in pion decay. By using a large enough magnetic field, the muons precess by a total of 90 degrees while travelling through the separator(s), and emerge with their spins perpendicular to their momentum. When operating the separator in "non spin-rotated mode" the ^ + spins are rotated by ~ 10 degrees from horizontal, antiparallel to their momentum, towards the vertical. In the L F configuration, the component of the muon polarization perpendicular to the field gives a precession signal in the side counters at the frequency corresponding to the total applied field. This signal 26 sometimes provided a useful check on field calibrations during the experiment. 2.1.3 Backward Muons: M 9 B Beamline At the end of this thesis project, some data was taken on the M9 channel with low energy backward muons (~ 60 MeV/c ) . The M9 beamline has two legs coming from the 1AT2 target. Leg A is dedicated to rare-event experiments. M9B is a backward channel with a superconducting solenoid for the decay section. It delivers a high flux beam over a momentum range approximately 20-100 M e V / c . The intensity of beam drops significantly at both the low and the high end of this momentum range. The relatively high muon momentum available on this channel enables the use of target vessels that have thicker windows necessary for high pressure experiments, such as the high pressure target (with a 1 mm T i window) used in the M u + C O study. 2.2 /iSR Setup The /zSR technique in general [119,133,138,141,210,211] and its application to the study of chemistry and physics in gases is well established [20-22,26,28,29,32,38,55,56,5.8,59,91,93,212,213]. The experiments reported in this dissertation were performed using the conventional Time-Differential (TD) fiSR technique, which measures the detailed time evolution of the muon polarization function as discussed in Chapter 3. In this type of measurement one records the arrival time of the fx+ and its subsequent decay at time te, and then constructs a time histogram for the intervals defined by At = (te-tn)- This technique requires the ability to unambiguously associate a given e + with the muon from which it was emitted. Normally, this requirement is satisfied by allowing only one muon at a time to be present in the target vessel (see Section 2.3). Typically, a muon enters the sample after passing through a thin scintillation counter, which signals the start of an event. The muon subsequently decays, emitting two neutrinos (which are not observed) and a positron which is detected in another scintillation counter. A histogram of detected positrons is kept for each detector as a function of the time difference between when the 27 muon enters the sample and when it decays. 2.2.1 Longitudinal Field (LF) The experimental geometry for longitudinal field experiments is shown in F ig . 1.1. The muons are initially polarized antiparallel to their momentum, along the magnetic field in the case of longitudinal field. The counters are placed in a forward-backward configuration. In practice, since one has to allow the beam to go through the window and allow some tubing connected to the target so that gases can be filled and evacuated, it is usually not possible to have the counters completely cover the forward and backward solid angles. This piping and tubing sometimes was positioned through a hole in the counter, and sometimes was surrounded by counters outside a target jacket designed to enclose the target vessel and its associated tubing. The histogram of numbers of positron decays is described by Eq. 1.8 as outlined earlier. The / iSR signal is given by In most cases, there is only one distinguishable component and the function (see Section 3.4, Eq. 3.34) is used in the fitting. To extract the correct amplitude A, both forward and backward counting rates have to be analyzed and fit simultaneously [214], but in the experiments conducted for this thesis research, only the relaxation rates are of interest and since the backward counter usually has much higher signal to noise ratio, the data were fit separately for each counter and weighted average of the relaxation rates were used. In some cases, the backward counters were not used at all due to very bad background. In the cases of multiple relaxing components in a L F , a general fitting procedure is not available. Currently, one has to consider each individual situation and hope to find some information (from elsewhere, e.g., T F results) so that certain parameters (e.g., the init ial asymmetry or rate for one component) of the fitting function can be fixed or to adjust the gas (2.2) j A(t) = At -xLt (2.3) 28 concentrations so that the relaxation rates are very different for all components (e.g., one very fast and one very slow). New fitting procedures are under development for multiple component relaxation in L F . Comparing to T F experiments, L F experiments have less demand for field stability, especially at high fields, but the absolute magnitude of the field is still important. Since one cannot obtain the field value from the / i S R signal obtained in L F , the field is usually calibrated using the T F technique. In principle, a hall probe (or even better a calibrated N M R probe) can be used to give a rough guide as well, but these were not practical in most of the setups used in this thesis. 2.2.2 Transverse Field (TF) The transverse field experimental geometry is also shown in F ig . 1.1. In this case muons are injected into the sample with their spins polarized perpendicular to the applied magnetic field. In low fields the signal in a given counter telescope array is generally given by A{t) = AMRX,M(t) cos(uiMt + <I>M) + AoRx,D(t) cos(wjof - <f>o) (2.4) where A are the effective asymmetries of the detector, w are the Larmor precession frequency and Rx(t) are the relaxation envelopes for muonium and diamagnetic muon respectively. The separate terms for diamagnetic muon and muonium are due to their different gyromagnetic ratios. In fact, they also have different signs, so that the spins wil l precess in the opposite directions. Since any bare muons, or muons in any diamagnetic environment in general, precess about 100-fold slower than muonium, it is very easy to separate the two terms. Chemical shifts are ~ 10's Hz between bare muons and any muon in a molecular diamagnetic environment [215,216] and, unlike in N M R , cannot be distinguished in fiSR experiment, even one employing a resonance technique [217]. They can, however, in principle be distinguished in a stroboscopic technique, in which the magnetic field is set to give exactly the same frequency as the cyclotron R F [216] but such high precision was not of interest in the present study. 29 2.2.3 ^ S R Spectrometer A typical fiSK spectrometer (e.g, Figs. 2.4-2.5) consists of certain sets of magnetic field coils to provide the desired magnetic field in different directions, a set of adjustable muon/positron counters to detect the arrival/emission of muon/positrons, other special equipment necessary for that spectrometer and a construction to hold all the above as well as the target vessel. A l l spectrometers can be rolled in the beam direction along tracks on the floor. The target position could be adjusted independently as well. Common for all spectrometers are the counters. • Muon counter A removable single thin (~ 0.4 mm) plastic scintillator for use with surface muons (P^ ~ 30 MeV) coupled to a photomultiplier tube (PMT) and placed between a beam collimator and the target served as the incident muon counter. This counter is thin enough so as to not stop the muon beam and gives only a very small signal from the ~ 30 — 50 MeV muon-decay positrons, which happens to hit this counter, since positrons are minimum-ionizing. This positron signal is discriminated against in the constant fraction discriminator, shown in Fig . 2.6. For backward muons with much higher momentum (P^ ~ 70MeV/c) , this counter is much thicker (~ 6 mm) but the principle is the same. • Positron counter array Decay positrons are detected with arrays of thick (~ 1 cm) plastic scintillator detectors coupled to photomultiplier tubes through lucite light guides, sometimes with graphite degrader between the counters. Generally 4 or 6 separate positron counters are used, typically covering on order ITT — 3TT steradian of solid angle (see also Fig. 1.1). The areas not covered are used to provide access to the target vessel as well as a beam entry point. Each counter array usually consists of two to three counters, though in some cases only one is used. Wi th an array of counters, one requires a coincidence in the detection of a positron in all three (two) counters of the array to count as a good event. This has the effect of eliminating one source of background, namely random triggering due to noise in the photomultipliers and/or 30 scattered photons. Frequently, this was not a problem, and single counters were used, as this simplified the electronics somewhat and was much simpler to align in the spectrometer. A brief description of the different spectrometers used in the course of gathering data for this thesis is given below. A . Gas Cart The Gas Cart was specially designed for gas chemistry experiments carried out at T R I U M F . A pair of main 1.5 m diameter Helmholtz coils can provide magnetic fields from 0 gauss to in excess of 300 gauss oriented horizontally, and homogeneous to ~ 1% over a 5 litre volume. There were also two other pairs of coils, capable of generating a few gauss, which could be used to accurately zero the field and to provide a weak vertical field when necessary. The Gas Cart uses a standard muon counter and two arrays of positron counters placed above (Top) and below (Bottom) the target—at right angles to both the field and the beam directions. Each array consisted of two 25 cm x 45 cm scintillators, followed by 2.5 cm of graphite degrader and one 41 cm x 45 cm scintillator. The upper scintillators were moveable vertically to accommodate varying sizes of target vessels. Sometimes, such as when using the high temperature target vessel (see below) which degraded the energy of emitted positrons, or when one counter was weak, only two counters of the three and no graphite degrader were used. The experimental setup with Gas Cart is shown in F ig. 2.3. The large sizes of the coils allow large gas vessels necessary for low pressure experiments to be used. But the use of this spectrometer is limited by its magnetic field to weak and intermediate (< 300 G) field measurements. B . O M N I and O M N I ' OMNI and O M N I ' , shown in F ig. 2.4, are two nearly identical spectrometers. Both employ multiple orthogonal Helmholtz coil designs, which allow a main field of 4.0(3.4) k G along the direction of beam travel and smaller fields up to approximately 100 G in perpendicular directions. O M N I " is also equipped with x-y-z tr im coils which allow compensation of stray fields in order to 31 A A C Figure 2.3: The fiSR gas chemistry spectrometer. The Helmholtz coils (A) are oriented for spin-rotated muons, giving a 300 G magnetic field in direction B. There are both upper and lower positron counter arrays (C), each with three scintillator and graphite degrader (£)). The target vessel (E) is shown without its variable-temperature insert. The spin polarized (fr) muon beam (F) traverses the evacuated beam-pipe (G), passes through a brass collimator (H), triggering the thin muon counter (/), before entering the target through a thin Kapton window (J). The muons stop near the center of the target, retaining most of their init ial polarization. Each muon precesses (out of the plane of the page) until it decays, emitting a positron which may be detected by one of the counter arrays (C). This diagram is taken from Ref. [28]. 32 Collimation Back e Counter Right e + Counter fi Counter Target Window Longitudinal Field Coils Vertical Field Coils Left e + Counter Target Vessel •Forward e Counter To Gas Handling System Figure 2.4: Schematic of the Top View of the O M N I ' spectrometers. Also shown is the beam pipe with some collimation and a target vessel. obtain zero field. Magnetic fields are generated by coils positioned in a Helmholtz arrangement. Again, a standard muon counter and two to six positron counters were used. The counter telescopes were either single counter or pairs. The arrangement of counters discussed in Sec. 1.2 is a reasonable representation of the actual situation for both O M N I and O M N I ' . In a L F experiment, the F and B counters (see Fig. 2.4) form matched pairs from which one can obtain the corrected asymmetry spectra. In a T F experiment in spin rotated mode, the L and R counters contain the relevant precession information. These instruments have also been discussed in detail elsewhere, such as in Refs. [218,219]. C. HELIOS The HEL IOS spectrometer, shown in F ig. 2.5, is equipped with a warm-bore superconducting magnet capable of generating up to a 70 kG magnetic field horizontally. This is the only 33 spectrometer that can provide high magnetic fields at T R I U M F . Geometrical restrictions imposed by the magnet's bore force cylindrical shapes to be adopted. The forward positron counters are made up of four segments of positron counters and are placed in the forward position to make up the Forward Counters and another set of four for the Backward Counters, not shown in Fig. 2.5. The backward counters always have a much higher background due to decay positrons from muons stopping in the collimator and positrons from the beamline, which manage to traverse the dc-separator. In a L F experiment, all forward (and backward) counter segments are logically O R ' d together to form a single counter. In a T F experiment, the forward counter is repositioned until there is some overlap over the target vessel and opposing segments are used to form the asymmetry spectra. The disadvantage of using HEL IOS is the unstable low field and narrow opening for insertion of target vessels. 2.3 Data Acquisition and Analysis In a T D fxSR experiment, data is collected at a reasonably high rate; up to 10 5 events/second are frequently accumulated. This necessitates the sorting of data as it is collected, to reduce the amount of storage needed. This data sorting is accomplished in hardware. The data acquisition system was logically equivalent to what had been used in earlier gas phase / /SR studies at T R I U M F [115] and the details are given therein, but some simplifications and improvements have taken place. It has been discussed elsewhere [28] so only a brief recount is given here on how the current data acquisition system operates. The information of interest in an experiment is the event rate in each scintillator and the direction and time of each decay positron, relative to the time of arrival of the corresponding / i + . The data acquisition system must therefore accomplish a number of tasks. It generates histograms of muon decays for each counter. It must also veto events where there were two muons in the sample at a time (otherwise one doesn't know which positron corresponds to which muon). In addition, it must veto events when two (arrays of) counters detect positrons when there was 34 Forward e + Counter Target Vessel TM Counter Forward e + Counter Col l imator /4+beam \ B e a m Vacuum HELIOS Magnet Bore 0150mm B , Figure 2.5: A cutaway showing the central portion of a high-field experimental arrangement with HEL IOS. 35 supposedly only one muon in the sample (double hits). F ig. 2.6 gives the logic diagram for the signal processing used to collect the /xSR data for this thesis; it is best understood with reference to the diagrams of the spectrometers, Figs. 2.3, 2.4 and 2.5. A muon, upon leaving the beamline, passes through a collimator in a 5 cm thick brass shield and then through the muon counter and gives a "signal". Each signal first passes through a variable delay, used to synchronize the various counters. Following this, each signal enters a constant fraction discriminator (CFD) , which, in the case of the muon counter, discriminates against noise (dark current) and positrons in the beam since these generally give lower light pulses. The output from the C F D is a well defined negative pulse of constant amplitude (—0.7 V) and width (~ 30 ns). The fi+ counter signal is scaled as the / i ; n c rate. It also generates the pileup gate. If another muon enters the sample during this gate, both events wil l be discarded. If the muon has arrived when the pileup gate was not triggered, then the high precision (125 ps time-resolution) LeCroy 4604 time-to-digital converter ( T D C ) or "clock" is started; a valid start has occurred. The start signal triggers the data gate, which is generally 500 ns shorter than the pileup. The data gate determines the time period when a valid clock stop can be detected. Wi th in about ten nanoseconds at pressure ~ 1 atm, the muon has thermalized and precesses in the applied magnetic field until it decays, a few microseconds later ( r p = 2.2/xs), emitting a positron as discussed before. This positron passes through a array of counters, which form a "telescope", and signals from these counters are logically A N D ' d in a coincidence circuit, thereby reducing the effects of noise in the individual detectors and giving a valid event, which stops the clock. If a positron is detected at a time when there is no pileup and during the data gate, its logic pulse is sent to the N I M / E C L converter, which routes the signal to the stop input of the clock (TDC) . To one input, not used for a telescope, is fed the pileup signal, which allows the T D C to reject these events and restart. Then the clock writes the time to a CAMAC-res iden t histogramming memory module (eg. C E S HM-2161), which increments the bin corresponding to the time of the decay in the histogram appropriate for the counter array. The valid events are stored in the C A M A C memory, which is periodically read by a data acquisition computer (a V A X workstation), that handles the display 36 and storage of the data. The clock resets if no positron is detected within a gate period of a few muon lifetimes, or if two (or more) positron "stop" signals are received before the data gate expires. The clock is programmed also through C A M A C from the data acquisition computer. The signals for the various positron counters are all delayed by approximately 200-300 ns relative to the fi+ counter. This permits the examination of times before the arrival of the muon, allowing one to measure the positron rates when it is known that no muons are in the sample. As a result, one is able to measure the time-independent background counting rate, "6" in Eq. 1.8, so that it can be subtracted from the data during analysis. A l l data analysis was performed on V A X computers at T R I U M F , using a version of the non-linear, multi-parameter optimization (x2 minimization) program M I N U I T [220]. (Although newer versions are available, the data were fit with the older (1971) implementation of the program.) A l l of the errors quoted from fits are full MINOS errors, which give the statistical "la" error, i.e., the change in the fit parameter such that the value of x2 increases by one, taking correlations between the various parameters into account. For weak T F experiments, the spectra were usually fit to Eq . (2.4), which is developed further in the next chapter (Eq. 3.31). The intermediate transverse field runs were analyzed according to Eq . 3.32; the longitudinal field runs by Eq. 2.2, also developed further in Eq. 3.34. The best fit is determined by minimizing the chi-square where f(t) is the appropriate expression for N(t), TV,- is the number of counts in bin i, and <r,- is its uncertainty, given by the Poisson counting statistics of these experiments, <Ji — y/ffi. In situations where the points were over-parameterized by the fitted function, one parameter was often fixed at its expected value. These were not the only fits necessary; the parameters determined by fitting a series of runs were subsequently analyzed in terms of various models, say y = f(x). These fits took account of (2.5) 37 Figure 2.6: The signals proceed generally from the top to the bottom of the picture, starting at the scintillation counters. Constant-fraction discriminators ( C F D ) are used for the thin counter (fx) and the first counter in each of the positron counter arrays (T or B) to give better t iming. Scalers are not strictly necessary but provide a quick indication of problems. The circles at the bottom are various inputs for the LeCroy T D C . A good event is recorded when all three counters of an array fire after there has been a muon counted (data gate, G) , but only one muon (no pileup, P) ; and no subsequent muon (bit 0) or positron (bits 1, 2) is detected within the gate. 38 both "x" and "y" uncertainties by minimizing the modified chi-square i < + ( / ' ( a ; , ) ( T x t ) where / ' represents the partial derivative of / and so converts <rx into an uncertainty in y uncorrelated with o~y. When necessary, these types of fits were performed with a version of M I N U I T [220], but the many linear fits were done with the modified linear regression outlined next. In the case of a straight line fit, for example, a fit of the reaction rate of a first order chemical reaction vs. the concentration of the reagent, y — f(x) = ax + b, the minimum of the x2 (and the best a and 6) could of course be determined by a general-purpose non-linear fitting program like M I N U I T , but it is often cumbersome and consuming too much computing resources unnecessarily. The approach used in this thesis was based on the analytical linear regression method described in Ref. [28]. The minimum x2 occurs where -2a?i(y,- - axj - b) _ 2a<r^(yt- - axt - 6) 2 < + (^x,)2 [< + K „ ) 2 ] 2 (2.7) n_dX2 _sr ~ 2 ( K ~ a X i ~ 6 ) to 8^ and  — _ db ^ <72,- + {a<rxi)2 are simultaneously satisfied. 2.4 Target Vessels A variety of target vessels were used in the experiments for this thesis research to accommodate the different requirements imposed by the varying reaction conditions, reagent gases and experimental objectives. A l l target vessels were cylindrical shaped and had a window on one end and at least one port for gas handling on the other. A brief description of each vessel used is given below. 2.4.1 High Pressure Targets Many experiments (Mu+CO, Mu+NO, and part of Mu + N 2 0 and M U + C 2 H 4 ) were conducted using a high pressure (< 60 atm) chamber (Fig. 2.7) which was an aluminum gas cell 39 to PMT Target Window Figure 2.7: High pressure target vessel. approximately 15.6 cm (10 cm inside) long with a 9.5 cm inside diameter. The muon beam entered the target cell through a 1.1 cm diameter, 100 / im thick window bored in an 2.7 cm thick titanium end flange. The cell was designed to be positioned in the center of the HEL IOS spectrometer, but it nevertheless can be used with all the other spectrometers. This target has its own special muon counter which was a small disc of scintillator positioned as close to the T i entrance window as possible inside the beam pipe (see Fig. 2.7) [221]. Decay positrons were detected by counters on the spectrometer in use. Wi th just one 1/4" port on the other end of the vessel for gas handling, this cell is capable of standing 60 atm pressure. At the end of this thesis work, some data were taken for M u + C O , using backward muons 40 and another target cell which can go to much higher pressures (500 atm). This cell was made from thick-walled stainless steel with a 1 mm thick T i entrance window. It was positioned in the center of the O M N I ' magnet for a few runs at high pressure (up to 250 atm) taken on the M9 beam channel. 2.4.2 High Temperature Target The reaction of Mu with N 2 0 was partly carried out in a thick-walled, cylindrical vessel (Fig. 2.8) of high-carbon 316 stainless steel, the same vessel used in the studies of the kinetics of the reaction M u + C 2 H 6 and Mu + CH4 [126]. The vessel is similar to that used in an earlier rate measurements of reactions M u + H2 and M u + D2 [31], with similar design considerations, but for safety constructed to obey the A S M E (American Society of Mechanical Engineers) pressure vessel code in every detail except the muon entry window. Briefly, the vessel has at one end a very thin (0.054 mm) muon entry window of the heat-resistant alloy inconel X-750, and at the other, plumbing to introduce and remove sample gas. The vessel was surrounded by thin metal cylindrical heat shields, with very thin, flat heat shields covering the window end. This assembly was placed in a larger stainless steel vessel, which was evacuated, to provide thermal insulation. The inner vessel was heated using tubular electrical heaters, purchased from Omega, Inc., welded to a thick stainless steel plate which was mounted on the plumbing end of the vessel. Flat electrical heaters would have been preferable but none could be found which could survive high enough temperatures in a vacuum. Heating was done at the plumbing end only, to avoid introducing a stray magnetic field in the muon stopping region. Large (1.5 m diameter) Helmholtz coils with their axis traverse to the muon beam, provided the desired 4 G field. 2.4.3 Target for Radical Studies Most experiments involving C 2 H 4 and C 4 H 8 for the study of free radicals from M u addition reactions used a gas cell constructed from an A l cylinder about 20 cm long with a ~ 3 cm inside diameter, mounted with a flange and a mylar entrance window (2 cm in diameter) to allow the muon beam to penetrate into the gas. The window thickness was adjusted according to the total 41 L M N 10 cm Figure 2.8: High temperature target vessel. Where A) thermal exchange gas inlet, B) vacuum jacket pumping port, C) sample gas inlet/outlet, D) thermocouple in pressure-tight tube, E) exchange gas outlet, F) Vi ton O-rings, G) 0.125 mm Kapton window, H) Teflon O-r ing, I) 0.025 mm stainless steel window, J) stainless steel wire support, K) aluminized mylar heat shield (perforated for easy pumping), L) insulating vacuum jacket, M) thermal exchange gas space, sample gas space. Taken from D. J . Arseneau's Ph . D thesis [28]. 42 pressure in the target, about 0.5 mm at the highest allowable pressure, ~ 15 atm, and about 0.1 mm at the lower pressures. Another similar target cell, but only 5 cm in length, was also used at the highest pressure in order to optimize the stopping distribution of the muon beam within the region of maximum homogeneity of the magnet. A special coil, made of copper tubing, can be attached to the cell to control the temperature if so desired. However, the temperature cannot exceed 100 °C. 2.4.4 Cs Target The measurement of the Mu+Cs spin exchange reaction was the most technically challenging experiment due to the extreme reactive nature of Cs and its concentration's ultra-sensitivity to the environment. To obtain and maintain a stable and suitable concentration of Cs a special target vessel was used. The target vessel is a nickel-plated stainless steel cylinder with inner dimensions of 20 x 63.5 c m 2 (total volume 20.3 litres) and a nickel window of 2.5 cm diameter and 0.05 mm thickness to allow the entrance of the muon beam. The window is positioned well inside the target vessel (2 cm from the flange) and aluminized Mylar is employed as a second window placed at the outside of the flange to ensure the temperature of the inside window is not lower than that of the system. There are two ports on the other end. One is connected to a Cs boiler, a container made of pure nickel, and used exclusively for Cs input. The other one is for sampling and gas handling. A schematic arrangement of the target and gas-handling system is shown in F ig. 2.9. Most measurements (e.g., optical pumping) involving Cs, and other alkali metals invariably use inert glass bulbs [76,78]. The target used in this thesis work may represent the first successful use of a metal target vessel. The vessel, the boiler and some portion of the plumbing system was wrapped in more than ten heating tapes and then in fabric insulator. Each heating tape was powered separately so that a uniform temperature distribution inside the target vessel could be achieved and that different parts of the system could have different temperatures. During the course of data taking, the temperature inside the reaction vessel was monitored and maintained at a constant and homogeneous value, which was significantly (> 50 K) higher than the boiler temperature to ensure 43 o TJ c Target Vesse l Inside Window L O Valve 4= Quick Connector Sampling Tube Figure 2.9: Gas handling system and target vessel for M u - C s spin exchange experiment. Area in the dashed line was kept above the boiler temperature. The bottom of the sampling tube was immersed in liquid Nitrogen during sampling. that the concentration of Cs inside the vessel was constant. 2.5 Reagent Gases The bulk of each sample mixture was, in most cases, the moderator gas, usually N2 or A r (as mentioned, to give large Mu signal). A l l of these were U H P grade, with quoted purity of 99.999%, and were used without further purification except where indicated otherwise. Most of the reactant gases were of research grade (>99.9% pure), obtained in lecture bottles, and were used without further treatment. Depending on the target vessel and gases involved, additional care was sometime required. These will be discussed later as the situation arises. Gas handling systems are generally consisted of one or two pumping stations capable of achieving better than 1 0 - 5 torr vacuum in the system, stainless steel or copper tubing, valves, cold traps, standard volume and pressure gauges (see Fig. 2.9). The whole system was always thoroughly tested before commencing an experiment to ensure there was no leakage or 44 contamination and that the materials were compatible with the gases involved. Parts of the system are connected either by welding or Swagelok connectors. The general procedure of filling a given target vessel is the following. The whole system is first purged by inert gases (eg. N 2 ) and evacuated to better than 1 0 - 5 torr. The target vessel is than filled with reagent gas to the desired pressure. For very low pressures, the reagent gas pressure is measured in a much smaller "standard volume" and the gets is then expanded (or flushed) into the target vessel. At this stage, if deemed necessary, a repeated freeze-pump-thaw (FPT) cycle 1 was carried out until primarily 0 2 impurity in the reagent gas was reduced to a satisfactory level so that measurements of relaxation rates showed that the 0 2 impurity could be ignored, e.g., a F P T cycle made no change to the rates. Then, an inert moderator gas, usually N 2 , Ar or He, is let into the system up to the desired total pressure. The target vessel is then valved off from the rest of the system. While good vacuum was important, these experiments did not require very high (or ultra high) vacuum. In many instances, the desired low level of impurities could be accomplished by frequent dilution-pumping cycle with inert (pure) moderator gases. The concentrations of reagents were calculated from partial pressures according to either the ideal gas law for most gases or the van der Waals equation for N 2 0 , which deviates from ideal gas significantly at high pressures. For this reason alone, the temperature inside the target vessel was of importance in all cases. A l l temperatures were measured with thermal couples ( O M E G A ) and in some cases temperature was controlled by an electronic temperature control device. In most cases, pressures under 10,000 torr were measured with two different M K S baratron capacitance manometers, depending on the range, one for 0-100 torr and one for 0-10,000 torr, with uncertainties less than 0.6%. Higher pressures, up to 60 atm, were measured with a "Marsh" gauge to an absolute accuracy of < 0.2 atm. Overall uncertainties in total pressure measurement were 'The gas in question must have a boiling point significantly different from that of O2 and higher than the tem-perature of liquid nitrogen (L-N2). If it is higher than O2, a freeze-pump-thaw cycle is carried out as the following: fill the system with the gas in question, condense the gas into a container immersed in L-N2 until the pressure in the system dropped below certain value, this is called "Freeze". Pump out the remaining gas in the system, separate the container from the L-N2 and warm it up to a temperature higher than the boiling temperature of O2 but lower than that of the gas in question, keep pumping, this is "Pump". Stop pumping and warm the container up to room temperature, this is "Thaw". If the boiling temperature of the gas is lower than that of O2, then O2 will condense first so: condense the gas only to the temperature below O2 boiling point, separate the container and the system which still contains a lot of gas. Warm the container up and pump out everything in it. This is actually freeze-thaw-pump. 45 less than 1%. Some detailed purity information is given below for all gases used, along with some brief description on handling of each gas. Special attention is given to Cs because it is the most difficult reagent to deal with. • C O Research grade (99.97% purity) C O from Canadian Liquid A i r was used as received without any further purification. The manufacture stated O2 impurity is less than 5 ppm. Some sample was analyzed with a mass spectrometer (a residual gas analyzer, R G A ) and O2 level was found to be low (on the order of ppm) but the result was not conclusive. F P T was not possible for this gas. • N 2 0 N2O gas was research grade (99.995% purity) from Alphagaz (Canadian Liquid Ai r ) . Since the reaction rate of N 2 0 with muonium is very slow even a small contamination (mostly O2) would cause significant errors in the result. F P T processes were repeated many times at the beginning of each experiment until the observed relaxation rate stayed constant and no spin exchange reaction was observable (there is no spin exchange between M u and N2O but all the possible impurity gases, O2, N O and NO2 can spin exchange with Mu). • C 2 H 4 and C 4 H 8 The pure ethylene (99.99%) and isobutene (99.9%) gases were obtained from Matheson as "research" grade and were typically F P T ' d several times in order to remove dissolved 0 2 , prior to charging the gas cell. Pressures were varied in the range 1-16 atm and recorded with either a simple (Bourdon) gauge, believed accurate to ~ 3 percent or a more accurate Baratron (MKS) capacitance manometer. However, accurate pressure readings were not of particular concern in these experiments, which can therefore be taken at the 3% level. • N O 46 C P . grade N O (Matheson, 99.0%) gases were used as received. From a mass spectrometric ( " R G A " ) analysis, 0 2 and NO2 were not detected in the N O at the sensitivity level of 100 ppm. The manufacture-stated impurities are mostly N2 (0.5%), C O (0.2%), N2O (< 0.05%) and NO2 (< 0.05%). No F P T was carried out since only very small amount of N O gas were used and small percentage of O2 in N O gas would have no effect on the measured rates. The only impurity of concern here is NO2 because of its fast chemical reaction rate with Mu atom [217,222]. This wil l be discussed in detail in Chapter 5. • Cs Vapor Since the muonium relaxation rate is dependent on the Cs number density, the concentration of Cs atoms in the target vessel must be known accurately in order to reliably extract the spin-flip cross sections of interest. Several intrusive methods have been developed to measure Cs vapor pressure (density) [223-229]. However they would all greatly complicate the present experiment and, moreover, in most cases require a calibration (by vapor pressure curves), so that their accuracy is no better than that of the known vapor pressure curve. These methods were deemed not worthwhile in this experiment. Instead the Cs vapor pressure curve of Taylor and Langmuir [223] was used directly for a given boiler temperature, to calculate the concentration of Cs inside the target vessel. Although 50 years old, this Cs vapor pressure curve has recently been verified as accurate to within a few percent near 500 K [224-228]. Vapor pressure curves have, in fact, been used in a variety of experimental situations [62,68-71,75,77], but it appears that the absolute accuracy of the curves for the alkali metals (including Cs) is not well established, particularly in a metal environment in the presence of a foreign gas, which is the present experimental situation. Independent determinations of the alkali-metal concentrations at given temperatures seem to be rarely carried out, although they could be essential to an accurate determination of the reaction rate studied. Consequently, a further means of determining the concentration of Cs atoms in the present experiment was necessary. 47 The Cs vapor pressure was established according to the following procedure. The system was first thoroughly cleaned and evacuated to better than 1 0 - 6 torr and heated (by heating tapes) above the operating temperature for more than 24 hours before an experiment. The system was then exposed to Cs vapor at the highest running temperature for some time so that any material absorbed on the inside walls of the system that could react with Cs had enough time to do so. The system was then again evacuated. The coldest spot in the system is purposely the Cs boiler, which is temperature controlled to stabilize at a set value determined by the desired Cs concentration in the reaction vessel. For each run the boiler, previously loaded with 99.95% pure liquid Cs (Strem Chemicals), is opened to the target vessel long enough to establish an equilibrium Cs vapor pressure in the system before it is closed. Temperature fluctuations during this period are less than 0.5 °C. Pure dry N2 gas (passing through a cold trap first) is then slowly let into the target vessel up to a certain total pressure, typically 2 atm in these experiments. As mentioned before, the inside of the target vessel was then maintained at a temperature about 50 K higher than the boiler temperature for the entire duration of data taking. To check the Cs density, a known volume of gas in the target vessel was pumped out through a sampling tube filled with glass wool and immersed in liquid nitrogen. During sampling the total pressure inside the target vessel was kept higher than 100 Torr to prevent Cs coming off the walls. Cs vapor condensed in the tube and was then washed out with distilled water and titrated against a standard HC1 solution. Results of these titrations were used to verify the Cs concentration predicated from the vapor pressure curve. Some results of these off-line tests are listed in Table 2.1. Two different moderator pressures, 1 atm and 2 atm, were employed in these off-line tests, though the actual data taking was carried out at 2 atm pressure. As can be seen from Table 2.1, the titration data agree with the vapor pressure curve to within about 10%. The differences are likely due to losses of Cs during sampling and washing. The slight decrease of Cs titrated at 1500 torr has been shown to be a consequence 48 Table 2.1: Cs Vapor Pressure Test Cs in the target (10" 5 mole) Tb (K) P (torr) Titration V P curve 452 ± 3 700 1.37 ± .31 1.75 ± .23 452 ± 3 1501 1.30 ± .31 1.75 ± .23 466 ± 3 700 3.06 ± .41 3.10 ± .37 466 ± 3 1503 2.80 ± .40 3.10 ± .37 482 ± 3 700 4.96 ± .61 5.70 ± .65 482 ± 3 1503 4.70 ± .60 5.70 ± .65 488 ± 3 702 7.00 ± .86 7.15 ± .78 488 ± 3 1500 6.61 ± .81 7.15 ± .78 507 ± 3 705 12.9 ± 1.5 13.9 ± 1.4 507 ± 3 701 13.1 ± 1.5 13.9 ± 1.4 507 ± 3 1500 11.9 ± 1.5 13.9 ± 1.4 First column: boiler temperature. Second column: total moderator (N2) pressure before sampling. Third column: total number of moles of Cs from titration. Errors are estimated systematic uncertainty of the technique. Last column: moles calculated from the Cs vapor pressure curve of Taylor and Langmuir [223]. Errors due to uncertainties in boiler temperature. of slower sampling. The comparisons in Table 2.1 shows that the vapor pressure curve [223] is indeed reliable for this system, and consequently Cs densities are calculated from these vapor pressures only. It is noted from the titration data that there could be a systematic error of ±10% (or less), possibly higher at the higher pressures. The uncertainty in the Cs concentration was primarily due then to an uncertainty in the boiler temperature, which, in the range of 440-510 K, is well below 3 K. Nevertheless, this still yields an appreciable (< 13%) uncertainty in the Cs concentration due to the exponential dependence of the number density with temperature. Thus the overall uncertainty in the data is at the < 18% level, which is discussed further in Chapter 5. 49 Chapter 3 Spin Dynamics of Muonium and Muonium-like systems As mentioned in Chapter 1, to accurately extract useful information from / /SR data requires the knowledge of how the muon spin will evolve with time in different magnetic environments. In other words, the time development of the muon spin polarization, leading to what has been referred to as the "relaxation function", Rx (t) and R2 (t) measured in T F and L F experiments respectively, has to be compared with some calculated theoretical relaxation functions in order to relate to the underlying physical mechanisms causing relaxation so that useful information about the system of interest can be extracted. Of particular interest here is how each interaction contributes to the relaxation rate of the muon spin in muonium. In this chapter, the current understanding of muon spin dynamics in different magnetic environments will be outlined. The approach here is to first concentrate on the solutions of the non-interacting part of the Schrodinger equation, i.e, the eigenstates, eigenenergies and transition frequencies, to obtain the non-relaxing part of the muon spin polarization functions and ignore the interaction terms contributing significantly only to relaxation, which are treated in the next chapter. Here relaxation is introduced phenomenologically only, insofar as it modifies the muon polarization (via transitions) with rate "A" . The following symbols and conventions wil l be used in this chapter: G(t) is the complex time evolution function of the muon spin polarization without any 50 depolarization from interactions with the medium; R(t) is the relaxation function describing the time dependence of the the polarization amplitude as a result of reacting with the medium; P(t) is the total theoretical polarization function describing the overall polarization time dependence with relaxation effects included, so P(t) = R(t) G(t); The subscripts D, M and R are used to indicate in which muon state or chemical environment (diamagnetic, muonium, or Mu-radical) the muon spin is incorporated. The actual init ial fraction (amplitude) of muons in a particular state is always fit empirically for the reasons given in Chapter 2 and is of no physical significance to the determination of relaxation rates, although there is a correlation in the fits. Hence, in this chapter, the initial polarization of the muon spin in each state is assumed to be unity. The final polarization function is the product of the relative initial (empirical) amplitude and the relaxation function given in the next chapter for each particular muon state summed over these states. For the experimental conditions which this work is concerned with, the diamagnetic muon spin always relaxes very slowly since the spin lattice relaxation time for diamagnetic muons (eg. MuH) is similar to that for diamagnetic protons-on the order of 10's of fis to ms [134,154,156,230], seen also from related / iSR studies [28,29,231,232], which is immeasurably slow on the //s time scale of the muon lifetime. Therefore any diamagnetic muon is considered nonrelaxing for all practical purposes unless otherwise stated, i.e., Ro(t) = 1-The direction of the magnetic field is always assumed to be in the z-direction. In T F , the signal is always assumed to be measured in the ^-direction (for simplicity) and in L F , obviously, in the z-direction. 3.1 Time Evolution of the Muon Spin in a Diamagnetic Environment Free muons, or muons in diamagnetic environments (eg. M u H , N 2 M u + ) are characterized by the spin Hamiltonian, H D / f t = - W D I • B , (3.1) 51 where the magnetic field is along the B direction, I is the spin operator of the muon and up = ojp = IKB^H is the muon Larmor frequency.1 In a transverse magnetic field, the muon precesses in the x-y plane and the transition between the two Zeeman states, ± | , is observed with the Larmor frequency up. The time evolution of the polarization function is given by the expectation value of the muon spin raising operator in unit of h [119,139,213,233-235], I + = IX + Hy, PD(t) = GD(t) = 2(l+) = e-iUDt. (3.2) In a longitudinal magnetic field, there is neither oscillation nor relaxation since in this case the muon spins are fixed in the intial direction with no induced transitions between eigenstates, so PD(t) = GD(t) = 2(I2) = l. 3.2 Polarization Function of the Muon Spin in Muonium 3.2.1 Time Evolution of the Muonium Spin State in Vacuum Consider first the time evolution of the muon spin polarization in a free muonium atom in "vacuum" (or any situation where there is no interaction between muonium and the media). The solution to this problem is perfectly analogous to that of the H atom and has been treated fully elsewhere [115,119,213,234,236,237], So in this section only the results and some physical interpretation are given. The formation of M u initiates a hyperfine interaction between the muon and electron magnetic moments, just as in the case of the proton and electron in the H atom. The non-relativistic spin Hamiltonian describing this spin system in an external magnetic field (B) consists of three terms: H M / f t = w e S - B - w ^ I - B + w 0 S - I , (3.3) where S is the spin operators of the electron with Larmor precession frequency, ue, and the hyperfine frequency is denoted w 0 . For muonium, UQ/2TT = i/0 = 4463.3 MHz . The eigenstates of 1 As noted earlier, the frequency of a muon in a diamagnetic environment is, in principle, different from that of a free muon [215,216] But this "chemical shift" is too small to be resolved in these experiments and can be ignored. 52 this Breit-Rabi Hamiltonian are explicitly given [119,213,235] in terms of the individual nuclear (cxp and Pp) and electron (ae and (3e) spin-up and -down states (in the field direction) by |1) = |a„ae), |2> = c\afil3e) + s\^ae), |3) = | « > , (3.4) |4) = c|a^/? e) - s\0pae), whose corresponding eigenenergies are Ei = ft(w0+4U>M)/4, E2 = fi(w0+4Q)/4, E3 = 7 » ( W o - 4 C J M ) / 4 , (3.5) E4 = -7i(3wo + 4f i) /4, with the transition frequencies observable in a T F environment being Wl2 = « M — fi, = w M + n, Wl4 = WAf + fi + Wo, (3.6) W43 = U>M — — Wo. These expressions involve the field-weighted amplitudes c = l / \ / 2 ( l + x/(\/l + x2))1/2, s = l / \ / 2 ( l — x/(y/l + x2))1!2, and three angular frequencies wo, u j f = (we — <Mfl)/2, and fi = | « 0 [ ( 1 + a ; 2 ) 1 / 2 - 1], wherein a; = ,(t»e + t0„)/wo = B/B0, and 5 0 = w 0 / [ 2 7 r ( 7 e + 7„)] = 1585 G is the contact field of the muon at the electron, 7 being the appropriate gyromagnetic ratio; 7 e = 2.803 M H z / G , 7^  = 0.01355 M H z / G , and, for comparison, -yM - 1.394 M H z / G . These eigenstates and transition frequencies (i>0 = u>o/2ir) are also illustrated in Fig. 3.1. Figure 3.1 can be understood by considering an expansion of the eigenstates of the Hamiltonian in terms of the basis of Zeeman states \F, M). In zero (or weak) field, the first three eigenstates in Eq. 3.4 are 53 5500 I I I ^ 4500 — 3500 -2500 — N X 1500 — c 500 & -500 i_ <u c -1500 LU --2500 — — -3500 --4500 --5500 I I I 1 ^ 0 600 1200 1800 2400 3000 Field in Gauss Figure 3.1: Breit-Rabi diagram for the isotropic magnetic hyperfine interaction A^"/h = 4463 MHz of the simple two-spin system consisting of a muon and an electron. The allowed transition frequencies in a transverse magnetic field are indicated. In the weak field conditions of the / iSR experiment ( » 8 G) , the experimental signature is due to the essentially degenerate frequencies ^12 and i>23- In the intermediate fields (30 G < B < 100G ), v\2 and 1/23 are slightly different hence a "beating" signal would be observed. 54 associated with a total spin F = I + S of unit magnitude (respectively |1,1), |1,0) and |1, —1) in \F, M) representation), while the last state has zero total spin (|0,0)). A nonzero magnetic field defines a preferred direction in space which removes the threefold degeneracy of the zero field F = l state and gives rise to the triplet family of energy states seen in F ig. 3.1. The eigenstates of the spin Hamiltonian in Eq. 3.4 will be used below as a basis for analyzing the time dependence of the muon spin polarization function. In the following sections, the initial polarization of the muon spin in muonium is assumed to be unity, which can always be corrected to satisfy real experimental situations by a time-independent asymmetry factor. Readers are reminded again that a M u atom is formed with the muon and electron spins aligned either parallel (o-Mu) or antiparallel (p-Mu) and that these two M u states form with equal probability because the muon is spin-polarized but the captured electron is not. Zero and Longitudinal Magnetic Field (LF) In zero field, o-Mu (lo^a,.)) corresponds to eigenstate |1) (or |1,1)) hence is a stationary state of the Hamiltonian. But p-Mu (|a^/?e)) is a superposition of eigenstates |2) and |4) (|1,0) and |0,0)). As a result, the |a,,/?e) M u state oscillates between the two eigenstates at the frequency, VQ. This frequency corresponds to the energy separation seen in F ig. 3.1 at zero field. The time resolution in the /zSR experiments in this thesis was typically a few ns, much longer than the hyperfine mixing time (1/wp = 0.035 ns) in muonium, and consequently the hyperfine oscillation cannot be observed so that the polarization of the p-Mu state, {a^Pe), is averaged to zero over the experimental observation time [56,87,90,113]. As a result, the total muon spin polarization in muonium is effectively reduced (depolarized) to half of its init ial value upon muonium formation. If a magnetic field is applied parallel to the initial muon polarization, i.e., a longitudinal field along the z-direction, muon spin precession is not observed. As with zero field, state \a^ae) is an eigenstate (corresponding to |1,1)) hence is a stationary state of the Hamiltonian, while \a^pe) is again a superposition of eigenstates |2) and |4) so that the fast oscillation (modulation) between 55 these two states remains unobserved in most fiSR experiments. In zero field, the muon spin in the p-Mu state spends half of its time in the |/3^ae) state and the other half in the lo^/Je) state with average polarization zero, therefore the overall polarization in M u is 1/2. In a nonzero longitudinal field, however, the time spent in each state is not equal and is dependent upon the strength of the magnetic field. The expectation value of the muon's Pauli spin operator in the .z-direction, ( I z ) , gives the time evolution function of the muon spin polarization in a L F field, as: G M ( o = 2 ( i , ) = i + i r 2 + 1 c ; ^ ) ] - (3-7) It should be pointed out that the muon polarization is not actually lost in muonium, but only shared with the electron to an extent that depends on the external field. If quantum irreversible processes cause relaxation of the electron spin by interaction with the medium, all the polarization eventually disappears; but without such a process, the muon polarization always returns to +1 periodically. The observed total muon spin polarization, after averaging over the experimental time resolution is, however, time independent: G M ( L F ) = G^T) = \ + \ - Y J ^ - (3-8) The apparent depolarization of the muon spin in muonium is "quenched" by a strong magnetic field (x ->• oo). Note that in zero field, GM(ZF) = 1/2. Transverse F i e l d ( T F ) In a nonzero field oriented perpendicular to the muon spin (in the x-direction), o-Mu is also a superposition of the eigenstates of the Hamiltonian and is no longer a stationary state. As shown in many references [119,139,213,233-235], the complex time evolution function of muon spin polarization in free muonium, in T F , is obtained from the expectation values of the raising operator, (I+), GM(t) = 2(1+) = i [(1 + S)eiu^ + (1 - &)eiunt + (1 + J ) c ' w » « + (1 - 8)eiu^} , (3.9) where S = x/y/l + x2. Each frequency carries muon polarization information; however, if the field is such that x is small (e.g., fields used in this thesis for Mu studies are B < 100 G) then, as can be 56 seen from Fig. 3.1, both frequencies 1/14 and 1/43 are of order of the hyperfine frequency (i/n), and are thus again too fast to be seen [56]. The muonium spin polarization function averaged over the hyperfine mixing time in a low T F (/TF) can then be simplified to: GM (/TF) = I[ E ' '(WM+n)t + e <(«„-n) t ] ^ A ^ where the two frequencies U>M and fi can be expressed in terms of the frequencies W12 and W23, as, UJM — | ( w i 2 + w 2 3), and fi = 5(0*12 — ^23)- One can interpret U M as the average of the two "triplet" muonium precession frequencies, and fi as the "beat" frequency. In magnetic fields less than 10 G , fi » 0 and hence W12 and W23 are indistinguishable from each other, W12 = w 23 = WM [31,32,56,58,119,238]. It is noteworthy that this frequency originates exclusively from the original o-Mu. Wi th this very weak field l imit, Eq . 3.10 further reduces to the very simple result: G M ( « T F ) = i e * * " ' . (3.11) In a sense this is analogous to the diamagnetic time evolution function of Eq . 3.2, with the init ial amplitude now reduced by 1/2 due to the fact that half of the M u formed are p-Mu which are "lost". In high fields, where 6 —> 1, the polarization evolution function is given by, GM(hTF) = I [e ' 'W l 3 t + eiw"1] . (3.12) Since x is now much larger than unity, the two frequencies are now approximately given by W12 = — {up — |wo) and W43 = —(w^ + |wo). Both of these are not observable in (iSR even though at certain fields (where u>u « |wo) u>i2, in principle, becomes small enough to be seen. However, for vacuum Mu , this requires a field around 164 kG which is not attainable from any existing pSR spectrometers. A t the highest field currently available, about 60 k G , w i 2 = 8.7 GHz . As wil l be discussed later, for some Mu-radicals which have smaller hyperfine frequencies, it is possible to observe these precessions; e.g., for the muonium ethyl radical, M u C H 2 C H 2 , the field required to bring w 1 2 to zero is only about 12.2 k G . It is also worthy of note that there are only two 57 frequencies of significant amplitude at high fields unlike at low fields where there are four frequencies of similar amplitude (but two of which are averaged to zero over time). 3.2.2 Polarization Function of Muonium in Matter The polarization functions given above do not include the effects of the interactions between muonium and the media. If the spin of the muon in M u is depolarized by some mechanism with a relaxation rate A, the amplitude of the spin polarization is reduced with time (e~xt), and the spin polarization function is then given by: S p i n E x c h a n g e : T F One example of an interaction causing muon spin relaxation is the electron spin exchange interaction, in which an a electron on one atom (e.g., Mu) is exchanged with a (3 electron on the other (e.g., Cs). In their brief encounter, the state vector |a e /? e ) in the atom-atom interaction is not an eigenstate of the spin Hamiltonian and as such evolves in time, emerging as the state \j3eae). In a T F , when the time between collisions is much longer than the hyperfine mixing time in muonium (0.035 ns), often referred to as the "slow" spin exchange limit, this electron spin flip may destroy the coherence of the muon precession in the muonium atom, causing a depolarization of the experimental precession signal. This well-known relaxation process is discussed in detail in Chapter 4 as well as in Appendix A . l and its effect on damping the / iSR signal is illustrated in F ig . 3.2. It can be shown [90,91] (Appendix A . l ) that, in the slow spin exchange l imit and a very weak field, in general, the muon spin polarization function is given by where ASF is the spin flip rate, defined by Eq. 4.58 on page 110. Note the two factors of one half. The one in front, as mentioned, is due to the loss of p-Mu. The one before ASF indicates that only half of the spin flip collisions are effective in leading to muon spin depolarization. One can think of the spin exchange process as a quick sequence of charge exchanges in which M u is first ionized to PM(t) = e-xt GM(t). (3.13) PM(WTF) (3.14) 58 0.10 0.5 1.0 1.5 2.0 Time in microseconds 0.5 1.0 1.5 2.0 Time in microseconds 2.5 Figure 3.2: The experimental jzSR signal after removal of normalization, decay, and background. The top spectrum was obtained in 1500 torr pure N 2 moderator at 566 K. The bottom spectrum was for the same conditions but with an added 9.1 x 10 1 4 molecules-cm - 3 of Cs. 59 / i + and then a new Mu atom is formed. It can be shown that the newly formed M u wil l populate the four eigenstates with equal probability and hence precess with one of the four possible frequencies in Eq. 3.6 also with equal probability. Since two out of the four frequencies are not observed and the two remaining frequencies are identical at low fields, a spin flip reaction wil l lead to spin depolarization half of the time and hence the factor of two in the SF rate. It is convenient to define an experimental relaxation rate A<j which is the rate of muon spin depolarization caused by electron spin flip interactions. The relation between A<j and ASF wil l depend upon the field condition by a factor / , A<f = / ASF- In the present case of a weak T F , / = | , as in Eq. 3.14. In intermediate fields, which are defined such that the field is small enough so that x2 0 but now large enough so that x2 is much larger than ASF/O>O (see Appendix A . l ) , the difference between frequencies W12 = U J J — fi and W23 = w « + fi cannot be ignored and both frequencies are observed. A typical "beating" signal in 96 G T F is shown in F ig . 3.3 In this case, Eq . 3.10 cannot be further simplified and shows a two-frequency precession. It can be shown for a spin 1/2 dopant [95,213,236] (see Appendix A . l for details of an alternate and more general treatment) that in this situation A<j = §ASF and where it is recalled that P(t) = R(t)G(t). Following the argument given in the last paragraph with regard to Eq. 3.6, only one out of four times the muon spin wil l precess with the same frequency before and after a spin flip collision. Since the interaction time (typically of order of picoseconds) is small compared to the hyperfine mixing time (35 ps), the signal will precess coherently with the / = f . This change of / factor from 1/2 to 3/4 has been dubbed and will be referred to later as the "3/4 effect". In high T F field (8 —• 1), there are only two frequencies and each SF collision leads to a loss of half of the polarization. The situation is therefore similar to that in weak field, £ i ( « M + f i ) « _j_ e > > M - n ) l e - 3 A s p t / 4 (3.15) same probability, 1/4. The effective rate of loss of the muonium signal is therefore A<j = | A S F or PM{hTF) = i [eiui3t + e'"- 3 '] e - * s p ' / 3 , (3.16) 60 0.00 0.07 0.14 0.21 0.28 0.35 T i m e i n m i c r o s e c o n d s Figure 3.3: The experimental /zSR signal at 96 G T F measured in 60 atm pure N 2 gas. The "beating" of two frequencies is clearly seen. The modulation of diamagnetic signal causes the apparent slope. and again / = i . If the spin flip rate is much faster than the muonium hyperfine frequency wo, known as the "fast" spin exchange l imit, the muon spin in muonium can not follow the rapid change in the electron spin. In this case the muon spin in muonium behaves as if the muon were in a diamagnetic environment. Factor / can no longer be used to relate the rates. The muon spin polarization function can also be explicitly calculated for this case [94], P M ( T F ) = i e - ' ^ e x p - ^ t . . (3.17) Note that now the observed relaxation rate Xj = 2 A ^ r * s inversely proportional to the SF rate ASF and the precession frequency is that of a muon in a diamagnetic environment. Data obtained in the course of this thesis work, for the Mu + 0 2 reaction at high 0 2 pressures, is explained by Eq. 3.17, but these results are not discussed herein. S p i n Exchange : L F Electron spin exchange in a longitudinal field also leads to a loss in the muon spin polarization, again, through the hyperfine interaction of the muonium electron and the muon spin. The effect of 61 this depolarization is "quenched" in the same way as for p-Mu discussed above because the polarization is again shared by both the electron and the muon. That is to say, ~ 2(i+r3)) - 2(i+x>) i n a longitudinal field. Or, the complete polarization function, recalling Eq. 3.8, is, 1 -4- 2rr2 * S F ' P " ( L F > = 2(lW~3(,+*3)- ( 3 > 1 8 ) This relation is formally derived in Refs. [90,91]. In the l imit of weak fields x —• 0 and thus P M ( ^ L F ) = i e - ^ . (3.19) The relaxation rate is half of the spin flip rate (/ = | ) just as in weak transverse field. In contrast, from Eq. 3.18, at very high fields, x ^> 1, there is no relaxation at al l . Physically, this means that the system remains ("locked") in the |11) = la^tte) eigenstate. C h e m i c a l R e a c t i o n : T F Another example of the interactions causing spin depolarization is a chemical reaction in which the muon in the product is either in a diamagnetic state, thus precessing in a T F with frequency U>D in the opposite direction, or in some Mu-containing (radical) species with a markedly different precession frequency or a fast spin relaxation. In both cases, the coherent free M u precession signal is lost and one observes a relaxation of the M u signal with the relaxation rate equal to the rate of chemical reaction, A c . It should be emphasized that in a chemical reaction, the muon may precess with different frequencies before and after the reaction but the muon spin direction is not altered during the rapid reaction time. The resulting polarization function is, (TF) = \e~Kt eiWMt. (3.20) Li The effect of what can be viewed as an abstraction reaction (Mu + N 2 0 —>• M u O + N2) on the M u signal can be seen in F ig. 3.4. In contrast to spin exchange, a chemical reaction can destroy the coherence of the muon precession completely if accompanied by a change in muon precession frequency, which is the usual case (e.g., diamagnetic product), regardless of the field. Thus, the polarization decay rate due to 62 E E CO < 0.05 0.00 f i I,, , i i h -0.05 --0.10 --0.15 i 0.0 0.5 1.0 1.5 2.0 2.5 Time in microseconds Time in microseconds Figure 3.4: The experimental / /SR signal after removal of normalization, decay, and background. The top spectrum was obtained in 15.7 atm pure N2 moderator at 298 K and 5G T F . The bottom spectrum was for the same conditions but 22 atm pure N2O. 63 chemical reaction is always equal to the chemical reaction rate and is not dependent upon the applied magnetic field. Chemical Reaction: L F Any chemical reaction in a longitudinal field that places the muon in a diamagnetic environment (such as MuNO) , merely breaks the fx — e hyperfine interaction without changing the muon spin polarization. Since, as noted at the beginning of this chapter, the diamagnetic muon relaxation is immeasurably slow on the microsecond time scale of the muon lifetime, any chemical reaction that results in the formation of a diamagnetic product not only causes no spin relaxation, but also prevents further spin relaxation in that diamagnetic product. 2 Thus chemical reaction alone is not observable in longitudinal field because it causes no relaxation. However, such reactions cannot be ignored since the ensemble polarization is weighted by the overall reactivity of muonium. In the presence of competing relaxation processes, chemical reaction affects the total relaxation rate measured in L F (see below). If the product of a chemical reaction is a paramagnetic radical (eg., MuO or MUCH2CH2), muon spin can relax, for example through spin-rotational coupling. In this case, a chemical reaction will lead to muon spin relaxation but the relaxation function now not only depends on the chemical reaction rate but also on the depolarization rate of the product (see below). 3.3 Polarization Function of the Muon Spin in Mu-Radical This section discusses the spin dynamics of muonium-substituted radicals in the gas phase, formed by addition reactions such as M u + C2H4 ^ MuCH 2 C H 2 . This is treated differently from the above since rigorous solution is not possible in this case, to M u is that the muon spin relaxation is due to intra-molecular As noted, the approach here is to obtain the non-relaxing part of the muon spin polarization functions and ignore the interaction terms contributing significantly only to relaxation, which are treated along with other "external" relaxation interactions in the next 2In molecular ion environments, eg. N 2 M U + , thermal electron exchange reactions with added dopants can cause relaxation but reactions of this nature are not important here. 64 chapter. From general considerations as well as the theory developed by Turner and Snider, the spin Hamiltonian for a Mu-radical in a laboratory fixed frame is given by [157] where, in addition to the symbols given in Eq. (3.3), J is the rotational angular momentum quadrupole [157,239], u>j and uip are the Zeeman precession frequencies for the molecular rotation and the protons; wo = ITA^, U>SP = 2nAp, WSR, WIR, and WPR are the isotropic hyperfine frequencies (A's are isotropic hyperfine constants) for the (electron) spin-muon, spin-proton, spin-rotation, muon-rotation, and proton-rotation couplings; U>A is the anisotropic Zeeman frequency for electron (Zeeman anisotropics for the muon and protons are not included on the basis of small magnetic moments); CA and PA are the anisotropic hyperfine coupling constants. Some insignificant terms (e.g., coupling between the muon and protons) are not included. The task of finding the eigenstates and associated eigenenergies of this Hamiltonian in order to describe motion of the states is a formidable one and no exact solution of this problem exists. Some additional simplifications have to be made. For instance, the anisotropic contributions to the hyperfine coupling arising from direct dipole-dipole interactions are averaged to zero due to rapid tumbling of the radicals in gas (and liquid) phase. Although couplings to the rotational angular momentum and the anisotropic couplings are extremely important in causing spin relaxation (i.e., depolarizing the muon spin), they are first ignored here in order to derive the (static) energy levels of a muonium-substituted radical in gases, where there is no evidence for anything other than the usual isotropic hyperfine couplings given by Eq. (3.22). Thus, Eq. (3.21) is reduced to the following truncated Hamiltonian: (3.21) operator; Ip is the spin operator of the protons in the radical; [J]^2^ is the rotational n/h = ueS • B — ui^I • B + WQS • I — J^WpIp • B + ]T)w S pS • Ip. (3.22) 65 Considering that in most cases the muon hyperfine coupling is larger than the coupling with other nuclei and that proton spins have a very small gyromagnetic moment, at high enough fields [uie W S P ) , one can treat all contributions from protons as perturbations to the Hamiltonian (Ho) given in Eq. (3.3), H/ft = Ho/ft + [- ] T > P I P • B + £ > S P S . i p ] . (3.23) The approximate solutions to the above are as following: the zero-order eigenstates are -|1> = \ailae)U\K,M), |2> = ( s | a ^ e ) - r c | ^ a e ) ) n | ^ , M ) , |3) = \/3^e)U\K,M), (3.24) |4) = ( c | a ^ e ) - s | ^ a e ) ) n | ^ , M ) ; the first-order eigenenergies are -Ei = h[(u>0 + 4 w M ) / 4 - Y1UPM + \ H w s p M ] , E2 = h[(u0 + 4n)/4-^wpM + ±(c2-s2)J2"spM], E3 = ft[(w0-4wM)/4,-X]wpM-i^wspM], (3.25) EA = -ft[(3w 0 + 4 n ) / 4 - ^ w p M - i ( c 2 - S 2 ) 2 u ; s p M ] ; and the /zSR-transition frequencies are -W12 = UM - fi + s2 y^u>spM, = W M + fi + w 0 + c ! ^ u S p M , (3.26) ^43 = W M - f i - W o + S 2 ^ ] U)SpM, where H\K, M) is the multiple of all nuclear states in a K , M representation with K=]H I p , the total nuclear angular momentum operator, and K = ip, ]fj / — 1, ...0 (or \ ) , the total nuclear 66 0 4 0 80 120 160 2 0 0 Field in Gauss Figure 3.5: Breit-Rabi diagram for three-spin-system. The values of Ap= 330 MHz and Ap = 67.1 MHz are the same as in M u C H 2 C H 2 [35]. spin, and M is the sum of all (including the electron and the muon) spin's Zeeman projection on the quantization axis (usually chosen as the field direction). Note that these give the same results as for vacuum-Mu (Eq. 3.6) when the nuclear spins are ignored. It should be emphasized that higher order corrections are required at low fields. To illustrate, a three spin system containing just one proton as well as a muon and a electron can be used. Numerical calculations produced the energy-levels plotted in the Breit-Rabi diagram of F ig . 3.5 for such a system with hyperfine constants set the same as those measured in the ethyl radical, M u C H 2 C H 2 [A^ = 330 MHz and APij3 = 67.1 MHz [35]). Even with this relatively simple system, at low fields, the myriad of couplings (transitions) among which the muon polarization is distributed (15 in F ig. 3.5, 765! in the complete M u C H 2 C H 2 radical itself [7,138,139]) renders the radical unobservable in a weak transverse field by the / /SR technique. It is worth noting that many of these couplings involve an electron "spin flip" which in turn are indirectly coupled to the muon 67 through the eigenstates of the isotropic / / + - e ~ hyperfine interaction. In a longitudinal field these transitions correspond to off-diagonal matrix elements and give rise to 7\ spin relaxation. A phenomenological model has been developed for such relaxation rates, valid over a wide range of fields and pressures [231,232], which will be dealt with in detail in the next chapter. On the other hand, since > Ap, even by fields ~ 200 G , this plethora of spin couplings has collapsed to a degenerate pair of transitions, labelled t / i 2 and 1/34 in Fig. 3.5 (Recall Eq . 3.9, with <5 —• 1 at high field, these two are the only significant transitions.), which can be readily measured in strong transverse field using a Fourier Transform (FT) technique [240]. From the high-field eigenstates, solutions to Eq. (3.23), it can be seen that these transitions correspond directly to muon spin flips, which in fact remains true regardless of the number of protons [7,133,138-140,241,242]. This " E N D O R " limit is the basis for the detection of muonated radicals by Fourier Transform / /SR [240]. Again these transitions, which appear as observable frequencies in a high T F , contribute to (Ti) relaxation in a L F . 3 For the relevant experiments in this thesis the high field condition is always satisfied and the discussion wil l be limited to this condition hereafter. At high fields, since proton spins contribute very little to the Hamiltonian, as seen in Fig. 3.5, Eq. (3.23) can be further reduced to the same form as Eq. (3.3) with the only difference being the magnitude of the hyperfine coupling constant (e.g., for MUCH2CH2, u>o/2n = 330 MHz) . A Breit-Rabi diagram for this reduced Hamiltonian is plotted in F ig . 3.6 for MuCH 2 CH2- A l l the results derived in the last section (except those specific for weak fields) can be directly applied to the muon spin in the radical, with some depolarization rate XR always present except at extremely high fields where all couplings are "quenched". So, in general, one expects the polarization function of the muon spin in a radical to have the form (also see Refs. [8,79]) PR(t) = ^ ^ e - A " t e ' ( w " t + * « ) . (3.27) Note, again, in a L F , the frequencies (and initial phases) in the above equation are always set to 3In fact, at ~ 200G, there would also be line splittings observable due to residual p + - e - hyperfine couplings [7,133,138-140], reflecting the fact that the correct solutions are not the high-field eigenstates of Eq. (3.23), but in fields B > 1 kG these proton couplings are fully "quenched". 68 zero. 3.4 Competition among Depolarizing Processes The next question to be asked is what happens if there are multiple relaxation processes, e.g., both a chemical reaction and an electron spin exchange interaction occurring with Mu . The essence of the mathematical model for this kind of problem was first treated in early "residual polarization" experiments [106,243,244] and a general solution, valid in both transverse field and longitudinal field is provided in Appendix A.3 ; only the results are outlined here. Consider a transition between two generalized "states" (not necessarily a quantum state) i and / of the muon spin in an T F . In the initial state, i, the depolarization rate is A< and precession 69 frequency w,-, and similarly A/ and u/ for the final state / . Assume the transition rate from i to / is Aj and that no polarization wil l be lost during the transition. The time dependent polarization is then, Pi(t) = eXUit e~Xit and Pj(t) = e* <"' t e~x'1 for states i and / respectively. Generally, spins of i and / could precess in the same or opposite direction (s). Wi th a transition distribution function for the rate of reaction D(t) = Xte~Xtt, it is shown in Appendix A.3 that P(t) = e-(\i+\i)t e i W _ A t + A,- - Xf + i(uj - u>f) ( A t + A , - A / ) 2 + ( w , - W / ) 2 ' \ , At + A,- - Xf + i(u{ -ut) x t , t + 77——7 . , rjAte ' e ' . (3.28) (At + A,- - Xf)1 + (w,- - uj)1 In a L F , there is no precession, just "modulation" of the muon spin, which, as noted, is averaged to zero by the experimental time resolution. Using equation (3.28), but setting all frequencies to zero, the rate equation in longitudinal field is given by The relevant situations in this thesis are where an initial M u state undergoes some chemical reaction (transition rate, A c ) to form either a diamagnetic or a paramagnetic final state; where the muon spin in M u relaxes through an electron spin exchange type interaction (initial relaxation rate, Aj); and/or where, in the paramagnetic free radical state, the muon spin can be relaxed by a spin-rotation interaction (A r ) , as well as electron spin exchange. Relaxation in a diamagnetic state is ignored as discussed before. Some special cases of these situations are considered below (and more detailed analysis is given in the next chapter). 3.4.1 Spin Exchange and Chemical Reaction with Diamagnetic Final State Transverse Field At weak fields where only one Mu precession frequency is observed, assume the depolarization rate in the M u state is A j , transition rate A c , and the final state is diamagnetic so that no further depolarization of the muon spin in final states occurs; from Eq . 3.28, one has P(t) - AM e-(Xc+Xi'>t C ' ^ W + ^ M ) * + A d ci(-ur>+*D)t (3 30) 70 where AM, AD, <J>M and (/>D are all functions of A c , A j , U>M and w D (see Appendix A.3 for detail). The important points here are: first, the total observed relaxation rate in M u is the sum of the chemical reaction rate and the spin exchange depolarization rate; second, the precession acquires a phase shift as the result of different precession frequencies before and after the chemical reaction. From Eq. (3.30) it follows that one cannot distinguish chemical reaction and spin exchange interaction from the relaxation rate in a weak transverse field alone. In principle, the asymmetry and phase contain this information but in practice these are affected by many experimental factors as well which makes it very difficult to extract accurate individual rates. Because relaxation processes for muons in diamagnetic environments are generally too slow to be observed, the explicit form of the real part of P(t) from Eq. (3.30), which is the projection of P(t) onto the z-axis and is the fitting function for A(t) used in the //SR. analysis, when only muonium and diamagnetic muon are considered, is Ax{t) = Re[P(t)] = AMe-XTtcos(ujMt + <f>M) +ADcos{wDt-<pD), • (3.31) where Ax = A c + Aj = A c + | A S F - Note that in the actual experiments, AM, AD, <J>M and <f>o all depend on experimental conditions and are only phenomenological parameters (to be differentiated from those in Eq. 3.30). The second term also includes those thermalized as diamagnetic muons initially. This is essentially the same form as given earlier, Eq . 2.4. At intermediate fields, 30 G < B < 100 G , one also has, Ax(t) = e~Xrt [AX cos (w12t + fa) + A2 cos (u23t + fa)] + AD cos (uDt - <f>D), (3.32) and . 3 . A T = A C + T A S F -4 where « 1 2 = U>M + fi and u>23 = U>M — fi, as discussed earlier (Eq. 3.15). Since the exponential decay factor (/) for A(t) due to spin exchange changes from 1/2 at weak fields to 3 /4 4 at intermediate fields, and does not change for chemical reactions, the 4 This ratio is actually independent of the spin of the paramagnetic dopant 71 I CO 3.0 2.4 1.8 < 1.2 0.6 0.0 • I Pure N 2 o 0 2 dopont X N 20 i _ Q • • 0 20 40 60 Field (G) 80 100 Figure 3.7: T F relaxation rates as a function of fields for M u + 0 2 and M u + N2O. For small O2 concentrations, the M u + O2 undergoes SE only. The solid line denotes the relaxation rate, Xj, predicated from the weak field value at 4.5 G (dotted line). In the M u + N 2 0 chemical reaction, at these pressures, there should be no spin exchange, so no field dependence. measurement of A(t) at both weak and intermediate fields under the same conditions wil l allow both the spin exchange and chemical reaction rates to be determined. This is the basis of a new experimental technique (first suggested by Senba [95,98]) developed during this thesis research, which can be used, in addition to the more "standard" technique to be discussed next, to detect, conveniently, the existence of any spin exchange (pr spin-exchange-like, see M u + C O result) reaction and to separately measure spin exchange reaction and chemical reaction rates in a T F . A plot comparing the rate dependence on the field for two systems, one undergoes SE and the other undergoes chemical reaction, is shown in F ig. 3.7. 72 L o n g i t u d i n a l F i e l d By substituting Xt = A c , A,- = A j , Xj = 0, and the proper asymmetry, the polarization function in longitudinal field with both chemical reaction and spin exchange operative becomes p ^ = Y ^ e _ ( A c + A d ) t + i r r T - ' <3-33) where A = 2 ( i + g 2 ) > * s the spin polarization function of muonium in the absence of relaxation as given earlier (Eq. 3.18). Eq. 3.33 contains both a relaxing and a non-relaxing component. In this case, the ^zSR signal, measured in the z direction, has the form A,(t) = AMe-XLt (3.34) where A S F A L = A c + 2(1 + x 2 ) ' The second term in Eq. 3.33 is time-independent and is blended into No of Eq . 1.8 in the fittings. The contributions from chemical reaction and spin exchange interaction are readily separated if one measures the relaxation rates over a range of fields. Specifically, a plot of XL vs. •q^y will be a straight line with the y-intercept giving A C and the slope giving A S F / 2 . This LF-/zSR technique is the "standard" method which has been used for years to measure both chemical and spin exchange reaction rates. It has also recently been employed in this thesis work, in a study of the M u + N O reaction. In contrast, the "3/4 effect" technique in T F mentioned earlier is new and still under development. 3.4.2 Chemical Reaction with Mu-Radical Final State In this case, relaxation is observed in both the initial state and the final state. T ransverse F i e l d Since A,- = d, Xt = Xc and Xj = A R , the polarization function is: P(t) = AM e _ ( A c + A d ) t e ' ( a ' M + * M ) t + AR e~Xrt ei<.-u"+<l"<)t + AD e ' X - ^ - r ^ ) ^ ( 3 35) 73 where, again, AM, ,AR, <J>M and <f>R are functions of A c , A r , U>M and U)R. A S mentioned before, in general, there are many different radical precession frequencies in T F , but they are too fast to be seen in weak field and hence the / iSR signal has the same form as Eq . 3.31. The diamagnetic term is still kept because some muons are in diamagnetic environments initially. L o n g i t u d i n a l F i e l d Similar to Eq. 3.33, but with \j = A r , the polarization function is P ® = A [ X * - X r e-<*-+*'>' + * c (3.36) Ac Ad — Ar Ac -f- Ad — Ar To conclude this chapter, in general, where there are multiple muon containing species (including diamagnetic muon, muonium and Mu-substituted radical), the total muon spin polarization function has the form, P{t) = ^ J 4 i e - A i t e , ' ( ^ t + ^ ) , (3.37) j where the jth species is characterized by an amplitude (Aj), a frequency (WJ), an initial phase ((/>j), and a relaxation rate (Aj). Of course, all modulations vanish upon averaging over experimental time in a L F . In typical experimental conditions, only one or two (at most three) terms will usually dominate. It should also be pointed out that, in practice, sometimes additional terms are required in fitting the data to account for signals originating from the surrounding environment which may have their own relaxation rate (e.g., muons stopped in the Cs target wall). 74 Chapter 4 Theoretical Background 4.1 Chemical Reaction The chemical reaction systems investigated in this thesis research are all with small molecules of interest to both current theoretical calculations and studies of (H-atom) combustion kinetics. They are all analogous to the corresponding H-atom systems and as such perhaps the most important aspect of these comparisons is the large kinetic isotope effects seen. A brief, selected review of some of the theoretical treatments of reaction kinetics and isotope effects in general, as well as the mechanisms of these particular reactions in the context of a pSR experiment, are discussed in this section. 4.1.1 Reaction Kinetics Potential Energy Surface The starting point for all reaction kinetics calculations is the determination of a potential energy surface (PES) to describe the interatomic potentials of the reacting atom/molecules. There are two major approaches to the problem of calculating reaction rate constants: dynamical calculations which follow the trajectories of specific reactions over potential surfaces and statistical calculations which average over some microcanonical rate constants according to certain distribution functions. A l l trajectory calculations require complete knowledge of the P E S which is not always (in fact, in most cases) attainable. Although statistical approaches are developed to sidestep this difficulty, some knowledge of the potential surface is still required in these calculations, particularly in the 75 region of the barriers. In this regard, chemical reactions involving M u are of principal interest and importance because its ultra-light mass is often sufficient to reveal features of reactions that are not experimentally detectable from reactions involving its much more massive cousins (H, D and T ) . In principle, it should be possible to determine potential energy surfaces from ab initio methods involving the solution of the Schrodinger equation, perhaps with the aid of approximations based on various quantum mechanical criteria. Advances in computational techniques for accurate quantal calculations have reached the stage where totally ab initio calculations of useful accuracy are now possible for simple enough chemical systems [179,245-252]. Unfortunately, it is still impossible to perform such calculations with sufficient accuracy to be of general use in the study of reaction kinetics[248,253-259], considering the computing resources required for truly accurate calculations. In the face of this obstacle, it is still useful, often necessary, to employ so-called "semi-empirical" potential energy surfaces which are distinguished from ab initio surfaces by the fact that parameters are left for adjustment based not on theoretical grounds, but rather on a posteriori experimental results[260]. The use of a semi-empirical P E S necessarily removes some (but certainly not all or even most) of the predicative utility of the theory. Indeed, while the accuracy of a kinetic calculation depends rather directly on the accuracy of the P E S [173,179,249,258,261-263], many qualitative predictions can and have been made from the consideration of inaccurate or even completely hypothetical PES[168,173,264-268]. In this way, chemical kinetic theory and experiments take on an explicit symbiotic relationship in a "bootstrap" procedure whereby experiments serve not only to test the accuracy of the calculations, but also to adjust the parameters of the P E S , which, in turn, leads to improved calculation [249,268]. To describe the concepts and define some of the terms used later in the discussion of the results, a simple collinear reaction system A + B-C -»• A - B + C (eg. M u + H 2 -> M u H + H) is first considered here. Calculations have shown that the general topology of a P E S for such a system has the form shown in Fig. 4.1, where potential energy is plotted against bond lengths. The course of the 76 Potential energy P (A t B-C) B-C distance Final state A-B • C Minimum-energy path Initial state A • B-C A-B distance Completely dissociated state A + B * C B-C distance Figure 4.1: On the left plot is a perspective view of a potential-energy surface. Plot on the right is a potential energy surface shown as a contour diagram. The dashed line shows the minimum-energy path (MEP) , corresponding to the paths of steepest descent from the saddle-point into the two valleys. Both plots are taken from Ref. [263] reaction is represented by a movement on the P E S from P to Q, and the system tends to travel along paths where the energy is not too high. The essential feature of such surfaces is that running in from the points P and Q there are two valleys, which meet at a saddle point (also called col). For the system to pass from P to Q, it travels up the first valley (the reactant valley) and passes over the saddle point into the product valley. The paths of steepest descent from the saddle point into the two valleys constitute a path that is known as the minimum-energy path (MEP) . A plane surface perpendicular to the M E P which should not be crossed twice during the reaction is called the dividing surface (DS). Of particular interest is the one at the saddle point. The actual reaction paths, or the trajectories, followed by individual reaction systems are not exactly the same as this minimum-energy path and depend on the energy states and other features of the colliding molecules. A curvilinear coordinate which links the reactant region of the potential surface to the product region is called the reaction coordinate. A section through the minimum-energy path (or along the reaction coordinate), known as a potential-energy profile, is shown in Fig. 4.2. The maximum energy in this profile, at the saddle 77 Activated complex A - B - C A + B - C Figure 4.2: Potential energy profile. The thick line is a section along the minimum-energy path in a potential energy surface. Zero-point-energies are also indicated. point in the P E S , gives the potential barrier and is a point of particular significance. It is not only a position of maximum energy along the minimum-energy path, but also a position of minimum energy with respect to motions at right angles to the path. Systems represented by a small region round this point are known as activated complexes, and their state is a barrier to reaction referred to as the transition state (TS), denoted by the symbol J , and often written as Vg. The width of this barrier region is also of considerable importance to the calculation of reaction rates. Classically speaking, for a reaction to proceed from reactants to the product side, the system has to overcome the energy barrier imposed by the transition state, i.e., the energy difference between the reactants and the activated complexes. The magnitude of this potential barrier is related to the activation energy, Ea, the additional energy required to activate the system. The line in Fig. 4.2 represents the classical ground state energy which is lower than the energy of the lowest quantum state of the systems. The difference between the classical and quantum energies of the system, c2, is called zero point energy (ZPE) since the higher quantum energy is due to the 0-th (the lowest) vibrational mode. 1 Wi th this in mind, the activation energy should actually be the difference between the Z P E of the reactants and the Z P E of the activated 'At 0 K , it is only this state that is populated. 78 complexes. Ea = Vl+e\-ez (4.1) R a t e of R e a c t i o n a n d R e a c t i o n D y n a m i c s Reaction rates usually increase with temperature. The fractional change in the rate coefficient k(T) with respect to reciprocal temperature gives the experimental activation energy, _ -kBd[\nk(T)] E a ~ —Kf)—• (4>2) A n interpretation of this effect was developed by Arrhenius [269] after init ial work of Van't Hoff [270], which gives the famous Arrhenius equation in its simplest form: k(T) = Ae-E°'kBT. (4.3) This equation can be derived directly from Eq. 4.2 if Ea and A are both assumed temperature independent. Suppose a reaction occurs with a reaction cross section <r(v) at relative velocity v, the rate constant at that velocity is k(v) — <r(y)v. At a given temperature T , the overall average rate constant is: / • O O Jfe(D = / f(T, v) <r(v) v dv, (4.4) Jo where f(T, v) is the distribution function of the relative speed at temperature T , which is the Maxwell-Boltzmann distribution expression if the system is at thermal equilibrium. It is usually convenient to integrate over the translational energy E instead of the speed v, giving the following "collision-theory" expression: (Note, this equation has not taken into account of any internal energy, only translational energy is considered.) A simple "line-of-centers" model [101,263] is instructive here, where, for reaction with barriers, <r(E) = <r0(l - ^-) (<r(E) = 0 when E < Et) where Et is the threshold energy and crQ is an 79 energy-independent (hard sphere) cross section. When inserted in Eq . 4.5, this gives the expression jfe(D = [ !*£l ] i/2 a o e-E''k°T (4.6) 7T/i which has the classic Arrhenius form, but note that the pre-exponential factor (A) is in fact temperature dependent. Generally, a thermally averaged rate constant is given by [271]: k(E) e-BlkBT k(T) = I Jo -dE. (4.7) Q where E is total energy, e~ElkBT is the Boltzmann weighting factor, Q is the partition function which normalizes the result, Q = QTQJ, where Qj is the partition function for internal degrees of freedom and QT is the translational partition function, QT = (2ir(ikBT)? /h3. k(E) is the microcanonical rate constant per unit energy interval and generally depends on the internal energy as well as translational energy. Considering translational energy only, k(E) = ^jfjajfs^) Eq. 4.7 gives again the expression in Eq . 4.5. Substitution of Eq . 4.7 into 4.2 yields: _ E k(E) dE fo°° e-E'kBT k{E) dE (4.8) The first term of this expression is clearly an average energy and it is interpreted as the average energy of those collisions which result in reaction, (E*). The second term is just the average energy of the reactants, (E). Thus, Eq . 4.8 is simply: Ea = (E*) - (E), (4.9) an improved form of the the original Tolman's theorem [272] due to Fowler and Guggenheim [273]; i.e., the activation energy is just the difference between the average energy of those collisions that actually result in reaction and the average energy of all reactants. This can also be interpreted in terms of Eq . 4.1 above where Ea is the difference between the energies of the activated complex and the reactant state. It is noted that Ea can be strongly temperature-dependent. In particular, in the presence of tunneling, Ea -> 0 in some cases, eg. M u -f F 2 [123]. 80 Recent developments in direct ab initio dynamics methods have opened up the possibility for detailed quantitative dynamical calculations of thermal rate constants and kinetic isotope effects of gas-phase chemical reactions from first principles. However, in order to obtain accurate results, extremely time-consuming computations are required for even simple systems [274] and most of al l , it sti l l demands detailed knowledge of the P E S . It is not yet practical to perform these direct calculations for most reaction systems. In many cases, there is insufficient knowledge of the P E S . This prompted the development of statistical treatments of reaction rate theories which generally don't require any detailed knowledge of the P E S . The most generally employed and widely accepted of these are various flavors (variations) of the transition-state theory (TST) which has been developed over the years following the pioneering works of Henry Eyring [275] and M . G . Evans and M. Polanyi [276,277] in 1935 (this is now referred to as conventional T S T , C T S T ) . The main assumptions/requirements of C T S T are: • Molecular systems that have surmounted the saddle point in the direction of products cannot turn back and form reactant molecules again (no "recrossings"). • The energy distribution among the reactant molecules is in accordance with the Maxwell-Boltzmann distribution. Furthermore, it is assumed that even when the whole system is not at equilibrium, the concentration of those activated complexes that are becoming products can also be calculated using equilibrium theory. • It is permissible to separate the motion of the system over the saddle point from the other motions associated with the activated complex. • A chemical reaction can be satisfactorily treated in terms of classical motion over the barrier, quantum effects being ignored (even if Z P E effects are included). From these assumptions one arrives at the following well-known equation [263], k(T) = T ^ e-E°/k°T (4.10) h qAqBC 81 where qA and qsc are partition functions related to the two reactants A and B C , g* is a special type of partition function for the activated complex, and EQ is the difference between the molar zero-point energy of the activated complexes and that of the reactants. Recall also Eq . 4.1. Note that Eo is the hypothetical activation energy at absolute zero. The partition functions in this expression must be evaluated with respect to the zero-point levels of the respective molecules. The transmission coefficient T is introduced in a completely ad-hoc manner and is used to account for multiple crossing or alternate reaction paths, as well as contributions from quantum tunneling, outlined below. Not all activated systems wil l form products. The first assumption mentioned above is not always true and some times a system would cross the DS multiple times or even find an alternate reaction path. To account for multiple crossing, other than including it in T, the variational transition-state theory (VTST) was developed, in which the position of the DS is varied so that the calculated rate is the minimum which is closest to the truth, ignoring quantum effects [278-280]. This approach gives an upper bound of the rate constant [281]. In order to minimize the effect of alternate reaction paths, even assuming a single P E S , another modification to C T S T was the vibrationally adiabatic treatment in which the reaction system was assumed to remain in quantized vibrational states during the course of reaction (typically, v = 0 to v = 0 in the TST) [192,196,282]. Since quantum features are missing in C T S T , various attempts have been made to introduce more quantum mechanics into the T S T formulation. One of the most serious omissions is the tunneling effect [191,194,280,283-288]. The importance of tunneling and the failure of C T S T to account for this effect is perhaps best illustrated in the Mu(H) + X 2 (halogen) reactions [26,115,123,289-291]. As mentioned in Chapter 1, the M u + F 2 reaction likely provides the most definitive example of tunneling in gas phase reactions, where, at low temperatures, Ea -> 0, a limit known as "Wigner threshold tunneling" [194,200]. Only when tunneling effects are correctly accounted for (often requires quantum-mechanical T S T ) , the experimental and theoretical results agree really well over a large temperature range. The fact that tunneling invariably shows 82 up most dramatically at low temperatures can also be appreciated from Eq . 4.9 in which (E*) is much reduced from its classical value. In the present thesis, for the first time, dramatic tunneling effects are seen near room temperature, for the M u + N 2 0 and M u + C O reactions. Although rigorous quantum formulations exist [287,292-296], transition state theory is essentially a classical theory and does not a priori include quantum tunneling. The transmission coefficient T is therefore introduced as an ad-hoc factor to incorporate quantum tunneling into the formalism. There has been significant debate on the methods of calculating these transmission coefficients; and measurements of M u isotope effects have been important in evaluating the accuracy of these methods [26,103,104,123,280,284,285,288,297]. A d d i t i o n , C o m b i n a t i o n , a n d U n i m o l e c u l a r Reac t i ons A l l chemical reaction systems studied in this thesis involve combination or addition reactions, which are usually considered in the context of discussions of the reverse dissociation processes, unimolecular reactions [298]. It is widely accepted that the rate constants of these two processes are related by an equilibrium constant [298-300]. So a brief description of some concepts and results of unimolecular theory, which will be later used to interpret and understand some of the experimental data, is given here. It is not intended to be a general review of these state-of-the-art theories, but rather to lay some foundation for later discussions in the next chapter. Only relevant concepts will be presented here. Reactions of this nature proceed via a series of energy transfer steps, which in their simplest form (so called strong collision limit) can be represented by : A + B *=^=i A B * , (4.11) fc_„ A B * + M A B + M , (4.12) where ka, k-a and ks represent rate constants for addition, unimolecular dissociation and stabilization, involving the intermediate-complex A B * , typically in some excited state, and moderator " M " . It is worth noting that dissociation can be much less important for polyatomic recombination reactions, unlike in atom recombinations, because the dissociation rate constant A;_0 83 High—pressure k a 10 -1 10° 10 10' Pressure (atm) Figure 4.3: Schematic illustration of the moderator pressure dependence of the thermal rate constant for recombination reactions. Broken lines indicate high- and low-pressure l imiting values. decreases with increasing number of oscillators of the forming molecule A B [298], a point referred to again below. It can also be noted that where A is an atom (Mu), A B is invariably a free radical, an important chemical species in the present studies. The only exception, discussed below, is M u N O . A n essential characteristic of unimolecular and combination/addition processes in the gas phase is the separation of the reaction and collision timescales [300]. One manifestation of this is that the rate constant for the overall reaction has the dependence on the pressure of the moderator gas illustrated in Fig. 4.3. This dependence involves a high pressure limit, in which the rate constant becomes independent of the pressure; a low pressure limit, in which the rate constant is proportional to the pressure, and a falloff regime which involves the transition from one l imit to the other. This means that at high pressures, a combination reaction is kinetically second order with respect to reactants; at low pressures, it is of third order overall, being second order with respect to reactants and first order with respect to moderator gas. This dependence can be seen from the differential rate equations that apply to the above set of reactions, which are d[A) = t _ „ [ A B * ] - * 0 [ A ] [ B ] , (4.13) dt 84 = fca[A]rB]-(*_a + * f [M])[AB*], (4.14) M = MM][AB*]. (4.15) These coupled differential equations can be solved exactly but in most cases, where valid, it is much simpler to invoke the steady-state approximation which states that the rate of change of the concentration of an intermediate, to a good approximation, can be set to zero whenever the intermediate is formed slowly and disappears rapidly, which is invariably the case in the reaction systems studied in this thesis. In the steady-state treatment, ^ = 0, and the concentration of the intermediate is then given by [AB*] = ^Wrci (4.16) L J k_a + k,[M] v ' with the overall second-order effective rate constant ' « I A B U = ,4.17) [A][B] dt ~ - k . . + t,{M\ At low pressures, M —> 0, the reaction is termolecular with a limiting ( third-order) rate constant kejjfl = h[M] k0 = p-k„ (4.18) where at high pressures, M —>• oo, the reaction is bimolecular with the l imiting (second-order) rate constant kef j ,<*, = ka, (4.19) as stated above and seen in Fig. 4.3. A point worth noting here is that the strong collision assumption implied here means that every collision causes complete energy deactivation, which is a good approximation for large (polyatomic) molecules as moderator. However, in many cases, especially for mono- and diatomic moderator molecules, this is a poor approximation. The general practice to get around any deviation from the strong collision l imit is to scale the stabilization rate constant k, by a factor /?, called the collisional efficiency. Generally speaking, /? increase with the number of atoms in a 85 Reactant AB Energized AB* Activated complex AB* Energy of energized molecule Energy in active modes Zero—point energy Classical ground state Zero—point Energy y-Classical ground state Figure 4.4: Schematic energy diagram for R R K ( M ) mechanisms. Reproduced from Ref. [263]. molecule because the more degrees of freedom the more likely that energies wil l be transferred to the moderator molecule. R R K and R R K M Among the current treatments of unimolecular reactions, R R K M theory is sti l l the most commonly used method of calculating microscopic rate constants for polyatomic molecules, despite its origins in the 1950's [301-304]. It is based on the application of transition state theory to a microcanonical ensemble of excited reactant molecules. It has proved very successful in interpreting a wide variety of unimolecular processes [168,170,264,300,305-311]. However, the theory is often formidable-looking, from which it is sometimes difficult to draw physical insight. The R R K [312-314] theory, from which Marcus developed R R K M , despite its considerable shortcomings, offers a much simpler expression, from which, useful and qualitatively correct conclusions can be drawn. This will be used later to interpret trends in some of the experimental results. Of particular interest here is the result for the dissociation rate constant, 86 In R R K M (and R R K ) theory, the dissociation process2 is schematically described as A B * -> A B * A + B. (4.20) A distinction is made between an excited molecule, represented as A * , and an activated molecule represented by A * . The latter is defined the same way as in T S T theory, that is, an activated molecule is passing directly through the DS of the P E S . A n excited molecule, on the other hand, is one that has acquired all the energy it needs to become an activated molecule; however, it must concentrate energy in specific vibrations before it does so. Suppose an amount of energy e* is distributed among the normal modes of vibration in an excited molecule. Different than in R R K , R R K M makes distinction between active and inactive energies. The inactive energy (f* n a c t i v e) * s energy that remains in the same quantum state during the course of reaction and that therefore cannot contribute to the breaking of bonds. Z P E and overall translational and rotational energies are inactive. Vibrational and internal rotational energies are active. Because the normal modes are coupled loosely, the active energy (e a c t ; v e ) can flow between them, and after a sufficient number of vibrations a critical amount of energy C Q (sufficient to overcome the barrier) may be concentrated in a particular normal mode and reaction can occur (see Fig. 4.4). In R R K M , the general form for the unimolecular dissociation rate constant, k(e*), for all reacting molecules of total energy e* is given by (ignoring all correctional factors [278,300,315]), where C?*(e* c t i v e ) represents the total number of vibrational/rotational states within the energy interval, and p(e*) is the density of states, i.e., the number of states of reactant per unit energy at energy e*, The technical details of calculating these two terms in general are beyond the scope of this work, but if a system can be described semi-classically and only vibrational degrees of freedom considered, then one gets Eqs. 4.22 and 4.23, for a reactant containing s classical degrees of vibrational freedom [315,316]. J = .^ H'' t (4-22) 2 A similar pathway exists for the product channel 87 and (€*)—i p(n =. LL. . , (4.23) (*- lr[li=ihvi where and u,- are the vibrational frequencies in the transition state and reactant molecule respectively. Since the transition state has one less degree of freedom, = s — 1, one has t ( , ) = to(£^)-\ ( 4 , 4 ) Note that to introduce some quantization, which is particularly important at low energies, zero-point energy is required, i.e., only energy above the zero-point energy (€ z, e|) is considered. A n essentially the same but much simpler expression is given by R R K theory [263,300,315,317], * _ a = ** ( ^ ^ P (4-25) using the conventional notation where c* is the energy of the excited molecule, 6*, is the critical energy, s is, in R R K , usually taken to be one-half the total number of normal modes in the molecule, and is the rate constant corresponding to the free passage of the system through the DS which can be directly related to the high-pressure limit pre-exponential (A) factor for the reaction, which is of the order of magnitude of a vibrational frequency (see Eq. 4.24) and is predicted to be about 10 1 3 s _ 1 [263,317] for most unimolecular reactions. 3 From Eq . 4.25 it can be seen that the lifetime rc ~ l / fc_ a is short, < ps, for molecules with few degree of freedom, which are all the cases of interest in this thesis. S A C M A different approach to unimolecular and recombination reactions is the statistical adiabatic channel model (SACM) developed by Quack and Troe [195,196,309,318-321], which differs from R R K M theory in that it replaces the transition state assumption with the assumption that one has rovibrational adiabaticity with product states (as illustrated in F ig . 4.5), i.e., there is no "hoping" between different channels leading to the product states. It also requires that for the reaction to 3 Not always true. Some reactions have much higher fc* which can be explained by CTST. See Ref. [263] 88 s Figure 4.5: Schematic energy diagram for S A C M mechanisms. Four rovibrationally adiabatic po-tential curves as function of reaction coordinate s. Numbers denote different quantum states. States ("channels") number i = l and 2 are "open" at energy Etot, while i=3 and 4 are not. Taken from Ref. [300] occur the potential maximum in the adiabatic potential for such a state is below the centrifugal barrier for the particular energy and angular momentum (therefore tunneling occurs with negligible rate). The S A C M gives the same result as R R K M theory for a reaction with a chemical barrier, since the channel maxima wil l then always be located at that barrier. However, for reactions with no or very small barriers, the S A C M and R R K M are qualitatively different. For simple-fission and (re)combination reactions, such as H(D) + N O , the S A C M provides a useful and acceptable alternative theory [168,192,282,309,322,323]. BAC-MP4 One of the T S T application to calculate rate constants is the B A C - M P 4 method, the Fourth-order Moller-Plesset perturbation theory (MP4) [324,325] with the bond additivity corrections (BAC) [174,255,256,326,327]. In this method geometries of saddle points and relative minima on a reaction P E S and the vibrational frequencies at these points are calculated (ab initio) at the Hartree-Fock level [328,329]. Using these geometries the potential energies at these points are refined first by an M P 4 calculation and a zero-point vibrational correction, and then by a series of semi-empirical corrections, the B A C based on the types of the bonds present. The resulting P E S 89 properties are generally quite accurate. The dynamics on these P E S are calculated statistically, which requires two fundamental assumptions, T S T and Strong-coupling of R R K M . Also, the reaction coordinate is assumed separable from the rest of the Hamiltonian which is further separated into Harmonic-oscillator and rigid-rotor parts. The tunneling effect can be calculated using Eckart [330] barriers. This method has been used successfully for many chemical reaction systems, including H(D) + N 2 0 [174,255,256,267,311,326,327,331-333]. Isotope Effects From Eq. 4.10, the kinetic isotope effect, the rate constant ratio of two isotopes 1 and 2 (for the sake of simplicity, let isotope 1 be the lighter of the two) can be expressed as Eq . 4.26 where AEV is the difference in the vibrational adiabatic barriers of the isotopes, which in turn are the differences in energy between the ground vibrational states of the transition state and that of the reactants [26]. t l = p . S l l e - ^ / k B T ( 4 2 ( «2 T 2 qi q\ By expanding the relevant partition functions and expressing AEva in terms of the vibrational frequencies of the transition state, vjs, then simplifying with the use of the Teller-Redlich product theorem [334], Eq.4.26 can be written as [26]: where fi* refers to the reduced mass of the reaction complexes at the geometry corresponding to the transition state and = fttij/fcflT. This equation is composed of three separate ratios, which can be considered independently. The first is the strongly temperature dependent ratio of the transmission coefficients, r i / r 2 , which includes contributions from barrier recrossing, alternate path ways and particularly quantum tunneling. At low temperatures, as noted, the tunneling effect is very important, especially for isotopes with large mass differences, but recrossing is not likely since there is little excess energy in most collisions that lead to an activated complex; therefore the overall ratio is greater typically *L = El k2 r 2 (4.27) 90 than one. A t high temperatures, or for reactions with low energy barriers, the tunneling effect is small but recrossing is sometimes important so that this term could be less than 1. The second ratio, (p\/'//f)1^2', is the square root of the reduced mass of the two transition states. It is important to emphasize that this term depends on the geometry of the transition state and therefore on the position of the reaction barrier. This term, therefore, contains information about the reaction dynamics and is often called the primary isotope effect [335]. For an endothermic reaction with a correspondingly late barrier, the reduced mass of the transition states resembles more closely that of the products and the ratio ( U ^ / M * ) 1 / 2 approaches 1. For an exothermic reaction, on the other hand, with an early barrier to reaction, the reduced mass of the transition state resembles that of the reactants. In this l imit, the ratio approaches that of the mean velocities, which can also be seen from the "collision theory" expression in Eq. 4.6. The third ratio, P[[p1,sinh(^2«/2)][/i2»sinh(ui,72)], appears complicated at first, but can be simply considered as containing all the information about the internal modes of the transition state other than the reaction coordinate (e.g., for a bimolecular exchange reaction, the symmetric stretch and bend modes). For this reason, it is often called a secondary isotope effect [335]. For an early barrier, where these internal modes have very low frequencies isotopic substitution has relatively little effect but, in contrast, for a late barrier where internal modes are strongly coupled, giving rise to high vibrational frequencies, these are strongly affected by isotopic substitution. The secondary isotope effect in this case approaches zero and thus can have a dramatic effect on the kinetics. In summary, for an endothermic reaction with a late barrier: the ratio of the transmission coefficients should approach 1 since tunneling is not expected to be important (barrier recrossing could reduce this ratio at higher temperatures, but at room temperature, recrossing is not significant [263]); the primary kinetic isotope effect approaches 1; and the secondary isotope effect approaches 0. The net effect can therefore be a pronounced negative isotope effect: Ari/Ar2 <S 1, as observed for Mu(H) + H 2 [31,122] and Mu(H) + C H 4 [125,126]. On the other hand, for an exothermic reaction with a correspondingly early barrier: the secondary isotope effect approaches 1; the primary isotope effect approaches the ratio of the mean velocities for the two isotopes; arid 91 the ratio of the transmission coefficients, T i / ^ , containing all the interesting information on the reaction dynamics, will be greater than or equal to 1. The net effect, which can be strongly temperature dependent due to the ratio of the transmission coefficients, is therefore a positive isotope effect: k\/k2 > ( /^/ / t i ) 1 / 2 - As already noted, where tunneling dominates in comparing M u and H-atom reactivity, & M U / & H ^> 1 is seen at low temperatures [124] and even at room temperatures in the present study of the M u +N2O reaction. For composite reactions, i.e., reactions consisting of multiple reaction steps, one must consider each reaction step/mechanism involved and no generalized prediction for the overall K I E can be given. 4.1.2 //SR and Reaction Mechanisms In /zSR, what one observes is the /x + polarization— the relaxation rate only tells us the rate of disappearance of the polarization of the fi+ in certain environment(s) depending on the situation. In T F , o-Mu and diamagnetic / i + can be observed separately since they have different precession frequencies. In L F , one can no longer distinguish the polarization of the muon in these different environments. The practical consequence is that in T F one can focus on one specific component, PMU (or P D , or P R ) , while in L F the total polarization P = PMU + PD + PR must always be taken into account. Therefore one must be careful when extracting rate constants from the relaxation rates measured in / iSR. It is necessary and crucial to identify all the possible muon states and their related rates (A,-, A t , and A/ ) discussed in the last chapter and to find out the relationship between these rates and all the rate constants involved which are determined from the concentration/pressure dependence of both the reagent gas and moderator, if present. Later in this thesis, A c wil l be used to represent the overall chemical reaction rate or the rate at which a chemical product is formed, which is not necessarily equal to either the rate, A0bs, at which / i + polarization is lost, or, A t , at which o-Mu is lost (transformed to another muon containing specie), as will be seen below. Consider then an initial 100% polarized o-Mu ensemble, init ial amplitude A0 = 1. After 92 some time t, the muons could be in one of the following three states of polarization, 1. Polarized muons in M u , amplitude A\. 2. Polarized muons in other magnetic environments, A2. 3. Unpolarized muons in any magnetic environment, A3. There are two alternate pathways to the unpolarized state, namely, direct depolarization of the Mu , or changing to an new environment 4 (without depolarization during the transition) first and then depolarization in the final environment, which is shown schematically below: Polarized At Polarized Unpolarized M u Other A l l Ai A2 A3 It is easy to identify the first two states with the initial and final states mentioned in Section 3.4 of Chapter 3, and the three rates are as defined therein. There is a very important difference between the signals observed in a T F and a L F . As noted, in a T F , one can follow polarizations of a particular frequency and focus on the polarization due to that particular component. Relaxation rates of ensembles with different frequencies are easily identified. For example, assume the observed signal is due to o-Mu in a weak T F , 5 then •fobs = PMU — A0bae'WMt and .<40bs = A%. In a L F , on the other hand, it is the sum of all polarized muons that is important, F 0 b s = PMU + PD + PR = -^ obs and AQbs = A\ + A2, and the polarizations (or relaxation rates) in different environments are not easily distinguished. In both cases, it is the decay of the amplitude that is causing the relaxation of muon spins 4 An environment is defined different if the muon spin in the new environment will precess with different frequency (direction) in a T F or be relaxed by some mechanism so that in a TF the transition leads to direct loss of polarization. 5 It is easier to consider just a weak TF here, involving only o-Mu. However, the argument would be similar in a TF of arbitrary strength, involving both o-Mu and p-Mu initial states. 93 in the initial states, necessitating a determination of the time dependence of these amplitudes. It is easy to solve the following equations, f ^ l = _ ( A t + A , - M i , (4.28) ^ = X t A i - X f A i , (4.29) giving the solutions, (since absolute amplitude is not relevant here, init ial o-Mu amplitude Ao is set to 1, and YJ.A,- = 1): Ai = e-<*« + A ' ) ' , (4.30) A. (e-*t*-e-(*i+>i)*)t (4.31) i — A/ \ / Therefore, A, + A,in T F : P o b s = Ax eiUMt = e-(A«+A«>' eiu,M\ Aobs = A t + A,-; (4.33) in L F : P0bs = Ai + A2 - A ' ~ A / e - ( A , + A i ) t + A t e ~ A / t . At + A,- — Xj Xt + A,- — A/ (4-34) These are identical to the results obtained earlier with the "residual polarization model" (Eq. 3.28). The next important step is to relate the three rates (Xt, A,-, Xj) to rate constants and gas concentrations pertaining to a particular reaction system. Of the three rates, Ay is non zero only if the product is a radical (in which case it equals the rotational relaxation rate of the radical plus the spin exchange rate between the radical and dopants with unpaired electron, if present), A< is often due to a spin exchange interaction (though not always, as wil l be discussed later), and A t is usually found to be not as straightforward as the other two. It requires a detailed solution of the reaction mechanisms. Some special cases of interest are discussed here: 94 Case A First consider a simple reaction, M u + X ' r A2 M u X •4 Unpolarized It is obvious that A ( = kc [X]. A n example of this is the M u + C2H4 reaction in the high pressure l imit. The final product in this reaction is the ethyl radical, MUCH2CH2, and the muon spin in the radical is totally depolarized by collisional relaxation interactions at low fields. In weak T F , A0bs = At = kc[X] (see discussion in Case B2); in L F , when kc[X] 3> A r , A O D S = A r . Case B Next consider a more complicated situation which is often found in a combination/addition reaction, here * denotes a vibrationally excited state, ! ! I Mu + X + M F=^=* M u X * + M M u X + M i i I t 1 A3 Unpolarized Now, for small molecules (e.g., MuNO* ) , the excited molecule (MuX*) is considered stil l in the Mu ensemble because the rapid dissociation makes the lifetime of this excited molecule so short that there is no change to either the muon spin or the muonium electron spin while M u is trapped in it. Upon dissociation back to a free M u , the M u appears to be no different than those that did not go through the radical formation. As in most cases, the important question is what is Xt. Apparently, A f = k5[M] [MuX*]. From discussions given earlier (see Eqs. 4.16 and 4.17) and 95 Appendix A.2, that [MuX*] = ka[X] *_« + * . [ M ] ' ka[X]k,[M] (4.35) (4.36) k.a + k,[M]' Again, the underlining assumption here is that since the lifetime of the excited state is very short (< ps), there is no muon spin depolarization nor electron spin flip in the excited state. Let's make further assumptions in order to look at some special cases which are related to reaction systems studied in this thesis. Case B l : A/ = 0 and A,- = Xj — / A S F • In both T F and L F , A 0 b s = At + / A S F - This is the case for the M u + N O reaction. Case B2: A, = 0, and A t » A/ = A r . The final radical product is depolarized by collisional relaxation and no spin exchange reaction is present. Again, Mu + C 2 H 4 is an example of this case. Note that at high pressures, k-a k„[M]. This gives the same result as in Case A : in L F , P(i) = e~Xrt so A 0 bs = A r ; in wTF , A 0 b s = At = &aP0- However, one has to take note that the upper l imit of observable relaxation rates in fiSR is about 10 / i s - 1 and if Xt is too high (likely at high [C2H4]) then the M u signal cannot be observed (it is relaxed instantly). Case C M j Mu + X + M MuX* + M I, )! MuY + Z + M k-a I I k.[M]\ M u X + M Comparing to Case B, one additional chemical reaction channel is considered here but there is no spin exchange. This is the situation found in the Mu + N 2 0 reaction. It is not difficult to 96 solve the coupled equations under steady-state assumption and show that (see Appendix A.2) A, = MM] + *,) [M , , * - ] = £ 2 ^ M L W . (4.37) In T F , A o b s = A { = ka[X]{kt + k.[M])/{k-a + *t + *«[Af]). Now in L F the result are not the same as in T F and the situation is quite complex. Generally, there are three relaxing components in the polarization function with rates At, A r i , and A r 2 . Furthermore, the relative amplitudes of all three terms are functions of these rate constants. A l l of these three rates are strongly dependent upon pressures of the reagent and moderator gases. Both of the latter two are also strongly dependent on the magnetic field. In practice, it is difficult to extract useful information from L F data in this case. This arises because of the following reasons. First, it is difficult to eliminate any one of the three rates since the order of magnitudes wil l change over the field range of experimental conditions. Secondly, related to the first, for some experimental conditions, two of these rates will be comparable in magnitude, which, coupled with the fact that the amplitudes are changing with pressure and field, makes the data analysis almost impossible. Unless one of the rates is much larger than the other two over a wide range of fields, one has yet to find a way to extract kinetic parameters from A0bs measured in L F in this case. For the last two examples, it is assumed that the muon spin in the (excited or stable) radical state behaves significantly different from that in a free M u state (at least in a T F ) . What if, one may ask, the muon spin of an (excited) radical behaves the same as in a free muonium? In T F , the loss of muon spin polarization in the radical states (excited or stable) now come directly into the Mu relaxation rate while it only influence the polarization indirectly through competition in the chemical process. Particularly, the loss of polarization of both muon spin and muonium electron spin in the excited radical state can no longer be ignored as in the previous examples where it has little effect on the Mu polarization. The most likely process to cause relaxation of electron spins in the radical is the spin rotational relaxation of the excited radical induced by collision. Of all possible spin flipping (flip-flop) terms, as discussed in Sections 4.3 and 5.2, at low field electron spin flip is much more likely. If the lifetime of the excited radical is so much shorter than the 97 hyperfine mixing time, which, from Eq . 4.25 can be expected for small radicals like M u C O * , one can assume that the muon spin would not be directly affected while in the excited state. However, if the excited radical dissociates with a flipped electron spin due to collisional relaxation (but muon spin intact) back to a free M u , it is as if the M u had undergone a spin exchange interaction, and the consequent hyperfine interaction will cause muon spin relaxation. It makes no difference, as far as the muon is concerned, how the electron spin got flipped. The equivalent "spin flip" rate is the rate at which p-Mu is formed in this process. This situation is shown in the following scheme. Case D In the following diagram, k$F is the apparent electron spin flip rate constant of the excited radical (its explicit expression is discussed later in this chapter). The " *" indicates that this is not due to a true (intermolecular) electron spin exchange interaction. The depolarization rate due to this apparent "spin flip" relaxation is and is related to the "spin flip" rate (k$F) the same way as in a real spin exchange interaction (see the next section). It is assumed here that all three M u states, Mu , M u X * and M u X , are indistinguishable and in both T F and L F the observed polarization is the sum of all three polarizations. The same relaxation function R(t) wil l be seen in both T F and L F . For simplicity, fc_0 is assumed to be much larger than any other rate constants. A3 | —T Unpolarized I i r i In this case, there is no intermediate A2 state and a different approach is more straightforward since A\ = [Mu] + [MuX*] + [MuX]. Again using steady state assumption and assuming [Mu]=l initially, one has (see Appendix 98 M | M u + X + M M u X * + M - ! ^ % M u X + M fe-o A.2) [Mu] = e-*'"1 (4.38) ^ = ( 4 ' 4 0 ) where A e / / - kefJ[X) - k_a + k m + fky[X\, (4-41) is the effective reaction rate, kejj, which has the usual meaning as defined by Eq. 4.17. It is likely that Ar*jF and A r are due to the same kind of mechanisms and are related. kgF is also likely to depend on [M]. At this stage, the question is how it wil l affect the overall polarization and the detailed discussion on its mechanism is deferred for the time being (see discussion on M u + C O result). Suffice it to say here that fc*.F and A r have the same order of magnitude (the former is always smaller though) and both increase with pressure at low pressures and decrease with pressure at high pressures (see Eqs. 4.85 and 4.86). As noted earlier, in this case, P„b, = R(t)e'UM in T F and P0bs = R(t) in L F . The observed relaxation function R(t) is just the time dependent amplitude Ai, given approximately by R(t) =Ai = - A r e-x'"1 + ^—e-^, (4.42) K — 1 1 — K where K~ ka[x}k,[My ( 4 > 4 3 ) Case D l : K -» 0 R{t) = e - A r t To satisfy this condition, it is necessary that either A r is very small or [X][M] —» oo. In any case, this requires high pressure dopant gas and/or high pressure moderator (or very high field). 99 Case D2: « —> oo R(t) = e - A " ' f Recognizing that £=g- = 1/fcn, where ICQ is the low pressure l imit rate constant for the addition reaction, which is usually small, on the order of 1 0 - 3 2 to 1 0 - 3 5 c m 6 m o l e c u l e - 2 s - 1 ) at room temperature, and that fcn[X][M] is simply the rate of addition reaction, as long as the rate of rotational relaxation A r is much larger than than the same addition rate, this condition is satisfied. This is true when either [M] ->• 0, or [X] -> 0, or A r is large enough. More precisely, when the addition rate is about 1 ~ 10 / x S - 1 , this requires that A r is on the order of 1 ~ 100/ is - 1 or faster, which is the usual rate of S R relaxation at low fields and low pressures. Case D3: K -> 1 R(t) = e~Kt = e~x'it Note that the polarization does not diverge since when « = 1, A r = A e / / . In L F experiments, the situation is extremely complicated since K wil l change over the range of fields used. At low pressures, the collisional relaxation rate is large at low field and decreases with field so the magnitudes of all these rates may be comparable at some (field) point and they are difficult to properly extract from the observed relaxation rates. However, at high enough pressures, the collisional relaxation rate is inversely proportional to the total pressure, and one can obtain the condition that K —t 0 so that A 0(„ = Ar, as in Case D l . It is only possible to observe the collisional relaxation rate alone at very high total and reagent pressures. More discussions of this case wil l be given with the M u + C O results. 100 4.2 Spin Exchange 4.2.1 Potential Energy, Phase Shift and Cross Section Spin exchange between paramagnetic species6 is one of the few elementary bimolecular processes that is readily accessible to detailed experimental studies and to rigorous theoretical description. It can be regarded as a change of spin states of particles with unpaired electron spins, in collisions, induced by an exchange interaction of partners that occurs when their electron orbitals overlap. A simple electron spin exchange collision between two atoms A and B, can be represented by the equation A(t) + B(4)->A(;) + B(t). (4.44) Before the collision, the spin of atom A is up, while the spin of atom B is down. After the collision the spin of atom A is down, and that of atom B is up; the spin orientations of the two atoms have been exchanged. The origin of the spin-exchange interaction is to be found in the difference between the lowest para- and ortho-spin potential energy curves of the molecular system A B . A n exchange interaction arises as a result of the dependence of the Coulombs potential of the electrons on their spin orientations. According to the Pauli-exclusion principle the spatial distribution of electrons are different depending on whether their spins are parallel or antiparallel in their atom-atom encounter. Two electrons with antiparallel spins (para) can be simultaneously found at the same point in space, while two electrons with parallel spins (ortho) can not. As a result the Coulombs repulsion is stronger for the first case. Note that the spin dependence of the interaction energy of paramagnetic species appears only when their electron orbitals overlap and that the spin state of electrons in a system influences both the interaction between them and their interaction with the nuclei. The difference between the singlet and triplet potentials causes the total scattering cross section of a colliding pair to depend strongly on their spin orientation. At large separations, the difference in potential energies approaches zero and hence so does the likelihood 6It is not the paramagnetism, per se, but the unpaired electron spins that is important. This is relevant in the,case of NO, which is actually diamagnetic at low temperatures, even though it has an unpaired electron. This is because the ground state is a 7Tlif2 and the orbital and spin magnetic moments cancel [146]. The first excited state, 2Yl3/2, is nearby so that NO is partially paramagnetic at room temperature. 101 for spin exchange. Spin exchange collisions of H isotopes with other atoms, particularly H-like atoms, are of intrinsic interest and are also of interest in astrophysics, in optical pumping and in other fields, partly due to the fact that H(isotope) + H-like systems are the only ones for which accurate potential energies can be calculated [78,249,336]. One of the spin exchange collisions studied in this thesis (Mu + Cs) is of this type. Since all of the spin exchange interactions studied in this thesis are between two spin | species collisions between an H isotope and another spin | species are considered below. The arguments present apply equally well to species with spins greater than | (eg., O2, S = 1), with some modification. The calculations of spin-exchange cross sections are carried out in two steps. First, the (singlet and triplet) interaction potentials have to be determined. Second, the integration of the potentials have to be performed to determine the phase shifts for elastic scattering on each potential, and hence the difference in phase shifts, followed by another integral or sum over some parameter to give the final result. Obviously, the accuracy of this calculation depends heavily on the accuracy of the interaction potentials. A typical set of potential curves is shown in F ig. 4.6. As outlined above, the triplet potential is repulsive, while the singlet potential is attractive over most of its range. These potential curves can be represented by an interaction of the form [76,78] V ( r ) = Vo{r) + S A • SBVi(r), (4.45) where r is the distance between the interacting paramagnetic species, Vn(r) is a spin independent term, and V i ( r ) is the so-called exchange integral. SA and S# are the spin operators of species A and B, respectively. Analogous but more complicated interaction potentials are required to describe the interaction between atoms with spins greater than | . Since the forces responsible for the spin-exchange interaction are of an electrostatic nature, potentials are on the order of electron-volts, and the cross sections for spin exchange are large (<r = 1 0 - 1 4 c r a 2 ) . A n important 102 Figure 4.6: The interaction energy of two hydrogen atoms as a function of the internuclear separation r. The data are taken from Refs. [251,337]. property of spin-exchange collisions is that even though the electron spin of an individual atom may flip during a collision, the total electron spin (S = S A + S B ) of a colliding pair of particles is The theory of spin exchange has been broadly developed in two alternate ways[76]. In the original work on spin-exchange scattering by Purcell and Field [45] and by Wittke and Dicke [46] the colliding atoms are assumed to follow classical paths. The spins of the two atoms rotate rapidly around the resultant spin, SA + SB, which is conserved during the course of the collision. The total angle <f> through which the spins rotate is given by the phase difference accumulated during the collision (also see Ref. [76,338]) The tota l 7 cross section a for spin flip8 can be obtained by averaging the exchange probability 7 Since the differential cross section are not measured in this thesis, all reference to cross sections will mean TOTAL (angle integrated, partial wave summed). 8 One could distinguish a spin-exchange interaction and a spin-flip cross section. A spin-exchange interaction could lead to the exchange of either like or unlike spins, only the latter giving rise to a measurable spin-flip cross section. conserved. (4.46) 103 |(1 — cos^) (see, e.g., Refs. [336,338]) over impact parameter R /•OO a- = n (1 - cos tf>)RdR. (4.47) Jo This method has also been used more recently by Swenson et. al. to calculate the spin flip cross sections of H - H and H - A atom collisions, at high energies [338]. In another approach, a partial-wave analysis of spin-exchange scattering can be carried out, as exemplified by the early calculations of Dalgarno [339], in which the /-wave phase shifts 6, and St for scattering from the singlet and triplet potentials, V, and Vt, were determined /oo V.dt, •oo /oo Vtdt. (4.48) •oo The spin-exchange cross section for a specific (hyperfine) state F' is then given by a{F'F>) = i{WT^ X > + X) S i n 2 ( < j » - 4), (4-49) where k is the propagation constant of the incident wave, F and F' denote the initial and the final values of the total spin of the hydrogen-isotope (F= I+S) and I is its nuclear spin. It is convenient and useful to define an intrinsic spin flip cross section for I = | (summing over F'), oo ^ F = ^ £ ( 2 / + l)sin2(<5i-,5{), (4- 5°) 1=0 which can be used to compare different theoretical and experimental results. Similar partial-wave analysis of spin-exchange scattering have been carried out by Ball ing, Hanson and Pipkin [340] and by Glassgold [341], who also studied the complications that arise when spin exchange occurs between identical atoms. Later, Cole and Olson [342] adopted this approach to calculate the cross sections for H-A systems. They also used the full quantal S-matrix technique of Johnson [343] in their calculations in addition to the straight-line trajectory integration and W K B approximation technique which were used in the previous studies. The treatments of spin exchange reactions have also been reviewed by Happer [76]. 104 Although the quantal S-matrix technique is required to reveal the quantum features of the cross sections and to give'accurate results, the straight-line technique correctly describes the general behavior and offers a more tractable physical picture of the process, which can be expected to be most accurate at higher (say > 0.1 eV, see Fig. 4.8) energies. To obtain the general behavior of the cross section in this method, one assumes the cross section is primarily determined by large-impact-parameter collisions. If one further assumes straight-line trajectories, regardless of the nature of the potentials used, it has been shown [342,344] that <rSF(v)1/2 = a - blnv, (4.51) where a and b are constants depending on the interaction potentials and v is the relative velocity of the colliding partners. Anticipating the use of Eq. 4.53, it is convenient to express total cross sections as a function of energy. The relation E = \nv2 can be used to convert the above equation to <TSF{E)1/2 = A- B\nE (4.52) where n is the reduced mass, A = a + \b\n ^ and B = | 6 . As discussed in the previous section, the cross sections measured in any "bulk" experiments are thermal averages, which, similar to the case of chemical kinetics, is related to the energy dependent cross section, <TSF{E), by 1 r°° ^ S F ( T ) = jo <rSF(E)Ee-E'k°TdE (4.53) where T is the temperature, ks is Boltzmann's constant. 4.2.2 Isotope Effects Though not explicitly stated, a view that often seems implicit in the literature [65] is that little or no isotope effect would be expected in the case of thermal electron spin exchange cross sections. However, this is not true as the following comparison between H and M u wil l show, where the isotopic mass range of a factor of 9 provides the required sensitivity, not offered by, for example, H and D atom spin exchange [61,81,345,346]. 105 There are a few factors need to be considered here. First, although Eq. 4.50 implies that the sum should be taken over all partial waves (/'s), up to infinity, the fact that the difference in phase shift disappears at large separations (see F ig . 4.6) indicates that there wil l be a maximum number of partial waves ( / m a x ) that wil l contribute to the cross section. Suppose Ro is the distance beyond which Sls — S[ = 0, then the largest contributing angular momentum is \AmaxCmax + = hkR0, (4.54) then for energy E = , W ( / m « + l ) = ^S^. (4.55) n The lighter isotopes wil l have less number of partial waves contributing at given velocity or energy. At high enough velocities, / m a x is roughly proportional to the reduced mass of the system. At the limit of large enough / m a x , one often-used assumption is the random phase approximation (RPA) [46,56] which can be stated, in the case where the difference in phase shift (Sa — 6*) is large and random, and the number of contributing partial waves is large, that sin2(<S" — S() in Eq. 4.50 is assumed to average to a value of \ up to some sharp "cutoff" maximum partial wave, beyond which they are zero. Thus, in the R P A , Eq . 4.50 can be written as l m a x <r S F(RPA) « ^ S ( 2 / m « + 1) = «I2 ('max + V? (4-56) (=0 This can be expected to be a better approximation for heavier isotopes and/or for large molecules. Thus, at high enough velocities, in the l imit of / m a x > 1, Eq. 4.54 gives / ^ a x « k2Rl and hence the R P A predicts that CTSF(RPA) — nRl/2, which is indeed both velocity (or energy) and mass independent (some weak energy-dependence in Ro could be expected, depending on the nature of the cutoff). It is noted that the thermal cross section (Eq. 4.53) would then be temperature independent. However, at the temperatures of interest in these experiments, the average energies are < 0.05 eV, and relatively few partial waves contribute (see Fig. 4.7). A small Z m a x means that s in 2 (J* - <$') is likely not random, particularly for Mu systems. In terms of an interaction time, if 6' - 6* can be assumed to be oc l/v, then for (6' - <£') 1, sin2(<$* - 8*) oc 1/v2, which is directly 106 proportional to the reduced mass ft at a given energy. Both / m a x and k are energy dependent but in such a way as to largely mutually compensate and the resulting cross section is also roughly proportional to 1/v2. So, in the l imit of small / m a x , f ° r a given energy, <TSF is proportional to the reduced mass of the system, or one may say (when / m a x is not too small), that the total cross section is roughly proportional to / m a x . Since less partial waves are sampled at each energy, the total cross section is expected to be less for lighter isotopes. This, and the fact that R P A is a poor approximation for Mu can be seen from Fig . 4.7, which plots the partial cross sections o~i from the calculations of Ref. [347,348] for Mu (triangles, top scale) and H (circles, bottom scale) versus I for the Mu(H )+02 spin exchange interactions. Though O2 has 5 = 1 , the same general features shown in F ig. 4.7 can be expected for any comparison of H(Mu) spin exchange interactions, in particular H(Mu) + Cs considered below. Figure 4.7: H(Mu) + O2 partial-wave spin-flip cross sections. The triangles are for M u + O2 and the circles are for H 4- O2. Taken from Ref. [56]. Secondly, in addition to the velocity effect on the total cross section, one must take into account the velocity distribution effect in the averaging process to calculate the thermal cross section which is what most experiments measure (Eq.4.53). Thus, e.g., according to the early calculations of Dalgarno and Rudge [344] the intrinsic cross section ((Tex(v)) depends only on the 107 ionization potentials of the colliding partners, so for hydrogen isotopes with similar IP's, there is very little isotope effect on crex(v). Since the P E S of two isotopes are the same (as a result of the Born-Oppenheimer approximation), the total cross sections are equal at a given velocity. But the thermal average wil l depend on the velocity distribution, which is mass dependent at certain temperature. This factor also favors the heavy isotopes since the sampled velocities are lower and the cross sections are higher at lower velocities. This effect wil l be much enhanced if one consider the fact that there are abundant resonances at lower velocities (see Fig. 4.8). Furthermore, as mentioned, there exist specific resonances which are not accounted for by straight-line trajectory methods. These resonances are evident in Cole and Olson's full quantal calculation (see Fig. 4.8). However, the origins and mass dependence of these resonances are not clear. One argument is that at certain velocities, a quasi-bound state will form so that the electrons spend longer time within the interactive range and the probability of exchange (or the difference in phase shifts) is greatly enhanced. Note that the formation of this kind quasi-bound state only depends on and is very sensitive to the singlet potential. Addit ional details on this subject will be discussed later with reference to specific systems. 4.2.3 yuSR Relaxation Rate The next question is how to relate the theoretical-thermal SF cross section to the relaxation rate measured in a /zSR experiment. As emphasized throughout in this thesis, in a fiSK experiment one measures the depolarization rate of the muon spin. The rate of the electron spin exchange interaction is measured indirectly through the hyperfine interaction. It is therefore crucial to establish the connection between the observed muon spin depolarization rate and the electron spin flip cross section. The spin polarization of the muon in muonium undergoing repeated spin exchange collisions has been investigated extensively using various methods such as Wangsness-Bloch equations [107,236], master equation [349], Boltzmann equations [90], a n d a time-ordered stochastic method [91,94,350], where the two most recent studies give the same results [90,91,94,350,351]. Turner and co-workers [90] used a Boltzmann-equation approach to 108 Figure 4.8: H + Cs spin exchange cross sections. Q is the state-to-state total cross section as defined in Eq. 4.49. The solid line is the result of the full quantal calculation and the dashed line is calculated with the straight-line trajectory method. Taken from Ref. [342]. derive the mathematical form of the observables for spin relaxation of hydrogen-atom isotopes by electron spin exchange with paramagnetic dopant gases. The results are the same as those of Senba et al [56] modelled from the density matrix calculations of Bal l ing, Hanson and Pipkin [75]. Turner and Snider also related different observed relaxation rates for different experimental techniques to the same intrinsic collision rate which is proportional to the same spin^flip cross section. This made it possible to compare experimental results obtained by different techniques. Senba [91,94,350] used a time-ordered approach to investigate the dynamics of the muon spin in muonium during repeated electron spin exchange with spin-^ paramagnetic species which has now been expanded to include dopants of arbitrary spin [351]. The latter approach also gave a more physical picture of time evolution of the muon spin and led to development of new /zSR techniques as well as improving the overall level of understanding of the spin dynamics of the muon in muonium, some aspects of which are discussed in this dissertation. Despite the fact that in Senba's approach the 109 relation between the spin-flip rate ( A S F ) and the intrinsic spin-flip cross section (<TSF) is not clearly defined, the present section follows mainly his approach since it is more physically transparent. One term in the total Hamiltonian of the systems under consideration is the interaction potential energy V , which depends on both the translational and spin degrees of freedom. For the latter, it is noted that because of the respective sizes of the gyromagnetic ratios, the magnitude of the interaction of the dopant's electron spin with the hydrogen electron spin is much stronger than with the hydrogen nuclear spins [62,76,148,149,352,353]. Thus it is practical to assume that the interaction potential is spin dependent only via the M u electron spin operator S M and the dopant electron spin operator S D , and that this variation occurs only through the combination S M • S o . Orbital angular momentum coupling is also ignored here, again on the basis of the relative sizes of the gyromagnetic ratio [75,76,352,353]. Thus the interaction potential given by Eq. 4.45, re-labeling the spin operators by S M and S#, is V = V Q + V ^ S M - S D , (4.57) with the distance variable r omitted here for simplicity. It is now necessary to define the spin-flip rate, A S F , which was mentioned in the previous chapters. There are two alternate (but equivalent in the sense that they both lead to the same expressions of final result) definitions: • Definition One: ^^Wrw"^- (4'58) where SD is the total electron spin of the dopant, n is the number density of the unpolarized dopant gas, v is the mean relative velocity. Definition Two: A S F = 3 ( 2 5 0 4 - l ) 2 A s m 2 <4-59> where A is some average phase shift for a SE collision, and A is the collision encounter rate. 110 In the case of a sp in - | dopant, the above equations simply give A S F = nv(TsF(T) and A S F = A s in 2 y . The So dependence is necessary though for the same results to be valid in cases where So is greater than 1/2. These are related to SF factors / introduced below. The following results can be derived rigorously by tracing over the density matrix of unpolarized dopant spin So as well as translational states [90,91] (also see Appendix A . l ) , but as mentioned, only a brief outline of the time-ordered approach is given below. One can follow the the spin direction of a muon after it enters the target and forms a muonium atom at time to(— 0). The spin of the muon wil l evolve according to time evolution function G(t) given in the last chapter if there is no spin exchange collision. What happens when there is some number of SF interactions at different times between to and t? By averaging over all possible numbers of collisions and times, taking appropriate weights, and assuming the collision process is Poissonian, Senba has shown [91] that the statistically averaged muon polarization at time t is expressed by P(t) = J2E~XSFTXSF d t l dt2... dtn x Pn(t) (4.60) n=0 " ' 0 "'° * ' 0 where Pnt is the muon spin polarization in a muonium atom initially formed at time to, after n spin-flip collisions at fixed times ti,t2, t3, ..., t„ (in chronological order): Pn(t) = G{t - tn)G(tn - tn-i) • • -G{t2 - U) G{h -10). (4.61) This also has the form of the "residual polarization" discussed earlier. The time evolution functions G(t) are given in the last chapter for various conditions. Next, some of these conditions wil l be examined and the expression P(t) wil l be explicitly given. It is useful to note that in many cases because Pn(t) does not depend on ti, t2, t3, ...,tn, Eq . 4.60 can be reduced to Pit) = ^ e - A S P « M : p n W ] ( 4 6 2 ) n=0 This is the result for a Poisson distribution. It is also useful to recall at this point the equation Eoo 1 n x n=0 Ti*. e • 111 4.2.4 Slow Spin Exchange In the case of slow spin exchange, meaning that the spin flip rate is much slower than the muonium hyperfine frequency, or the time between SF collisions are much larger than the hyperfine mixing time ( 1 / A S F S» l /^o = 0.035 ps), which is the case as for all relevant studies in this thesis research, the muon spin time evolution depends on the field and has the following forms: Transverse F i e l d • W e a k F i e l d In weak transverse fields (B < 10 G) so that 6 in Eq. 3.9 is small, G(t) takes the form of Eq. 3.11, and Pn(t) = (!)"+! eiUMt. (4.63) P(t) is then given by P(t) = ^ e - A s P t ( ^ i r ( l ) » + l e . a ) M t n=0 1 2 Note that \ D = ^ \ S F , f = \-• I n te rmed ia te F i e l d At a field B > 30G, but still much smaller than Bo so that 6 is still small, one can observe two frequencies with typically about 1 ns experimental resolution, and then the time evolution of the muon spin can be approximately expressed by Eq. 3.10. Thus, Pn(t) = ( ± ) " + 1 eH"M+n)t + ( i ) „ + i ei(«M-n)« +C(t1,t3,...,tn,t,n) (4.65) The cross term C(<i,<2,.. .,tn,t,il) can be shown to vanish after statistical averaging if 40 > A S F (true for fields >30G) [95,98] (also see Appendix A . l ) . The polarization at t is then P(t) = ^ e ~ A s F t ^ l l ) ! l ( I ) n + i |g«(«;M+n.)« + e <(w M -n ) t j n=0 112 - i „ - 3 W / 4 — 4 e ei(wM+n)t _j_ e i (w M -n) t j (4.66) and A d = f A S P , / = f. The same result can be obtained using the Boltzmann-equation approach given in Ref.[90] (see Appendix A . l ) . • H i g h F i e l d Although no high field data were taken for this thesis, the result is given here for completeness. In this case, c = 1, s = 0, and S = 1, so that from Eq. 3.12, G(t) = | eiw"1 + i e-iU3tt, (4.67) and Pn{t) = {\)n+1 e^'* + ( ^ ) n + 1 c ' w « ' +C(tl,t2, • ••,*„, *,«„). (4.68) Similar to in an intermediate field, the cross term wil l be averaged to zero after integrations[95,96], therefore, one obtains n=0 = \ [eiu,13t + eiw^] exp ( - A S F t / 2 ) . (4.69) The relaxation rates are the same as in low fields, A<f = | A S F , / = \ -L o n g i t u d i n a l F i e l d For slow spin exchange, the time scale is such that the time evolution of the muon spin in a longitudinal field is not time-dependent (Eq. 3.8) and Pn(t) is expressed by ™ = <WT7rf+1 (4-70) and one immediately gets P l t ) - 1 + ^ ^ C _ W ( A S P ^ ) " n t ) ~ 2( l + x2) n ! 1 + 2*2 / \SFt \ 2TiT^exH"2TTT^)J ( 4 J 1 ) 113 and the relaxation rate is given by Xfi = —— x ~ fa —— x TTT-O (4-72) 2 1 + z 2 2 l + u ^ / w 2 , v ; where x = (uie + u>lt)/u>o « u>e/u)o, is used. The field required to bring down the relaxation rate to 1 /2 of the zero field value is B = Bo. There is a similar effect seen in the amplitude itself. This field corresponds to the field at which the Larmor precession frequency of the electron is the same as the muonium hyperfine frequency. 4.2.5 Fast Spin Exchange If the spin flip rate is much faster than the muonium hyperfine frequency wo, the muon spin in muonium can not follow the rapid change in the electron spin. In this case the muon spin in muonium behaves as if the muon were in a diamagnetic environment. The polarization function can be calculated using Eq. 4.60 [94] but since fast spin flip is not studied in this thesis and the calculation is rather involved, only the results are given below for completeness. In a longitudinal field, P(t) = e~XLt, (4.73) where In a transverse field, X L = SAI X TT^TAI;- ( 4 J 4 ) P(t) = e~iUDt exp{-\Tt), (4.75) where The above equations are valid at all fields. Note that in the T F case, both the precession direction of the muon spin and the dependence of the observed relaxation rate on the spin flip rate are opposite to that in a slow spin flip process (see Eq.4.64). The muon in this case behaves like a diamagnetic muon, even though it is in a paramagnetic environment. Evidence is seen in the reaction Mu + O2 at high 0—2 pressure, but these data are not considered herein. 114 4.3 Collisional Relaxation (Spin Relaxation of Free Radicals) This section discusses the spin relaxation of Mu-radicals in the gas phase. There are two approaches to treat this problem [157,231]. Both will be referred to in this section but the emphasis is on the phenomenological treatment [231], which, although not derived from first principles, offers a good description and a more intuitive picture of the relaxation process. Attempts will be made to give some justification to this phenomenological model. However, the init ial, and primary motivation in this thesis work, for studying radical relaxation in the context of /zSR technique is that, in order to extract useful kinetic information from some fJ.SK experiments, one must know how the radical relaxation affects the observed muon polarization. In this regard, what is required is not a detailed understanding of the relaxation mechanisms involved but a model that properly and correctly describes the field and pressure dependence of the observed relaxation rate. So the goal in this thesis is to develop a phenomenological model, and to show that the model works for polyatomic molecules, e.g., C 2 H 4 and C4H8, and then to use this model to explain experimental results for chemical reactions of M u with other small molecules (CO and N2O), which are the primary interest of this work. 4.3.1 A Phenomenological Model Recall the Hamiltonian (Eq. 3.21) introduced in Chapter 3. Though the SR interaction(s) and hence the rotational (J) levels of the Mu-radical are not included in the calculation of eigenstates and eigenenergies (e.g., see F ig . 3.6), they nevertheless play a major role in determining the relaxation of the muon spin by collisions (see also Ref. [157]). The relevant experiments of this thesis refer to a regime in which neither a given rotational state nor an instantaneous orientation can be well specified. Mean collisional frequencies between 10 1 0 and 1 0 1 2 s - 1 are anticipated so that rotational levels appropriate to the isolated molecule (low pressure l imit) are broadened by amounts comparable to their separation over much of the pressure range employed. Such a situation is commonly encountered in magnetic resonance; namely that fluctuations of a given 115 interaction induce relaxation even when the spectral features associated with that interaction are unobservable, being averaged out when the fluctuation frequency exceeds the splitting which the static interaction would cause. In the present case, the SR interaction must be thought of as a coupling between the unpaired electron (or muon) spin and the molecular framework, which is modulated as the radical collides with neighboring molecules (all diamagnetic in the present studies). Spin relaxation (Ti) in a multilevel system such as M U C H 2 C H 2 is usually [8,79,145] described by the superposition of a small number of exponential terms corresponding to contributions from (off-diagonal) transitions between pairs of levels as in Eq . 3.27, rewritten here to emphasize the different transitions, PR(t) = Y,Aiexp(-\it) (4.77) where the At's are proportional to the populations of the states involved and the A,'s are eigenvalues of the relaxation matrix, linear combination of transition probabilities. In practice the relaxation function invariably approximates closely to a single exponential, which is also observed experimentally (see Chapter 5) [155,354]. It can also be noted that Turner and Snider in their recent (first-principles) calculation of the relaxation of the muon spin in M u C H 2 C H 2 , also found that the matrix of relaxations was dominated by a single solution (the longest-lived one) [157]. In the approximations inherent in F ig. 3.6 there are only a few dominant contributions, which are expected to have the classic form of Eq. (4.78), A, = i = £ wu,M = £ ^ irf^F (4J8) where Wij gives the strength of the fluctuation and J(uij) is the usual spectral density term, and the J2 indicates all possible contributions of different Wij and r c is the appropriate motional correlation time, which will assumed below to be given by some mean time between collisions in the gas. In the phenomenological model approach here then, it is simply a question of identifying the appropriate transitions and their contributions to the muon spin relaxation via their various allowed matrix elements. 116 In standard magnetic resonance treatments leading to the form of Eq. (4.78), spin relaxation is due to fluctuations caused by a time-dependent potential, V(t), which in the present case causes a change in orientation of the molecular framework, represented by J( t ) , whose off-diagonal elements fluctuate according to the spectral density function, J(wy) . Those that oscillate at a particular resonance frequency induce (non-secular) transitions between states, effecting a change of populations and causing T\ (spin-lattice) relaxation; fluctuations caused by the diagonal (secular) elements of V(t) cause direct dephasing (u> = 0), defined by 1/T2* in Eq. 4.80 below, and also broaden energy levels, both contributions then giving rise to T 2 relaxation. Under conditions where local magnetic fields are weak enough and also fluctuate many times during T\ (or T 2 ) , the usual (NMR) definitions 9 can be written Ai = = A 2 J{UiJ) (4.79) which is just the form of Eq. (4.78) and with where Ai and A 2 correspond to the measured relaxation rates (A/i) for the Mu-radical previously defined. The physical interpretation of Eq. 4.80 is one where population relaxation (1/Ti) is always accompanied by loss of phase coherence, regardless of the nature of the interaction, but there can also be additional mechanisms (secular relaxation) producing dephasing (1 /T 2 ) that do not affect populations. Inhomogeneous line broadening [42,43,356,357] is expected to be a major contributor to this term in the present study. In Eqs. 4.79 and 4.81, A 2 represents contributions from matrix elements due to the (spin-dependent) rms average of particular local field components 9Strictly, these expressions are rigorous only for small disturbances from equilibrium, so they must then be used with caution when the initial polarization is high, such as in a Mu-radical. Higher order couplings can also lead to deviations from exponential relaxation [355]. However, one can at least expect these expressions to give qualitative guidance. It should also be noted that the classic expression of Eq. (4.78) is only valid when the "observation" times are much longer than the "correlation" time, T c . This is certainly valid in the present case though where the experimental relaxation times are typically ~ 1 us, whereas T c < 100 ps. Also, Eq. (4.80) is strictly true in the limit of rapid correlation times, urc 1. 117 (Wij in Eq. 4.78) and J(u>ij) contains a difference in precessional Larmor frequencies between, for example, electron and muon moments. It is the quantities A 2 and uiij that must be calculated or found from specific models, whereas r c is strictly determined by gas collisions through the iniermolecular potential. In the kinetic-molecular theory of spin relaxation [90,148-151,157], r c involves a collision integral of the density operator, which is naturally proportional to the time between collisions. Typically, in conventional magnetic resonance (NMR) studies in gases [79,145,147,230], at high enough pressures, where the time between collisions gets to be very short,(u;,jr c) 2 1, T i = T2, assuming further isotropic fluctuations so that A 2 = 2 A 2 . A t the opposite l imit, in the "slow" collision regime of low pressures, rc —¥ 00 and hence —¥ 0, but now Ti approaches the equivalent of its fast collision (high pressure) l imiting value again. The condition when cjjjr c = 1 gives rise to the well-known "X i minimum", from which the correlation time can immediately be determined. Such minima have been observed at specific pressures in a variety of N M R studies in gases [134,147,152-155,230], and are seen here as well (Table 5.4). However, the present situation of spin relaxation of Mu-radicals in gases is quite different from this conventional picture because of the fact that it is the muon relaxation that is monitored by the / /SR technique but it is the electron that is most affected by collisions. This gives rise to a largely indirect coupling between muon and electron, which, as shown below, means that T\ « T2 only at very low fields (and high pressures). Related effects have been seen at specific temperatures in some recent / iSR studies of muon and Mu diffusion in solids [358-361]. There are a large number of possible transitions corresponding to off-diagonal matrix elements of the above states which can account for these different contributions, which can be described as follows, with reference to the Hamiltonian given by Eq. 3.21: (i) Direct Muon Spin Relaxation Depending on the importance of a particular fluctuating part of the interaction, the matrix elements may have the form <L4(I±iS z) (where SA = CA/2IT), arising from the fx-e dipole-dipole 118 coupling (hyperfine anisotropy) and/or C/( I • J ) (where Ci = WJH/27T), arising from modulation of the muon spin through its coupling with the rotational angular momentum of the molecule. Thus, for example, between states |1) and |2), 6A(I+SZ) = cSA and since the transition probability, and hence the relaxation rate is proportional to the matrix element squared, one can expect contributions to the muon relaxation oc c2(SA)2, or simply ~ (SA)2 in high fields. Similarly from the nuclear SR. interaction, one can expect contributions oc C2 in high fields. The spectral density contribution J(u\2), corresponds to the transition W i 2 = l/2wrj — w^, from Eq . 3.26 and Fig. 3.6. There is an analogous matrix element for the transition |3) —»• |4) with W34 = l/2uift + w^. Both of these transitions correspond to the observed " E N D O R " lines seen in F T - / / S R [231,240]. Thus, labelling the strength of these fluctuations by the parameter A M , one expects direct contributions to the muon relaxation of the form It's worth noting that a "resonant" contribution to the direct muon relaxation rate could occur in the first term of this expression, which is essentially due to the ^ - e dipole-dipole "level crossing" term, at a field of 12.2kG for Mu-substituted ethyl [143,231,240]. However, this particular field region was not well sampled in the present study (Fig. 5.3). ( i i ) Ind i rec t R e l a x a t i o n Transitions of this nature would enter directly into E S R , since it is electron transitions which are monitored in E S R , but in ^ S R electron relaxation can affect the muon only indirectly through its (isotropic) hyperfine coupling with the muon. Thus, in complete analogy with the above, one could expect transitions of the form (1|S + I Z |4) ) with strength again oc (SA)2 in high fields, arising from the muon-electron anisotropic hyperfine coupling or with strength oc C j (where Cs = wsfl/2?r) from the same kind of matrix element but involving the appropriate part of the SR interaction, ( 1 | S + J Z | 4 ) . Here the spectral density contribution, J (w i 4 ) corresponds to the transition |1) —»• |4) in F ig. 3.6 with w i 4 = w e + l/2u>0 ~ w e , from Eqs. (3.26), over the field range of interest. The (indirect) coupling to the muon, through the isotropic hyperfine interaction, is not 119 revealed by matrix elements of this nature though which involve only (S± i ) for the electron. This requires a solution of the complete coupled equations resulting from consideration of the time-dependence of the populations of these states, which are not considered in the present study. Relaxation processes of this nature have been considered by Celio and by Celio and Meier [362,363] and by Cox and Sivia [354] with reference to Mu-radical. A more rigorous treatment solving the full equations of motion of the density matrix, including nuclear degrees of freedom, has also recently reported by Turner and Snider for the M U C H 2 C H 2 radical [157]. Most transparent here, are the matrix elements of (S z ) , corresponding to a modulation of Sz by collisions, which induces transitions between levels |2) and |4). In particular, from the electron SR interaction, C*sS • J , the matrix element C s ( 2 | S z J z | ) 4 = 2scCs, with transition probability W24 oc C | / ( l + x2). Since there are no contributions from 7-states to the energy levels in this model approach, ( J z ) has no effect on the transition. The "decoupling factor" 1 ^ x 3 is noteworthy, and is the same factor seen in the case of "slow" SE discussed earl ier. 1 0 There can be no contributions to muon relaxation from electron coupling at high fields. The transition |2) —• |4) involves the energy difference a/24 = w e + w,,, from Eqs. (3.26). However, again at the fields of interest, these are all ~ w e to a high degree of accuracy (for fields B > 500G). Thus, from earlier equations, in terms of the parameter A ^ , one may expect contributions to the relaxation of the muon from electron relaxation of the form 1/Tf = A | x 1/(1 + x2) x 2 r e / ( l + (u>ere)2) (4.83) (iii) Cross Relaxation Here there can be contributions due to "flip-flop" ( A M f t e = 0) or "flip-flip" ( A M ^ e = ±2) . There are two possible source of p-e flip-flop, both involving the transition |2) —»• |4), arising either from the anisotropy in the muon-electron hyperfine interaction (<L4(2|I+S_ + I_S + |4) ) or from a 1 0 It is worth remarking that this factor is only rigorously true for the case of "slow" SE, which means when the time between collisions is much longer than the hyperfine mixing time; ie, T c S> l/wo. Such a condition would seemingly not be well obeyed by the present experiment, since wo is relatively small for the M U C H 2 C H 2 radical (I/Q ~ 330MHz [35]), in marked contrast to that for the Mu atom itself (1/0 ~4463 MHz). However, the SE interaction per se is actually an intermolecular one whereas the SR interaction of interest here, while admittedly involving an electron "spin flip" process (and muon spin-flip as well), is an intramolecular one. It is not clear that these two situations are parallel. Nevertheless, this concept forms the basis of an alternate model, which is under development and has been discussed briefly in Ref.[232]. It is noted that the data argues strongly for the presence of such a decoupling factor. 120 modulation of the isotropic hyperfine coupling itself (A P (2| I • S|4)). Both of these matrix elements give the result c 2 — s 2 = x/^l + x2, albeit with different weightings (the isotropic contribution is weighted by 1/2), and, significantly, the transition probability and hence the relaxation rate is oc x2/(l + x 2 ) in both cases. It's interesting to note this actually increases slightly with field, at weak fields, an effect which has been reported elsewhere [36,364]. In the present study, x > 5 even at the weakest fields, so this effect is not really observable. Again, at a given pressure, the spectral density J(w 24) is essentially just determined by the frequency w24 ~ w e-As can be seen from the usual contributions to the p-e dipole-dipole coupling [79,145,365] and if hyperfine anisotropy is important, then a muon-electron "flip-flip" term (AA/^e = ±2) should also be included, corresponding, for example, to the matrix element <L4(1|I+S+|3) = 6A and hence a contribution to the measured relaxation rate oc (SA)2, at a frequency W 1 3 = w e — w^, or again ~ uie at fields in these experiments. One cannot distinguish these contributions, AM = 0 and A M = ±2 , on the basis of the present model, since they enter with essentially the same spectral density response. Hence, adding both together, in terms of the parameter A M E , one expects additional contributions to the muon relaxation rate in the radicals of the form Discrimination between these flip-flop and flip-flip contributions may be possible in some cases though, on the basis of physical arguments. Thus, as discussed below, in the Mu-t-butyl radical, which is a pure /3-hyperfine coupling, it is likely that the modulation of A^ is most important, based on evidence from current E S R studies in the liquid phase [366]. On the assumption that one can treat the processes above as individual contributions to the measured relaxation rates, as implied in Eq. 4.78, all terms are simply summed to give a model describing the gas phase relaxation rates of a Mu-radical. Though, in principle, each of these separate processes could be identified with a different correlation time, in practice it would be difficult to meaningfully extract these from fits to the data and hence a single correlation time, rc, is assumed for all contributions. While this would not be true in the case of a fully quantum 121 mechanical treatment of spin relaxation, where dipolar and spin-rotation relaxation times in the gas are generally different [147-150,157], (H2 being a celebrated case [147,230]), for multilevel relaxation of a polyatomic like MUCH2CH2 at ambient temperatures, the assumption of a single correlation time is reasonable, where it is some average large value of (J) which is being perturbed by collisions. This assumption of an essentially J-independent relaxation rate follows as well from classical treatments of spin relaxation [79,145,151]. It is further expected that r c would simply scale inversely with the gas pressure, 1 1 as seen in a number of N M R studies [134,147,152-156] and demonstrated below as well. Although the treatment given here is entirely phenomenological, it is not inconsistent with the results of the aforementioned and much more rigorous analysis of the spin Hamiltonian [157]. 4.3.2 Ti Spin Relaxation Thus, from the present phenomenological model, in a L F , the gas phase relaxation rates in these experiments are expected to have the form: 1 .2 1 2r c E" 2r« Ti. *l + (B/Bo)2 l + w f r | + ^ M E 1 , 7,7"" \2 ( 4 ' 8 5 ) 1 + K r c ) 2 + M l l + (w 1 2 r c ) 2 + H - ( W 3 4 r c ) 2 J ' where, the (fitted) parameters A # , A M E and A M have been introduced above and w i 2 = TvAft — uift and W 3 4 = •KA^ + w^, as given earlier. Note that the first two terms involve just the electron Larmor frequency, in the high-field approximation utilized here, whereas the A M term involves only the muon Larmor frequency, which will only be important at the highest fields.12 1 1 This is effectively an ideal gas result, which one can expect to be valid at pressures up to a few atm. Expected modifications at higher pressures are not considered in this thesis. 12Though the energy denominators in the first two terms of Eq. (4.85) are written in an approximate form involving only we = 2irve, which can be seen from Eqs. (3.26) with t'e^f/j, in the actual fitting of the data, complete energy expressions [90,91,94,133,140,231] were utilized. In like manner, the factor x 2 /(l+ x2) discussed above, multiplying the AME term, is also included. In practice, neither of these effects makes any real difference to the results in the field range of interest here, B> 500G. 122 4.3.3 T 2 Spin Relaxation In a T F , electron (Ti) relaxation contributes as well to muon (T2) relaxation, but in this case there should be no additional (1/(1 + x2)) decoupling effect on the relaxation rate as previously demonstrated in the case of electron spin flip processes related to M u atom spin exchange [90,91,94,95,235,350,367-369]. This is because, in a T F , none of the states in Eqs. (3.26) are eigenstates due to the precessing nature of the muon spin. A similar situation prevails in L C R , when the muon experiences a transverse field (T2) component [370]. In addition, from Eq. (4.80), one can expect a "secular" contribution to the relaxation rate, corresponding to fluctuations of (Hz)2 at J(0). This term will also contain effects due to "inhomogeneous line broadening", which will be most severe at low pressures and high magnetic fields [42,43]. Both contributions are described here by the parameter A s - Thus from Eqs. (4.80) and (4.85) one can wr i te 1 3 A 2 = ± = ( A | + * ? M E ) * . ! ' , 2 + A | r c (4.86) It should be noted that Eq. (4.86), while motivated by standard N M R expressions (4.80), departs from this form in that As is introduced here as a different parameter than those used to characterize the T i relaxation rates. This is partly due to expected contributions from inhomogeneous line broadening, as mentioned, but also because of the fact that, in contrast to E S R studies of free radicals, where electron relaxation would directly influence the signal, in / /SR the electron is only indirectly coupled to the muon, as represented by the A# parameter. Thus the true "secular" contribution to T2 relaxation rates of Mu-radicals can be expected to arise primarily from the A M E term (or A M E + A M ) - It is also an approximation here to include both contributions from line broadening and secular relaxation in the same parameter. The basis for this is the expectation that line broadening, like r c itself, wil l scale inversely as the pressure over the pressure range of these experiments [42,43], since the muon will sample a larger region of field inhomogeneity at lower pressures. 1 3 The AM terms are not shown here but are included in the fits, discussed below. However, in contrast to the fitting of the T\ data, where these contributions are important at high fields, they make essentially no contribution to the T2 fits, being overwhelmed by the size of the As parameter. 123 Eqs. 4.85 and 4.86 have been motivated by studies of the spin relaxation of largish-polyatomic free radicals (e.g., M u C H 2 C H 2 , M U C D 2 C D 2 , M U C 2 F 4 , t-butyl M u and MuCeHs) and the model fits for these system are not treated in detail herein but are discussed elsewhere [34,231,232,354,364]. Rather, as will be shown later, the quality of global fits for T i and T 2 data obtained for M U C H 2 C H 2 (and T\ for Mu-t-butyl) wil l be demonstrated and then this model will be applied to two small free radical system of interest: M u N 2 0 (or MuO) and M u C O . 124 Chapter 5 Results and Their Interpretation 5.1 Mu + Cs: Spin Exchange The M u + Cs reaction was measured in (6 G , 50 G) T F at various temperatures. The primary motivation for this experiment was to compare the experimental electron spin exchange cross sections for M u - C s with available theoretical calculations for H -Cs [342,344] and to investigate the isotopic mass dependence, if any, of the spin-exchange process for collisions of H-isotopes with alkali metals (A), which are probably the next simplest systems to that of H + H spin exchange. Before this experiment was carried out, there had been no reported studies of any H-atom isotope effects in spin exchange other than the comparisons between H [47,52,53] and M u [55,56,58,59] spin exchange with O2 and N O molecules. Studies of H(D) with A were reportedly underway [61,62], but here the mass difference is only a factor of 2. After the experiment was finished, an optical pumping study utilizing the spin exchange interaction of ( / i ~He) + e - + Rb and Mu + Rb, with an amazing mass ratio of 35, was published [65], but no isotope effect in the spin-flip cross sections was reported. On the other hand, in a completely different temperature regime, a study of H - D mixture at about 1 k found huge isotope effect in spin exchange cross section, favoring H - D over H - H [51]. In the earlier / iSR studies it was found that the experimental M U - O 2 and M u - N O spin-flip cross sections at room temperatures were considerably reduced compared to the corresponding experimental H-atom cross sections [53,55,56,90], by almost a factor of 3 on average. Other than the theoretical comparisons of H - H and M u - H by Shizgal [346], in which large isotope effects favoring M u - H scattering at low 125 temperatures were predicted, and those of Aquilanti and co-workers [56,347] comparing M U - O 2 and H-O2 spin-flip cross sections, there are no calculations of isotope effects in H-atom spin exchange. Moreover, theoretical calculations failed completely to account for the experimental H-molecule cross sections [56,347]. The calculation of spin-flip cross sections for atom-molecule scattering (H -NO and H-O2) is complicated by poorly known potential energy surfaces. The simplest spin-exchange process is unquestionably that between two spin 1 /2 atoms, the most fundamental of which is H-H(Mu,D) scattering. Calculations of M u - H and H - H electron spin-flip cross sections have been performed by Shizgal [346], Koyama and Baird [371], Berlinsky and Shizgal [372] and All ison [373]. Impressive agreement is obtained with the experimental results for H - H over a wide range of temperatures [48]. Large isotope effects are also predicted at low temperatures, due to specific resonances favoring the spin-flip cross section of M u - H over that of H - H by a factor of 5 at 50 K [346]. As mentioned, experimental results of H - D spin exchange are also available at very low temperatures [51]. However, the experimental determination of spin-flip cross sections for M u - H is complicated by the difficulty of producing a known concentration of H atoms, even with a radical redesign of current /zSR gas phase reaction vessels. This is primarily the reason why the study of electron spin exchange processes with the alkali metals, notably M u - C s interactions in the present experiment was chosen. There have been two theoretical calculations of H - A spin-flip cross sections, O~SF(E): an early calculation by Dalgarno and Rudge [344] and a more recent one by Cole and Olson [342]. In both cases, the alkali-metal atom is assumed to have an inert core. Wi th the exceptions of reports of H - N a spin exchange [63], the work cited in Ref [61,64] and the previously mentioned work of Barton et al. [65], no other H-atom experiments have been carried out on these simple spin 1/2 systems. Ueno et al. [63] measured the H - N a spin-flip cross sections as part of a development program to produce polarized proton beams by optical pumping. These authors were able to measure the forward-scattering (< 1.1°) differential cross section for H - N a spin-exchange. They also calculated both total and differential cross sections with their total cross section agreeing with that of Cole and Olson, but their calculated value for the differential 126 cross sections were a factor of 7 lower than their experimental values. Since their total cross section calculation agrees with those of Cole and Olson, the level of disagreement between theory and experiment for the differential cross section is surprising, since the theory should be highly accurate, at least given the approximation of no core-excitation channels. What are needed as well therefore are measurements of total cross section. It could be that a core-excitation process is more important than heretofore realized, likely manifest in the repulsive part of the interaction potential, which could show up dramatically in the forward-scattering amplitude. In this context, it can be remarked that similar contributions to the (exact) H - H scattering potential are non-existent, possibly accounting for the aforementioned exemplary agreement between theory and experiment [48,346,373]. The discrepancy between experiment and theory for H - N a spin-flip cross sections is an important motivation then for the study of total M u - C s spin-flip cross sections, as reported herein. 5.1.1 Results The fiSR relaxation rates at varies Cs concentrations, measured in ~ 6 G T F with N 2 moderator at 543, 566 and 643 K, are listed in Appendix B. These experiments were carried out on the M20 beam channel. The Mu + Cs system is a simple system as far as the the /xSR technique is concerned since, as discussed below, the only significant interaction involved is the electron spin exchange interaction. In this case, recalling Eq. 4.33, A t = A/ = 0 and, in transverse field, XT = A,- = A<j. Moreover, the previously-described experimental procedure for determining the Cs density, while demanding, is in principle straightforward. Nevertheless, there are possible source of systematic error: Cs dimers, total pressure dependence or even Cs chemical reactions. A l l are considered negligible. The presence of Cs dimers, C s 2 , inside the reaction vessel could contribute to an error in the Cs density as well as contributing to a damping of the /zSR signal due to chemical reactions, forming CsMu. However, the dimer is only about 0.5% of [Cs] at 560 K under saturated pressure 127 without a foreign gas [374,375]. It would be less in the target vessel since Cs is not saturated there, although a three-body association reaction could conceivably occur at higher moderator pressures, 2 atm in this experiment. Nevertheless, it was determined that the contribution from dimers was minimal and probably non-existent here. There is some discussion in the literature on the total pressure dependence of saturated Cs vapor pressure, which causes some concern in experiments relying on data from vapor pressure curves [68-71]. In the current study, however, in the experimental procedure, the Cs is saturated when there is no moderator gas present. The vessel is pressurized only after the boiler (and the coldest spot) is isolated from it (see F ig . 2.9). Therefore the Cs is no longer saturated when the total pressure is changed and hence the Cs density should not be affected by a change in the moderator pressure. The off-line tests (Table 2.1) also showed very little total pressure dependence. The possibility that CsMu could be stabilized by collision with N2, is also considered unlikely, as shown below. Another possible, albeit very unlikely, source of error could be a slow reaction of CS+N2. However, while there is some evidence of a very slow reaction of L i and Mg with N2, there appears to be no information on any reactivity of Cs with N 2 [376]. It is worth noting here that the boiler and the target vessel are free of N2 when loading Cs, so any reaction involving N 2 can only occur after the boiler is closed, which could conceivably reduce the Cs density in the vessel, though this is very unlikely over the approximately 2-h time period when Cs is in contact with N2. Furthermore, the product of this reaction would be expected to be CS3N [376], a virtual impossibility at the Cs density of the present experiment. Moreover, the time-dependence studies (i.e. consecutive runs on the same gas over a time period of 4 h) showed that the relaxation rates were independent of time, indicating that the effect of any competing reactions is insignificant. No time-dependence was detected in off-line titration tests either. Generally, the titration results gave good agreement with expectations from the vapor pressure curve (Table 2.1). Thus, the uncertainty in the Cs number density wil l be primarily due to an uncertainty in the boiler temperature. As discussed in Chapter 2, at 440-510 K, the uncertainty in the Cs density 128 is estimated to be < 13% (Table 2.1). The seemingly very large uncertainty is the result of the exponential dependence of density on temperature. The experimentally observed depolarization rate (XT) of the / iSR signal (Fig. 3.2) is related to the thermal spin-flip cross section as given in Chapter 3: XT = Xd + X0 (5.1) Here An is again some background relaxation due to contributions such as field inhomogeneityand chemical impurities while Aj is proportional to the thermal spin-flip cross section at temperature T, (TSF(T), which is defined in Eq. 4.53. In a weak transverse field (< 10 G) and with an S = | (Cs) collision partner, Xj is given by (see Eqs. 3.14 and 4.58): Xd = ^ A S F = \nvaSF(T) (5.2) where, again, A S F is the rate of spin-flip interaction, n is the number density of Cs atoms, and v is the thermal velocity (v = \j8kBT/nfx, where ks is Boltzmann's constant and // is the reduced mass of the colliding atoms). Experimentally, (TSF(T) is determined by measuring A at several values of n at a given temperature and fitting to Eqs. 5.1 and 5.2. This is illustrated in Fig. 5.1 for data taken at 566K. Each data point in Fig. 5.1 took one or two hours to measure. Two positron counter arrays were used to give two histograms of positron events (two to three mill ion counts in each to give less than 10% statistical error in the relaxation rate), which were fit separately. The average of the two fits is taken as the relaxation rate and plotted in F ig. 5.1. Results for O -SF (T ' ) at different temperatures are recorded in Table 5.1, which compares the data as well with the theoretical calculations of Dalgarno and Rudge [344] and Cole and Olson [342]. The total cross sections are calculated using Eq. (5.4) and the parameters given in Ref.[344] for the former and estimated from the reaction rate curve given in Ref.[342] for the latter. The values given in Table 5.1 are four times those in Ref. [342], which are defined as \ < r S F ( T ) . This comparison is shown in F ig . 5.2 as well. The three higher-temperature data points (Table 5.1) agree within (random) errors (note the reproducibility at 565 and 566 K ) , but the lowest temperature point at 543 K seems considerably below the trend 129 7 . 0 O 10 2 0 3 0 C s D e n s i t y ( 1 0 M m o l e c u l e / c m 3 ) Figure 5.1: Spin relaxation rate A vs. Cs number density at 566 K and an N2 moderator pressure 2 atm. The vertical error bars are due to counting statistics; the horizontals are due to uncertainty in Cs density (see discussion in the text). The spin flip cross section, CTSF(T), is obtained from the slope and Eq. (5.2) and is equal to (42±2) x l O - 1 6 c m 2 . Table 5.1: Thermal Spin-Flip Cross Sections for M u + Cs T (K) C ( M u ) (10 - 1 6 cm 2 ) *s hF €°(H) Ref. [344] Ref. [342]c 543 ± 3° 35.9 ± 1.8" 59.4 99 565 ± 3 41.2 ± 1.6 59.4 98 566 ± 3 42.3 ± 1.8 59.2 98 643 ± 3 39.5 ± 2.6 58.6 97 a. T uncertainty is < ± 3 K, which has little effect on the calculation of the Cs number density in the target vessel. b. Quoted errors are one standard deviation from the fits of Eq. (5.1) and (5.2) to the data. Estimated systematic errors are about 13%, mainly due to uncertainties in the Cs density from the boiler temperature (Table 2.1). c. These are total cross sections, 4 times of the state-to-state cross sections given in Ref. [342]. 130 1 2 0 i 1 1 r C o l e &c O l s o n 9 0 E o (0 T o 6 0 D a l g a r n o & R u d g e to 3 0 0 i 1 1 1 1 1 5 3 0 5 5 6 5 8 2 6 0 8 6 3 4 6 6 0 T e m p e r a t u r e ( K ) Figure 5.2: Spin flip cross sections <7SF(T) for Mu-Cs. The lines are calculations for H-Cs. The vertical errors on the experimental points are one standard deviation for the fits of Eq . (5.2) to the data. Systematic errors are 13%. The horizontal errors are estimated temperature uncertainties, as discussed in the text. in the data. However, considering the possibility of a 13% systematic uncertainty in the Cs density, as described earlier, giving rise then, to an overall uncertainty of < ±18% (i.e. ±7 x 1 0 - 1 6 c m 2 ) , one cannot state conclusively that the lowest temperature data point is anomalously low but rather that the data exhibit a temperature dependence not inconsistent with the trend in the theoretical calculations (both calculated temperature dependences give rise to only about a 2% decrease in cross sections over the range of the data). It can be noted that the possible decrease in ^ S F ( T ) at the lower temperatures in Table 5.1 is also seen in the M u - 0 2 data [56]. A simple average of the M u - C s results in Table 5.1 gives <TSF = 39.7 ± 7.1 x 1 0 - 1 6 c m 2 , taking into account the overall uncertainty. The analysis above assumes that only spin flip contributes to relaxation of the M u precession signal. Chemical reactions, in which the fi+ is placed in a different magnetic environment, would also contribute to the transverse field relaxation rate. A possible diamagnetic 131 channel in these experiments is the formation of CsMu ('muide'), the muon analog of CsH. This could form either through a termolecular collision with Cs and a moderator (N2) molecule or directly via reaction with Cs dimers, though, as stated, the concentration of the latter is believed to be negligible. Here the "3/4 effect" discussed in Chapter 3 also comes into play. From measurements conducted at both weak and intermediate fields, it was found that there is no significant chemical reaction contribution to the M u - C s relaxation rate. Hence it is concluded that only spin exchange contributes to the relaxation of the M u signal. As a side note, it was the first time that this technique was used in any experiment. 5.1.2 Comparison of Cross Sections At present, there are no theoretical calculations available for M u - C s spin-exchange collisions. As noted, spin-flip cross sections for H -Cs have been calculated by Dalgarno and Rudge [344] and by Cole and Olson [342] giving values of 59 x 1 0 - 1 6 c m 2 and 98 x 1 0 - 1 6 c m 2 at 560 K, respectively (see Table 5.1 and the footnote therein). Theoretical calculations of the spin-flip cross sections for H - A collisions have traditionally assumed that the core electrons can be treated as closed shells [342,344,377]. The first calculations of spin exchange for H - A as well as for A - A were those of Dalgarno and Rudge [344], who used asymptotic expansion methods to determine the difference potential between the X 1 ! ] and a 3 E molecular states and the straight-line trajectory method to estimate the cross sections. Chang and Walker [377], using semiclassical and partial wave analysis, also calculated the spin-flip cross sections for A - A collisions. The more recent calculations by Cole and Olson [342] for H - A collisions have used a pseudopotential molecular-structure method to calculate the interaction potentials and a full quantal S-matrix technique, as well as a straight-line trajectory method, to determine the spin-flip cross sections. The quantal method included contributions from the quasibound states (or "orbiting resonances"), which are absent in the straight-line method. Both the Chang and Walker [377] and the Cole and Olson [342] calculations find that Dalgarno and Rudge underestimate the spin flip cross sections by 10% ( H - L i and L i -L i ) to 40% (H-Cs and Cs-Cs) . 132 There are a number of experimental results available for C s - C s collisions [68,71,72], which mostly range over (2.0 — 2.4) x 1 0 - 1 4 c m 2 with more recent data [68,71] giving values of 1.4 and 1.5 x l 0 - 1 4 c m 2 , respectively. A l l these values are in relatively good agreement with Dalgarno and Rudge's calculation of 2.0 x 1 0 _ 1 4 c m 2 , a lower value than that of Chang and Walker's calculation which gives 2.8 x 1 0 - 1 4 c m 2 , as noted, though this difference is not large. Since Dalgarno and Rudge's method requires that the difference between the ionization potentials (IP) of the colliding partners is small, it is expected that their results for H-Cs (it has the largest IP difference among all H - A collision partners) are not nearly as accurate as for C s - C s collisions. Thus Cole and Olson's calculation for H-Cs collisions should be the most accurate. However, as mentioned earlier, the experimental differential spin flip cross section of H-Na at 300 K were found to be seven times larger than the theoretical calculations in Ref. [63]. Since the total cross section from this same calculation agreed with that of Cole and Olson, this again suggests that the latter calculations underestimate the true values. To the author's knowledge, to the present time, there is no direct measurement of H - A total spin flip cross sections that have been published. However, a few relevant experiments exist. The spin-flip reaction rates for H - R b collisions have been estimated from H-atom polarizations in an optical pumping experiment [61]. It is suggested therein that the Cole and Olson calculations underestimate the reaction rate by a factor of 2, in the same direction as the results of Ref.[63]. In another experiment, total spin-flip cross sections of H - N a for a fast (2 keV) H beam incident on a N a target were also evaluated from proton polarizations and found to be in reasonably good agreement with extrapolations of Cole and Olson's calculations [64,338]. This agreement may be fortuitous, though, since the method used in the calculations is presumably not accurate at large (> 2.5eV) collisional energies because core-core interactions are not treated correctly. The most relevant data of all in the present context is the recent work of Barton et al. [65] on the He/i + Rb system. The neutral species, He/i ( H e + + / i ~ e _ ) , can be viewed as an isotopic analogue of muonium, with an amazing 35 times heavier mass. In the / iSR realm, the principal difference between M u and He/i is that the / i ~ polarization in the ground state of the thermalized He/i atom 133 is essentially zero, whereas, for Mu , in a weak magnetic field the / i + polarization is 50%. In their elegant experiment, Barton et al. have succeeded in repolarizing the p~ in He/i by spin exchange collisions with optically-pumped (polarized) Rb. The motivation for the experiment was to obtain an accurate measurement of the induced pseudoscalar form factor in the weak neutral current for 3 H e + p~ nuclear capture. The interest here is that their repolarization curves provide a measure of the SF cross section for the process Their experiment measured the p~ asymmetry from decay electrons in a manner similar to that for M u in L F described in the previous chapters. A related set of experiments, using unpolarized positive muons, in fact measured the equivalent repolarization of muonium in a L F where the Rb is polarized along the field direction. From global fits to both the He^i and M u + Rb repolarization data, under the (apparent) assumption that the cross section <TSF(T) is the same for all three isotopic species Hep, H and Mu -f Rb spin exchange, Barton et al. obtained the result <TSF(T)= 136 ±30 x 1 0 _ 1 6 c m 2 at about 475 K. (Separate errors are given in the reference but are added in quadrature here.) Cole and Olson's calculations give O~SF(T)= 91 x 1 0 - 1 6 c m 2 at 500 K for In conclusion, it appears that the agreement between experimental and theoretical values of thermal H - A spin-flip cross sections, insofar as comparisons can be made [61,63,65] seems generally to be poor, with the calculations of Cole and Olson in particular underestimating the cross sections. As can be seen as well from Fig. 5.2, the measured M u - C s thermal spin-flip cross sections are significantly lower than the calculated H -Cs values: two thirds of the lower value of Dalgarno and Rudge, and less than half of the presumably more accurate value of Cole and Olson, indicating a significant isotopic effect. As mentioned above, if Cole and Olson's calculation do in fact underestimate the H - A spin-flip cross sections, then the magnitude of this isotope effect could be greater sti l l . As noted earlier, the same trend has also been seen in the cases of M u - 0 2 versus (5.3) H + Rb.) 134 H-O2 and M u - N O versus H - N O spin-exchange scattering [53,55,56,90] (also see Section 5.4), where the ratio of OSF(Mu) : <TSF(H) is about 1 : 3 for both cases. A theoretical value for O"SF U (T ) can be estimated from the straight-line trajectory methods [342,344]. In the calculation by Dalgarno and Rudge, the difference potential (between singlet and triplet) is only dependent on the ionization potentials of the colliding partners, and since M u has essentially the same IP as H, M u - C s should have roughly the same spin-flip cross sections as H-Cs at the same velocity. Although in Cole and Olson's calculation a more realistic potential is assumed, the spin-flip cross section in both cases is represented by Eq . 4.51. If H -Cs and M u - C s would have the same parameters a and 6, then the thermally averaged cross sections of M u - C s would be expected to be less than that of H-Cs at the same temperature because its velocity distribution is shifted to higher values ( J 7 M u ^ 3 V H ) . If for H -Cs one can write, at energy " E", [<&(£)]1/2 =A-Bln(E) (5.4) where A and B are constants depending on a, b, and / / H , the reduced mass of H -Cs , then for M u - C s , [a^(E)}1/2 = A - B l n ^ H / Z ' M u ) - B ln (£) (5.5) with Liyiu being the reduced mass of M u - C s . Using Eqs. (4.53) and (5.5) and Dalgarno and Rudge's parameters for H -Cs , 0 S F U ( T ) is calculated to be 52 x 1 0 - 1 6 c m 2 at 560 K. Though this is within 20% of the experimental value (Table 5.1), it is recalled that their calculations are likely to underestimate this cross section. Similarly, from considerations of velocity distribution effects only, from Cole and Olson's results, it is estimated that 0 S F " ( T ) « 85 x 10~ 1 6 cm 2 at the same temperature, in considerable disagreement with the experimental data (Table 5.1). In comparison with both the H-atom and Mu-atom values in Table 5.1, then, it seems clear that these calculations do not account for the isotope effect seen here, though a trend to a decreased spin-flip cross section for Mu is indicated. Recall the discussion in Chapter 4 on spin exchange isotope effects, specifically Eq. 4.55. 135 Since M u is lighter than H atom by a factor of 9, the maximum number of partial waves for M u is about three times less than H at a given energy, assuming both have the same interaction potentials. The isotope effect discussed above is likely a reflection of the sampling of fewer partial waves in the M u scattering process [56,90], as well as different resonance structures, but accurate theoretical calculations are required to confirm this. Now consider again also the experimental measurements of He/i + Rb SF cross sections, aHen+Rb _ 1 3 6 ± 30 x i o - i6 c m 2 ) vs_ t h e p r e s e n t determination of cf^0' = 40 ± 7 x 10~ 1 6 c m 2 . In the light of the above discussion, this factor of about four enhancement (or possibly two, allowing for differences due to polarized Rb [342]) favoring the much heavier He/J isotope seems perfectly reasonable with perhaps as much as a factor of six enhancement in lmax expected, consistent with the aforementioned trend in the measured SF cross sections for Mu(H) + N O and C>2- In short, isotope effects are expected in thermal SF cross sections, at least at typical thermal energies, ~ 100 — 1000-ftT. (At much higher energies, where an R P A approximation (see Chapter 4, Eq. 4.56) to Eq. 4.50 may be appropriate, the SF cross sections would indeed be isotopically-mass (and temperature) independent.) The puzzling aspect in the treatment of Barton et al. [65] is their seeming assumption that there should be no isotope effect in spin exchange. It may be that the experimental techniques are more different than appears at first glance; for example, the polarized Rb target used in Ref. [65] could account for a factor of two (see Ref. [342]). It can be also noted that the calculated H-Rb spin-flip cross section is actually smaller (~ 15 %) than that of H-Cs [342] and that the He/i + Rb SF cross sections were measured at temperatures about 100K lower than the Mu + Cs data (which in fact only tends to exacerbate the difference). At thermal energies, the contributions from orbiting resonances are significant [342] (See Fig. 4.8). Furthermore, these contributions only depend on singlet potentials and are very sensitive to small changes in potential curves and reduced masses [372,378,379]. Therefore, an accurate calculation of the M u - C s potential energy curve is necessary for a precise evaluation of these spin-flip cross sections. Comparisons of this nature between M u - and H-atom spin-flip cross sections could prove 136 valuable in determining intermolecular potentials of short and intermediate range between H atoms and various dopants, since only the H - H spin-exchange potential is accurately known [251,252,378]. Interestingly, in the calculations of Ref. [346], a dramatic isotope effect is predicted in M u - H versus H - H spin-exchange at low temperatures presumably due to specific partial-wave scattering resonances, but it is one in which <r^ pu is actually considerably larger than for <Tc?F, just opposite to the situation reported here (and in Refs. [56,58,59]). 5.1.3 Summary The thermally-averaged electron spin-flip cross sections between M u and Cs have been measured by the LISR technique to an overall uncertainty of less than 15%. The measured M u - C s cross sections are found to be less than one-half of the most recent calculated value for H -Cs [342] indicating a significant isotope effect. The actual isotope effect may be even greater if the calculated H - A spin-flip cross sections [342,344] are underestimated, as some experiments seem to indicate [61,63,65]. This interesting isotope effect may originate from the different sampling of partial waves and the different resonance structures of the two systems, which are very sensitive to potential energy curves and reduced masses. Theoretical calculations of this cross section are required to compare with the experimental data and to explain the origins of the isotope effect. 5.2 Spin Relaxation of Free Radicals The experimental studies of the spin relaxation of Mu-radicals in the gas phase are an important contribution to the field of spin relaxation in gases, since E S R is extremely difficult in the study of polyatomic free radicals in gases, due to the multitude of allowable transitions, all broadened by the spin rotation interaction. There has only been one gas phase E S R study reported for a polyatomic, ( C F 3 ) 2 N O [9], though studies in the liquid phase are quite common [132,365,366,380-382]. More relevant here, the phenomenological model developed during these studies provides a foundation for further experimental studies of smaller molecules (MuCO, MuO, M u N 2 0 ) . As mentioned, this thesis research is primarily concerned with reactions of Mu with such small (mono, di and 137 tri-atomic) molecules and the subsequent spin relaxation of the radical so formed. The results of large polyatomic molecules (Mu-ethyl and Mu-t-butyl) are discussed in this section mainly to demonstrate that the model developed in Chapter 4 for collisional spin relaxation free of radicals can fit the experimental data very well and the parameters of these fits are reasonable and in many cases are consistent with theoretical predictions. In this respect, included here as well are fitting results of the data taken in previous experiments ( M u C D 2 C D 2 , M u 1 3 C H ^ 3 C H 2 , and some of M U C H 2 C H 2 data) [232] conducted by Dr. Fleming's group, a complete analysis of which is under development and presented elsewhere [231]. These data and their interpretation provides a basis then for the interpretation of other radicals, of interest to the present kinetics studies, notably, M u C O , MuO, M U N 2 O , as well as Mu02 , though the latter is not discussed in this thesis. 5.2.1 Results Eqs. 4.85 and 4.86 give the general forms used to fit the X i and X 2 spin relaxation rates of the Mu-radicals investigated. Since the same parameters (except A s ) should fit both the T i and T 2 data, a global fit, in which data at all fields and pressures are fit to Eq . 4.85 (LF data) and Eq . 4.86 (TF data) simultaneously, for each radical was first carried out. Very good fits to the T\ data and acceptable fits to the X 2 data, were achieved with a common set of parameters for AE, A M B , AM, AS and r c. Fit t ing the T\ and T 2 data separately does improve the \ 2 m e a c n case, more so for X 2 , since the parameters are mainly determined by the X i data, which has smaller error bars. Even so, there is not much change outside uncertainties (~ ±5%), seen mainly in the AE and TC parameters. Both approaches to fitting the data are addressed in the discussion to follow. Longitudinal (Ti) Relaxation Relaxation rate in L F were obtained by fitting the / /SR signal of each counter to Eq. 2.3 since in all systems under study in this section the chemical reaction rates are much faster than the radical relaxation rates and Eq. 3.36 is reduced to one term ( e - A r t ) , as discussed in Chapter 4, Case A and Case B2. Table 5.2 shows representative data for the L F relaxation rates Ai for M U C H 2 C H 2 at pressures of 1, 3.8 and 11.9 atm and for different applied fields in the range ~ 0.5 — 35 k G . These 138 Table 5.2: Relaxation Rates (Ti) for M U C 2 H 4 1 atm 3.8 atm 11. ) atm B(kG) Ai (fis-1) B(kG) Ai (fis~l) B(kG) Ai (us-1) 0.47 11.5±0.9 0.47 3.90±0.12 1.0 10.5±2.0 1.0 5.71 ±0.55 1.0 1.91 ±0.06 1.5 3.22±0.13 2.0 2.60 ±0 .3 2.1 2.9±0.3 2.1 1.43 ±0.05 3.0 1.1±0.3 3.2 1.25 ±0.09 3.2 1.95 ±0.05 3.2 1.30±0.05 5.0 0.52±0.05 5.0 1.12±0.07 5.5 0.45 ±0.06 5.5 0:96±0.03 5.5 0.94 ±0.04 7.7 0.27±0.07 7.7 0.59±0.03 13.2 0.15±0.04 13.2 0.24±0.03 13.2 0.43 ±0.03 15.0 0.10±0.03 16.5 0.14±0.08 16.5 0.20±0.07 16.6 0.36 ±0.03 19.9 0.20 ±0.05 21.0 0.31 ±0.03 22.0 0.08 ±0.02 22.1 0.065 ±0.008 35.0 0.06 ±0.02 35.0 0.14±0.05 same data are plotted at the top of Fig. 5.3. These are weighted averages of the rates obtained from the forward and the backward counters. The curves are calculated with Eq. (4.85) using parameters of global fits of all T i and T 2 data, employing the non-linear-least-squares fitting program M I N U I T . Since an appreciation of the goodness of fit at high fields is of subsequent interest, the T i data is plotted on a logarithmic scale. Addit ional results (tabulated in Appendix B) were obtained at pressures of 2, 6.5 and 10 atm. The data were taken in several different run periods (and on different beam lines) over a three year period. Good agreement was obtained throughout. The left plot of F ig. 5.4 presents results and a fit of L F (Ti) data at 2.6 atm pressure for the Mu-t-butyl radical to Eq . 4.85. The right plot of the same figure shows the T F (T 2) data obtained in a mixture of 2.6 atm Mu-t-butyl and 4.5 atm N 2 (total 7.1 atm) and a fit of Eq . 4.86 to the data. The reason that a global fit was not carried out is that the two sets of data have very different rc's since, unlike in the other cases, N 2 and C H 2 C ( C H 3 ) 2 are quite different as colliding partners. Included here as well are fits of Eqs. 4.85 and 4.86 to the data for M u 1 3 C H 2 3 C H 2 (Fig. 5.5) and M u C D 2 C D 2 (Fig. 5.6) from Refs. [231,232], in support of the model. The parameters of the fits (including those for previously obtained data) are listed in Table 5.3, 139 10' 10' w 3 10" -10" 10 40 30 A20 10 -2 B (kG) \ 1 "T 1 1 \ \ \ -\ \ l \ \ \ - -\ \ ' \ ^ « \ N N 1 1 1 1 12 18 B (kG) 24 30 Figure 5.3: Top: M U C H 2 C H 2 relaxation rates in L F at 3 different pressures, 1.0 (dotted line, squares), 3.8 (dashed line, circles) and 11.9 atm (solid line, triangles). The lines are calculated with Eq . 4.85 using the parameters obtained in the global fit. Bottom: M U C H 2 C H 2 relaxation rates in T F at two different pressures, 3.8 (dashed line, circles) and 10 (solid line, triangles) atm. The lines are calculated with Eq. 4.86 using the same parameters as above. 140 B (kG) 10 15 B (kG) Figure 5.4: M u C H 2 C ( C H 3 ) 2 relaxation rates. Left: L F with 2.6 atm M u C H 2 C ( C H 3 ) 2 . Right: T F with 2.6 atm MuCH 2 C ( C H 3 ) 2 and 4.5 atm N 2 (total 7.1 atm). The lines are separate fits of the data to Eqs. 4.85 and 4.86, respectively. 8 (kG) Figure 5.5: M u 1 3 C 2 H 4 relaxation rates in L F at 2.6 atm. The curve is a fit of Eq. 4.85 to the data excluding the two on-resonance points (open diamonds). These are due to the p-e dipole-dipole "level crossing" (AM=2) contribution in the direct muon relaxation ( A M ) term [35]. 141 10 2 10' i to 3. X io° 10" 1 10° 10 1 10 2 B (kG) Figure 5.6: M U C D 2 C D 2 relaxation rates in L F at 3 different pressures, 3.0 (dotted line, squares), 6.5 (dashed line, circles) and 16.0 atm (solid line, triangles). The lines are calculated with Eq . 4.85 using the parameters obtained in the global fit (T2 data are not shown here). along with the values of the isotropic hyperfine constants for each radical. A separate set of parameters denoted by asterisks was obtained from a fit to the T2 data alone and is discussed later. The (1 o~) uncertainties listed in Table 5.3 arise from the statistical errors derived from the M I N U I T fits. Some systematic errors are also be expected, particularly in the case of the A M parameter, which is mainly determined by the T\ relaxation rates at the highest fields, which are too slow to give unambiguous, reproducible fits. A systematic error ±30% here is quite possible and in some cases it could conceivably be as much as a factor of two. In fact, a value of A M arbitrarily set as low as 5 / i s - 1 gave acceptable global fits (~ 10% increase in Y 2 ) to the present data for M U C H 2 C H 2 (Fig. 5.3), with little change in the other parameters. However, such a fit is significantly worse at the lowest pressures and highest fields, falling below the data points by more than a factor of two. It is clear that a third term in Eq. (4.85) is required, a point which is most dramatically illustrated in F ig. 5.4 (top) for the Mu-t-butyl radical: the dashed line is the result with A M = 0 (and the other parameters refitted). The marked change in shape in the Ti relaxations as a function of both field and pressure, 142 Table 5.3: Isotropic Hyperfine Coupling Constant (A^) and Relaxation Parameters Parameter" M u C 2 H 4 M u C 2 D 4 M u 1 3 C 2 H ^ M u C 4 H g A» (MHz) 330 340 331 290 AE (fis~l) 2120±20 1323 ± 3 7 2473 ± 5 5 405 ± 3 6 AME{^S~X) 328 ± 1 339 ± 2 8 321 ± 1 0 158 ± 7 A M ( / ^ S - 1 ) Asips-1) 2 3 ± 2 3 0 ± 4 5 0 ± 1 1 8 8 ± 5 443 ± 9 899 ± 4 3 rc (latm)(ps) 6 5 ± 1 9 4 ± 2 54 ± 8 6 115±3 A*E (v*-1)* 2254 ± 5 3 1405 ± 5 0 As (ps-1)" 409 ± 1 6 843 ± 4 1 T c *( latm)(ps) r f 7 4 ± 7 9 3 ± 1 a The parameter values given here are essentially y/2 times larger than those quoted in Ref.[232], due to a change in the definition of J(u) in Eq. (4.85). 6 Data for only one pressure, ~ 2.6 atm. Values taken from Ref. [232]. c F i t to T i data only (cf. text). d F i t to T 2 data with AME and AM fixed at the above values obtained from global fits of T i and T 2 . illustrated by the data in Figs. 5.3—5.6 is noteworthy. These rates are always highest at the lowest fields, most clearly at the lowest pressures, reflecting the dominance of the first term in Eq. 4.85, as can be seen from the relative sizes of the parameters listed in Table 5.3. It is worth remarking that a SR interaction corresponding to a ~ 2 0 0 0 / i s - 1 , would render E S R linewidths far too broad (~ 200 G) to be seen in the gas phase at normal pressures, consistent with the results reported by Schaafsma and Kivelson for (CF3) 2NO many years ago [9]. The same effect would be manifest in /zSR at very low fields, x -> 0, resulting in extremely fast depolarization. This aspect facilitates the study of gas phase Mu addition kinetics in dilute systems [32,41] including those studied in this thesis (Mu + N 2 0 , M u + CO) . Although AE is by far the largest interaction parameter in Table 5.3, its contribution to l / 2 i at the highest fields is negligible, due to the combined effect of the spectral density response and, specially, the decoupling factor l / ( l + x 2 ) in Eq. 4.85. In contrast, the contribution from the AM term at these fields becomes more important, and eventually dominates at the highest fields (and lowest pressures). This is noticeable in Fig. 5.5 for the one pressure of 1 3 C 2 H 4 measured 143 previously and particularly in Fig. 5.4 for the Mu-t-butyl radical at this same pressure. (For T 2 relaxation, A# is relatively more important at higher fields since there is no decoupling factor, but the A M parameter is quite negligible, being overwhelmed by A s . ) The facility of the pSR technique in allowing the direct observation of such free radicals in gases, in marked contrast to the difficulties encountered in E S R , is an important development in the field. It is also worth emphasizing that radical-radical interactions can also considerably complicate the interpretation of E S R spectra [9], an effect which is totally avoided in TD-/zSR, where there is effectively only one Mu-radical in the system at any given observation time. For fields < 1 k G , the AE and AME terms in Eq . (4.85) dominate, and since (uieTc)2 < 1 at these fields, at all pressures, Ai is oc r c a 1/P, giving the fastest relaxations at the lowest pressures. In contrast, at higher fields, although ( w e T c ) 2 > 1, the AME term can stil l dominate at the highest pressures (smallest r c 's) , giving the opposite dependence: Ai oc l / r c oc P. On the other hand, at the highest fields but at the lowest pressures, the A M term becomes relatively more important, again giving rise to the dependence Ai oc 1/P. Even so since the lowest pressure in the present study is only 1 atm, at the highest fields the fastest relaxations are invariably seen at the highest pressures.1 These features can be seen in the data plotted in Figs. 5.3 and 5.6. In the intermediate field region (~ 2 — 3 kG) a maximum in Ai (minimum in T\) can be discerned as a function of pressure. This is clearer in Table 5.4 which gives the T\ rates for M U C H 2 C H 2 at a field of 2 k G , along with the correlation times r c derived from the fits. There is a minimum in T\ at a pressure near 3 atm. This observation is unique in the gas phase and provides an absolute calibration of the correlation time for reorientation: such a minimum occurs when w e r c = 1, from which the correlation time can immediately be found ( « 30 ps), consistent with the fitted values given in Table 5.4. Similar results are seen for M U C D 2 C D 2 , which exhibits a minimum at a somewhat lower pressure. The observation of a T i minimum here nicely supports the analogy between N M R [147,152,153,155,230] and pSR spin relaxation phenomena in gases. 1 Accordingly, the conventional interpretation of low and high-P limits, or, alternatively, slow and fast exchange, respectively, have to be interpreted with caution in gas-phase /iSR at high fields. 144 Table 5.4: Pressure Dependence of Relaxation and Correlation Times. P (atm) Ai (ps" 1 ) T i (fis) r c (ps) 1.0 2.6 ±0.03 0.40 65 2.0 2.9 ± 0 . 2 0.29 33 3.8 3.1 ± 0.1 0.32 17 6.5 2.3 ±0.08 0.42 10 11.9 1.4 ±0.05 0.67 5.5 M u C 2 H 4 at B = 2 k G . Transverse (T 2 ) R e l a x a t i o n : The T 2 relaxation rates were obtained directly from the time histograms, from fitting the data to the fiSR signal of Eq. 3.35. The rates could also be obtained from the widths of Fourier Transform peaks [235,369,383,384] as done for earlier data [231,232]. Good agreement was obtained with both methods. The T 2 data are not as extensive as the T i data, and in general exhibit considerably more scatter and hence result in poorer quality fits. However, these are still of considerable importance, since they provide a further test of the model. The results for M u C H 2 C H 2 at representative pressures and fields are listed in Table 5.5 in order of decreasing pressure and are plotted earlier in F ig . 5.3 (bottom) for pressures of 3.8 and 10 atm, which represent the most extensive sets of data points. The fitted curves were obtained with the same global-fit parameters mentioned earlier as listed in Table 5.3. Alternate fitting procedures were considered to account for the T 2 data. In order to distinguish true secular relaxation from contributions due to magnetic field inhomogeneities (see Eq. 4.81 and subsequent discussion in Chapter 4), which will be most important at high fields and low pressures (longer r c 's) , the A s term was split into a field-independent and a field-dependent part, by A s r c -> A | ( w = 0)r c + XA'£T'C (5.6) where A ' 5 represents contributions from inhomogeneous line broadening, which is assumed to scale with the applied field (x = B/BQ). Fitt ing the M u C H 2 C H 2 data of F ig. 5.3 to include the form of Eq. (5.6) makes very little difference in x2 but returns values of As(w = 0) and A ' 5 of 340 and 24.1 / i s - 1 , respectively. 145 Table 5.5: Representative T F Relaxation Rates for M U C H 2 C H 2 Field (kG) Pressure (atm) A2 ( ps~L) — n x — — 9 . 8 ±0 .9 13.2 14.6 13.0±1 25.0 10.0 6 .0±0 20.6 10.0 7.8 ± 0 19.2 10.0 7.0 ± 0 15.0 10.0 9 .5±0 14.4 10.0 9.3 ± 0 12.9 10.0 11.6±2 11.0 10.0 16.6±3 10.0 10.0 17.0±3 9.5 10.0 15.3±0 21.1 6.5 7 .0±0 14.4 6.5 6 .9±0 13.2 6.5 9 .1±0 9.5 6.5 13.7±1 22.2 3.8 5 .6±0 19.2 3.8 5.4 ± 0 16.6 3.8 5.8 ± 0 14.4 3.8 7.3 ± 0 11.0 3.8 9.0 ± 0 5.5 3.8 24.2±3 22.2 2.0 7 . 2 ± 1 14.4 2.0 8 .4±0 11.0 2.0 12.5±1 9.5 2.0 9 . 6 ± 1 Though the sizes of these parameters are very different, with secular relaxation dominant, nevertheless, at the highest fields (~ 25 kG) both terms in Eq . (5.6) contribute about equally to the T2 relaxation rate, ~ 2.5 fis-1 each at 3.8 atm and ~ 1.0 / x s - 1 each at 10 atm. For reference, a relaxation rate ~ 2.5 / x s - 1 at 25 k G would correspond to a AB ~ 30 G (A2 « 2 T T 7 ^ A 5 ) , or A B / B 0.1%, a reasonable value over the ( « 500 cm 3 ) stopping volume of the target. Though less reliable, a similar fit to the M u C D 2 C D 2 data gives a much smaller change from the previous value, with As(u> = 0)= 845 ps~l and A ^ = 17 fis-1, which means a line-broadening contribution about half that for M U C H 2 C H 2 , a result which is at least consistent with the fact that C 2 D 4 should have the smaller contribution of this nature, due to its somewhat (w 15/%) higher stopping density. Fit t ing the data to include the form of Eq. (5.6) does suggest then that most of the effect of A 5 in these X2 relaxations is due to true secular contributions, but this needs to be confirmed by rigorous theoretical calculations (as in Ref. [157]). The conclusion of a predominant secular component to the X2 relaxation rates is based on incorporating the same TC for both contributions in Eq. (5.6), 146 B (kG) Figure 5.7: T2 relaxation of M U C H 2 C H 2 radical. Similar to the bottom plot of F ig . 5.3 but with lines calculated using the * parameters as well as AME and A M in Table 5.3. Also included are results from Ref. [33] at 25 (squires), 35 (circles) and 50 (triangles) atm. The solid lines are predictions of the model for respective pressures. whereas these actually represent quite physically distinct processes. In principle, inhomogeneous line broadening should not depend on a r c at al l , but line broadening effects in a gas do scale with the muon stopping distribution and hence inversely with pressure, sampling a greater region of field inhomogeneity at the lower pressures, so that these contributions to the A's term also scale with 1/P. Thus both processes are assumed here to depend on TC in the same way. A n alternative approach is to use a single As parameter but allow the A # (and r c) parameters to float from their global fit values. F ig. 5.7 shows the results of such a fit to the M U C H 2 C H 2 data where the parameters AE and r c were allowed to vary from global fit values, with AME and A M held fixed. The new values are also listed in Table 5.3: A*E = 2 2 5 4 p s - 1 , A*s = 4 0 8 p s - 1 and T* = 74.3ps (at 1 atm). Both A*E and r* are increased slightly from their earlier values while A*s exhibits a slight decrease ( ( A 5 ) 2 r* « A | r c ) . These small changes are acceptable and do not alter the interpretation given below. Also shown in F ig . 5.7 are the data points reported by Roduner and Garner for M u C H 2 C H 2 at much lower fields and also higher pressures [33]. Here the dotted lines are the calculated predictions of the model using the * 147 parameters and the values of AME and A M from Table 5.3. The agreement is gratifying, since these data represent a very different field (~ 1 kG) and pressure (25-50 atm) dependence than reported herein and yet the present model also accounts very well for all the data with the same parameters. It is worth noting that the data of Ref. [33] are truly in the "high-pressure" l imit (high pressures and low fields) so that A 2 oc r c oc 1/P, a regime which is clearly seen in the present data only at the lowest fields (Figs. 5.3 and 5.6), as commented earlier. A similar improvement in fits to the MUCD2CD2 relaxations was also found, giving similar 10% variations in the values of AE, AS and r c, but in view of the limited data, this is not discussed further. 5.2.2 Interpretation of the Parameters Comparison of the values of the fitted interaction parameters listed in Table 5.3 reveals several important features. First, the relative sizes of AE {A*E), AME and A M show that electron spin relaxation is much more efficient than muon relaxation. This is not surprising, given the relative sizes of the magnetic moments, but it is significant that the muon spin relaxation experiments are sensitive to (and thus provide information on) electron spin relaxation. The importance of electron spin relaxation extends to electron-nuclear cross relaxation. This is evident from comparison of AME and A M - However, cross relaxation of the electron and some nucleus other than the muon cannot be distinguished from pure electron relaxation in this study; it is therefore included in the AE parameter. The AMJS specifically corresponds to simultaneous electron-muon spin flips. Comparison of values for different radicals is also revealing. It is noteworthy that, for the ethyl isotopomers, AE for the D-radical is the smallest by about a factor of two. The obvious inference is that electron-nuclear interactions play a significant role. Even so, the still relatively large value of A B for MUCD2CD2 points to an additional major contribution, which is common to the ethyl radicals but considerably smaller for Mu-t-butyl, strongly suggesting that the electron spin rotation (SR) interaction, which depends on the moment of inertia of the molecule, plays a dominant role. The relative contributions of the electron-nuclear dipolar and spin rotation interactions to spin relaxation in the radicals studied are discussed below, but on general grounds 148 one can expect the SR interaction to dominate. There are two types of magnetic dipole-dipole coupling: electron-muon and electron-nuclear (here, and in the following, nuclear means nucleus other than muon). Although both give rise to hyperfine anisotropy, the fluctuations of which cause muon spin relaxation, there is an important distinction in the context of / iSR: the electron-muon dipole-dipole coupling directly affects the muon and induces transitions of muon spin states (contributions to AM and AME terms) while the nuclei-electron coupling only induces transitions of electron spin states (contributions to term), which, in turn, indirectly relaxes the muon spin through muon-electron hyperfine coupling. The transition probability for each of these transitions can be explicitly calculated but the details of these calculations are beyond the scope of this thesis and wil l not be discussed here. It is, however, noted that contributions of these dipolar couplings to muon relaxation rate are functions of the distance between the two coupling dipoles as well as the gyromagnetic ratios and spins of the two coupling particles [8,79,145,231]. Of most relevance to the later discussion in this thesis is the electron-spin rotation (SR) coupling contribution to the A ^ term. The SR interaction has the form J • C • S, where C is a coupling tensor that depends on the moment of inertia Ir of the rotating molecule and on the deviation of the electron g factor, Age, from its free spin value (g„ = 2.00023). This interaction leads to electron spin relaxation, because both J and C are modulated by molecular collisions in the gas and hence are time dependent. As noted, relaxation due to the SR interaction is mainly expected to contribute to the AE term and has the general form [8,33,79,145,147-149,151,380] ^ - = (J(J + 1)>C| • J(Uij) =kjfhCl J(UiJ) (5.7) The replacement of (J(J + 1)) by kTIr/h2 is a classical approximation for its high temperature average, consistent with the assumption that relaxation of many-(J)-level radical can be thought of in terms of some single average (J) value (an assumption also employed in the much more rigorous calculations of Ref. [157]), characterized by a single r c. This can also be justified in terms of the strong-coupling model introduced some years ago by Freed [385]. In Eq. 5.7 the spin rotation 149 coupling constant for a polyatomic, assuming axial symmetry, has the form C J = | ( 2 C i + C | ) [9,33,146,147,380,381]; for a diatomic, C | = 2/3C£, giving rise to a standard form for T i relaxation due to the SR interaction[8,79,145,149,151] (recall again here the form of Eq. 4.78). For M u C H 2 C H 2 , Cs is of order ~ 100 fxs'1 [9,33,146], which, from Eq. 5.7, gives l / X i ( S R ) ~ 4 x 1 0 1 8 s - 2 x J(u>ij). Comparing to the electron-muon dipolar contribution, 1/Ti(dipolar) ~ 10 1 6 s - 2 x J(w,j), calculated using the spectroscopic data for M u C H 2 C H 2 (r ~ 0.2 nm) [33,231], it is clear that the SR interaction dominates that from dipole-dipole coupling, at least for the muon. The spin rotation constants needed to evaluate Eq. 5.7 are not generally known, but on the assumption that they can be treated by the same (average) moment of inertia as in Eq. 5.7, they can be estimated from the deviation of the principal elements of the electron <7-tensor from the free spin value [8,146,231,380,386,387]: d = -(h/Ir)Agi where = g« - 2.00231 . (5.8) By substituting Eq. 5.8 into Eq. 5.7 it can be seen that the spin rotation contribution to the electron relaxation rate should be inversely proportional to the moment of inertia for radicals with the same ^-tensor, such as the ethyl isotopomers. However, the empirical values of AE given in Table 5.3 conform only qualitatively to this, as can be seen by comparing them with the estimated moments of inertia listed in Table 5.6. Thus, AE for D-ethyl should be reduced by only 15% while Mu-t-butyl would be expected to be reduced by a factor of 2.2, giving an expected AE ~ 900 ^ s _ 1 , much larger than observed. This simply means that other relaxation mechanisms compete with spin rotation. In addition to muon (nuclear)-electron dipolar coupling, isotropic hyperfine couplings if modulated at some appropriate frequency can also cause muon or electron (then indirectly muon) spin relaxation. The importance of cross relaxation (AM=0) induced by fluctuations in isotropic hyperfine coupling (6A) has been demonstrated in E N D O R studies of powders [388] and chemically induced electron spin polarization in liquids [366]. In the model described herein, it is 150 Table 5.6: Estimated" Moments of Inertia Ir Molecule h h U (Ir) CH3CH2 8 40 43 30 M u C H 2 C H 2 7 35 39 27 M u C D 2 C D 2 13 46 54 37 M u 1 3 C H 2 1 3 C H 2 7 37 40 28 (CH3)2CCH3 108 108 201 139 M u C H 2 C ( C H 3 ) 2 101 106 195 134 A l l values are in units of 1 0 - 4 7 kgm' 2. 0 Calculated from a desktop molecular modelling program. For isotopic substitution, the same structure was assumed; only the mass was changed. necessary, however, to postulate that the hyperfine interaction is modulated by molecular collisions so that the "global" correlation time rc introduced above can apply. The disparity between the molecular collision rate obtained from the fits (typically sa 10 1 1 s - 1 , Table 5.4) and the intramolecular motion ( « 1 0 1 3 s _ 1 ) suggests that contributions to the empirical parameters AE (arising from modulation of /3 proton hyperfine coupling) and AME (from modulation of the corresponding muon hyperfine coupling) may not properly incorporate this effect. On the other hand, these disparate time scales mainly indicate that rotational "fine structure" is not observed and certainly rotational relaxation is effected by collisions in the gas. While it is possible to try and distinguish different relaxation mechanisms by different correlation times, this would be difficult to do in practice, within the framework of the present model. In presenting quantitative estimates of the contributions from the various relaxation mechanisms to the parameters in Table 5.3 it is convenient to consider the empirical relaxation parameters in turn: • The AE Parameter: There are four major contributions to the empirical A J S parameter determined in these studies for the Mu-ethyl radicals (Table 5.3): the most important is the electron S R interaction, followed by coupling to the central atom, with contributions from modulation of the isotropic proton-electron hyperfine coupling perhaps comparable in strength, and with the a — p electron dipolar coupling the weakest. The only contribution that is important in 151 the case of M 1 1 C D 2 C D 2 is the SR one, explaining the fact that AE for the D-radical is much smaller than that of the other ethyl isotopomers. A l l of these contributions cause electron relaxation via matrix elements of the form (S z ) or (S±), multiplying those of ( I p ) or (J) but these are simply "spectators", with no effect on the relaxation rate in the present model. This is because in the model invoked herein, specific nuclear and rotational (J) angular momentum are not included in the Hamiltonian, which is one of an isotropic hyperfine interaction (pseudo-Mu atom), as given by Eq. (3.3). It is important to note that electron relaxations of this nature would be observed directly in E S R , which is the reason why the corresponding E S R line widths are so broad in the gas phase, rendering such studies notoriously difficult to carry out (and in fact, as mentioned, few have been reported for polyatomics [8,9,146]). Their indirect coupling to the muon via the isotropic hyperfine interaction in pSR, accounting for the decoupling factor 1/(1 + X2) seen in Eq . 4.85, is the principle reason why electron SR relaxation is measurable in / iSR, an important development in the spin-relaxation field.2 From Eq . 5.8, the spin rotation contribution to AE can be written A\{SR) = (kBT/Ir)A~g2 , (5.9) where Ag2 represents the average square deviation from the free spin (/-value. By adopting a value of (0.0004)2 for this parameter, based on the ^-tensor for ethyl in an Argon matrix [136], it is estimated that AE(SR) = 1570, 1340, and 1540 ps~l for M u C H 2 C H 2 , M U C D 2 C D 2 and M u 1 3 C H 2 1 3 C H 2 , respectively. The excellent agreement between this value for M u C D 2 C D 2 and the empirical result (1323 ps-1) from Table 5.3 is probably fortuitous, given the uncertainty in the <;-tensor and the approximations inherent in Eqs. 5.7 and 5.8. Though this likely leads to a lower-limit estimate for the value of A B , it is adopted here principally on the grounds that the evidence argues in favour of additional contributions from other sources in the case of the Mu-ethyl and 1 3 C radicals. 2 As will be shown in the section on Mu + CO results, very fast electron SR relaxation in excited radicals can also be measured in wTF in some conditions. This is possible because SR relaxation manifests itself through the dynamic equilibrium between addition and unimolecular dissociation, which can also reduce the observed rate drastically. 152 This is apparent when the same calculation is applied to the Mu-t-butyl radical. If, for lack of better information, it is assumed that the (/-tensor is similar to that of the ethyl radicals, then AE{SR) « 700 / i s - 1 is estimated, a factor of two higher than the empirical value as anticipated earlier. Clearly, t-butyl is different to ethyl in regard to the spin rotation interaction and/or (/-tensor anisotropy. It is perhaps significant that the ethyl radical has roughly cylindrical symmetry about the C - C axis, i.e. perpendicular to the orbital containing the unpaired electron, whereas t-butyl is pyramidal with its principal symmetry axis coincident with the radical centre. Both dipolar coupling (hyperfine anisotropy) and modulation of the isotropic hyperfine coupling also contribute to AE, as noted. The strong dependence of the dipolar coupling on r, varying as 1/r 6 , means that such coupling between the electron and /?-substituted nuclei can be neglected in comparison to a-nuclear coupling, and thus cannot contribute to A B for the t-butyl radical. Calculations show that the contribution from o-substituted nuclei is significant (much less for a protons than for a - 1 3 C ) partially explaining the difference in the empirical values of AE for MUCH2CH2 and MUCD2CD2 [231]. In general, modulation of the isotropic hyperfine coupling is significant. This latter contribution to A B has two sources. The muon-electron interaction contributes to A A / B , since a combined muon-electron flip-flop occurs. However, the ^ -proton coupling applies to A B - T O estimate the maximum contribution, one can set A B ( / ? - H ) = \A\, arising from the matrix element A^(2|I • S |4), where A\ is the torsional hyperfine parameter, the main contribution to the isotropic hyperfine coupling A^ (or Ap).3 A\ has been determined from the temperature dependence of the muon and proton hyperfine couplings in MUCH2CH2 [389] and has the value 185 MHz . Including the effect of the second j3 proton, one finds A B ( / ? - H ) < 820 / / s - 1 . This is substantially greater than the effect of dipolar coupling of the electron to the a protons. Unfortunately, the two contributions can not be separated on the basis of the experimental 3This is estimated by the temperature dependence of isotropic hyperfine coupling constant, which has the general term A)i(Ap) = Ao + A\ cos2 <j>, where <j> is the dihedral angle between the /3-muon(p) and the C-C bond axis. 153 data. Both are reduced by a factor of 1 6 (in relaxation rate; 4 in AE) in the M U C D 2 C D 2 radical, on account of the factor 1\I(I + 1 ) and is the principle reason why the AE parameter is so much smaller for the deuterated ethyl radical. • The AME Parameter: There are two possible contributions to this term, electron-muon dipolar couplings and modulation of the isotropic electron-muon hyperfine coupling. The similarity of the empirical AME parameters for the ethyl isotopomers is not surprising, since the relevant term in Eq. 4 . 8 5 involves simultaneous electron-muon spin flips. The potential contribution of C H 2 M U rotation to the muon-electron cross relaxation can be estimated as mentioned above, which are much larger than contributions to AE by nuclear-electron hyperfine coupling since the muon-electron hyperfine constant are much larger [ 3 8 9 ] . this gives a maximum contribution to A M £ ( J 8 - M U ) ~ 1 9 0 0 / i s - 1 [ 2 3 1 ] . The empirical values are about six times smaller, suggesting that, near room temperature, relaxation caused by the fluctuations in A^ are (An) w 1 / 6 ^ , where A^ is the average (observed) value reported in Table 5 . 3 . Since the muon-electron hyperfine anisotropy is small ( « 1 0 MHz , based on the proton hyperfine tensor [ 1 3 7 ] ) its estimated contribution to AME is about ~ 4 5 / i s - 1 [ 2 3 1 ] , almost an order of magnitude smaller than the measured parameters (Table 5 . 3 ) . It would appear then that one can account for essentially all of the size of the AME parameter in the Mu-ethyl radicals from the modulation of A^. The value of the AME parameter found for the Mu-t-butyl radical, 1 5 6 / / s - 1 (Fig. 5 . 4 ) is also relevant. Though possible contributions to muon relaxation rates from hyperfine anisotropy ( A M = 0 , 2 ) and modulation of A^ ( A M = 0 ) cannot be distinguished in /zSR , these can be distinguished in the equivalent (liquid phase) E S R experiment, as discussed recently by Goudschmidt and Paul for the C 4 H 9 radical [ 3 6 6 ] . In that study, it was established that modulation of Ap indeed dominates the spin relaxation of C 4 H 9 , based largely on the 1 5 4 observation that the A M = 2 (flip-flip) transition was much weaker than the A M = 0 (flip-flop) one. (As established elsewhere [231,365,366], if hyperfine anisotropy is an important source of relaxation, then the A M = 2 contribution should actually be stronger than that from A M = 0, by a factor of six!) It is reasonable to expect then that this would also be the case for the corresponding Mu-t-butyl radical. Accordingly, as determined for the Mu-ethyl isotopomers, AME ^ 1/6-A^, and hence, from the data in Table 2 (A^ = 290 MHz = 1820/xs - 1 ) , one would expect AME ~ 300 / x s - 1 , in only qualitative agreement with the experimental value of 156 / x s - 1 , though not inconsistent with related studies reported elsewhere [36]. It can also be recalled here that the present model assuming the same TC for all interaction terms may be less reliable for describing the internal rotation of the /? — CHiMu group, which would likely show up most dramatically for the t-butyl radical. • The AM Parameter: From the aforementioned proton hyperfine tensor, the contributions to A M by muon-electron dipolar coupling has been estimated to be A M = 25 / x s - 1 , in good agreement with the empirical values for MUCH2CH2 and M u C D 2 C D 2 [231]. The same contribution should apply to all the radicals studied, with only small variation, so the higher values of A M found for the Mu-t-butyl and the 1 3 C-label led ethyl (5.3) are puzzling. The size of A M is essentially determined by the high field asymptotic value of the measured longitudinal relaxation rate; the AE and AME terms are suppressed at high field by the condition UJ2T2 » 1 (and the additional effect of 1/(1 + X2) for the AE term). It may be that other possible contributions to A M are also important. These include the nuclear spin-rotation interaction [156,157,231], which has similar matrix elements to those from hyperfine anisotropy, and avoided level crossing resonance [231,390]. Although the former is small [231,366] (though, curiously, can be source for enhanced line widths in 1 3 C N M R [134,152]) the latter contribution can be comparable to that from hyperfine anisotropy [231,390]. It is also noted that from the rotational diffusion model of Ref. [390], the relaxation rate is predicted to be oc Ir, which is 155 at least consistent with the trend in data, notably the much larger value found for the A M parameter for the Mu-t-butyl radical. It remains, however, to be established whether this is just fortuitous and/or whether this picture can account for the A M parameter in a wider variety of muonium-substituted radicals. • The A s (A*s) Parameter As can be seen from Table 5.3, there is little difference between the fitted parameters AE, A s and r c and their * counterparts optimized for the T 2 data alone. The As(A*s) parameter accounts for contributions to T 2 relaxation from both true secular relaxation (J(u — 0)) and inhomogeneous line broadening [42,43,356,357], the latter proportional to the applied field, as defined by Eq. (4.84). From fits to the data, A s appears to be mainly due to secular relaxation, though contributions from inhomogeneous line broadening become equally important at the highest fields. The size of this parameter (or As) is comparable to AME, which we have interpreted as arising from fi-e flip-flop (and/or flip-flip) transitions. From (NMR) arguments given earlier and recalling that only the AME and A M parameters are indicative of processes that are directly coupled to the muon in the T i relaxation, one would expect the true secular contribution to the T 2 relaxation to be given by As(w = 0) « ( A M £ + A2M)ll2 « AME since AME ^ AM- The difference ( A | — Ame)TC would then represent contributions from inhomogeneous line broadening. A t 3.8 atm, for example, this difference is ~ 1.3 / / s - 1 for M u C H 2 C H 2 , consistent with the estimates given earlier, from Eq. 4.84. As already pointed out, such effects are much less important at higher pressures and lower fields and in fact are completely negligible in fitting the data of Roduner and Garner [33] (lowest field points in Fig. 5.7. It wil l be interesting to see if ongoing calculations of the T 2 relaxation rates, as in Ref. [157], confirm our expectation that secular relaxation is largely driven by muon-electron flip-flop (or flip-flip) processes. 156 5.2.3 Summary Relaxation rates of Mu-substituted radicals in both T F (1.5-35 kG) and L F (0.03-35 kG) have been measured for M U C H 2 C H 2 , M u C H 2 C ( C H 3 ) 2 at room temperature over a pressure range of 1-12 atm and compared with results for M U C D 2 C D 2 and M u 1 3 C H 2 3 C H 2 reported elsewhere [231]. The T\ (LF) and T2 (TF) muon spin relaxation rates can be described very well by a purely phenomenological theory based on a spin Hamiltonian which includes Zeeman and hyperfine terms. Fluctuations in the muon and the electron spin-rotation interactions, and in the muon-electron hyperfine interactions induce transitions between spin levels. This model can account for flourinated ethyl radical as well [232]. Three distinct terms are invoked to account for muon spin relaxation in a L F (Ti): indirect, represented by the parameter A # ; coupled, represented by the parameter AME] and direct, represented by the parameter A M , each of which is multiplied by a spectral density term, J(w,j), the complete expression being defined by Eq. (4.85). The values of these parameters found from the fits, are listed in Table 5.3. The expression for T2 spin relaxation, Eq . (4.86), contains these same parameters with the addition of a "secular" term, A s , which is intended to account for both "secular" relaxation and additional spin-dephasing due to inhomogeneous line broadening. Separate contributions from each were also explored (Eq. 5.6) revealing, as expected, that line-broadening effects were most important at the highest fields (and low pressures). In fitting both the T i (Figs. 5.3-5.5) and T 2 data (Figs. 5.3-5.7) to Eqs. (4.85) and (4.86), each term is represented by the same correlation time, r c, which is assumed to be oc 1/P, where P is the absolute pressure. This assumption is consistent with that of some single average (large) J value for each molecule, given by the Boltzmann distribution ((J) ~ 15 for M U C H 2 C H 2 ) , which is being reoriented by collisions. The sizes of the fitted parameters are consistent with the physical basis for their introduction: AE (or A*E) is given primarily by the electron SR interaction. The calculated value for M U C H 2 C H 2 , using the reported electron anisotropy from E S R and moment of inertia for the 157 ethyl radical [33,136,231], agrees qualitatively well with experiment and scales in a meaningful way with both changes in isotopic mass and/or changes in molecular size. Addit ional contributions, from nuclear hyperfine interactions are also indicated. Even so, the (electron) S R interaction is likely of prime importance. Thus, for example, the Mu-t-butyl radical, with its much larger moment of inertia (Table 5.6), has a correspondingly much smaller AJS than found for M U C H 2 C H 2 . The AME derives contributions from both the anisotropy in the muon-electron hyperfine coupling and the modulation of the isotropic hyperfine coupling. The interpretation of A M is less clear. It is concluded that this is largely due to the anisotropic (muon-electron) hyperfine coupling but other contributions (e.g., level crossing resonance) are also possible. The A s (A*s) parameter in the fits to the T2 relaxations is comparable in magnitude to AME and appears to be largely due to true "secular" (w=0) spin relaxation, consistent with the fact that only the A M E parameter (and A M , but this is much smaller) is a measure of (diagonal) matrix elements which directly couple to the muon in a X i process (and thus should also contribute to secular (X2) relaxation.) Further experimental studies over a wider range of molecules including a study of the temperature dependence of these relaxation rates is called for in order to clarify some of the different possibilities discussed above. Nevertheless, principal results of this study are very important to studies of other systems (Mu + C O and M u + N 2 0 ) in this thesis: first, the indirect relaxation term A ^ is much larger than the other terms; second, electron S R interaction contributes significantly to this electron relaxation; third, the electron S R contribution calculated from the "Cur l " formula of Eq. 5.8 is consistent with the experimental results. 5.3 Mu + N 2 O : Addition/Decomposition and Collisional Relaxation The Mu + N 2 0 reaction was measured both in T F (at fields of 6, 8, 40 and 100 G) and L F ( in a range of 10 to 19200 G) on the M15 beam channel. The analogous reaction, H + N 2 O , is quite interesting in combustion chemistry [391]. For example, N 2 0 is an important intermediate formed during propellant combustion [392] and is 158 known to contribute to the depletion of stratospheric ozone [255]. The title reaction is also a key reaction in N2O flames and one of the few that can convert N2O into N2 thus avoiding production of undesirable nitrogen oxides in the atmosphere [393-395]. In addition, N 2 0 is an important greenhouse gas. The development of chemical kinetic models to control N2O formation is therefore highly desirable. To this end, it is important to understand the temperature and pressure dependence of these reactions so that appropriate rate constants can be included in combustion models. The H + N2O reaction has a very large activation barrier despite being highly exothermic, spin allowed, and symmetry allowed [396-398] and therefore is of fundamental interest. Many experiments measuring the reaction rates of H(D) + N2O have been conducted [174,391,399-422]. A l l of these were done at high temperatures (400 - 3000 K, some involved hot H / D atoms) and low pressures (mostly less than 1 atm). There are also several theoretical calculations of the rate constants for this reaction with different techniques [174,256,264,311,399,421]. There exists some disagreement on the mechanisms (dominate channel) of the overall reaction and thus the dependence of the rate on temperature and pressure. A t the relatively low pressures that have characterized the H(D) + N2O experiments to date, no pressure dependence has been observed. However, this is not conclusive since any pressure dependence would have been obscured by the small pressure ranges covered, especially in light of the tunneling effect which exhibits the opposite pressure dependence [311]. Furthermore, although significant isotope effects ascribed to tunneling effects were observed with H and D [174,399], these differ only by a factor of two in mass. A much greater effect can be expected for the M u + N2O reaction due to the ultra-light Mu-atom mass, which can also be measured up to very high pressures (up to 60 atm in this research, but experiments at much higher pressures, with an appropriate target vessel are planned), therefore providing an invaluable probe of the pressure dependence in H(Mu) + N2O kinetics. It can be noted that N 2 0 has no unpaired electrons, so a spin exchange interaction is not expected. Previous measurements of the M u + N2O reaction have been carried out in the liquid phase ( N 2 0 saturated in H 2 0 ) where, at room temperature, a kinetic isotope effect on the order of 1000 was reported in favor of the M u atom [423]. However, the mechanism is not all clear in this study, 159 which can be expected to be dominated by M U N 2 O formation, based on the results presented below. Formation of N2+MUO is also possible in solution. Moreover, the data is based on only one (indirectly determined) concentration point, casting doubt on the size of the reported K I E . Nevertheless, despite these uncertainties, the trend to a large K I E , indicating pronounced M u tunneling, seems reasonable and is in fact reproduced below. The present studies are the first of their kind in the gas phase and are carried out over a wide range of temperature and pressure, providing unique insight into the reaction mechanisms involved. 5.3.1 Results In 6 ~ 8 G T F , the reaction rates for M u + N 2 O reaction were measured at 303 K, 403 K, 496 K and 593 K, under total pressures from 2 atm to 60 atm. The results are listed in table 5.7 (and Appendix B)and plotted in Figs. 5.9 and 5.8. Since these data were taken at different run periods (over a span of two years) and with different target vessels and fxSR spectrometers, which resulted in different background relaxation rates (An), the values given here are weighted average for the two counters and are background-corrected (XT = XEXP — An, as discussed in the previous section). As outlined previously (see the experimental section), the N 2 O gas was freeze-pump-thawed at the beginning of each run period until the impurities (mainly O2 and NO) fell below a level at which no significant contributions from impurities to the relaxation rate could be observed by both repeated runs with the same N 2 O pressure before and after a F P T . This was also checked by the "3/4 effect" described in Chapter 3 and 4, based on the fact that impurity N O , O2 and N O 2 wil l relax the M u signal by spin exchange reactions, but N 2 O itself only undergoes chemical reactions with Mu [98]. The thermal decomposition of N 2 0 ( N 2 O + M -»• N 2 + O + M) is very slow over the temperature range concerned [424] and the experimental data showed that this reaction had no significance at the experimental conditions, both from the absence of any "3/4 effect" (Fig. 3.7) and reproducible rates measured over long time spans. It can also be emphasized that the results were reproduced very well with completely different setups (different target vessels, gas'bottles, and spectrometers). 160 Table 5.7: Transverse Field N 2 0 Results [N 2 0] Total Pressure Temperature XT (1020 molecule c m - 3 ) (atm) (K) (ps - 1 ) 0.4888 ± 0.0099 2.00 303 0.1075 ± 0.0087 0.976 ± 0.020 4.00 303 0.266 ± 0.013 1.688 ± 0.035 6.83 303 0.476 ± 0.031 1.691 ± 0.035 6.84 303 0.416 ± 0.032 1.691 ± 0.035 6.84 303 0.484 ± 0.024 1.695 ± 0.035 6.85 303 0.492 ± 0.021 1.712 ± 0.036 6.86 303 0.487 ± 0.017 2.333 ± 0.050 9.33 303 0.624 ± 0.033 2.449 ± 0.053 9.50 303 0.714 ± 0.022 3.872 ± 0.088 14.6 303 1.228 ± 0.041 4.039 ± 0.092 15.3 303 1.278 ± 0.063 a 4.039 ± 0.092 15.3 303 1.321 ± 0.059 5.65 ± 0.14 21.0 303 2.001 ± 0.061 5.96 ± 0.15 21.6 303 2.037 ± 0.058 8.75 ± 0.24 30.6 303 3.71 ± 0.14 12.46 ± 0.40 40.4 303 6.02 ± 0.26 17.76 ± 0.77 51.4 303 10.26 ± 0.35 0.2265 ± 0.0039 1.24 403 0.125 ± 0.016 0.4799 ± 0.0085 2.62 403 0.248 ± 0.016 0.720 ± 0.013 3.92 403 0.396 ± 0.028 0.921 ± 0.016 5.00 403 0.514 ± 0.024 1.339 ± 0.024 7.24 403 0.928 ± 0.036 2.289 ± 0.042 12.2 403 1.856 ± 0.11 0.1857 ± 0.0030 1.25 496 0.207 ± 0.016 0.3719 ± 0.0060 2.50 496 0.384 ± 0.016 0.785 ± 0.013 5.26 496 1.012 ± 0.039 1.033 ± 0.017 6.93 496 1.420 ± 0.042 1.317 ± 0.022 8.79 496 1.945 ± 0.099 1.530 ± 0.025 10.2 496 2.285 ± 0.076 2.006 ± 0.033 13.3 496 3.12 ± 0.19 0.2499 ± 0.0038 2.02 593 0.600 ± 0.026 0.6539 ± 0.0099 5.26 593 1.792 ± 0.069 0.824 ± 0.013 6.63 593 2.62 ± 0.13 1.049 ± 0.016 8.43 593 2.90 ± 0.14 1.418 ± 0.022 11.4 593 4.34 ± 0.29 1.958 ± 0.030 15.6 593 6.91 ± 0.59 2.087 ± 0.032 16.7 593 7.21 ± 0.65 a. Intermediate Magnetic fields (40 - 120 G) 161 Higher temperature (and some room temperature) runs were measured in the High Temperature target vessel and room temperature (especially high pressure) runs were measured in the High Pressure target vessel both described in Chapter 2. In both cases, the target vessels were temperature controlled by heating tapes and a temperature controller with thermocouple readings of the temperature at various locations of the target as described in Chapter 2. The source of uncertainty in [N2O] is mainly the uncertainty in measuring the pressure and the temperature. As mentioned in Chapter 2, temperature uncertainty is well below 3 K but considering the slow drifting over the runs, 3 K is used in data analysis as the upper l imit of absolute uncertainty in temperature. A n upper l imit of 1% error in pressure is also used. Another, perhaps not minor, source of error lies in the conversion of pressure to concentration. N2O deviates from an ideal gas at pressures higher than about 10 atm at room temperature (less so at higher temperatures) and the van der Waals equation was used to calculate the concentration from measured pressures. The values of the van der Waals parameters were taken from Ref. [425]. The concentrations so obtained were in fact essentially the same as those found from compressibility curves. A generous allowance for the combined error in the N2O concentration, [N 2 0] , from all sources is estimated to be ±5%. Measured relaxation rates at each temperature were first corrected for background since the background relaxation rates (Ao) at different pressures were different due to different stopping ranges, and then fit to the quadratic equation which is expected from the overall reaction mechanism, as discussed below; ki is the bimolecular rate constant for the decomposition reaction of Mu +N2O to form final products (N2 + MuO) , and &2 is the termolecular rate constant for the addition reaction to form the stabilized adduct (MuNNO), while c is used here to offset any uncertainties in the background subtraction. The rates at lower pressures in fact obey a linear equation: Results of these fits are listed in Table 5.8 and a typical plot is shown in F ig . 5.8 at 303K, up to 60 A T = * i [N 2 0 ] + * 2 [ N 2 0 ] 2 + c. (5.10) A t = fci[N20] + c. (5.11) 162 0 5 10 15 2 0 N 2 0 ( 1 0 2 0 m o l e c u l e s c c " 1 ) Figure 5.8: T F relaxation rates at different [N 2 0] and 303 K for M u + N 2 0 . The solid line is the fit of the data to Eq. 5.10. Backgrounds have been corrected. Most of the data were measured using the high pressure target vessel while the high temperature target vessel was used for some low pressure points. The reproducibility is very good. Table 5.8: Rate Constants of the M u + N 2 Q Reaction. T (K) * i ( 1 0 - 1 4 c m 3 m o l e c - 1 s _ 1 ) * i ( 1 0 _ 1 4 c m 3 m o l e c - 1 s - 1 ) *2 ( l O - ^ c n ^ m o l e c - V 1 ) 303 0.2817 ± 0.0094 0.2465 ± 0.0067 0.0187 ± 0.0010 403 0.546 ±0.012 0.431 ±0 .037 0.171 ± 0.030 496 1.046 ±0.036 0.981 ±0.059 0.337 ±0 .052 593 2.69 ± 0.12 2.310 ± 0.099 0.57 ± 0.11 atm N 2 pressures. The temperature dependence is shown in F ig . 5.9 at pressures up to ~ 15 atm while F ig. 5.10 plots low pressure data at these same temperatures, with fits to Eq. 5.11. Note the linear dependence, at pressures up to < 10 atm. Some moderator-dependence studies were also carried out. The third body effects were measured using N 2 and Ar gases. The results are listed in Table 5.9 and selectively plotted in Fig. 5.11. The collisional efficiencies (/?) appear to be ordered in the same way as the number of atoms in the moderator molecules, as expected (and discussed in chapter 4), i.e., /?N3O > /?N3 > PAT- A t 163 0 1 2 3 4 5 N 2 0 ( 1 0 2 0 m o l e c u l e s c c " 1 ) Figure 5.9: T F relaxation rates at different temperatures and [N 2 0] for M u + N 2 0 . The solid lines are fits of the data to Eq. 5.10. Backgrounds have been corrected. These data were measured with the high temperature target vessel except a few points at 300 K. High pressure data at 303 K are not included (see Fig. 5.8). The fit, however, includes all data points. 0.0 0 .5 1.0 1.5 2 .0 2 .5 N 2 0 ( 1 0 2 0 m o l e c u l e s c c " 1 ) Figure 5.10: T F relaxation rates at different temperatures and low [N 2 0] for M u + N 2 0 . The solid lines are fits of the data to Eq. 5.11. Note the linear dependence. The slopes are k[s. 164 0 3 6 9 12 15 [M] (1020 molec/cc) Figure 5.11: Pressure dependence at 300 K for Mu + N 2 0 reaction. N 2 0 concentrations are: 0.490 (squares, dash line), 1.93 (diamonds, solid line) and 3.71 (circles, dash-dot line) x 10 2 0 molecule c m - 3 . The asterisk is the pure N 2 0 (0.490) data point. This positive linear dependence was observed at all but the highest N 2 0 concentrations for N 2 moderator. For Ar , though, the pressure dependence was negative (cf. text and Table 5.9). 165 Table 5.9: M u + N2O Results, Moderator Dependence. [N 2 0] Total Pressure Temperature (102 0 molecule c m - 3 ) (atm) (K) (ps" 1 ) M = N 2 0.4888 ± 0.0099 10.0 303 0.1081 ± 0.0082 0.4888 ± 0.0099 30.0 303 0.145 ± 0.013 0.4888 ± 0.0099 60.0 303 0.200 ± 0.016 1.929 ± 0.041 15.0 303 0.638 ± 0.022 1.929 ± 0.041 30.0 303 0.706 ± 0.030 1.929 ± 0.041 60.0 303 0.885 ± 0.035 3.705 ± 0.067 29.25 303 1.419 ± 0.040 3.705 ± 0.067 59.52 303 1.922 ± 0.093 8.75 ± 0.24 40.41 303 4.95 ± 0.30 8.75 ± 0.24 59.86 303 4.29 ± 0.17 0.2187 ± 0.0038 12.24 403 0.1254 ± 0.0073 0.2499 ± 0.0038 12.9 593 1.056 ± 0.034 M = A r 1.929 ± 0.041 30.0 303 0.571 ± 0.021 1.929 ± 0.041 60.0 303 0.4551 ± 0.024 0.6539 ± 0.0099 9.57 593 2.144 ± 0.093 0.824 ± 0.013 9.26 593 3.61 ± 0.18 0.824 ± 0.013 12.0 593 3.38 ± 0.17 0.824 ± 0.013 15.0 593 3.09 ± 0.20 M = N 2 / A r 0.2499 ± 0.0038 13.0/10.0 593 0.855 ± 0.033 0.6539 ± 0.0099 15.0/9.57 593 2.372 ± 0.099 low total pressures (< 2 atm) no significant moderator pressure dependence was observed. At higher total pressures and room temperature, the observed relaxation rates increased with total pressures linearly (see Fig. 5.11), as expected. However, at the highest N2O concentrations and highest moderator pressures, rates decreased with increasing pressure. It also appears, from the data in Table 5.9, that A r "quenches" the relaxation rate much earlier and stronger. In L F , rates were measured only at room temperature. A t each pressure (mixture), rates were measured at many fields from 10 G up to 20 k G . The relaxation rates appear to contain more than one relaxing component, due to a combination of MuO and MUN2O, and possibly free M u relaxation as well. A n example is shown in the asymmetry plot in F ig. 5.12 where at least two relaxations are indicated. This situation could be expected from the results given in Chapter 4, case C . It can be remarked that the large relaxation rates seen in weak T F are the consequences and evidence of spin rotational relaxation in the product radicals, a point referred to in the 166 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 Time in microseconds Figure 5.12: Mu + N 2 0 Relaxation rates with 38.4 atm N 2 0 at 300 K in 96 G L F . The solid line is single relaxation exponential fit of the data to Eq. 3.34. The fit missed the data points clearly when it approaches the base line, indicating the existence of a second but slower relaxing component. A two exponential fitting function produces much better fits but, in general, the parameters are strongly correlated and similar \ 2 c a n be obtained for different combinations of the parameters since there are too many degrees of freedom. previous discussion. It can also be remarked that in a T F there is only one relaxing component, since any M u O (or M u N 2 0 ) formation wil l effect essentially instantaneous relaxation on the time scale of M u precession. This is not the case in a L F environment though which is a rather similar case to the Mu + C O reaction discussed later. The trend in the L F rates with pressure and magnetic field are, nevertheless, of interest and these can be obtained by a fit to a single exponential relaxation model (Eq. 2.3). The results so obtained are listed in Appendix B and are selectively plotted in F ig . 5.13. Clearly, the partial pressure dependence and the weak (or lack of) field dependence at low fields indicate that the depolarization are not completely due to radical relaxation. As wil l be shown in the next section, the low field rates are actually mostly due to chemical reaction rates which are functions of N 2 0 partial pressure but not field. Recall that the radical relaxation rate is only dependent on total pressure not partial pressure. The high field rates, on the other hand, show that the depolarization is quenched by high field which, as discussed earlier, cannot happen in a chemical reaction, indicating that radical relaxation does contribute significantly at high fields. A l l of these trends are consistent with the scheme introduced in Chapter 4, Case D and discussed in 167 CO 10' I I • 10" 10 -1 5 5 5 • 38.4/38.4 atm 0 7.7/38.4 atm X 2.0/38.4 atm I s5 *n11 10 - 2 10 -1 10" Fie ld (kG) 10' 10' Figure 5.13: M u + N 2 0 Relaxation rates at 38.4 atm total pressure in a L F . N 2 0 partial pressures are indicated in the legend. detail in Section 5.5. No detailed discussion of the L F fits, in terms of the model parameters introduced earlier, is given here, however, because a proper fitting routine has yet to be established to account for the multiple-exponential relaxation seen in these results. The development of new fitting procedures is underway. The following discussion relates primarily to the T F results where only one exponential decay due to chemical reaction is present. 5.3.2 Reaction Pathways of H (Mu ,D) + N 2 0 For the sake of simplicity, the H-atom wil l be used in the following discussion. The arguments can be equally applied to other H isotopes and in particular M u , of interest here, with some adjustment of the numbers, notably the heat of reaction. The reaction, H + N 2 0 , has the following four possible products, also shown in the potential energy diagram of F ig. 5.14, H + N 2 0 -!+ N2 + O H (5.12) H + N 2 0 A H N N O (5.13) H + N 2 0 N H + N O (5.14) 168 NNH+O NH+NO NNOH* 200 -150 -100 -50 -0 -- 5 0 --100 --150--200--250-Figure 5.14: Reaction coordinate for H + N 2 0 , adopted from Refs. [174,264,311,399] (cf. text). The Z P E is included [311]. O E C P l_ CD C U J H + N 2 0 N N H + O. (5.15) Fig. 5.14 suggests that the major channels, particularly at lower temperatures, would be N 2 + O H or H N N O , based on energetic grounds. There are several pathway considerations: • Pathway a: H + N 2 0 — • N 2 + O H This is highly exothermic overall, AH°9SK = —261kJ m o l - 1 , but also with high reaction barriers. There are two pathways for this reaction to proceed, a "direct" pathway and an "indirect" one as shown in Fig. 5.14. The direct pathway is the addition of an H atom to the oxygen end of N 2 0 to form the unstable N N O H * intermediate, which immediately dissociates to N 2 and O H . The calculated enthalpy barrier relative to the reactants for this direct process is 76 kJ m o l - 1 at 300 K. Note that this direct mechanism is distinct from simple abstraction. Rather than initial H attack at the O atom with simultaneous weakening of the N-0 bond, which would lead to a large pre-exponential factor (the " A " factor in the Arrhenius expression, Eq. 4.3 because of the loose TS) , the H atom first adds to the multiple N-0 bond followed by rearrangement to N N O H which falls apart [399]. This channel has no pressure dependence since N N O H is not a stable product. At temperatures above 1000K, 169 calculations show that this channel contributes significantly to the overall reaction rate but it does not account for the observed rate constant at lower temperatures [174,264]. The alternate indirect mechanism involves addition of H to the nitrogen end of N2O to form the H N N O * species (TS1) followed by a 1,3-hydrogen shift to make again the unstable N N O H intermediate (via TS2); the addition process has an enthalpy barrier of 38 kJ m o l - 1 at 300 K while the 1,3-hydrogen shift has an overall enthalpy of 64 kJ m o l - 1 at 300 K, both barriers calculated relative to N2O + H. At lower temperatures, the indirect pathway is favoured over the direct one since it is much easier to tunnel through the lower and/or narrower barriers of the indirect process. The H data agree much better with the indirect model than the direct model calculations below 1000 K [264,399]. The tunneling effect is dramatic because the stable intermediate (HNNO) precursor to the 1,3-hydrogen shift transition state gives rise to a large (but narrow) internal barrier of 126 kJ m o l - 1 relative to H N N O . This indirect channel is dependent on the total pressure because it is competing with the stabilization of H N N O . • Pathway b: H + N 2 0 ^ H N N O * A H N N O This pathway shares the same addition step as in pathway a, again giving the precursor to the 1,3-H shift above, but in this pathway the intermediate is stabilized by collisions to give the product H N N O and thus is expected to be strongly pressure-dependent. The addition is also exothermic, AH°QSK = —61kJ m o l - 1 , but this stabilization channel is not important at lower pressures and/or high temperatures. However, this reaction path is unusual in that the addition reaction to form H N N O has a higher effective A factor than the reverse unimolecular dissociation, which is a result of the low entropy of the reactant H-atom, combined with loss of a rotor in the dissociation [264,399]. Thus, the H N N O adduct can be expected to have a relatively long lifetime, leading to a higher probability of stabilization (and tunneling). Theoretical calculations have shown that this channel is important at even 200 torr total pressures [264,311], in contrast to earlier reports that this channel is not a major pathway based on observed pressure independence of the reaction [174]. Here it is 170 worthwhile recalling Fig. 5.8 for the M u + N 2 0 reaction, which clearly establishes the importance of pressure-dependent pathways in the overall mechanism. • Pathway c: H + N 2 0 — • N H + N 0 This reaction is highly endothermic with reaction enthalpy about 146 kJ m o l - 1 at 300 K, and thus is not an important contribution to thermal reaction rates. It can however be an important channel in "hot atom" reactions [255,256,264,421,426] • Pathway d: H + N 2 0 — • N N H + 0 This pathway is also highly endothermic with an enthalpy of reaction about 203 kJ m o l - 1 at 300 K. This channel, like pathway c, has no pressure dependence, but again would not be expected to contribute to thermal rates. Although the last two reaction pathways, especially path-c forming N H + N O , are important at very high temperatures [255,256,264,421,426-428], they are very unlikely at the temperature range of interest here and no further discussion wil l be given. The reaction M u + N 2 0 can be expected to proceed in a similar fashion with pathways a and b described by the following scheme: Mu + N 2 0 M u O N N * • M u O + Na (5.16) Mu + N 2 0 M u N N O * (5.17) M u N N O * M u O N N * • M u O + Na (5.18) M u N N O * + M M u N 2 0 + M (5.19) where the rate constants have the same meaning as the more general expression of Eq. 4.37, and /? is the efficiency of collisional stabilization in the "strong collision" model. As discussed before (see case C of Chapter 4, Eq. 4.37) and in Appendix A.2, both the steady state approximation and the eigenvalue solution give the same total thermal rate constant for the overall reaction 171 where ktotai is defined by — d ^ u ^ = «totaJ [N20][Mu]. It can be remarked that since the temperatures involved were well below 1000K, the direct pathway (kdi) is not likely to contribute significantly. Furthermore, the moderator dependence of the experimental data showed that the addition reaction is still in the low pressure regime (see F ig . 5.11). When k-a is much larger than /?fcs[M] + kdi (the low pressure l imit, indicated over the range of pressures run) and kdi is small (low temperature l imit, expected to be true over the temperature range run), the above expression reduces to ktotai = ki-rk2[M], (5.21) with *x = (5.22) k-a k2 = ^BL, (5.23) where ki is the overall bimolecular rate constant for MuO+N2 formation and k2 the termolecular rate constant for M u N N O stabilization, or, using the notation of Ref. [311], k\ = kdec a n d k2[M] = kadd- Note that the moderator " M " here can be either N2O itself or some added Moderator (N2 or A r in this study) and that Eq. 5.21 gives Eq . 5.10 for M = N 2 0 (pure N 2 0 runs), and ktotai is a "pseudo rate constant" . According to Ref. [311], kdec is not a true "constant" but one that decreases with total pressure, presumably due to competition with H N N O stabilization. It is also predicted in Ref. [311] that for H(D) systems with A r as moderator at pressures above a few 100 torr, the addition reaction is in the fall-off region and the rate constant (kadd) does not increase with pressure linearly as implied by Eqs. 5.21 and 5.23. However, the Mu data, in pure N 2 0 or N2 moderator (Fig. 5.9 and Fig. 5.11), do not show any deviation from a linear dependence except at the highest N 2 0 pressures.4 The linear N 2 moderator dependence seen in F ig . 5.11 can only come about if kdec does not decrease with total pressure and kadd increases linearly with moderator pressure. In other words, except at the highest pressures run (50 atm at 300k), the addition reaction is still in the low pressure regime. The linear moderator dependence also suggests that the 4 For pure N 2 O , a linear moderator dependence means a quadratic dependence on [NjO]. 172 stabilization is not competitive enough to significantly affect the "indirect" decomposition reactions, hence the pressure-independence of kdec In transverse field, one measures the disappearance rate of Mu . As noted, N 2 0 has no unpaired electron, there is no spin exchange interaction. The two possible M u containing reaction products (MuO and M u N 2 0 ) are both radicals and wil l undergo rapid spin relaxation, primarily due to the electron SR interaction. Recall that if there is no spin relaxation and no spin exchange, no Mu relaxation wil l be observed in a L F . The large relaxation rate observed in L F (see Figs. 5.12 and 5.13 and Appendix B) indicate that the product, whether M u O or M u N 2 0 (more likely), is indeed relaxing. It is well known that the O H radical is difficult to observe in liquid phase E S R due to its large spin rotation interaction [8,146]. It can also be noted that, in the addition step forming M u N 2 0 , the nitrogen nuclear and quadrupole moment could cause some dephasing in a T F (unlike the M u C O case discussed below), similar, for example, to the situation described earlier for M u C 2 H 4 formation with ethylene. However, such dephasing is unlikely to be a major effect since it could not contribute to relaxation in a L F , contrary to the results (Fig. 5.12 and 5.13). The formation and subsequent fast, essentially instantaneous, relaxation of MuO (or M u N 2 0 ) in a weak field, make it possible to measure the chemical reaction rate by T F / /SR. Recall Chapter 4, case C: using the notations therein, A,- = 0, A t = fctotaj[N20] so that X0b, simply equals A; t o t a i [N 20] or more accurately Ao6, = A 0 + (*i + * 2 [M]) [N 2 0] (5.24) where Ao is due to background relaxation as mentioned before. Again, this gives Eq . 5.10 for pure N 2 0 with the background (Ao) removed (as noted, the constant c is introduced there as well). At very low total pressures, /?/ts[M] <§; k^, the relaxation rates are first order with respect to N 2 0 concentration, as shown by the linear dependence of relaxation rate on [N 2 0] at low pressures in F ig. 5.10. In this case, fci » k2[M] and A 0 t , = ki [N 2 0] +A 0 (the notation k[ is used in Eq . 5.11 and Table 5.8 to avoid confusion with the linear term of the quadratic fit, which is slightly different than the linear fit. In principle, k[ > ki). These low-pressure rate constants at different 173 temperatures are most important in comparing with the experimental data of H(D) + N 2 0 by Marshall et al [174], measured at even lower pressures, < 1 atm, and all are plotted in F ig . 5.15. There are other important experimental points to note with respect to Eq . 5.24 and the data: i) At higher pressures, but not too high so that «_<, is still much larger than kdi + 0k, [M], the dependence on [N 2 0] becomes quadratic, as shown earlier in F ig. 5.8. ii) A t fixed [N 2 0] , X0b> depends linearly on [N2] (Fig. 5.11), consistent with Eq . 5.24. iii) However, with high pressure A r as moderator (or N 2 moderator but at very high [N 2 0] as well), ktotai actually decreased with increasing moderator pressure (see Appendix B) . This effect is puzzling and not understood. Even at the highest pressure, at the high-pressure limit, the rate constant should stay constant with pressure, not decrease! This could be an experimental artifact which requires additional data to either dismiss or confirm. It will not be discussed further in this thesis. 5.3.3 Comparison of Reaction Rate Constants Among the many experimental studies of the kinetics of H(D) -f N 2 0 reaction, the result of Marshall et al. [174,399] is the most recent and the most relevant. It is one of the few that has covered lower temperature ranges (390 - 1310 K) and the only isotope-effect study under thermal conditions similar to the /iSR experiments conducted here. It also offered some theoretical calculations and considerations. They employed a high-temperature photochemistry technique in which time-resolved resonance fluorescence spectroscopy was used to monitor the reduction in concentration of H (D) generated by flash photolysis of NH3 (ND3) in a reactor containing mixtures of N 2 0 and Ar . The total pressure was between 55 to 430 torr (Ar as moderator). They found that the overall reaction rate constants were pressure-independent and given by the empirical equations: kH = 4.2 x l 0 - 1 4 e - 2 2 9 ° / T + 3.7 x 1 0 " 1 V 8 4 3 0 ' V m o l e c u l e - V 1 , (5.25) 174 and * H = 3.5 x i o - 1 3 e - 3 6 0 ° / T + 5.3 x 1 0 - 1 0 e - 9 1 7 0 / T c m 3 m o l e c u l e - 1 s - 1 (5.26) over the temperature ranges 410-1230 and 450-1210 K, respectively. These results are plotted in Fig. 5.15 (200 torr). They also carried out B A C - M P 4 calculations (see Chapter 4) using a model involving rearrangement of an H N N O intermediate coupled with tunneling through an Eckart potential barrier [330]. The distinct curvature of the Arrhenius plot (see Fig. 5.15, 200 torr) at lower temperatures was attributed to the effect of quantum-mechanical tunneling due to the H-atom (1,3) migration process (pathway a above), following intial addition to the thermal N atom. They ruled out channel (d) completely and argued that channel (b) was not a major pathway either. They also neglected any tunneling through the first barrier in the indirect pathway. However, two recent theoretical calculations indicated that the addition channel is important, even dominant, at lower temperatures over the pressure range of their experiment [264,311]. The strong pressure dependence of the Mu data at all temperatures (Figs. 5.8, 5.9 and 5.11) clearly supports the theoretical predictions that the addition channel plays an important role in the overall reaction kinetics and that the "indirect" pathway is more dominant at lower temperatures. To compare with the H(D) data, which were obtained at much lower pressures, one could use the value of k[ obtained from Eq. 5.11 but it is more meaningful to calculate ktotai from Eq. 5.21 with [M] being the value of that used in the H(D) experiment. This comparison is shown in F ig. 5.15, where the temperature dependence of the total rate constants at different temperatures and pressures are plotted along with H(D) rate constants [174,311] for comparison. What is plotted at 200 torr are the experimental results from Ref. [174], but the theoretical calculation of Ref. [311] also gave a very good account of these data. The higher pressures indicated in F ig. 5.15 are all from these calculations (the data does not extend beyond 430 torr). Strong curvature of the Arrhenius plots is seen in the temperature range of the data for all three isotopes, particularly at low pressures and particularly for the M u data. At pressure less than 1 atm, the decomposition channel dominates. At higher pressures, there are no experimental H data 175 I CO u o E >o E o 10 10' -12 -13 10' - H 10 -15 10 -16 V 1 r \ V \, \ V \. V \ V T 1 200 torr i \ V 0.5 1.0 1.5 2.0 2.5 3.0 3.5 1000/T (K"1) i co V 3 O 0) o E to E o 1 10 10 -12 10 -14 10" 10 -18 ^—i 1 1 r V V \ 1 atm j L 0.5 1.0 1.5 2.0 2.5 3.0 3.5 1000/T (K"1) i (0 v 3 O _0) o E KI E o P 10 10 10 -13 10 10" -15 1—~I 1 1 10 atm \ • j • \ ' ' 0.5 1.0 1.5 2.0 2.5 3.0 3.5 1000/T (K_1) i CO T JD . 3 O o £ to E o •B 10 10 -12 -13 10 10' -15 10 i r 50 atm \ . \ ' , i i i i v! 0.5 1.0 1.5 2.0 2.5 3.0 3.5 1000/T (K"1) Figure 5.15: Arrhenius plot of total rate constant for M u -f N 2 0 reaction. Total ( = N 2 0 ) pressure is indicated on the top-right corner of each plot. The dot and dash lines are H and D data [174] (for the 200 torr plot) or calculations [311] (for all the higher pressures where experimental data are not available), respectively. The solid lines are not fits but are simply to guide the eye. Note that the calculation shown in the 50 atm plot is the high pressure limit (see also F ig . 5.16) while those for 1 and 10 atm plots are calculated for corresponding pressures. It should also be pointed out that the H-atom calculations are done for an A r moderator. 176 but theoretical calculations indicate that both addition and decomposition contribute [264,311]. The total reaction rates for M u + N 2 0 are much higher than the corresponding H atom reaction rates at all pressures. At the lowest pressure, the KIE(Mu,H) = 120 and 20 at 303 K and 593 K respectively, the largest seen in comparative studies of M u and H in the gas phase above room temperature. It can be noted that the KIE(H,D) is only about 2.3 at 450 K. The aforementioned KIE(Mu,H) reported in liquid water is on the order of 1000 at 300 K, almost an order of magnitude higher than seen here in gas phase, indicating a large uncertainty in the liquid phase data [423]. The overall Mu + N 2 0 reaction rate constant is much higher than that of the H(D) reaction systems at all pressures (Fig. 5.15). This dramatic isotope effect cannot be explained at all by Z P E effects, which are most important for endothermic reactions ("tight" TS) and any way lead to an "inverse" K I E , as seen for example in Mu + H 2 [122] and M u + C H 4 [125,126]. This has been outlined earlier in Chapter 4. According to C T S T , the kinetics of O H (MuO) formation is controlled by the second, higher barrier at TS2 (see reaction coordinate diagram Fig 5.14), and any earlier wells or lower barriers are irrelevant. Under these assumptions, from Eq . 4.26 (ignoring the transmission coefficients for the moment), the kinetic isotope effect is KIE(1,2) = 4 - eto-'M**. (5.27) At lower temperatures where the difference in Z P E of the transition state overcomes any effect from the translational partition functions (translational isotope effects), the heavier atom could always be favored due to its lower Z P E , but, as noted, this cannot account for the isotope effect observed here. Such a huge effect has to come from the ratio of the transmission coefficients (r in Eq. 4.10) and can be explained by pronounced tunneling through both barriers for M u O formation and tunneling through the first barrier for M u N N O stabilization. This tunneling effect can also explain the strong pressure dependence observed in the the Mu system. A t low pressures, where the decomposition channel dominates, the upward curvature at low temperature is characteristic of tunneling since at these temperatures the tunneling effect is always more prominent, which can be seen as well from the Tolman definition of the activation energy in Eq. 4.9 and its relationship to 177 the potential barrier in Eq. 4.1. It is important to note that a difference between the current M u study and the previous H(D) experimental studies is that in the current work the contributions from addition and decomposition can be distinguished by the unprecedented pressure range investigated, so that there is little ambiguity as to which channel is contributing to the total rate constant. Theories [264,311] have predicted that addition is very important even at 200 torr for low temperature H(D) systems and that the curvature in the Arrhenius plot of the H(D) experimental data (Fig. 5.15) is mainly due to the change of dominant mechanism from decomposition to addition. Data from this thesis establish that the addition channel is surely not the reason for the curvature at low temperatures for M u , as suggested in Refs. [264,311] for H(D) reactions, since the addition reaction is negligible at pressures less than a few atm for all temperatures. This can be seen in F ig. 5.17, which gives the separate Arrhenius plots for ki = kdec and fc2[M] = kadd (from Eqs. 5.22 and 5.23). It is only at high pressures, > 10 atm, that the addition channel becomes important (seen also in the curvatures in Figs. 5.8 and 5.9). Although this channel is also subject to tunneling, the enormous temperature dependence of the dissociation channel (k-a) in the addition reaction causes the Arrhenius plot to curve downward at higher temperatures [311], as can be clearly seen in F ig. 5.16 which plots the Arrhenius dependence of kadd for different pressures. The H(D)-atom data is compared again as well. Note that at the higher temperatures, and particularly lower pressures, the isotope effect is reversed for the addition channel. This is a particularly interesting dynamical mass effect and is reported here in the M u case for the first time. This downward curvature of the addition rate constant cancels the tunneling effect and makes the total rate constant almost linear at higher pressures (Fig. 5.15, 50 atm plot). The overall large isotope effect is a result of the much enhanced tunneling in the M u + N 2 0 reaction compared to H(D) + N 2 0 . It is important to note that tunneling through both barriers of the indirect pathway are contributing, primarily via ka and fcj2 in the definition of k\. A t high temperatures, the isotope effect is much smaller since the tunneling is much less important there. At higher pressures, isotope effect due to tunneling is also less important because of the increased 178 Table 5.10: Activation Energy for Mu(H,D) + N 2 0 Ea (kJ/mol) Total Pressure M u H " D a (atm) 300-400 K 500-600 K <500 K >700 K <500 K >700 K 0.263 5.8 21 19 70 30 76 1.0 6.2 21 10 9.5 18 30 13 14 50 14 14 a From Ref. [174]. stabilization probability. It is noted, in the high pressure l imit ([M] —> oo), ktotai = ka, from Eq. 5.20. This is shown in the 50 atm plots for the H(D) calculation in Figs. 5.15 and 5.16. Since the Arrhenius plots are clearly curved (Fig. 5.15), it is not possible to fit to a simple activation energy. Thus, fitting the two higher temperature points and the two lower temperature points of the total rate constants respectively to Eq. 4.3 at a given total pressure, activation energies for the two temperature ranges can be found and these are listed in Table 5.10. The experimentally measured activation energies (calculated from Eqs. 5.25 and 5.26) of the corresponding H(D) + N 2 0 reactions at low pressures are also listed in Table 5.10 [174]. The activation energies for M u + N 2 0 are much smaller than those for H(D) + N 2 0 over both temperature ranges. A t lower pressures, the Ea's at lower temperatures in all three reactions are much smaller than those at higher temperatures and the relative magnitudes of temperature dependence of Ea increase with decreasing mass. A t higher pressures, however, the temperature dependence is much smaller as can be seen in the M u data (Table 5.10). A l l of these trends are consistent with the fact that the tunneling effect is more important at lower pressures and lower temperatures for lighter isotopes, a trend noted in Ref. [311] as well. It is also important to note that the much reduced B 0 ( M u ) in comparison with ^ ( H ) is an immediate indication of the dominance of quantum tunneling in the M u reaction, as outlined earlier and seen as well in Eqs. 4.9 and 4.1, where Ea < Vj| corresponding to (E*) = (E). The small difference between Ea(H) and Ea(D) at high temperature is also an indication that Z P E shifts are not so strong, as expected for an earlier barrier. 179 10 -14 10 o I 10 -16 E o •o o 10 10 -17 -18 i r "i 1 200 torr S \ i i i 0.5 1.0 1.5 2.0 2.5 3.0 3.5 1000/T ( K _ 1 ) 10 -14 10 -15 10 -16 10 -17 10 -18 n 1 1 1 r 1 atm \ A I I I I L 0.5 1.0 1.5 2.0 2.5 3.0 3.5 1000/T ( K _ 1 ) 10 -13 -14 2 1 cu 3 10" 1 5 JD O E o 10 E o -16 ! 10 -17 10 -18 i 1 1 r 10 atm V \ *. \ \ \ J I ' 0.5 1.0 1.5 2.0 2.5 3.0 3.5 1000/T ( K " 1 ) 0.5 1.0 1.5 2.0 2.5 3.0 3.5 1000/T ( K - 1 ) Figure 5.16: Arrhenius plot of the addition rate constant for the Mu -I- N 2 0 reaction, compared with the total rate constants for H, D. Total pressure is indicated on the top-right corner of each plot. The squares are M u data. The lines are not fits but are simply to guide the eye. The triangles (dotted lines) and diamonds (dashed lines) are calculations for the H and D reactions [311], respectively. Note that the calculation shown in the 50 atm plot is again the high pressure l imit but all others are calculated for the corresponding pressures. It should also be pointed out that the calculations for H(D) are done for A r moderator and the M u experiments plotted here used pure N 2 0 (unmoderated). 180 , - 1 3 w D U V o E 10 E u 1.5 1.9 2.3 2.7 3.1 3.5 1000/T (K _ 1 ) £ o 10 10 10 -16 -17 J I L 1.5 1.9 2.3 2.7 3.1 3.5 1000/T (K _ 1 ) I CO J) o _g> o E 10 E u 1.9 2.3 2.7 3.1 1000/T (K" 1) 3.5 i 0) 10 -is 10 -2 10' £ 3 10 -16 10 -17 1 1 1 " S^lfc-^ 50 1 atm ^ • —B -- X -i i i 1.5 1.9 2.3 2.7 3.1 1000/T ( K - 1 ) 3.5 Figure 5.17: Comparison of contributions from addition and decomposition channels to the total rate constant for the M u -f N 2 0 reaction only. Total pressure is indicated on the top-right corner of each plot. The symbols are as shown in the legend of the 50 atm plot. 181 Note that the curvatures in the Arrhenius plots of both the addition and decomposition reactions of the Mu system are consistent with the trend in the theoretical calculations (Figs. 5.16 and 5.17). Namely, the addition reaction curves downward and the decomposition upward; but the ratio of kdec/kadd did not increase as quickly with temperature as calculated in Ref. [311] (see Fig. 5.17 and Table 5.8). In ascertaining all of the above mentioned "discrepancies" between the theory [264,311] and the M u experimental data are based on specific calculations of the H(D) + N 2 0 reactions and are likely the result of a much enhanced tunneling over the temperature range of the measurements. There are no specific calculations for M u + N 2 0 at present but indications are that the theoretical calculations may have underestimated tunneling effect. Another explanation that needs to be considered is the possibility that the direct pathway may contribute more than expected for M u systems and that this contribution does not depend on the total pressure in the same way as the indirect one does. Answers to these questions as well as definite explanation of the K IE ' s seen in comparison of Mu(H) + N 2 0 wil l only be forthcoming after theoretical calculations of the Mu + N 2 0 kinetics are carried out. 5.3.4 Summary Rate constants for the M u + N 2 0 reaction, separately determined for both addition and decomposition channels have been measured over a range of temperatures (300 ~ 593 K) and pressures (6.5 ~ 60 atm) in T F / /SR experiments. This is the first experimental study of the important (H-isotope + N 2 0 ) reaction system over such a wide range of pressure. Comparing with the H(D) reaction system, pronounced kinetic isotope effects are evident and their pressure and temperature dependences are, qualitatively, consistent with trends established in theoretical predictions for H(D) + N 2 0 [264,311]. Although the addition reaction forming M u N N O is important at high pressures, as predicted by theory, the decomposition channel dominates at pressures below a few atm at all temperatures, contrary to these same calculations for H(D) [311] which predicts that addition channel dominates at low temperatures. Pronounced quantum 182 tunneling is evident in the upward curvature seen in the Arrhenius plot at lower temperatures and particularly low pressures (Fig. 5.15) which is much more dramatic than that seen in the corresponding H + N 2 0 studies, indicating that the tunneling effect is much more important than the stabilization of M u N N O adduct. Nevertheless, high pressure M u data provides the first experimental confirmation of the theoretical prediction that the formation of H ( D ) N 2 0 can be important at lower temperatures in the H(D) + N 2 0 reactions, where one would expect that tunneling is much less important (compared to Mu) and the addition channel wil l be more dominant than in the M u + N 2 0 reaction. The data also show (Fig. 5.11) that the addition channel is, as intial calculations showed for H(D) [399], in the low pressure regime over the pressure range of the M u experiment. Though reasonable extrapolations of the theoretical calculations for H(D) + N 2 0 can be made for the present M u data, a confident assessment of the dramatic K IE 's established in this thesis must await specific calculation of the M u + N 2 0 reaction dynamics. Longitudinal field data were measured as well and it is evident that spin relaxation due to collisional interactions of a radical species (MuO or M u N 2 0 ) is large. However these data cannot be properly analyzed in the absence of a model which can correctly distinguish the relaxation rates due to chemical reaction and spin relaxation between these two possible radical channels. Such a fitting procedure is currently under development but is not included in this thesis. When available, further constraints on the measured values of ki and fc2 a n d hence on the kinetics mechanisms wil l be possible. 5.4 Mu 4- NO: Addition and Spin Exchange The thermal (re)combination reaction of H atoms with N O , involving the unstable intermediate H N O * , is important in the field of combustion kinetics [429-431] and can be thought of as well as a type of radical-radical reaction [165,170,282,315]. It has been extensively investigated over a range of pressures and the kinetic data for thermal addition has been reviewed by Tsang and Herron [432]. This reaction has also been studied at much higher energies, where the abstraction reaction becomes important [246,433], but such an energy regime is not of interest here. 183 The mechanism of thermal kinetics for the H + N O combination/addition reaction is well established and has the form k. k. H + N O + M # H N O * + M - • H N O + M , (5.28) k_. where ka, k-a and k, are rate constants for addition, (unimolecular) dissociation and stabilization with moderator " M " , respectively. Here it is the formation of H N O * by H-atom addition to the N-atom in N O that is important, not the formation of N O H * . The latter has a large activation barrier, ~ 54 kJ /mo l , in contrast to the zero (electronic) barrier for H N O * formation[168,190,434]. The kinetic scheme in reaction (5.28) can be described in terms of an effective overall rate constant, as shown earlier in Chapter 4, Eq. 4.17, which is rewritten here, **' = E7OT <5'29> which exhibits the usual limits: at low pressures, when k-a 3> &»[M], «e//,o = ^o[M], where ko = k a k , / k - a is the low pressure termolecular rate constant; at high pressures, when re_0 •etc «»[M], the high pressure l imit koa = kejj><x> = ka is reached. Obtaining experimental values in recombination reactions for both of these limits, as well as for the "falloff" region in between, is important both as a test of reaction theory and to give a better understanding of the nature of the underlying P E S , for H + N O , particularly in the region of the transition state [168,190,246,434]. Most of the thermal data to date for H(D) + N O have been obtained at relatively low total pressures (< 2 atm), in the termolecular regime [430,432]. The early work of Hartley and Thrush [431] is representative and relevant here as well, since these authors have measured the recombination rates of both H and D-atom with N O . There appears to be only one brief report of a direct measurement of the high-pressure-limiting rate constant, from measurements carried out at pressures up to 1200 atm [435]. The onset of the high pressure regime is reported to occur at ~ 400 atm (with M = N 2 ) with a transition pressure beginning at ~ 60 atm. Pressures of this order are indeed indicated in order to achieve a direct measurement of koQ, judging from data obtained for a variety of related reaction systems [168,323,436,437] Establishing this l imit by experiments is 184 important, since it can be accurately calculated, for example from (variational) Transition State Theory (VTST) [41,170,180,438] or the Statistical Adiabatic Channel Model (SACM) [168,282,309,323], thereby helping to determine the value of Aro in the low pressure regime. The direct experimental measurement of koo for H + N O by Forte is seemingly in quite poor agreement with recent theoretical calculations of this same reaction [168]. There have only been a few reports to date of isotope effects comparing H and D-atom reactivity in the study of recombination reactions. Of particular relevance here is the aforementioned work of Hartley and Thrush in the termolecular regime for reaction (5.28), with which comparisons of the present M u + N O study can be made directly, but also of interest is the work of Pi l l ing et al. of H(D) + CH3 recombination [165,439]. Closely related to (re)combination, are addition reactions to unsaturated bonds, exemplified by H(D and Mu) + C 2 H 4 (and C 2 D 4 ) [32,41,177]. In this particular case, on an early-barrier P E S , the high pressure l imit is attained at ~ 1 atm and a K I E of k^/kQ ~ 100 is indicated at the lowest temperatures (~ 100K), due to the dominance of M u tunnelling; whereas at the highest temperatures (~ 500K), this ratio approaches the classical result due simply to the difference in mean velocities, o^oV^oo ~ 3. In comparing just H and D-atom reactivity with C 2 H 4 , however, the classical ratio ^OATO = 1-4 w a s observed at all temperatures (~ 100-500K), as is generally the case seen in a variety of H(D)-atom addition reactions in the high pressure l imit [182,440]. In contrast to the situation for combination/addition reactions in the high pressure regime, there are few measurements of K IE 's at low pressures. In the reaction under study here, M u + N O , comparisons with corresponding H and Diatom combination results in the termolecular regime [430-432] is the first of its kind and represents then an important contribution to the field in the interpretation of K IE 's in unimolecular dissociation, as reflected in differing values for the l imiting rate constant ko • Results are reported for the reaction kinetics of M u + N O with up to 60 atm moderator pressure, at room temperature. In addition reactions of this nature, K IE 's are well known to be both pressure and temperature dependent; in particular, changing from "inverse" to "normal" with increasing pressure [315,317,441]. Thus, for example, in the H(D) + N O study of 185 interest here, a small inverse isotope effect, k$/k® ~ 0.9, has been reported at low pressures [431] whereas in the case of H(D) + CH3 combination, in the high pressure regime, kH/k® ~ 4 is reported [165,439,442]. This latter result is in fact not well understood, providing an important and vexing test of T S T [165,170,192,282,439] which has prompted interest to measure the kinetics of M u + C H 3 at the T R I U M F cyclotron. Though there is not really any fundamental reason why (for example, see the work of Troe and coworkers [436]) generally most kinetics experiments are not carried out over wide pressure ranges. The / iSR technique, with which the reaction kinetics of the muonium isotopic analog of reaction (5.28) are studied, has the distinct advantage that, in principle, it can be carried out at any (high) pressure, the only l imiting factor being the thickness of the entrance window for the muon beam. It also has the advantage over most established H-atom techniques of having neither self-reaction nor wall reactions as mentioned before. In analogy with reaction (5.28), and also consistent with previous discussion of the M u -I- N2O reaction, assuming a strong collision (/? = 1) mechanism, the association reaction of M u with N O can be treated in a simplified way in terms of two-body steps[298], as: M u + N O -+ M u N O * , k-a M u N O * -J - M u + N O , k, MuNO* + M -»• M u N O + M where M represents an added inert gas moderator, [MuNO]* a vibrationally-excited unstable molecule, which will re-dissociate in the absence of collision with M , and M u N O is the stable product. As shown in Chapter 4, assuming steady state kinetics (d[MuNO*]/di = 0), the overall rate is given by -d[Mu]/<ft = fce//[Mu][NO], (5.33) where kejj is given by Eq. (4.17). The kinetics are then termolecular at low pressures, with k0 = kaks/k-a, but become bimolecular at higher pressures, koo = ka, with an overlap of both processes in the intermediate falloff region. The K I E , (kMu/kH), can then be viewed from the 186 (5.30) (5.31) (5.32) perspective of these two limits. In the low pressure regime, the rate d[Mu]/dt and hence the rate constant ko can be thought of as proportional to the equilibrium constant (K — ka/k^a) for the formation of Mu(H)NO* and to the stabilization rate constant (ks) of the complex. Both factors should be smaller for the M u reaction, due to the higher vibrational frequencies and lower density of states of the [MuNO]* complex, so an inverse kinetic isotope effect with kg1" < k^ can be expected. It is noted that quantum tunneling should have no effect on K, if it in fact represents a true equilibrium constant. In the high pressure l imit of Eq. (5.33), M u N O formation depends only upon the rate constant for the addition process (koo). Since this is an exothermic reaction but on an essentially zero (electronic) barrier P E S [190,434] the K I E should approach, at only moderately high temperatures, > 300K), the classical velocity dependence, given simply by the difference in reduced masses between M u and H, k^/k^ ~ 3 [26,32,122,441] A t lower temperatures, quantum tunneling via vibrationally-enhanced reaction barriers could be important, as discussed recently by Cobos for H + N O [168], with the result that k^u > k^ may be expected, possibly even at room temperature. It is relevant to note in the case of the aforementioned M u addition to C 2 H 4 , which actually has a small (~ 8 kJ/mol) electronic barrier, that the classical kinetic isotope effect of ^ u / & a = 2.8 was found at the highest temperature studied, near 500 K, but this increases to ~ 5 at 300 K due to tunneling contributions [32]. 5.4.1 Results In the / iSR experiment of M u + N O , collisions with N O contribute to the loss of M u signal by two competing processes: first, direct spin exchange,5 k s p Mu(t) + NO(4.) -)• Mu(4.) + NO( t ) , (5.34) 5 It has been incorrectly assumed in the past that ground-state NO, with its single unpaired electron, was naturally paramagnetic, but in fact it is (well-known) diamagnetic in its ground state, due to a cancellation of orbital and spin magnetic moments [146], with the first excited-sate being paramagnetic. However, this does not change the fact that spin exchange interaction happens and the reported values for the cross section, since it is the unpaired electron that is important in spin exchange, not the paramagnetism (or diamagnetism) of the molecule (in contrast, for example, to magnetic susceptibility measurements). 187 causes spin relaxation due to a modulation of the p+ — e hyperfine interaction[56,59,443]; second, the combination reaction, Mu + N O + M -)• M u N O + M , (5.35) reduces the M u amplitude by forming diamagnetic M u N O at random times, a process which becomes more important at higher pressures. In the present case it is important to realize that the M u N O adduct formed from M u + N O combination is diamagnetic, in contrast to most M u addition-type reactions which give rise to free radical environments, eg., MuC 2H*,[32,231,232], where a competition between dissociation and spin relaxation ensues [41,232]. If the latter process is fast enough, Mu regenerated by dissociation could be completely depolarized [32,41], much as if it had undergone a spin-flip encounter, as in Eq. (5.34).6 In the present case, in a L F , muon T i relaxation of gas phase M u N O can be 10's of ps, as noted in Chapter 3, so that given the expected ~ 1-10 ps lifetime of M u N O * , which is justified below, there can be no mechanism for spin relaxation other than the competitive process of electron spin flip, represented by Eq. (5.34). For each mixture of N O and N 2 (A r ) , at least four L F measurements from 2.5 to 15 k G were performed and the results (A/,) were plotted as a function of 1/(1 + X 2 ) in order to separate the relaxation rates due to chemical reaction and spin exchange, from Eq. (3.34). Two representative data sets are presented in F ig . 5.18, for [NO] = 1.07 x l O 1 7 mo lecu lecm - 3 at 20 and 60 atm pressure, to show the effect of total pressure, which can be seen in the differing intercepts. Alternatively, at fixed total pressure, the N O concentration can be varied as in Fig. 5.19. Here both the intercepts and the slopes differ, the latter due to differing rates for spin exchange. 5.4.2 Combination Reaction The chemical reaction rate constants for reaction (5.35), A c , are obtained from fits of Eq . (3.34) to the field(X)-dependence of the data, given by the intercepts as in Fig. 5.18 and 5.19 and are listed in Table 5.11 as a function of N O concentration at different total (moderator) pressures. Most of the data was taken with N 2 moderator but a few points were also taken with Ar , at 20 atm 6 See the result on Mu + CO reaction. 188 o.o 1 1 1 1 1 O . O O 0 . 1 0 0 . 2 0 0 . 3 0 0 . 4 - 0 1 / O - t - x 2 ) Figure 5.18: Longitudinal relaxation rate (AL) VS. 1/(1 + X2) for [NO] =1.07 x 10 1 7 mo lecu lecm - 3 at 20 (squares) and 60 (circles) atm N 2 moderator pressure. The solid (straight) lines are fits of Eq. (3.34) to the data. Note the difference in y-intercepts (A C ) but similar slopes (ASF) for these two cases. 1 8 . 0 O . O O 0 . 3 0 0 . 6 0 0 . 9 0 1 . 2 0 V O - r - x 2 ) Figure 5.19: Longitudinal relaxation rate ( A / J vs. 1/(1 + X2) for [NO]=0.086 (triangles), 0.35 (squares), 0.71 (circles) and 1.07 (diamonds) x l O 1 7 mo lecu lecm - 3 at 40 atm N 2 moderator pressure. Note that both the intercepts and slopes increase with [NO]. The intercepts give directly the chemical addition rate, A C . 189 pressure. For completeness, Table 5.11 also lists an earlier set of data points taken at much lower L F strengths[55]; note the concomitantly larger errors. Those data were in fact not included in the present analysis. To correctly extract the addition rates, effects of impurities in the gases must be considered. As noted in Chapter 2, the only impurity of concern here is NO2 because of its fast chemical reaction rate with the M u atom [217]At the highest N O concentration used, the upper l imit of [NO2] is 1.8 x 10 1 4 molecules c m - 3 which, in conjunction with the rate constant given in Ref. [217] for MU+ N O 2 , could, at most, give a chemical relaxation rate of 0.085 / i s - 1 [217]a contribution which would be a function of [NO] but independent of [M]. Though this effect, conceivably, could be as much as 8% of the relaxation rate measured, it can easily be accounted for in the data analysis, as discussed below. Any O2 impurity present in the moderator gas could also be of some concern since the termolecular reaction 2NO + O2 — > 2NO2 is fast enough (rc298 = 2.0 x 1 0 - 3 8 c m 6 mo lecu le - 2 s - 1 [432]) to generate some NO2, particularly at the highest pressures run. However, any relaxation rate due to N 0 2 produced in this way would be dependent on the moderator pressure, [N 2], but independent of [NO] and would show up as a non-zero y-intercept in plots of A c (from Eq. 3.34) such as Fig. 5.20 below. In fact A c goes through the origin at all pressures, indicating that any NO2 formed from 0 2 or O2 itself had no effect on the measured chemical relaxation rates. Impurity NO2 and O2 from any source can also undergo fast intrinsic SE relaxation with Mu , but not at a rate appreciably different than with N O itself [59,90,217] which is present in much greater amounts. Again, plots similar to F ig. 5.20 but for A s e also showed zero-intercept at all pressures. Thus we conclude that impurities in the moderator gas are not a problem. Accordingly, since impurities either have a negligible effect or can be corrected for, the N O concentrations are taken initially "as given". According to the gas-filling procedures described in Chapter 2, the overall uncertainties in total pressures are less than 1%. Considering the filling procedure, the uncertainty of [NO] in the target vessel is estimated to be less than 2% but an uncertainty of 5% was actually used in the fits to account for any possible change in [NO] during the experiment. A l l data were obtained at ambient temperature (~ 298 K ) . 190 Table 5.11: Chemical Relaxation Rates of Mu + N O at 298 K. [N0] a Total Pressure A c (1017 molecule cm" 3 ) (atm) ( p s - 1 ) M = N 2 0.351 60 0.532 ± 0.040 0.702 60 0.978 ±0.085 1.067 60 1.58 ±0.13 0.086 58 0.181 ±0.050 0.631 58 0.99 ± 0 . 2 7 6 0.963 58 1.42 ± 0 . 3 6 6 0.086 40 0.108 ±0.047 0.351 40 0.334 ±0.024 0.710 40 0.716 ±0.065 1.067 40 1.115 ±0.083 0.086 20 0.070 ±0.040 0.351 20 0.197 ±0.029 0.710 20 0.358 ± 0.040 1.068 20 0.635 ±0.053 0.086 5 0.062 ±0.035 0.738 5 0.147 ±0.035 1.634 5 0.443 ± 0.067 2.142 5 0.494 ±0.067 3.614 5 1.13 ±0.15 M = A r 0.7114 20 1.049 ±0.051 1.065 20 3.11 ±0.15 1.431 20 1.95 ±0.12 a. Uncertainties estimated to be < 2%, but 5% actually used in the fits to give the error limits in column 3. b. The previously published data [55]. Not included in fitting. Allowing for possible impurities in the added N O , as described above for NO2, A c is related to the total chemical reaction rate constant, A;c, by A c = k c[NO] = (k 0[M] + k i m p ) [NO] , (5.36) where fc,mp is a bimolecular rate constant (or sum of rate constants) for an impurity, weighted by its percent amount in the N O . In Eq. (5.36), the low-pressure (termolecular) l imit is used for ke/f in the Mu + N O combination reaction, which is justified below. Fig. 5.20 gives a typical plot of kc vs. [NO], at 40 atm N 2 . The solid line is a fit of Eq. (5.36) to the points, taken from Table 5.11. The quality of the fit is excellent, confirming that Figure 5.20: Longitudinal relaxation rate (A c) vs. [NO] for 40 atm N 2 moderator pressure. See also Fig. 5.19 (intercepts). The slope gives the total chemical reaction rate constant, kc — (1.00±0.05) x 1 0 - 1 1 c m 3 m o l e c u l e - 1 s - 1 . M u N O formation is linearly dependent on [NO], as expected; it is also noteworthy that the fitted line passes through the origin, indicating that 0 2 impurity in the moderator is not important as discussed above. The value of kc are obtained from the slope of plots such as F ig . 5.20 and are given in Table 5.12 along with a result from Ref. [55] at 20 atm. Though these values are not immediately useful, since they contain contributions from re,mp, it was felt worthwhile that they still be recorded here. F ig. 5.21 plots the values in Table 5.12 vs. N 2 moderator pressure, from which the rate constant of interest, kMu = (8.76± 0.46) x 1 0 - 3 3 c m 6 mo lecu le - 2 s - 1 with M = N 2 , is obtained from the slope, while the value of rc,mp = (1.4 ± 0.2) x 1 0 - 1 2 c m 3 mo lecu le - 1 s - 1 is found from the intercept.7 Significantly, the reaction rate increases completely linearly with increasing moderator pressure, demonstrating that the Mu + N O addition reaction remains well in the termolecular regime, even up to 60 atm. Though not of much importance here, the relatively large value of fc,mp obtained from the intercept in F ig . 5.21 is some what puzzling. As argued above, the main impurity contribution is expected to be from N 0 2 , but if « , m p is due to M u + N 0 2 , then 7 An alternate way is to plot kc(= A c / [ N O ] ) vs. [M] for all points and fit to Eq. (5.36), the slope is fco and j/-intercept kimp- The results are the same. 192 2 . 0 O . O 1 1 ' ' 1 O . O 4 . 0 8 . 0 1 2 . 0 1 6 . 0 T o t a l P r e s s u r e ( 1 0 m o l e c u l e s c m ) Figure 5.21: Chemical reaction (total) rate constant (A:c) vs. N 2 moderator pressure at ~ 298 K. See also Fig. 5.20. The slope here gives the termolecular rate constant fc^" = (8.76 ± 0.46) x 1 0 - 3 3 c m 6 mo lecu le - 2 s - 1 . The solid square indicates our previously published rate constant at 58 atm, based on lower field data[55]. It is shown here for comparison but is not included in the fit to give the termolecular rate constant, fen. Table 5.12: Bimolecular Chemical Reaction Rate Constants for M u + N O . Total Pressure kc (atm) ( 1 0 - 1 1 cm 3 molecule - 1 s - 1 ) 5 0.255 ± 0.021 20 0.558 ± 0.038 40 1.002 ± 0.054 58 1.63 ± 0.26° 60 1.464 ± 0.081 20 1.39 ± 0.076 a. The previously published data [55]. Not included in fitting. b. Ar as moderator. Only two points and the intercept were used in fitting. 193 kvvo3 x [NO2] = 4.8 x 1 0 - 1 0 cm 3 mo lecu le - 1 s - 1 from the value in Ref. [217], in which case, CNO3, the percent amount of NO2 in the N O , would be 0.3%, an order of magnitude higher than stated by the manufacturer. However, this is entirely possible since manufactures' labels are often unreliable and it is not unusual to find N O containing 0.1-1% N 0 2 . It may also be that « , m p is actually due to a sum of several (unknown) impurities. Regardless, this result is not of further interest since it affects the intercept only and not the measured rate constant, ko, obtained from the slope. The few data points obtained in A r at 20 atm pressure are internally inconsistent (see Table 5.11) with the likely result that the point at [NO] = 1.06 is simply an experimental error. The remaining two points do lie on a straight line (including the zero intercept), but give the surprising result that the third body efficiency of A r appears to be at least three times higher than N 2 . From H + N O combination kinetics, it is established that kH(N2)/kH(AT)= 1.6 ~ 1.9 [431,432,444,445], and thus one would expect a similar trend for M u + N O . The reversal of M efficiencies may indicate a change of mechanism of the reaction from H to Mu[446], but it cannot be claimed with certainty since clearly there is insufficient data to draw any firm conclusions here. Tsang and Herron have reviewed most of the published data on H + N O + M reactions[432]. A comparison with their recommended rate constant for H atom addition in N2, kH = 3.9 ± 1.8 x l O - 3 2 c m 6 mo lecu le - 2 s - 1 indicates k^u/kH = 0.23 ± 0.12, giving an "inverse" K I E in the expected direction for the low pressure regime of this reaction, as outlined earlier. In the aforementioned study of Hartley and Thrush, the total rate constants for H + N O + Ar and D + N O + A r have been measured,using a discharge flow technique with values kH = 3.1 ± 0.4 x 10~ 3 2 c m 6 m o l e c u l e - 2 s - 1 and A# = 3.5 ± 0.6 x 1 0 - 3 2 c m 6 mo lecu le - 2 s - 1 [431] . Their value for kH is in good agreement with that recommended by Tsang and Herron in N 2 moderator. 8 The ratio, kH/k® = 0.9 ± 0.3, also gives an "inverse" K I E (though the error should 8 Tsang and Herron in fact did not give fco for Ar, and Hartley and Thrush did not measure the rate in N2 moderator, so no direct comparison can be made. However, it is found experimentally that the third body efficiencies are similar for diatomic molecules. Therefore if one assumes the rate in H2 moderator [431] is the same as in N2, this should be ~ 1.9 times the rate in Ar, which would mean a value in Ar of 3.1 (Hartley and Thrush) vs. 2.1 (Tsang and Herron). Clyne and Thrush also found this value to be 2.3 in Ar[444], which has been recommended by Baulch[430]. Although the absolute rates are different, both Hartely and Clyne found the efficiency ratios to be around 1.7 ~ 1.9. 194 be kept in mind) and is of interest here in comparison with the much larger effect seen for M u and H. Curiously, in both cases the ratios k^/k® and k^/k^ are consistent with the inverse of that expected classically for an early-barrier P E S , where ka for addition simply scales with the square root of the mean velocity: k™/k® = 1.4 and k^/k^ = 2.9. It can be commented that the experimental rate constants, for both H(D) and M u + N O are total rate constants in that they are a measure of both ground-state and excited-state electronic H N O * (MuNO*) populations, which were actually measured separately for H and D by Hartley and Thrush[431]. The formation of ground state ( M ' ) M u N O is highly exothermic, though considerably less so than for H N O , perhaps by ~ 33 kJ /mo l from arguments given elsewhere[26, 31,122,125], or a A H ^ U 167 kJ /mo l , based on the theoretical values for H + N O of Guadagnini et al., of Cobos and of Walch et al.[168,190,434]. This could have some bearing on the K IE 's mentioned above, though it is unlikely to be of much importance since the first excited state (3A") of H N O * lies at ~ 75 kJ /mo l and has an activation barrier ~ 20 kJ /mo l . Indeed, including the PES 's of excited states for H + N O has been shown to have only a small effect on the reaction kinetics[168]. It is noted earlier that the transition pressure between the low and falloff regime for H N O formation appears to be approximately 60 atm with the high pressure l imit not being reached until ~ 400 atm[435], although no details of the experiment have been given. Cobos [168] has calculated the high pressure rate constant but no information on the transition pressure was given. His calculated value, A:" = fc" = 2 x 1 0 - 1 0 c m 3 m o l e c u l e - 1 s _ 1 is almost a factor of 4 lower than the limit reported in Ref. [435] but is of interest to the discussion below, as is his calculated K I E , k^/k® = 1.9, at room temperature. That the corresponding transition pressure for M u + N O appears to be considerably higher than 60 atm (Fig. 5.21) is not surprising, considering the few vibrational degrees of freedom available in the [Mu(H)NO]* complex and the isotope effect mandating higher vibrational frequencies of the M u complex, making it even more difficult to stabilize. We conclude that for either N2 or Ar moderator, all values reported seem well within their respective error ranges. 195 Besides the aforementioned (trivial) isotopic effect arising from differences in mean velocity (due to differences in the translational partition function), there are two more important mass effects that can influence the reaction kinetics, which have also been mentioned previously but deserve additional comment here: vibrational energy shifts (changes in the vibrational partition function) and quantum tunneling. Changes in vibrational energy levels due to differences in Z P E (recall m M u / m n ~ | ) , though generally most important for late barrier (endothermal) PES 's , can have essentially two consequences: first, there are changes in the critical energy for bond dissociation, en, affecting &_„; secondly, there are statistical-weight effects due to changes in the state densities of reactant and activated complex. It is well known that these Z P E effects tend to act in opposition with any tunneling contribution, the latter favoring the lighter isotopic rate constant of a single step reaction in chemically activated systems. [26,122,175,315,317,441,447] Thus, at high pressures, the lighter isotope is favored in (exothermal) bimolecular combination reactions since kejj = k^ = ka. Indeed, the S A C M calculations mentioned above [168] for the thermal kinetics of H(D) + N O , in this l imit, give k^/kW = 1.9 with tunneling corrections included, and, interestingly, 1.6 (not 1.4) in the absence of tunneling. As there is essentially no (ground state) electronic barrier to addition[168,190,434], any effect of tunneling on koo arises through the rovibrational-baxner for H N O , treated as being primarily due to the bending degree of freedom in Ref. [168]. On this quite exothermic surface {AHH 200 kJ/mol[168,190,434],) the calculated barrier is both early and small, ~ 2 kJ /mo l , and would likely not be appreciably enhanced by Mu substitution (in contrast, eg., to endothermic reactions, such as discussed in Refs. [26,31,125], and [122]). Hence dramatic tunneling effects in M u + N O are not likely, even at the lowest temperatures. Nevertheless, some enhancement may be expected. Around ~ 300 K, Cobos finds a tunneling factor for H + N O of T H = 1.7 (recall Eq . 4.10) and the corresponding enhancement for Mu + N O , T\fu, could perhaps be twice, thus ~ 3.5, based also on previously-cited results for Mu(H) + C 2 H 4 addition at room temperature(though, the appreciably higher barrier surface, ~ 8 kJ/mol[32], should be kept in mind.) In the low pressure region, of principle interest in the present study, the net isotope effect on 196 ko = kaks/k-a depends on the magnitudes of the isotope effects for three separate rate constants. How does one expect a change in isotopic mass to affect these individual rates? A n estimate of the effect of difference in vibrational zero point energy at the transition state on the critical energy for dissociation (en), can be appreciated with reference to the dissociation channel Eq . (5.31) by expressing the unimolecular rate constant Ar_0 by its simplistic (RRK) form, Eq. 4.25. Recall that s* represents a parameter that is typically about | to | the number of normal modes [32,317,448-451] (so about 1.5 for M u N O * or H N O * ) . Though for Mu(H,D) + N O the ground state is so exothermic ( A H Q ~ —200 kJ/mol) that only rather small isotopic effects on /t_a are likely at thermal energies, nevertheless, for D N O * , the dissociation energy eft wil l be larger than for H N O * due to its larger mass and hence k®a < k^a is expected; whereas for M u N O * , eg wil l be much smaller leading to a much larger shift in the same direction, k^a < Estimating that eo(MuNO) is less than C Q ( H N O ) by about 0.3 eV, from a harmonic Z P E correction, which accounts as well for the aforementioned difference in bond enthalpies, one could expect fcMu ^ 2 x 10 1 1 s - 1 > 3Ar"0. This can certainly explain the trend in the K IE 's given above, since it is k-a in the definition ko = kak,/k-a that is directly affected. Dissociation lifetimes kzl ~ 5 ps (for MuNO*) or ~ 15 ps (for HNO*) would correspond to high pressure limits ~ 1000 atm (300 atm), consistent with the pressure regimes in the aforementioned data of Forte [435] and Troe and co-workers[436,452]. This also explains the fact that, even at 60 atm, there is no indication of any departure from termolecular kinetics for Mu -f N O addition (Fig. 5.21). However, to put even this simple result in proper perspective, some important further questions need to be raised. First, what is the isotopic effect for vibrational de-excitation on the collisional stabilization (ks) of MuNO*? Second, what is the effect of the possibly much reduced density of states expected for MuNO* vs. H N O * , due to the much lighter Mu-atom mass and/or, how non-RRK(M) is the dissociation of MuNO*? Thi rd, what is the effect, if any, of quantum tunneling in establishing the value of the low pressure rate constant k0 for M u -f N O and/or H + NO? The effects of Z P E could mean up to a factor of three reduction in density of states for the 197 Mu analog and concomitantly an increased AE for V - T energy transfer by this same factor, which could give a reduction at low pressures in the cross section and hence the rate for vibrational deactivation by ~ exp - 2 ( A B ) / ' ' ! T [101,315,317,453,454] , thereby directly affecting k, and hence ko. If one assumes that AE is of the order of the bend frequency at the transition state calculated by Cobos for H N O * , ~ 190 cm _ 1 [168], then the above factor would mean an isotopic effect k^a/k^ < 0.2, with larger (stretching) frequencies [168,246] giving concomitantly sharper reductions. On the other hand, it may well be that the strong collision assumption invoked here is much less valid for M u N O * , with its higher vibrational energy levels, or, stated otherwise, perhaps the collisional stabilization factor "/?" often introduced in such a model is <^ 1 for M u + N O . At this point it is worth recalling the basic experimental result of the present study. For M u + N O in the termolecular regime, k^u = (8.76 ± 0.46) x 1 0 - 3 3 c m 6 mo lecu le - 2 s _ 1 with M = N 2 , which in comparison with the recommended result for H + N O [432] gives for the ratio k^/k^ ~ 0.2, consistent as well with the ratio K g 1 / * ! ? ~ 0.9 from the earlier study of Hartley and Thrush. The arguments presented above are qualitatively consistent with the trend. Specifically, since ko = kaks/k-a and since from Eq. (4.25) and the subsequent discussion, k^ ~ 3k-a and k™u < 0.2kf, one can write for the ratio k^u/kH < 0.07 i t ^ u / « " , which then depends directly on the high-pressure constants, koo = ka for both isotopes. The only recent experimental determination of this is from the previously noted very brief report of Forte[435], giving k\\ = kH = 8.3 x 1 0 - 1 0 c m 3 s - 1 at pressures higher than 400 atm. A n extrapolated value, based on Troe's falloff curve [455] with data obtained at pressures less than 100 atm, is also quoted, kH = 2.5 x 1 0 - 1 0 c m 3 s - 1 (no explanation for this difference is offered). In Cobos' S A C M calculation[168], kH = 2.0 x 1 0 - 1 0 c m 3 s - 1 , which is in good agreement with the lower experimental value and which, as mentioned above, includes a tunneling enhancement of TH = 1.7. If taking this value, include the (trivial) mean velocity effect and the aforementioned estimate for M u tunneling of T M u = 3.5, one obtains an estimate for w 12 x 10~ 1 0 c m 3 s - 1 for the Mu + N O reaction. Though such a value seem anomalously large in comparison with other studies of Mu reactivity[26,32,124,217], and much faster also than the competitive process of SE 198 (see below) or indeed of other M u SE reactions such as M u + Cs described earlier and Mu + O2 [56], it is not inconsistent with general expectations for highly exothermic radical-radical reactions on zero barrier surfaces (which are often similar to the rates of ion-molecule reactions [28,29]), where the reaction cross section is known to increase with decreasing energy. This can be seen also from the recent computations of the P E S for H + N O by Guadagnini et al [190] from which one could expect a thermal "capture" cross section ac » ITR^^ ~ 30A 2, corresponding to the long range part of the interaction potential at Rmax ~ 3A. For M u + N O at ~ 300 K, this value would give a thermal (capture) rate constant of fcj1" ~ 2 x 1 0 - 9 c m 3 s - 1 , in qualitative agreement with the above estimate of 1.2 x l O - 9 c m 3 s - 1 based on the calculations of Ref. [168]. Adopting this latter value, gives k^/k^ < 0.4, about a factor of two larger than the experimental ratio. If Forte's experimental high pressure rate constant is accepted, this ratio becomes < 0.1, assuming the same tunneling factor. However, it is not clear that any tunneling enhancement, assumed here to be F M U / F H ~ 2, should be carried over into the ratio k^/k^. Indeed, if one writes ko = K k,, where K = ka/k-a is a true equilibrium constant, then there should be no tunneling (or other dynamical mass effects), reducing the above estimate based on Cobos's calculation for k^/k^ to ~ 0.2, in good agreement with the experimental number. (Forte's high pressure experimental value would then give ~ 0.05, in poor agreement with the present data.) There are a number of uncertainties raised in the questions stated above which bear on the interpretation of the experimental result for the ratio of termolecular rate constants, k^/k^ ~ 0.2 , for the combination reactions, M u (H) + N O . It is concluded that tunneling plays a minor role in establishing this ratio, as would be expected if the ratio ka/k-a can be viewed as a true equilibrium constant. On the other hand, given the relatively few degrees of freedom and hence possibly much reduced density of states in MuNO* compared to H N O * , resulting from the pronounced isotopic mass ratio /TIMU/"*H ~ 1/3, it may be that there is insufficient time to establish a true equilibrium in the M u N O * adduct, corresponding to a thermal distribution of allowed energy levels, in which case tunneling could well be of some importance. This in turn would imply n o n - R R K M 199 dissociation of M u N O * , possibly via the formation of specific resonance states, thereby affecting k-a in a particular way. It is well known in some related (small-molecule) systems, for example HCO[193], that pronounced resonance structure is indeed important. Answers to these questions wil l only be forthcoming after: i) theoretical calculations for the thermal M u + N O addition rate on the same P E S as for H + N O [168,246,434] are carried out; and ii) a direct experimental measurement of the high pressure rate constant, « £ o U = k™u is accomplished. 5.4.3 Spin Exchange The spin exchange (or "spin flip") rate constants («SF) for M u + N O at different total pressures, calculated from the slopes of the lines in plots of \ L vs 1/(1 + X2) (as in F ig . 5.19) are given in Table 5.13. It is noteworthy that these are independent of moderator pressure, as expected from Eq. (3.34) and from the experimental condition that the time between (SE) collisions is much longer than the hyperfine mixing time ("slow" SE limit)[94]. The point of note here is, if there were additional contributions to M u "spin flip", arising from MuNO* dissociation, some pressure dependence would be expected, thus supporting that feature of the analysis described above where it is only the chemical (combination) reaction that is pressure dependent. The SE process is a direct reaction and is much faster than the termolecular combination reaction forming M u N O * (though not so in comparison with the expected high pressure l imit, &J 1"), with, on average, every SE collision leading to a loss of about half of the muon polarization [22,98] . The average SE rate constant from the determinations in Table 5.13 is &SF = (6.00 ± 0.24) x 1 0 - 1 0 c m 3 mo lecu le - 1 s - 1 . This corresponds to a spin flip cross section CTSF = (8.0 ± 0.3) x 1 0 " 1 6 c m 2 , in good agreement with the earlier reported value CT S F = (8.3 ± 0.9) x 10~ 1 6 c m 2 obtained by transverse field pSR and at moderator pressures ~ 1 atm [58,59,90]. As discussed in these earlier papers, the corresponding cross sections for H atom were reported as (26 ± 2) x 1 0 - 1 6 c m 2 [53,90] and (25 ± 2) x 1 0 - 1 6 c m 2 [47,90], and the ratio of the cross sections is about 3 in favor of H atom. This is similar to what was observed in atom-atom (Mu + Cs) spin exchange collisions and is probably due to the same reasons, i.e., the reduction in contributing partial waves and the increase in 200 Table 5.13: Spin Fl ip Rate Constants for M u + N O at 298 K. Total Pressure fcsF atm (10 - 1 0 cm 3 molecule - 1 s - 1 ) 5 6.03 ± 0.18 20 6.09 ± 0.20 40 6.16 ± 0 . 2 2 58 5.84 ± 0.28" 60 5.72 ± 0.24 a. The previously published data[55]. velocity (see discussion on M u + Cs). 5.4.4 S u m m a r y The Mu atom addition/combination and spin exchange rate constants are reported for the M u + N O reaction at room temperature and over a range of (N 2 ) moderator pressures from 2.5 -60 atm. The prime motivation for the experiment was to obtain the chemical rate constant for the combination process, separate from the contribution due to spin exchange. This has been achieved by varying the magnetic field up to 15kG, much higher than in an earlier study[55]. The present data (Fig. 5.21) are the first detailed results of their kind and clearly show that the addition rate remains well in the low pressure regime, up to the highest pressures measured (60 atm). The corresponding termolecular rate constant, ko= (8.76±0.46) x 1 0 - 3 3 c m 6 mo lecu le - 2 s - 1 exhibits an inverse kinetic isotope effect in comparison with the corresponding H + N O studies [431,432], with k^/k^ ~ | , the largest isotope effect yet reported for experimental (H-atom) dissociation reactions, at least near room temperature. Both quantum tunneling and vibrational (ZPE) shifts could be important in establishing this effect, though tunneling is unlikely to be of major consequence. It is likely that Z P E shifts, particularly affecting the dissociation rate &_<, in the definition of ko (= kaks/k-a), constitute the main isotopic effects but theoretical (possibly non -RRKM) calculations are required in order to assess this. Further experiments at higher pressures are underway in order to directly measure the l imiting high pressure rate constant koo = ka for Mu + N O . 201 5.5 Mu + CO: Addition, Collisional Relaxation and "Spin Flip" The M u + C O reaction was first investigated about 10 years ago at T R I U M F [456,457]. In a weak T F , the relaxation rate increased linearly with [CO], much as in Fig. 5.10 for M u + N2O, indicating a chemical reaction, likely M u C O formation. That investigation and later measurements in both T F and L F revealed a few peculiarities of this system which could not be explained at the time and no results of chemical kinetics were published. This was one of the reaction systems (another being M u + O2) that prompted current investigations of collisional spin relaxation in free radicals [34,231,232,458]. The initial motivation was to compare this system with the H + C O reaction, which produces the H C O formyl radical. Since the formyl radical plays a key role in the combustion of hydrocarbons and in atmospheric oxidation process, the collision dynamics of H with C O and the spectroscopy as well as the molecular dissociation of H C O have been studied extensively by both experiment and theory over the last two decades, with renewed interest recently [187,188,427,428,445,459-473]. As a triatomic radical, H C O is considered an excellent candidate for high-level ab initio P E S calculations because it is composed of light atoms. Earlier ab initio P E S calculations had to be empirically adjusted to better match the experimental data for H + C O = H C O reaction system [474,475]. Very recently, a new ab initio P E S is presented, which holds up much better in comparison with experimental results but discrepancies, particularly in isotope effect studies of resonances energies and widths for H(D)CO, stil l exist [187,471]. Further experimental as well as theoretical results are required to understand and reconcile differences between theory and experiment, as well as to establish the correct model in the theoretical treatments. As discussed previously, analogous isotopic reactions often provides important input in this kind of situation. Investigation of the D C O system is expected, and has been requested by several authors, to provide important data in testing the P E S as well as the theoretical dynamics treatment [187,188,459] and a new. study of H(D)CO is reportedly underway [187,476]. However, 202 as emphasized in this thesis, the isotope effects expected in H / D are small and it is difficult to experimentally obtain the level of precision that is required to make a sound theoretical assessment [188]. In this regard, the study of the Mu + C O reaction with the / iSR technique, despite its limitation as a "bulk" experiment, wil l definitely provide valuable information. Furthermore, the formyl radical is one of the few triatomic radicals that have been detected by E S R in the gas phase [477] and its magnetic properties have been the subject of many studies in both gases and in liquids [478-480]. The pSR studies of nuclear spin relaxation phenomena in M u C O will not only provide an important extension of isotopic effect studies in this area but also, in light of the previously mentioned difficulties in the E S R technique, may provide, in some cases, an additional handle for the investigation of spin relaxation mechanisms in the H C O isotopomers. There is now renewed interest in the kinetics of the Mu + C O system (and experiments are still underway at T R I U M F ) because with the phenomenological radical relaxation model [231,232] introduced above and developed further herein, a framework for an interpretation of the results is now at hand and with earful planning useful kinetic information can be extracted from measured relaxation rates. Since the study of this system is still underway and only a limited set of data is currently available, notably in transverse fields where the data can only be properly fit at present. Consequently, in this section, rather than attempting a complete data analysis, it wil l be demonstrated how the data can be best interpreted in terms of the SR relaxation model developed in previous sections, illustrating the feasibility of tackling this rather complicated system in the context of the / /SR technique. In this regard, no detailed error analysis wil l be given. 5.5.1 E x p e r i m e n t a l R e s u l t s t o D a t e The M u + C O reaction was measured in T F on the M15 and M9 beamlines, and in L F on M15, using the high pressure target vessels discussed earlier, with N 2 or A r as moderator at room temperature. As mentioned in Chapter 2, the uncertainties in concentrations and pressures of both C O and moderator gases are about 1%. T F relaxation rates were measured over the pressure range 6.5 ~ 60 atm N 2 and 6.5 ~ 272 203 0.00 0.25 0.50 0.75 1.00 Time in microseconds Figure 5.22: Observed relaxation signal for M u + C O in 8 G (top) and 37 G (bottom) T F with 2.25 x l O 1 8 mo lecu lecm - 3 C O at 6.5 atm N 2 pressure. The relaxation rates are, from the fits, 0.93 ± 0.05 and 1.4 ± 0.1 / J S - 1 , respectively, giving the ratio 1.5 ± 0.1. atm A r total pressures. Two typical signal histograms are shown in F ig 5.22. The weak field (< 10 G) data (top) was fit to Eq. 3.31 and the intermediate field (> 30 G) data (bottom) to Eq. 3.32 to obtain the relaxation rates which are listed in Appendix B. Some representative points are plotted as a function of pressure in Fig. 5.23. A l l background errors have been corrected as in the earlier sections. At low pressures, the ratios of relaxation rates at intermediate and weak T F for the same [CO] and [M] combination were 1.5, but slowly dropped to about 1.1 at higher pressures. These ratios are plotted as a function of total pressure in Fig. 5.24. This changing ratio indicates a changing mechanism for spin relaxation with pressure. 204 20 40 60 Pressure (atm) 80 Figure 5.23: T F relaxation rates of M u + C O at room temperature are plotted vs. moderator (N 2 ) pressures. Two [CO] concentrations are represented, diamonds for 2.25 and squares for 6.44 x 10 1 8 mo lecu lecm - 3 . The open symbols represent data measured at weak (6-8 G) T F and the solid ones were measured at intermediate (37 or 95 G) fields. 100 150 200 Pressure (atm) 250 300 Figure 5.24: Ratio of M u + C O relaxation rates in weak (6-8 G) and intermediate (37 or 100 G) T F at 300K. The squares are data in an A r moderator and the crosses are in N 2 moderator. 205 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 Time in microseconds Figure 5.25: Relaxation signal observed in 19.2 kG L F with 15 atm pure C O . The relaxation rate is 3.3 p s " 1 . A typical L F signal is shown in Fig. 5.25, note the relatively large asymmetry, in comparing with a similar plot for Mu + N O in Fig. 1.4. Fit t ing such signal to a simple single-exponential relaxation function (Eq. 2.3), allows a measure of relaxation rates which illustrate the trend with pressure and field as plotted in Figs 5.26 and 5.27. These rates are functions of both the chemical addition rates of M u + C O and SR relaxation rates of the radical M u C O (see later discussion). Without additional information, one cannot extract individual rates from these data. However, it is clear that relaxation rates in L F are very large at low fields and still relatively large even at fields near 60 kG with values ~ 5 p s - 1 , about an order of magnitude faster than in the case of M U C 2 H 4 addition for example (Fig. 5.3 top). The weak-field pressure dependence is also completely different than that seen in other radicals such as M U C 2 H 4 , showing slower relaxation rates at the lower pressure. The following discussion will focus on T F data and no further analysis on L F data will be made except using the trend shown in Figs. 5.26 and 5.27 to support some arguments given below. In this regard, the L F data suffers some of the same complication as that for N2O described earlier, requiring further development of the correct fitting function. It can be noted here that further T F experiments conducted with the same gas mixtures as used in the L F measurements are necessary to properly extract information from these L F data. 206 CO 3. 10' 10' -10 -1 10' X o 3 atm x 7 atm H 15 atm + 40 atm 10' B (kG) 10' Figure 5 . 2 6 : Relaxation rates in L F for the M u + C O reaction. 4 sets of pure C O data are shown. C O pressures are given in the legend. 10' CO a. 10' -10° 10" B (kG) Figure 5 . 2 7 : Addit ional Relaxation rates in L F for the M u + C O reaction. The open squares are mixtures of 2 0 0 torr C O and 6 . 5 atm N 2 . The crosses are pure C O (high concentration) at similar ( 7 atm) total pressure. Note that the low [CO] data at low field ( 7 5 G ) gives similar rate (~ 4 . 5 / i s - 1 ) to that measured in T F (Fig. 5 . 2 3 , solid square, at 3 7 G , A T = 4.1 / i s - 1 ) . 2 0 7 5.5.2 Interpretat ion Unlike the M u +N2O system, the chemical aspect of the M u 4- C O reaction is fairly simple as far as the reaction mechanism is concerned. There can be no abstraction reaction at ordinary temperature so the only chemical reaction channel is the formation of the Mu-substituted formyl radical in the addition reaction M u + C O + M ^====± M u C O * M u C O + M . (5.37) k — a in like manner to M u N O formation discussed earlier. In fact, on first look this is simpler than M u + N O since there is no spin exchange interaction (in the conventional sense, see below) in this system because C O has no unpaired electrons. Like the M u + C 2 H 4 addition reaction [32,41] though, M u C O can undergo collisional spin relaxation and would be expected to behave similarly. However, as advertized previously, the determination of rate constants from relaxation data for this chemical reaction turned out to be rather complex, for the reasons given below. The initially unexpected, "peculiar" results obtained include: i) at low L F field and pure high concentrations of C O , the observed relaxation rate increases with pressure first (expected to decrease from the relaxation model, see Figs. 5.3 and 5.6) and the becomes somewhat independent of pressure at about 25 atm up to 60 atm (see Fig. 5.26); ii) in T F , a "3/4 effect" is observed, which, as expected, is not seen for other addition reactions forming free radicals, where no SE is involved (e.g., Mu + C 2 H 4 and M u + N2O, see Appendix B and Fig. 3.7); iii) in w T F at low pressures (6 ~ 50 atm), there is a region over which the rate is proportional to [CO] but somewhat independent of total pressure, also never seen in other addition reactions studied by fiSR. A l l of these behaviours can, however, be qualitatively explained. First, unlike most M u radicals, M u C O has a very high hyperfine coupling constant (A^ > 1100 MHz) [232,458] (seen also for H C O [480]) but has no nuclear moments other than that of the muon. Most other radicals have either nuclear moments and/or small A^'s which generate a multitude of transition frequencies (792 in M U C 2 H 4 ) in weak T F , with the result that, on average, a Mu-substituted free radical precess much faster than free Mu . This results in an easily 208 measurable relaxation rate in a weak T F which can be thought of either as a depolarizing due to a marked difference in precession frequency [41] or, more correctly, to the fact that the myriad of p + - / z + - e~ hyperfine coupling in weak field, among which the muon polarization is distributed, renders it unobservable. However, this is not the case for M u C O . Recalling Eqs.3.26, the summation term is zero (no nuclear moments) and Cl is negligible below 6 G (Bo is large), so as in free M u , u>i2 = w23 — ^>M- Consequently, at very weak T F (< 6 G) , M u C O (MuCO*) wil l precess just as Mu itself. It is therefore difficult, if not impossible, to distinguish polarizations (and relaxation contributions) of Mu in the three (Mu, M u C O * and MuCO) different environments. For instance, at weak T F , if there was no spin relaxation mechanism in the radical, no relaxation of Mu signal would be observed, at any pressure. In contrast, for M u + N O the reaction product M u N O is diamagnetic and would precess with a 100-fold slower frequency; or, in M u + C 2 H 4 , the product M U C 2 H 4 precesses with much higher (order of a magnitude) frequencies. Both result in a loss of coherent M u signal even if the spin polarization in the radical is preserved. It is for this reason that any depolarization observed in M u + C O (in any field) must be the result of spin relaxation either of the excited radical, M u C O * , or of stabilized M u C O itself. This relaxation in turn is due to an expected strong spin rotation interaction in the M u C O radical. A significant spin rotation interaction for the formyl radical H C O has been noted by various authors, deduced from E S R measurements of the g-tensor in a solid matrix [478] and more directly from microwave rotational spectroscopy in the gas phase [479] as well as by liquid phase E S R studies where the associated spin relaxation has been invoked to account for the broad E S R lines observed [480]. In terms of the SR relaxation model and the "Cur l formula" [8,386,387] described earlier (Eq. 4.85 and Eq.5.8), H (Mu)CO has a very large shift in the electron g-factor, Age [480], as well as a relatively small moment of inertia, J r , both giving rise to a large A E parameter as seen from Eq. 5.9. One can estimate the magnitude of A# for M u C O from the H C O data [480]. For H C O , Ir = 1.2 x 10~ kg-m : 2 , C S R = 51 n s " 1 A 2 1(H) = kBT 1.3 x 10 2 2 s - 2 . 209 Since the rotation is about the C - 0 bond, the moment of inertia is roughly proportional to the mass of the H atom 9 and consequently, 7 r (Mu) = 0.1121 x / r (H ) which then gives A | ( M u ) = 1.2 x 10 2 3 s - 2 , or A B ( M u ) = 350,000/zs - 1 ! Though the simple scaling of Ir (about the bond axis) for M u C O may be in error here, it is clear that A E for M u C O is much larger than the same parameters measured for all the other (larger) radicals (Table 5.3). A t zero (or very weak field) and assuming r c = 100 ps at 1 atm, this leads to a relaxation rate of ~ 10 4 /p ps-1, where p is the total pressure in atm. So at the low pressure characteristic of the early experiments [456,457], ~ 1 atm, this gives a relaxation rate ~ 10 4 pa~lm, even at 60 atm this is ~ 170 ps-1, sti l l too fast to be seen by pSR. Although the M u C O radical has not been directly observed (due to the fast relaxation), 1 0 the fact that high relaxation rates were observed in both low T F and L F (Figs. 5.22 and 5.25-5.27) is sufficient evidence that M u C O does form and the observed muon spin relaxation is induced by collisional interactions. Secondly, it is well known that, although H + C O is very similar to the H + N O reaction, in the sense that the reactants of both systems have similar masses, structures and frequencies, the addition rate constant of the former reaction is anomalously low, a full 2 orders of magnitude smaller than the latter reaction [445]. This is also seen in this thesis work. There are two reasons why the reaction rate of H(Mu) + C O is so much slower than that of H(Mu) + N O . One is that the H(Mu) + C O reaction has a relatively large reaction barrier compared to H(Mu) + N O reaction which has essentially zero electronic reaction barrier. The other reason is that the exothermicity of reaction is much less for H(Mu) + C O , —59 kJ /mo l vs. —200 kJ /mo l . A t pressures below a few 10's of atm, the H + C O addition reaction proceeds very slowly because the lifetime of the excited radical is expected to be very short (order of 1 ps or less) [187,188,193,459,469,471,475,480], which can be seen as well from Eq. 4.25, requiring pressures on the order of 1000 atm to effectively stabilize the radical [188]. Recall that M u + N O is stil l in the low pressure regime at 60 atm (Fig. 5.21). 9 Assuming a distance between H-atom and the bond axis to be r = 1 x 10 - 8 cm 2, then Ir (H) = 1.67 x 10~4 7 kg m 2 , which is close to the value given in Ref. [480]. 1 0 A n indirect evidence is also presented in Ref. [36], though this is not entirely convincing. 210 This question of the actual lifetime of M u C O * (HCO*) is of central interest to studies of Mu(H) + C O addition reactions. If the lifetime were long enough then two things could happen, as discussed in Chapter 4: first, the muon spin can flip due to modulations of the anisotropic hyperfine coupling, second and much more likely, the electron spin correlation can be lost due to its SR coupling to the rotational motion of the radical which indirectly relaxes the muon. Conversely, if the lifetime of the radical were extremely short, <§; r c, a naive expectation would be that no significant relaxation could be observed in weak T F (at low pressures), even if there exist some spin relaxing mechanisms in the excited radical as long as the relaxation rate is not astronomical, because the muon spin or the electron spin of the excited radical would then have no time to relax before returning to the free Mu state. In like manner, one would not expect a large component (asymmetry) of a fast relaxing signal at low pressures because the fraction of any stabilized radical (MuCO) would be small. These expectations are, however, over-simplified. As will be shown below, the test is not just the direct competition between dissociation (k-a) and SR relaxation (fcgF, see Section 4.1, Case D). As long as the collisional relaxation rate itself is not too low (which is certainly the case for M u C O , as mentioned above. Also see below), some relaxation of the signal can be observable, at least at high enough pressures (a few atm is sufficient). As a matter of fact, the very large dissociation rate (fc_a) makes the analysis of the Mu + C O data much simpler than would be expected otherwise (see below). This situation is exactly the Case D discussed in Section 4.1. When k-a is very large, the relaxation function is given by Eq. 4.42, rewritten here, R(t) = K 1 (5.38) K - 1 1 - K where in the present context K ka[CO]f3k,[M] (5.39) 211 _ ka[C0](f3k.[M] + fk'SF) , , 4 m A e / / = 7 . (5.40) K-a On the assumption that spin relaxation is the same in M u C O * and stabilized M u C O , several l imiting situations can be considered [recall that P(t) = R(t)G(t)]. i. First consider the case when A r = 0 . R(t) = 1, as expected from the arguments above. The chemical reaction is then transparent to / i S R in weak T F as well as in L F . i i . Second, consider the high pressure case. Noting that A r is proportional to 1 / [M] at high pressures (see below), and hence that K is proportional to 1 / [ M ] 2 ( 1 / [ M ] 3 in pure CO) , it goes more quickly to zero than A r when pressure increases. When [M] is large enough so that K - ¥ 0 and A r is still large enough to be observed by / /SR, R(t) = e~Xrt. In both L F and T F , only the collisional relaxation rate is observed. For M u C O , using the above estimated value, AE ~ 10 5 n s - 2 , assuming a reasonable value for the correlation time at 1 atm, T° = 100 ps (see Table 5.3), and taking the contribution from the electron SR coupling only, the rate due to collisional relaxation at zero (and weak) field is (see Eq. 4.85) A r ~ ^ . (5.41) P At the highest pressure run so far, 272 atm, the rate is sti l l 72 p s - 1 , too fast to measure. However, in large L F , the rate is quenched by the field, 2 0 4 1 Ar ~ — p 1 + X* Scaling from the H C O value [480], the hyperfine field of M u C O is about 500 G [231,232,458], so at, say 5 k G and 10 atm total pressure, x = 10, A r = 10 ps~x which is within the measurable range by pSR. One has to keep im mind, however, whether this is observable or not also depends on K and other contributing terms to the total relaxation rate. i i i . At pressures between 1 and a few hundred atm, where A r is large at low fields, the second term in Eq. 5.38 is not measurable at weak T F , and the relaxation is instantaneous. The 212 observed rate would be then the slower one that comes from the first term. Xoos = A e / / . One can also see this from examining K (see below). This is the pressure range in which the current experimental data was obtained. One can usually choose the right combination of [CO] and [M] to reach this regime, but this was not a priori known at the time this thesis research was started. Of course, to see a reasonable signal, the amplitude must be reasonably large, or K cannot be too small. The value of K from Eq. 5.39 can be estimated as following (recall Eq . 4.18, k0 = kaf3kt/k-a): assuming that we choose the right combination of [CO] and [M] so that the overall addition reaction rate (reo[CO][M]) is 2 ps'1 (see Fig. 5.28 and 5.23), then K = A r /& 0 [CO][M]= 1 x 0 4 / p . At the highest pressure run at weak T F , 272 atm, K - 37. The amplitude of the signal is ~ 1. In all conditions currently under study, even at the upper l imit of observable rate (20 ps~l), the amplitude is always large at weak fields. Now let us consider the expression for A e / / , or particularly the expression for «g F . Even though the lifetime of the excited radical may be rather short, there is still a finite probability that the unpaired electron spin could be flipped while in this excited state, due to the strong SR interaction in M u C O * . As discussed in chapter 4, in the eyes of the muon there is no difference how the electron spin was flipped. It has no bearing on the consequent hyperfine interaction which depolarizes the muon spin. For any mechanism that relaxes the muon spin indirectly through the hyperfine interaction, the factor / discussed in Chapter 4 and shown in Eq. 5.40 should apply. The same arguments given in Chapter 4 for spin exchange are valid for any electron spin relaxation mechanism that does not relax the muon spin directly. In other words, the factor of / comes about not because of the nature of spin exchange but because that the muon spin is "indirectly" relaxed via the hyperfine interaction. The "3/4 effect" introduced earlier is then expected for Mu + C O if the addition rate is much smaller than the rate due to SR relaxation (or ks[M] ^ S F ) - F ° r reference, it is noted that this effect is not seen in the M u + C 2 H 4 and 213 Mu + N2O reactions, since in both cases the addition rate is much larger compared to that due to the SR relaxation, so kejj is dominated by the addition rate. In case iii) then, on the assumption that the excited radical undergoes similar spin rotation interactions and only the indirect term (which dominates at weak field) of Eq. 4.85 contributes, one can expect the SR relaxation rate constant k$F to be simply proportional to AE J(w,j). In low fields, x —r 0, the transition frequencies (see Eq. 3.6) are no longer equal to w e but either U>M or W Q * (the M u C O hyperfine frequency), i.e., 1 1 + A£ rc° [M] 1 + W l + (w«r c )2 j 1 .[Af] 2 + (uM r°Y + [M]2 + K T°) 2 J where [M] is the total pressure and r ° is the correlation time at unit concentration (pressure). Inserting this into Eq . 5.40, and recalling Eq. 4.41, (5.42) h , fki , fh (5.43) [ M ] 2 + Ci [ M ] 2 + C 2J where ko = kapks/k-a is the low pressure l imit in the usual sense (recall Section 4.1, Eq. 4.18 and discussion in M u + N O results) and , kaAE rc° k l = Hfe ' Ci = ( W M r c 0 ) 2 , C2 = ( u , * r p 0 ) 2 . (5.44) Of course, it could be that excited radicals (MuCO*) have somewhat different frequencies or frequency responses than the corresponding stable radicals, or that some scaling factor has to be considered. The point here, however, is that the effective rate constant has the form of Eq. 5.43 which enables one to extract the important quantity A;n from the measured relaxation rates. Eq. 5.43 is also consistent with the "3/4 effect" at low [CO] and [M]. To illustrate, the T F M u + C O data described earlier (see Appendix B) were fit to Eq . 5.43. Since A r and N2 have different collisional efficiencies (/?), and higher pressure rates were only 214 0 50 100 150 200 250 300 Pressure (atm) Figure 5.28: Effective rate constants for the Mu + C O reaction as a function of total pressure. kejj is defined by A T / [ C O ] . If more than one rate constant is available at a certain total pressure, the weighted average is used. The thick solid line is the fit of Eq. 5.43 to the A r data (open circles). The thin solid line is calculated using the same equation and parameters but with ko scaled up by a factor of 1.5 (cf. text) to compare with the N 2 data (solid diamonds). The dotted line is ko for Ar . The dash-dotted lines are the 2nd and 3rd terms, respectively in Eq . 5.40. Only statistical errors (la) are included and actual errors could be somewhat larger. 215 measured in A r moderator, only a fit of all the A r data in w T F was carried out. The results are shown in F ig. 5.28, which yields: k^1" = (2.1 ±0 .1) x 1 0 - 3 2 c m 6 mo lecu le - 2 s - 1 . Using the same parameters, but increasing ko by a factor of 1.5, based on ratios of /?(iV2)/P(Ar) measured in H + C O reactions [428,445,462,464,468], Eq. 5.43 can also account for the N 2 data, which is also plotted in F ig . 5.28 (thin solid line). The formula fits the N 2 data rather well considering the fact that the low pressure regime, where the N 2 data lie, were poorly sampled in the A r data. This is a rather crude fit since only a single rate ([CO]) was measured at most total pressures, and yet the quality of the fit is gratifying and strongly indicates that the model describes the overall mechanism very well. The relaxation parameters from the fit ( A B and r c) are not trustworthy, though, since only two data points are available and are not quoted here. In principle, one could also find tau° if w's are known and more data points are taken. Now, consider again the "3/4 effect" plot of F ig. 5.24. It can be argued that the drop in this ratio at high pressures can be attributed to the fact that stabilization is becoming more and more important and SR in the excited radical is contributing less and less. Note that the drop-off for the A r data starts above about 60 atm, where the addition becomes dominant (Figs. 5.28 and 5.24) and that the drop-off for N 2 begins at a lower pressure, consistent with its larger collisional efficiency. It would appear from the data that the ratio is leveling off at 1.1 but this is not firmly established. Expectations are that it should go down to 1 at higher pressures. The large uncertainties contained in these data make it difficult to judge. Experiments to reduce the uncertainty and measurements at much higher pressures (up to 500 atm) are planned. The aforementioned "anomalous" pressure dependence in low L F (Figs. 5.26 and 5.27) can also be explained. Again, at weak field, A r is too fast to be seen, only A e / / is observed. A t relatively low pressures, as in wTF , the rate increase with both pressures of dopant and moderator (Figs. 5.26 and 5.27) since the rotational relaxation rate constant (k^F) is proportional to total pressure and the rate is the product of the rate constant (k$F) and K a [CO] / f c_ a . At higher total pressures the rate constant decreases with increasing pressure and in pure C O , the total rate is independent of pressure as seen in F ig. 5.26 and Appendix B. A similar but not entirely the same 216 pressure dependence of relaxation rate seen in the Mu -f N 2 0 reaction at weak L F (Fig. 5.13) is probably also due to the fact that it is A e / / that is observed. However, in the Mu -f N 2 0 reaction, (rcgF) is relatively smaller so it is the chemical reaction rate that dominates A e / / . In this case it needs to be pointed out that the SR relaxation contribution to Xeff can only come from the M u N 2 0 * excited radical (which is the intermediate complex of both product channels) not MuO. Although MuO (the main product at low pressures) is expected to have a large AE parameter, judging from the coupling constant known for the analogous O H radical [146,481], the M u N 2 0 radical is likely to have a much smaller AE due to its much larger moment of inertia [399]. It is also likely that the anisotropy in the g-tensor, Age, is considerably reduced in comparison with that for M u O . 1 1 Cautiously, from the fit given in Fig. 5.28, the low pressure l imit for the addition rate constant in A r can be tentatively taken as stated above, kMu = (2.1 ± 0.5) x 1 0 - 3 2 c m 6 mo lecu le - 2 s - 1 . The large error is assigned somewhat arbitrarily but it is felt necessary, considering the small number of data points measured. This rate constant is much faster than the corresponding constant for H + C O , k" = 1.5 x 1 0 - 3 4 c m 6 molecu le - 2 s " 1 , recommended by Baulch [428] for A r at 300K, indicating a large kinetic isotope effect KIE(Mu,H) « 140 in the low-pressure l imiting rate constant, ko. This is in marked contrast with that reported in Section 5.4 in this thesis for the M u + N O reaction, where this same K I E is 0.23. However, this difference is not surprising since the electronic barrier to reaction in the the H(Mu) + N O reaction is essentially zero and hence tunneling has little effect on the overall reaction rate. In contrast, for H(Mu) + C O , the reaction barrier is much higher, about 6 ± 2 kJ /mo l [249,474] and the probability of tunneling is considerably enhanced, seemingly consistent with the large K I E observed. However, this interpretation is not so straightforward. As pointed out earlier in the discussion of H(Mu) + N O addition, k0 = kak,/k-a and, in principle, kinetic mass effects can affect "Contrary to both Mu + CO and Mu + N 2 0 , in the Mu + C 2 H 4 reaction, the fc0 is so large (compared to A r) that K is very small therefore only one relaxation term (e*rt) is observed through out the field range studied. 217 all three rate constants. If the addition (ka) and dissociation (k-a) channels can be represented as a true equilibrium, as in H(Mu)NO, then ko = Kks and all the dynamical mass effects would originate with k,. In the case of H(Mu) + N O treated earlier, estimates give k^u < 0 . 2 K " and this may also be reasonable here, depending on the frequency of the bending degree of freedom. In any event, the dramatic K I E seen here for Mu(H) + C O cannot be understood on this basis meaning either that k^ <g. k^a since it is strongly due to specific resonances, or that k™" ^ k^, or both, the latter bearing directly on the importance of quantum tunneling. Wagner and Bowman [188] have calculated the rate constants for H(D) + C O addition reactions with the isolated resonance ( R R K M ) model, and predicted that the KIE(H,D) in the low pressure l imit is 1.4 at room temperature. The same calculation also predicted the onset of the high pressure l imit to be above 100 atm He for H + C O , while their more conventional R R K M model predicts it to be around 10,000 atm, consistent with estimates mentioned earlier. Although the trend in the low-pressure K I E is correct, the magnitude is obviously far too small. The M u experimental result, even allowing for a considerable systematic error, indicates that the tunneling effect, which was not included in the isolated resonance model, cannot be ignored. The linear pressure dependence seen in the effective rate constant (« e / / , F ig . 5.28) up to 270 atm (scaling from the above calculated H value, this pressure should be close to the high pressure limit) means that the M u reaction is still in the low pressure regime and the high pressure l imit must be at much higher pressures, indicated in the case of M u + N O as well. The above discrepancies between theory and experiment is perhaps the result of overlooking some resonances, especially those below the barrier to addition [193]. Such effects wil l be of paramount importance to M u + C O addition. It is also pointed out by Gazdy et al in Ref. [193] that the two lowest resonances have widths 3 orders of magnitude lower than any other resonances, which is probably related to the fact that substantial tunneling is involved. Putting this another way, the dissociation channel and hence the magnitude of «_„ may be highly n o n - R R K M , thus possibly also explaining the magnitude of the K I E seen in k^/k^ ~ 140. Further experimental work is required but theoretical (resonance) calculations of M u + C O addition are also urgently required. 218 5.5.3 S u m m a r y and Future Exper iments In conclusion, addition and SR relaxation rates of M u + C O in both T F and L F at A r (N2) moderate pressures from 6.5 to 272 atm and room temperature were reported. These preliminary results show that the phenomenological collisional relaxation model ( 4.85 and 4.86) can be used to explain the peculiarities of this system but further studies of both the chemical reaction kinetics and the SR interaction dynamics are required. At the highest pressure measured (272 atm), the addition reaction is still in the low pressure regime. The measured low pressure termolecular addition rate constant is, tentatively, (2.1 ±0 .5) x 1 0 - 3 2 c m 6 m o l e c u l e - 2 s - 1 . The K I E between Mu and H is about 140 in favor of M u , suggesting substantial tunneling in the addition step (ka), but this is not clearly established since ko = kak,/k-a and mass effects on all three rate constants need to be considered, as in the H(Mu) + N O case previously discussed. Even though theoretical predictions for H(D) + C O are qualitatively consistent with the M u results, quantitatively, the magnitudes of both the K I E and the range of low pressure regime seem much larger than expected from theory, notwithstanding some uncertainties both in the theory and the experiments. Certainly any discrepancies between theory and experiment must be confirmed by more reliable data, but just as important, accurate theoretical calculations of the M u + C O reaction are required. The model for fitting the T F relaxation rate given herein also needs to be checked against different experimental conditions. As mentioned, new measurements at higher pressures and more detailed measurements at lower pressures in T F as well as development of fitting procedures for L F data are currently underway at T R I U M F . This may allow a direct measure of the high pressure l imit, k<x> = ka, which can be more easily compared with theory and should reveal in the clearest way the importance of quantum tunneling. Studies of the temperature dependence are also important. 219 Chapter 6 Summary and Conclusions This thesis describes, in considerable detail, present understanding of the three underlying mechanisms of muonium spin relaxation in the gas phase: namely, spin exchange, chemical reaction and collisional spin relaxation, with emphasis on their effects on the muon spin polarization, the primary experimental signature in pSR studies of this nature. Models and formulae describing the effects of these processes on the muon spin polarization were developed and employed to extract reaction rate constants and cross sections as well as other kinetic parameters in reactions of M u with small molecules, in both longitudinal (LF) and transverse (TF) magnetic fields. A new experimental technique, the "3/4 effect" of spin exchange is presented and its usefulness is demonstrated with experimental examples. The experimental data obtained in this thesis work are consistent with theory and the models so introduced and it has been shown that pSR can indeed be used as a very powerful and important tool to investigate reaction processes involving small molecules and radicals, particularly in the realm of isotopic-effect studies. Not merely a by-product of testing theoretical models, the experimental studies of M u and Mu-radical interactions in the gas phase are themselves very important. The kinetics information obtained in these studies are in all cases useful and unique and, in some cases, highly desirable contributions to studies of chemical kinetics and spin dynamics in general. The primary goal of examining the isotope effects in these reaction processes has been met, with measurements of spin exchange interactions, addition/combination reactions, abstraction reactions and collisional spin 220 relaxation of M u and Mu-radicals with a variety of small molecules. In each case, significant isotope effects were observed and important theoretical developments considered, which wil l be summarized below. 6.1 Spin Exchange Interaction Two systems involving direct (inter molecular) spin exchange interactions have been studied: Mu + N O and M u + Cs. The M u + N O system was studied using the L F technique and the relaxation rates were measured over a wide range of magnetic fields (~ 1 to 15 kG) and pressures (up to 60 atm moderator) at room temperature. The cross sections obtained agree well with a previously reported value measured in T F [58]. The M u + Cs system was studied using the T F technique and the relaxation rates were measured over the temperature range of 543 to 643 K. Very little temperature dependence was observed. This is the first detailed direct measurement of H-isotope-Alkali-metal atom spin exchange interactions. Comparing with experimental and/or theoretical studies of corresponding H atom systems, the kinetic isotope effects (KIE) in both systems are about 0.3, in favor of the heavier atom, H. This is consistent with previous experimental measurements for Mu + N O and Mu 4- 0 2 systems [56,58,90]. Since spin exchange is a quasi-elastic process, any isotope effect is a manifestation of differing partial-wave scattering amplitudes, likely sensitive to the intermediate or even short-range nature of the scattering potential. In the case of M u + Cs, the effect of specific resonances or orbiting is perhaps important. Both of these results invite theoretical calculations of these cross sections in order to further investigate the nature of the isotope effect seen. It is noted that such isotope effects are most likely revealed in a comparison between H and Mu , with their factor of nine difference in mass, than any other isotopic comparison. In general though there are almost no reports of isotopic effects on spin exchange in the literature. As an unusual but powerful technique for studying chemical reactions and spin dynamics, (iSR is itself sti l l evolving. The new "3/4-effect" technique has proven to be very useful and promising for further studies. Although, in principle, one can deduce both chemical reaction and 221 spin exchange rates from this technique, further refinement and exploration of this technique are required before more accurate quantitative kinetic information can be obtained from it. 6.2 Chemical Reactions Most systems studied in this thesis involve chemical reactions, particularly, addition/combination reactions, which include: N O , N 2 0 and C O (also C 2 H 4 and C 4 H 8 , but the chemical reaction aspect of these systems was not discussed in this thesis). Reaction rates in all cases are measured over a wide range of pressures (1 to 60 atm, up to 272 atm for CO) and in one case (N2O) also over a range of temperatures (300-600 K ) . Dramatic isotope effects are observed in every one of these systems. The relatively small "inverse" K I E in an essentially fission-like reaction found in M u + N O was expected since the tunneling effect is not important here and both zero-point-energy and density-of-states effects favor the heavier (H) atom. In cases where large reaction barriers are involved (Mu + N2O, M u -I- CO) , quantum tunneling of the much lighter Mu-atom dominates, and the K IE 's are always "normal" (&MU > &H) but of unprecedented magnitude, on the order of 100 even at room temperature. These data present the first observation of such a large tunneling effect at room temperature in addition reaction systems in the gas phase. The available experimental H(D) isotope effects also exhibit the same trend as observed here, though on a much smaller scale. Most of these results seem to be consistent at least with the trends in theoretical studies of these reactions, but these have only been carried out to date for H(D) systems [168,174,187,188,190,193,264,311,421,434,470,471,482]. New calculations both of the potential energy surface and dynamics of these reaction systems are needed to help answer some of the questions raised in this thesis and to reconcile many of the discrepancies between experiment and current theory: e.g., the relative importance of each channel hence the dependence of total reaction rate constants on both temperature and pressure in the M u + N 2 0 reaction, the magnitude of the K IE 's observed in the Mu + N2O reaction as well as in the M u + C O reaction, the onset of the high pressure regime in the M u + N O and M u + C O addition reactions. In addition, it is expected that such calculations wil l reveal defects in the theory not detectable in a 222 comparison of H(D) reactions. 6.3 Collisional Relaxation of Mu-radicals Collisional spin relaxation phenomena of Mu-radical plays an important role in fiSK studies of reaction systems and has to be understood before useful and reliable information can be extracted from the measured relaxation rates. A phenomenological model is presented which describes the spin relaxation of the muon in a Mu-substituted radical due to collisional relaxation processes. Some experimental results for larger molecules ( M U C 2 H 4 and MUC4H .8) were obtained which are well described by this model in terms of three parameters: AE, accounting for electron spin relaxation processes which are indirectly coupled to the muon, and AME and AM, which are directly coupled but with different frequency responses. Furthermore, this model was employed to, successfully, explain and analyze the small molecule systems studied in this thesis that also form Mu-radicals, namely, M u + N 2 0 and particularly M u + C O . Two distinct relaxation components were observed in the L F Mu + N 2 0 reactions, consistent with the fact that two different M u radicals (MuO and M U N 2 O ) are formed. Although the analysis of these L F data is yet to be complete, preliminary results show that these data show the pressure and field dependence expected from the relaxation model. It is because the muon spins in these radicals relax quickly at low fields by collisional relaxation so that w T F measurements of the chemical reaction rates are possible. The M u C O radical has no nuclear moment but the electron spin rotation coupling is very strong because it has both a small moment of inertia and a large deviation in electron g-factor. The observed fast relaxation in w T F and in L F are consistent with this expectation. Although a more rigorous derivation of the formulas, from first principles, is desirable, the current model does serve as a useful bridge linking the observed relaxation rates and the physical, chemical and magnetic properties of the reactants and offers invaluable insight into the underlying mechanisms causing muon spin relaxation. Of course, further development of this model is necessary and interpretations of the parameters in this model are also still under study [231]. A much preferred result would be from an exact Hamiltonian and Boltzmann equation approach in this regard, 223 which is also currently under development [157]. 6.4 Future As in many thesis studies, often as many questions are raised as answered, indicating the need for further studies. There are still many aspects of the reactions studied in this thesis that need to be addressed. More data are required in many cases before a definite conclusion can be drawn. The spin exchange of M u + Cs, is self-contained but theoretical calculations are lacking. In the M u + N 2 0 study, the pronounced K I E seen is the largest ever seen at room temperature in gases and again requires new theoretical studies, extending those of H + N 2 0 of Ref. [311]. It is also important to establish the high pressure limit for Mu + N2O formation, which, in principle, can be seen in a L F study. For Mu + N O , only room temperature and low pressure regime data are available. Both high temperature and particularly high pressure studies are required, again, in order to establish the high pressure l imit, which will allow a direct comparison with theory. And the study of the Mu 4- C O reaction is far from complete. I have, however, spent more than my share of time in school and have written, as it is now, perhaps the longest thesis in 1995 at U B C . It is therefore sensible to stop here and look forward. What to do next? Find a job, pal! Seriously, though, the most immediate goal should be to finish the study of the M u 4- C O reaction, in light of the understanding brought forward by the models described in this thesis. Then, the more complex system, M u 4- O2, not addressed herein, can be studied to compare with the H 4- O2 reaction which is one of the single most important and studied reaction systems in chemistry, to provide invaluable information on kinetic isotope effects. It is also possible now, with the new high pressure target described, to measure the fall-off region and the high pressure l imit for addition rate constants over a wide range of pressures, up to > 500 atm, for all of the reaction systems described in this thesis, particularly, M u 4- N O , M u 4- C O and M u 4- N 2 0 , all of which are important from a theoretical point of view, and to the study of combustion kinetics in general. On a broader outlook, the reaction kinetics studies using fiSK has been demonstrated again as a useful and complementary technique to other techniques in chemical kinetics studies. 224 Probably the single greatest contribution of pSR to the chemical kinetics community in general is its ability to facilitate a measurement of large kinetic isotope effects, shown again in this thesis, some of which are of unprecedented magnitude, due to the remarkable mass ratio, puu/^H ~ 1/9. These results hopefully will spur new and detailed theoretical studies to account for the enormous KIE 's observed here. 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