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Application of the muonium spin rotation technique to a study of the gas phase chemical kinetics of muonium… Garner, David Michael 1979

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APPLICATION OF THE MUONIUM SPIN ROTATION TECHNIQUE TO A STUDY OF THE GAS PHASE CHEMICAL KINETICS OF MUONIUM REACTIONS WITH THE HALOGENS AND HYDROGEN HALIDES by DAVID MICHAEL GARNER A THESIS SUBMITTED IN THE REQUIREMENTS DOCTOR OF PARTIAL FULFILLMENT OF FOR THE DEGREE OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Department of Chemistry) We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA June, 1979 (c) David Michael Garner, 197 9 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree a t the U n i v e r s i t y of B r i t i s h C o l u m b i a , I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by the Head of my Department o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department nf Chemistry  The U n i v e r s i t y o f B r i t i s h Columbia 2075 Wesbrook P l a c e V a n c o u v e r , Canada V6T 1W5 24 September, 1979 ABSTRACT Muonium (Mu) i s the atom formed by an electron bound to a pos i t i v e muon "nucleus" (charge:+l, spin:l/2, l i f e t i m e : 2.2 ys). Since muons are 207 times as massive as electrons, the reduced mass of Mu i s 0.996 that of the hydrogen atom, and the Bohr r a d i i and i o n i z a t i o n potentials of Mu and H are e s s e n t i a l l y the same. Therefore, the chemical behaviour of the Mu atom i s that of a l i g h t H isotope (m M u = 1/9 niH) with a greatly enhanced s e n s i t i v i t y to H isotope"effects. Mu reaction rates are measured by a method c a l l e d "Muonium Spin Rotation" (MSR) which resembles conventional resonance techniques such as NMR or ESR i n that i t monitors the character-i s t i c Larmor precession of the Mu atom. However, unlike NMR or ESR, the MSR method does not detect the Mu Larmor precession by resonant power absorption, but rather through the peculiar spin dependent radioactive decay of the muon i t s e l f . The t h e o r e t i c a l basis for the application of the MSR technique to the measurement of muonium reaction rates i s derived. An extensive discussion i s given to the p r a c t i c a l aspects of the experimental implementation of the MSR technique. Rate constants and a c t i v a t i o n energies are reported for the gas phase reactions: Mu + F ^ MuF + F and Mu + C ^ + MuCl + Cl between 3 00 and 4 0OK, and room temperature rate constants are reported for the reactions: Mu + Br 2 MuBr + Br and • -Mu + HX ? * * / X = C l , Br, I. While i n most of these MuX + H systems Mu reacts considerably faster than the heavier H isotopes, attention i s focussed on hydrogen isotope e f f e c t s i n the Mu + F 0 and Mu + Cl,, reactions. This discussion i s based on the extensive t h e o r e t i c a l investigations of Connor et aJL. , which show the Mu + reaction to be dominated by quantum mechanical tunnelling at room temperature. Experimentally, quantum tunnelling manifests i t s e l f i n t h i s reaction by producing two dramatic isotope effects at 30OK: (1) the b i -molecular rate constant for the Mu reaction (1.4 x 10"*"^  1/mole-s) i s at least six times that for the analogous H atom reaction, and (2) the apparent Arrhenius ac t i v a t i o n energy of t h i s Mu reaction (0.9 kcal/mole) i s less than half of that for H + F In contrast, the Mu + reaction does not show any such strong isotope e f f e c t s at 300K: (1) the bimolecular rate constant for Mu + (5.1 x l O 1 ^ 1/mole-s) i s no more than four times that of the analogous H reaction, and (2) the apparent ac t i v a t i o n energies for both Mu and H reactions are the same (1.4 kcal/mole) Preliminary calculations of Connor et a l . on Mu + C ^ suggest that c l a s s i c a l "wall r e f l e c t i o n " p a r t i a l l y o f f s e t s any rate enhancement due to quantum tunnelling. Quantitative isotope effects cannot be defined for the Mu + B r 2 and Mu + HX reactions and t h e i r hydrogen isotopic analogues because of the absence of s u f f i c i e n t experimental and t h e o r e t i c a l data; these reactions are discussed in terms of the general theory of isotope e f f e c t s . - i v -Table of Contents Chapter I - Introduction 1 A Positive Muons and the y SR Method 1 B Muonium and the MSR Method 10 C Muonium Chemistry - An H i s t o r i c a l Background 15 D Organization of the Dissertation 20 Chapter II - Experimental Details 23 A TRIUMF and the M2 0 Muon Beam Line 23 B The Gas Target, Counters, and Magnetic F i e l d 3 2 C Data Ac q u i s i t i o n 37 D Data Analysis 49 Chapter III - Theoretical Background 58 A Introduction 58 B Potential Energy Surfaces 60 (i) Semi-Empirical Potential Energy Surfaces 60 (i i ) Contour Plots of the Potential Energy Surface for the Reaction A + B C + A B + C 65 ( i i i ) Potential Energy Surfaces for the Reactions: Y + X„ ->• YX + X, Y = Mu, H, D, T; X = F, C l , Br 7 3 C Energy 82 (i) C l a s s i c a l Trajectories 83 ( i i ) Q u a s i c l a s s i c a l and Quantum Mechanical Trajectories 87 ( i i i ) Transition State Theory 90 (iv) Reaction Enthalpy 92 (v) Reaction Activation Energy 94 (vi) Potential Energy Surfaces for the Reactions: YX + TT Y + HX YH + X ' Y = M U ' H ' °' T ; X = C 1 ' B r ' 1 9 7 D Trajectory Calculations 109 E Transition State Theory 118 F Tunnelling 122 Chapter IV - Experimental Results and Their Interpretation 127 A Mu + F 2 -> MuF + F 12 9 B Mu + C l 2 -> MuCl + C l 157 C Mu + B r 2 -y MuBr + Br 171 D M U + HC1 < J J ^ V c ? 17 8 T-. > TTT, MuBr + H . _ _ E Mu + HBr < M u H + B r 185 F Mu + HI < M U I + H 190 Chapter V - Summary and Conclusions 194 A Summary 194 B Past Perspective 195 C Future Perspective 198 D Closing Remarks 2 01 -v-Table of Contents (Cont'd) Literature Cited 2 03 Chapter I 2 0 3 Chapter II 2 0 6 Chapter III 2 0 7 Chapter IV 2 1 2 Chapter V 2 1 6 Appendix I 2 ^ 7 Appendix II 218 Appendix III 2 1 ^ Appendix I - The Time Evolution of the u + Spin Pol a r i z a t i o n i n Muonium i n a Transverse Magnetic F i e l d 220 A State Vectors 220 B Time Evolution of the Mu^  States 223 C Time Evolution of the y Spin p o l a r i z a t i o n i n Mu 228 D Experimental Implications of <P (t)^ > 230 (i) Very Weak Fie l d s (<_ 10 gauss) - the Standard MSR Signal 231 ( i i ) Intermediate F i e l d s (10 < B < 150 gauss) -Two Frequency Muonium 23 4 ( i i i ) High F i e l d s (_> 150 gauss) 236 Appendix II - The E f f e c t of Chemical Reaction on the Muon Po l a r i z a t i o n 23 9 A General 23 9 B Generation of a Coherent Diamagnetic Muon Background: X ->- 15 y s ~ l , B >^ 10 gauss 242 Appendix III - Data Acquisition with High Current Muon Beams: Theory and Practice 250 A The Optimal Good Event Rate 2 50 B Spectral Distortions due to Muon Pile-up 2 53 (i) Pre-y^ Muons and x^: .100% Decay Positron Detection E f f i c i e n c y . . 257 ( i i ) Pre-y. Muons and x : e Decay Positron . I y -l Detection E f f i c i e n c y 261 ( i i i ) Pre-y^ Muons and the MSR Signal: e Decay Positron Detection E f f i c i e n c y 269 (iv) Post-y^ Muons and x : 100% Decay Positron Detection E f f i c i e n c y 278 (v) Post-y. Muons and x : e Decay Positron H i y J Detection E f f i c i e n c y 280 (vi) Post-y^ Muons and the MSR Signal: e ;Decay Positron' Detection E f f i c i e n c y 284 C The MSR Data Acquisition. ;System 2 94 (i) The Electronic Logic 294 ( i i ) The Microprogrammed Branch Driver 2 98 - v i -L i s t of F i g u r e s Chapter I 1 Energy spectrum of p o s i t r o n s from muon decay and the energy dependence of the asymmetry parameter 5 2 A t y p i c a l y SR time histogram 9 3 A t y p i c a l MSR time histogram 13 4 The MSR s i g n a l 16 Chapter I I 5 The TRIUMF C y c l o t r o n and experimental f a c i l i t i e s (1977) 24 6 The M20 beamline ( d e t a i l ) 27 7 The gas phase MSR t a r g e t apparatus 33 8 N i t r o g e n versus argon as moderator gases 3 6 9 MSR d a t a a c q u i s i t i o n l o g i c ( s i m p l i f i e d ) 3 9 10 The s p e c t r a l d i s t o r t i o n due to " e a r l y " second y r e j e c t i o n 45 Chapter I I I 11 A p o t e n t i a l contour map f o r the exothermic c o l l i n e a r A + BC AB + C r e a c t i o n 61 12 P o t e n t i a l energy s u r f a c e f o r the Y + r e a c t i o n 71 13 P o t e n t i a l energy s u r f a c e s f o r the c o l l i n e a r Y + F 2 r e a c t i o n p l o t t e d i n mass weighted c o o r d i n a t e s 74 14 The b o t t l e n e c k e f f e c t 77 15 Mass weighted p o t e n t i a l energy s u r f a c e f o r the c o l l i n e a r Mu + r e a c t i o n 86 16 P o t e n t i a l s u r f a c e s f o r the c o l l i n e a r H + HC1 -> H 2 + C l r e a c t i o n 100 17 Isotope e f f e c t s i n t r a n s i t i o n s t a t e v i b r a t i o n s from mass v a r i a t i o n s of atom Y f o r the r e a c t i o n Y + AB -> YA + B 12 3 18 T u n n e l l i n g t r a n s m i s s i o n c o e f f i c i e n t s f o r the tr u n c a t e d B e l l p a r a b o l a and the E c k a r t b a r r i e r 12 6 Chapter IV 19 The e f f e c t of temperature on the Mu + F 0 MSR r e l a x a t i o n r a t e s *" 132 20 Experimental A r r h e n i u s p l o t s f o r Y + F_ r e a c t i o n s , Y = Mu, H 13 3 21 C o l l i n e a r quantum and q u a s i c l a s s i c a l t o t a l r e a c t i o n p r o b a b i l i t i e s f o r Y + F 2 ( v = 0 , l ) 142 22 Integrand of the quantum c o l l i n e a r r a t e constant at 300 and 900K f o r Y + F 2(v=0) 144 2 3 Integrand f o r the c o l l i n e a r quantum and q u a s i -c l a s s i c a l r a t e constant a t 300K f o r Y + F 2 ( v = l ) 145 24 Non-reactive q u a s i c l a s s i c a l t r a j e c t o r i e s f o r Mu + F 2 (v=0) on the mass weighted LEPS s u r f a c e 147 25 A r r h e n i u s p l o t s f o r the c o l l i n e a r quantum, q u a s i -c l a s s i c a l , and t r a n s i t i o n s t a t e theory r a t e con-s t a n t s f o r Y + F 2 151 2 6 Comparison of quantum and t u n n e l l i n g c o r r e c t e d t r a n s i t i o n s t a t e theory r e a c t i o n p r o b a b i l i t i e s f o r Y + F 2(v=0) 153 - v i i -L i s t of Figures (Cont'd) Chapter IV (Cont'd) 27 Comparison of quantum and t r a n s i t i o n state theory Arrhenius plots for the c o l l i n e a r Mu + F 2(v=0,l) 154 28 Collin e a r quantum r e l a t i v e population d i s t r i b u t i o n s of product v i b r a t i o n a l states for Y + F„(v=0) 156 2 9 The e f f e c t of temperature on the Mu + C I 2 MSR relaxation rates 159 30 Experimental Arrhenius plots for Y + C l 2 reactions, Y = Mu, H, D 16 0 31 Tunnelling corrected t r a n s i t i o n state theory Arrhenius plots for c o l l i n e a r Y + F„ and Y + Cl„ 169 3 2 MSR relaxation rates as a function or Br„ concen-t r a t i o n i n argon moderator at 2 95K 173 33 MSR relaxation rates as a function of HCl concen-t r a t i o n in N 2 moderator at 295K 180 34 The MSR signals in pure N 2 versus pure HCl at 295K 181 3 5 MSR relaxation rates as a function of HI concen-t r a t i o n in argon and N 2 moderator at 295K 192 Chapter V 3 6 MSR signals i n pure argon and i n B r ? i n argon at 1.3 gauss and 295K; data obtained at LBL 196 Appendix I 1-1 Breit-Rabi diagram of the energy eigenstates of muonium i n a magnetic f i e l d + 2 26 1-2 The time evolution of the y spin p o l a r i z a t i o n in a 100 gauss transverse magnetic f i e l d 235 1-3 "Two frequency precession" of the muon i n muonium in fused quartz at 95 gauss 237 Appendix II I I - l The e f f e c t of chemical reaction on the muonium signal 241 II-2 The lin e a r dependence of the relaxation rate of the muonium signal on reagent concentration 243 II-3 The generation of a coherent diamagnetic muon back-ground signal by f a s t chemical reactions of muonium 247 II-4 The dependence of the amplitude of the "residual muon p o l a r i z a t i o n " on muonium reaction rate at 7.5 gauss 248 Appendix III I I I - l The net good event rate as a function of beam current for various muon decay gates 2 52 III-2 The ef f e c t s of pre-y. muons on the apparent muon li f e t i m e with e = 100%. 262 III-3 Logarithmic plot of Figure III-2 263 III-4 The e f f e c t s of pre-y. muons on the apparent muon li f e t i m e with e = 10%" 267 III-5 Logarithmic plots of Figure III-4 268 III-6 The ef f e c t s of pre-y. muons on the MSR signal with e = 10% 1 273 - v i i i -L i s t of Figures (Cont'd) Appendix III (Cont'd) III-7 The eff e c t s of pre-y. muons on the MSR signal with e = 10% (detail}" 274 III-8 The origi n s of the l i f e t i m e d i s t o r t i o n s due to pre-y. muons 27 6 III-9 A possible example of the e f f e c t of pre-y. muons 277 111-10 The eff e c t s of post-y. muons on the apparent muon li f e t i m e with e= 100% 1 281 I I I - l l Logarithmic plots of Figure 111-10 282 111-12 The ef f e c t s of post-y. muons on the apparent muon li f e t i m e with e = 10% 1 285 111-13 Logarithmic plots of Figure 111-12 286 111-14 The eff e c t s of post-y. muons on the MSR signal with e = 10% 1 288 111-15 The eff e c t s of post-y. muons on the MSR signal with e = 10% ( d e t a i l ) 1 28 9 111-16 The origi n s of l i f e t i m e d i s t o r t i o n s due to post-y. muons 2 92 111-17 Relaxation e f f e c t s i n the MSR signal due to muon pile-up 293 111-18 The TRIUMF MSR data acquisition l o g i c (detail) 2 95 111-19 Pulse timing and event i d e n t i f i c a t i o n for the l o g i c of Figure 111-18 297 111-2 0 Flow diagram of the TRIUMF MBD programme 3 07 - i x -L i s t o f T a b l e s C h a p t e r I I P r o p e r t i e s o f p o s i t i v e muons 2 I I P r o p e r t i e s o f muonium 11 C h a p t e r I I I I I N o m i n a l f o r w a r d and backward y momenta and v e l o c i t i e s a s a f u n c t i o n o f d e c a y i n g TT momentum 2 8 C h a p t e r I I I IV E n e r g y d e f i n i t i o n s f o r t h e Y + F» r e a c t i o n s , Y = Mu, H 93 V Bond d i s s o c i a t i o n e n e r g i e s , z e r o p o i n t e n e r g i e s , and r e a c t i o n e n t h a l p i e s 95 C h a p t e r IV VI Summary o f t h e e x p e r i m e n t a l r a t e p a r a m e t e r s f o r Mu and H r e a c t i o n s i n t h e g a s p h a s e 128 V I I MSR r e l a x a t i o n r a t e s f o r t h e r e a c t i o n Mu + F„ -> MuF + F 131 V I I I C a l c u l a t e d r a t e c o n s t a n t s f o r t h e c o l l i n e a r Y + F„ -> YF + F r e a c t i o n s 13 6 IX C a l c u l a t e d r a t e c o n s t a n t r a t i o s f o r t h e c o l l i n e a r Y + F« + YF + F r e a c t i o n s 137 X C a l c u l a t e d a c t i v a t i o n e n e r g i e s f o r t h e c o l l i n e a r Y + F 2 + YF + F r e a c t i o n s 138 XI MSR r e l a x a t i o n r a t e s f o r t h e r e a c t i o n Mu + Cl„ -> MuCl + C l 158 X I I E x p e r i m e n t a l r a t e p a r a m e t e r s f o r t h e r e a c t i o n s Y + Cl„ -* YC1 + C l , Y = Mu, H, D 162 X I I I MSR r e l a x a t i o n r a t e s f o r t h e r e a c t i o n Mu + Br„ -> MuBr + B r 17 2 XIV MSR r e l a x a t i o n r a t e s f o r t h e t o t a l Mu + H C l r e a c t i o n a t 295K 17 9 XV MSR r e l a x a t i o n r a t e s f o r t h e t o t a l Mu + HBr r e a c t i o n a t 295K 186 XIV E x p e r i m e n t a l r e a c t i o n r a t e p a r a m e t e r s f o r Y + Y 1 Br ^ Y Y ' + B r ~*YBr + Y 1 , Y = Mu, H, and D 187 XVII MSR r e l a x a t i o n r a t e s f o r t h e t o t a l Mu + HI r e a c t i o n a t 295K 191 A p p e n d i x I 1-1 V a l u e s o f m a g n e t i c f i e l d d e p e n d e n t v a r i a b l e s i n e q u a t i o n s 1(8) and 1(12) 232 -x-Acknowledgement I t i s a p l e a s u r e to acknowledge the support of my r e s e a r c h d i r e c t o r , Dr. Don Fleming, who i s always more of a " c o l l e a g u e " than a " d i r e c t o r . " His a d v i c e , always given w i t h h i s c h a r a c t e r i s t i c enthusiasm, i s u s u a l l y accompanied by a p a t i e n t r e s p e c t f o r my independence. Research a t a meson f a c t o r y i s o n l y p o s s i b l e w i t h the e f f o r t s of an enormous number of people - u n f o r t u n a t e l y i t i s o n l y p o s s i b l e to mention a few. In l e a r n i n g the "ropes" of experimental n u c l e a r p h y s i c s , i t seems to me t h a t my main teach e r s were Burt P i f e r of the U n i v e r s i t y of A r i z o n a , Glen M a r s h a l l , J e s s Brewer, and P r o f e s s o r John Warren, of the UBC P h y s i c s Department. A s p e c i a l acknowledgement should be given to Ryu Hayano of the U n i v e r s i t y of Tokyo; the e n t i r e TRIUMF ySR group i s indebted to h i s genius i n c r e a t i n g our e x c e e d i n g l y powerful data a q u i s i t i o n system which has helped advance ySR i n t o a new g e n e r a t i o n . I a l s o wish to thank my p a r e n t s , which I have never before p r o p e r l y done, f o r encouraging and s u p p o r t i n g my e d u c a t i o n , even a f t e r i t became f a r removed from t h e i r e xperience. F i n a l l y , I thank Rosa Ho (who, q u i t e unsuspec-t i n g l y , became a muonium chemist h e r s e l f , by marriage) f o r keeping me on course w i t h her l o v e and support whenever t h i s t h e s i s work f e l l i n t o a s t a t e of c r i s i s . As a muonium chemist she i s more of a " d i r e c t o r " than a " c o l l e a g u e . " -1-CHAPTER I - INTRODUCTION _A Positive Muons and the u +SR Method The muon i s an unstable elementary p a r t i c l e that was f i r s t observed as a component of cosmic rays [Anderson (37), Street (3-7)] and which i s now a r t i f i c i a l l y produced with high energy p a r t i c l e accelerators. Some of the properties of positi v e muons (y +) are summarized i n Table I and include: unit charge, spin H, and a mean l i f e t i m e of 2.2 ys. Muons are decay products of pions, which, i n turn, are produced i n the nuclear interactions that take place when a nucleus i s bombarded with high energy p a r t i c l e s such as protons. Typical nuclei used for pion production at accelerators are copper and beryllium, and the minimum proton k i n e t i c energy required for pion production i n such a nucleus i s about 145 MeV, the threshold energy. Positive pions ( T T + ) decay with a mean l i f e t i m e of 26 ns i n the p a r i t y v i o l a t i n g process [Bjorken (64)]: TT -> li + V y (1) which i s exoergic by about 3 4 MeV and produces 4.1 MeV y + . The muon neutrino, v , i s a spin h, p a r t i c l e with zero re s t mass and 100% negative h e l i c i t y . The h e l i c i t y operator i s defined as the dot product of the spin and momentum y\ -— d i r e c t i o n h = o_JB, and has eigenvalues of +1 (positive lei h e l i c i t y ) i n which the spin i s p a r a l l e l to the momentum, and -1 (negative h e l i c i t y ) in which the spin i s a n t i p a r a l l e l to the momentum. In order to conserve linear and angular momentum, the muon formed i n pion (spin 0) decay comes off i n the -2-TABLE I : PROPERTIES OF POSITIVE MUONS CHARGE: +1 SPIN: \ MASS: 105.6596 MeV/c2 =206.7685 me = .0.1126 m P - 0.7570 m + MAGNETIC MOMENT: 4.49048 x 10~ 2 3 erg G _ 1 = 3.18334 y P = 0.004836 y e g-FACTOR: 2.0023318 = 1.000006 g ^ ^e MEAN LIFETIME: 2.1971 ys Y GYROMAGNETIC RATIO, 13.5544 kHz G " 1 2TT COMPTON WAVELENGTH, k: 1.86758 fm =- ft m c y CHARGE RADIUS: <0.01 fm -3-d i r e c t i o n opposite to the neutrino with 100% negative h e l i c i t y as well. This decay process i s s p a t i a l l y i s o t r o p i c i n the rest frame of the pion. That p a r i t y i s viol a t e d i n pion decay i s seen from the fac t that under par i t y , the a x i a l vector a i s unchanged, while the polar vector p becomes -p and thus h, a pseudoscalar, becomes -h. In p r a c t i c a l terms, the muon h e l i c i t y created with the p a r t i c l e ' s b i r t h can be trans-lated into the design of y + beams i n which the muons have a net longitudinal spin p o l a r i z a t i o n . A more detailed dicussion of muon beams i s given i n Chapter I I . When high energy muons interact with matter, they — 9 ' thermalize primarily by io n i z a t i o n processes i n about 10 s and r e t a i n t h e i r spin p o l a r i z a t i o n [Hughes (66), Weissenberg (67), Brewer (75)]. In metals [Brewer (75), Grebinnik (76)] and gases such as He with large i o n i z a t i o n potentials [Stambaugh (74)], y + thermalize as "free" y + ions; i n many other materials, y + end up chemically bound i n diamagnetic environments. The spins of such muons w i l l precess i n a transverse magnetic f i e l d at a frequency which i s proportional + Y y to the y gyro-magnetic r a t i o , ^ = 13.55 kHz/gauss. Because muon beams have a longitudinal spin p o l a r i z a t i o n , a l l of the muons thermalize with the same i n i t i a l phase with respect to spin precession i n a transverse magnetic f i e l d . The unstable y + decays by another p a r i t y v i o l a t i n g process: y + -> e + + u + u (2) K e y where u i s an electron neutrino with negative h e l i c i t y , u e ^ y i s a muon a n t i n e u t r i n o w i t h p o s i t i v e h e l i c i t y , and e' i s a p o s i t r o n w i t h p o s i t i v e h e l i c i t y . Weak i n t e r a c t i o n theory p r e d i c t s t h a t the three-body decay of the y + i s s p a t i a l l y a n i s o t r o p i c with r e s p e c t t o p o s i t r o n emission, which i s p r e f e r e n t i a l l y along the d i r e c t i o n of the y + s p i n . T h i s q u a l i t a t i v e behavior was f i r s t confirmed e x p e r i m e n t a l l y by Garwin (57). The t h e o r e t i c a l p o s i t r o n decay spectrum i s give n by the e x p r e s s i o n [Sachs (75)]: dR(w,6) = w2 {(3-2w) - P(l-2w)cos6} dwdfi 2TT = C {1 + DcosG} where w = E / E M a x i - s the p o s i t r o n energy i n u n i t s of the maximal p o s s i b l e energy, E M a x = k™-^ = 52.8 MeV, 9 i s the angle between the s p i n of the decaying muon and i t s p o s i t r o n t r a j e c t o r y , and P i s the degree of s p i n p o l a r i z a t i o n of the decaying muons. The p o s i t r o n energy spectrum and the asymmetry parameter, D, f o r P=l are shown i n F i g u r e 1. In p r a c t i c e , the p o s i t r o n s are d e t e c t e d w i t h an e f f i c i e n c y e (w) which i s not constant over t h e i r energy range. The observed p r o b a b i l i t y d i s t r i b u t i o n then becomes [Brewer (75), Weissenberg (67)] d-R f 1 d-R (w.>80 e(w)dw d!T ^0 dwdQ = l _ e (1 + Acos6) 4TT I f p o s i t r o n s of a l l e n e r g i e s were d e t e c t e d w i t h the same e f f i c i e n c y , the observed average asymmetry, A, would be I P . In p r a c t i c e , the d e t e c t i o n e f f i c i e n c y of low energy T -5-w= E / E m a FIGURE 1: Energy spectrum of positrons from muon decay (upper curve) and energy dependance of the asymmetry parameter for 100% beam p o l a r i z a t i o n (P=l; lower curve). The energy i s given as a f r a c t i o n of the maximal possible energy, E =52.8 MeV. ^ max -6-e + i s reduced and the lowest energy e + are absorbed by matter before reaching the detectors r e s u l t i n g i n an observed A larger than 1 P. This e f f e c t i s o f f s e t , however, by the 3 reduction i n P due to kinematic depolarization (real muon beams are not 100% polarized) and due to averaging over f i n i t e detector s o l i d angle. In most muon experiments, the beam p o l a r i z a t i o n , positron detection e f f i c i e n c y and s o l i d angle corrections are not e x p l i c i t l y known and the resultant e f f e c t i v e muon asymmetry, A , i s treated empirically i n the expression: R(9) = 1 + A cos9 (3) Since the average decay positron energy i s about 3 5 MeV, _2 corresponding to a radiation length of 15 g. cm i n Pb,, most positrons are observable even i f the muon decay occurs deep inside a substantial target. The time d i f f e r e n t i a l measurement of the asymmetric decay of a spin polarized ensemble of pos i t i v e muons precessing i n a transverse magnetic f i e l d forms the basis of the y +SR technique. The acronym, y +SR, stands for "muon spin rotation" and was coined to draw attention to the strong resemblance i n information content that t h i s method bears to the fa m i l i a r resonance techniques of NMR and ESR. Except i n special variations such as the stroboscopic method [Schenck (76)], y +SR examines one muon at a time using counting techniques common to experimental nuclear physics. The phrase "muon ensemble" i n the present discussion, then, refers to an ensemble i n time rather than i n space. In a y +SR experiment, a lo n g i t u d i n a l l y spin polarized - v -+ y passes from the beam channel through a p l a s t i c s c i n t i l l a t o r counter a r r a y and t h e r m a l i z e s i n a t a r g e t m a t e r i a l of i n t e r e s t . The counters are arranged t o i d e n t i f y muons which stop i n the t a r g e t ; when such an event o c c u r s , an e l e c t r o n i c p u l s e i s generated which s t a r t s some k i n d of high p r e c i s i o n c l o c k . The muon precesses i n the t a r g e t at a frequency = y^B where i s i t s gyromagnetic r a t i o and B i s the t r a n s v e r s e magnetic f i e l d experienced by the y + . Noting t h a t 9=OJ t , the p o s i t r o n decay spectrum ( 3 ) becomes: R(t) = 1 + A cosw t y y The beam p o l a r i z a t i o n ensures t h a t a l l muons have the same i n i t i a l p r e c e s s i o n phase. A t r a n s v e r s e magnetic f i e l d of from 50 gauss to s e v e r a l kgauss i s e x t e r n a l l y a p p l i e d i n the case of non-magnetic t a r g e t s ; f o r ferromagnetic t a r g e t s , a s u b s t a n t i a l t r a n s v e r s e magnetic f i e l d may be i n t r i n s i c to the m a t e r i a l , i n which case the muon p r e c e s s i o n frequency i s a d i r e c t measure of the i n t e r n a l f i e l d a t the y + i n t h a t magnetic m a t e r i a l J N i s h i d a (77)]. A p o s i t r o n counter a r r a y p l a c e d i n the plane of muon p r e c e s s i o n a t an angle <j> to the i n i t i a l muon beam monitors the y + decay and generates an e l e c t r o n i c p u l s e t o stop the c l o c k p r e v i o u s l y s t a r t e d by the muon e n t e r i n g the t a r g e t . The measured time i n t e r v a l i s i n c r e m e n t a l l y binned i n a time histogram, the c l o c k i s r e s e t , and the process i s repeated, o" 7 t y p i c a l l y 10 - 10 times. Since y + decay i s s p a t i a l l y asymmetric, the p r o b a b i l i t y of d e t e c t i n g the p o s i t r o n from muon decay r i s e s and f a l l s as the p r e c e s s m g y s p i n swings p a s t the f i x e d e d e t e c t o r s . Because the s o l i d angle subtended by the p o s i t r o n -8-counters i s s m a l l , most muon decays are not d e t e c t e d , i n which case the c l o c k i s r e s e t a f t e r some a r b i t r a r y "time out" p e r i o d of s e v e r a l muon l i f e t i m e s . The r e s u l t a n t u +SR time histogram has the form: N(c(>,t) = N e - t ^ T y { l + A (t)cos(w t + d>) j +. Bg (4) where N i s the number of counts i n a histogram time b i n , N Q i s a n o r m a l i z a t i o n f a c t o r , T i s the y + l i f e t i m e of 2.2 us, y A (t) i s the muon asymmetry which i s u s u a l l y time dependent, to i s the muon p r e c e s s i o n frequency, and Bg i s a time independent background due to a c c i d e n t a l events. A t y p i c a l y +SR time spectrum i s shown i n F i g u r e 2; i t s most dominant f e a t u r e s are the e x p o n e n t i a l muon l i f e t i m e upon which i s superimposed the o s c i l l a t i n g asymmetrical muon decay. The asymmetry, A ^ ( t ) , o f t e n decays w i t h time due to s p i n dephasing phenomena i d e n t i c a l to r e l a x a t i o n i n N M R a n c ^ commonly has the form: A (t) = A e " X t (5) y y where X = l / T ^ • An example of such a r e l a x a t i o n mechanism i s y + s p i n dephasing due to l o c a l f l u c t u a t i o n s i n the i n t e r n a l magnetic f i e l d experienced by a y + d i f f u s i n g between d i f f e r e n t i n t e r s t i t i a l s i t e s i n a ferromagnetic c r y s t a l of Fe [ N i s h i d a y +SR i s a p a s s i v e non-resonance analogue of proton NMR i n which the s p e c i a l p r o p e r t i e s of the muons are r e s p o n s i b l e f o r s i g n a l g e n e r a t i o n , e l i m i n a t i n g the requirement f o r c o n v e n t i o n a l power a b s o r p t i o n d e t e c t i o n . A l l of the i n f o r m a t i o n contained i n NMR s p e c t r a transformed i n t o the time domain i s , i n p r i n c i p l e , c o n tained i n y +SR s p e c t r a . Of course, the time L I + S R : LL+ IN R L U M I N I U M . 6 9 GAUSS 6 0 0 0 5 0 0 0 -CO LU 4 0 0 0 -LU 3 0 0 0 CL LU £ 2 0 0 0 1000 h 0 i I 0 . 0 0 . 5 1 .0 1 .5 2 . 0 2 . 5 3 . 0 3 . 5 4 . 0 T I M E IN LLSEC (20 N S E C / B I N ) FIGURE 2: A t y p i c a l y SR time histogram (data points) and x -minimum f i t to equation (4). The error bars (on every 10th point) are due to counting s t a t i s t i c s only. The histogram contains about 5 x 10 5 events. The y + asymmetry i s about 35% and X = 0.03 y s - 1 . - 1 0 -s c a l e o f p h e n o m e n a d e t e c t a b l e b y y + S R i s f i x e d b y t h e y + l i f e t i m e . T h e y + S R m e t h o d h a s a n u m b e r o f p o t e n t i a l a d v a n t a g e s o v e r NMR; o n e m u o n a t a t i m e i s p r e s e n t i n t h e s a m p l e t h e r e b y e l i m i n a t i n g i n t e r f e r e n c e s d u e t o i n t e r a c t i o n o f t h e y + w i t h t h e m s e l v e s ; t h e y + S R s i g n a l i s m e a s u r a b l e f r o m v / i t h i n b u l k m a g n e t i c m a t e r i a l s w h i l e t h e r , f , r e q u i r e d f o r NMR w i l l o n l y p e n e t r a t e t h e s k i n o f t h e s a m p l e ; t h e y + i s a s i m p l e p o i n t c h a r g e w i t h o u t a c o m p l i c a t i n g s t r u c t u r e ; a n d , i n c r y s t a l s , y + p r o b e t h e i n t e r s t i t i a l r e g i o n w h i l e NMR i s o f t e n c o n s t r a i n e d t o e x a m i n a t i o n o f t h e l a t t i c e s i t e s t h e m s e l v e s . B_ M u o n i u m a n d t h e MSR M e t h o d I n m o s t g a s e s , l i q u i d s a n d n o n - m e t a l l i c s o l i d s , t h e y + c a p t u r e s a n e l e c t r o n f r o m t h e m e d i u m d u r i n g t h e f i n a l s t a g e s o f i t s t h e r m a l i z a t i o n p r o c e s s t o f o r m t h e h y d r o g e n - l i k e a t o m , m u o n i u m (Mu) [ H u g h e s ( 6 6 ) , M o b l e y - 1 - ( 6 7 ) , B r e w e r ( 7 5 ) , F l e m i n g ( 7 9 ) 1 . Some o f t h e p r o p e r t i e s o f Mu a r e g i v e n i n T a b l e I I . S i n c e t h e m u o n i s 2 0 7 t i m e s a s m a s s i v e a s t h e e l e c t r o n , t h e r e d u c e d m a s s o f Mu i s 0 . 9 9 6 t h a t o f H a n d c o n s e q u e n t l y t h e B o h r r a d i u s a n d i o n i z a t i o n p o t e n t i a l s o f H a n d Mu a r e e s s e n t i a l l y t h e s a m e . M u , t h e r e f o r e , b e h a v e s c h e m i c a l l y l i k e a l i g h t i s o t o p e o f H [ G o l d a n s k i i ( 7 1 ) , B r e w e r ( 7 5 ) , J e a n ( 7 8 ) ] w i t h a m a s s 1_ t h a t o f n o r m a l H , T h i s s u b s t a n t i a l m a s s 9 d i f f e r e n c e p o t e n t i a l l y m a k e s Mu a n e x c e p t i o n a l l y s e n s i t i v e p r o b e o f i s o t o p e e f f e c t s i n c h e m i c a l r e a c t i o n s o f H . I n m u o n i u m , t h e y + s p i n i s n o t o n l y c o u p l e d t o a n e x t e r n a l m a g n e t i c f i e l d b u t a l s o t o t h e e l e c t r o n s p i n v i a t h e h y p e r f i n e i n t e r a c t i o n . S i n c e t h e m u o n s a r e p o l a r i z e d w h i l e t h e -11-TABLE II: PROPERTIES OF MUONIUM MASS; 2 0 7 . 8 m 0 . 1 1 3 1 m, H REDUCED MASS: 0 . 9 9 5 6 y H FIRST BOHR RADIUS, (a ) : ' o Mu 0 . 5 3 1 5 x 10 -8 cm 1 . 0 0 4 4 ( a Q ) H "H FIRST IONIZATION POTENTIAL: 1 3 . 5 4 eV = 0 . 9 9 5 6 I.P THERMAL DEBROGLIE WAVELENGTH:(300K): 2 . 9 7 9 X 10 = 2 . 9 6 7 X u r i -8 cm HYPERFINE FREQUENCY, co 72TTk~Tm a 2 . 8 0 4 4 x 1 0 1 0 rad s _ 1 MEAN THERMAL VELOCITY (3 0 0 K ) 0 . 7 5 x 1 0 ^ cm s = 2 . 9 7 v H 8 ^ ^ 1 / 2 um -12-captured e l e c t r o n s are u n p o l a r i z e d , the i n i t i a l s p i n s t a t e s of Mu are 5 0% la a ) and 50% la 3 ^ , where the muon 1 u e 1 y er ' p o l a r i z a t i o n d i r e c t i o n i s the q u a n t i z a t i o n a x i s . In f i n i t e t r a n s v e r s e magnetic f i e l d , the time e v o l u t i o n of the y + s p i n p o l a r i z a t i o n i n Mu i s 'quite complicated; the d e t a i l e d c a l c u l a t i o n i s g i v e n i n Appendix I. The upshot of t h i s c a l c u l a t i o n i n the weak t r a n s v e r s e magnetic f i e l d l i m i t s (< 10 gauss) i s , however, simple: y + i n h a l f of the Mu ensemble ( |a a ) ) precess at the muonium Larmor frequency, y e GJMU = 103 OJ^ , i n the sense o p p o s i t e to " f r e e " y + p r e c e s s i o n ; y + i n the other h a l f of the Mu ensemble ( l a y £ ^ ) o s c i l l a t e at the h y p e r f i n e frequency, OJ o = 2.8 x 10"^ rad s . Since the experimental time r e s o l u t i o n i s about 1 nanosecond, the h y p e r f i n e o s c i l l a t i o n i s not observable and t h i s h a l f of the Mu ensemble appears to be t o t a l l y d e p o l a r i z e d . M o n i t o r i n g the time e v o l u t i o n of the y + s p i n i n Mu i n weak t r a n s v e r s e magnetic f i e l d v i a the asymmetric y + decay forms the b a s i s of the MSR method f o r s t u d y i n g muonium. The MSR acronym stands f o r "muonium s p i n r o t a t i o n " and the method + + i s i d e n t i c a l with y SR w i t h the exceptions t h a t the y p r e c e s s i o n frequency i n Mu i s 103 times t h a t f o r " f r e e " y + and t h a t the y + asymmetry i n Mu i s reduced by h a l f . F i g u r e 3 shows a t y p i c a l MSR spectrum which has e s s e n t i a l l y the same appearance as a y +SR spectrum i n a magnetic f i e l d 103 times s t r o n g e r . In p r a c t i c e , MSR histograms have the form: N((J,,t)=N oe^" t / Ty{l + A ^ (t) cos ( u ^ t + <J>Mu) + A cos (co t -.• <J> ) } + Bg (6) y y Y y ^ where OJ», and to' are the muonium and muon p r e c e s s i o n f r e q u e n c i e s , Mu y ^ MSR: MU IN 780 TORR N 2 RT 6 .9 GAUSS CO LU O cm LU CD 12000 10000 8000 6000 -4000 h 2000 0 0.0 0 .5 1.0 1.5 2.0 2 .5 3.0 3 .5 4 .0 T I M E IN LLSEC (20 N S E C / B I N ) 2 FIGURE 3: A t y p i c a l MSR time' histogram (data points) and x -minimum f i t to equation (6). The histogram contains about 10^ events and the error bars are due to counting s t a t i s t i c s only. The Mu asymmetry i s about 11% and the background u + asymmetry i s about 5.5%. 4>Mu and <f) are the i n i t i a l muonium and muon phases, while A M u ( t ) and A are the muonium and muon asymmetries. The "free" muon y - 2 precession term may come from several sources: Mu formation i n some materials i s incomplete; i n gas targets, muons may scatter into the walls of the gas vessel where they may not form Mu; muonium atoms may react chemically on the very short time scale of less than one hyperfine period of 0.225 ns (probably by epithermal reaction) i n which case the hyperfine int e r a c t i o n i s broken, r e s u l t i n g i n coherent precession of the polarized y + ensemble residing i n diamagnetic environments. In muonium experiments, the asymmetry of the free y + term i s usually independent of time while the muonium asymmetry generally has a form si m i l a r to equation (5): A., (t) = A e ~ X t (7) Mu Mu The muonium relaxation rate, A, i s due to eff e c t s such as pressure broadening in gases, chemical reaction and magnetic f i e l d inhomogeneity. The effects of chemical reactions are discussed i n some d e t a i l i n Appendix I I . F i n a l l y , i t should be noted that the phases, a n <3 <f>^f have opposite signs to account for the fact that the free y and y i n Mu precess i n opposite d i r e c t i o n s . In condensed media with a well defined y + stopping region, the magnitudes of and <f>^' are the same; however, i n low pressure gases, the y stopping region i s smeared out enough that s i g n i f i c a n t differences i n the magnitudes of <j>Mu and <J> may appear. Throughout th i s thesis, reference w i l l be made to the MSR "signal," S(c|),t), which i s defined as: -15-S(<f>,t) = A,, e~ X tcos(w., t + ch ) + A cos (w t - cf> ) (8) Y ' Mu Mu vMu y y y Figure 4 shows the MSR signal corresponding to the time histogram of Figure 3. The slow free muon precession appears as an approximately li n e a r background at the weak magnetic f i e l d s used i n MSR. _C Muonium Chemistry - An H i s t o r i c a l Background Muonium formation was f i r s t proposed as an explanation for the observation that the "residual muon p o l a r i z a t i o n " (see Appendix II) i s not the same i n a l l condensed media [Swanson (58)] . Nosov and Yakovleva (63) and Ivanter and Smilga (68) derived a detailed model for muon depolarization phenomena, i n s o l i d s . Firsov and Byakov (65) attempted to r e l a t e the residual p o l a r i z a t i o n to Mu chemistry i n l i q u i d s with, a model i n terms of which the l a t e r r e s u l t s of Babaev (66). .were ' .misinterpreted. In 1969 , Ivanter and Smilga (.69) extended the formalism to c o r r e c t l y treat Mu chemistry i n l i q u i d s for simple reactive systems. The f i r s t extensive experimental study of thermal Mu reactions was by Brewer (72) who applied a modified form of the muonium mechanism of Ivanter and Smilga to the measurement of bimolecular rate constants of simple Mu reactions i n l i q u i d s . These experiments, conducted at the 184" Cyclotron at the Lawrence Berkeley Laboratory (LBL), grew out of an experiment to determine" the muon's magnetic-moment pr e c i s e l y • [Hague (70)] which required small corrections due to chemical e f f e c t s . Brewer also found i t necessary to extend the model to include epithermal reactions of Mu as well as reactions i n which 0 . 1 5 MSR: MU IN 780 TORR N 2 RT 6 . 9 GRUSS >-I— >-CO d 0 . 1 0 0 . 0 5 0 . 0 0 - 0 . 0 5 - 0 . 1 0 • 0 . 1 5 : i ii 0 . 0 0 . 5 1.0 1 . 5 2 . 0 2 . 5 3 . 0 3 . 5 4.0 T I M E IN LISEC (20 N S E C / B I N ) FIGURE 4: The MSR signal, S(<J>,t), corresponding to the histogram shown in Figure 3. The l i n e i s a x 2 -minimum f i t to equation (8). The rapid o s c i l l a t i o n s are due to y+ precession in Mu, while the slowly curving d r i f t i s due to "free" y + precession. -17-transient muonic radi c a l s are formed. Although Brewer's work established the foundations of experimental Mu chemistry, i t suffered from a number of serious shortcomings, the most notable of which was the f a i l u r e to detect muonium i n l i q u i d s d i r e c t l y by the MSR method. Muonium reactions were measured by the i n d i r e c t residual p o l a r i z a t i o n method described i n Appendix II using y +SR techniques. With t h i s method, d e t a i l s of the reaction mechanisms had to be inferred and the rate constants obtained were largely model dependent. In reactions involving several rate processes, such as those due to intermediate r a d i c a l formation, the extracted rate constants were highly correlated and of questionable absolute accuracy [Percival 1- (76) , Per c i v a l (77)]. While th i s i n d i r e c t method has been substantially•replaced with more d i r e c t methods described below, Brewer's pioneering work provided a valuable preliminary insight into the d e t a i l s of muonium chemistry i n l i q u i d s . For example, i n spite of tentative r e s u l t s suggesting the p o s s i b i l i t y of d i r e c t detection of muonic r a d i c a l s i n condensed media [Kent (77), Bucci (78)3 ^ t Brewer's work s t i l l provides the most convincing demonstration of the importance of such r a d i c a l s i n l i q u i d phase reactions of Mu [Brewer (73)]. Although the subject of some contention [Percival (78).], the residual p o l a r i z a t i o n method might well prove to be the one most amenable to the study of epithermal Mu reactions i n l i q u i d s . During preparation of t h i s thesis, the d i r e c t observation of several muonic radi c a l s was confirmed at SIN .{Roduner^l (78)] . -18-The study of muonium chemistry i n l i q u i d s i s complicated by several processes such as s o l v o l y s i s and "spur" reactions [see, for example, Gold (78)]. From the viewpoint of understanding elementary chemical rate processes, the gas phase provides a physical context which i s more t h e o r e t i c a l l y tractable than the l i q u i d phase. The only gas phase Mu chemistry studies p r i o r to the work i n t h i s thesis were a series of experiments conducted by Mobl-ey .e_t a l . [Mobley (66) , l-(67) , ,2-(67)] i n argon gas at high pressure (40 atmospheres) at the Nevis Laboratory at Columbia University, A variety of techniques including d i r e c t observation of Mu by the MSR method were employed to examine the interactions of Mu with C>2, C2 H4 a n <^ CH-3CI a n (3 a number of other reagents. Unfortunately, the conventional muon beam available to Mobley was of such high momentum that very high pressure gas targets were required to thermalize a useful f r a c t i o n of the beam. At 40 atmospheres, three body . processes play an important role i n the chemical reactions; i t i s preferable to use low pressure gas targets at about 1 atmosphere to measure bimolecular Mu reaction rates. Motivated by a proposal to measure the conversion of muonium (y +e ) to antimuonium (y e +) I see, for example, Lederman (77)] which requires the production of thermal Mu i n vacuum, a group from the University of Arizona designed a new kind of low momentum muon beam l i n e at the 184" Cyclotron at LBL IPifer (76)]. Some of the d e t a i l s of t h i s new "surface" muon beam (sometimes c a l l e d an "Arizona" muon beam) are given in Chapter I I . In collaboration with the Arizona group, th i s thesis work was started at t h e i r surface muon f a c i l i t y at -19-LBL during 1974-75 . when the f i r s t low pressure gas phase Mu bimolecular reaction rate constant was determined by the MSR technique,for the Mu + B r 2 reaction at 295 K i n 1 atmosphere of Ar [Fleming (76)]. The collaboration with the Arizona group was continued u n t i l July, 1975, when support for physics experiments at the 184" Cyclotron ceased and the machine became a dedicated medical f a c i l i t y . I t i s , perhaps, an h i s t o r i c a l footnote to remark that the Mu + C l 2 reaction rate measurement [Fleming 1-(77)] was the l a s t non-medical experiment executed on that machine. A major technological advance i n the study of muon physics and chemistry i n recent years i s the development of a new generation of a very high current "intermediate" energy p a r t i c l e accelerators, the so-called "meson f a c t o r i e s " . These machines produce meson beams with i n t e n s i t i e s that are two or more orders of magnitude greater than those previously available. At present, there are three such f a c i l i t i e s operat-i o n a l i n the world: the Schweizerisches I n s t i t u t fur Nuklearforschung (SIN) near Zurich', the Clinton P. Anderson Meson Physics F a c i l i t y (LAMPF) at Los Alamos, and the Tri - U n i v e r s i t y Meson F a c i l i t y (TRIUMF) i n Vancouver. In 1976, gas phase Mu reaction rate measurements at low pressure were f i r s t performed at TRIUMF on the M20 y +SR F a c i l i t y operating i n surface muon mode. Although surface muon beam li n e s required for low pressure gas phase targets are currently being commissioned at LAMPF and under construction at SIN, at present TRIUMF i s the only meson factory with an operational f a c i l i t y of this kind. Recently, another surface muon f a c i l i t y (using the beam components from the o r i g i n a l Arizona beam l i n e at Berkeley) was de-commisioried' at the 6 00 MeV synchrocyclotron of the Space Radiation Effects Laboratory (SREL) i n Vir g i n a . With the advent of meson fac t o r i e s came a number of advances i n Mu chemistry, among the most important of which was the d i r e c t observation of Mu i n water by the MSR techniqu at SIN [Percival 2-(76)] . This discovery, recently confirmed at TRIUMF [Jean - ; (78)], has large l y rendered obsolete the residual p o l a r i z a t i o n method used by Brewer i n the study of thermal chemical reactions of Mu i n the l i q u i d phase and plac l i q u i d Mu chemistry on the firmer experimental footing previously enjoyed only by gas phase studies. I t should be remarked that l i q u i d phase MSR signals are much weaker than gas phase signals. To date, the SIN group have applied MSR to the study of a number of chemical reactions i n a variety of l i q u i d media [Percival (77), Roduner 2-(78)]. Further impetus was given to gas phase Mu chemistry when the f i r s t d etailed t h e o r e t i c a l c a l c u l a t i o n of a Mu reaction rate was performed by a group i n Europe [Connor 1-(77)] for the reaction: Mu + F2->MuF + F. Considerable attention w i l l be given to th i s and subsequent calculations i n Chapter IV. ID Organization of the Dissertation This thesis reports-the f i r s t , measurements of Mu reaction rates i n low pressure gases (^1 atmosphere), for the reactions: Mu + .X9. •>•• MuX + X, X= F, C l , Br -21-and Mu + HX ->- MuH + X, X = C l , Br, I at 295K. In addition, a c t i v a t i o n energies are reported for the F 2 and C l 2 reactions between 300 and 400K. As detailed i n Chapters III and IV, the motivation for t h i s study i s twofold: (1) as a l i g h t isotope of H, Mu provides a remarkably sensitive probe of mass ef f e c t s i n H atom reactions, and (2) unlike the techniques of H atom chemistry, MSR i s l i t e r a l l y a one-atom-at-a-time method, unencumbered by interactions of the Mu atoms with themselves. With s u f f i c i e n t understanding of the f i r s t point, i t may be possible to exploit the second point to obtain accurate values of H atom reaction rates for systems where they are not measurable by other methods. This thesis i s composed of three main parts. Chapter II describes how the MSR technique i s applied to the measurement of gas phase chemical reaction: rates. Included are descriptions of the surface muon beam, gas target apparatus, counting procedures, electronic l o g i c and data acquisition, and methods of data analysis. Chapter III presents a b r i e f general t h e o r e t i c a l discussion of gas phase reactions of Mu as an H isotope. Mu and H are compared in terms of the k i n e t i c isotope e f f e c t ; possible implications of d i f f e r i n g energy dispositions in the t r a n s i t i o n state and among reaction products of Mu and H reactions are discussed; and, f i n a l l y , some dynamical isotope -22-e f f e c t s are examined with p a r t i c u l a r attention to quantum mechanical tunnelling. In Chapter IV, the experimental gas phase Mu reaction rate measurements are compared with experimental values for the analogous H atom reactions and with t h e o r e t i c a l predictions. CHAPTER II - EXPERIMENTAL DETAILS It was mentioned i n Chapter I that the f i r s t of the experiments described i n thi s thesis were conducted at the 184" Cyclotron at Berkeley; d e t a i l s of those experiments are not given here but may be found i n several references [Pifer (76), Fleming (76), Fleming l - ( 7 7 ) ] . Like most technologies, MSR methods are constantly evolving. Rather than attempting to provide a history of MSR development at LBL and TRIUMF, th i s Chapter w i l l only describe the "state of the art" techniques as practiced at TRIUMF i n 1978. Some s p e c i f i c suggestions for future improvements, p a r t i c u l a r l y i n the electronic l o g i c system, are included. A. TRIUMF and the M20 Muon Beam Line The TRIUMF Annual Reports 1972-76 are a good source for detailed information on the many TRIUMF f a c i l i t i e s ; only a few central points are given here. The TRIUMF cyclotron and experimental areas are shown i n Figure 5. TRIUMF i s a sector-focussed H cyclotron that delivers protons of continuously variable energy ranging from 185-520 MeV at maximum design currents of 100 yA, at 500 MeV and 450 yA at 450 MeV. Most of the experiments described i n thi s thesis were conducted with a 5-10 yA proton beam at 500 MeV. One of the most at t r a c t i v e features of the TRIUMF cyclotron from the viewpoint of MSR i s i t s 100% macroscopic duty cycle: seen on a macroscopic time scale (as short as microseconds), the proton beam appears to be a continuous current without a time E X I S T I N G P R O P O S E D FIGURE 5: The TRIUMF C y c l o t r o n and experimental f a c i l i t i e s (1977). structure. The microscopic duty cycle i s a 5 nanosecond burst of protons every 43 ns. The MSR method requires that at most one muon be i n the target at a time. Thus, the instantaneous muon stopping rate i n the target has an 5 "+ -1 absolute upper l i m i t of the order of 10 y s (the inverse of a few muon l i f e t i m e s ) . At pulsed beam f a c i l i t i e s such as LAMPF, which has a 6% duty cycle (a 500 ys burst every 8 ms), the maximum allowable average counting rate i s decreased by exactly the duty factor of the machine. This l i m i t a t i o n occasionally, can be side-stepped by the use of special multiple muon techniques such as the stroboscopic method [Schenck (76)], but the severe r e s t r i c t i o n s placed on the experiments that use thi s method greatly l i m i t i t s a p p l i c a b i l i t y . Even with i t s 100% duty cycle, the intense beams available at TRIUMF are capable of implanting more than one muon i n a target at a time; a detailed discussion of t h i s problem of muon "pile-up" i s given i n Section C and Appendix II I . A proton beam, extracted from the cyclotron by stripping the electrons from the H ions, passes down beamline-1 (BL-1) i n the "meson h a l l " and strikes a pion production target, T2. The target used i n t h i s work consists of a water-cooled beryllium s t r i p , 10 cm long i n the beam d i r e c t i o n , and 5mm by 15 mm i n cross section; pions are produced here v i a nuclear reactions such as: 9Be (p, T T +) 1 0Be . Three secondary beamlines simultaneously extract mesons (TT or y) produced at T2: M8, primarily intended for use i n IT cancer therapy; M9, a "stopped" T T - or y~ beamline used for a variety of experiments (the modifer "stopped" indicates that the TT or y beam i s of s u f f i c i e n t l y low energy to stop i n small experimental targets, i n contrast to TT or y beams used for scattering experiments) ; and M20, a stopped beamline which i s e s s e n t i a l l y dedicated to y +SR. The experiments described i n t h i s thesis were performed on M20 which generally operates p a r a s i t i c a l l y , d e l i v e r i n g muons whenever there i s beam on T2. M2 0 (shown i n Figure 6) transports a muon beam i n vacuum to the experimental target i n one of three operating modes: "conventional," "cloud""' or "surface" 1 muon mode. In conventional mode, pos i t i v e pions produced from T2 at 55° to the proton beam are coll e c t e d into M20 by the quadrupole doublet Q1-Q2 and then momentum selected ( £E - 20%, p <_ 170 MeV/c) by the f i r s t bending magnet, Bl ("Patty-Jane"). Of course, Bl also charge selects p a r t i c l e s since neutrals (Y,fT°,n) pass straight through Bl and negative p a r t i c l e s are bent out of the beamline. Some f r a c t i o n of the T T + ( T = 26ns) decay i n f l i g h t between Bl and B2 ("Cal-Tech") i n quadrupoles Q3-Q7 (the "straight section"). Seen i n the pion rest frame, T T + decay i s s p a t i a l l y i s o t r o p i c and the y + formed have a momentum of 29.8 MeV/c. Thus y + that are formed i n the momentum d i r e c t i o n of the pion beam ("forward" muons) have <_2 9.8MeV/c more momentum than that selected by Bl and y + that are formed opposite to the momentum d i r e c t i o n of the pion beam ("backward" muons) have >29.8MeV/c less momentum than that selected by B l . Nominal forward and backward y + momenta are given as a function of decaying T T + momentum i n Table I I I . FIGURE 6: The M2 0 beamline ( d e t a i l ) . -28-TABLE III: NOMINAL FORWARD AND BACKWARD y + MOMENTA AND VELOCITIES AS A FUNCTION OF DECAYING T T + MOMENTUM Pions p +(MeV/c) 3 + TT / TT Forward Muons p^+(MeV/c) 3 y + Backward p^+(MeV/c) Muons V 170 0. 77 180. 6 0. 86 86. 7 0. 63 160 0. 75 171. 2 0. 85 80. 5 0. 60 150 0. 73 161. 8 0. 84 74. 2 0. 57 140 0. 71 152 . 4 0. 82 67. 9 0. 54 130 0. 68 143. 0 0. 80 61. 5 0. 50 120 0. 65 133. 7 0. 78 55. 1 0. 46 110 0. 62 124. 5 0. 76 48. 5 0. 42 100 0. 58 115. 4 0. 74 42. 0 0. 37 90 0. 54 106 3 0 71 35 3 0 32 80 0 50 97 3 0. 68 28 5 0. 26 70 0 45 88 .5 0 64 21 7 0. 20 60 0 39 79 .7 0 60 14 .7 0 14 50 0 34 71 . 0 0 56 7 . 6 0 07 40 0 .27 62 .5 0 .51 0 .4 0 .00 30 0 .21 54 .1 0 .45 -7 .0 -0 . 07 20 0 .14 45 . 9 0 .40 -14 .4 -0 .13 10 0 . 07 37 .8 0 .34 -22 .1 -0 .20 0 0 .00 29 * .8 0 .27 -29 .8 -0 .27 T p and 3 are given with respect to the TT beam d i r e c t i o n with * "surface" or "Arizona" muons Notice that the momentum separation between y T and TT"1" i s excellent for backward y + but rather poor for forward y + . The second bending magnet, B2, i s tuned to momentum select either forward or backward muons from the other p a r t i c l e s + + + + in the beam (e , TT , y , p ) . The resultant high momentum polarized y + beam i s delivered to the y +SR apparatus at the end of the beamline through the l a s t quadrupole doublet, Q8-Q9. In "cloud" muon mode, high momentum (<170 MeV/c) y + produced from the "cloud" of pions decaying i n f l i g h t between T2 and Bl are coll e c t e d and transported through M2 0 with Bl and B2 both set at the same momentum-selecting f i e l d s . The ess e n t i a l difference between conventional and cloud muon modes, then, i s that i n the former case, muons are produced from pions decaying i n f l i g h t after the f i r s t bending magnet, while i n the l a t t e r case, muons come from T T + decaying before the f i r s t bending magnet near T2. In general, cloud muon mode produces higher fluxes of y + by a factor of as much as 4 but with a lower p o l a r i z a t i o n than conventional forward muon mode (50-60% p o l a r i z a t i o n compared to 70-80%) and with much worse contamination with protons, pions and positrons. When M20 i s tuned for backward conventional muons, the flux i s lower by about a factor of 10 compared to forward muons but beam contamination i s lower by several orders of magnitude (see Table I I I ) . In practice, with proton currents of <10yA, beam qual i t y i s often s a c r i f i c e d for flux by operating M20 i n cloud muon 4 + -1 mode, y i e l d i n g about 10 y s over a 10 cm x 10 cm area per - 3 0 -yA of protons. The beam delivered i n thi s mode i s contaminated with positrons and pions i n a 100:3:1 r a t i o to muons; there i s also some contamination from forward scattered protons from T2. y +SR experiments u t i l i z i n g cloud muons are complicated by the necessity to work around t h i s contamination. Since a l l p a r t i c l e s delivered by the beamline are of the same momentum d i s t r i b u t i o n , the slower, more massive protons and pions may be eliminated by d i f f e r e n t i a l absorption i n degrader placed upstream of the y +SR target. Positrons are separated from muons l o g i c a l l y , rather than p h y s i c a l l y , by placing a veto counter downstream of the muon target; the positrons are s u f f i c i e n t l y energetic to pass through the target i n which most of the muons stop. None of these methods are completely successful i n removing beam contamination r e s u l t i n g i n the appearance of various background signals i n the y +SR time spectrum. In addition to these drawbacks, cloud or conventional muons are unsatisfactory for gas phase targets since t h e i r high momentum and concomitant long range require a high pressure stopping target [Mobley 2-(67)]. Surface or Arizona muon mode [Pifer (76)] i s si m i l a r to cloud muon mode inasmuch as muons are coll e c t e d by M20 d i r e c t l y from T2. The difference i s that surface muons come from pions that decay at res t on the surface of the pion production target (see Table III).whereas cloud muons come from pions decaying i n f l i g h t between T2 and B l . Since cloud muons include both forward and backward muons, the beam p o l a r i z a t i o n i s low; v i r t u a l l y a l l surface muons, however, arise from T T + decaying i n the forward d i r e c t i o n , giving them a very high p o l a r i z a t i o n (>95%) . Surface muons are nearly monoenergetic (4.lMeV) with a nominal momentum of 2 9.8MeV/c corresponding to a range of -2 only 148 mg cm of CH 2 or about 130 cm i n argon gas at one atmosphere. Contamination of the surface muon beam with pions and protons i s n e g l i g i b l e . The v e l o c i t y of 30 MeV/c pions i s about 0.2c. corresponding to a beamline t r a n s i t time of 160 ns or 6 pion l i f e t i m e s ; thus only about 0.3% of the small number of pions i n i t i a l l y produced at 3 0 MeV/c survive to reach the end of the beamline. Protons at 3 0 MeV/c have i n s u f f i c i e n t range to penetrate the thin (0.05 mm Mylar) beamline vacuum window. However, as i n other modes of operation, there are about 100 times more positrons than muons i n the surface muon beam. Fortunately, the positrons can be l o g i c a l l y distinguished from u + by pulse height discrimination, as described i n Section C below [Marshall (76)] which eliminates the complication of a positron veto counter requirement. Unfortunately, t h i s t i d y , l o g i c a l removal of the contamination requires physical placement of the counters used to monitor positrons from muon decay at 9 0° to the beam to prevent t h e i r saturation by beam positrons; for many experiments, t h i s r e s t r i c t i o n i s p r o h i b i t i v e . In the near future, some of the M20 positron contamination w i l l be removed by adding a very thin degrader i n the straight section. This w i l l reduce the y + momentum much more than the positron momentum, so that by tuning Cal-Tech to the lower y + momentum, the positron contamination w i l l be considerably reduced. The disadvantage of thi s simple separation technique i s that i t s a c r i f i c e s both u + flux and range, the l a t t e r being at a premium for surface muons. However, i t should be possible to p a r t i a l l y compensate for t h i s loss by redesigning the surface y +SR targets to incorporate fewer and thinner windows. At present, M20 delivers about 6xl0 3' surface \i+s over a 10 cm x 10 cm area per uA of 500 MeV protons incident on the 10 cm Be target of T2. B_ . The Gas Target, Counters and Magnetic F i e l d The gas target and counter configuration are i l l u s t r a t e d i n Figure 7. The gas target vessel i s an aluminum cylinder 75 cm i n length with a 25 cm inner diameter. One end of the gas can i s f i t t e d with a thin (0.13 - 0.25 mm Mylar) window 20 cm i n diameter which i s capable of supporting a vacuum of 5 x 10 ^ to r r or an absolute pressure of £2.5 atmospheres. The volume of the target vessel i s 36,65 ± 0.34 1 at an absolute pressure of 800 t o r r , accounting for the volume displacement caused by pressure d i s t o r t i o n of the Mylar window. The other end of the aluminum cylinder i s closed with a flange housing pressure gauges and a stainless s t e e l vacuum rack. The gas can i s wrapped i n heating tape and in s u l a t i o n providing the target with an operational temperature range from ^300 to^400K. The temperature i s monitored with a copper-constantan thermocouple which i s placed to probe the muon stopping region; the temperature variation across the diameter of the vessel i s less than 2K. Cooling c o i l s are mounted at both ends of the vessel to accelerate s t a b i l i z a t i o n of the target temperature. This rudimentary target system w i l l be redesigned i n the near future to allow a much wider working temperature range and -33-•TO GAS RACK FIGURE 7: The gas phase MSR target apparatus (top view). The counter telescopes are designated l e f t and r i g h t according to the convention: muon's eye view. -34-somewhat larger pressure range. The gas target vessel i s mounted on a portable cart between dual Helmholtz c o i l s driven by a current regulated Hewlett-Packard Harrison 6268A power supply. These c o i l s provide a variable magnetic f i e l d from ^ 1 to 75. gauss which 3 i s homogeneous to better than 0.1% over a volume of 400 cm . A single thin (40 mg cm ) 10 cm x 10 cm NE102 p l a s t i c s c i n t i l l a t o r [Marshall (76)] serves as a beam defining and y - stop timing counter. Two positron telescopes are placed at ±90° to both the beam d i r e c t i o n and the transverse magnetic f i e l d . Each telescope consists of one 20 cm x 20 cm x 0.6 cm (closest to the target) and two 20 cm x 40 cm x 0.6 cm p l a s t i c s c i n t i l l a t o r s , as shown i n Figure 7, which normally operate with 1" of graphite degrader between the f i r s t and second counters. This degrader serves to reduce scattered beam positron background and to absorb low energy positrons from muon decay, thereby enhancing the empirical muon asymmetry (see Figure 1). The " l e f t " and "righ t " positron telescopes are designated by the mnemonic convention: muon1s eye view. The gas phase targets consist of chemically i n e r t moderator gas containing small concentrations of the reagent of i n t e r e s t . The moderator gas serves not only to thermalize the incoming muons, but also to provide the i o n i z a t i o n processes etc. for the formation of Mu [Stambaugh (74)]. In the e a r l i e r experiments, Ar was employed as the moderator gas; recently, N 0 has been used because i t has been found to be about 1.5 times more e f f i c i e n t than Ar at producing Mu without causing a s i g n i f i c a n t l y d i f f e r e n t background Mu relaxation, Aq (see Appendix I I ) , as i l l u s t r a t e d i n Figure 8. Furthermore, N 2 has a lower muon stopping density than Ar, thereby providing a longer muon range that affords greater f l e x i b i l i t y i n the design of windows and counters for optimizing the location of the muon stopping region with respect to the positron counters. The experimental operating pressure i s usually chosen to be about 780 to r r i n order to reduce possible 0 2 leakage into the reaction vessel. In N 2 at t h i s pressure, the residual range of surface muons, which have been degraded by passing through the thin counter and two windows, i s about 30 ± 5 cm at 300K, At higher temperatures, t h i s operating pressure i s maintained; the subsequent lower density of the gas target i s compensated for by using a thicker window (required for high temperatures) and additional sheets of Mylar degrader, A muon range curve i s taken at each temperature to ensure optimal location of the muon stopping region. Occasionally,, the reaction rate of Mu with a reagent i s s u f f i c i e n t l y slow (eg. HCl) that such large concentrations of that reagent are required that i t must then also serve as the moderator. In recent experiments, the high purity moderator gases are further p u r i f i e d by passing them through activated charcoal or a Dow Chemical G.C. c a r r i e r p u r i f i e r , reducing 0 2 contamination to less than 1 ppm. This results i n a reduced background Mu relaxation rate, A.q, although the e f f e c t i s not dramatic. Measured concentrations of reactant gas are added to - 3 6 -0 . 1 5 0 . 1 0 0 . 0 5 0 . 0 0 - 0 . 0 5 - 0 . 1 0 -0 . 1 5 0 . 1 5 0 , 1 0 0 . 0 5 0 . 0 0 - 0 . 0 5 - 0 . 1 0 - 0 . 1 5 MU IN 780 TORR N2 AT 6 . 9 GAUSS i ; i : t " MU IN 670 TORR RR AT 6 . 9 GAUSS J ii I _L _L 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 3 . 0 TIME IN uSEC (20 NSEC/BIN) 3 . 5 4 . 0 F I G U R E 8: N i t r o g e n v e r s u s a r g o n a s m o d e r a t o r g a s e s . B o t h s p e c t r a w e r e t a k e n u n d e r i d e n t i c a l c o n d i t i o n s , e x c e p t t h a t t h e g a s p r e s s u r e s w e r e c h o s e n t o o p t i m i z e t h e l o c a t i o n o f t h e m u o n s t o p p i n g r e g i o n . I n n i t r o g e n ( t o p ) , t h e m u o n i u m s i g n a l a m p l i t u d e i s l l . _ £ ± 0 . 3 % w i t h a r e l a x a t i o n r a t e o f 0 . 3 4 ± 0 . 0 2 y s . I n a r g o n ( b o t t o m ) , t h e m u o n i u m s i g n a l a m p l i t u d e i s 7.6J612% w i t h a r e l a x a t i o n r a t e o f 0 . 3 3 ± 0 . 0 3 y s the reaction vessel by f i l l i n g a small bulb of known volume to a measured pressure, then flushing i t into the evacuated target can with moderator. In t h i s way, concentrations of 15 19 reactant may be conveniently varied from 10 - 10 - 3 - 6 _ 2 molecules cm (10 - 10 M). Reagents which are condensed at S.T.P., such as bromine, are introduced by f i l l i n g the bulb with the equilibrium vapour pressure at a known temperature, as described i n Fleming (76). At each temperature, the gas vessel i s "conditioned" with 300 torr of reactant for about 1 hour before any experiments are run. This ensures that s u f f i c i e n t quantities of reagents l i k e F^ have enough time to form i n e r t compounds on the surfaces of any components of the target vessel that are chemically reactive with that reagent. Interestingly, the Mylar window has proven to be i n e r t even to 300 t o r r of F^ at 400K, In order to v e r i f y the inertness of the target vessel surfaces to very reactive chemicals l i k e F^, the reagent concentrations are varied randomly from one Mu rate measurement to .the next. Since reactions with metal surfaces tend to follow -1 order k i n e t i c s [Frost (61)], i t should be possible to i d e n t i f y any ongoing interference reactions from the systematics of the Mu rate measurements. This serves to check the v a l i d i t y of the i m p l i c i t assumption that the concentration of reagent remains constant during the experimental runs which t y p i c a l l y take 1 to 2 hours each. C_ . Data Ac q u i s i t i o n Before describing the data acquisition system, i t i s useful, perhaps, to r e i t e r a t e the e s s e n t i a l features of an -38-MSR experiment. Upon leaving the beamline, a muon passes through a counter (designated 'D' i n Figure 7) which generates - 9 a s t a r t pulse for a high precision clock. Within >10 seconds of reaching the stopping region of the target, the muon thermalizes as Mu and precesses i n a weak transverse magnetic f i e l d . At some l a t e r time (up to several microseconds) the muon decays, emitting a positron p r e f e r e n t i a l l y along i t s spin d i r e c t i o n at the moment of decay. If the spin vector of the muon happens to point toward the positron telescope when i t decays, there i s a high p r o b a b i l i t y that the decay positron w i l l be detected, generating a stop pulse for the clock. The resu l t i n g time i n t e r v a l i s incrementally binned i n a histogram, 6 7 the clock i s reset and the entire process i s repeated 10 - 10 times. Should no decay positron be detected during some adjustable "time-out" period of several muon l i f e t i m e s , the clock i s automatically reset. A s i m p l i f i e d diagram of the TRIUMF MSR data a c q u i s i t i o n system i s shown i n Figure 9 (taken from [Marshall (76)]). Pulses from the counters at the top of the diagram (corresponding to those i n Figure 7) are time adjusted by variable delays (denoted ^ i n the diagram) before being input to discriminators where " r e a l " signals are distinguished from noise. The thin 'D' counter used for detecting surface muons also serves to discriminate muons from positrons: at 30 MeV/c, positrons t r a v e l e s s e n t i a l l y at c and are minimum i o n i z i n g , depositing very l i t t l e energy i n the thin counter, i n contrast to the slower muons t r a v e l l i n g at < 0.3c which are many times more i o n i z i n g . By adjusting the voltage on the D counter vDISC tLl'L2'L3= e(left) COIN. PDP-11/40 FAN IN L route c .DISC y-stop COIN. stop start TDC 100 reset MBD-11 branch highway CAMAC FIGURE 9: MSR data acquisition l o g i c (simplified) R route [Rl-R2'R3: e (right)j COIN. -P 03 w 1 I U) I start s t oP TAC (time-to-amplitudel convertgrft PHA (pulse height analyzer) UBC computer -40-photomultiplier, positron signals can be made to form a band with a pulse height of ^ 50 mV, while muon signals form a band with pulse heights of 300 - 400 mV. Adjustment of the D discriminator threshold to greater than 50 mV e f f e c t i v e l y makes the muon "trigger" transparent to positrons while retaining a high e f f i c i e n c y (>95%) for muons. The muon pulses are input to the s t a r t of an E.G. & G. Model TDC-100 ti m e - t o - d i g i t a l converter, which has a nominal time resolution of 0.125 ns and an adjustable range from 4 us to 34 ms. The TDC-100 also activates a fast "time-out" reset i f no stop pulse i s accepted during the pre-selectable time range. Discriminated pulses from the l e f t and r i g h t positron telescopes are input into separate coincidence units ( l o g i c a l "and's") which i d e n t i f y positrons by the Boolean l o g i c a l expressions: e T=Ll*L2«L3 or e =R1•R2•R3. The threefold coincidence requirement ensures that accepted events correspond to positrons that pass through a l l three counters and the carbon degrader. This defines the acceptance s o l i d angle and eliminates low energy positrons from the muon decay, thereby enhancing the empirical u + asymmetry; more importantly, the degrader absorbs scattered positrons from the beam, thereby reducing background which has the time structure of the TRIUMF cyclotron (23.3 MHz), Accepted positron events are l o g i c a l l y "or-ed" from the l e f t and r i g h t telescopes with a "fan-in" unit and input to the stop of the TDC. Simultaneous with stopping the clock, the l e f t or r i g h t positron pulses set a telescope i d e n t i f i c a t i o n b i t i n a pattern recognition unit mounted i n the CAMAC computer-logic i n t e r f a c e . -41-Upon completion of the d i g i t i z a t i o n , the TDC writes the measured time i n t e r v a l into a CAMAC input r e g i s t e r which generates a "look-at-me" (LAM) signal to activate the Bi-Ra Microprogrammed Branch Driver (MBD-11, Model 2) which services the data stored i n CAMAC. The MBD i s a fast micro computer which i s interfaced v i a UNIBUS to the main data a q u i s i t i o n computer (a D i g i t a l Equipment Corporation PDP-11/40) and controls the CAMAC crate(s) v i a a Branch Highway. Although under the ultimate control of the main computer, the MBD's operation i s functionally independent of and simultaneous with that of the PDP-11, thereby r e l i e v i n g the l a t t e r from time-consuming data ac q u i s i t i o n tasks, l i b e r a t i n g i t for more sophisticated on-line data analysis. The MBD reads the CAMAC data and resets the electronics i n preparation for acceptance of a new event. The MBD i d e n t i f i e s the positron telescope that generated the event, and performs the necessary s h i f t i n g , subtracting and base-addition functions required to increment the address i n the PDP memory representing the histogram bin corresponding to the measured time i n t e r v a l . Thus separate l e f t and r i g h t histograms are co l l e c t e d simultaneously, each normally consisting of 2000 bins of 2 ns each, giving a t o t a l range of 4 ys. The system i s capable of supporting almost any number of histograms of any size with a maximum time resolution of 0.125 ns. However, at present the time-resolution of the counters i s about 1.5 ns. The data acquisition hardware and software i s interfaced to the experimenter through the PDP-11 computer, executing a sophisticated programme written primarily by R.S. Hayano of -42-the University of Tokyo [Hayano 1-(76), Hayano. 2-(76) ] with help from J.H. Brewer of U.B.C. This data a q u i s i t i o n programme w i l l support several independent experiments running simultaneously and i s completely f l e x i b l e with respect to the number, size and time resolution of histograms required for each experiment. Many experiment-monitoring features are b u i l t - i n , including provision to display a l l or part of any histogram on a graphics terminal under l i g h t pen control. The programme provides a high l e v e l of data protection by regularly updating histogrammed data on permanent disk f i l e s ; a powerful "crash recovery" f a c i l i t y minimizes data loss due to computer problems. Many levels of redundancy ensure continued data a q u i s i t i o n c a p a b i l i t y i n the face of non-pathological hardware f a i l u r e - for example, breakdown of a disk drive, the graphics terminal, or other control terminal w i l l not cr i p p l e the computer's data taking functions. Several on-line analysis routines such as fast fourier transforms (FFT) are available for monitoring an experiment. At present, the PDP-11 does not support data analysis programmes of s u f f i c i e n t capabil-i t y to perform " f i n a l " data analysis, although implementation of such programmes w i l l be made i n the near future. U n t i l then, data i s written on a 9-track magnetic tape and analyzed o f f -l i n e on the UBC computer center IBM 37 0/16 8 Michigan Terminal System (MTS) as described i n the next Section. The foregoing description of the MSR data, a c q u i s i t i o n system i s a s i m p l i f i e d overview; the serious problem of "muon pile-up" has been ignored and only s u p e r f i c i a l treatment has been given to the intertwined data processing relationships between the elec t r o n i c l o g i c , MBD and PDP. The q u a l i t a t i v e problems of muon pile-up and hardware-independent solutions are i d e n t i f i e d below. A numerical assessment of these problems as a function of muon beam current i s l e f t to Appendix I I I , which also provides a detailed description of the MSR data acquisition system. In the following discussion, i t i s convenient to define a fixed muon decay gate or maximum muon l i f e expectancy time range, T, which i n practice i s set to a few muon l i f e t i m e s . A muon entering the target at the onset of t h i s time i n t e r v a l i s assumed to have decayed by the expiration of T, correspond-ing to the TDC "time-out" period mentioned above. For instance, i f an experimenter sets T = 4^ u-> , then the assumption that the muon has decayed during T i s good to better than 2%. Following the entry of a muon, u^, that opens the T-gate and starts the TDC, a second "pile-up" muon, y .;__,_-]_> m a Y enter the target before the expiration of T and before any decay positron i s detected. When such an event sequence occurs, an ambiguity i s created since there i s no way to id e n t i f y which muon i s associated with any subsequently detected decay positron. Since there i s a high p r o b a b i l i t y that any decay positron detected during T w i l l belong to y^ +^ rather than to JJN , the time c o r r e l a t i o n between a u and i t s decay e i s l o s t i f t h i s positron i s allowed to stop the clock. Acceptance of these events at s u f f i c i e n t l y high muon beam currents w i l l r e s u l t i n a time histogram containing a reduced MSR signal and a- distorted background. To zeroth -44-order, i t i s necessary to l o g i c a l l y r e j e c t t h i s event sequence (called "early second y" events) represented schematically by: where time moves from l e f t to ri g h t and —//— indicates some arbitrary time. Rejection of early second y events i s not, however, a complete solution to the problem of multiple muons. In fa c t , a l i n e a r d i s t o r t i o n . •. of the time spectrum (with a negative slope) i s generated at s u f f i c i e n t l y large y-stop rates when only early second y are rejected, as i l l u s t r a t e d i n Figure 10. This comes from the fact that, given a constant beam current with muons a r r i v i n g at times given by a Poisson d i s t r i b u t i o n (see Appendix I I I ) , there i s a higher p r o b a b i l i t y that an event w i l l be rejected due to an early second y i f y^ decays at late times than i f i t decays at early times. There i s simply a greater opportunity for an early second muon to enter the target i f the f i r s t muon survives a long time before decaying. It should be noted that the presence of i n the target merely creates an ambiguity i n the association of any detected e with i t s decaying y; sometimes the detected e does correspond to y^. That i s , some early second y events are "good" events i n the sense that the decay e that stopped the - 4 5 -0 . 1 5 0 . 1 0 0 . 0 5 0 . 0 0 >— CO 0 1 - 0 . 0 5 >— Ql - 0 . 1 0 - 0 . 1 5 0 . 2 5 0 . 1 5 £ 0 . 0 5 £ - 0 . 0 5 ex - 0 . 2 5 - 0 . 1 5 h MU IN 780 TORR N2 RT 6 . 9 GRUSS I _L _L i i i i i 1 r MU IN 1140 TORR N2 . 7 . 8 G. ERRLY -2ND u _L 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 3 . 0 TIME IN uSEC (20 NSEC/BIN) 3 . 5 4 . 0 FIGURE 10: The spectral d i s t o r t i o n due to "early" second y r e j e c t i o n : the top spectrum has both "early" and " l a t e " ( i . e . "post-y.") second y re j e c t i o n , while the bottom spectrum has "early" second y r e j e c t i o n only, giving r i s e to a large back-ground d i s t o r t i o n with a negative slope.It should be noted that the asymmetry scales are di f f e r e n t for the two spectra. -46-clock corresponds to u^, the muon that started the clock, even though they are not i d e n t i f i a b l e as such. Another way of looking at the a r t i f i c i a l d i s t o r t i o n , then, i s that because there i s much time available for an early second u to enter the target and cause r e j e c t i o n of an event i f T_K decays at late times, the e f f i c i e n c y of event acceptance (that i s , the number of "good" events accepted r e l a t i v e to the t o t a l number of good events) i s small at late times; conversely, i f decays at early times, early second u have l i t t l e opportunity to lower the e f f i c i e n c y of event acceptance. The res u l t i s that the normalization of equation (6) (Chapter I) decreases with time. If f i t t i n g procedures assume time independence of the normalization, i t s a r t i f i c i a l time dependence expresses i t s e l f i n an erroneously small apparent muon l i f e t i m e and an art i f i c i a l l y large apparent muonium relaxation rate. It i s , therefore, e s s e n t i a l that a constant f r a c t i o n of events per histogram time increment be rejected i n order to avoid generation of the a r t i f i c i a l backgrounds described above. This i s accomplished by not only r e j e c t i n g early second u events, but also r e j e c t i n g what are c a l l e d "late second u" events; that i s , events i n which enters the target before the expiration of T, but after a decay positron i s detected: T >| - 4 7 -Thus, an accepted event i s one i n which no second y arrives during T: < - T > Higher order corrections for multiple p a r t i c l e events can be made, but the rej e c t i o n of early and late second y ( c o l l e c t i v -ely c a l l e d "post-y^ second y") i s the most important, both i n terms of absolute numbers (which are muon rate dependent) but also i n terms of the spectral d i s t o r t i o n s introduced by f a i l -ure, to rej e c t these events (see Appendix I I I ) . One higher order correction comes from consideration of the time i n t e r v a l preceding the entry of y^ into the target. In the foregoing discussion, i t was assumed that, upon entry, y^ i s the only muon resident i n the target; but thi s may not be the case. Even with post-y^ second y r e j e c t i o n , there are two situations i n which a muon may already be resident i n the target when y^ enters. (1) If yj__ 2 w a s t n e previous T-gate opening muon and U^_-j. w a s a n early or late second y, then aft e r the T-gate has closed i t may be assumed that y^_ 2 has decayed, but V-^_-^ may s t i l l be present when y^ opens the next T-gate. (2) Whenever there i s an accepted event, there i s an i n t r i n s i c electronics "deadtime" during which ti m e - d i g i t i z a t i o n occurs and the event i s transferred to the histogram. During t h i s deadtime, the experiment i s e f f e c t i v e l y "turned o f f " and a l l event monitoring i s suppressed (reasons for t h i s are detailed i n Appendix I I I ; t h i s i s a general feature of the lo g i c whenever more than one electronics module, such as the TDC and pattern recognition u n i t , must be read and reset by the computer. To preserve the i n t e g r i t y of the next event, i t i s es s e n t i a l that a l l such modules be available for new data at exactly the same time). Consequently, when the experiment i s "turned on" again at some arbit r a r y l a t e r time, the lo g i c i s unaware of the presence of any muons i n the target. To cor r e c t l y deal with muon pile-up, the only acceptable events are those where no muon enters the target during a time T either before or after y^ entered the target: -^ K T 1^ However, i t i s shown i n Appendix III that "pre-y multiple muon events are several orders of magnitude less frequent than "post-y^" multiple muon events. Furthermore, while such muons do lower the apparent Mu asymmetry, they d i s t o r t the histogram less - s i g n i f i c a n t l y than po'st'-y-- multiple muon events. Another higher order correction can be made for events with more than one positron detected during T afte r y^ enters the target, creating an obvious ambiguity. Possible sources of "extra" e include accidental counts (possibly related to beam contamination and therefore beam current dependent) or e -49-from muons that happen to survive T (in the example above where T = 4 T , 2% of the muons survive T). If the positron telescopes,- are properly shielded against accidentals from the beam, multiple-e events are extremely rare and can be ignored for a l l but the most precise work. The most s i g n i f i c a n t d i s t o r t i o n introduced i n the histogram by multiple-e i s l i k e l y to be the microscopic time structure of the cyclotron beam. The experiments described i n t h i s thesis employed "post-y^" multiple muon rej e c t i o n and multiple-e r e j e c t i o n only. "Pre-y^" multiple muon re j e c t i o n w i l l be incorporated into the data acquisition l o g i c at higher beam currents. A f i n a l high event rate consideration of relevance to very high precision work i s associated with counter response c h a r a c t e r i s t i c s [Hague (70)]. Counter photomultipliers have a minimum recovery time of about 20 ns. Signals produced by a photomultiplier which i s not f u l l y recovered are reduced i n amplitude and may be rejected by discriminators. This may be prevented by additional gating to ensure that the events of i n t e r e s t were counted by f u l l y recovered photomultipliers. This consideration i s an argument for improving the beam qual i t y of M20 to ensure that the severe positron contamination does not d i s t o r t y + signals from the D counter. D_, Data Analysis Figures often beguile me, p a r t i c u l a r i l y when I have the arranging of them myself. The remark attributed -50-to D i s r a e l i would apply - "There are three kinds of l i e s - l i e s , damned l i e s , and s t a t i s t i c s . " Mark Twain's Autobiography (Vol. I , p. 246). Most data a n a l y s i s i s p r e s e n t l y performed o f f - l i n e on the IBM 370/168 using multiparameter chisquared m i n i m i z a t i o n performed by a powerful, general m i n i m i z a t i o n r o u t i n e c a l l e d MINUIT [James (71)] t h a t was adapted from the C o n t r o l Data Corporation (CDC) 76 00 computer l i b r a r y at the European Organization f o r Nuclear Research (CERN), i n Geneva. MINUIT i s an easy-to-use programme w i t h enough f l e x i b i l i t y to allow the user to devise a wide v a r i e t y of f i t t i n g s t r a t e g i e s ; only a few of i t s c a p a b i l i t i e s are mentioned here. Two minimi-z a t i o n algorithms are normally used: the simplex method of Nelder and Mead [Nelder (67)] and a v a r i a t i o n of the Davidon (6 8) v a r i a b l e m etric method c a l l e d MIGRAD, The l a t t e r method, which i s p a r t i c u l a r l y e f f i c i e n t given a good set of i n i t i a l parameter guesses, r e q u i r e s f i r s t p a r t i a l d e r i v a t i v e s of the f u n c t i o n being minimized; these may be provided a n a l y t i c a l l y by the user or may be c a l c u l a t e d numerically by MINUIT. MINUIT. w i l l accomodate up to 50 v a r i a b l e parameters, any number of which may be FIXed at any time and RESTOREd at any l a t e r time. Parameters may be constrained to any p h y s i c a l l y meaningful numerical range. Covariance matrices and c o r r e l a t i o n c o e f f i c i e n t s are c a l c u l a t e d by MINUIT, e i t h e r as an estimate generated by MIGRAD or from the s o - c a l l e d hessian m a t r i x , which i s exact f o r a Gaussian parent d i s t r i b u t i o n . D e t a i l e d non-symmetric e r r o r estimates of parameters f o r non-parabolic minima may be c a l c u l a t e d by a search -51-raethod c a l l e d MINOS. A number of checks for the presence of l o c a l minima are also made by MINUIT. The gas phase Mu data analysis i n t h i s thesis has been performed i n three stages: raw histograms are analyzed to -extract pseudo-first order rate constants; the l i n e a r dependence of these pseudo-first order rate constants on reactant concentration yi e l d s bimoleeular : rate constants at a given temperature; and, f i n a l l y Arrhenius f i t s of the temperature dependent bimoleeular rate constants provide values of a ctivation energies and pre-exponential factors. Raw histograms are f i t t e d to a model of the basic form of equation (6), Chapter I: N (<)>•, t) = N e ~ t / T y [1 + A., e " A t c o s ( y M Bt + d>„ ) o K Mu 1 Mu YMu + A cos(y Bt - (j) )] + Bg y y r y ' 1 ^ where t i s the independent variable, and eight unknown parameters are sought: N , A,, , X, B, , A , <$> , and Bg f n o' Mu' ' ' KMu' y' y ^ with Y „ B = u as defined i n Appendix I, and y B = to . Of 1Mu - L ^ ' 'y y these, X i s the parameter of central i n t e r e s t , although A M and A^ provide information about fast epithermal Mu reactions (see Appendix I I , Section B). As explained i n Chapter I, experience has shown that <b„ and d>" cannot be assumed to be c Mu y equal, p o s s i b l y ; because some • f r a c t i o n of the "free" y + signal comes from muons stopped i n the walls of the gas target vessel which are geometrically inequivalent to the ensemble of Mu stopped i n the gas. In a l l cases, the muon lif e t i m e i s assumed, to be fixed at 2.1971 ys. This assumption i s p h y s i c a l l y v a l i d since y + l i f e t i m e s are independent of t h e i r environment to at least a few parts per m i l l i o n [Sachs (75)]; p r a c t i c a l l y , the v a l i d i t y of t h i s assumption depends very strongly upon the i n t e g r i t y of the multiple muon rej e c t i o n l o g i c described i n the preceding section. The empirical value of A normally ranges between 10% and 15% and A^ i s generally less than 5%, depending upon the stopping medium and detailed counter configuration; conveniently measurable values of X range between 0.1 ys and 1 r " I 15 ys Left and r i g h t histograms are analyzed independently, y i e l d i n g two redundant values of X at each reagent concentra-t i o n . The time bin corresponding to "time zero" i s estimated to a precision of about 2 ns by performing a b r i e f measurement of the time required for beam positrons to scatter between the 'D' counter and positron telescopes. To accomplish t h i s , the photomultiplier voltage on the 'D' counter i s increased, thereby increasing the positron pulse height above the 'D1 discriminator threshold, and the positron telescope threefold coincidence requirement i s reduced to the single counter closest to the target, which always defines the timing of the coincidence output. Each 2000 bin histogram of 2 ns bins i s normally rebinned to 4 or 8 ns/bin depending upon the Mu precession frequency, r e s u l t i n g i n an e f f e c t i v e histogram size of about 1000 or 500 bins containing about 10 events (some of the o r i g i n a l bins, are eliminated because they precede t=0<) . V a l i d data i s normally contained i n the histograms within about t = 10 ns after time zero, but careful adjustment of the lo g i c timing can reduce t h i s to about t = 3 ns. In contrast, i t may be noted that experiments requiring a positron veto seldom contain v a l i d data before t = 25 ns, and often not before t = 100 ns, due to the width of the anti-coincidence requirement. An eight parameter f i t to a 450 bin histogram by MINUIT consumes about 2 to 2 5 seconds of CPU time, depending on the quality of the i n i t i a l guesses to the parameters. 2 The f a m i l i a r d e f i n i t i o n of x i s given by X 2(x) = I k=l K -fY. - T.(x) k _k 2 (9) where x = x^, i = l , n , are the variable parameters, K i s the 2 number of data points, Yfc and a f c are the measured values and th e i r variances and T k(x) are the values predicted by the model. Since counting s t a t i s t i c s generally follow a Poisson 2 d i s t r i b u t i o n , the variance i s just Y^ for large Y^, the number of events per f i t t e d time bin. For histogram analysis, 2 the d e f i n i t i o n of x i s modified to o K fY, - T (x))2 2 / \ v ^ k k — ; (in) X (x) = 2 Tlx)— k=l 2 2 where = Y^ i s replaced by = T^(x). This modification i n weighting i s made to eliminate extraordinary weighting of un-usually low points and can be seen as follows: consider a si t u a t i o n i n which one datum i s unusually high and another i s correspondingly low; d e f i n i t i o n (9) provides the high point with 1 2 a smaller weighting factor (— ) than the low point thereby k biasing the f i t to the lower point; d e f i n i t i o n (10) weights both points equally. For most histograms, model (10) provides -54-2 2 a x P e r degree of freedom of 0.95 < x 1.05. Individual pseudo-first order rate constants from l e f t and r i g h t counter telescopes are simultaneously f i t t e d to equation 11(2), Appendix I I : X = k[X] + x 0 to y i e l d the bimolecular rate constant, k. X i s determined for at least f i v e concentrations of X including [X] = 0. 2 The true d e f i n i t i o n of x " from (9) above i s used for these f i t s , but i n t h i s case i s given from the errors i n X calulated by MINUIT. More experimental data i s accumulated i n histograms with fast relaxations i n order to reduce the r e l a t i v e uncertainty i n the determination of X. Plots of X versus [X] i n t h i s thesis show - weight averaged X's from l e f t and r i g h t telescopes for graphical c l a r i t y , but the f i t t e d l i n e s correspond to simultaneously f i t t e d l e f t and r i g h t values of X. For temperature dependent k's, f i t s are made to the f a m i l i a r expression: k = A e " E a / R T (11) where E , the Arrhenius ac t i v a t i o n energy, and A, the c l pre-exponential factor, are the parameters of i n t e r e s t . Again, 2 the true d e f i n i t i o n of x from (9) i s used for f i t s of the logarithmic form of (11): Ink = -E /RT + InA (12) c l Cvetanovic and Singleton (77) have pointed out that the proper -55-weighting factors of the k's i n equation (12) must be obtained by the i t e r a t i v e procedure k.* (k. - k.*) w- ' = T ,v * / w - i ( 1 3 ) l l n ( k i / k i * ) l where w. 1 = ( )^ i - s the exact s t a t i s t i c a l weight for an ex-o. I perimehtal In k. i n (12), w. = ( i s the s t a t i s t i c a l weight 1 1 a. l of an experimental k. i n (11), and k. i s the best f i t * prediction of k^. Since k_^  are unknown, w^1 are obtained by * i t e r a t i o n of current MINUIT values of k^. Most of the gas phase Mu measurements reported i n t h i s thesis have been taken at a magnetic f i e l d of 7 to 8 gauss. Fields greater than 10 gauss are complicated by the beat frequency r 2 wo,h u o u + 2 (see Appendix I) , Q = {(D + j-^) - o~ . - —-.', , _ ' K + 4 ^ ( i ) 0 , At 10 gauss, 5 -1 the envelope of cos fit (ft = 2.8 x 10 s ) reduces A., to Mu 0.4 A M u after 4 ys, a t y p i c a l experimental time range. When f i t t e d to equation (6), t h i s leads to an apparent relaxation rate of 0.2 ys ^ for a stable, long-lived Mu s i g n a l . This bogus "relaxation" rate increases as the square of the applied-f i e l d and has the appearance of a Gaussian relaxation. In p r i n c i p l e , a f i t t i n g function can be devised to include the beat without introducing any new parameters since 0 depends only on known constants and B, which i s a parameter anyway. However, inclu s i o n of t h i s complication to the f i t t i n g function increases computational cost and the beat i s highly correlated to A and X at low f i e l d s . C l e a r l y , these factors y are not p r o h i b i t i v e , but they are e a s i l y avoidable complications. -56-Fail u r e to account for the beat envelope using model (6) as the f i t t i n g function w i l l generate a systematic error reducing k. This arises since, at 10 gauss for example, a non-relaxing signal w i l l appear to relax at ^ 0.2 ys while a fa s t relaxing signal (A = 15 ys \ say) i s unaffected by the cosSlt envelope which i s almost f l a t near t = 0. Thus, the systematic error introduced to A decreases with increasing reagent concentration. For f a s t relaxations, higher f i e l d s are preferable, i n p r i n c i p l e , for reasons i l l u s t r a t e d i n the figures of Appendix II and also because a large number of o s c i l l a t i o n s i n the short-lived Mu signal produce more r e l i a b l e f i t s to the f i e l d and phases. It would then seem to be optimal to increase B as a function of A. There i s a serious p r a c t i c a l objection to t h i s proposal, however, in that relaxation rates of 10 - 15 ys are generally d i f f i c u l t to f i t . It i s often necessary to FIX several parameters i n order to reduce model (6) to a function that i s sensitive to the data and A, The important candidates f o r Fixing are B and A M u which requires accurate foreknowledge of these parameters. In practice, i t i s not reasonable to p r e c i s e l y c a l i b r a t e the magnetic f i e l d produced i n the Helmholtz c o i l s as a function of e l e c t r i c current because non-reproducable background contributions to B fluctuate over time periods of days, making constant r e c a l i b r a t i o n necessary. These unreproducible contributions to B can be traced to such events as movement of the TRIUMF 50 ton crane over the experimental area, changes i n magnetic f i e l d settings of beam l i n e components i n adjacent beamlines, and constant re-stacking of s t e e l and iron neutron shielding -57-around adjacent experiments. Such effects constructively or destructively add s i g n i f i c a n t , though homogeneous, components to the experimental magnetic f i e l d . Fortunately, the experimental f i e l d i s generally constant over the time during which a series of concentrations of a reactant are examined, Without c a l i b r a t i o n , i t i s impossible to set B reproducably to better than a few percent. Experience has shown that f i t t e d values of weak f i e l d s from experimental data taken over 24 - 48 hours are constant within a standard deviation of less than 1%, giving a more accurate measure of the f i e l d than c a l i b r a t i o n would give. In the near future, the f i e l d w i l l be s t a b i l i z e d by a continuous monitor i n a feedback loop, which w i l l allow r e l i a b l e and consistent f i e l d settings. The f i t t i n g procedure adopted i s to f i r s t f i t B (in the 7-8 G rangey for a series of runs at low reagent concentration and then to FIX t h i s value of the f i e l d to f i t the fas t relaxation runs. -58-CHAPTER III - THEORETICAL BACKGROUND A Introduction The i n i t i a l motivation for undertaking the experimental study of the chemical reaction rates of Mu was to examine the behaviour of Mu as a l i g h t isotope of hydrogen (see Chapter I, Section B). I t was hoped that the substantial mass difference between Mu and H would provide an exacting t e s t of modern c a l -culations of H atom reaction k i n e t i c s , p a r t i c u l a r l y with respect to quantum mechanical e f f e c t s such as tunnelling [Fleming (76)]. It was expected that comparison of both t h e o r e t i c a l and experimental r e s u l t s for reactions of Mu with those of the other H isotopes might not only lead to improved t h e o r e t i c a l methods for treating H atom reactions, but i t might also provide new information about such computational tools as potential energy surfaces. It i s shown i n t h i s Chapter and the next that many of these objectives have already reached a high l e v e l of r e a l i z a t i o n . In the course of studying Mu reaction k i n e t i c s , a second motivation for the experiments became clear; t h i s i s discussed i n Chapter IV. •The selection of chemical systems for study, namely Mu with the halogen and hydrogen halide f a m i l i e s , was based p a r t l y on the considerable t h e o r e t i c a l and experimental inte r e s t i n the H analogue reactions, and p a r t l y on the experimental .. •. compatability of these reagents with the MSR method: gas phase targets at about 1 atmosphere may be r e a d i l y prepared with a wide range of reactant concentrations; and the reactions are s u f f i c i e n t l y fast at or near room temperature to consume Mu during i t s 2.2 ys l i f e t i m e . As noted i n Chapter I, the experiments have been confined to the measurement of thermally averaged rate constants and a c t i v a t i o n energies. Modern state-to-state techniques employing lasers, atomic beams, and infared chemiluminescence are not yet a v a i l a b l e to Mu studies. I t would be inappropriate to attempt to present a comprehensive review of the theory of chemical k i n e t i c s i n t h i s thesis which i s e s s e n t i a l l y experimental i n content (indeed, the pace of development of chemical reaction rate theory i s so f r e n e t i c that such a review would be impossible). On the other hand, the debut of gas phase Mu reaction rate data has sparked considerable t h e o r e t i c a l a c t i v i t y , notably by Connor, Jakubetz, Manz, and Lagana who have performed quantum mechan-i c a l (QMT) [Connor l-(77), l-(78), l-(79) ], q u a s i c l a s s i c a l (QCT) [Connor 1-(79) ] > and c l a s s i c a l (CT) [Jakubetz ' (79)] t r a j e c t o r y calculations, as well as t r a n s i t i o n state theory (TST) [Connor i -(79)] Jakubetz 1-(78), (79) ]calculations on the reactions of Mu, H, D, and T with F^ and who are presently performing similar calculations on the C l 2 reactions; other authors have done state-to-state calculations of Mu reaction rates which are of less d i r e c t relevance to the present experimental work (see eg. [Fischer 1,2-(77), Korsch (78)]). Since most of the experi-mental r e s u l t s of t h i s thesis are interpreted i n terms of the calculations of Connor et_ aJ_. (Chapter IV) , one of the aims of the present Chapter i s to outline t h e i r various t h e o r e t i c a l _. approaches. The primary aim of t h i s Chapter though, i s to explore some q u a l i t a t i v e predictions of the reaction rates of Mu versus H based both on the calculations mentioned above and -60-on selected considerations from the theory of chemical k i n e t i c s . B Potential Energy Surfaces Most chemists l i k e l y have at least some f a m i l i a r i t y with the notions of a potential energy surface and an associated reaction path. These concepts are i l l u s t r a t e d i n Figure 11. The determination of a potential energy surface to describe the interatomic potentials of the reacting atoms i s the s t a r t i n g point for a l l trajectory calculations [Johnston(66), L a i d l e r (65)], and, to a lesser extent, i t i s a requirement for TST calculations as well. Depending on the d e t a i l s of the s p e c i f i c c a l c u l a t i o n , TST may not require the complete pote n t i a l surface, but only the minimum energy path for the reaction. Before considering p a r t i c u l a r potential energy surfaces for the reactions studied i n t h i s thesis, a few comments should be made about such surfaces i n general. (i) Semi-Empirical Potential Energy Surfaces In p r i n c i p l e , i t should be possible to determine po t e n t i a l energy surfaces from ab i n i t i o methods involving the solution of the Schrodinger equation, perhaps with the aid of approxima-tions based on various quantum mechanical c r i t e r i a [Laidler (65), Jakubetz l - ( 7 8 ) ] . Unfortunately, i t i s s t i l l impossible to perform such calculations with s u f f i c i e n t accuracy to be of general use to reaction k i n e t i c s [Van Hook (70), Jakubetz 1- . (78)], with the possible exception of the H + system (see eg. [Liu (78)]). ' i n the face of t h i s obstacle, i t i s customary to -61-FIGURE 11: A poten t i a l contour map for the exothermic c o l l i n e a r A + BC -> AB + C reaction. The minimum energy path through the saddle point (+) i s denoted by the dashed l i n e . The entrance v a l l e y depth i s -D (BC) measured from the v a l l e y f l o o r , and the ex i t valfey depth i s -D (AB). The saddle point i s above the v a l l e y s , but befow the plateau. -62-employ so-called "semi-empirical" potential energy surfaces (although, the degree of empiricism actually employed often blurs any distinction between "semi-empirical" and "wholly" empirical methods) which are distinguished from ab i n i t i o surfaces by the fac t that parameters are l e f t for adjustment based not on t h e o r e t i c a l grounds, but rather on a p o s t e r i o r i experimental results [Laidler (65)]. The use of a semi-empirical surface necessarily removes some (but c e r t a i n l y not a l l or even most) of the predictive u t i l i t y of the theory. Indeed, many reviewers (see eg. [Johnston (66), L a i d l e r (65), Thompson (76)]) have pointed out that while the accuracy of a ki n e t i c c a l c u l a t i o n depends rather d i r e c t l y on the accuracy of the potential energy surface, many q u a l i t a t i v e predictions can and have been made from consideration of inaccurate or even completely hypothetical p o t e n t i a l energy surfaces (eg. [Kuntz (65), Polanyi (69), Mok (69), Polanyi (78)]).. In t h i s way, chemical k i n e t i c theory and experiments take on an e x p l i c i t symbiotic r e l a t i o n s h i p i n a "bootstrap"• procedure whereby experiments serve not only to test the accuracy of the c a l -culations, but also to adjust the parameters of the poten t i a l energy surface, which, i n turn, leads to improved ca l c u l a t i o n s . Probably the most commonly used semi-empirical methods of determining a po t e n t i a l energy surface are variations of the method due to London, Eyring, Polanyi and Sato (LEPS). A good discussion of the development of semi-empirical p o t e n t i a l energy surfaces i s given i n Laidl e r and Polanyi (65). The surfaces considered i n t h i s thesis are a l l e s s e n t i a l l y v a r i a -tions of the LEPS surface. The LEPS method i s a modification -63-of the Heitler-London approximation for the el e c t r o n i c energy of H 2 using the (unnormalized) wavefunction $ = ^ ( 1 ) ^ ( 2 ) ± ^ ( 1 ) ^ ( 2 ) which has H-H int e r a c t i o n eigenvalues of V ( r ) = = - T - . — A where Q . J , and S are the Coulomb, exchange, and overlap integrals which are functions of r, the internuclear separation. In equation (14), the plus sign refers to the si n g l e t (bound) state and the minus sign refers to the t r i p l e t (repulsive) state. I t may be noted that J i s negative and | j | > Q near r , the equilibrium internuclear separation. London, Eyring, and Polanyi extended t h i s treatment to the system, to give a potential energy expression V ( rAB' rBC' rAC ) = QAB + QBC + QAC ± 4 <JAB " JBcJ 2 1 2 1 2 1/2 where they i d e n t i f i e d the s i n g l e t state with the Morse func-tions of the diatomic p a i r s : Q. . + J. . = ^ ( A r . . ) = 1D, , ( e " 2 3 i j A r i j - 2 e ~ 3 i j A r i j ) (16) 1 j 1 3 I D 1 3 where ^ D ^ j and are constants obtained spectroscopically and Ar^j = r^_. - r g with i , j r e f e r r i n g to appropriate combinations of the atomic labels A,B, and C. "*"D. . i s related to 6 . . by 1 3 1 3 8 . . = TTV 1 3 0 2 y 1 3 1/2 where v i s a fundamental v i b r a t i o n a l 0 frequency and y i s the reduced mass of the diatomic p a i r . The 2 LEP treatment neglects the overlap i n t e g r a l S , and assumes a t °A , constant f r a c t i o n of coulombic binding energy (——+ ^ J -A. A -64-independent of r. The Sato modification [Sato (55)] of equation (16) i s : V ( rAB' rBC' rAC ) = 377^2 [ QAB + QBC + QAC ± 4 ( J A B " JBC ) ' 1 2 1 2 1/2 + I ( J B C - J A C > + I ( J A C " J A B ' > ' 1 where the t r i p l e t (anti-bonding) state i s i d e n t i f i e d with a modified Morse function (anti-Morse or Sato-Morse): Q i j ~ J i j 3„,. , ^D, . , -2 3 . .Ar. . „ - 3 . .Ar. .. = V(Ar. .) = 1 3 (e 1 3 1 3 + 2e " 1 3 1 3 ) (18) (1 - S ) 3 2 and the si n g l e t state (equation (16)) becomes: Q. . + J. . 1 T T /. , 1^ , -23. .Ar. . _ - 6 . .Ar. 1 3 1 3 - V(Ar..) = D..(e ij 1 3 - 2e ^ 1 3 1 3 ) (19) (1 + sz) The overlap i n t e g r a l i n the LEPS formulation (equation (17)) i s 2 l e f t as an adjustable parameter (A = S i s c a l l e d the Sato parameter) which i s normally found to be much smaller than the true overlap i n t e g r a l . I t should be noted that the Sato 2 modification set S constant over a l l internuclear separations and independent of atomic labels. Most authors use empirical variations of the LEPS formulation such as: replacement of the 2 2 constant S with S^^ terms for each atomic pair [Kuntz (66), Jonathan (7 2)] which may or may not be dependent upon the internuclear distance (Jonathan et a l . examined both cases, 2 Kuntz et a l . used S^j independent of r ^ j ) ; empirical adjustment of the t r i p l e t anti-Morse function (equation (18)) by forms XD 3 which replace i j by an adjustable D.. ([Jonathan (72)] also examined t h i s ) ; or replacement of the anti-Morse function by, for example: -65-3„ /» » 3_ , -2$; .Ar. . , _ -8. .Ar. . . . ^ * V i j ( A r i j ) = i j ( 1 D 1 : 1 1 D 1 D ) f o r r i j - r —or * = C(r. . + A)e i j for r. . > r 1 3 I D -3 * where D, B, C, A, a, and r are adjustable parameters ([White (73)]; i n f a c t , White used an empirical valence bond v a r i a t i o n of the LEP surface). Clearly, a l l LEPS formulations mentioned require experi-mental input. LEPS surfaces are usually "optimized" by adjusting the variable parameters u n t i l some type of trajectory calculations performed on the surface reproduce a set of experimental r e s u l t s . For example, Jonathan et a_l. (72), t a i l o r e d the LEPS surface for the reaction H + F 2 -»- HF + F such that three dimensional q u a s i c l a s s i c a l trajectory calculations (see Section D below) give an HF v i b r a t i o n a l energy d i s t r i b u -t i o n , reaction a c t i v a t i o n energy, rate constant and reaction enthalpy i n agreement with experiment. Clearly, t h i s procedure i s not l i k e l y to converge to a unique "correct" surface; i t i s only hoped that i t produces a useful surface. ( i i ) Contour Plots of the Potential Energy Surface for the  Reaction A + BC •> AB + C In t h i s Section, i t i s shown how contour p l o t s , such as i n Figure 11, provide the basis for setting up a conceptually and computationally simple picture of the atom-diatom c o l l i s i o n process. For the three atom A + BC system, the interatomic pot e n t i a l energy i s a function of the positions of the three nuclei and therefore a function of nine coordinates. However, i t i s only the r e l a t i v e motion of the nuclei within t h e i r -66-center of mass frame that i s of relevance to the c o l l i s i o n since t r a n s l a t i o n or rotation of the three atoms together as a r i g i d body w i l l not a f f e c t t h i s interatomic p o t e n t i a l . Thus, the interatomic p o t e n t i a l energy i s a function of three coordinates: r . ^ , r , and r or, more commonly, r , r , and A B B C A C A B B C 6)ABC' the ABC bond angle. Consequently, contour plots of the pot e n t i a l energy function must be drawn with respect to two coordinates, with the t h i r d coordinate fixed. Usually, such contour plots show the p o t e n t i a l energy as a function of r A B and r^^, (as i n Figure 11) with 0 fixed; i n fact, most B C A B C commonly, 9AB(~. i s fixed at 180° describing the c o l l i n e a r configuration of the three atoms. There are at l e a s t two good reasons why the c o l l i n e a r p o t e n t i a l surface i s the one most often considered: f i r s t l y , as discussed i n Section D below, the degree of complexity of trajectory calculations . increases tremendously from the c o l l i n e a r to the coplanar to the three dimensional cases, not only because of the need to consider more surfaces, but also because of the increased number of i n t e r n a l degrees of freedom of the system with the corresponding increase i n the number of reaction channels available (more product v i b r a t i o n a l and r o t a t i o n a l states are included); and, secondly, the simpler c o l l i n e a r calculations often (but by no means always) provide a reasonably accurate description of the reaction, p a r t l y because i t i s usually the case that the c o l l i n e a r reaction geometry i s the energetically favored one. In f a c t , Jonathan et a l . (72) have pointed out that an energetically favored c o l l i n e a r configuration seems to be a general feature of LEPS surfaces; however, t h i s general--67-i z a t i o n does not i n i t s e l f imply that the c o l l i n e a r con-fi g u r a t i o n w i l l necessarily dominate the reaction dynamics because i t neglects other topological features of the potential surface as well as the role played by multi-dimensional internal, energy modes of .the- target molecule, which may be available for promoting reaction. The chemical reaction A + BC -* AB + C i s envisioned as the movement of a representative mass point (the features of which are detailed below) along the poten t i a l energy surface through the reactant v a l l e y , across the saddle point, and f i n a l l y e x i t i n g along the product v a l l e y . This notion that the ele c t r o n i c p o t e n t i a l energy surface mediates the motion of the nuclei of the atoms i s an i m p l i c i t statement of the Born-Oppenheimer (BO) approximation: the elec t r o n i c energy of the atoms i s separable from t h e i r nuclear energy. This approx-imation, sometimes c a l l e d the low k i n e t i c energy approximation [Schatz (77) , Levine (74)] , i s v a l i d for most atoms at normal temperatures, where nuclear v e l o c i t i e s are much less than the electron v e l o c i t i e s . Since the mean v e l o c i t y of an ensemble of atoms at a given temperature i s inversely proportional to the square root of t h e i r masses (see Section D below), i t i s expected that the BO approximation w i l l break down at lower temperatures for Mu than for H. To date, the BO approximation has always been invoked i n calculations of the reactions of Mu, although i t s v a l i d i t y i n these cases has not yet been examined [Jakubetz l - ( 7 8 ) ] ; a discussion of the v a l i d i t y of the BO approximation with application to, among others, the H + HD reaction i s given by Bardo and Wolfsberg (78) who f i n d i t to be -68-accurate to within a few percent i n the cases studied. Correlated with the BO approximation i s the assumption of electronic adiabaticity [ N i k i t i n (74)] : the system remains i n the ground ele c t r o n i c state throughout the reactive c o l l i s i o n . Besides depending on the c o l l i s i o n time, t h i s assumption depends upon the electron angular momentum and the c o r r e l a t i o n of electronic states [Smith (77)]. The assumption of electronic -adiabaticity i s the standard procedure for both trajectory and TST treatments of the reactions considered i n th i s thesis; the p r a c t i c a l consequence of t h i s assumption i s that only ground state p o t e n t i a l energy surfaces need be considered. The f i n a l , and, from the point of view of t h i s thesis, most important consequence of the BO approximation i s that the poten t i a l energy surface i s invariant to isotopic substitution [Van Hook (70), N i k i t i n (74)]; that i s , i d e n t i c a l potential energy surfaces are applicable to reactions of Mu, H, D and T. I t has already been stated that reactive atom-diatom c o l l i s i o n s can be pictured as the transmission of some kind of p a r t i c l e across the ba r r i e r of a pot e n t i a l energy surface. Levine and Bernstein (72) c a l l t h i s picture the "analogue" formulation of the problem and i t i s necessary to determine the ide n t i t y of these p a r t i c l e s moving on the poten t i a l surface. To th i s end, i t i s useful to consider the freshman physics problem of the c o l l i s i o n of two structureless b i l l i a r d b a l l s . Although t h i s problem i s treated c o r r e c t l y by separately solving the equations of motion for each b a l l , i n some ways i t i s more useful to consider the equivalent problem i n which the motion of the center of mass i t s e l f i s partitioned from the r e l a t i v e motion within the center of mass frame. This procedure s i m p l i f i e s both the problem and the inter p r e t a t i o n of i t s solution by eliminating the motion of the center of mass which i s extraneous to the c o l l i s i o n i t s e l f . For example, one finds that the system k i n e t i c energy i n the center of mass frame i s given by = i y v 2 (20) trans 2 M r where v r i s the r e l a t i v e v e l o c i t y of the two b a l l s and y i s th e i r reduced mass. Equation (20) i s remarkable i n that i t looks just l i k e an equation of motion for a single p a r t i c l e of mass y. Given t h i s i n t e r p r e t a t i o n , some of the properties of the two body c o l l i s i o n are described by the analogue equations of motion of a single representative point with some e f f e c t i v e mass. The use of a poten t i a l energy contour p l o t to describe the atom-diatom c o l l i s i o n i s a generalization of t h i s procedure to three bodies. Points along the reaction path on the potent i a l energy surface describe the configuration of the three atoms at various stages of c o l l i s i o n . Instead of solving the equations of motion for a l l three atoms (which i s occasionally done), the reactive c o l l i s i o n i s described by solving the equations of motion of a single representative mass point moving along the potential energy surface, situated in the center of mass frame. The problem now arises as to what e f f e c t i v e mass to assign to t h i s representative point. To i l l u s t r a t e t h i s problem, consider the c o l l i n e a r LEPS poten t i a l energy surface -70-due to Jonathan et a l . (72) shown i n Figure 12 for the reaction Y + F 2->YF + F, Y = Mu, H, D, T (adapted from [Connor l-(78)]) . As the representative point moves along the reactant v a l l e y towards the potential saddle on a l i n e p a r a l l e l to the r^p axis, i t s motion simply describes a two body c o l l i s i o n as i n equation (20) with H (say) as one body, and F 2 a s the other. . n ^ ( m + m ) Thus, the e f f e c t i v e mass i s y„ „ = ; ; - 1 amu, and '2 mH F m F the r e l a t i v e t r a n s l a t i o n a l k i n e t i c energy i s given by 1 • 2 dr Ej. = -KV.„ m r T T„, where r E -g^. On the other hand, aft e r trans 2 MH,F 2 HF' dt reaction, the representative point moves along the product va l l e y away from the saddle point on a l i n e p a r a l l e l to the r„„ axis, thereby describing another two body system, t h i s time (mR + mp)mF of HF and F. Here the e f f e c t i v e mass i s y ~ HF,F mH + mp + mp - 9 amu and the r e l a t i v e t r a n s l a t i o n a l k i n e t i c energy i s given 1 * 2 Y E t r a n s = " ^ H F ^ F F " I n 9 e n e r a 1 ' i t c a n b e shown that the -effective mass •'-."•of- the. representative .point is. a function of the di r e c t i o n of i t s motion along the pot e n t i a l energy surface (an excellent discussion of t h i s subject i s found i n Johnston (66)), varying continuously from 1 to 9 amu i n the present example as the slope of i t s tra j e c t o r y varies from 0 to 0 0. When the vi b r a t i o n a l motion of the target F 2 and product HF are taken into account, t r a j e c t o r i e s of the representative point are o s c i l l a t o r y so that t h e i r e f f e c t i v e masses are also o s c i l l a t o r y , thereby making t h i s analogue picture both computationally and conceptually complicated. This complication i s removed by representing the poten-t i a l energy surface with a mass weighted coordinate system (for FIGURE 12: Potential energy surface for Y + F 2 due to Jonathan (7 2) , adapted from Connor 1-(78). * Solid contours are labelled i n eV from the F^ d i s s o c i a t i o n l i m i t . Dashed contour, B, i s at ^ ^ a n s = 0.087 eV, the q u a s i c l a s s i c a l threshold. a derivation, see Johnston (66)). For the c o l l i n e a r configuration of atoms A, B, and C, i t may be shown that the ki n e t i c energy i n the center of mass frame i s given by: E t r a n s = ^M [ mA ( mB + V ^ B + 2 m A m c W B C .2 ( 2 1> + mc(mA + m B ) r B C ] where M = m, + nu + m_,. The f i r s t and l a s t terms have been A B C previewed i n the above discussions of pure A-BC and pure AB-C motion. The middle cross-product term provides the continuous v a r i a t i o n between the two motional extremes and anticipates that any new coordinate system, q- D and q , that diagonalizes A B B C equation (21) w i l l be skewed with respect to the cartesian r ^ B and r D„. In general, there may be more than one coordinate B C transform that diagonalizes the k i n e t i c energy [Johnston (66)]; for the c o l l i n e a r case, a common mass weighted coordinate transformation i s [Marcus (77)] : rAB = qAB " qBC C t n a ( 2 2 ) rBC = S qBC ° S C a in (itv + m_) where s -m C ( m A + 1/2 and cosa = mm A C (m, + m_) (m + m ) J 1/2 and a i s the skewing angle. Equations (22) give the ki n e t i c energy expression: 1 • 2 • 2 E t r a n s = 2 yA,BC ( qAB + qBC ) and define the constant e f f e c t i v e mass of the representative point as V . For other than the c o l l i n e a r configuration, d i f f e r e n t but similar expressions to equations (22) are required to diagonalize the k i n e t i c energy (see eg. [Gatz (66)]) With the transformation of the potential energy surface into a -73-mass weighted coordinate system, the atom-diatom collision can be completely understood c l a s s i c a l l y i n terms of the traj e c t o r y of a b a l l r o l l i n g along a physical surface under the influence of gravity. Although the foregoing discussion e x p l i c i t l y refers to t r a j e c t o r i e s , i t i s also applicable to TST c a l c u l a -tions which can be viewed as a s t a t i s t i c a l treatment of t r a j e c t o r i e s not requiring t h e i r i n d i v i d u a l c a l c u l a t i o n (there are also s t a t i s t i c a l dynamical theories [Connor l-(7 6)]). ( i i i ) Potential Energy Surfaces For the Reactions: Y + ->  YX + X, Y = Mu, H, D, T; X = F, C l , Br, I While invariance of a poten t i a l energy surface under isotopic substitution i s a consequence of the BO approximation, the above discussion c l e a r l y shows that the e f f e c t i v e p o t e n t i a l energy surface ( i . e . mass weighted) displays no such invariance. This i s i l l u s t r a t e d i n the mass weighted LEPS surface for the Y + F 2 + Y F + F , Y=Mu, H, D, T, reaction shown i n Figure 13 (adapted from [Connor l-(78)]>, corresponding to the LEPS surface of Figure 12. In t h i s Figure, the mass weighting scheme used i s [Connor (75)]: a/2 YF (r. YF + 0 . 5 r ) FF l l / 2 (23) r where R^  i s the distance from Y to the center of mass of F 2 , T,F 2 and the skewing angle a i s given by: - 7 4 -Y = M u . H . D , T 11 : : L _ 0 1 2 ( M Y , F 2 / M F / 2 R X F / A FIGURE 13: Potential energy surfaces of Jonathan (72) for the c o l l i n e a r Y + F 2 reaction, plotted i n mass weighted coordinates, adapted from Connor l-(78). The mass weighting scheme i s described i n the text. The single contours are for E t r a n s = 0.087 eV(2.01 k c a l / mole) (contour B of Figure 12). Saddle points are indicated by crosses. tana = fitly + 2 n v i l / 2 -75-, 1 /-). F mY In t h i s mass weighting scheme, the k i n e t i c energy- i n the center of mass frame i s given by: E t r a n s = I ^ F ^ (*YF + with an e f f e c t i v e mass of y^ for the representative point for F2 a l l isotopic forms of Y.. The single contours shown i n Figure 13 correspond to a r e l a t i v e t r a n s l a t i o n a l k i n e t i c energy, E =2.01 kcal/mole (or the equivalent p o t e n t i a l energy trans r e l a t i v e to zero as defined i n Section C below), which i s approximately the c l a s s i c a l threshold for the reaction. The skewing angles of the Mu, H, D, and T surfaces are 86.9, 80.9, 77.4, and 74.9 degrees respectively. Besides showing that the skewing angle approaches 90° as the isotopic mass decreases, Figure 13 shows a pronounced contraction of the e x i t v a l l e y and a sharp ' c o n s t r i c t i o n (or "bottleneck") i n the entrance v a l l e y near the saddle point for the l i g h t e r isotope. Another feature of t h i s surface to note i s that the po t e n t i a l energy ba r r i e r or saddle point i s "early"; that i s , i t i s located along the entrance v a l l e y for a l l isotopic variations of the reaction. Although the LEPS surfaces for the reactions of H i s o -topes with C l 2 , B r 2 and I 2 are not as well known as those for the F 2 reaction [Jakubetz l-(78), Connor 2-(77)], i t i s expect-ed that t h e i r topological features should be sim i l a r to the F 2 surface [Bauer l-(78), P a t t e n g i l l (76), B l a i s (74)] with two notable exceptions: (1) while the c o l l i n e a r reactive geometry i s strongly favored for the F 2 and C l 2 reactions [Jakubetz l-(78), -76-Polanyi (75)], t h i s i s probably not the case for the B r 2 and 1^ reactions [Baybutt (78), Bauer l-(78), B l a i s (74)]; and (2), the existence of a s t a t i c potential energy b a r r i e r (saddle point on the potential energy surface) i s not c l e a r l y established for the B r 2 and I 2 reactions [White (73), Baybutt (78), B l a i s (74)]. For the sake of argument, i n the following discussion, i t i s assumed that the LEPS surfaces for the halogen homologous series are s i m i l a r , with potential energy barri e r s decreasing i n the order Cl > Br > I (the F 2 reaction has an anomalously smaller b a r r i e r than the C l 2 reaction [Pattengill (76), Anlauf (72)]); furthermore, i t i s assumed that the Mok-Polanyi [Mok (69)] c o r r e l a t i o n holds: the poten-t i a l energy b a r r i e r moves to consecutively e a r l i e r positions as the b a r r i e r height decreases i n a series of exothermic reac-tions . Within the assumptions made above, i t seems clear that the mass weighted LEPS surfaces for the H isotope reactions with the halogen series should display the same es s e n t i a l behaviour as that shown i n Figure 13, from which a number of generaliza-tions can be made concerning the reaction dynamics. A substantial "bottleneck e f f e c t , " whimsically i l l u s t r a t e d i n Figure 14 (adapted from Connor 2-(77)), was predicted by Manz (76) on the basis of Figure 13 before being v e r i f i e d by d e t a i l -ed trajectory c a l c u l a t i o n s . This e f f e c t suggests that c l a s s i -c a l contributions to the reaction p r o b a b i l i t y favor the heavy H isotopes for which the notion of the representative points through the bottleneck displays e s s e n t i a l l y laminar flow i n contrast to the turbulent flow exhibited i n the Mu reaction. F+ YF Y+ F2 The bottle-neck effect FIGURE 14: Adapted from Connor 2-(77). -7 8-On t h i s basis, i t i s predicted that c l a s s i c a l l y the reaction p r o b a b i l i t y w i l l follow the order T > D > H > Mu. S i m i l a r i l y , the contraction of the product v a l l e y i n the Mu case presents a greater p r o b a b i l i t y for c l a s s i c a l non-reactive "back-r e f l e c t i o n " of the representative points o f f the strongly repulsive wall i n the product v a l l e y d i r e c t l y opposite the saddle point, corresponding to high energy c o l l i s i o n s ; that i s , the contracted product v a l l e y makes i t much more d i f f i c u l t for the reaction to "turn the corner" i n the Mu case than for the other H isotopes. This e f f e c t , which has been t h e o r e t i c a l l y v e r i f i e d [Connor 2-(78)], again suggests a reaction p r o b a b i l i t y •order of T > D > H > Mu, since the c l a s s i c a l Mu reaction p r o b a b i l i t y (or cross section) w i l l f a l l o ff at lower energies i n the high temperature regime than for the other H isotopes. A much less important but correlated e f f e c t i s that due to the skewing angle of the po t e n t i a l surface: the smaller the skewing angle, the less prone to back-reflection i s the reaction. Again, t h i s favors the order T > D > H > Mu. The existence of an "early" p o t e n t i a l energy b a r r i e r has a number of implications. From the point of view of t h i s thesis, perhaps the most important implication of t h i s topological feature i s the fac t that the representative point crosses the poten t i a l b a r r i e r while i t i s s t i l l on a trajectory that i s more or less p a r a l l e l to the r axis. As shown i n Part ( i i ) above, t h i s means that the e f f e c t i v e mass of the representative point, on either the unweighted surface or on the surface that i s mass weighted according to equations (22), i s p , which i s e s s e n t i a l l y equal to the H iso t o p i c mass Y, X 9 -79-(a l t e r n a t i v e l y , i n the Connor mass weighting scheme of equation (23) i n which the e f f e c t i v e mass of the representative point i s the same for a l l H isotopes, the width of the energy b a r r i e r i s 1 /o proportional to (u ) ). Consequently, the f u l l mass Y,x 2 e f f e c t of H isotopic substitution i s u t i l i z e d i n b a r r i e r penetration (quantum mechanical tunnelling) [Jakubetz l - ( 7 8 ) ] . This would not be the case i f the potential energy surface had a symmetrically placed or l a t e b a r r i e r . From these considera-tions, i t i s expected that the Mu reaction w i l l be subject to much greater tunnelling than the other H isotopes, tending to order the quantum mechanical reaction p r o b a b i l i t i e s i n the low temperature regime Mu > H > D > T. Furthermore, shown to scale on the left of Figure 12 are the thermal de Broglie wave-lengths (X - , . ,1/2) of the e f f e c t i v e masses of the repre-B sentative points u Y p , corresponding to Y-F 2 motion, and u X F F 2 1 (- constant for a l l Y), corresponding to YF-F motion ; substi-tution of X = C l , Br, or I for F does not a f f e c t the u, Y,X2 l . 1 rough rule, i f the thermal de Broglie wavelength of a p a r t i c l e wavelength^ but further contracts the u Y X x wavelength. As a I S t r i c t l y speaking, the representative point slows down as i t encounters the p o t e n t i a l energy b a r r i e r so that i t s de Broglie wavelength i s a function of i t s coordinates on the p o t e n t i a l energy surface. Denoting q^ as a general reaction coordinate, the de Broglie wavelength of the representative point i s given by X(q r) = ( 2 y [ E V(q ) ] y 1 / 2 ' w h e r e u i s i t s e f f e c t i v e trans r mass, E. i s the i n i t i a l r e l a t i v e t r a n s l a t i o n a l k i n e t i c trans energy, and V(q r) i s the height of the p o t e n t i a l surface at q^ above the asymptotic reactant v a l l e y [Nikitin (74)]. Conse-quently, the thermal de Broglie wavelengths shown i n Figure 12 are minimum thermal averages. This point does not fundamen-1 t a l l y a f f e c t the arguments made above. -80-i s shorter than the width of a b a r r i e r , tunnelling i s minimal; conversely, i f i t s thermal de Broglie wavelength i s longer than the b a r r i e r width, tunnelling i s expected to be important [N i k i t i n (74)]. These considerations suggest that quantum mechanical reaction p r o b a b i l i t i e s may be accurately estimated by applying some sort of one dimensional b a r r i e r penetration correction to c l a s s i c a l or TST calculations [Jakubetz (79)] without the need to consider such complications as alternate tunnelling paths ("corner cutting") (see eg. [Marcus (78), 2 Johnston (61)]). Another implication of the early potential b a r r i e r con-cerns the f i n a l state v i b r a t i o n a l energy d i s t r i b u t i o n s of the reaction products (rotational energy transfer cannot take place i n c o l l i n e a r c o l l i s i o n s ) . Although f i n a l state d i s t r i b u t i o n s are as yet experimentally inaccessible to Mu studies, they are of s u f f i c i e n t t h e o r e t i c a l i n t e r e s t to warrant a b r i e f discus-sion. For exothermic reactions, late p o t e n t i a l energy b a r r i -ers are associated with repulsive energy release i n which the reaction exoergicity i s released as the reacting atoms separate; early p o t e n t i a l energy b a r r i e r s are often associated with mixed energy release, that is, part of i t i s a t t r a c t i v e (released as the p r o j e c t i l e atom approaches the target molecule), and part of i t . i s repulsive. Although the c o l l i n e a r LEPS surfaces for the F^ and reactions are known to be predominantly repulsive 2 Actually the ba r r i e r s to tunnelling considered i n most trajectory or TST calculations are. not i d e n t i c a l to the s t a t i c b a r r i e r s described by the potential energy surface. Never-theless, the present discussion i s v a l i d because the location and shape (but not height) of these b a r r i e r s are e s s e n t i a l l y the same as the s t a t i c p o t e n t i a l energy b a r r i e r s . This point i s discussed i n Section C below. -81-[Polanyi (75) , P a t t e n g i l l (76) , Wilkins (75)] , i t i s not clear i f t h i s i s the case for the B r 2 and surfaces [Polanyi (75), Bla i s (74), Baybutt (78)]. In any case, a c o r o l l a r y of the Mok-Polanyi c o r r e l a t i o n i s expected to hold: " i n a homologous series i n which a f a l l i n g b a r r i e r i s not accompanied by an increase i n exothermicity, the increase i n a t t r a c t i v e energy release w i l l be accompanied by a decrease i n repulsive energy release [Mok (69)]." Roughly speaking, the a t t r a c t i v e part of a mixed energy release i s transformed into v i b r a t i o n a l energy of the products, while the repulsive part i s transformed into t r a n s l a t i o n a l k i n e t i c energy of the products [Polanyi (72)]. Since the skewing angles of the Y + poten t i a l energy surfaces are approximately the same, i t i s expected that as X changes from F to I, the increase i n a t t r a c t i v e energy release w i l l be accompanied by an increase i n the v i b r a t i o n a l energy of the products [Wilkins (75)] . On the other hand, as Y varies from T to Mu, the product energy d i s t r i b u t i o n s should display the " l i g h t atom anomaly [Polanyi (75)]:" on repulsive surfaces, when the mass of the attacking atom i s much less than those of the target molecule, less reaction exoergicity i s channelled into product v i b r a t i o n a l energy as the mass of the attacking atom decreases. This may be pictured as an i n e r t i a l e f f e c t i n which the rapid release of the reaction exoergicity on the dominant repulsive part of the surface imparts such momentum to the separating heavy atoms (B-C) that the r e l a t i v e l y i n s i g n i f i -cant momentum of the l i g h t attacking atom (A) i s overwhelmed. On the/other hand, i f A were of a comparable mass to the atoms of the target molecule, i t would have such i n e r t i a that when -82-the repulsive reaction exoergicity slammed B into i t , the r e s u l t would be a v i b r a t i o n a l l y excited A-B molecule. Since the dynamics of the reactions of H isotopes with the hydrogen halides are probably influenced more by the d i s p o s i t i o n of energy among in t e r n a l molecular modes than by the topology of the p o t e n t i a l energy surfaces, the discussion of H-HX LEPS surfaces i s deferred u n t i l the next Section where these energy e f f e c t s are taken into account. C Energy To t h i s point, H isotope e f f e c t s have been discussed on the basis of i n t u i t i v e predictions of the behaviour of t r a j e c t o r i e s of a representative point encountering character-i s t i c topological features of e l e c t r o n i c a l l y adiabatic potential energy surfaces. Besides the p o t e n t i a l energy surface and the t r a n s l a t i o n a l k i n e t i c energy of the represent-ative point, reference has been made to other energies such as the reaction a c t i v a t i o n energy and enthalpy, c l a s s i c a l thres-hold energy, and i n t e r n a l energy of the target and product molecules. The task of t h i s Section i s to define these forms of energy and interpret t h e i r roles i n the reaction process. F i n a l l y , some of these ideas are applied i n a discussion of H isotope - hydrogen halide reactions. Energy d e f i n i t i o n s used i n conjunction with potential energy surfaces depend upon the choice of an a r b i t r a r y reference point of zero energy for which, unfortunately, there i s no single convention. For example, i n Figure 12, a l l of the contours of the LEPS surface are drawn with respect to -83-zero defined as the d i s s o c i a t i o n l i m i t of F^r except for the dashed contour representing a poten t i a l energy equivalent to the r e l a t i v e t r a n s l a t i o n a l k i n e t i c energy of the c l a s s i c a l threshold, which i s drawn with respect to a d i f f e r e n t zero as defined below. This confusion i s further aggravated by the fact that various authors often use the same name to refer to d i f f e r e n t energies. For example, i n a discussion of the reactions of Y + F 2 + YF + F, Y = Mu, H, D, T, what Connor et al.l-(79) c a l l the "barrier height" i s quoted with values of both 2.35 and 1.08 kcal/mole, the former r e f e r r i n g to the height of the saddle point r e l a t i v e to the bottom of the asymptotic reactant v a l l e y , while the l a t t e r refers to t h i s value less the zero point v i b r a t i o n a l energy of the F^ molecule This m u l t i p l i c i t y of d e f i n i t i o n s has i t s genesis i n the multitude of approaches to the c a l c u l a t i o n of reaction k i n e t i c s for example, c l a s s i c a l trajectory calculations apply a d i f f e r -ent meaning to the "barrier height" than q u a s i c l a s s i c a l or quantum mechanical trajectory c a l c u l a t i o n s . Clearly, there i s a need for considerable care i n defining the various energy terms. (i) C l a s s i c a l Trajectories The picture of a b a l l r o l l i n g along the minimum energy path of the potential energy surface corresponds to a purely c l a s s i c a l trajectory i n which the in t e r n a l v i b r a t i o n a l and ro t a t i o n a l energies of the target molecule are i n i t i a l l y zero. In t h i s case, i t i s useful to consider the pot e n t i a l b a r r i e r Cl height, denoted here as E h and defined as the elevation of the -84-saddle point above the bottom of the asymptotic reactant v a l l e y . Cl Within the BO approximation, E^ i s the same for a l l i s o t o p i c variants of the H atom reactions. Since there i s no i n t e r n a l energy i n the target molecule at the onset of c o l l i s i o n i n t h i s picture, the r e l a t i v e t r a n s l a t i o n a l k i n e t i c energy of the representative point i s the only energy available to propel i t over the potential b a r r i e r to bring about reaction. Therefore, C l the r e l a t i v e t r a n s l a t i o n a l k i n e t i c energy, E. , i s also "cr an s measured with respect to zero taken as the bottom of the asymptotic reactant v a l l e y . I t i s useful to picture the r e l a t i o n s h i p between the k i n e t i c energy of the representative point and the potential energy surface i n terms of an airplane f l y i n g through the v a l l e y at a constant a l t i t u d e measured from the asymptotic minimum of the reactant v a l l e y . In t h i s picture, the height of the plane above the v a l l e y f l o o r corresponds to the k i n e t i c energy of the representative point, and i t i s clear that i f the a l t i t u d e of the airplane does not exceed the elevation of the saddle point, a non-reactive crash w i l l occur. The u t i l i t y of t h i s pedantic analogy w i l l become evident i n the l a t e r discussion of q u a s i c l a s s i c a l and quantum mechanical t r a j e c t o r i e s . Closely related to the potential energy b a r r i e r height Cl i s the notion of a c l a s s i c a l threshold energy, denoted E^ , and defined as the minimum t r a n s l a t i o n a l k i n e t i c energy of the representative point required for reaction. In terms of the purely c l a s s i c a l picture discussed i n the previous paragraph, i t might seem that the threshold energy i s i d e n t i c a l to the potential b a r r i e r height, but t h i s i s generally not the case. -85-To understand the difference, i t must be noted that although the c l a s s i c a l picture under consideration assumes that i n i t i a l -l y the target molecule possesses no i n t e r n a l energy, i t does not prohibit the transfer of c o l l i s i o n a l k i n e t i c energy into i n t e r n a l energy of the target molecule. As implied i n the discussions i n Section B on back-reflection and the l i g h t atom anomaly, i n e r t i a w i l l cause the trajectory of the repre-sentative point to deviate from the minimum energy path as i t attempts to "turn the corner" of the potential energy surface [ N i k i t i n ( 7 4 )]. Not even i n the case of early b a r r i e r s can i t be assumed that the saddle point i s c o l l i n e a r with the incident minimum energy path, as i l l u s t r a t e d i n Figure 15 for the Mu + F2 -> MuF + F reaction (adapted from Connor l - ( 7 7 ) ) . Consequently, the representative point generally attempts to cross the potential b a r r i e r at a point other than the minimum ba r r i e r height. For surfaces with symmetrically placed, or l a t e b a r r i e r s , the representative point w i l l possess a r e l a -t i v e l y large component of v e l o c i t y perpendicular to the minimum energy path as i t attempts to cross the b a r r i e r , corresponding to conversion of some of the i n i t i a l t r a n s l a t i o n -a l k i n e t i c energy into v i b r a t i o n a l energy of the reacting species. In short, threshold energies are dynamical quanti-t i e s while energy b a r r i e r s are s t a t i c . From these considera-tions of the c l a s s i c a l trajectory, i t can be seen that the c l a s s i c a l threshold energy must be greater than or equal to the pot e n t i a l energy b a r r i e r height. In the case of quasiclas-s i c a l t r a j e c t o r i e s , i t may happen that the reaction threshold C l energy i s less than , as discussed l a t e r i n t h i s Section. -86-1.0 R M U F / A 30 P 0.2 0.3 0.4 x/A FIGURE 15: Mass weighted potential energy surface for the c o l -l i n e a r Mu + F 0 reaction, adapted from Connor l-(77); x i s defined i n the text. The dash-dot l i n e i s the minimum energy path. The dashed l i n e s indicate con-tours where the reactant and product t r a n s l a t i o n a l energy i s zero. Contours A, B, and C are at E ° ^ a n s = 0.08, 0.16, and 0.24 eV. Line P i s the " l i n e of no return" mentioned i n Chapter IV. -87-For surfaces possessing early b a r r i e r s , i t i s expected that threshold energies for the various H isotopes w i l l be similar (but not identical) to each other due to the r e l a t i v e l y mild d i s t o r t i o n s of the reactant v a l l e y under the transformation to mass weighted coordinates (eg. Figure 13). As discussed i n the previous Section, H isotope reactions with the halogens Cl display a "bottleneck" e f f e c t which tends to order E T : Mu > H > D > T. I t should be noted that the commonly used phrase, "the c l a s s i c a l threshold energy of the reaction," implying the existence of a unique value, often represents a misuse of the language. Certainly, i n the case of purely c l a s s i c a l t r a j e c t o r i e s , there i s a unique threshold energy for each surface. However, multidimensional trajectory calculations must be performed on several surfaces, each with i t s own threshold energy. Thus, while one may speak of the c l a s s i c a l threshold energy for a one dimensional trajectory, three dimensional t r a j e c t o r i e s have a range of threshold energies over the various ABC bond angles and impact parameters (that i s , the minimum distance between the approach trajectory and the center of mass of the target molecule). The notion of a single reaction threshold energy i s even less precise i n the case of q u a s i c l a s s i c a l t r a j e c t o r i e s , discussed next. ( i i ) Q u a s i c l a s s i c a l and Quantum Mechanical Trajectories Although i t was shown i n the previous Section that c l a s s i c a l t r a j e c t o r i e s provide a q u a l i t a t i v e l y useful picture of the dynamics of a reaction, i t i s u n r e a l i s t i c to ignore the -88-i n i t i a l i n t e r n a l v i b r a t i o n a l and r o t a t i o n a l energy of the target molecule. In p r i n c i p l e , a l l of the v i b r a t i o n a l energy of the target molecule i s available to promote reaction since the B-C stretch i n the reaction A + BC -> AB + C corresponds to the reaction coordinate along the product v a l l e y . Quasi-c l a s s i c a l trajectory calculations are formulated such that before any i n t e r a c t i o n of the c o l l i s i o n partners occurs, the i n t e r n a l states of the target molecule are described by quantum mechanical p r o b a b i l i t y density functions; but once the t r a j e c -tory begins, a l l of the motion i s c l a s s i c a l [Thompson (76)]. Of course, quantum mechanical trajectory calculations involve quantum state d i s t r i b u t i o n s throughout the reaction. A more detailed discussion of the various types of t r a j e c t o r i e s i s given i n Section D. For QCT and QMT c a l c u l a t i o n s , i t i s customary to define a number of energies r e l a t i v e to zero taken as the height of the v i b r a t i o n a l energy of the target molecule (denoted above the bottom of the asymptotic reactant v a l l e y , thereby assuming that a l l of t h i s v i b r a t i o n a l energy i s available for reaction. For example, t h i s zero energy i s shown as the dashed contour i n Figure 15 for the reaction Mu + F 2 MuF + F with F 2 i n the v = 0 state. Based on t h i s energy zero, the physical b a r r i e r height, E ^ h ^ s , i s defined as p h y s = C l _ ( 2 4 ) b to v I t may be noted that a l l of these quantities are invariant under isotopic substitution of the p r o j e c t i l e atom. E ^ ^ S serves as the boundary that d i f f e r e n t i a t e s dynamical tunnelling from s t a t i c tunnelling as discussed i n Section F below. Since -89-a l l of the v i b r a t i o n a l energy of the target molecule may be available to promote reaction, the t r a n s l a t i o n a l k i n e t i c energy of the representative point i s also measured r e l a t i v e QC to t h i s zero contour and i s denoted E. . In terms of the trans picture of an airplane f l y i n g up the reactant v a l l e y , t h i s new zero energy corresponds to a flooded v a l l e y with a shoreline corresponding to the dashed contour of Figure 15, for example. The i n i t i a l t r a n s l a t i o n a l k i n e t i c energy of the representative point i s equivalent to the a l t i t u d e of the airplane above sea l e v e l , and i f t h i s a l t i t u d e does not exceed the elevation of the physical b a r r i e r , a non-reactive crash w i l l occur. For C l the reaction of Mu with F 2 i n the v = 1 state, E^ > E^ so there i s no physical b a r r i e r to reaction. For QCT, i t i s also common to define a threshold energy, OC E^ , corresponding to the minimum t r a n s l a t i o n a l k i n e t i c energy required for reaction. While the notion of a threshold energy has l i t t l e meaning i n terms of QMT c a l c u l a t i o n s , the quasi-c l a s s i c a l threshold energy i s useful for p a r t i t i o n i n g QMT r e s u l t s into c l a s s i c a l l y allowed and purely quantum mechanical processes, as discussed i n Section F below. For the same reasons mentioned i n the case of c l a s s i c a l threshold energies, q u a s i c l a s s i c a l threshold energies must be greater than or equal to the physical b a r r i e r height, E J ^ y S . However, because any amount of the v i b r a t i o n a l energy of the target molecule, E , may be available to promote reaction, the q u a s i c l a s s i c a l threshold energy may not only be greater than or equal to, but C l also less than the potential b a r r i e r height, E^ [ N i k i t i n (74)]. A good discussion of the o r i g i n s and interpretation of quasi--90-c l a s s i c a l threshold energies i s given i n Porter et a l . (73). A major problem i n defining q u a s i c l a s s i c a l thresholds for even a single surface i s the fact that the reaction p r o b a b i l i t y i s not only energy dependent but i t also depends on the phase of o s c i l l a t i o n of the target molecule. Q u a s i c l a s s i c a l threshold d e f i n i t i o n s are obtained by some kind of averaging process (such as Monte Carlo averaging) over the o s c i l l a t o r phase; how-ever, d i f f e r e n t procedures r e s u l t i n s l i g h t l y d i f f e r e n t thresh-old energies [Connor l-(76)]. As i n the discussion of c l a s s i c a l thresholds above, the mass d i s t o r t i o n s of the e f f e c t i v e p o t e n t i a l energy surfaces for H isotope - halogen reactions suggest that the q u a s i c l a s s i c a l threshold energies are also ordered Mu > H > D > T. ( i i i ) Transition State Theory While TST calculations may be based on the pot e n t i a l C l energy b a r r i e r , E^ [Persky (77), Jakubetz (7.9)], i t i s more common to make the assumption of v i b r a t i o n a l adiabaticity(VA) : "the reactant vibrations (except for the one that becomes the reaction coordinate) evolve smoothly into those of the activated complex, and f i n a l l y into those of the product, with-out any change i n v i b r a t i o n a l quantum numbers [Weston (72)]." Of course, the amount of v i b r a t i o n a l energy i s not constant because the v i b r a t i o n a l force constants (or curvature of the pote n t i a l surface) change during the progress of the reaction. For the reaction A + BC -> AB + C, the VA b a r r i e r heights are defined as [Connor 1-(79), Jakubetz (79)]: E™(A) = E^ 1 + E+(A) - E v (25) -91-where E^(A) i s the energy of the bound normal mode(s) of (ABC)^ Cl and E, and E have been defined above. The double dagger b v refers to the t r a n s i t i o n state. The values of E^ and hence v VA E depend on the H isotopic mass and are ordered: Mu > H > D > T, predicting an inverse isotope e f f e c t for a l l isotopic H atom reactions which orders the reaction rates: T > D > H > Mu (this i s often referred to as the "secondary isotope e f f e c t " [ N i k i t i n (74), Van Hook (70)]). In many cases, such as Y + X 2 -> YX + X, Y = 'Mu, H, D, T; X = F, C l ; the presence of Y i n the t r a n s i t i o n state weakens the X-X bond without complete compensation from the formation of the Y-X bond [Jakubetz (79), Connor l-(79)] with two r e s u l t s : (1) E v > ± VA C l E'(Y) and thus E (Y) < E, and (2) due to v i b r a t i o n a l v v ^ D anharmonicity, higher energy v i b r a t i o n a l states are more 1 X c l o s e l y spaced than lower ones so that E^(Y) - E Q > E|(Y) - E^ ^ VA VA and thus E Q (Y) > E^ (Y) . Linear t r a n s i t i o n state triatomics: have four normal modes of v i b r a t i o n : two bound bending modes, the bound symmetric stretch corresponding to motion along a l i n e perpendicular to the reaction path at the saddle point, and the unbound asymmetric stretch corresponding to motion along the reaction path i t s e l f (this mode has an imaginary frequency). For the c o l l i n e a r reaction, the symmetric stretch i s the only bound normal mode, uniquely defining E^(A) . In general, the assumption of v i b r a t i o n a l adiabaticity i s approximately v a l i d at normal temperatures [Levine (74)]. In the p a r t i c u l a r cases of H isotope reactions with halogens, the early b a r r i e r s are expected to favor the VA assumption because the t r a n s i t i o n state corresponds to an only s l i g h t l y perturbed - 9 2 -target molecule [Connor l - ( 7 9 ) , Jakubetz ( 7 9 ) ] . Although q u a s i c l a s s i c a l trajectory calculations do not assume v i b r a t i o n -a l adiaba1d.city (allowing a continuous energy transfer between the v i b r a t i o n a l and t r a n s l a t i o n a l modes i n either d i r e c t i o n ) , the preceding arguments on the expected v a l i d i t y of VA provide the basis for making the " f i r s t guess" prediction that the OC q u a s i c l a s s i c a l thresholds, , for these reactions w i l l be very s i m i l a r to the VA b a r r i e r heights. Table IV compares the values of the energy d e f i n i t i o n s made so far for the c o l l i n e a r reaction Y + F 2 YF + F, Y = Mu, H, based on the LEPS surface due to Jonathan et a l . (72) . (iv) Reaction Enthalpy Thermodynamic reaction enthalpies are calculated with Hess's Law by summing the heats of formation of reactants and products under isothermal standard state conditions at 298K, AH^. . The r e s u l t s are averaged over Maxwell-Boltzmann in t e r n a l energy state d i s t r i b u t i o n s at 298K and also include any contributions due to physical state changes (heats of vaporization, s o l i d i f i c a t i o n , e t c . ) . From the viewpoint of calculations of the rates of i s o l a t e d atom-diatom reactions, i t i s more useful to consider reaction enthalpies as the difference between the bond d i s s o c i a t i o n energies, DQ, of the product and reactant molecules. The bond d i s s o c i a t i o n energy i s defined by D = D - E~, where D i s the equilibrium 2 0 e 0 e ^ d i s s o c i a t i o n energy (depth of the Morse p o t e n t i a l ) , and E Q i s the zero point energy (ZPE). Since t h i s d e f i n i t i o n of reac-- 9 3 -TABLE I V : ENERGY1" DEFINITIONS FOR THE COLLINEAR Y + REACTION  Y = Mu, H, FOR THE LEPS SURFACE OF JONATHAN (72) Mu H C L A S S I C A L B A R R I E R H E I G H T , E ^ 1 2 . 3 5 2 . 3 5 P H Y S I C A L B A R R I E R H E I G H T , E^ yS 1 . 0 8 1 . 0 8 § Q U A S I C L A S S I C A L T H R E S H O L D E N E R G Y , E ^ C 1 . 8 0 2 . 0 6 Z E R O P O I N T E N E R G Y O F F 2 , E Q ( F 2 ) 1 . 2 7 1 . 2 7 Z E R O P O I N T E N E R G Y O F A C T I V A T E D C O M P L E X , E J ( Y F F ) 1 . 2 0 1 . 1 2 V I B R A T I O N A L L Y A D I A B A T I C B A R R I E R H E I G H T , , V ' 0 , V J l A E Q 2 . 2 8 2 . 2 0 A E , 2 . 1 5 1 . 9 1 kcal/mole, taken from Connor (7 9 ) OC the o r i g i n of the lower value of E ^ for Mu i s explained i n Chapter I V , p. 1 4 8 . -94-t i o n enthalpy i s based on ZPE 1s, i t corresponds to the Maxwell-Boltzmann population at O K and i s often denoted A H Q [Wolfrum ( 7 7 ) , Douglas ( 7 6 ) ] . A H Q i s also independent of physical state changes. The bond d i s s o c i a t i o n energies, ZPE's, and reaction enthalpies of the molecules and reactions studied i n th i s thesis are summarized i n Table v. This table shows that some of the reactions of Mu with the hydrogen halides are endo-thermic; some implications of t h i s are discussed below i n t h i s Section. In general, because of the larger ZPE of products containing l i g h t e r H isotopes, the exothermicity of H isotope reactions based on A H ^ are ordered: T > D > H > Mu. (v) Reaction Ac t i v a t i o n Energy Although a general discussion of trajectory methods i s l e f t to the next Section, i t i s useful at t h i s point to anticipate one of the major concepts common to those methods, i n order to derive the Tolman interpretation of the acti v a t i o n energy (following [Levine ( 7 4 ) ] ) . A l l trajectory calculations provide values of some form of reaction rate constant that i s a function of the r e l a t i v e t r a n s l a t i o n a l k i n e t i c energy of the c o l l i d i n g species. In order to calculate thermally averaged rate constants from these r e s u l t s , i t i s necessary to compute an i n t e g r a l of the following general form: k(T-) = °° -E/k_T e B k(E)dE ( 2 6 ) o where e E / / k B T i s the Boltzmann weighting factor and Q i s the p a r t i t i o n function which normalizes the r e s u l t . P a r t i t i o n TABLE V: BOND DISSOCIATION ENERGIES, ZERO POINT ENERGIES, AND REACTION ENTHAPLIES r e a c t a n t molecule F 2 c i 2 B r 2 HCl HBr HI D product molecule D E 0 e u • -37.59 1. 27 MuF 141.13 16.61 -88.2 HF 141.13 5.78 -9 9.0 DF 141.13 4.25 -100.6 TF 141.13 3.56 -101.2 57.88 0.81 MuCl 106.43 12.24 -37.1 HCl 106.43 4.27 -45.1 DC1 106.43 3.07 -46.3 TCI 106.43 2.52 -46.8 45.92 0.46 MuBr 90.36 10.92 -34 .0 HBr 90.36 3.79 -41.1 DBr 90.36 2.68 -42.2 TBr 90.36 2.20 -42.8 106.43 4.27 MuH(MuCl) 109.46(106.43) 13 .53(12. 24) + 6.2 ( + 7 .9) HH(HCl) 109.46(106.43) 6 .23 (4. 47) - 1 . 1 ( 0 .0) DH(DCl) 109.46(106.43) 5 .38 (3. 04) -1.9(-1 .2) TH(TCI) 109.46(106.43) 5 .07 (2. 52) -2.2(-1 .8) 90.36 3.79 MuH(MuBr) 109.46(90.36) 13 .53(10. 92) -9.4(+7 .1) HH(HBr) 109.46 (90.36) 6 .23 (3. 79) -16.7 (0 .0) DH(DBr) 109.46(90.36) 5 .38 (2. 68) -17.5(-1 .1) TH(TBr) 109.46(90.36) 5 .07 (2. 20) -17.8(-1 • 7) 73.66 3.27 MuH(Mul) 109.46(73.66) 13 .53 (9. 52) -25.5(+6 • 3) HH (HI) 109.46(73.66) 6 .23 (3. 27) -32.8 (0 .0) DH (DI) 109.46(73.66) 5 .38 (2. 33) -33.7(-0 .9) TH(TI) 109.46(73.66) 5 .07 (1. 91) -34.0(-1 .4) a l l v alues are i n kcal/mole c a l c u l a t e d from s p e c t r o s c o p i c data from G. Herzberg, Sp e c t r a of Diatomic Molecules, 2nd ed., Van Nostrand, P r i n c e t o n , 1950. -96-functions have the general form Q = E e " e i / k B T (27) where i s the energy of the i t h state. On the other hand, —E /k T the Arrhenius a c t i v a t i o n energy expression, k(T) = Ae a B , i s a measure of the rate of change of the rate c o e f f i c i e n t as a function of inverse temperature. Assuming temperature i n -dependence ofthe pre-exponential factor (which i s , i n fact, weakly temperature dependent), th i s may be re-written: -k_ d [ l n k(T) ] E = — , (28 Substitution of equation (26) into (28) y i e l d s : / b E e - E / V k(E) dE _ f kB d [ l n Q ( T ) ] E -a " r e-E/k BT k ( E ) d E [ d ( l } (29) o The f i r s t term of t h i s expression i s c l e a r l y an average energy and i t i s interpreted as the average energy of those c o l l i s i o n s * which r e s u l t i n reaction, <E >. D i f f e r e n t i a t i o n of the second term of equation (29) followed by substitution of equation (27) y i e l d s : k B T 2 d [Q (T) ] _ leiQ 1 Q(T) S e ~ e i / k B T i which i s just the average energy of the reactants, <E>. Thus, equation (29) i s simply: * E = <E > - <E> (30) a that i s , the a c t i v a t i o n energy i s just the difference between the average energy of those c o l l i s i o n s that actually r e s u l t i n reaction and the average energy of a l l c o l l i s i o n s . This -97-conceptually useful r e s u l t i s due to R. C. Tolman (27). * The process of quantum mechanical tunnelling reduces <E > from the value i t would have c l a s s i c a l l y , thereby lowering E . a In the high temperature regime, the r e l a t i v e contribution of tunnelling to the reaction rate i s diminished from that of the low temperature regime since a higher f r a c t i o n of c o l l i s i o n s are energ e t i c a l l y capable of reacting c l a s s i c a l l y . These considerations predict that the tunnelling process w i l l manifest i t s e l f experimentally i n terms of the temperature dependence of equation (30): E w i l l decrease with decreasing a temperature. S i m i l a r i l y , i t has already been mentioned that Mu i s expected to tunnel more e a s i l y than H i n reactions with halogens due to the smaller e f f e c t i v e mass of the representa-t i v e point. Thus, i n the same temperature range (fixed <E>), one expects to fi n d experimental values of E & to be reduced for the Mu reactions. Other dynamical e f f e c t s besides tunnelling * may contribute to <E >; for example, a l l of the c l a s s i c a l dynamical e f f e c t s discussed so far i n the H isotope - halogen * reactions tend to raise <E > for the Mu reaction, thereby possibly o f f s e t t i n g any tunnelling e f f e c t s . These considera-tions c l e a r l y show that a c t i v a t i o n energies are not just energy averages but also dynamical averages. ^YH + X (vi) Potential Energy Surfaces for the Reactions Y + HX -> YX + H  Y = Mu, H, D, T; X = C l , Br, I. Before considering s p e c i f i c p o t e n t i a l energy surfaces for the hydrogen - hydrogen halide (HX) reactions, i t should be noted that t h i s seemingly simple substitution of the x„ mole--98-cule by an HX molecule greatly complicates both the experiment-a l and t h e o r e t i c a l studies of t h i s series of reactions. Test-imony to t h i s i s the vast amount of c o n f l i c t i n g l i t e r a t u r e published i n the past twenty years on these reactions; as an example, a good summary of the t h e o r e t i c a l and experimental debate on the H + HC1 reaction may be found i n [Bauer 2-(78), Weston (79)]. The Y + HX systems have two reaction channels: hydrogen abstraction (Y + HX -> YH + X) and hydrogen exchange (Y + HX -> YX + H) , each with i t s own potential energy surfaces. Experimentally, t h i s means that rate data for the i n d i v i d u a l reaction channels must probably be obtained v i a measurements of product formation rather than reactant depletion. Since the MSR method i s of the l a t t e r v a riety, i t has so far only been possible to measure the totaK.Mu reaction rates (k , + k ) c abs exc and, i n f a c t , only the room temperature reaction rates have been measured to date. i n p r i n c i p l e , i t may be possible to determine the Arrhenius parameters for the i n d i v i d u a l reaction channels of Mu by simply measuring the temperature dependence of the t o t a l reaction rates i n the usual way. If both the Mu exchange and abstraction reactions display Arrhenius straight-l i n e behaviour over a wide temperature range (which, i n view of the r e s u l t s discussed i n Chapter IV, might not be the case, and, i n f a c t i t i s not clear that even H displays t h i s behav-iour [Bauer 2-(78), Clyne (66)]), and i f the a c t i v a t i o n ener-gies for the two reaction channels are substantially d i f f e r e n t (this i s probably true; see eg. [Bauer 2-(78)]), then the Arrhenius p l o t for the t o t a l reaction could show a break with the high a c t i v a t i o n energy reaction described by the high -99-temperature part of the p l o t and the low a c t i v a t i o n energy reaction described by the low energy part. Reactions of HF are not considered .in t h i s thesis because i t i s expected that the reaction rates of Mu with HF are so slow that they are immeasurable by the MSR technique. Accurate experimentally optimized LEPS surfaces do not exi s t for either the exchange or abstraction reactions of H isotopes with the hydrogen halides because many of the experimental r e s u l t s are "equivocal or contradictory [McDonald (75)]." Two t o p o l o g i c a l l y d i f f e r e n t surfaces have been recently considered for the abstraction reactions (Figure 16 shows the H + HCl -> H 2 + Cl examples) : (1) the simple LEPS surface shown at the top of the Figure (adapted from [Persky (78)], optimized for the reverse reaction: H 2 + Cl -»- HCl + H) which has an early b a r r i e r to H abstraction and the same ess e n t i a l features as the H isotope - halogen surfaces (Figure 12), and (2) the valence bond modified LEP surface mentioned i n Section B, shown at the bottom of the Figure (adapted from [Porter (73)]; also optimized for the reverse reaction) which, besides having an early b a r r i e r to H abstraction, shows shallow pote n t i a l wells i n both the reactant and product v a l l e y s . Although three dimensional QCT calculations performed on the Persky surface are i n very good agreement with experimental re s u l t s of C l + H 2 versus C l + D 2 isotope e f f e c t s and the absolute rate constant for the C l + H 2 reaction [Persky (78)] , and, although a t r u l y accurate surface w i l l provide the basis for accurate descriptions of a chemical reaction i n both d i r e c t i o n s , surfaces that have been optimized with respect to a -100-thermal (300K) de Broglie wavelengths R , / «.u. FIGURE 16: Potential surfaces for the c o l l i n e a r H + HC1 ->• H„ + Cl reaction, adapted from Persky (78) (top) and Porter (73) (bottom). Note that the Persky surface i s drawn reversed from the other contour plots shown i n t h i s thesis; i t has i t s reactant v a l l e y at the top and i t s product v a l l e y at the r i g h t . The Persky surface contours are l a b e l l e d i n kcal/mole r e l a t i v e to the d i s s o c i a t i o n l i m i t of H^. The 3 00K thermal de Broglie wavelengths of the representative points on t r a j e c -t o r i e s p a r a l l e l to the axes are shown at the l e f t of each surface. -101-reaction i n one d i r e c t i o n must be viewed with caution when applied to the reverse reaction [Heidner (76)]. In fac t , a general f a i l i n g of LEPS surfaces i s "that semi-empirical surface parameters obtained by c a l i b r a t i o n on one reaction are often not transferrable to another reaction involving the same atoms [White (73)];" (in his paper, White discusses some successful exceptions to th i s generalization). This problem sometimes appears as a general constraint i n the consideration of H isotope exchange versus abstration reactions with the hydrogen halides, although many authors simply treat each reaction channel independently of the other (eg. [Klein (78)]). Porter et al_. predict that the abstraction reaction surfaces for the li n e a r configuration possess wells corre-sponding to weakly stable H 2—X and H—HX, X = C l , Br, I, with depths ranging from 1-4 kcal/mole. Furthermore, they have found that some of these minima are s u f f i c i e n t l y deep to accomodate one or more v i b r a t i o n a l l e v e l s for the H, D, or T isotopic variations of the complexes and that these complexes should be stable enough to permit i s o l a t i o n at low temperatures. However, there i s not yet any experimental evidence available to support these predictions. On the other hand, experimental data exists to suggest that HX2 species have been isol a t e d [Noble (68), Bondy&ey (71) , Noble (72)] and Porter et a l . found that by using t h e i r parameters for the H2~X surfaces, they can construct H-X2 surfaces that q u a l i t a t i v e l y agree with these experimental observations. From t h i s f a c t , the c r e d i b i l i t y of the presence of wells i n the H2~X surfaces might be inferred. However, a number of counter-arguments on the question of the -102-existence of stable H - X 2 species are given i n Bauer et a l . 2-(78) and references therein. If the existence of pot e n t i a l wells l i k e those of the Porter surface i s assumed, i t i s i n t e r e s t i n g to speculate on what e f f e c t they would have on the reactions. I t i s expected that the Mok-Polanyi c o r r e l a t i o n holds for t h i s series of reactions [Porter (73)]: as X changes from Cl to I, the b a r r i e r height decreases and moves to progressively e a r l i e r positions. Unfortunately, the b a r r i e r heights for these reactions are not well known, but they appear to be about twice as high as those for the corresponding Y + reactions and range from about 5 kcal/mole for H + HC1 to about 0.5 kcal/mole for H + HI [Klein (78), Persky (77), White (73), Bauer 2-(78)]. Conse-quently, for the H + HC1 and H + HBr reactions at least, reac-t i v e c o l l i s i o n s require such energy that i t i s u n l i k e l y that either the reactants or products w i l l be trapped or even much affected by the wells, unless an extremely e f f i c i e n t energy transfer mechanism e x i s t s . For the H + HI system, r e l a t i v e l y long-lived complexes may e x i s t even for reactive c o l l i s i o n s . In that case, the reaction would no longer be "d i r e c t " [Levine (74)] with a c o l l i s i o n time shorter than one v i b r a t i o n a l period -13 (^ 10 s ) , but would be "compound" or "complex", with very complicated t r a j e c t o r i e s . S i m i l a r i l y , low temperature (low r e l a t i v e velocity) non-reactive c o l l i s i o n s of H with HX, X = C l , Br, I, may be expected to be of a compound nature, rather than d i r e c t . This has i n t e r e s t i n g implications for the measurements of Mu reaction rates with HX molecules with the MSR method. Direct non-reactive c o l l i s i o n s of Mu with target -103-molecules are not expected to cause much depolarization of the -14 muon because the in t e r a c t i o n times are short, t y p i c a l l y <10 s, compared to the hyperfine frequency of Mu, <10 s " S the fact that a long-lived Mu signal i s observed i n N 2 i s experi-mental evidence of t h i s . On the other hand, compound non-reactive c o l l i s i o n s may be s u f f i c i e n t l y intimate that the muon w i l l be e f f i c i e n t l y depolarized by the quasi-stable muonic r a d i c a l formed i n the c o l l i s i o n . From these considerations, one might expect Mu reactions to display apparent inverse Arrhenius behaviour at low temperatures where the r e l a t i v e numbers of t r u l y reactive c o l l i s i o n s are few; as the tempera-ture i s lowered, the e f f i c i e n c y of non-reactive depolarization of the muon increases, thereby increasing the apparent Mu reaction rate. In t h i s way, the MSR method may present an experimental means of tes t i n g the existence of poten t i a l wells i n these reaction surfaces. One f i n a l consideration on t h i s subject i s the large ZPE of Mu-containing molecular bonds: since the wells are r e l a t i v e l y shallow, they may not be capable of supporting any bound v i b r a t i o n a l states of the muon-i c complex molecules, i n which case the complexes would not be long-lived. A detailed consideration of the potential wells i s required to c l a r i f y t h i s question. Besides having wells, the Porter surfaces d i f f e r from Persky's surface by the fac t that t h e i r valleys possess bottlenecks near the saddle point, p a r t i c u l a r l y i n the H-HBr and H-HI surfaces [Porter (73)]. Other than these d i f f e r -ences, the main topological features of the two types of surfaces are si m i l a r . The reaction dynamics for abstraction -104-are dominated by the c o l l i n e a r reaction for both surfaces [Klein (78), Thompson (75)], mainly because the ba r r i e r increases by a factor of about six as the H-H-Y bond angle changes from 180° to 90°. Although these surfaces possess early b a r r i e r s , they are not as early as the corresponding Y-X^ surface b a r r i e r s , so that the saddle points are more displaced from the l i n e along the reactant approach v a l l e y . The r e l a t i v e "lateness" of the barri e r s suggest that the Cl c l a s s i c a l thresholds, E^ , w i l l be subs t a n t i a l l y greater than Cl the p o t e n t i a l b a r r i e r heights, E^ , and that the quasiclas-OC s i c a l thresholds, E^ , w i l l be substantially greater than the physical b a r r i e r heights, E J ^ V S . Furthermore, the activated complexes of Y-HX are expected to have r e l a t i v e l y strong Y-H and H-X bonds, unlike the Y-X2 activated complexes which are just s l i g h t l y perturbed X 2 molecules. This e f f e c t , combined with the r e l a t i v e l y large ZPE's of Y-H molecular bonds, sug-gest that E^(Y) > E ., so that:(l) E V A(Y) > E ? 1 and (2) E^ A(Y) 3 v a* v v ^ b 0 VA < E, (Y), opposite to the Y-X„ case. These considerations, plus the r e l a t i v e heights of the poten t i a l b a r r i e r s , suggest that the abstraction reactions, Y + HX, w i l l be slower than the corresponding Y + X^ reactions. Although the assumption of v i b r a t i o n a l adiabaticity i s not l i k e l y to be as v a l i d for the Y + HX abstraction reactions as for the Y-X2 reactions because of t h e i r b a r r i e r locations, i f the VA b a r r i e r heights are taken as " f i r s t guesses" of the q u a s i c l a s s i c a l thresholds, then i t i s expected that the inverse isotope e f f e c t that orders the reac-t i o n rates T > D > H > Mu w i l l be much more severe for the Y + HX abstraction reactions. The VA ba r r i e r s also suggest - 1 0 5 -that v i b r a t i o n a l e x c i t a t i o n of the target molecules w i l l be less e f f e c t i v e i n promoting the abstraction reactions than i n the halogen molecule reactions; t h i s prediction has been experimentally confirmed [Wolfrum ( 7 7 ) , Arnoldi ( 7 6 ) ] . I t may also be expected that the r e l a t i v e l y large v i b r a t i o n a l non-adiabaticity of the abstraction reactions causes VA- TST. to over-estimate the e f f e c t i v e b a r r i e r height, thereby making i t s predicted reaction rates erroneously small. The displacement of the saddle points from the reactant approach va l l e y s also has a number of ef f e c t s on the tunnelling process. I t i s less l i k e l y that a single one dimensional b a r r i e r penetration correction applied to QCT or TST c a l c u l a -tions of Y + HX abstraction w i l l provide an accurate approx-imation of the QMT r e s u l t s as i n the case of Y + X^ reactions because: the e f f e c t i v e mass of the representative point changes during the tunnelling process; there i s no obvious single tunnelling path or b a r r i e r due to the i n e r t i a l e f f e c t s that cause the representative point to deviate from the minimum reaction path; and "corner cutting" tunnelling paths are l i k e l y to be important [Marcus ( 7 7 ) , Johnston ( 6 1 ) ] . The l a s t point i s r e a d i l y appreciated when i t i s noted that the skewing angles of the mass weighted coordinate system for the li n e a r Y-HCl surfaces are 7 1 . 6 ° , 4 5 . 8 ° , 3 6 . 4 ° , and 3 1 . 4 ° for Y = Mu, H, D , and T respectively; skewing angles for Y-HBr and Y-HI surfaces are quite s i m i l a r . Consequently, corner cutting tunnelling path lengths are ordered Mu > H > D > T, p a r t i a l l y o f f s e t t i n g the tunnelling advantage Mu enjoys due to the e f f e c t i v e mass of the representative point, y H y ( e q u a t i o n s ( 2 2 ) ) . Figure 16 -106-shows the thermal de Broglie wavelengths of the represent-ative points for the i s o t o p i c variants of H corresponding to motion p a r a l l e l to the unweighted surface axes. The p a r t i c -ular mass combinations for Y + HX abstraction do not contract the product path de Broglie wavelengths as much as i n the case of Y + X 2 reactions (c.f. Figure 12). Since the represent-ative point crosses the p o t e n t i a l b a r r i e r on a trajectory that i s between the asymptotic reactant and product t r a j e c t o r i e s , Mu holds a smaller tunnelling advantage i n these reactions than i n the Y + X 2 reactions. Thus, while tunnelling s t i l l orders the reaction rates Mu > H > D > T, tunnelling i s not expected to greatly favor Mu over the other H isotopes i n these reactions. The A H Q endothermicity of the Mu + HC1 abstraction reaction (Table 5) due to the ZPE of Mu-H also r e s t r i c t s Mu tunnelling by rendering a substantial part of the b a r r i e r inaccessible as a tunnelling path. It has already been stated that the Mok-Polanyi r e l a t i o n holds for the Y-HX abstraction reactions and thus i t i s expect-ed that as X changes from C l to I, more reaction energy i s transferred into product v i b r a t i o n . However, the l i g h t atom anomaly i s not expected to operate strongly on these reactions since the reaction exoergicity slams an H atom into an atom of comparable mass, r e s u l t i n g i n a v i b r a t i o n a l l y excited product. Thus, i t i s expected that a much greater f r a c t i o n of the reaction energy appears as product v i b r a t i o n i n the Y-HX abstraction reactions than i n the Y-X2 reactions. The l i g h t atom anomaly s t i l l predicts that the products of the Mu + HX abstraction w i l l have less reaction energy channelled into -107-v i b r a t i o n than i s the case i n the H + HX reactions. The potential energy surfaces for the Y + HX hydrogen exchange reactions are much more poorly known than those for the hydrogen abstraction reactions. For the H + HCl exchange reaction, for example, proposed surfaces range from those with symmetrically placed potential wells (instead of barriers) of 5-9 kcal/mole, to those with pot e n t i a l b a r r i e r s of 15-25 kcal/mole ([Bauer 2-(78)] and the references therein). The LEPS formulation has been declared "too i n f l e x i b l e " to model these pot e n t i a l surfaces [Valencich (77)]. Consequently, i t i s of l i t t l e use to consider any s p e c i f i c examples of exchange reaction surfaces. Nonetheless, i t i s possible to comment on some of the gross topological features of the surfaces for these reactions. The bulk of the post-1970 t h e o r e t i c a l and experimental papers on these reactions agree that Y + HX exchange reactions possess pot e n t i a l b a r r i e r s , rather than wells, i n the symmetrical Y-X-H configuration [Klein (78), Bauer 2-(78), Endo (76), Botschwina (77), Dunning (77), Wolfrum (77), Valencich (77)], and several of these authors believe that the exchange ba r r i e r s exceed the corre-sponding abstraction b a r r i e r s [Bauer 2-(78), Endo (76), Botschwina (77), Dunning (77), Wolfrum (77)]. Unlike the abstraction reaction surfaces, the exchange reaction surfaces do not seem to be very sensitive to the bond angle [Klein (78), Thompson (75)], and, i n f a c t , Klein and Veltman's (78) LEPS surface s l i g h t l y favors a bond angle of 90° over the c o l l i n e a r surface. I t has also been suggested that the exchange reac-t i o n surfaces possess pot e n t i a l wells i n the product and reac--108-tant v a l l e y s similar to those proposed for the abstraction reaction surfaces [Thompson.'(75) ] . A l l c o l l i n e a r exchange reaction p o t e n t i a l surfaces-are p e r f e c t l y symmetrical about a l i n e drawn through the o r i g i n and saddle point at 45° to each axis. If i t i s assumed that the exchange reaction surfaces possess b a r r i e r s rather than wells, then the saddle point corresponds to a complex with equally strong Y-X and X-H bonds and a l l of the energy threshold and b a r r i e r r e l a t i o n s predicted i n the abstraction reactions w i l l also hold for the exchange reactions, except that the inequality r e l a t i o n s may be even stronger. Although surfaces with symmetrically placed b a r r i e r s are expected to be prone to "corner cutting" tunnelling paths [Marcus (78), Johnston (61)], the strong e f f e c t due to the sharp skewing angles for the mass weighted abstraction surfaces i s absent i n the exchange surfaces where the skewing angles for Y + C1H, for example, are: 89.5°, 88.4°, 87.8°, and 87.3° for Mu, H, D and T respectively. The importance of tunnelling i s , however, diminished for Mu + XH exchange because of the endothermicity of the reactions (see Table 5) which r e s t r i c t s tunnelling to the top part of the b a r r i e r s . Without knowledge of the potential energy b a r r i e r s , i t i s impossible to predict which.reaction channel -is faster: hydrogen,atom abstraction or exchange. Most experimental evidence suggests that abstraction i s faster than exchange at ordinary temperatures, but that the reverse i s true at high temperatures (>2000K) [Endo (76), Bauer 2-(78)]; these r e s u l t s have been interpreted as evidence that hydrogen exchange has -109-an unusually small s t e r i c factor [Thompson (75)] or that exchange has a much higher a c t i v a t i o n energy than abstraction [Endo (76), Bauer 2-(78)]. A dynamical argument has been proposed to explain these experimental observations [Bauer 2-(78), K l e i n (78)]. At room temperature, the most populated r o t a t i o n a l states of hydrogen halide molecules are 2 or 3 12 -1 corresponding to r o t a t i o n a l frequencies of about 2 x 10 s Since the HX center of mass i s almost coincident with the X nucleus, to a slowly approaching atom, the HX molecule looks l i k e a sphere with an H atom "crust" covering the larger X atom. Consequently, the c o l l i s i o n takes place i n the abstraction reaction configuration: Y-H-X. As the r e l a t i v e v e l o c i t y of the c o l l i s i o n partners increases, t h i s r o t a t i o n a l screening of the halogen atom diminishes, increasing the opportunity for the exchange reaction configuration to occur: Y-X-H. This e f f e c t has not been predicted i n three dimension-a l QCT calculations [Thompson (75), White (73)] which show l i t t l e s e n s i t i v i t y to the target molecule r o t a t i o n a l states; s t a t i s t i c a l phase space ca l c u l a t i o n s , however, are i n q u a l i t a -t i v e agreement with t h i s e f f e c t [Truhlar (69)]. If i t i s assumed that t h i s e f f e c t i s important, then i t i s expected that the branching r a t i o s for abstraction to exchange are smaller for Mu reactions than for other H isotope reactions at the same temperatures, since the mean v e l o c i t y of the l i g h t e r Mu atom i s three times that of H. D Trajectory Calculations The f i r s t objective of a l l trajectory calculations, i s -110-to determine the state-to-state reaction probabilites as a function of the r e l a t i v e t r a n s l a t i o n a l k i n e t i c energy or ve l o c i t y of the c o l l i d i n g species. These p r o b a b i l i t i e s are denoted P . (E, ), where s 1 refers to product molecule s' «-s trans ^ states and s refers to reactant molecule states. The number and type of quantum states included i n s and s 1 depends on the dimensionality of the calculations and the l e v e l of approxima-t i o n to which the calculations are carr i e d out. As mentioned i n Section B, c o l l i n e a r c o l l i s i o n s are d i r e c t "knock out" processes where the attacking atom approaches the center of mass of the target molecule end on. For such a c o l l i s i o n , the ro t a t i o n a l states of the target molecule are ignored and there i s no c o l l i s i o n a l o r b i t a l angular momentum transferred since the impact parameter, b = 0. Furthermore, i t has already been noted that the bending vibrations of the activated complex are also ignored i n c o l l i n e a r c o l l i s i o n s . These considerations, plus the fact that t r a j e c t o r i e s need only be performed on the one potential energy surface corresponding to a bond angle of 180°, indicates that c o l l i n e a r state-to-state reaction proba-b i l i t i e s may be determined with r e l a t i v e ease. Two dimension-a l (coplanar) tr a j e c t o r y calculations include the c o l l i n e a r case as well as t r a j e c t o r i e s on a l l surfaces corresponding to bond angles ranging from 0° to 180°. A l l in-plane rotations and vibrations are included i n the target, product, and activated complex molecules. Because the impact parameters are not constrained to zero, o r b i t a l angular momentum transfer may occur. A l l i n t e r n a l states and impact parameters are included i n three dimensional ca l c u l a t i o n s , thereby requiring -111-the sampling of a continuum of possible t r a j e c t o r i e s . State-to-state reaction p r o b a b i l i t i e s are given by [Persky (77) ] : N R, (E. ) P (E ) = 5 5 trans S ' * B trans ( } s trans where N , i s the number of reactive state-to-state t r a j e c -t o r i e s at a given energy and N g i s the t o t a l number of t r a j e c t o r i e s calculated at that energy for i n i t i a l state, s. For three dimensional trajectory calculations, i t i s common to define a t o t a l reaction cross section [Persky (77), White (73)] : a , (E ) = T T b 2 (E^ )P , (E ) (31) s -<-s trans Max trans s'-<-s trans where b i s the largest impact parameter that y i e l d s an appreciable reaction p r o b a b i l i t y ; i n order to compare the calculations with experimental atomic and molecular beam data, i t i s useful to define a d i f f e r e n t i a l cross section [Persky (77)] : d a s ^ s ( E t r a n s ) = u b M a x N s ^ s ( E t r a n S ' 9 ) dfi 2 T r s i n e N s ( E t r a n s ) A 9 (32) where N , (E^ _ ,6) i s the number of reactive c o l l i s i o n s with s'<-s trans scattering angle between 9 and 9 + A9, and dfi i s an increment of s o l i d angle. Of course, three dimensional trajectory cross sections have the units of area and d i f f e r e n t i a l cross sections have the units of area/solid angle. Occasionally, so-called "cross sections" are defined for coplanar c a l c u l a -tions [Baer (76)] analogous to equations (31) and (32): 0 , (E ) = 2b • (E. )P , (E. ) (33) s'-<-s trans Max trans s'«-s trans -112-and da , (E. .), 2b N R, (E ,9) s -«-s trans _ Max s'^s trans ^ 4 ) dfi 2N (E.- )A9 s trans These "cross sections" have units of length and length/unit angle respectively. State-to-state reaction p r o b a b i l i t i e s or cross sections are often summed over a l l f i n a l states, s', to give a t o t a l reaction cross section of an i n i t i a l state, s [Connor l - ( 7 8 ) ] : o t{Ex_ • ) =• E, a , (E^ ) s trans s' s ' - f-s v trans Equation (26) (Section C) gives a general i n t e g r a l for convert-ing energy dependent rate constants to thermally averaged rate constants; p a r t i c u l a r i n t e g r a l expressions equivalent to equation (26) are [Eliason (59), Connor l-(78), Weston (72)]: (ID) k (T) = — h ^ ) 1 / 2 P'NE. )e E t r a n s / k B T dE,. (35) v ' s ^2Tryk T ; J 0 s trans trans B (2D) k (T) = ( 2 } l / 2 1 r a t 1/2 e - E t r a n s A B T v ' s k RT J 0 s trans trans trans (36) (3D. k . ( T , , r a B t ( B t r a n . » E t r a n . « - B t r „ ^ B I p B 0 • d E t r a n s ( 3 7 ) where u i s the reduced mass of the reactants and k (T) i s an s i n i t i a l state thermal reaction rate constant with units of cm 1 — 1 — 1 2 . . -1 -1 , 3 , , -1 -1 , molecule s , cm molecule s , and cm molecule s for the one, two, and three dimensional cases respectively. F i n a l l y , t o t a l thermal reaction rate constants are obtained by averaging k g(T) over the Boltzmann d i s t r i b u t i o n of i n i t i a l states, s: -113-k(T) = Ef (T) k (T) s s s (38) where f g ( T ) i s the f r a c t i o n of target molecules i n state s given by: where g i s a degeneracy factor and e i s the energy of state, s s s [Weston (72)]. An important feature of equations (35) -(37) i s that the thermal rate constants are functions of the -1/2 temperature independent reactant reduced mass term, u , regardless of the t r a j e c t o r y dimensionality. For reactions of H isotopes with r e l a t i v e l y heavy molecules (M.W. >_ 35 amu) , t h i s mass factor predicts that k„ :k-k :k - 2.9:1.0:0.72:0.59 ^ Mu H D T [Connor l - ( 7 8 ) ] . This isotope e f f e c t may be simply i n t e r -preted as the mass dependence of the mean r e l a t i v e v e l o c i t y of c a l l e d the " t r i v i a l " isotope e f f e c t [Fleming (76), Fleming 1-(77) , Jakubetz l-(78)] because i t i s not dependent on the reaction dynamics; a l l dynamical information about the reaction, as well as the e f f e c t s of the mass weighting of the p o t e n t i a l energy surface, are contained i n the reaction cross section. State-to-state reaction p r o b a b i l i t i e s are a r t i f i c i a l constructs i n the cases of c l a s s i c a l and q u a s i c l a s s i c a l tr a j e c t o r y calculations which, by d e f i n i t i o n , have access to a continuous rather than quantal range of energy transfer. In both cases, the f i n a l state energies are related to quantum states by some a r b i t r a r y binning procedure which assigns a range of f i n a l state energies extending above and below a given f s ( T ) the reacting species, [Weston (72)] and i s often -114-quantum energy state to that state [Thompson (76)]. I t may be r e c a l l e d (Section C) that i n the case of purely c l a s s i c a l t r a j e c t o r i e s , the target molecules i n i t i a l l y possess no int e r n a l energy (in v i o l a t i o n of the zero point energy) while i n the case of q u a s i c l a s s i c a l t r a j e c t o r i e s , the target molecule i n i t i a l l y possesses proper quanta of int e r n a l energy, C l a s s i c a l and q u a s i c l a s s i c a l t r a j e c t o r i e s are often calculated by solving Lagrange's or Hamilton's equations of motion. Lagrange's equations are given by [Messiah (58)] d r 3 L ^ f | •= 0 (r = 1,2,...,R) q r dt with the Lagrangian function given by L(q ±,q 2,••.q R,q ±,q 2,••• , 4 R;t) = T(q ±,q 2,...,q R) - V(q ±,q 2,...,q R) where q^ are generalized coordinates, T i s the k i n e t i c energy and V i s the potent i a l energy of the system. The c l a s s i c a l Hamiltonian function which spans the 2R dimensional coordinate and momentum space (phase space) i s given by [Messiah (58)]: R # 9L H(q ±, . . . ^ ^ p . ^ . . . ,p R;t) = I_qr ^ - L = T(p ±, . . . ,pR) + V(q 1, . . . ,qR) and the equations of motion are given by _ 9H 8H q r 9p r ; P r ~3q r (r = 1,2,...,R) In p r i n c i p l e , these equations of motion can be solved exactly to obtain the completely deterministic c l a s s i c a l t r a j e c t o r i e s . S t a t i s t i c a l l y averaged reaction p r o b a b i l i t i e s must be obtained by multiple integration over c o l l i s i o n variables such as impact -115-parameter, molecular orientation and v i b r a t i o n a l phase. Generally, i t i s d i f f i c u l t to determine the functional form of the reaction p r o b a b i l i t y dependence on these c o l l i s i o n vari-. , ables which i s required to perform t h i s integration. Consequent-l y , s t a t i s t i c a l averaging i s often accomplished by a procedure such as Monte Carlo integration which determines the reaction p r o b a b i l i t y from a s t a t i s t i c a l l y s i g n i f i c a n t sample of t r a j e c t o r i e s computed from values of the c o l l i s i o n variables selected randomly from a weighted d i s t r i b u t i o n . This proce-dure, which normally requires the c a l c u l a t i o n of several thousand t r a j e c t o r i e s , has the p h y s i c a l l y appealing- feature "that i t simulates the random process by which c o l l i s i o n s i n the laboratory actually occur [Thompson (76)]." QCT c a l c u l a -tions are often c a l l e d "Monte Carlo" calculations (eg. [Blais (74)]) or simply " c l a s s i c a l " calculations to d i s t i n g u i s h them from quantum mechanical treatments. The conventional quantum mechanical approach to atom-diatom reaction t r a j e c t o r i e s i s to solve Schrddinger's equation, HT = ET, H = T + V, to evaluate the scattering matrix (S-mat-rix) elements which lead d i r e c t l y to the quantized reaction 2 p r o b a b i l i t i e s , p s i ^ _ s = l s s ' + - s l [Manz (75)]. This procedure, which i s almost always car r i e d out by approximation methods, only gives the net r e s u l t of the c o l l i s i o n i n terms of a reaction p r o b a b i l i t y , since the entire potential i s inserted into Schrddinger's equation and only the asymptotic reactant and product wavefunctions are determined while the i n t e r a c t i o n region i s treated l i k e a "black box." C l a s s i c a l variables such as the phase of the harmonic o s c i l l a t o r appear i n ampli--116-tudes of p r o b a b i l i t y functions quantum mechanically and may give r i s e to such purely quantum mechanical e f f e c t s as wave interference. Although the quantum mechanical treatment may represent an exact formulation of the reaction problem, i n general, only approximate solutions can be found for i t , whereas.the approxi-mate c l a s s i c a l formulation of the reaction can usually be solved exactly. From the viewpoint of understanding the reaction dynamics i n d e t a i l , c l a s s i c a l r e s u l t s provide a valuable i n s i g h t . For example, i t i s not possible to d i s -tinguish purely quantum mechanical e f f e c t s such as tunnelling from c l a s s i c a l e f f e c t s on the basis of quantum mechanical re-sul t s alone (this point i s discussed further i n Section F). More importantly, c l a s s i c a l t r a j e c t o r i e s are t o t a l l y deter-m i n i s t i c and provide a detailed picture of the reactive or non-reactive scattering processes; because S-matrix quantum mechanical r e s u l t s do not r e a l l y define any t r a j e c t o r i e s , the e f f e c t of s p e c i f i c topological features on the potential cannot be determined d i r e c t l y . A number of methods have been devised to obtain more dynamical information from quantum mechanical c a l c u l a t i o n s . One approach i s to monitor the S-matrix as the pote n t i a l energy function i s changed to i n d i r e c t l y i n f e r the e f f e c t s of various pote n t i a l features. A more d i r e c t approach i s to calculate the flow of the quantum mechanical p r o b a b i l i t y d i s t r i b u t i o n s through the inte r a c t i o n region (so-called "streamlines," the quantum analogue of c l a s s i c a l t r a j e c t o r i e s ) by formulating the reaction as a quantum hydrodynamics problem (see eg. [Hirsch--117-felder (76)]). Another approach i s an ingenious modification of the conventional S-matrix calculations known as the "state path sum method [Manz (74), (75)]." This method i s b r i e f l y described here since i t was employed by Connor et a l . l-(78) , l-(77) i n t h e i r quantum mechanical calculations of the Mu reaction rates. Normal S-matrix calculations can be thought of as d i v i d i n g the p o t e n t i a l energy surface into three regions: the asymptotic reactant and product regions, separated by the i n t e r a c t i o n region of the p o t e n t i a l . At the boundary between the reactant and i n t e r a c t i o n regions, the quantum state p r o b a b i l i t y d i s t r i b u t i o n s are known; the solution of the Schrodinger equation gives the corresponding quantum state -p r o b a b i l i t y d i s t r i b u t i o n s at the boundary of the i n t e r a c t i o n and product regions, the connection being made v i a the "black box" S-matrix. With the state path sum method, the i n t e r -action region i s subdivided into an a r b i t r a r y number of sectors and an S-matrix i s calculated for each. As a r e s u l t , v i r t u a l quantum state p r o b a b i l i t y d i s t r i b u t i o n s are known at each sector boundary of the i n t e r a c t i o n region of the potential surface. Consequently, i f , for example, a p a r t i c u l a r c o l l i -sion has a low net reaction p r o b a b i l i t y , i t i s possible to determine which part of the potential energy surface i s responsible for t h i s r e s u l t . A "state path" i s a complete l i n e connecting an asymptotic reactant state, s, to an asymptotic product state, s', through the various sectors of the i n t e r a c t i o n region. Manz has devised algorithms for sectoring the i n t e r a c t i o n region and i d e n t i f y i n g the dominant state paths. -118-E Transition State Theory No attempt i s made to derive TST i n t h i s Section; i t merely examines TST predictions for H isotope e f f e c t s . As normally formulated [Johnston (66) , Weston (72) , Kuppeirmann (79)], the TST expression for the rate constant of the reaction Y + AB + YA + B i s : k = r V t h QY QAB Q+ - E V A A T e ' B (39) where i s a tunnelling correction factor, h i s Planck's constant, c T , Q , and Q are the products of the t r a n s l a t i o n a l , Y A B v i b r a t i o n a l , e l e c t r o n i c , and r o t a t i o n a l p a r t i t i o n functions of VA the activated complex and reactants respectively, and E i s an energy b a r r i e r , taken as the VA b a r r i e r i n t h i s treatment (Section (C) , Part ( i i i ) , and equation (25)). Kuppeririann' (7 9) has shown that equation (39) applies -to both the c o l l i n e a r and three dimensional-reactions when the appropriate partition-functions and VA ba r r i e r s are-used. -Equation (39)' only assumes the existence of- thermal equilibrium among the reactants ( i m p l i c i t i n the d e f i n i t i o n of a thermal reaction rate constant), v i b r a t i o n a l a d i a b a t i c i t y , and the absence of effects due to the curvature of the reaction path [Kuppermann (7 9)]. Since the target molecule i s the same for H isotope reactions of the type, Y + AB YA + B, the rate constant r a t i o s for the Y and Y' isotopic reactions do not include C L p a r t i t i o n functions: A B l Q v i trans Q T T T rot Q Q 7 ¥ vib Q + Q TT trans ,e-[E+(Y) - E+(Y') ]/k BT (40) -119-where the elec t r o n i c p a r t i t i o n functions are assumed to cancel due to the BO approximation. For a non-linear YAB molecule the three dimensional t r a n s l a t i o n a l and r o t a t i o n a l p a r t i t i o n functions are given by [Van Hook (70)] : Q. = ( 2 T r M k T ) 3 / 2 V/h 3 trans B Qrot = i*h)l**\W1/2 k BT h 2J 3/2 (41) (42) where M i s the molecular mass, V i s the container volume, T_A , I B and 3^-, are the moments of i n e r t i a of the molecule about i t s three p r i n c i p a l axes, and s i s a symmetry factor. The quan-ti z e d harmonic o s c i l l a t o r v i b r a t i o n a l p a r t i t i o n functions are given by [Frost .(61).]'": i n t e r n a l modes Q v i b n i ... -hv./k T -1 (1 - e I B ) (43) where i s a normal mode v i b r a t i o n a l frequency Substituting equations (41) to (43) into (40) and expanding the VA ba r r i e r term y i e l d s : k k' r t y i 3/2 3/2 I' I • I' A B C 3n-7 - i _ - h v | ( Y - ) A R T ) , n 1 ? 1 B i - h M * (44) ! _ e-hvT(Y)/k BT e - 2 k T T ^ 4 ( Y ) " V i ( Y ' ) ] B i where m^  and nu., are the atomic masses of the Y and Y1 isotopes, M and M1 are the molecular masses of the activated complexes and vf are the bound normal v i b r a t i o n a l energies of the activated complexes (the unbound v i b r a t i o n corresponding to the reaction coordinate i s excluded). This expression may be further s i m p l i f i e d by substitution of the Redlich-Teller -120-product theorem [Johnston (66) , Van Hook (70)] M > 3/2 f — 1 VB 1^ 1 / 2 I' I' I' ABC n = II. i m, >, 3/2 3n-6 V n - 4 l I (45) where are the masses of the atoms comprising the molecule of mass M, to y i e l d : k k' J . )n-7 v t T v^(Y) 3 -7 vT(Y) ^ - i n - 4 — i vl(Y') I 1 - e - h v t ( Y ' ) / k B T  x _ e-hvJ(Y)/k BT H 3 n " 7 i ± • e-2kTT ?-.:tvJ(y) - v f ( Y ' ) ] B l where the imaginary (unbound) frequencies, v T , corresponding to the reaction coordinate, have been factored out. Denoting ut" = h v i , the l a s t three factors i n (46) may be combined to give k BT T v T(Y) 3n-7 n ut(Y) .sinh ut(Y«) l k ' r t V+(Y') sinh uJ(Y) f u J ( Y ' ) l (46) Noting that the isotopic frequency r a t i o s are related to the isotopic masses according to [Weston (72), N i k i t i n (74), Karplus (70)] , v(Y) v(Y') y ( Y ' ) ^ 1 / 2 (Y) the rate constant r a t i o becomes .1/2 3n-7 k k' y (Y') y u+ (Y) sinh uf (Y') l (Y) J n i sinh ( i r f e(Y) l uj (Y') (47) where u denotes the e f f e c t i v e masses possessed by the representative point on the barrier-crossing trajectory. Equation (47) contains a number of i n t e r e s t i n g terms, Unlike the temperature independent term of c o l l i s i o n theory (Section D) which only depends on the reduced mass of the y * Y' u* Y 1/2 reactants, the TST temperature independent term, depends on the e f f e c t i v e mass of the representative point as i t crosses the poten t i a l b a r r i e r ; that i s , i t depends on the location of the poten t i a l b a r r i e r . In t h i s sense, the TST temperature independent term contains dynamical information, i n contrast to the corresponding c o l l i s i o n theory term. In the l i m i t i n g case of a very early b a r r i e r (such as i n the Mu Y + F 0 reactions) , t h i s term does predict n — ^ 2.9, i n 2 k R accord with the c o l l i s i o n theory r e s u l t ; t h i s temperature independent mass e f f e c t i s often c a l l e d the "primary" isotope e f f e c t [ N i k i t i n (74)] . I t i s customary to denote T v = — : — U 4 — t o indicate the quantum nature of the v i b r a t i o n a l smh(u/2) ^ p a r t i t i o n functions [Johnston (66)]. In the l i m i t of low v i b r a t i o n a l frequencies and high temperatures [Johnston (66), Weston (72)] \ u 2 u 4 u 6 -1 r v = "TTTuT * ( 1 + 24 + T 9 2 0 + 77^ 6 + •••) sir nh ( ^ - J 7 ! 2 •2 and l i m r = 1 u^O v Conversely, i n the l i m i t of low temperature and high frequen-cies , -u/2 T - ue ' v and l i m r = 0 v u-> 0 0 Clearly, the exponential dependence of on u indicates that i f the is o t o p i c s ubstitution of Mu r e s u l t s i n a substantial increase i n u^, there w i l l be a very strong reduction i n -122-(Mu)/ r ^(H). In general, i f the barr i e r i s early, the a c t i -vated complex corresponds to a very s l i g h t l y perturbed target . molecule with symmetric stretch vibrations that display a very weak dependence on isotopic substitution, and thus r^(Mu)/r^(H) 1; conversely, as the barr i e r becomes progressively l a t e r , the values of u^ increase and take on strong isotopic depen-i dences, and thus r^(Mu)/r^(H) 0. Stretching vibrations are usually stronger than bending vibrations [Johnston (66)] and thus they might be expected to have a stronger influence on (Mu)/r^(H); Figure 17 shows t h i s to be the case. The Figure plots r ^ / r j , a s a function of the percent of u/u' ( i . e . u cor-responds to a l i g h t e r isotope than u 1) for various values of u 1 . From the Figure, i t i s clear that a small increase i n u over u' for a strong stretching v i b r a t i o n ( t y p i c a l l y >300 cm for an early barrier) reduces r ^ / r ^ , more than a large increase i n u/u 1 for a weak bending v i b r a t i o n ( t y p i c a l l y <50 cm for an early b a r r i e r ) . Isotope e f f e c t s due to r^/T^ are referred to as "secondary" isotope e f f e c t s [ N i k i t i n (74)]. F Tunnelling There are two d e f i n i t i o n s of tunnelling applicable to chemical reactions [Connor l - ( 7 6 ) ] . The f i r s t i s the standard " s t a t i c " or "energetic" d e f i n i t i o n associated with b a r r i e r penetration; the model of nuclear alpha decay i s one of the more celebrated examples. In a chemical reaction, s t a t i c tunnelling occurs when there i s a non-zero reaction p r o b a b i l i t y despite the fact that the t o t a l energy of the c o l l i d i n g species ( i . e . the r e l a t i v e t r a n s l a t i o n a l k i n e t i c -123-I S O T O P E E F F E C T DUE TO f * / f * . v ( C M " 1 ] 1 0 0 1 2 5 1 5 0 1 7 5 2 0 0 2 2 5 2 5 0 2 7 5 3 0 0 U/U' {%) FIGURE 17: Isotope e f f e c t s i n t r a n s i t i o n state vibrations from mass variations of atom Y for the reaction Y + AB -> YA + B. r . / r * , i s plotted as a function of the per cent increase Y n the isotope-dependent.vibrational frequency of v over v 1 for the various values of v 1 indicated on the r i g h t . It i s assumed that the frequencies of v and v 1 correspond to t r a n s i t i o n state molecules containing l i g h t and heavy isotopes of atom Y respectively,, at 300K. energy plus the i n t e r n a l v i b r a t i o n a l energy of the target molecule) i s less than the p o t e n t i a l b a r r i e r height. In terms of the energy d e f i n i t i o n s of Section C, the physical b a r r i e r height, E^* 1^ 3, defined by equation (24) , i s the s t a t i c tunnelling b a r r i e r ; reactive c o l l i s i o n s with less r e l a t i v e t r a n s l a t i o n a l k i n e t i c energy that E ^ y S occur by s t a t i c tunnelling. As discussed i n the next Chapter, although Mu reactions show an appreciable amount of s t a t i c tunnelling, t h i s form of tunnelling i s r e l a t i v e l y unimportant at normal temperatures (>200K) [Connor l-(77), l-(76)]. The second form of tunnelling i s "dynamic." This refers to reactive c o l l i s i o n s that are e n e r g e t i c a l l y allowed and which do occur quantum mechanically but which are c l a s s i c a l l y forbidden, not because of energy, but because of the reaction dynamics. According to the d e f i n i t i o n s of Section C, the q u a s i c l a s s i c a l OC threshold energy, E^ , , i s the dynamic tunnelling b a r r i e r ; reactive c o l l i s i o n s with less r e l a t i v e t r a n s l a t i o n a l k i n e t i c energy than E^ C but more than E P ^ ^ S occur by dynamic tunnel-l i n g . Dynamic tunnelling i s by far the most dominant form of tunnelling i n chemical reactions [Connor l-(7.6)]. In Section C i t was noted that q u a s i c l a s s i c a l threshold energies are d i f f i c u l t to define p r e c i s e l y because of Monte Carlo averaging; thus, dynamic tunnelling may be somewhat QC ambiguous since i t i s defined i n terms of E T . I t i s simply noted here that Connor l-(76) has shown that tunnelling may be unambiguously defined i n terms of complex-valued c l a s s i c a l t r a j e c t o r i e s a r i s i n g from semiclassical scattering theory. The tunnelling correction term, V , applied to TST - 1 2 5 -(c.f. equation ( 3 9 ) ) i s normally calculated as a quantum bar r i e r penetration c o e f f i c i e n t for a mathematically one dimensional b a r r i e r [Johnston ( 6 6 ) , Jakubetz ( 7 9 ) ] . Johnston ( 6 6 ) . notes that t h i s approach may be "better chemical engineering than natural philosophy;" nonetheless, t h i s a t t r a c t i v e l y simple quantum correction to the c l a s s i c a l rate expression i s often remarkably successful. Three one dimensional b a r r i e r penetration correction models are most commonly used for r , . The f i r s t order Wigner correction 2 [Johnston ( 6 6 ) , N i k i t i n ( 7 4 ) ] i s Y. = 1 + where | v * | = 1 / 2 . Z 2 4 J V 1 _ 2TT i s the imaginary frequency corresponding to the i s the force constant V reaction coordinate and | F * | = d2V(g) dq 2 (curvature) of the pot e n t i a l surface at the saddle point; t h i s h I v * I f i r s t order correction i s v a l i d for J _ 1 << 1 . While the B Wigner expansion may be applied to any shape of one dimension-a l b a r r i e r , exact tunnelling corrections have been worked out for two s t y l i z e d reaction b a r r i e r s [Johnston ( 6 6 )] : the truncated, inverted parabola, calculated by R. P. B e l l ; and the b a r r i e r due to C. Eckart. The Eckart b a r r i e r , the most r e a l i s t i c of the two since i t has a smooth, continuous base unlike the truncated parabola, may be symmetrical, correspond-ing to a thermoneutral reaction, or unsymmetrical, correspond-ing to an exothermic reaction. The parameterization of the B e l l and Eckart b a r r i e r s and the a n a l y t i c a l forms of the transmission c o e f f i c i e n t s may be found i n Johnston ( 6 6 ) or Jakubetz ( 7 9 ) . Figure 18 (adapted from Johnston ( 6 6 ) ) compares the transmission p r o b a b i l i t i e s for the two b a r r i e r s as a function of energy at various values of the b a r r i e r height. -126-FIGURE 18: Tunnelling transmission c o e f f i c i e n t s as a function of energy normalized to the ba r r i e r height for the truncated B e l l parabola (top) and Eckart barrier (bottom), adapted from Johnston (61). The a para-meter describes the shape of the b a r r i e r : large a -> high, wide barri e r ; small a -> short, narrow b a r r i e r . Johnston has noted that the B e l l transmission c o e f f i c i e n t s are symmetrical to inversion about K = 0.5, £ = 1. Furthermore, because of the B e l l truncation, K does not approach zero at £ = 0 for low values of a. I t i s also noted that quantum mechanical r e f l e c t i o n as well as penetration occurs with these b a r r i e r s . -127-CHAPTER IV - EXPERIMENTAL RESULTS AND THEIR INTERPRETATION In t h i s Chapter, the experimental re s u l t s for each Mu reaction are reported and compared with recent experimental res u l t s for the analogous H atom reactions and with t h e o r e t i c a l predictions, where available; Table VI summarizes the r e s u l t s . For some reactions, several H atom reaction rate parameters are reported with rate constants varying by factors of three or more and a c t i v a t i o n energies varying by 50%. This underscores the fact that " the wealth of data on bimoleeular reactions that involve free r a d i c a l s or atoms i s more testimony to the growing awareness of the importance of these intermediates i n k i n e t i c systems and the frequency of t h e i r occurence than to the great accuracy of the r e s u l t s [Benson (60)]." The r e l a t i v e l y poor knowledge of gas phase H atom reaction rates available today i s due to two experimental l i m i t a t i o n s : (1) u n t i l about a decade ago, there were few techniques available to measure gas phase H atom reaction rates d i r e c t l y , either by monitoring reactant depletion v i a some observable of H atoms or i t s reactant partner, or by monitoring product formation; rather, H atom reaction rates were i n d i r e c t l y inferred from a postulated reaction mechanism and associated "steady state" approximations, thereby making the res u l t s model dependent; (2) while the advent of modern tech-niques such as mass spectrometric fast flow sampling of product formation or ESR detection of H atoms i n d i l u t e gases has made rate measurements d i r e c t , i t has not completely removed the systematic errors due to competitive reactions among the r e l a -t i v e l y large concentrations of highly reactive atomic and molecular species simultaneously present i n the experimental TABLE VI: SUMMARY OF THE REACTION RATE PARAMETERS FOR Mu AND H IN THE GAS PHASE Muonium , Hydrogen , k Reaction E (kcal/mole) k(295K) T E (kcal/mole) k(295K) T r-—(295) r e f . a a K Y + F 2 YF+F 0. 92 + 0.23 1.4 + 0. 1 2.4 + 0. 2 0.20 + 0. 05 6.8 + 1.5 Dodonov(7 0) 2.2 + 0. 1 0. 09 + 0. 01 14.6 + 1.6 Homann(77) Y + C l 2 YC1+C1 1. 36 + 0.21 5.1 + 0. 2 1.8 + 0. 6 1.7 + 0. 6 2.9 + 1.0 Dodonov(7 0) 1.4 + 0. 2 0.41 + 0. 04 13 + 1.2 Ambidge (7 6) 1.20 + 0. 14 1.2 + 0. 1 4.4 + 0.4 Wagner(7 6) 1.14 + 0. 17 1.3 + 0. 1 4.1 + 0.3 Bemand(77) Y+Br2 -> YBr+Br 24 + 3 2.2 + 1. * 5 * 11 + 8 Fleming(7 6) 1.8 + 0. 4 5.1 + 0. ic 6 k 4.7 + 0.8 Fass (70) + Endo(7 6) Y+HC1 products <0.000034 0.000005 + 0. 009 + 0. 004 <0.004 + 0.002 Bott (7 6) Y+HC1 YH + Cl 3.18 + 0. 17 0. + 0. 0021 0002 <0.016* + 0.003 Weston (79) Y+HBr products 0.91 + 0. 10 2. 57 + 0. 11 0.21 + 0. 02 4.4 + 0.6 Endo (7 6) Y+ HI ->- products 2.53 + 0. 13 0.00 + 0. 25 0.11 + 0. 02 23 + 4 Sullivan(62 0.70 + 0. 25 1.5 + 0. 5 1.7 + 0.6* Jones(73) 10 1/mole-s * estimates only -129-apparatus. For example, the very fast reaction of H + C l 2 * HCl +C1, where * denotes a v i b r a t i o n a l l y excited molecule, may be accompanied by the following side reactions [Wagner (76)] : H + HCl* + H 2 + Cl C l + wall + 1 / 2 C 1 2 which consume additional H and C l atoms and regenerate C l 2 , thereby a l t e r i n g the reaction stoichiometry. To reduce these interferences, H atom k i n e t i c i s t s are constantly s t r i v i n g (with considerable success) to perform rate measurements under more d i l u t e conditions, but an impairing reduction i n the observable signal inevitably accompanies these e f f o r t s . In Chapter I I I , i t was noted that the f i r s t motivation for undertaking the k i n e t i c study of the reactions of Mu was to investigate isotope e f f e c t s i n H atom reactions. The second motivation for the study arises from the fact that MSR measure-ments are l i t e r a l l y one-atom-at-a-time experiments which are not susceptible to the kinds of interferences that plague H atom measurements as outlined above. As a r e s u l t , the MSR method might well provide the most accurate (isotopic) values of H atom reaction rates. This i s not to say that MSR measurements are necessarily unambiguous - since the method simply measures the relaxation of the MSR signal, care must be taken i n i d e n t i -fying the source of t h i s relaxation which need not be chemical reaction (see Appendix I I ) . A Mu + F 2 ->• MuF + F The MSR relaxation rates at various F 2 concentrations, measured i n N 0 moderator between 295 and 383K, are l i s t e d i n -130-Table VII. The i n f l u e n c e of temperature on the r e a c t i o n r a t e s i s i l l u s t r a t e d i n Figure 19 [adapted from Garner (78)] which p l o t s the MSR r e l a x a t i o n r a t e data at 295 and 383K. The b i -2 molecular r a t e constants determined by x minimum f i t s of the r e l a x a t i o n r a t e data to equation II (3) are a l s o given i n the Table and i l l u s t r a t e d i n the Arrhenius p l o t of Figure 20. 2 [Garner (78)]. The x minimum f i t of these data to the l o g a -r i t h m i c Arrhenius expression (equation (12)) y i e l d s : log 1 Qk(1/mole-s) = (10.83 + 0.20) - (200 + 50/T) , (la) w i t h k(300K) = (1.46 + 0.11) x 1 0 1 0 1/mole-s and E = (0.92 + 0.23) kcal/mole. The experimental r a t e parameters of the r e a c t i o n : H + F^ ~+ HF + F have been reviewed by Jones et a l . (73) and Foon and Kaufman (75).. These authors recommend the d i r e c t mass specto-metric probe measurements of a f a s t flow system by A l b r i g h t et a l . (69) and Dodonov et a l . (70.) from 294 to 565K'.which-yielded k(300) = (2.15 + 0.46) x 10 9 1/mole-s w i t h E = 2.4 + 0.2 k c a l / c l mole and log 1 QA(1/mole-s) = (11.079 + 0.035). These r e s u l t s are i n good agreement w i t h the more recent EPR flow system measure-ments of Rabideau et a l . (72) who determined k(300K) = (2.5 + 0.2) x 10 1/mole-s and estimated E & = 2.6 kcal/mole, and w i t h the e a r l i e r i n d i r e c t r e s u l t s of Levy and Copeland (68) obtained by thermal, 0^ - i n h i b i t e d E^ - F^ r e a c t i o n , which gave k(288K) = 9 1.8 x 10 1/mole-s. However, the most recent measurement of t h i s r e a c t i o n r a t e i s the flow system mass spectrometric d e t e r -mination from 224 to 493K by Homann et a_l. (77) which y i e l d e d E, = 2.2 + 0.1 kcal/mole, log. nA(1/mole-s) = (10.6 + 0.1) and 3 . — _|_ U — 9 k(300K) = (1.00 + 0.08) x 10 1/mole-s. While the a c t i v a t i o n -131-TABLE VII: MSR RELAXATION RATES FOR THE REACTION Mu + F 2 -» MuF + F Bimoleeular Temperature Rate Constant [F,,] Relaxation (K) k ( 1 0 1 0 M " 1 s" 1) (10~ 4 M) Rate A (us" 1) + 295 + 2 1.42 + 0.07 0. 0 0. 68 + 0. 06 0. 40 + 0. 02 1. 27 + 0. 11 0. 69 + 0. 04 1. 63 + 0. 12 1. 08 + 0. 05 1. 74 + 0. 14 1. 23 + 0. 03 2. 18 + 0. 23 1. 43 + 0. 06 2. 25 + 0. 18 1. 93 + 0. 04 3. 56 + 0. 34 2. 33 + 0. 06 4. 25 + 0. 38 2. 98 + 0. 07 5. 66 + 0. 47 327 + 3 1.63 + 0.10 0.0 0. 64 + 0. 04 0.59 + 0. 02 1.32 + 0.14 1.16 + 0. 03 2.73 + 0.41 1. 68 + 0. 04 3.33 + 0.41 2.11 + 0. 05 3.55 + 0.47 2. 67 + 0. 06 6. 52 + 0. 62 353 + 4 1.84 + 0.13 0.0 0.72 + 0. 07 0.48 + 0. 02 1. 55 + 0. 08 0. 99 + 0. 03 2.44 + 0.23 1.40 + 0. 03 3. 53 + 0. 42 1. 83 + 0. 05 4 . 34 + 0.40 383 + 2 2.03 + 0.14 0.0 0.72 + 0.08 0. 91 + 0. 02 2.41 + 0.31 1.24 + 0. 03 3. 95 + 0.47 1.82 + 0. 05 4.16 + 0.42 2. 46 + 0. 06 5.87 + 0.59 Relaxation rates reported are weighted averages of the rig h t positron telescope histograms. l e f t and -132-MU I N F 2 / N 2 : ± = 2 9 5 K . • = 3 8 3 K 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 3 . 0 3 . 5 F 2 C O N C E N T R A T I O N • . ( 1 0 " 4 M) FIGURE 19: The e f f e c t of temperature on the Mu + F 2 MSR relaxation rates. The l i n e s are x 2 minimum f i t s of the data to equation 11(3) corresponding to k = (1.42+0.07) x 1 0 1 0 1/mole-s at 295 K (triangles) and k = (2.03 + 0.14) x 1 0 1 0 1/mole-s at 3 83K (squares). Experimental points shown are weighted averages from the l e f t and r i g h t telescope histograms. -133-0.01 1 1 1 1 I I I I 1 . 5 2 . 0 2 . 5 3 . 0 3 . 5 4 . 0 4 . 5 5 . 0 1 0 0 0 / T E M P ( K _ 1 J FIGURE 20: Experimental Arrhenius plot for the Y + F 2 reactions, Y = Mu, H. The Mu data i s on the top l i n e (this work). The H data i s due to Rabideau(72)(diamond), Levy(68)(octagon), Dodonov(70)(squares), and Homann(77)(triangles). The error bars on the Mu data are s t a t i s t i c a l only; the error bars on the H data are estim-ates given by the authors which apparently . include systematic errors. energy i s i n agreement with the previous determinations, k(300K) i s a factor of two smaller. Homann et aJL. (77) c i t e several possible reasons for t h i s discrepancy. The experimental re s u l t s for the H atom reaction from a l l of these authors are also shown i n the Arrhenius plot of Figure 20. Clearly, the acti v a t i o n energy for the Mu reaction with F^ i s less than half of that for the analogous H atom reaction at 300K, ind i c a t i n g that the average energy of reactive Mu c o l l i s i o n s i s much less than that of reactive H c o l l i s i o n s , according to the Tolman interpretation of ac t i v a t i o n energy (see Chapter I I I , Section C). Furthermore, the Mu:H rate constant r a t i o i s kM either r—^(300K) = 6.8 + 1.5, using the H atom res u l t s of KH Albright et a l . , or 14.6 + 1.6, using the res u l t s of Homann et a l . Certainly, the Mu reaction at 3 0OK i s much faster than the H atom reaction; i t i s at least (2.3 + 0.5) times faster than predicted by the temperature independent mass factor of 2.9. This extra rate enhancement must be due to dynamical e f f e c t s , and, as discussed throughout Chapter I I I , the only such e f f e c t l i k e l y to enhance the rate of Mu reaction with F^ i s quantum tunnelling. The measured reduction i n the Mu ac t i v a t i o n energy, r e l a t i v e to the H atom values, i s also consistent with t h i s tunnelling interpretation. The only "stand alone" experimental indicator of the presence of tunnelling i n thermally averaged reactions i s curvature i n Arrhenius plots (Chapter I I I , Section C, [Laidler (65)]), but t h i s test i s not unambiguous since the preexponential factor i s also weakly temperature dependent. Besides, i t i s d i f f i c u l t to obtain s u f f i c i e n t experimental precision over a wide -135-enough temperature range to demonstrate s i g n i f i c a n t Arrhenius plot curvature, p a r t i c u l a r l y for reactions of gases [Laidler (65), Jakubetz (7 9)]. Consequently, the absence of curvature i n the limited Mu data of Figure 20 i s more l i k e l y a manifestation of the i n s u f f i c i e n t temperature range of the measurements than an i n d i c a t i o n of the absence of tunnelling. On the other hand, the H atom data of Albright et a_l. does show a s l i g h t curvature although i t may not be s i g n i f i c a n t given t h e i r estimated rate constant uncertainties of 25 to 3 0%. In any case, these data are suggestive of a tunnelling contribution to the H + F2 reaction and give apparent ac t i v a t i o n energies of about 2.2 kcal/mole from 300-- 4 0OK and 3.3 kcal/mole from 450 - 57OK. A number of QMT, QCT, and TST investigations have been performed on the reactions of H isotopes, including Mu, with F 2. The c o l l i n e a r modified LEPS surface of Jonathan et a l . (72) (shown in Figure 12) has been used by Connor et aJL. to calculate exact c o l l i n e a r quantum mechanical t r a j e c t o r i e s by the state path sum method [Connor l-(77), l-(78), l-(79)]. c o l l i n e a r quasi-c l a s s i c a l t r a j e c t o r i e s [Connor 2-(78), Connor IT(-79)]. and c o l l i n e -ar v i b r a t i o n a l l y adiabatic TST calculations [Connor 1- (7-9) ] ; Jakubetz l-(78),(7 9) also used t h i s surface to investigate tunnel-l i n g corrections to TST c a l c u l a t i o n s . The c o l l i n e a r reaction rate constants, isotopic rate constant r a t i o s , and apparent ac t i v a t i o n energies calculated by these authors for the reactions Y + F2 (v = 0, 1) -> YF + F, Y = Mu, H, D, T between 200 and HOOK are l i s t e d i n Tables VIII, IX, and X respectively. The v i b r a t i o n a l populations of F 2 at thermal equilibrium at 3 00, 550, and 900K are 98%, 89%, and 74% for v = 0, and 2%, 9%, and 19% for TABLE V I I I : CALCULATED RATE CONSTANTS FOR THE COLLINEAR Y + F_ -»- YF + F REACTIONS (a) k n(Y) (cm s molecule ) (b) k (Y) (cm s molecule ) T/K Mu H D T T/K Mu H . D . T Quantum^ Quantum^ 300 1.5(4) 2.3(3) 1.4 (3) 1.1(3) 300 1.9(4) 3.3(3) 2.2(3) 1.8 (3) 550 4.7(4) 1.3 (4). 8.9(3) 7.3(3) 550 5.5(4) 1.6(4) 1.2(4) 9.6 (3) 900 1. 0(5) 3.4 (4) 2.4(4) 2.0(4) 900 1.1(5) 3.9(4) 2.8 (4) 2.3(4) t Q u a s i c l a s s i c a l t Q u a s i c l a s s i c a l 300 4.5(3) 1.6(3) 1.2 (3) 9.8 (2) 300 7.2(3) 2.5(3) 1.9(3) 1.7 (3) 550 3.1 (4) 1.2 (4) 8.4 (3) 7.0 (3) 550 3.6(4) 1.5(4) 1.1(4) 9.3(3) 900 8.8(4) 3.2 (4) 2.3(4) 1.9(4) 900 8.7(4) 3.8(4) 2.8 (4) 2.3(4) + TST (no t u n n e l l i n g ) TST t (no t u n n e l l i n g ) 300 4.1 (3) 1.6 (3) 1.2(3) 1.0 (3) 300 5.1(3) 2.6(3) 2.1(3) 1.8 (3) 550 3. 1 (4) 1.2 (4) 8.5 (3) 7.1(3) 550 3.5(4) 1.5(4) 1-2 (4) 9.9 (3) 900 9.1 (4) 3.2 (4) 2.4(4) 2.0 (4) 900 9.7(4) 3.8(4) 2.8(4) 2.4 (4) TST (Eckart t u n n e l l i n g c o r r e c t i o n ) ^ TST (Eckart t u n n e l l i n g c o r r e c t i o n ) ^ 300 1.6(4) 2.1(3) 1.4(3) 1.1(3) 300 1.9(4) 3.3(3) 2.4(3) 2.0(3) 550~ 5.0(4) 1.3(4) * 8.9(3) 7.3 (3) 550 5.5(4) 1.6(4) 1.2(4) 1.0(4) Y = Mu, H, D, or T. The number i n p a r e n t h e s i s i n d i c a t e s the power of 10 by which the e n t r y should be m u l t i p i e d . from Connor l-(79) from. Jakubetz (79) TABLE IX: CALCULATED RATE CONSTANT RATIOS FOR THE COLLINEAR Y + F„ YF + F REACTIONS (a) k Q(Y)/k (H) (b) k 1(Y)/k 1(H) T/K Mu H D T T/K Mu H D T Quantum^ Quantum^" 300 6.6 1 0. 63 0.50 300 5.7 1 0. 68 0.55 550 3.7 1 0. 69 0.56 550 3.4 1 0.71 0.59 900 3.1 1 0.70 0.58 900 2.9 1 0.72 0. 60 4 -Q u a s i c l a s s i c a l 1 t Q u a s i c l a s s i c a l 300 2.8 1 0.74 0.62 300 2.9 1 0.77 0.66 550 2.7 1 0.73 0.61 550 2.4 1 0.75 0. 63 900 2.7 1 0.72 0.60 900 2.3 1 0.74 0.61 TST (no f tunnelling) TST (no t tunnelling) 300 2.6 1 0.75 0.63 300 2.0 1 0.81 0.71 550 2.7 1 0.74 0.61 550 2.3 1 0.76 0. 65 900 2.8 1 0.73 0.61 900 2.6 1 0.74 0. 63 TST (Eckart tunnelling correction)^ TST (Eckart tunnelling correction)^ 300 7.6 1 0. 66 0.53 300 5.6 1 0.72 0.60 550 4.0 1 0.71 0.58 550 3.4 1 0.73 0. 62 * Temperature Independent Factor. 2.9 1 0.72 0.59 Y = Mu, H, D, or T. from Connor l-(79) from Jakubetz (7 9) TABLE X: CALCULATED ACTIVATION ENERGIES FOR THE COLLINEAR Y + F „ - * Y F + F REACTIONS (a) E ( 0 ) ( Y ) a (kcal mole " V (b) E( 1 ) (Y) a i. •• --1 * (kcal mole ) T/K Mu H D T T/K Mu H D T Quantum^ Quantum^ 300 1.2 2.1 2.3 2.3 300 1.1 1.9 2.0 2.0 550 1.9 2.5 2.6 2.6 550 1.8 2.3 2.3 2.3 900 2.6 2.9 3.0 3.0 900 2.4 2.7 2.7 2.7 f Quasiclassical t Q u a s i c l a s s i c a l 300 2.4 2.5 2.5 2.5 300 2.0 2.2 2.2 2.2 550 2.7 2.7 2.7 2.7 550 2.3 2.5 2.4 2.4 900 3.1 3.1 3.1 3.1 900 2.7 2.8 2.8 2.8 TST t (no tunnelling) TST t (no tunnelling) 300 2.6 2.5 2.5 2.5 300 2.5 2.2 2.1 2.1 550 2.8 2.7 2.7 2.7 550 2.7 2.5 2.4 2.3 900 3.2 3.1 3.1 3.1 900 3.0 2.8 2.7 2.7 TST (Eckart tunnelling correction) § TST (Eckart tunnelling correction) § 300 1.2 2.1 2.3 2.3 300 1.1 1.9 2.0 2.0 550 1.9 2.6 2.6 2.6 550 1.8 2.3 2.3 2.3 U ) CO I Y = Mu, H, D, or T. + from Connor l-(79) from Jakubetz (7 9) -139-for v = 1 respectively; consequently, (T) approximates k(T) to better than 0.5% at 300K and better than 2.8% at 900K. I t may be noted that the o r i g i n a l QMT reaction p r o b a b i l i t i e s for the Mu reaction [Connor l-(77), Connor l-(78)] were recalculated and found to be about 12% larger than f i r s t reported [Connor 1(79)]; the new r e s u l t s are thought to be accurate to better than 3%. I t should also be noted that the quantum calculations for the Mu reactions were calculated by defining a " l i n e of no return" on the p o t e n t i a l surface such that the reaction i s presumed to proceed once a given c o l l i s i o n crosses t h i s l i n e ; t h i s procedure does not allow for r e f l e c t i o n of the representative point from the repulsive product v a l l e y wall and therefore over-estimates the reaction p r o b a b i l i t y for c o l l i s i o n s at very high r e l a t i v e t r a n s l a t i o n a l k i n e t i c energy. However, QCT calculations show that t h i s e f f e c t should not influence the rate constants i n the temperature range below 1000K [Connor 2-(78), l,-(:79)]. At 300K, the predicted c o l l i n e a r QMT a c t i v a t i o n energies for the Mu and H reactions, 1.2 and 2.1 kcal/mole respectively, are i n good agreement with the experimental values, 0.9 + 0.2 and 2.3 + 0.2 kcal/mole respectively. Furthermore, at 300K, comparison of the predicted Mu:H rate constant r a t i o of 6.6 with the experimental values shows remarkable agreement with the value of 6.8 + 1.5 obtained from k^ measured by Albright et a l . , but clear disagreement with the value of 14.6 + 1.6 obtained from the r e s u l t s of Homann et al_. That these c o l l i n e a r calculations apparently agree well with most of the experimental r e s u l t s i s , i n i t s e l f , an i n t e r e s t i n g f a c t . The quantitative agreement might well be fortuitous since: (1) "the sucessful -140-t h e o r e t i c a l prediction of an energy of act i v a t i o n does not imply that the d e t a i l s of the theory are even q u a l i t a t i v e l y correct [Truhlar (78)]," and (2) despite the agreement between the predicted Mu:H rate constant r a t i o and the experimental value that uses the re s u l t s of Albright et a l . , t h i s cannot be i n t e r -preted as removing the ambiguity of the experimental rate con- -stant r a t i o s since the Jonathan et a l . surface used i n the calculations was optimized by q u a s i c l a s s i c a l t r a j e c t o r i e s which did not incorporate tunnelling, and may, therefore, be inaccurate [Connor 2-(78)]. On the other hand, there are a number of reasons to suppose that the c o l l i n e a r calculations do f a i t h f u l l y describe the reaction, at least q u a l i t a t i v e l y [Connor l-(79), Jakubetz (79)]: (1) as mentioned i n Chapter I I I , the c o l l i n e a r configuration i s energetically favored for the Jonathan et a l . surface, (2) three dimensional tra j e c t o r y calculations show that the reaction i s c o l l i n e a r l y dominated due to the reaction dynamics [Polanyi (75)], and (3) because the saddle point i s very early, the t r a n s i t i o n state i s just a s l i g h t perturbation of the target molecule and thus the three dimensional bending vibrations of the t r a n s i t i o n triatomic should not greatly a l t e r the c o l l i n e a r potential [Connor 1-(79)]. Like the experimental r e s u l t s , the QMT calculations are strongly suggestive of tunnelling i n the Mu reaction. The quantum k(Y)/k(H) rate constant r a t i o s of Table IX for both the F^ (v = 0) and F^ (v •'= 1) reactions approach the l i m i t i n g tempera-ture independent mass factor r a t i o s of 2.9:1.0:0.72:0.59 (see Chapter I I I , Section D, p. 113) as the temperature approaches 900K, ind i c a t i n g that the large dynamical e f f e c t s that enhance -141-the room temperature Mu reaction cease to operate i n the high temperature " c l a s s i c a l " regime. The dramatic increase i n the Mu reaction a c t i v a t i o n energy (Table X), which approaches an isotope independent value near 900K, i s also consistent with tunnelling i n the context of the Tolman interpretation of act i v a t i o n energy (Chapter I I I , p. 97). A revealing i n d i c a t i o n of the dynamics of the Y + F^ reactions i s i l l u s t r a t e d i n Figure 21 (adapted from [Connor l-(79)]) which compares the energy dependence of the t o t a l reaction p r o b a b i l i t i e s , P1" = E P t 1 (see Chapter I I I , Section D) , for S i s s s the q u a s i c l a s s i c a l and quantum mechanical t r a j e c t o r i e s . In both OC t cases, at lower values of , P i s ordered Mu>H>D>T, while trans s OC t at higher values of E^ , P displays the opposite behaviour; trans s the curves cross near P^ = 0.5. In the Figure, the physical ba r r i e r height, E ^ ^ S , for the F^ (v = 0) reaction i s indicated by OC an arrow at E^ = 1.08 kcal/mole (there i s no physical barrier trans to the F^(v = 1) reaction). It may be r e c a l l e d from Chapter I I I , Section F, that E ^ y S represents the " s t a t i c " tunnelling b a r r i e r : c o l l i s i o n s with less r e l a t i v e t r a n s l a t i o n a l k i n e t i c energy than EPhys no^_ a l l o w e c j t o r e a c t c l a s s i c a l l y due to t h e i r energy d e f i c i t . From Figure 21, i t i s clear that only the Mu reaction displays considerable s t a t i c tunnelling. I t w i l l also be . OC r e c a l l e d that the q u a s i c l a s s i c a l threshold energies, E~ , repre-sent the barriers to "dynamic" tunnelling: c o l l i s i o n s with less OC r e l a t i v e t r a n s l a t i o n a l k i n e t i c energy than E~ , but with more than E ^ ^ S , are c l a s s i c a l l y forbidden, not because of the energy balance as i n the " s t a t i c " case, but because of the a v a i l a b i l i t y of energy, as governed by the reaction dynamics. The Figure 0.8 0.4 1 1 1 ; Quasiclassical ! i I 1 1 1 / : / / : 1 / ' / / _ i i i — i j / / / / / j / i; ll A ' / i / i ' /> 1 ' 1 / / ' / •• / i i 1 / 1 i ^ans A a l mof1 0.8 0.4 t O . O t F Quasic lass ica l 0.4 0.0 lc Quantum 0.8 E i r L / k c a l m o f 1 FIGURE 21: Collinear quantum and q u a s i c l a s s i c a l t o t a l reaction p r o b a b i l i t i e s as a function c o l l i s i o n energy for Y + F 2(v=0>l) •*• YF + F, Y = Mu ( ), H( ), D(---), or T (• adapted from [Connor 1-(-79.)}-. The significance of the arrows i s described i n the of -•-) text -143-c l e a r l y shows that a l l isotopic variants of the reaction display considerable dynamic tunnelling which i s much more dominant than s t a t i c tunnelling, even i n the case of muonium. The Boltzmann d i s t r i b u t i o n must be considered i n order to appreciate the importance of the tunnelling-enhanced reaction p r o b a b i l i t y on the ensemble reaction process. Figure 22 plots t -E /k_T the integrand of equation (35) (P (E )e trans B ) as a function of for the quantum mechanical reaction p r o b a b i l i -trans ^ c t i e s at 300 and 900K for the Y + F ^  (v = 0) reaction (adapted from [Connor l - ( 7 8 ) ] ) . In the Figure, arrow A indicates the physical b a r r i e r height at 0.0472 eV (1.08 kcal/mole), arrow B, at 0.087 eV (2.01 kcal/mole), indicates the q u a s i c l a s s i c a l threshold for the H atom reaction (from [Jonathan (72)]), and arrow C indicates kgT at (a) 300K and (b) 900K. A comparison of arrow B with the q u a s i c l a s s i c a l thresholds of Figure 21 reveals that i t i s approx-imately the average of the q u a s i c l a s s i c a l thresholds for the isotopic variants of t h i s reaction. For the sake of i l l u s t r a t i o n , OC arrow B, the average E^ , divides the "tunnelling" reaction region from the " c l a s s i c a l " region. Thus, Figure 22 c l e a r l y demon-states that the room temperature muonium reaction i s dominated by tunnelling, which also contributes s i g n i f i c a n t l y to the room temperature H atom reaction, whereas, at 900K, c l a s s i c a l processes dominate the reactions for a l l H isotopes. Figure 23 (adapted from [Connor 1- (79) ]),. a p l o t similar to Figure 22, compares the rate constant integrand for the q u a s i c l a s s i c a l and quantum mechanical reactions of the F^ (v = 1) state at 3 00K. Once OC .again, i f the average' E j ^ i s taken as the l i n e that approximately separates c l a s s i c a l from tunnelling processes, i t i s seen that H G M M M integrand for rate constant /[orb. units} O r t • fD \ -t>t>T-0.04 o § 0 . 0 3 o c c n CD 0.02 0.01 0 1 - 0 -Quasiclassical Q u a n t u m E t r a n s /kcaL mol 6 I FIGURE 23: Integrand for the c o l l i n e a r quantum and q u a s i c l a s s i c a l rate constant k.. (T) at 3 00K for the reaction: Y + F, (v=l) YF + F, Y = Mu ( ) , H ( ) , D( - • •) , and T( ) , adapted from [Connor l-'(79),]. The arrows indicate kgT at 300K. -146-tunnelling completely dominates the muonium reaction at room temperature and contributes s i g n i f i c a n t l y to the H atom reaction rate as well. The calculations of Connor et a l . also reveal a great deal about the high temperature " c l a s s i c a l " behaviour of the Y + reactions. Figure 21 shows that P^ for high energy c o l l i s i o n s i s ordered T>D>H>Mu for both the QCT and QMT cal c u l a t i o n s . From the discussion i n Chapter I I I , Section B, p.76, t h i s behaviour may be explained i n terms of the c l a s s i c a l "bottleneck" e f f e c t a r i s i n g from the sharper c o n s t r i c t i o n in the reaction v a l l e y for the l i g h t e r H isotopes. V e r i f i c a t i o n of t h i s e f f e c t i s given i n Figure 24 (adapted from [Connor 2-(78)]) which shows non-reactive q u a s i c l a s s i c a l t r a j e c t o r i e s at various c o l l i s i o n energies on the mass weighted muonium potential energy surface OC with F 0 i n i t i a l l y i n the v = 0 state. The plot at =1.6 A trans kcal/mole, which i s greater than the physical barrier height, but less than the q u a s i c l a s s i c a l threshold, shows c o l l i s i o n s at a l l v i b r a t i o n a l phases of F^ to be not only non-reactive, but also e l a s t i c since the v i b r a t i o n a l frequency of F 2 i s not altered by the c o l l i s i o n s . The other plots are for values of OC OC E ^ > E „ and show ranges of v i b r a t i o n a l phase for which the trans T ^ ^ c o l l i s i o n s are non-reactive; i t may be noted that some of the non-reactive c o l l i s i o n s are i n e l a s t i c , p a r t i c u l a r l y at high OC values of ET; . A l l four plots of Figure 24 show the quasi-t3T3.ll S c l a s s i c a l non-reactivity of the Mu + F 2 c o l l i s i o n s at moderately high energies to be due to the bottleneck e f f e c t . Figure 21 indicates the importance of t h i s e f f e c t i n the r e l a t i v e l y slow r i s e of the q u a s i c l a s s i c a l P^ curves from 0.5 to 1 for the Mu -147-FIGURE 24: Non-reactive q u a s i c l a s s i c a l t r a j e c t o r i e s for the reaction: Mu + F2(v=0) ->- MuF + F on the mass weighted LEPS poten t i a l energy surface of [Jonathan (72)], adapted from [Connor 2-(78)]. In the notation of the text, E^ = E§ c . The q u a s i c l a s s i c a l threshold t trans ^ energy for the reaction i s 1.8 0 kcal/mole. -148-reaction, p a r t i c u l a r l y for the F_(v=l) c o l l i s i o n s for which P t £ s shows some structure. The o r i g i n of t h i s structure i s non-reactive back r e f l e c t i o n of the representative point o f f the strongly repulsive wall of the product v a l l e y , as discussed i n Chapter I I I , p. 78. The QCT calculations of Connor 2-(79) confirm that the onset of t h i s e f f e c t occurs at lower c o l l i s i o n energies for Mu than for the other H isotopes due to the extreme contraction of the mass weighted product v a l l e y for the l i g h t e r isotope. For the v=0 reaction, the onset of wall r e f l e c t i o n OC occurs at Er; = 7 and 4 0 kcal/mole for Mu and H respectively. *cr an s OC For Mu + F_(v=l) t wall r e f l e c t i o n begins at =2 kcal/mole, 2 trans thereby competing with bottleneck r e f l e c t i o n and giving r i s e to the observed structure, s Arrows i n Figure 21 indicate the v i b r a t i o n a l l y adiabatic VA barriers (see Chapter I I I , Section C) E Q = 2.28, 2.20, 2.17, and VA 2.16 kcal/mole and E * = 2.15, 1.91, 1.84, and 1.80 kcal/mole for Mu, H, D, and T respectively. In discussing VA i n the context of an early b a r r i e r , i t was pointed out that when the VA VA assumption holds, E v gives good " f i r s t guess" values for the q u a s i c l a s s i c a l threshold energies. Figure 21 shows t h i s to be an excellent approximation for the Y + reactions, with the exceptions of the muonium reaction and the fact that E V A ( Y ) have v OC the opposite ordering to E ^ ( Y ). In other words, the VA assumption, which has general v a l i d i t y for these reactions, i s better for the heavier H isotopes than for the l i g h t e r ones. This may also be understood i n terms of the bottleneck. Besides r e f l e c t i n g representative points non-reactively (the normal bottleneck e f f e c t ) , t h i s c o n s t r i c t i o n i n the saddle point region -149-promotes v i b r a t i o n a l non-odiabaticity- by presenting a potential surface geometry that greatly perturbs the q u a s i c l a s s i c a l t r a j e c t o r i e s . For a r e s t r i c t e d range of v i b r a t i o n a l phases, the conversion of v i b r a t i o n a l energy of the F^ molecule to t r a n s l a t i o n a l energy of the representative point may help propel the system to reaction, thereby reducing the quasi-c l a s s i c a l threshold energy. Evidently, the sharper bottlenecks of the l i g h t e r H isotopes cause greater v i b r a t i o n a l - t r a n s l a t i o n a l energy transfer since they cause a more dramatic perturbation i n the q u a s i c l a s s i c a l trajectory. Expressed i n the jargon of molecular dynamics, the bottleneck encounter for the l i g h t isotope takes place i n the "sudden" regime, while the heavy isotope encounter i s i n the "adiabatic" regime [Levine (74)]. The foregoing discussion of the general v a l i d i t y of the v i b r a t i o n a l adiabaticity assumption suggests that simple TST calculations of the Y + F^ reaction rates using a v i b r a t i o n a l l y adiabatic b a r r i e r should provide f a i r l y accurate estimates of the q u a s i c l a s s i c a l reaction rates, with the possible exception of the muonium reaction. Indeed, Connor et a l . l-(79) have found t h i s to be the case. Tables VIII - X show the VA-TST rate constants, rate constant r a t i o s , and ac t i v a t i o n energies for the H, D, and T reactions to be within 5% of the q u a s i c l a s s i c a l r e s u l t s i n most cases; for muonium, the somewhat less spectacular agreement i s t y p i c a l l y i n the 10 - 2 0% range, except for the case of the F^(v = 1) rate constant at 300K which d i f f e r s by about 40%. Clearly, the e a s i l y calculated VA-TST rate constants are s u f f i c i e n t l y accurate, i n general, to be used as substitutes for the much more laborious q u a s i c l a s s i c a l rate constant c a l c u l a t i o n s . -150-Thus, VA-TST may be used to economically optimize pot e n t i a l energy surfaces, not only for t h i s reaction, but also for reactions of the same general type ( i . e . exothermic, light-heavy-heavy atom reactions with early b a r r i e r s that are dominated by the c o l l i n e a r reaction geometry). Figure 25 (adapted from [Connor 1-(.79),]) displays the TST, QCT, and QMT rate constants as Arrhenius plots (for H, D, and T, the TST r e s u l t s are e s s e n t i a l l y coincident with the QCT r e s u l t s and therefore are not shown). As expected, the quantum Arrhenius plots show noticeable curvature due to tunnelling, but i t should be noted that the q u a s i c l a s s i c a l plots are also weakly curved. Although the curvature i n the quantum Arrhenius plot for the Mu reaction i s s i g n i f i c a n t , i t i s not dramatic; i f the t h e o r e t i c a l plot proves to be phy s i c a l l y accurate, then the experimental demonstration of the Arrhenius plot curvature, even for the case of the Mu reaction, w i l l require that the experi-ment be conducted over a wide temperature range, ^2 00 - 600K [Jakubetz (7 9)]. Having noted the r e l a t i v e success of c o l l i n e a r v i b r a t i o n a l l y adiabatic TST calculations i n reproducing the q u a s i c l a s s i c a l rate constants for the Y + Y^ reaction and having noted the s t r i k i n g resemblance between the quantum mechanical reaction p r o b a b i l i t y curves of Figure 21 with the one dimensional tunnelling trans-mission c o e f f i c i e n t s of Figure 18, Jakubetz l-(78),(79) i n v e s t i -gated the application of one dimensional tunnelling corrections to VA-TST calculations for the reactions: Y + and Y + Cl,,. Three tunnelling corrections were investigated: the Wigner correction, and corrections for the truncated B e l l parabola and 1000 o T / K o If) o o r O I I I 1 1 1 1 1 Quasiclassical (TST ) •-— -2 3 1 0 0 0 K / T 10 \> 8 O £ 6 000 I o o in T / K a o r O I I I 1 ^ ^ ^ ^ i i 1 I Quasic lassical Mu (TST ) -T ^ ^ ^ • ^ """^ ^ ^ ^ ^ - - - ^ ^ ^ ^ * * ^ ^ ^ ^ ^ ^ ^ * ^ ~ - -1 0 0 0 K / T 25: Arrhenius plots for the c o l l i n e a r quantum, q u a s i c l a s s i c a l , and t r a n s i t i o n state theory rate constants for the reaction: Y + F 2 + Y F + F , Y=Mu, H, D, and T, adapted from [Connor l-(79). The TST results for the H, D, and T isotopes are es s e n t i a l l y coincident with the q u a s i c l a s s i c a l r e s u l t s , and therefore are not i l l u s t r a t e d . -152-the unsymmetrical Eckart b a r r i e r s (see Chapter I I I , Section F). Figure 26 (adapted from [Jakubetz (79)]) compares the (v = 0) t o t a l reaction p r o b a b i l i t i e s r e s u l t i n g from the various tunnel-l i n g corrections with the exact quantum mechanical re s u l t s of Connor et aJL. and with uncorrected TST r e s u l t s for the Y + F^ reactions. In a l l cases, the Eckart barrier-TST curves are i n excellent agreement with the exact quantum re s u l t s - the agree-ment i s almost perfect for the D and T reactions. The Arrhenius plots of Figure 27 (adapted from [Jakubetz (79)]) show that the B e l l and Eckart corrections provide e s s e n t i a l l y the same excel-lent agreement with the quantum r e s u l t s for the H, D, and T reactions; indeed, even the very much simpler Wigner correction (not shown in Figure 27) provides good " f i r s t guess" approx-imations to the quantum r e s u l t s for these reactions. However, Figure 27 also shows that t h i s i s not the case for the muonium reaction which i s only well described by the Eckart correction. Jakubetz has pointed out that the f a i l u r e of the B e l l correction for Mu i s due to the u n r e a l i s t i c truncation of the parabolic bar r i e r which re s u l t s i n an over estimate of the low c o l l i s i o n energy reaction p r o b a b i l i t i e s (c.f. Figure 26). The general success of these tunnelling corrections can be largely a t t r i -buted to the early ba r r i e r location in these reactions as discussed i n Chapter II I , p. 78. Figure 26(d) also shows an Eckart f i t to the "conservation of v i b r a t i o n a l energy" (CVE) Cl b a r r i e r , which i s just the c l a s s i c a l barrier height, E^ , as defined on page 83; the f a i l u r e of t h i s b a r r i e r supports the assumption of v i b r a t i o n a l a d i a b a t i c i t y . The l a s t point of discussion on the Y + F 9 reaction i s FIGURE 26: Comparison of quantum and t r a n s i t i o n state theory reaction p r o b a b i l i t i e s for the reaction: Y + F 0 -> YF + F, Y = Mu, H, D, or T, adapted from [Jakubetz (79)]; quantum ( -) [Connor 1- (78)]; tunnelling corrected Eckart b a r r i e r VA-TST ( ), tunnelling corrected B e l l barrier • VA-TST (•••), uncorrected VA-TST' (•-•-); plo t (d) also shows tunnelling corrected Eckart barrier CVE-TST (•••)• I n t n e notation of the text, E^ = E g C t trans. 1.0 -1 r " (a) Mu + F 2(v=0) — M u F + F E t / k J mo t 12 E | / k J m o r 1 Mu+F2(v)-*MuF + F v--0 2 3 1 0 0 0 K / T 27: Comparison of quantum and t r a n s i t i o n state theory Arrhenius plots for c o l l i n e a r Mu + F 2(v=0,l) (left) and II + F 2 (v=0) and T + F 2 (v=0) ( r i g h t ) , adapted from [Jakubetz (79)]. Results shown are exact quantum (Q) [Connor l - ( 7 8 ) ] , Eckart VA-TST (E), B e l l VA-TST (B), and uncorrected VA-TST (CL). For the Mu reaction, kQ and k^ coincide over the whole temperature range, while for the H and T reactions, k°j and k^ coincide over the whole range. -155-the product v i b r a t i o n a l state d i s t r i b u t i o n . For the Y + F 2 reaction, Figure 28 (adapted from [Connor l-(78)]) plots the calculated r e l a t i v e population d i s t r i b u t i o n of product v i b r a t i o n a l states, normalized to the most populated state, s 1 , as a function of the f r a c t i o n of product v i b r a t i o n a l energy,'f , = E^/Dg, where D n i s the d i s s o c i a t i o n energy of YF and E , i s the energy u s of the s 1 l e v e l , both measured r e l a t i v e to s' = 0. The Figure OC shows the re s u l t s at E^ =2.45 kcal/mole, but the v i b r a t i o n a l trans d i s t r i b u t i o n s are r e l a t i v e l y i n s e n s i t i v e to the c o l l i s i o n energy [Connor l - ( 7 8 ) ] . The most populated l e v e l has the values s 1 = 1, 6, 9, and 12 for Mu, H, D, and T respectively; the H atom r e s u l t i s i n agreement with the infared chemilluminescence re s u l t s of Jonathan et al . (72) and Polanyi et al_. (72) . From these c a l c u l a t i o n s , the average f r a c t i o n of one dimensional t t product v i b r a t i o n a l energy, defined by < C f 0> = £ Ps'«-0 ^s l / / P0 ' s' i s 0.40, 0.58, 0.64, and 0.68 for Mu, H, D, and T respectively; again, the H atom res u l t s are i n agreement with the corrected [Jakubetz 2-(78)] experimental values of 0.55 due to Jonathan et a l . and 0.62 due to Polanyi et a l . The order Mu<H<D<T for < C f s V > i s i n q u a l i t a t i v e agreement with the l i g h t atom anomaly (Chapter I I I , p. 81), i n which less reaction exoergicity i s transformed into product v i b r a t i o n as the mass of the attacking atom decreases. Fischer and Venzl (78) derived an analytic expression that succeeds well i n ca l c u l a t i n g the product v i b r a t i o n a l energy d i s t r i b u t i o n for exothermic light-heavy-heavy atom reactions and which i s sensitive to the inte r a c t i o n length (saddle point location) and the r e l a t i v e attractiveness of the potential energy surface (see Chapter I I I , p. 80). This expression may be used -156-FIGURE 28: Collin e a r quantum mechanical r e l a t i v e population d i s t r i b u t i o n of product v i b r a t i o n a l states normalized to the most populated state s 1 at E =2.45 kc a l / mole for the Y + F 2(v=0) ^ YF + F reactions, Y = Mu, H, D, and T, adapted from [Connor l - ( 7 8 ) ] . -157-to economically narrow the parameter range for LEPS surfaces by f i t t i n g the experimental r e s u l t s for the v i b r a t i o n a l d i s t r i b u t i o n . Korsch (7 8) derived a s i m i l a r , but simpler, expression which only requires a hand calculator to compute. Although i t i s not yet possible to experimentally measure product v i b r a t i o n a l energy d i s t r i b u t i o n s for Mu reactions, the present work has i n d i r e c t l y aided i n the development of the computational tools described above since i t prompted the exact quantum mechanical c a l c u l a -tions of these d i s t r i b u t i o n s which were then used as a c r i t i c a l t est of the analytic expressions subsequently developed [Jakubetz 1-(78)]. B Mu + C l 2 -> MuCl + Cl The MSR relaxation rates at various C l 2 concentrations, measured between 29 5 and 38IK, are l i s t e d i n Table XI. To i l l u s t r a t e the influence of temperature on the reaction rate, Figure 2 9 plots the MSR relaxation data obtained at 2 95 and 2 3 84K. The bimolecular rate constants, determined by x minimum f i t s of the relaxation rate data to equation 11(3), are also l i s t e d i n the Table and i l l u s t r a t e d i n the Arrhenius plot of 2 Figure 30. The x minimum f i t of these data to the logarithmic Arrhenius expression (equation (12)) y i e l d s : log 1 Qk(1/mole-s) = (11.72 + 0.14) - (300 + 50/T), (la) with k(300K) = (5.29 + 0.14) x 1 0 1 0 1/mole-s and E = (1.36 + — a — 0.21) kcal/mole. As indicated i n the Table, the moderator for three of the rate constant measurements i s argon, while N 2 i s the moderator for the 37OK measurements. Other than the MSR signal enhancement due to N 9 moderator (see Figure 8), no - 1 5 8 -TABLE XI: MSR RELAXATION RATES FOR THE REACTION: Mu + C l 2 MuCl + Cl Bimolecular Temperature (K) Rate Constant Relaxation [Moderator gas] k ( 1 0 1 0 M - 1 S " 1 ) (10~ 5 M) Rate XCys" 1) 1" 2 9 5 + 2 5 . 1 7 + 0 . 2 4 * * [Argon] ( 5 . 4 5 + 0 . 1 9 ) 0 . 0 0 . 1 3 + 0 . 0 2 * 0 . 6 6 + 0 . 1 2 * 1 . 4 5 + 0 . 2 2 2.36 + 0.17* 2.49 + 0.27* 3. 07 + 0.17 (4.73 + 0.56) 0.0 0.29 + 0.07** 1.09 + 0.15** 2.32 + 0.25** 3.03 + 1.11 336 + 2 6.83 + 0.59 [Argon] 370 + 3 7.27 + 0.72 [Nitrogen] 381 + 2 9.22 + 0.65 [Argon] .  0 . 98 + 0 . 04 1 . 58 + 0 . 05 1 . 80 + 0 . 07 2 . 4 5 + 0 . 08 3 . 12 + 0 . 07 3 . 71 + 0 . 11 4 . 90 + 0 . 11 6 . 85 + 0 . 1 5   1 . 89 + 0 . 05 3 . 92 + 0 . 10 7 . 4 0 + o . 17 0 . 0 2 . 2 3 + 0 . 07 3 . 84 + 0 . 1 1 0 . 0 2 . 19 + 0 . 07 3 . 40 + 0 . 10 4 . 47 + 0 . 13 5 . 8 1 + 0 . 16 5 . 94 + 0 . 16 6 . 79 + 0 . 19 0 . 0 0 . 74 + 0 . 05 1 . 94 + 0 . 07 6 . 69 + 0 . 18 1.09 + 0.03* ' * r 1 . 90 + 0 . 03! 1 . 6 5 + 0 . 1 7 * 0 . 42 0 . 02 1 . 82 + 0 . 19 3 . 40 + 0 . 40 0 . 00 + 0 . 23 1 . 58 + 0 . 43 3 . 62 + 0 . 77 3 . 64 + 0 . 92 5 . 23 + 1 . 05 3 . 83 + 1 . 7 5 4 . 09 + 0 . 91 0 . 47 + 0 . 03 1 . 07 + 0 . 09 2 . 2 1 + 0 . 27 7 . 2 5 + 0 . 64 Relaxation rates reported are weighted averages of the l e f t and r i g h t positron telescope histograms. Room temperature data obtained at LBL (1975). Room temperature data obtained at TRIUMF (1976). MSR relaxation rates i n l e f t and r i g h t histograms d i f f e r sys-tematically because of the use of a d i f f e r e n t geometry for each telescope r e s u l t i n g i n d i f f e r e n t X n's for l e f t and r i g h t . Relaxation rates reported are weighted averages of (X - X n ) . -159-0 10 2 0 3 0 4 0 5 0 6 0 7 0 8 0 CL 2 CONCENTRATION (uM) FIGURE 29: The e f f e c t of temperature on the Mu + C I 2 MSR r e l a x a t i o n r a t e s . The l i n e s are x minimum f i t s to the p s e u d o - f i r s t order k i n e t i c expression of equation I I (3) corresponding to k = (5.17 + 0.24) x 1 0 1 0 1/mole-s at 295K (squares and t r i -angles) and k = (9.22 + 0.65) x 1 0 1 0 1/mole-s at 381 K (diamonds). The t r i a n g l e s represent data taken at LBL during 197 5, while the squares and diamonds represent data taken at TRIUMF. The 295 K data represent (X - X ) i n order t o account f o r the d i f f e r e n t . . X ' s obtained at LBL and TRIUMF. -160-1.0 1.5 2.0 2 . 5 3.0 3 . 5 4.0 1 0 0 0 / T E M P ( K " 1 ] FIGURE 30: Experimental Arrhenius plot for the Y + CI2 reactions, Y = Mu, H, D. The Mu data i s on the top l i n e (this work). The H data i s due to Stedman(70)(open octagon), Dodonov(70)(squares), Ambidge(76) (triangles) , Wagner(76) ( + ) , and Bemand(77)(x). The D datum i s due to Stedman (7 0)(diamond). The error bars on the Mu data are s t a t i s t i c a l only. -161-moderator e f f e c t s are detectable i n the rate constant measure-2 ments. A x minimum f i t to equation (12) using only the Ar moderator data y i e l d s log^k(1/mole-s) = (11.78 +0.14) -(320 + 50/T), consistent with the r e s u l t reported above. As discussed i n [Fleming l - ( 7 7 ) ] , the 295 K reaction rate was measured both at LBL and TRIUMF using the same method but completely d i f f e r e n t equipment, y i e l d i n g rate constants of (5.4 + 0.2) and (4.7 + 0.6) x 1 0 1 0 1/mole-s respectively. To date, t h i s i s the only measurement of the r e p r o d u c i b i l i t y of the MSR method i n determining gas phase Mu reaction rates. The H + + HC1 + C l reaction rate has been measured d i r e c t l y by several authors i n the past 10 years; t h e i r r e s u l t s are compared with the Mu rates i n Table XII and Figure 30. From the Table, the a c t i v a t i o n energies determined by the various investigators are i n reasonable agreement; however, the rate constants and preexponential factors show some serious discrep-ancies. The two most recent r e s u l t s , due to Bemand et a i . (77) and Wagner et a l . (76), using Lyman-a fluorescence, are e s s e n t i a l l y i d e n t i c a l to each other. The recent ESR measurements of Ambidge et a l . (76), however, give a value of k(300K) a factor of three smaller and a value of A a factor of two smaller that the Lyman-a fluorescence r e s u l t s . Bemand et a i . and Wagner et a l . discuss the possible o r i g i n s of the various experimental discrepancies and suggest that the ESR r e s u l t s of Ambidge et a l . may be i n error due to t h e i r use of a nearly equal H/C12 stoichiometry which may lead to interferences due to C l atom-wall reactions. In contrast, the Lyman-a fluorescence experiments used a Cl 2/H r a t i o varying from 5 - 1 5 . The less s e n s i t i v e , e a r l i e r mass spectrometrie TABLE X I I : EXPERIMENTAL RATE PARAMETERS FOR THE REACTION: Y + C]_ 2 -> YC1 + C l , Y = Mu, H, D H Isotope Method Temp(K) (kcal/mole) l o g 1 Q A (M _ 1 s - 1 ) k(300K) ,, A10 -1 -1. (10 M s ) Mu kH(D) (300K) r e f e r e n c e Mu H H H H H D MSR t ms t ms ESR L f § L f § ms f 295 - 384 294 - 565 300 292 - 434 252 - 458 300 - 750 300 1.36 + 0.21 1.8 + 0.3 1.4 + 0.2 1.20 + 0.14 1.14 + 0.17 11.72 + 0.14 11.57 + 0.04 10.66 + 0.11 10.94 + 0.08 10.93 + 0.07 5.3 + 0.1 1.8 + 0.6 2.1 + 0.7 0.42 + 0.04 ;i.2 + 0.1 1.3 + 0.1 0.72 + 0.30 2 . 9 + 1.0 2.5 + 0.8 13 + 1.2 4.4 + 0.4 4.1 + 0.3 7.4 + 3.1 present work A l b r i g h t ( 6 9 ) Stedman (7 0) Ambidge(7 6) Wagner(7 6) Bemand(77) Stedman(7 0) mass sp e c t r o m e t r i c f a s t flow Lyman-a f l u o r e s c e n c e i I -163-measurements agree with the Lyman-a fluorescence r e s u l t s within the experimental error. I t may also be noted from Table XII that Stedman et a_l. (70) measured the D + C l 2 reaction rate at 300K mass spectrometrically. Despite the v a r i a t i o n i n the measured H atom reaction rate data, two points c l e a r l y emerge from t h e i r comparison with the Mu reaction rate parameters: (1) i n a l l cases, the apparent activa-t i o n energies are the same within the experimental uncertainties, and (2) compared with the mass spectrometric H atom reaction rate constants at 300K, there i s no rate enhancement for the Mu reaction beyond the temperature independent mass factor of 2.9 (Chapter I I I , p 113), or, i f compared with the Lyman-a fluores-cence re s u l t s at 300K, the rate constant i s enhanced by a factor of only 1.48 + 0.14 beyond the factor of 2.9. (As previously mentioned, the anomalously large Mu:H rate constant r a t i o due to Ambidge et a l . (76) appears to be i n e r r o r ) . Furthermore, the rate constant r a t i o s , Mu:H:D at 3 00K, are 2.7 + 0.9:1.0:0.34 + 0.18 using the mass spectrometric H atom rate constants, or 4.3 + 0.4:1.0:0.58 + 0.24 using the Lyman-a fluorescence H atom rate constants; these may be compared with the temperature independent mass factors of 2.9:1.0:0.72. Clear l y , there i s no experimental evidence to indicate that Mu exhibits a substantial tunnelling advantage over the other H isotopes when reacting with C l 2 , i n sharp contrast to the Y + F 2 reaction. One possible explanation for t h i s r e s u l t i s that none of the isotopic versions of the reaction display s i g n i f i c a n t tunnelling at 300K. This p o s s i b i l i t y i s supported by the Lyman-a fluorescence Mu:H:D rate constant r a t i o s which -164-are very close to the c l a s s i c a l mass factor r a t i o s . This apparent c l a s s i c a l behaviour can be understood i n terms of the experimental indications of a r e l a t i v e l y low q u a s i c l a s s i c a l threshold energy. Although there i s no simple analytic r e l a t i o n s h i p between ac t i v a t i o n energies and q u a s i c l a s s i c a l threshold energies, in the absence of strong dynamical c o n t r i -butions, such as tunnelling or the bottleneck e f f e c t , the a c t i v a t i o n energy i s very nearly equal to the q u a s i c l a s s i c a l threshold energy [Levine (74)]. Thus, a threshold energy of about 1.4 kcal/mole i s indicated for the Y + C l 2 reactions, which may be compared with the value of 2 kcal/mole for the Y + F„ reactions and with k_T(300K) = 0.6 kcal/mole. Clearly, z a the Y + C l 2 reactions are much more capable of reacting c l a s s i c -a l l y at 300K than the Y + F 2 reactions, thereby minimizing the importance of tunnelling. Indeed, the above hypothesis i s confirmed by recent preliminary QCT ca l c u l a t i o n s , performed on a newly optimized LEPS surface, which gives q u a s i c l a s s i c a l threshold energies for the c o l l i n e a r Mu and H reactions with Cl 2(v=0) of 1.2 and 1.4 kcal/mole respectively [Lagana (79)]. Preliminary quantum calculations indicate a large reduction i n tunnelling for the Y + C l 2 reactions compared with the Y + F 2 reactions at 300K [Lagana (7 9)]. Topologically, the Y + C l 2 LEPS surface used i n these calculations c l o s e l y resembles the Y + F 2 LEPS surface of Jonathan (72) with t h e i r saddle points placed i n almost exactly the same r e l a t i v e positions. Of course, the Y + C l 2 surface has a deeper reactant v a l l e y and a shallower product va l l e y (see Table V) than the Y + F„ reaction. The most -165-s i g n i f i c a n t difference between these Y + C l 2 and Y + surfaces i s t h e i r c l a s s i c a l b a r r i e r heights: ^1.5 kcal/mole for the Y + C l 2 surface compared with 2.35 kcal/mole for the Y + F 2 surface [Jakubetz l-(78). Again, the reduced tunnelling enhancement for the Y + C l 2 reactions at 300K i s consistent with t h i s low reaction b a r r i e r which allows a large f r a c t i o n of t r a j e c t o r i e s to proceed to reaction c l a s s i c a l l y . It should be noted that t h i s smaller Y + C l 2 b a r r i e r also explains the observation that the Y + C l 2 reactions proceed faster than the corresponding Y + F 2 reactions. The preliminary calculations of Lagana (79) also indicate that the onset of non-reactive back r e f l e c t i o n of the represen-OC t a t i v e points off the product v a l l e y wall occurs at < 2.0 kcal/mole for the Mu + Cl 2(v=0) reaction, whereas for the other OC H isotopes i t occurs at >> 3.5 kcal/mole. This i s an c trans int e r e s t i n g contrast to the Y + F 2(v=0) reaction where t h i s QC phenomenon does not occur u n t i l E7_ -7 and 40 kcal/mole for trans Mu and H respectively - c o l l i s i o n energies which are c e r t a i n l y unimportant even at 1000K. The e f f e c t of wall r e f l e c t i o n i s to reduce the reaction p r o b a b i l i t y , P*", from unity. Calcula-tions are presently being undertaken to determine how much the Mu + C l 2 reaction rate i s reduced due to wall r e f l e c t i o n [Lagana (79)]. Q u a l i t a t i v e l y , i t i s clear that for t h i s reaction, wall r e f l e c t i o n w i l l tend to o f f s e t tunnelling more and more with increasing temperature. Thus, i t appears that the value of k„„ /kTT at 300K i s less for the Cl„ reaction than Mu H 2 for the F 2 reaction, not only because of the reduced importance of tunnelling due to the smaller reaction b a r r i e r , but also because some of the tunnelling that does occur i s cancelled due to c l a s s i c a l wall r e f l e c t i o n . There are other interesting consequences of t h i s wall r e f l e c t i o n phenomenon. As the temperature i s raised and the centroid of the Boltzmann d i s -t r i b u t i o n s h i f t s toward energies where wall r e f l e c t i o n domin-ates, the rate of increase i n k (Mu + C l 2 ) w i l l f a l l and even-t u a l l y , at s u f f i c i e n t l y high temperatures (perhaps 1000K), k(Mu + C l ^ ) i t s e l f w i l l actually decrease. One might there-fore expect the act i v a t i o n energy of t h i s reaction to pass through a maximum as i t passes from the low temperature tunnel-ling-dominated region to the high temperature wall r e f l e c t i o n -dominated region. It i s int e r e s t i n g to speculate on the reason for the dramatic reduction i n the minimum c o l l i s i o n energy for the onset of wall r e f l e c t i o n i n going from to Cl,,. According to the Connor mass weighting scheme (p 73), changing the mass of X 2 from F 2 to C l 2 further contracts the product v a l l e y by about 26% for each H isotope. While t h i s greater contraction i n the Y + C l 2 product v a l l e y undoubtably enhances wall r e f l e c t i o n quite s i g n i f i c a n t l y , i t seems un l i k e l y that t h i s alone accounts for the reduction i n the wall r e f l e c t i o n "threshold", from 7 kcal/mole for Mu + to 2 kcal/mole for Mu + C l 2 , for example. It i s l i k e l y that the exothermicity of the reactions also plays an important r o l e . For the Y + F 2 reaction, the bottom of the product v a l l e y l i e s about 106 kcal/mole below the c l a s s i c a l b a r r i e r , whereas for the Y + C l 2 reaction the product v a l l e y i s only about 50 kcal/mole below the c l a s s i c a l b a r r i e r . Since the saddle points for the two reactions l i e -167-at about the same positions r e l a t i v e to the reactant and product v a l l e y s , i t i s clear that the force tending to make the representative point "round the corner" and "bobsled" down the product v a l l e y i s greater on the steeper Y + surface than on the more gently sloped Y + Cl^ surface. F i n a l l y , since the angle of r e f l e c t i o n i s equal to the angle of incidence, i t i s l i k e l y that the fact, that the skewing angle for the mass weighted Mu + C l 2 surface i s about 1 degree more than for the Y + F^ surface also makes a minor contribution i n reducing the Mu + C I 2 wall r e f l e c t i o n threshold. Before closing t h i s Section, i t should be remarked that the role played by the experimental Mu reaction rate measure-ments i n the development of the t h e o r e t i c a l calculations of Connor et a l . i s quite d i f f e r e n t for the F2 and C l 2 reactions. In the former case, the reasonably accurate potential energy surface due to Jonathan (72) , optimized for the H + F2 reaction, existed before calculations were performed for the Mu + F2 reaction. Thus, i n t h i s case, the experimental Mu reaction rate data provided a test of the qual i t y of the t h e o r e t i c a l predictions based on t h i s surface. The f i r s t c a l c u l a t i o n s , which were c o l l i n e a r QMT c a l c u l a t i o n s , gave two main r e s u l t s : (1) the Mu + F2 reaction i s dominated by quantum tunnelling, and (2) despite the facts that the Mu + F^ reaction rates were calculated using a " l i n e of no return" method (p 139) and the calculations were only c o l l i n e a r , the QMT r e s u l t s seemed to predict the r a t i o k M u/k^ and the act i v a t i o n energies quite accurately. Next, QCT calculations showed that (1) the c l a s s i c a l (high temperature) Y + F„ reaction rates are governed -168-by the bottleneck e f f e c t and (2) wall r e f l e c t i o n i s unimportant up to 1000K, thereby explaining the success of the " l i n e of no return" method. F i n a l l y , i t was discovered that simple Eckart tunnelling corrected VA-TST worked well for the Y + F 2 reactions, and, more importantly, the success of t h i s method could be understood i n terms of the very early reaction b a r r i e r and the favored c o l l i n e a r geometry. The natural next t h e o r e t i c a l step was to make similar calculations for the Y + C l 2 reactions to determine i f the conclusions previously drawn could be generalized for l i g h t -heavy-heavy atom reactions with early b a r r i e r s . Unfortunately, unlike the Jonathan surface for the Y + F 2 reactions, no accurate experimentally optimized p o t e n t i a l energy surface existed for the Y + C l 2 reactions. Two very similar LEPS surfaces due to Kuntz et a l . (66) and Baer (74) have been used i n several investigations of the H + C l 2 reaction (eg. [Wilkins (75), [Essen (76)], [Truhlar (78)]), but the primary aim of these studies has been to compare computational methods [Truhlar (79)], such as TST versus QMT, rather than to model experimental r e s u l t s . Jakubetz (79) used the Kuntz surface to t e s t i f tunnelling corrected VA-TST gives comparable estimates for. the QMT calculations for the H + C l 2 reaction and for the H + F 2 reaction, or i f there are some unforeseen kinematic e f f e c t s due to the change i n X 2 mass from F 2 to C l 2 . Arrhenius plots of the tunnelling corrected VA-TST rate constants for the Y + C l 2 reaction, calculated on the Kuntz surface, are compared with those for the Y + F 2 reaction, calculated on the Jonathan surface, i n Figure 31. As expected for the H + Cl„ reaction, i 1 % r FIGURE 31: Arrhenius plots for c o l l i n e a r Y + F 2 and Y + C l 2 , for Boltzmann d i s t r i b u t e d reactants, calculated by tunnelling corrected Eckart b a r r i e r VA-TST, adapted from Jakubetz (79). LEPS surface due to Jonathan (72) used for Y + F 2 , LEPS surface due to Kuntz (66) used for Y + Cl„. -170-tunnelling corrected VA-TST i s as good as QMT. However, these calculations do rather poorly i n reproducing the ex-perimental r e s u l t s . For example, the predicted rate constant r a t i o , kMu+Cl ^H+Cl ' l s ^'^ c o m P a r e < i with the experimental value of 4.3 + 0.4 at 3 0OK. Furthermore, at 30OK the Mu + C±2 and H + C l 2 a c t i v a t i o n energies are calculated to be 1.4 and 2.4 kcal/mole respectively, compared with the experimental values of about 1.4 kcal/mole for both. In f a c t / t h e s e predictions are quite similar to those for the Y + reaction. This r e s u l t i s not unexpected since the Kuntz surface, which i s known to be inaccurate, i s very similar to the Jonathan surface, with c l a s s i c a l barriers of 2.42 and 2.35 kcal/mole for the Kuntz and Jonathan surfaces respectively. Besides underlining the deficiency of the Kuntz surface, the calculations of Jakubetz suggest that tunnelling corrected VA-TST i s also e s s e n t i a l l y applicable to the Y + C l ^ reactions, where Y = H, D, and T; for Y = Mu, t h i s inference could not be made since no quantum calculations were available for comparison. From t h i s point, the calculations on the Y + C l ^ reactions have been proceeding i n the reverse order to the Y + F^ reactions. F i r s t , Jakubetz used tunnelling corrected VA-TST to "tune" the c a l c u l a t i o n of an optimized Y + C l 2 surface. Jakubetz found that by reducing the b a r r i e r height from 2.4 to 1.5 kcal/mole, the rate constant r a t i o , k,„ /kTT , at 3 00K Mu H became 4.1, i n good agreement with the experimental r e s u l t . I t has been noted by Connor l-(78), that a much stronger constraint i s placed on the choice of a potential energy surface by the k- M u/k H r a t i o than by the a c t i v a t i o n energies. The d e t a i l s of the newly optimized Y + C l 2 surface are currently i n press [Connor 3-(79)]. In t h i s way, the experimental rate constant measurements of the Mu + C l 2 reaction have been used to optimize the Y + C l 2 surface, rather than to test the qu a l i t y of the t h e o r e t i c a l predictions. Consequently, i t i s inappropriate to c a l l any of the subsequently calculated values of the Mu + C l 2 reaction rates "predictions." However, as t h i s Section has shown, t h e o r e t i c a l calculations using t h i s new surface are able to explain the o r i g i n s of the experimental res u l t s by such e f f e c t s as the reduction i n tunnelling and increase i n wall r e f l e c t i o n . It may be noted that the preliminary r e s u l t s discussed i n t h i s Section on wall r e f l e c -t i o n suggest that the " l i n e of no return" method may not be applicable to the Mu + C l 2 reaction. Since TST i m p l i c i t l y also makes use of a " l i n e of no return" (the so-called "dividing surface"), TST may also f a i l for muonium at high temperatures. C Mu + Br^ -»• MuBr + Br The MSR relaxation rates at various Br 2 concentrations, measured i n argon moderator at 295K, are l i s t e d i n Table XIII and plotted i n Figure 32. The bimolecular rate constant at 2 295K, determined by x minimum f i t s of the relaxation rate data to equation 11(3), i s [Fleming (76), 2-(77)] k(295K) = (2.4 + 0.3) x 1 0 1 1 1/mole-s. (la) Details of t h i s Mu reaction rate measurement, which was conducted at LBL, are given i n Fleming (76). As shown i n the Figure and Table, two of the MSR relaxation rates are anomal-ously large; these points were taken during a time of known -172-TABLE X I I I : MSR RELAXATION RATES FOR THE REACTION Mu + B r 2 ->• MuBr + B r § [Br 2] R e l a x a t i o n Rate (10~6M) A ( u s ' V 0.0 0.19 + 0.03 0.0 0.17 + 0. 04 1.74 + 0.23 0.72 + 0.12 2.19 + 0.24 0.71 + 0.06 2. 95 + 0.32 0. 98 + 0.10 3.45 + 0.44 1.21 + 0.15 4.13 + 0.45 1.11 + 0.10 5.75 + 0. 61 1.43 + 0.16 7.73 + 0.81 2.22 + 0.25 1. 02 + 0. 09 1. 46 + 0.20 4.25 + 0.55 3. 47 + 0.58 data from [Fleming (76)]. r e l a x a t i o n r a t e s reported are weighted averages of the l e f t and r i g h t p o s i t r o n telescope histograms. p o i n t s taken w i t h poor magnetic f i e l d r e g u l a t i o n (see text) -173-5 i 1 i i r o 4-B R O M I N E C O N C E N T R A T I O N {fx. M O L E S / L I T E R ) FIGURE 32: MSR relaxation rates as a function of Br~ concentra-t i o n i n argon moderator at 295K; data taken at LBL during 1975 [Fleming (76)]. The high points are discussed i n the text. - 1 7 4 -poor magnetic f i e l d regulation due to an unstable power supply. This serves to i l l u s t r a t e the fact that magnetic f i e l d inhomogeneities contribute to the background relaxation rate, A Q , i n pure inert moderator gas. To ensure that such e f f e c t s do not int e r f e r e with the Mu reaction rate measure-ments, A Q i s p e r i o d i c a l l y checked during the experiments. It i s also noted that the B r 2 concentrations are not as prec i s e l y determined as the concentrations of the other gases studied in t h i s thesis; these were determined from B r 2 ( 1 ) vapour pressures, for which reported values vary by up to 3 0 % [Nesmeinov ( 6 3 ) ] . Insofar as i t i s known, the bimoleeular thermal rate constant for the H + B r 2 reaction has never been d i r e c t l y measured, although i t i s currently being investigated with Lyman-a fluorescence [Clyne ( 7 9 ) ] . From a l i t e r a t u r e survey [Fleming ( 7 6 ) ] , i t has been estimated that k U ( 2 9 5 ) = ( 2 . 2 + 1 . 5 ) x 1 0 1 0 1/mole-s and k D ( 2 9 5 ) = ( 6 . 1 + 3 . 2 ) x 1 0 9 1/mole-s, which gives rate constant r a t i o s at 2 9 5 K of Mu:H:D = 1 1 + 8 : 1 . 0 : 0 . 3 + 0 . 2 . Perhaps more r e l i a b l e estimates of these reaction rate constants can be obtained by combining recent d i r e c t ESR measurements of the rate constants for H + HBr -> H 2 + Br and D + DBr -> D 2 + Br [Endo ( 7 6 ) , Takacs ( 7 3 ) ] with e a r l i e r photolysis measurements of the r a t i o s of these rate constants to those for the H + B r 2 and D + B r 2 reactions [Fass ( 7 0 ) , ( 7 2 ) ] . For H at 2 9 5 K , k(H + HBr) = ( 2 . 2 + 0 . 2 ) x 1 0 9 1/mole-s [Endo ( 7 6 ) ] and k (II + Br2)/k (H + HBr) = 2 2 . 7 + 2.3 [Fass ( 7 0 ) ] , which gives k(H + Br 2) = ( 5 . 1 + 0 . 6 ) x 1 0 1 0 1/mole-s. For D at 2 9 5 K , k(D + DBr) = ( 8 . 0 + 1 . 0 ) x 1 0 8 1/mole-s [Endo ( 7 6 ) ] and k(D + Br 2)/k(D + DBr) = 5 8 + 1 . 7 [Fass (72)] which gives k(D + Br 2) = (4.6 + 0.6) x 10 1/mole-s. Using these r e s u l t s , the rate constant r a t i o s at 2 95K are Mu:H:D = 4.7 + 0.8:1.0:0.9 + 0.2 compared with the temperature independent values of 2.9:1.0:0.72. Like the rate constants, the ac t i v a t i o n energies for the H + Br2 and D + B r 2 reactions are not well-known, though they are known to be small [Blais (74)] . Again, the d i r e c t ESR measurements of Endo may be combined with the photolysis r e s u l t s of Fass to give estimates of the H + Br 2 and D + B r 2 activation energies. For H, E (H + HBr) = 2.6 + 0.1 kcal/mole [Endo (7 6)] c l and E (H4 HBr) - E (H + Br„) = 0.8 + 0.3 kcal/mole [Fass (70)] a a z. — which gives E (H + Br„) = 1.8 + 0.4 kcal/mole. S i m i l a r i l y c l A — for D, E (D + DBr) = 1.7 + 0.1 kcal/mole [Endo (76)] and E (D + a — a DBr) - E (D + Br„) = 0.9 + 0.2 kcal/mole [Fass (72)], which 3. Z — gives E (D + Br„) = 0.8 + 0.3 kcal/mole. However, recent molecular beam re s u l t s of Hepburn et a l . (78) suggest E & 1 kcal/mole for both the H and D reactions with B r 2 . Without d i r e c t l y determined rate parameters for the H + Br2 and D + B r 2 reactions and an a c t i v a t i o n energy measure-ment for the Mu + B r 2 reaction, i t i s d i f f i c u l t to speculate on isotope e f f e c t s i n t h i s reaction family. From the estimat-ed' D:H rate constant r a t i o of 0.9 + 0.2, which i s 1.3 + 0.3 times the temperature independent mass factor of 0.72, there appears to be an "inverse" isotope e f f e c t at room temperature, i f , i n fact, there i s any difference at a l l . At c o l l i s i o n energies greater than the c l a s s i c a l b a r r i e r height (estimated to be about 1 kcal/mole [Hepburn (78)]), t h i s e f f e c t seems to be well-established i n the reaction cross section measurements -176-of Hepburn et a l . (78) and i n the trajectory calculations of Malcolme-Lawes (78) and White (73). This has been explained c l a s s i c a l l y [White (73), Hepburn (78)] i n terms of non-reactive back reflection of the representative points off the repulsive wall of the contracted product v a l l e y , which has been discussed for the Mu reactions with F 2 and Cl,,. An equivalent way of pic t u r i n g t h i s e f f e c t without e x p l i c i t l y r e f e r r i n g to poten t i a l surfaces, i s to note that chemical reaction requires momentum transfer between the l i g h t attacking atom and the heavy parting product molecule; but in the case of H + Br,, t h i s has l i t t l e time to occur because the H atom moves much faster than the heavy Br atoms. At a given value of E , the H - Br„ trans 2. c o l l i s i o n i s about /2~ faster than the D - Br 2 c o l l i s i o n , and thus H + B r 2 has a lower reaction cross section. Indeed, Hepburn et a l . (78) found that for E^_ > 1 kcal/mole, the H + * trans — ' Br„ and D + Br cross sections are coincident when plotted as a ^ 2 function of r e l a t i v e c o l l i s i o n v e l o c i t y rather than energy. As discussed i n the preceding Section, the reduction i n the wall r e f l e c t i o n threshold for the H - halogen reaction, from 40 to <1 kcal/mole as X^ changes from F 2 to B r 2 , can probably be attributed to three factors: (1) according to the Connor mass weighting scheme, the product v a l l e y s for the Y + Br 2 reactions are about 50% narrower than for the Y + F 2 reactions, (2) the "down h i l l " part of the poten t i a l surface i s about 106 kcal/mole for Y + F 2 compared with about 45 k c a l / mole for Y + Br 2, and (3) the skewing angle for H + B r 2 i s 85° 'compared with.81° for_the H + F 2 reaction. Also, i f the Mok-Polanyi rel a t i o n s h i p holds (p 76), then the H + B r 2 b a r r i e r -177-i s e a r l i e r than the Y + F^ and Y + Cl^ b a r r i e r s , and thus the slope of the down h i l l part of the surface at the corner i s probably less than i t would be i f the barrier were l a t e r ( i . e . the H + B r 2 surface i s more a t t r a c t i v e , p 81); thus, there i s even less of a tendancy for the representative point to "round the corner." Certainly, extrapolation of the t h e o r e t i c a l predictions of the wall r e f l e c t i o n thresholds for Mu + F^ and Mu + C l 2 to Mu + B r 2 predicts that t h i s e f f e c t w i l l dominate the reaction rate for the l a t t e r system. In fact, one might expect the Mu + Br 2 reaction to be slower than the H + Br 2 reaction at 300K and that the estimated r a t i o : k__ /kTT(300K) i s i n error. How-Mu H ever, i t must be cautioned that when the Boltzmann d i s t r i b u t i o n i s taken into account, the reaction cross sections at c o l l i s i o n energies less than 1 kcal/mole (where there may be no wall r e f l e c t i o n ) have a strong influence on the thermally averaged reaction rates at room temperature. While these low energy cross sections for Y + B r 2 are as yet unknown, they should be much larger than for the F^ or C l 2 reactions because of the very low c l a s s i c a l b a r r i e r and the fact that the c o l l i n e a r reaction geometry does not dominate t h i s reaction [Baybutt (78) Bauer l-(78), B l a i s (74)]. On the other hand, i t cannot be expected that quantum tunnelling greatly enhances the Mu + Br 2 reaction rate at 300K since the b a r r i e r i s so low. As discuss for the Mu + C l 2 reaction, one might predict that the Y + Br 2 reaction apparent ac t i v a t i o n energies pass through a maximum as they change from the low temperature tunnelling region ( i f one exists) to the high temperature wall r e f l e c t i o n region. -178-If i t turns out that the apparent a c t i v a t i o n energy for the Mu + B r 2 reaction i s less than that for H + Br 2 at 300K, then t h i s would not necessarily indicate tunnelling, unlike the case of the F^ reaction. ^ MuCl + H D Mu + HCl •> MuH + C l In Chapter I I I , p 98, i t i s noted that i t i s not yet possible to determine the i n d i v i d u a l reaction rate constants for the exchange and abstraction reaction channels for the Mu + HX reactions by the MSR method - only the t o t a l reaction rate constants are measured. The MSR relaxation rates at various HCl concentrations, measured i n N 2 moderator at 295K, 2 are l i s t e d i n Table XIV and i l l u s t r a t e d i n Figure 33. A x minimum f i t : of the relaxation rate data to equation 11(3) y i e l d s a bimolecular rate constant k(295K) < (3.41 + 0.46) x 10 5 1/mole-s (la) The rate constant for t h i s very slow reaction i s written as an inequality to emphasize the fact that i t only represents an upper l i m i t . There are two reasons why t h i s experiment only gives an upper l i m i t to the rate constant. (1) MSR relaxation rates are known for only two HCl concentrations, one of which i s pure HCl at one atmosphere. Moreover, because of the reduced MSR signal amplitude (Table XIV and Figure 34) for these data and t h e i r small MSR relaxation rates, any systematic errors introduced to the measurements could e a s i l y exceed the s t a t i s t i c a l error i n the rate constants. Thus, the data i s both sparse and inherently unreliable. (2) More importantly, a known, estimable systematic error can account for half of -179-TABLE XIV: MSR RELAXATION RATES FOR THE TOTAL Mu + HC1 REACTION AT 2 95K [HC1] Relaxation Rate Total' (10"2M) A ( y s _ 1 ) A (%) A+(%) (%) -Mu-1—• y -0.0 0.40 + 0.04 12.2 + 0.4 4.0 + 0.4 28.4 + 1.2 2.92 + 0.11 1.66 + 0.39 8.9 + 1.3 7.0 + 0.7 24.8 + 3.3 4.33 + 0.12 1.79 + 0.26 5.9 + 0.6 9.7 + 0.3 21.5 + 1.4 relaxation rates reported are weighted averages of the l e f t and r i g h t positron telescope histograms. t o t a l asymmetry = 2A M u + A^+ (see Appendix II) -180-MU IN HCL WITH N 2 MODERATOR. 295 K 0 . 2 h 0 .0 1 1 1 1 I I 0 .00 0 .01 0 . 0 2 0 . 0 3 0-.04 0 . 0 5 HCL CONCENTRATION (M) FIGURE 33: MSR relaxation rates as a function of HCl concentration i n N 2 at 2 95K. The high concentration point represents/1 atmosphere of pure HCl. The l i n e i s a x 2 minimum f i t of the data y i e l d i n g k(295K) £ (3.41 + 0.46) x 10 M s . These data represent only an upper l i m i t , as described i n the text. -181-CO 0 1 - 0 . 0 5 >— t— C O c r - 0 . 1 5 0 - 0 0 - 5 1 . 0 1 . 5 2 . 0 2 . 5 3 . 0 3 . 5 4 . 0 TIME IN M SEC (20 NSEC/BIN) FIGURE 34 The MSR signals i n pure N 2 versus pure HCl. Both spectra represent about 10 events. The li n e s are x 2 minimum f i t s to equation (8). For N 2, A M u = 12.2 %, A = 4.0%, X = 0.4 y s - 1 ; for HCl, A M u = 5.9%, A = 9.7%, and X = 1.8 y s - 1 . -182-the observed MSR relaxation. The anhydrous HCl reagent used i n t h i s experiment, obtained from Canadian Liquid A i r Ltd., has a t y p i c a l 0^ impurity of <100 ppm. Mu undergoes spin exchange with paramagnetic oxygen which relaxes the MSR signal with a bimolecular rate constant of (1.6 + 0.1) x 10"'""'" 1/mole-s i n the gas phase [Fleming (79), Marshall (78)]. An 0^ concentration of 10 0 ppm i n one atmosphere of HCl gives a spin exchange MSR relaxation rate of about 0.7 ys "*", half of the observed e f f e c t . The experiment must be repeated with electronic grade HCl which has an 0^ concentration of <4 ppm. I t i s important to emphasize that a similar systematic error due to -contaminated reagents does not arise i n the other systems studied i n t h i s t h e s i s . In the f i r s t place, the 0^ contamination of the other gases used i s <10 ppm and, secondly, a l l of the other reaction rate constants are within two orders of magnitude of the spin exchange rate constant. Consequently, the systematic error due to contamination i n the X^ and HX gases (other than HCl) i s less than 0.1%. It may be noted that i n addition to providing an estimate of the thermal Mu + HCl reaction rate constant, t h i s experiment possibly gives some information, about the muonium formation process or the role played by "hot" atom reactions of Mu. As i l l u s t r a t e d i n Figure 34, the muonium signal amplitude in pure HCl i s about half of that i n pure under i d e n t i c a l conditions. As shown i n the data of Table XIV, the reduction i n the muonium signal with increasing HCl concentration i s accompanied by an increase i n the background "free" y + signal amplitude. These data may be explained i n two ways: (1) Mu may undergo fast -183-hot atom reactions with HCl before the muon spin has time to precess s i g n i f i c a n t l y , thereby placing muons into diamagnetic product molecules where they precess coherently l i k e "free" y + ions (see Appendix I I ) , or (2) the high energy charge exchange cross sections of Mu with HCl may be such that a large f r a c t i o n of the muons thermalize as free y + ions, rather than as Mu atoms. The t o t a l signal amplitude, given by 2A M u + A^-, appears to decrease with increasing HCl concentration; unfortunately, these data cannot be treated quantitatively, since the "free" y + signal amplitudes r e s u l t from f i t s of data covering only about one period of the slow y + precession at 6.9 gauss. Again, further experiments are required to c l e a r l y interpret these e f f e c t s . In p r i n c i p l e , i t i s possible to dis t i n g u i s h the hot atom process from the charge exchange process by the use of the residual p o l a r i z a t i o n method (Appendix I I , and [Brewer (72)]). I t must be emphasized that the l i m i t i n g rate constant for Mu + HCl reported i n t h i s Section i s for the thermal reaction, not the hot atom reaction. Hot atom reaction processes take place during the f i r s t several nanoseconds of the muon's entry into the target, whereas the MSR signal relaxation i s measured over several microseconds. As discussed i n Chapter I I I , the experimental and th e o r e t i c a l s i t u a t i o n with respect to the H + HCl abstraction and exchange reactions i s rather confused. An excellent review of these reactions has recently been prepared by Weston (79). For the H + HCl + H 2 + C l reaction, Weston (79) recommends values of E =3.18+0.17 kcal/mole, log. nA(1/mole-s) a — i u = 9.87 + 0.11 and k(295K) = (2.1 + 0.2) x 10 7 1/mole-s. -184-Although most experimental evidence indicates that the abstraction reaction i s faster than the exchange reaction, t h i s question remains unresolved [Weston (79)]. Bott and Heidner (76) measured the t o t a l reaction rate for H + HCl d i r e c t l y by 7 laser induced fluorescence and found k(295K) = ( 9 + 4 ) x 10 1/mole-s. Since the rate constant reported here for the Mu + HCl reaction i s also a t o t a l rate constant, the Bott and . Heidner H atom rate constant provides the most useful comparison giving k M u / k H at 2 95K <_Q"'.004 + 0.002. Even with the very large uncertainties i n the H and Mu data, i t i s certain that the t o t a l Mu reaction rate with HCl i s at least 100 times slower than the corresponding H atom rate! It i s unnecessary to turn to fancy detailed t h e o r e t i c a l c a l c u l a -ions to explain t h i s r e s u l t . From Table V, i t i s seen that both reaction channels for Mu with HCl are endothermic: A H Q = +6.2 and +7.9 kcal/mole for abstraction and exchange respectively. In contrast, the H + HCl reactions give A H Q = -1.1 and 0 kcal/mole for abstraction and exchange respectively. The c l a s s i c a l b a r r i e r s for these reactions, though poorly known, are thought to be ^4 kcal/mole for the abstraction reaction [Thompson (75)] and even more for the exchange reaction [Weston (79)]. However, even i f the classical barriers were zero, Mu must overcome an enormous zero point energy barrier of at least 6.2 kcal/mole in order to react with HCl; a b a r r i e r which i s at least 6.2 kcal/mole greater than the reaction b a r r i e r for H + HCl. In the absence of r e l i a b l e experimental data or t h e o r e t i c a l c a l c u l a t i o n s , i t i s impossible to check the general predictions on the Y + Cl„ -185-reactions given i n Chapter I I I , p 97-109. ^rMuBr + H E Mu + HBr -> MuH + Br . The MSR relaxation rates at various HBr concentrations, measured i n Ar moderator at 295K, are l i s t e d i n Table XV and 2 i l l u s t r a t e d in Figure II-2 (Appendix I I ) . A x minimum f i t of these relaxation rate data to equation 11(3) y i e l d s a t o t a l bimoleeular rate constant for the exchange plus abstrac-t i o n reactions of k(295K) = (9.09 + 0.97) x 10 9 1/mole-s (la) Insofar as. i t is known, Endo et al. (76) and Takacs et a l v (73) have made the only d i r e c t measurements of the analogous H reaction rate using ESR detection of H atoms i n a flow system. Their measurements, summarized i n Table XVI, gave the rate parameters for the sum of the exchange plus abstraction channels for the four H and D variations of the Y + Y'Br reaction between 230 and 318K, as well as the rate constants for the H + DBr and D + HBr reactions at 2 95K from which the abstraction:exchange branching r a t i o s are obtained. These measurements c l e a r l y show that for H and D, the abstraction reaction channels are much faster than the exchange reaction channels at room temperature. From the measured abstraction reaction a c t i v a t i o n energies and the estimated exchange reac-t i o n a c t i v a t i o n energy, i t can be inferred that the c l a s s i c a l b arrier to abstraction i s about l.,5-3 kcal/mole whereas the ba r r i e r to exchange i s about 5 kcal/mole. In drawing t h i s inference- i t must be cautioned that a c t i v a t i o n energies are approximately equal to the c l a s s i c a l b a r r i e r heights only i n -186-TABLE XV: MSR RELAXATION RATES FOR THE TOTAL Mu + HBr REACTION AT 2 95K [HBr] (10~4M) 0.00 0.77 + 0.07 1.50 + 0.03 2.85 + 0.08 4.49 + 0.10 5.92 + 0.13 Relaxation Rate A (ys ) 0.27 + 0.04 0.92 + 0.09 1.60 + 0.26 2.56 + 0.71 4.76 + 0.94 6.66 + 1.20 relaxation rates reported are weighted averages of the l e f t and r i g h t positron telescope histograms. TABLE XVI: EXPERIMENTAL RATE PARAMETERS FOR THE REACTIONS: Y + Y'Br^T C-r> I v i , Y = Mu, YBr + Y H, D; Y' = H,D Reaction E ' a (kcal/mole) log 1 QA(M 1 s ~ 1 ) k(295K) Y+HBr (10 9 1/mole-s) kH+HBr (295K) AH° [type] (kcal/mole) Mu+HBr -»• products H+HBr products D+HBr •> products 2.57 + 0.11 11.22 + 0.05 2.13 + 0.08 10.59 + 0.03 H+DBr ->• products 2.19+0.11 10.82+0.04 D+DBr products 1.69 + 0.13 10.14 + 0.03 D+HBr -> DBr+H H+DBr -* HBr+D 5.2 10 9. 09 + 0.97 2.08 + 0.16 1.02 + 0.05 1.57 + 0.18 0.78 + 0.11 0.0078 + 0.0024 <0.023 4.4 + 0.6 1.0 0.49 + 0.04 Y+DBr (295K) H+DBr 1.0 0.50 + 0.09 "abs (295K) exc 137 + 60 -32 >69 -9.4[abs] +7.1[exc] -16.7 [abs] 0 [exc] -17.5[abs] -1.1 [exc] -16.4[abs] +1.1 [exc] -17.4[abs] 0 [exc] between 23 0 and 318K estimates based on the measured rate constant at 2 95K and high temperature molecular beam and photolysis data [Endo (7 6) and references therein]. -188-the absence of strong dynamical e f f e c t s ; as discussed on p 109, i t i s possible that the a c t i v a t i o n energies are govern-ed by r o t a t i o n a l screening of the H isotope. In f a c t , from the Table i t appears that at 295K, the abstraction:exchange branching r a t i o for D + HBr i s greater than for H + DBr. This i s consistent with the r o t a t i o n a l screening hypothesis since the D atom approaches the more quickly rotating HBr about /2 times slower than the H atom approaches the more slowly rotating DBr. As remarked on p 109, i f r o t a t i o n a l screening i s important, i t i s expected that for Mu, the abstraction:exchange branching r a t i o w i l l be smaller than for the other H isotopes at 2 95K due to the greater mean v e l o c i t y of the l i g h t e r muonium atoms. On the other hand, i t can be argued that the exchange channel for Mu + HBr should be very much suppressed because AHg = +7.1 kcal/mole ; t h i s i s the minimum exchange bar r i e r for Mu + HBr. If the ba r r i e r heights estimated from the a c t i v a t i o n energies are correct, then the e f f e c t i v e Mu.+ HBr exchange b a r r i e r i s at least twice as large as the abstraction barrier; therefore, i t seems reasonable to assume that the measured value of k for Mu + HBr e s s e n t i a l l y corresponds to the abstraction reaction channel only. Table XVI gives the rate constant r a t i o s Mu:H:D at 300K of 4.4 + 0.6:1.0:0.49 + 0.04, which exceeds the temperature independent mass r a t i o s by 1.52 + 0.21 and 1.47 + 0.12 for Mu:H and H:D respectively. Truhlar (79) has pointed out that tunnelling i s expected to be more important for a given H + HX abstraction reaction than for the corresponding H + reactions, because the imaginary frequency of the unbound normal -189-mode vi b r a t i o n of the t r a n s i t i o n state tends to be much larger for the H + HX systems than for the H + X^ systems (see Figure 18; i n t h i s Figure, the a parameter i s inversely proportional to the imaginary frequency [Johnston ( 6 1 ) ] ) . This i s equivalent to saying that H + HX ba r r i e r s tend to be narrower than the H + X^ b a r r i e r s . Assuming that the rate constant r a t i o s reported above do correspond to abstraction, then they may be interpreted as an indicati o n of tunnelling i n th i s reaction. Since the ba r r i e r height for abstraction appears to be comparable to the H + F^ b a r r i e r (^ 2 kcal/mole), by analogy i t may also be expected that tunnelling w i l l be important at 3 00K. Although the degree of tunnelling cannot be inferred from a set of rate constant r a t i o s at one tempera-ture, i t appears that Mu and H tunnel comparable amounts when reacting with HBr at 295K. This i s consistent with the discussion on p 105 where i t i s suggested that "corner cutting" might equalize the tunnelling advantage of the various isotopes due to the mass weighted coordinate skewing angles of 72°, 45°, 36°, and 31° for Mu, H, D and T respectively. Table XVI shows the ac t i v a t i o n energies for the H reactions with HBr and DBr to be larger than those for D by about 25-30%. This r e s u l t may be explained i n terms of the v i b r a t i o n a l l y adiabatic b a r r i e r s which should be considerably larger for the l i g h t e r H isotopes (see p 104 and Table V). On the other hand, t h i s seems to contradict the tunnelling hypothesis just discussed, since, despite corner cutting, H i s expected to tunnel more than D, thereby predicting smaller act i v a t i o n energies for H + HBr than for D + HBr. Of course, -190-the p o s s i b i l i t y always exists that the experimental data are in error. If i t i s assumed that these data are correct, then, in order to r a t i o n a l i z e the two seemingly contradictory observa-tions that even though H + HBr has a larger a c t i v a t i o n energy than D + HBr, H + HBr s t i l l reacts faster at 295K, i t seems necessary to postulate that these systems have unusual exc i t a -t i o n functions ( i . e . cross section versus c o l l i s i o n energy curves). I t w i l l be i n t e r e s t i n g to see i f Mu + HBr also follows t h i s trend by having a larger a c t i v a t i o n energy than H + HBr. ^ Mul + H F Mu + HI MuH + I The MSR relaxation rates at various HI concentrations, measured i n both argon and N 2 moderator gases at 295K, are 2 l i s t e d i n Table XVII and i l l u s t r a t e d i n Figure 35. A x minimum f i t of these relaxation rate data to equation 11(3) y i e l d s a t o t a l bimoleeular rate constant for the exchange plus abstraction reactions of k(295K) = (2.53 + 0.12) x 1 0 1 0 1/mole-s (la) Insofar as i t i s known, the analogous H atom reaction rate has never been d i r e c t l y measured, and the i n d i r e c t measurements that have been made are sparse and unreliable [Bauer 2-(78)]. For the abstraction reaction, Jones et a l . (73) report E & = 0.7 + 0.25 kcal/mole, log 1 Q A (1/mole-s) = 10.7, and k(295K) = (1.5+0.5) x 1 0 1 0 1/mole-s, based on the H 2/I 2 thermal reaction experiments of S u l l i v a n (62) between 6 67 and 800K, which gives k M u / k H = 1.7 + 0.6 when extrapolated to 295K. However, t h i s estimate must represent some type of re-analysis of TABLE XVII: MSR RELAXATION RATES FOR THE TOTAL Mu + HI REACTION AT 2 95K [HI] Relaxation Rate (10 _ 4M) A - A ( u s - 1 ) + * 0 . 00 0 . 00 0 . 3 6 + 0 . 0 1 0 . 9 6 + 0 . 1 3 * 1 . 03 + 0 . 03 2 . 3 5 + 0 . 2 0 * 1 . 3 9 + 0 . 0 4 4 . 1 9 +• 0 . 5 5 * 2 . 0 3 + 0 . 06 5 . 1 1 + * 0 . 67 0 . 0 0 0 . 0 0 § § 0 . 42 0 . 4 6 + 0 . 0 2 1 . 2 9 + 0 . 98 + 0 . 03 3 . 5 8 + 0 . 6 3 § 1 . 2 5 + 0 . 0 3 3 . 8 4 + 0 . 5 2 § relaxation rates reported are weighted averages of the l e f t and r i g h t telescope histograms, given as A - A q to account for the s l i g h t l y d i f f e r e n t background relaxation rates i n argon and N 2. j argon moderator t N~ moderator -192-M U / H I . •= N 2 . A = RR MODERATOR. 295 K • 50 100 150 200 250 HI CONCENTRATION (uM) FIGURE 35: MSR relaxation rates as a function of HI concentration i n argon (diamonds) and N2 (squares). The data are plotted as A - A Q to account for the small differences i n A Q for each moderator gas. The l i n e i s a X 2 minimum f i t of the data y i e l d i n g k(295K) = (2.5 + 0.1) x 1 0 1 0 M"1,'s'1. -193-Sullivan's data since the paper referenced reports E & = 0.0 + 0.25 kcal/mole, log 1 QA(1/mole-s) = 9.05 + 0.07 and k(295K) = (1.1 + 0.2) x 10 9 1/mole-s which gives k.. /kTT = 23 + 4 when — ^ Mu H — extrapolated to 2 95K. Photolysis experiments of Persky and Kuppermann (74) give abstraction fractions (k , /[k , + k ].) = 0.95 + 0.04 and 0.88 + 0.08 for H + DI and D + HI respec-t i v e l y , which again, indicates that abstraction i s much faster than exchange for these H-HX reactions. However, i t should be cautioned that i n an analogous experiment with HBr, the abstraction fractions i n d i r e c t l y obtained by Persky and Kuppermann (74) have the opposite ordering to the d i r e c t measurements of Endo (76). Given the t e r r i b l e experimental s i t u a t i o n with these reactions, l i t t l e can be said about the Mu + HI reaction and isotope e f f e c t s . It appears that the reaction b a r r i e r for abstraction i s very small, which explains why the Mu + HI rate constant i s larger than the Mu + HBr rate constant. It also appears that k„„ /kTT at 295K for t h i s reaction i s greater c c Mu .H than one, as expected, though t h i s estimate i s based on an extrapolation of very questionable data. Finally,, based on the experimental re s u l t s of Persky and Kupperman and invoking the endothermicity arguments of the previous Sections, i t seems reasonable to again suggest that the abstraction reaction channel dominates, the Mu + HI reaction at 3 0OK. A measure-ment of the ac t i v a t i o n energy of Mu + HI at 3 0OK would ce r t a i n l y represent a substantial increase i n the available data on the Y + HI system. -194-CHAPTER V - SUMMARY AND CONCLUSIONS A Summary This thesis describes, i n considerable d e t a i l , the present experimental and t h e o r e t i c a l state of the study of gas phase muonium reaction k i n e t i c s . On the experimental side, i t out-l i n e s most of the s i g n i f i c a n t p r a c t i c a l problems encountered i n t h i s study, d e t a i l s the currently implemented solutions to these problems, and makes some s p e c i f i c suggestions for further improvements. P a r t i c u l a r attention i s paid to coping with the data acquisition problems that arise i n handling the high current beams produced by meson f a c t o r i e s . On the t h e o r e t i c a l side, the remarkably large body of calc u l a t i o n s , mainly due to Connor, Jakubetz, Manz, and Lagana, provide detailed interpretations of the experimental r e s u l t s and esta b l i s h the relevance of gas phase muonium reaction k i n e t i c s to the more conventional and more general f i e l d s of chemical k i n e t i c s and molecular reaction dynamics. Two main contentions are made: (1) that muonium provides an unusually useful tool with which to investigate hydrogen isotope e f f e c t s , s p e c i f i c a l l y i n terms of the dynamics of the Y + X 2 and Y + HX reactions, and (2) that the peculiar property of the MSR technique - that i t l i t e r a l l y examines one atom at a time - blesses i t with some d i s t i n c t advantages over conventional H atom studies. Thanks to the th e o r e t i c a l work of Connor et a_l. , the experimental study of gas phase muonium reactions seems to have sparked progress i n both the understand-ing of the Y + F^ and Y + C l 2 reactions and i n the development of useful computational tools for dealing with them. However, more experimental data on the reactions of Mu and H are re--195-quired i n order to firmly e s t a b l i s h the MSR method. B Past Perspective Chapter I includes a b r i e f h i s t o r i c a l summary that sketches the development of muonium chemistry in gases up to 197 5; the discussion then takes a quantum leap by describing the present status of the subject. In many ways, t h i s creates a distorted perspective which t h i s Section s h a l l attempt to correct. The work i n t h i s thesis took place during a time when the use of muons as probes of physical phenomena matured from seed to seedling - from well-demonstrated p o s s i b i l i t y to a serious, a l b e i t s t i l l - d e v e l o p i n g , study. This i s p a r t i c u l a r l y true i n the case of muonium chemistry which, in 1975, was rather neglected compared with the application of u +SR to s o l i d state physics. Indications of the maturation of gas phase muonium chemistry from 1975 to the present are many; Figure 36 provides a graphic example of i t s experimental development. The Figure i s a reproduction of t y p i c a l MSR signals obtained during the study of the Mu + Br^ reaction at the Lawrence Berkeley Labor-atory i n 1975 [Fleming (76)]. With an average muon stopping 3 - 1 . . . rate of 2 x 10 s , these were necessarily low s t a t i s t i c s runs 5 of t y p i c a l l y 1.5 x 10 events. Figure 36 may be compared with the TRIUMF spectra i l l u s t r a t e d i n Figure I I - l , taken under much less primitive conditions. The early MSR experiments were characterized by an almost t o t a l preoccupation with gadgets and gizmos required to obtain muons and ultimately to obtain data -196-LxJ O CL < 0.20 0.I0H O.OOH -0.I0H - 0 . 2 0 H 1 1 1 1 1 1 - r — - r _ 0 25 5 0 75 100 125 150 175 2 0 0 225 BIN NUMBER (20 NANOSECONDS / BIN) 0.20-O.IOH LU Q Z> • O.OOH -o.ioH - 0 . 2 0 0 25 5 0 75 100 125 150 175 2 0 0 225 BIN NUMBER (20 NANOSECONDS / BIN) FIGURE 36: Gas phase MSR signals i n a magnetic f i e l d of 1.8 gauss obtained at LBL i n 1975. The target con-tained pure Ar (top) and Ar with ^10 ppm Br,, (bottom) at 295K and one atmosphere pressure. The error bars are due to counting s t a t i s t i c s ^ only. Each histogram contains about 150 x 10 events. The l i n e i s a x minimum f i t to an approximation of equation (.8) • [Fleming (7 6) ] . -197-from them; "doing physics " seemed to play a subordinate r o l e . At times i t appeared t h a t the p o s s i b i l i t y of having the c y c l o -t r o n , beam'.lines, counters, e l e c t r o n i c s , and data a c q u i s i t i o n computers a l l f u n c t i o n i n g simultaneously was l i t t l e more than a f a n c i f u l dream. Today, the p r i o r i t i e s are u s u a l l y reversed. G e t t i n g muons and t a k i n g data are more-or-less r o u t i n e ; equip-ment breakdowns are l e s s frequent and tend to be i r r i t a t i o n s r a t h e r than catastrophes. The bulk of the experimental e f f o r t now goes i n t o designing more s o p h i s t i c a t e d t a r g e t s w i t h which to explore new and o f t e n more su b t l e phenomena. This t h e s i s work leaned h e a v i l y on the t h e o r e t i c a l work of Connor et al_. , not only i n order to e x p l a i n the r e s u l t s , but a l s o as a guide w i t h which to formulate an experimental s t r a t e g y . This happy symbiosis of theory and experiment came about by a f o r t u n a t e chain of circumstance. The f i r s t pub-l i c a t i o n of a low pressure gas phase r e a c t i o n r a t e constant, f o r the Mu + B r 2 r e a c t i o n [Fleming (76)], set f o r t h the b a s i s of the MSR technique and o p t i m i s t i c a l l y o f f e r e d an e x p e r i m e n t a l i s t ' s view of i t s prospectus. Connor et al_. , who had j u s t completed a t h e o r e t i c a l QMT study of the c o l l i n e a r H + F^ r e a c t i o n [Connor 2-(76)], picked up on t h i s Mu paper, decided to extend t h e i r c a l c u l a t i o n s to i n c l u d e the Mu, H, D, and T r e a c t i o n s , and suggested t h a t an experimental study of the Mu + F^ r e a c t i o n be c a r r i e d out. Although experimental work was i n progress on the other r e a c t i o n s reported i n t h i s t h e s i s , there was reason to ^ Around c y c l o t r o n f a c i l i t i e s , one speaks of "doing p h y s i c s " -saying "doing chemistry" i n v a r i a b l y has an u n s e t t l i n g e f f e c t on the l i s t e n e r . " S c i e n t i s t s have odious manners, except when you prop up t h e i r theory; then you can borrow money from them." Mark Twain, What i s a Man and Other Essays, p 283. -198-b e l i e v e t h a t the Mu + F^ r e a c t i o n would be immeasurably slow, and, besides, the use of F^ w i t h the thin-windowed t a r g e t apparatus would create formidable (probably insurmountable) problems. F o r t u n a t e l y , the t h e o r i s t s ' judgement p r e v a i l e d , and the experiment proved to be f e a s i b l e . The p r e l i m i n a r y t h e o r e t -i c a l c a l c u l a t i o n s on the Mu + F^ r e a c t i o n [Connor 1-(77)] were completed a few months before the experiment [Garner (78)]. As discussed i n Chapter IV, the t h e o r e t i c a l c a l c u l a t i o n s have since been r e v i s e d and supplemented. Subsequently, the experimental and t h e o r e t i c a l work has proceeded i n p a r a l l e l . Experimental r e s u l t s on the Mu + Cl,, r e a c t i o n [Fleming (79) ] have been followed by TST c a l c u l a t i o n s of Jakubetz (7 9), w hile QMT and QCT c a l c u l a t i o n s are p r e s e n t l y underway [Connor 2-(78)]. C Future P e r s p e c t i v e The. .experimental i n t e r e s t i n gas phase muonium i s by no means confined to the study of i t s thermal chemical r e a c t i o n s . At TRIUMF, programmes are i n , progress to examine the muonium formation process ( u + charge exchange) i n v a r i o u s gases [R.J. Mikula and D.G. Fleming], muonium spi n exchange w i t h paramagnetic species [D.G. Fleming, R.J. M i k u l a , and D.M. Garner], and the production of thermal muonium i n vacuum through the use of f i n e powdered i n s u l a t o r s as a stopping medium [G.M. M a r s h a l l , R. K i e f e l , and J.B. Warren]. I t i s h i g h l y l i k e l y t h a t both the under-standing of these phenomena and the development of experimental techniques w i t h which to study them w i l l have a mutual impact on the f u t u r e s t u d i e s of Mu chemical r e a c t i o n r a t e s . I t seems tha t the immediate o b j e c t i v e , of f u t u r e gas phase -199-Mu reaction rate experiments ought to be to complete the present study. In p a r t i c u l a r , i t i s desirable to develop a new target reaction vessel that provides an operational temperature range of from about 200 - 600K, i f possible, and to extend the rate measurements of the reactions studied i n t h i s thesis to span that temperature range. In conjunction with the analogous H atom reaction rate data, t h i s would provide a very complete set of isotopic rate parameters for two dynamically d i f f e r e n t classes of elementary chemical reactions. For the Y + X^ reactions ( i f possible, X should be extended to include I ) , t h i s temperature range should be s u f f i c i e n t to check the predicted Arrhenius plo t curvature, thereby placing a firmer experimental handle on the reaction dynamics. In changing X from F to I, the reaction dynamics should gradually s h i f t from c o l l i n e a r domination to the f u l l three dimensional reaction which may dramatically a f f e c t the H isotope e f f e c t s . As already described, the Y + HX reactions are dynamically much more complicated than the Y + X^ reactions, and, i n fact, the Mu + HX reactions may be quite d i f f e r e n t from the H + HX reactions. In Chapter I I I , p.98, i t i s described how i t may be possible to determine the exchange:abstraction branching r a t i o s for Mu + HX reactions by simply obtaining Arrhenius data over a wide temperature range. Chapter I I I , p. 102, also describes how low temperature rate data on these reactions might provide information about the presence or absence of wells i n the Mu + HX potential surfaces. I t seems probable that t h i s temperature range extension of the present experiments w i l l provide new dynamical information on these important elementary reactions. -200-For many years, the ultimate reactions for experimental k i n e t i c s study have been the isotopic variations of the H + E^ reactions, mainly because they have been the subject of exhaustive t h e o r e t i c a l investigation. Preliminary data on the Mu + E^ reaction [Mikula (7 9)] indicates that at room temperature, 5 the rate i s at or near the lower l i m i t of the MSR method (-10 1 mole ^ s "S . However, t h i s reaction, which has a high a c t i v a t i o n energy (_> 7 kcal/mole [Jones (73)]), may be measurable between 400 - 600K where i t i s at least from 20 - 350 times faster than at 3 00K. Certainly, the experimental investigation of t h i s reaction should have a high p r i o r i t y in the immediate future. Looking deeper into the c r y s t a l b a l l , where should gas phase Mu reaction k i n e t i c s go after the programme outlined above i s completed? This may depend strongly on the p r e v a i l i n g technology and theory at the time. One d i r e c t i o n that i t could c e r t a i n l y take, i s simply to measure the reaction rates of muonium with other molecules. The d e s i r a b i l i t y of such a programme w i l l depend larg e l y on the understanding of Mu isotope e f f e c t s . If the differences and s i m i l a r i t i e s between Mu and H reaction rates could be predicted with confidence, then the MSR method may provide a more accurate means of measuring H atom reaction rates. More importantly, MSR may also be applied to chemical systems where d i r e c t measurements of the H atom reaction rates are not experimentally possible. One very i n t e r e s t i n g d i r e c t i o n gas phase muonium ki n e t i c s might take i s to venture into the realm of state-to-state chemistry. Unfortunately, the experimental obstacles to such -201-a study presently appear to be p r o h i b i t i v e . In the f i r s t place, muonium atomic beams do not exist; only very high energy muon ion beams do. Some progress has been made i n producing thermal muonium i n vacuum [Marshall (78)]; however, even i f an id e a l "muon to muonium converter" existed, the use of such an atomic beam would be severely r e s t r i c t e d due to the muon l i f e t i m e of 2.2 ys. At 300K, a Mu atom travels about one inch during i t s l i f e t i m e . Associated with the l i f e t i m e problem i s the . i n t r i n s i c beam in t e n s i t y problem. State-to-state experiments would l i k e l y not be MSR experiments, but would employ some other detection technique such as infared chemiluminescence. Even at meson f a c t o r i e s , the most intense muon beam i d e a l l y available 8 + —1 2 would d e l i v e r no more than 10 y s over a 1 cm area. According to Appendix I I I , the average number of muons in a target at any time, given a 100% duty cycle beam, i s ^ T u » where 8 — 1 71 i s the beam current; with 91 = 10 s , the target would have no more than 2 00 muons i n i t at a time. This may be improved by using a low duty cycle accelerator which could d e l i v e r bursts of 4 5 10 - 10 muons at any instant. S t i l l , one i s faced wxth the formidable problem of measuring an observable at such i n t e n s i t i e s on the time scale of the muon l i f e t i m e . For example, i t appears that Mu infared chemiluminescence experiments would require infared, energy s e l e c t i v e , single photon counters - the infared analogue of such gamma ray detectors as sodium iodide c r y s t a l s . D Concluding Remarks "He i s not a l i a r , but he w i l l become one i f he keeps on." Mark Twain, Following the Equator, p. 2 91 Two aspects of t h i s thesis work have, I think, given me an -202-unusual and unique view of gas phase chemical k i n e t i c s i n genera l , and muonium chemistry i n p a r t i c u l a r : (1) the inherent i n t e r d i s c i p l i n a r y nature of a subject which e x p l o i t s p a r t i c l e p hysics as a chemical t o o l , and (2) the t i m i n g of my involvement i n the programme, which spanned the t w i l i g h t days of the 18 4" Cyc l o t r o n at Berkeley to the e a r l y stages of the operation of the TRIUMF meson f a c t o r y . On the f i r s t p o i n t , I w i l l simply s t a t e t h a t I began i n a s t a t e of innocence - as I r e c a l l , the f i r s t question I ever asked my research supervisor was, " What i s a muon?" To t h i s day, my w i f e , an a r t h i s t o r i a n , s t i l l questions (quite s e n s i b l y , I think) the s a n i t y of people who cl a i m to study unseen p a r t i c l e s t h a t l a s t f o r two m i l l i o n t h s of a second. As f o r my involvement i n the i n f a n t research p r o j e c t , t h i s gave me a range of experience i n MSR th a t w i l l not o r d i n a r i l y be a v a i l a b l e to f u t u r e MSR workers. As an o l d t i m e r , I have helped b u i l d the c y c l o t r o n and beamlines (which are now buried i n r a d i a t i o n s h i e l d i n g , and seldom a c c e s s i b l e ) ; I was invo l v e d i n beam l i n e tuning and the design and implementation of data acquisition and a n a l y s i s systems. Of course, I d i d not always c h e r i s h t h i s experience which seemed at many times to be f r u s t r a t i n g drudgery. Nonetheless, I t h i n k t h i s ground f l o o r experience has provided me w i t h a reasonable understanding of most of the experimental paraphernalia', and, i n some cases, a f a i r l y i n t i m a t e understanding, which d e - m y s t i f i e s the many black boxes of MSR, thereby a f f o r d i n g greater c o n t r o l over the experiments. -203-L i t e r a t u r e C i t e d - Chapter I Anderson (37) Babaev (66) Bjorken (64) Brewer (72) Brewer (73) Brewer (75) Bu c c i (78) Connor l-(77) F i r s o v . (65) Fleming (76) Fleming l-(77) Fleming (7 9) Garwin (57) Gold (78) C.C. Anderson and S.H. Neddermeyer, Phys. Rev. 51 (1937) 884. A.I. Babaev, M.Ya. B a l a t s , G.G. Myasishcheva, Yu.V. Obukhov, V.S. Roganov, and V.G. F i r s o v , Sov. Phys. JETP 23_ (1966) 583. J.D. Bjorken and S.D. D r e l l , R e l a t i v i s t i c  Quantum Mechanics, McGraw-Hill, New York, (1964). J.H. Brewer, Ph.D. T h e s i s , Lawrence Berkeley Laboratory Report LBL-950, (1972). J.H. Brewer, F.N. Gygax, and D.G. Fleming, Phys. Rev. A8 (1973) 77. J.H. Brewer, K.M. Crowe, F.N. Gygax, and A. Schenck, i n Muon P h y s i c s , V o l . I l l , Eds. C.S. Wu and V.W. Hughes, Academic P r e s s , New York, (1975), Chapter 7. C. B u c c i , G.-Guidi, G.M. de'Munari, M. Manfredi, P. P o d i n i , R. Tedeschi, P.R. C r i p p a , and A. V e c l i , Chem. Phys. L e t t e r s 57 (1978) 634. J.N.L. Connor, W. Jakubetz, and J . Manz, Chem. Phys. L e t t e r s 45_ (1977) 265. V.G. F i r s o v and V.M. Byakov, Sov. Phys. JETP 2_0 (1965) 719. D. G. Fleming, J.H. Brewer, D.M. Garner, A.E. P i f e r , T. Bowen, D.A. D e l i s e , and K.M. Crowe, J . Chem. Phys. 6 ± (1976) 1281. D.G. Fleming, D.M. Garner, J.H. Brewer, J.B. Warren, G.M. M a r s h a l l , G. C l a r k , A.E. P i f e r , and T. Bowen, Chem. Phys. L e t t e r s 4_8 (1977) 393. D.G. Fleming, D.M. Garner, L.C. Vaz, D.C. Walker, J.H. Brewer, and K.M. Crowe, i n P o s i t r o n i u m and Muonium Chemistry, Ed. H.J. Ache, American Chemical S o c i e t y Advances i n Chemistry S e r i e s , i n p r e s s . R.L. Garwin, L.M. Lederman, and M. Weinrich, Phys. Rev. 105_ (1957) 1415. V. Gold and M.E. McAdam, Accounts of Chemical Res. 11 (1978) 36. -204-Literature Cited - Chapter I (Cont'd.) Goldanskii (71) Grebinnik (76) Hague (70) Hughes (66) Ivanter (68) Ivanter (69) Jean (78) Kent (77) Lederman (77) Mobley (6 6) Mobley l-(67) Mobley 2-(67) Nishida (77) Nosov (63) Perciv a l l-(76) V.I. Goldanskii and V.G. Firsov, Ann. Rev. Phys. Chem. 22 (1971) 209. V.G. Grebinnik, I.I. Gurevich, V.A. Zhukov, I.G. Ivanter, A.P. Manych, B.A. N i k o l ' s k i i , V.I. Selivanov, and V.A. Suetin, Sov. Phys. JETP Letters 2J3 (1976) 8. J.F. Hague, J.E. Rothberg, A. Schenck, D.L. Williams, R.W. Williams, K.K. Young, and K.M. Crowe, Phys. Rev. Letters 2_5 (1970) 628 V.W. Hughes, Ann. Rev. Nucl. S c i . 16_ (1966) 445. I.G. Ivanter and V.P. Smilga, Sov. Phys. JETP 27 (1968) 301. I.G. Ivanter and V.P. Smilga, Sov. Phys. JETP 28^  (1969) 796. Y.C. Jean, J.H. Brewer, D.G. Fleming, D.M. Garner, R.J. Mikula, L.C. Vaz, and D.C. Walker, Chem. Phys. Letters 57 (1978) 293. M. Kent, M.Sc. Thesis, University of B r i t i s h Columbia, (1977), unpublished. L.M. Lederman, Comments on Nucl. and Part. Phys. VII (1977) 89. R.M. Mobley, J.M. Bailey, W.E. Cleland, V.W. Hughes, and J.E. Rothberg, J. Chem. Phys. 44_ (1966) 4354. R.M. Mobley, J.J. Amato, V.W. Hughes, J.E. Rothbefcr, and P.A. Thompson, J . Chem. Phys. 47_ (1967) 3074. R.M. Mobley, Ph.D. Thesis, Yale University, (1967), unpublished. N. Nishida, R.S. Hayano, K. Nagamine, T. Yamazaki, J.H. Brewer, D.M. Garner, D.G. Fleming, T. Takeuchi, and Y. Ishikawa, Solid State Comm. 2_2 (1977) 235. V.G. Nosov and I.V. Yakovleva, Sov. Phys. JETP 16 (1963) 1236. P.W. Per c i v a l and H. Fischer, Chem. Phys. 16_ (1976) 89. -205-L i t e r a t u r e C i t e d - Chapter I (Cont'd) P e r c i v a l 2-(76) Perci v a l (77) Perci v a l (78) P i f e r (76) Roduner l-(78) Roduner 2-(78) Sachs (75) Schenck (76) Stambaugh (7 4) Street (37) Swanson (58) Weissenberg (67) P.W. P e r c i v a l , H. F i s c h e r , M. Camani, F.N. Gygax, W. Ruegg, A. Schenck, H. S c h i l l i n g , and H. Graf, Chem. Phys. L e t t e r s 3_9 (1976) 333. P.W. P e r c i v a l , E. Roduner, H. F i s c h e r , M. Camani, F.N. Gygax, and A. Schenck, Chem. Phys. L e t t e r s 47_ (1977) 11. P.W. P e r c i v a l , ySR Newsletter _16 (1978) 409. A.E. P i f e r , T. Bowen, and K.R. K e n d a l l , Nucl. I n s t r . Methods 135 (1976) 39. E. Roduner, P.W. P e r c i v a l , D.G. Fleming, J . Hochmann, artd H. F i s c h e r , Chem. Phys. L e t t e r s 57 (1978) 37. E. Roduner, ySR Newsletter 16 (1978) 407. A.M. Sachs and A. S i r l i n , i n Muon P h y s i c s , V o l . I I , Eds. C.S. Wu and V.W. Hughes, Academic Press, New York, (1975). A. Schenck, i n Nuclear and P a r t i c l e Physics  at Intermediate Energies, Ed. J.B. Warren, Plenum, New York, (1976) 159. R.D. Stambaugh, D.E. Casperson, T.W. Crane, V.W. Hughes, H.F. Kaspar, P. Souder, P.A. Thompson, H. Orth, G. zu P u l i t z , and A.B. Dennison, Phys. Rev. L e t t e r s 3_3 (1974) 568. J.C. S t r e e t and E.C. Stevenson, Phys. Rev. 5_2 (1937) 1003. R.A. Swanson, Phys. Rev. 112 (1958) 580. A.O. Weissenberg, Muons, North-Holland, Amsterdam, (1967), published i n Russian i n 1964. Literature Cited - Chapter II Cvetanovic (77) R.J. Cvetanovic and D.L. Singleton, Int. J. Chem. Kinet. IX (1977) 481; i b i d , IX (1977) 1007. Davidon (68) W.C. Davidon, Computer Journal 10_ (1968) 406. Fleming (76) D.G. Fleming, J.H. Brewer, D.M. Garner, A.E. P i f e r , T. Bowen, D.A. Delise, and K.M. Crowe, J . Chem. Phys. 6_4 (1976) 1281. Fleming l-(77) D.G. Fleming, D.M. Garner, J.H. Brewer, J.B. Warren, G.M. Marshall, G. Clark, A.E. P i f e r , and T. Bowen, Chem. Phys. Letters •48 (1977) 393. Frost (61) A.A. Frost and R.G. Pearson, Kinetics and Mechanism, 2nd E d i t i o n , Wiley, New York, (1961). Hague (7 0) J.F. Hague, J.E. Rothberg, A. Schenck, D.L. Williams, K.K. Young, and K.M. Crowe, Phys. Rev. Letters 25_ (1970) 628. Hayano l-(76) R.S. Hayano, University of Tokyo Nuclear Physics Report UTPN-61, (1976). Hayano 2-(76) R.S. Hayano, University of Tokyo Nuclear Physics Report UTPN-64, M.Sc. ; Thesis, (1976). James (71) F. James and M. Roos, MINUIT, CERN Computer 7600 Interim Programme Library, (1971). Marshall (76) G.M. Marshall, M.Sc. Thesis, University of B r i t i s h Columbia, (1976), unpublished. Mobley 2-(67) R.M. Mobley, Ph.D. Thesis, Yale University, (1967), unpublished. Nelder (67) J.A. Nelder and R. Mead, Computer Journal 7 (1967) 308. P i f e r (76) A.E. P i f e r , T. Bowen, and K.R. Kendall, Nucl. Instr. Methods 135 (1976) 39. Sachs (75) A.M. Sachs and A. S i r l i n , i n Muon Physics, Vol. I I , Eds. C.S. Wu and V.W. Hughes, Academic Press, New York, (1975). Schenck (76) A. Schenck, i n Nuclear and P a r t i c l e Physics at Intermediate Energies, Ed. J.B. Warren, Plenum, New York, (1976), 159. Stambaugh (74) R.D. Stambaugh, D.E. Casperson, T.W. Crane, V.w. Hughes, H.F. Kaspar, P. Souder, P.A. Thompson, H. Orth, G. zu P u l i t z and A.B. Dennison, Phys. Rev. Letters 33 (1974) 568. -207-Literature Cited - Chanter III Anlauf (72) Arnoldi (76) Baer (76) Bardo (78) Bauer l-(78) Bauer 2-(78) Baybutt (78) Bl a i s (74) Bondybey (71) Botschwina (77) Clyne (66) Connor (75) Connor l-(76) Connor l-(77) Connor 2-(77) Connor l-(7 8) Connor 2-(78) Connor i'-.(7 9) Douglas (7 6) K.G. Anlauf, D.S. Home, R.G. MacDonald, vT.C. Polanyi, and K.B. Woodall, J. Chem. Phys. 57 (1972) 1561. D. Arnoldi and J. Wolfrum, Ber. Bunsenges. physik. Chem. 8_0 (1976) 892. M. Baer, J. Chem. Phys. 65_ (1976) 493. R.D. Bardo and M. Wolfsberg, J. Chem. Phys. 68^  (1978) 2686. W. Bauer, K. Shobatake, J.P. Toennies, and K. Walaschewski, J. Chem. Phys. 68^  (1978) 3413. W. Bauer, L.Y. Rusin, and J.P. Toennies, J. Chem. Phys. 68^  (1978) 4490. P. Baybutt, F.W. Bobrowicz, L.R. Kahn, and D.G. Truhlar, J. Chem. Phys. ^8 (1978) 4809. N.C. B l a i s and D.G. Truhlar, J . Chem. Phys. 61 (1974) 4186; i b i d 65 (1976) 3303. V. Bondybey, G.C. Pimentel, and P.N. Noble, J. Chem. Phys. 5_5 (1971) 540. P. Botschwina and W.Meyer, J. Chem. Phys. 67 (1977) 2390. M.A.A. Clyne and D.H. Stedman, J. Chem Soc. Faraday Trans. 62 (1966) 2164. J.N.L. Connor, W. Jakubetz, and J. Manz, Mol. Phys. 2_9 (1975) 347. J.N.L. Connor, Chem. Soc. Rev. 5_ (1976) 125. J.N.L. Connor, W. Jakubetz, and J. Manz, Chem. Phys. Letters, 45_ (1977) 265. J.N.L. Connor, private communication, (1977). J.N.L. Connor, W. Jakubetz, and J. Manz, Chem. Phys. 28 (1978) 219. J.N.L. Connor, private communication, (1978). J.N.L. Connor, W. Jakubetz, and A. Lagana, J. Phys. Chem. 83^  (1979) 73. D.J. Douglas, J.C. Polanyi, and J.J. Sloan, Chem. Phys. 13 (1976) 15. -208-Literature Cited - Chapter III (Cont'd) Dunning (77) T.H. Dunning, J. Chem. Phys. 66 (1977) 2752. Eliason (59) Endo (76) Fischer l-(77) Fischer 2-(77) Fleming (76) Fleming l-(77) Frost (61) Gatz (66) Heidner (76) M.A. Eliason and J.O. Hirschfelder, J. Chem. Phys. 3_0 (1959) 1426. H. Endo and G.P. Glass, Chem. Phys. Letters 44 (1976) 180. S. Fischer and G. Venzl, J. Chem. Phys, (1977) 1335. 67 S. Fischer, Ber. Bunsenges. physik. Chem. 81 (1977) 197. D.G. Fleming, J.H. Brewer, D.M. Garner, A.E. P i f e r , T. Bowen, D.A. Delise, and K.M. Crowe, J. Chem. Phys. 64^  (1976) 1281. D.G. Fleming, D.M. Garner, J.H. Brewer, J.B. Warren, G.M. Marshall, G. Clark, A.E. P i f e r , and T. Bowen, Chem. Phys. Letters 4_8 (1977) 393. A.A. Frost and R.G. Pearson, Kinetics and  Mechanism, 2nd Edition, Wiley, New York, (1961) CR. Gatz, J. Chem. Phys. 4_4 (1966) 1861. R.F. Heidner and J.F. Bott, J . Chem. Phys. 64 (1976) 2267. Hirschfelder (76) J.O. Hirschfelder and K.T. Tang, J. Chem. Phys. 64 (1976) 760. Jakubetz l-(78) Jakubetz (7 9) Johnston (61) Johnston (66) W. Jakubetz, "Gas Phase Muonium Chemistry, Isotope E f f e c t s , and C o l l i s i o n Theory: Theoretical Investigations of the Mu + and Mu + C l 2 Reactions and th e i r Isotopic Counter-parts," pre-print submitted to Hyperfine Interactions, from Proceedings of the F i r s t International Conference on uSR, Rorshack, Switzerland, September, (1978). W. Jakubetz, J. Am. Chem. Soc. 101 (1979) 298. H.S. Johnston and D.L. Rapp, J . Am. Chem. Soc. 83 (1961) 1. H.S. Johnston, Gas Phase Reaction Rate Theory, Ronald, New York, (1966). Jonathan (72) N. Jonathan, S. Okuda, and D. Timlin, Mol. Phys. 24 (1972) 1143. -209-Literature Cited - Chapter III (Cont'd) Karplus (7 0) Klein (78) Korsch (78) Kuntz (66) Kuppermann (7 9) Laidl e r (65) Levine (74) Liu (78) Manz (74) Manz (75) Manz (76) Marcus (77) McDonald (75) Messiah (58) Mok (69) N i k i t i n (74) Noble (68) Noble (72) P a t t e n g i l l (76) M. Karplus and R.N. Porter, Atoms and Molecules, Benjamin, New York, (1970). F.S. Klein and I. Veltman, J. Chem. Soc. Faraday II 74 (1978) 17. H.J. Korsch, Chem. Phys. 33 (1978) 313. P.J. Kuntz, E.M. Nemeth, J.C. Polanyi, S.D. Rosner, and C.E. Young, J. Chem. Phys. 44^  (1966) 1168. A. Kuppermann, J. Phys. Chem. 8_3 (1979) 171. K.J. L a i d l e r and J.C. Polanyi, Progress i n Reaction Ki n e t i c s , Vol. 3, Ed. G. Porter, Pergamon, New York (1965), 1. R.D. Levine and R.B. Bernstein, Molecular  Reaction Dynamics, Oxford University Press, (1974). B. L i u and P. Siegbahn, J. Chem. Phys. 68_ (1978) 2457. J. Manz, Mol. Phys. 28^  (1974) 399. J. Manz, Mol. Phys. 3_0 (1975) 899. J. Manz, private communication, (1976). R.A. Marcus and M.E. C o l t i n , J. Chem. Phys. 67_ (1977) 2609. J.D. McDonald and D.R. Herschbach, J. Chem. Phys. 62^  (1975) 4740. A. Messiah, Quantum Mechanics, Vol. I, translated by G.M. Temmer, North-Holland, Amsterdam, (1958). M.H. Mok and J.C. Polanyi, J. Chem. Phys. 51 (1969) 1451. E.E. N i k i t i n , Theory of Elementary Atomic and  Molecular Processes i n Gases, M.J. Kearsley, translator, Clarendon, Oxford, (1974). P.N. Noble and G.C. Pimentel, J . Chem. Phys. 49 (1968) 3165. P.N. Noble, J . Chem. Phys. 56 (1972) 2088. M.D. P a t t e n g i l l , J.C. Polanyi, and J.L. Schreiber, J. Chem. Soc. Faraday II 7_2 (1976) 897. -210-Literature Cited - Chapter III (Cont'd) A. Persky, J. Chem. Phys. 6J5 (1977) 2932. A. Persky, J. Chem. Phys. 68 (1978) 2411. Persky (77) Persky (78) Polanyi (69) Polanyi (72) Polanyi (75) Polanyi (78) Porter (73) Sato (55) Schatz (77) Smith (77) Thompson (75) Thompson (7 6) Tolman (27) Truhlar (.69) Valencich (77) Van Hook (7 0) Weston (72) Weston (7 9) J.C. Polanyi and W.H. Wong, J. Chem. Phys. 51 (1969) 1439. J.C. Polanyi, Accounts of Chem. Res. 5_ (1972) 161. J.C. Polanyi, J.L. Schreiber, and J.J. Sloan, Chem. Phys. 9 (1975) 403. J.C. Polanyi and N. Sathyamurthy Chem. Phys. 33_ (1978) 287. R.N. Porter, L.B. Sims, D.L. Thompson, and L.M. Raff, J. Chem. Phys 58^  (1973) 2855. S. Sato, J. Chem. Phys 23_ (1955) 2465. G.C. Schatz and J. Ross, J. Chem. Phys. 66^  (1977) 1021. I.W.M. Smith, i n Gas Kinetics and Energy  Transfer, Vol. 2, Ed. P.G. Ashmore and R.J. Donovan, B i l l i n g s , London, (1977). D.L. Thompson, H.H. Suzukawa, and L.M. Raff, J. Chem. Phys. 62^  (1975) 4727. D.L. Thompson, Accounts of Chem. Res. 9^  (1976) 338. R.C. Tolman, S t a t i s t i c a l Mechanics With  Applications to Physics and Chemistry, Chem. Catalog, New York, (1927), 260. D.G. Truhlar and A. Kuppermann, J. Phys. Chem. 73_ (1969) 1722. T. Valencich, J. Hsieh, J. Kwan, T. Stewart, and T. Lenhardt, Ber. Bunsenges. physik. Chem. 81 (1977) 131. W.A. Van Hook, i n Isotope Effects i n Chemical  Reactions, ACS Monograph, C.J. C o l l i n s and N.S. Bowman, Eds., Van Nostrand Reinhold, Toronto, (1970), 1. R.E. Weston and H.A. Schwarz, Chemical  Kinetics , Prentice H a l l , Toronto, (1972). R.E. Weston, J. Phys. Chem. 83 (1979) 61. -211-Literature Cited - Chapter III (Cont'd) White (73) J.M. White, J. Chem. Phys. 58_ (1973) 4482. Wilkins (75) R.L. Wilkins, J . Chem. Phys. 63 (1975) 2963. Wolfrum (77) J. Wolfrum, Ber. Bunsenges. physik. Chem. 81 (1977) 114. -212-Literature Cited - Chapter IV Albright (69) Ambidge (7 6) Baer (74) Bauer 1-(78) Bauer 2-(78) Baybutt (78) Bemand (77) Benson (60) Bla i s (74) Brewer (72) Clyne (79) Connor l-(77) Connor l-(78) Connor 2-(78) Connor l-(7 9) Connor 2-(79) Connor 3-(7 9) R.G. Albright, A.F. Dodonov, G.K. Lavrovskaya, I.I. Morosov, and V.L. Tal'roze, J. Chem. Phys. 5_0 (1969) 3632. P.F. Ambidge, J.N. Bradley and D.A. Whytock, J. Chem. Soc. Faraday I 7_2 (1976) 1157. M. Baer, J. Chem. Phys. 60 (1974) 1057. W. Bauer, K. Shobatake, J.P. Toennies, and K. Walaschewski, J. Chem. Phys. 68_ (1978) 3413. W. Bauer, L.Y. Rusin, and J.P. Toennies, J. Chem. Phys. 68^  (1978) 4490. P. Baybutt, I.W. Bobrowicz, L.R. Kahn, and D.G. Truhlar, J. Chem. Phys. 6_8 (1978) 4809. P.P. Bemand and M.A.A. Clyne, J. Chem. Soc. Faraday II 73(1977) 394. S.W. Benson, The Foundations of Chemical Kinetics , McGraw-Hill, New York, (1960). N.C. B l a i s and D.G. Truhlar, J. Chem. Phys., 61 (1974) 4186; i b i d 65_ (1976) 3303. J.H. Brewer, Ph.D Thesis, Lawrence Berkeley Laboratory Report LBL-950, (1972). M.A.A. Clyne, private communication, (1979). J.N.L. Connor, W. Jakubetz, and J. Manz, Chem. Phys. Letters 45 (1977) 265. J.N.L. Connor, W. Jakubetz, and J . Manz, Chem. Phys. 28 (1978) 219. J.N.L. Connor, private communication, (1978). J.N.L. Connor, W. Jakubetz, and A. Lagana, J. Phys. Chem. 83_ (1979) 73. J.N.L. Connor and A. Lagana, "Quasiclassical dynamics of l i g h t heavy heavy atom reactions: The reaction X + F 2 -> XF + F, X = (Mu, H, D, T)", preprint, (1979). J.N.L. Connor, W. Jakubetz, J . Manz, and J.C. Whitehead, "The reaction X + C l 2 ->•• XCl + C l (X = H, D). I. A new inversion procedure for obtaining potential energy surfaces from experimental rate c o e f f i c i e n t data," to be published, (1979). -213-L i t e r a t u r e C i t e d - Chapter IV (Cont'd) Dodonov (7 0) Endo (76) Essen (7 6) Fass (70) Fass (72) Fleming (7 6) Fleming l-(77) Fleming 2-(77) Fleming (7 9) Foon (75) Garner (7 8) Hepburn (7 8) A.F. Dodonov, G.K. Lavrovskaya, I.I. Morosov, R.G. A l b r i g h t , V.L. T a l ' r o z e , and A.K. Lyubimova, K i n e t i c a i K a t a l i z , 1_1 (1970); e n g l i s h t r a n s l a t i o n : K i n e t i c s and C a t a l y s i s 11 (1970) 677. H. Endo and G.P. Glass, J . Phys. Chem. 8_0 (1976) 1519. H. Essen, G.D. B i l l i n g , and M. Baer, Chem. Phys. 17 (1976) 443. R.A. Fass, J . Phys. Chem. 7_4 (1970) 984. R.A. Fass, J.W. Hoover, and L.M. Simpson, J . Phys. Chem. 76^  (1972) 2801. D.G. Fleming, J.H. Brewer, D.M. Garner, A.E. P i f e r , T. Bowen, D.A. D e l i s e and K.M. Crowe, J . Chem. Phys. 6£ (1976) 1281. D.G. Fleming, D.M. Garner, J.H. Brewer, J.B. Warren, G.M. M a r s h a l l , G. C l a r k , A.E. P i f e r , and T. Bowen, Chem. Phys. L e t t e r s 48_ (1977) 3 93, D.G. Fleming, J.H. Brewer, and D.M. Garner, Ber. Bunsenges. physik. Chem. 81 (1977) 159. D.G. Fleming, D.M. Garner, L.C. Vaz, D.C. Walker, J.H. Brewer, and K.M. Crowe, i n Positronium and Muonium Chemistry, Ed. H.J. Ache, American Chemical So c i e t y Advances i n Chemistry S e r i e s , i n press. R. Foon and K. Kaufman, Progress i n Reaction K i n e t i c s , V o l . 8, Ed. G. P o r t e r , Pergamon, New York, (1975) 85. D.M. Garner, D.G. Fleming, and J.H. Brewer, Chem. Phys. L e t t e r s , 55 (1978) 163. J.W. Hepburn, D. Klimek, K. L i u , J^C.:Polanyi, and S.C. Wallace, J . Chem. Phys. 69^  (1978) 4311. Homann (77) K.H. Homann, H. Schweinfurth, and J. Warnatz, Ber. Bunsenges. physik. Chem. 8_1 (1977) 724. Jakubetz l-(78) W. Jakubetz, "Gas phase muonium chemistry, isotope e f f e c t s , and c o l l i s i o n theory: t h e o r e t i c a l investigations of the Mu + F^ and Mu + C l 2 reactions and the i r isotopic counter-parts", preprint submitted to Hyperfine Inter-actions , from proceedings of the F i r s t -214-Literature Cited - Chapter IV (Cont'd) Jakubetz l-(7 8) (Cont'd) Jakubetz 2-(78) Jakubetz (7 9) Johnston (61) Jonathan (72) Jones (73) Korsch (78) Kuntz (66) Lagana (7 9) La i d l e r (65) Levine (7 4) Levy (68) Malcolme-Lawes (78) Marshall (78) Nesmeinov (63) Persky (74) Polanyi (72) Polanyi (75) International Conference on uSR, Rorshack, Switzerland, September, 1978. W. Jakubetz, J. Chem Phys. 69 (1978) 1783. W. Jakubetz, J. Am. Chem. Soc. 101 (1979) 298. H.S. Johnston and D.L. Rapp, J. Am. Chem. Soc. 83_ (1961) 1. N. Jonathan, S. Okuda, and D. Timlin, Mol. Phys. 24 (1972) 1143. W.E. Jones, S.D. MacKnight, and L. Teng, Chem. Rev. 73_ (1973) 407. H.J. Korsch, Chem. Phys. 3_3 (1978) 313. P.J. Kuntz, E.M. Nemeth, J.C. Polanyi, S.D. Rosner, and C.E. Young, J. Chem. Phys. 4J_ (1966) 1168 A. Lagana, private communication, (1979). K.J. L a i d l e r and J.C. Polanyi, Progress in Reaction Kinetics , Vol. 3, Ed. G. Porter, Pergamon, New York, (19 65) 1. R.D. Levine and R.B. Bernstein, Molecular  Reaction Dynamics, Oxford University Press, (1974). J.B. Levy and B.K. Copeland, J. Phys. Chem., 72 (1968) 3168. D.J. Malcolme-Lawes, J. Chem. Soc. Faraday II 7_4 (1978) 182 G.M. Marshall, J.B. Warren, D.M. Garner, G.S. Clark, J.H. Brewer, and D.G. Fleming, Phys. Letters 65A (1978) 351. A.N. Nesmeinov, in Vapor Pressure of the  Chemical Elements, Ed. R. Gary, E l s e v i e r , New York, (1963). A. Persky and A. Kuppermann, J. Chem. Phys. 61 (1974) 5035. J.C. Polanyi and J.J. Sloan, J. Chem. Phys. 57 (1972) 4988. J.C. Polanyi, J.L. Schreiber, and J.J. Sloan, Chem. Phys. 9 (1975) 403. -215-Literature Cited - Chapter IV (Cont'd) Rabideau (7 2) Stedman (7 0) Sulliv a n (62) Takacs (73) Thompson (7 5) Truhlar (7 8) Truhlar (7 9) Venzl (78) Wagner (7 6) Weston (79) White (73) Wilkins (75) S.W. Rabideau, H.G. Hecht, and W.B. Lewis, J. Magn. Reson. 6^  (1972) 384. D.H. Stedman, D. Steffeson, and H. N i k i , Chem. Phys. Letters 7 (1970) 173. J.H. S u l l i v a n , J. Chem Phys. 3_6 (1962) 1925. G.A. Takacs and G.P. Glass, J. Phys. Chem. 77 (1973) 1060. D.L. Thompson, H.H. Suzukawa, and L.M. Raff, J. Chem. Phys. 62 (1975) 4727. D.G. Truhlar and J.C. Gray, Chem. Phys. Letters 57 (1978) 93. D.G. Truhlar, J. Phys. Chem. 8_3 (1979) 188. G. Venzl and S.F. Fischer, Chem. Phys. 3_3_ (1978) 305. H. Gg. Wagner, U. Welzbacker, and R. Zellner, Ber. Bunsenges. physik. Chem. 8_0 (1976) 902. R.E. Weston, J. Phys. 8_3 (1979) 61. J.M. White, J . Chem. Phys. 58 (1973) 4482. R.L. Wilkins, J. Chem. Phys. 63 (1975) 2963. -216-Literature Cited - Chapter V Connor 2- (76) Connor l-(77) Connor 2-(78) Fleming (76) Fleming (79) Garner (7 8) Jakubetz (79) Jones (73) Marshall (78) J.N.L. Connor, W. Jakubetz, and J . Manz, Chem. Phys. 17 (1976) 501. J.N.L. Connor, W. Jakubetz, and J. Manz, Chem. Phys. Letters 45 (1977) 265. J.N.L. Connor, private communication, (1973). D.G. Fleming, J.H. Brewer, D.M. Garner, A.E. P i f e r , T. Bowen, D.A. Delise, and K.M. Crowe, J. Chem. Phys. 6^4 (1976) 1281. D.G. Fleming, D.M. Garner, L.C. Vaz, D.C. Walker, J.H. Brewer, and K.M. Crowe, i n Positronium and Muonium Chemistry, Ed. H.J. Ache, American Chemical Society Advances i n Chemistry Series, i n press. D.M. Garner, D.G. Fleming, and J.H. Brewer, Chem. Phys. Letters 55 (1978) 163. W. Jakubetz, J. Am. Chem. S o c , 101 (1979) 29i W.E. Jones, S.D. MacKnight, and L. Teng, Chem. Rev. J! (1973) 407. G.M. Marshall, J.B. Warren, D.M. Garner, G.S. Clark, J.H. Brewer, and D.G. Fleming, Phys. Letters 65A (1978) 351. Mikula (79) R.J. Mikula, private cummunication, (1979) -217-Literature Cited - Appendix I Bre i t (31) G. Br e i t and I, 2082. Rabi, Phys. Rev. 38 (1931) Brewer (7 5) Carrington (67) Fleming (7 9) Gurevich (71) Perciv a l l-(76) Schenck (76) J.H. Brewer, K.M. Crowe, F.N. Gygax, and A. Schenck, i n Muon Physics, Vol. I l l , Eds., C. S. Wu and V.W. Hughes, Academic Press, New York, (1975), Chapter 7. A. Carrington and A.D. McLachlan, Introduction  to Magnetic Resonance, Harper and Row, New York, (1967), 14. D. G. Fleming, D.M. Garner, L.C. Vaz, D.C. Walker, J.H. Brewer, and K.M. Crowe, i n Positronium and Muonium Chemistry, H.J. Ache, Ed., American Chemical Society Advances i n Chemistry Series, i n press. I.I. Gurevich, I.G. Ivanter, E.A. Meleshko, B. A. N i k o l ' s k i i , V.S. Roganov, V.I. Selivanov, V.P. Smilga, B.V. Sokolov, and V.D. Shestakov, Sov. Phys. JETP 33 (1971) 253. P.W. Perci v a l and H. Fischer, Chem. Phys. 1(5 (1976) 89. A. Schenck, i n Nuclear and P a r t i c l e Physics  at Intermediate Energies, Ed. J.B. Warren, Plenum, New York, (1976) 159. Tinkham (64) M. Tinkham, Group Theory and Quantum  Mechanics, McGraw-Hill, New York, (1964), Chapter 5. -218-Literature Cited - Appendix II Arnold (68) Brewer (7 2) Brewer (75) Fleming (7 9) V.I. Arnold and A. Avez, Ergodic Problems i n Quantum Mechanics, Benjamin, New York, (19 68) 16. J.H. Brewer, Ph.D. Thesis, Lawrence Berkeley Laboratory Report, LBL-950, (1972). J.H. Brewer, K.M. Crowe, F.N. Gygax, and A. Schenck, i n Muon Physics, Vol. I l l , Eds., C. S. Wu and V.W. Hughes, Academic Press, New York, (1975), Chapter 7. D. G. Fleming, D.M. Garner, L.C. vaz, D.C. Walker, J.H. Brewer, and K.M. Crowe, i n Positronium and Muonium Chemistry, Ed., H.J. Ache, American Chemical Society Advances in Chemistry Series, i n press. Schenck (71) A. Schenck and K.M. Crowe, Phys. Rev. Letters 26 (1971) 57. -219-L i t e r a t u r e C i t e d - Appendix I I I B i s w e l l (73) L.R. B i s w e l l and R.E. R a j a l a , Los Alamos S c i e n t i f i c Laboratory Report LA-5144, (1973) . F e l l e r (50) W. F e l l e r , An I n t r o d u c t i o n to P r o b a b i l i t y Theory and i t s A p p l i c a t i o n s , Wiley, London, (1950), 337. Shlaer (74) S. Shlaer, Los Alamos S c i e n t i f i c Laboratory Report LA-511-MS, (1974). Thomas (73) R.F. Thomas, Los Alamos S c i e n t i f i c Laboratory Report LA-5404-MS, (1973). -220-Appendix I - The Time Evolution of the u Spin P o l a r i z a t i o n i n Muonium i n a Transverse Magnetic F i e l d . Solutions to the problem treated i n th i s Appendix may be found i n several references which use the density matrix formalism [Gurevich (71), Brewer (75), Schenck (76), Perci v a l l - (76)] . In th i s Appendix, the approach to the problem follows that of Fleming (79) which i s , perhaps, more phys i c a l l y transparent than the density matrix approach. A. State Vectors The i n i t i a l states of the system are most e a s i l y described using the u + spin p o l a r i z a t i o n d i r e c t i o n as the quantization axis. Since a l l of the muons are polarized while the electrons captured to form Mu are unpolarized, the i n i t i a l states are = \a^>± and ^ B ( 0 ) = |a$>j_ using the standard convention i n which the f i r s t a or 3 refers to u + spin and the second refers to e spin. The subscripts j_ indicate that the quantization axis i s perpendicular to the applied magnetic f i e l d . The application of a magnetic f i e l d transverse to the i n i t i a l muon p o l a r i z a t i o n d i r e c t i o n defines a new quantization axis and i t i s the task of th i s section to show the appropriate transformation of the i n i t i a l Mu state functions into t h i s new coordinate system. -221-The required transformation of state vectors i s i l l u s t r a t e d below: B + y spin y/^ p o l a r i z a t i o n 1 - 7 7 rotation 2 > about Y-axis old system Z ^ P o l + y spin a r i z a t i o n new system The rotation i s most e a s i l y applied to spin states labled with respect to the t o t a l Mu spin angular momenta, | JM>j_ , where J i s the t o t a l Mu spin and M i s i t s projection on the o r i g i n a l quantization axis, as usual. The i n i t i a l spin states can be expressed i n t h i s way by the appropriate manipulation of Wigner or Clebsch-Gordon c o e f f i c i e n t s [Tinkham (64)] , rM m mM-mMv Ym KM-m from which one obtains: |1, - t > ± = |6B > L |1,0>. = JL( | a B > L + I Ba>, ) I 0,0>, = 1 ( |aB>, - I Ba>. ) /2 X K D Thus, i K(0) = I a a ) . 1,1> I{J b(0) = I a3>,_ = 1 ( 1 1 , 0 ^ + |0,0>L ) /2* A rotation of a state function which i s a li n e a r combination of basis vectors, - 2 2 2 -m m m — — about Eulerian angles a,B, and y, i s accomplished with the application of the rotation operator [Tinkham (64)] R(a,B,y)^ = Z.<f>J [D (j) m m m' Following the Condon and Shortley phase convention given i n IT •' Tinkham, the required rotation i s a=0, 3 = — y = 0 , where 3 i s negative since i t would advance a r i g h t handed screw i n the negative y - d i r e c t i o n , The matrix elements are defined as n(j) f„ R v^ _ -im'g -imy v (-l ) V ( J+m) i (J-m) ! (J+.m') 1 (J-m' ) ,1  U I A ' ^ ' Y V M E E K K I ( j+m - K ) I(j-m'-K)!(K+m'-m) I . B > 2 j - 2 K-m1+m , . B\2K+m'-m • (cos|) J (-sin|) Since we are rotating the coordinate system (basis vectors) rather than functions, for which the above expression for E) was calculated, the required rotation corresponds to The required matrix i s , then, (j) D ( j ) (-Y,-B,-cO -1 -1_ / 2 1 / 2 Thus and , R ( 0 , | , 0 ) V 0 ) = i | l , ^ + /J|l,0j> + || !,-]>> R(0,|,0)^ B(0) = ~|.|1,1^ > + 1|1,-1^ + /||0,0^ where the subscripts || indicate that the states are quantized with respect to the magnetic f i e l d and where i t i s noted that the t o t a l l y symmetric basis function | 0, (£> i s unaffected by the r o t a t i o n . Using 1(1) to transform back into uncoupled spin -223-K2) states, [ro m ) , we obtain: ' ' y e ' ij^(0) = j(|aoCj[ + |33^ j> + |a3^ j> + | 3aj>) ^ B(0) = |.(-|acxj> + |3fcCj| + |aB>. - |Bc^) B Time Evolution of the Mu States In order to fin d the expectation value of the y + p o l a r i z a t i o n i n the Schrodinger picture, i t i s necessary to determine the time evolution of the Mu spin states. These are simply given by Tp (t) = U(t)^(0) where U(t) i s the time -i#t/n * evolution operator: U(t) = e ' where- 3X = 0 and (0) are the 9t eigenvectors of X . The Hamiltonian i s the sum of the y and e Zeeman terms and the y +-e hyperfine i n t e r a c t i o n : s\ /\ /\ X = g 3 BS - g 3 Bl + aS«I ^ e z ^ y z where g i s the electron or muon g-factor (which are e s s e n t i a l l y the same) , 3 e and 3 ^ are the e and y magnetons, S and I are. the. e and y spin operators, and 'a' i s the Fermi contact constant.-The electron and muon magnetons are equal to eh where m i s the 2mc corresponding electron or muon mass. The 'a' term has been calculated to be [see, for example, Carrington (67)] a = 1p g 3 e g 3 y k ( 0 ) | 2 i 2 where |^(0)| i s the pr o b a b i l i t y density of the electron at the muon. For the Is electron o r b i t a l ^ ( r ) - - I ' e - r / % -224-with * 2 / 2 a Q = ft /ye which i s the f i r s t Bohr radius; here y represents the reduced mass of the electron and muon. The eigenvalues of t h i s Hamiltonian are most e a s i l y determined by re-expressing the spin vector operators of the hyperfine term i n t h e i r vector components: X- = g 3 BS - g$ Bl + a (S I + S I + S I ) e z ^ y z x x y y z z K 3 ) Defining the r a i s i n g and lowering operators i n the usual way and S = s + i s x y S = S - iS x y i t i s r e a d i l y shown that S I„ + S I = h (S'I + S I ' ) x y y y Substitution of t h i s expression into 1(3) gives y\ /\ / \ ( S\ /\ s\ /\ /\ 7t = g 3 BS,, - g 3 BI + a{%(S I~ + S~I ) + S I } ^ e z ^ y z z z The matrix representation of ^ given by H^ ^ = < ^ J K |^| cf>^ > with respect to the basis set Ira m ) i s : H = ft CO CO 03 CO -225-where = k ( a B B/ft ± g3 B/ft) and to = — i s the hyperfine ± 2 - e ^ y o f t frequency. The eigenvalues of the Harailtonian are: 03 E./fi = to + 2 03 o 03 , V f t = — I ' + ( w+ + 4^ I ( 4 ) 03 E 0/h = - o o + -rE 3/ 4 03 • 0 03. , O , 2 . 0 X *5 2 The eigenvectors of H may be obtained by inspection for the l x l submatrices and by a tedious but straightforward c a l c u l a t i o n for the 2x2 submatrix: |,1> - |ac£| I 2> = s I a3> + c I f?a> I f |3> = | B B ^ | 4> = c|aB> - s| 3a> * II where 1(5) ~ - _L " • 2 A J + sh 1,, , x , h Q ~ / 2 ( 1 + / T — 2 > = / 2 ( 1 + / T — 1 } / o o + 4OJ , / l + X o + and 1 ( 1 •_. - 2 1 ± ,h _ l n _ x % 03 + 403 , ' 0 + 2.to. + 2 2 with x = . I t should be noted that c + s = 1. 03 0 The above res u l t s are i l l u s t r a t e d i n the fam i l i a r Breit-Rabi diagram i n Figure 1-1 [Breit (31)] . The zero f i e l d -226-F I E L D FIGURE: 1-1: Breit-Rabi diagram of the energy eigenstates of muonium i n an external magnetic f i e l d . The four allowed t r a n s i t i o n s (Am=+1) are indicated. In weak f i e l d s (B<L0 gauss), the t r a n s i t i o n s , C O 1 2 and o o 2 3 / are degenerate and provide the c h a r a c t e r i s t i c muonium frequency, = w = 8.76 x 10 *B rad s -227-s p i t t i n g or hyperfine frequency, to f for Mu i s 2.8 x 1 0 ^ rad s ^. The cross-over of u and u may be calculated bv setting 1 2 2 Denoting A g = g3 e/ft and A^ = g3 /ft and r e c a l l i n g the d e f i n i t i o n s of u and w_, t h i s cross-over i s found to be at A - A -i m- ft e y 1 • :y 2A - A o 2A o 2gm 3 o e y y ^ e e which corresponds to about 160 kgauss. It i s now possible to express the i n i t i a l state vectors 1(2) i n terms of the eigenvectors of the Hamiltonian I (5) : iDA(0) = ||1> + p|2> + |;|3> - q|4> and ^ B(0).= -||1> + q|2> + || 3> + p | 4> s + c s - c where p = 2 and q = 2 The time dependent Mu states are computed from the eigenvalues 1(4): • ,.x 1.-icoiti.v . -io), 11 „\ , 1 —iu), 11 _\ --iwut i .\ ^ A ( t ) = -je 1 11/ + pe 2 | 2/ + -e 3 | 3> - pe * | 4/ K 6 ) and * B ( t ) = ~\ e" i a 3 l tU> +qe- i a , 2 t|2> + \ ^ ^ \ ^ + p e ^ ^ l 4 > where O J . = E . /ft Since the c a l c u l a t i o n of the time evolution of the y + spin p o l a r i z a t i o n w i l l require computation of the expectation value of the muon spin operator, the c a l c u l a t i o n i s most e a s i l y done i n the basis |m in ) . In thi s basis, equations 1(6) ' y e ' become: -228-I (7) y ( t) = l e - i a 3 l t | a a > + ( p s e - i a j 2 t - q c e _ i u 4 fc) | a 3 > A 2 + | e - i a , 3 t | 3 3 > + ( P c e - i w 2 t + q s e " i W l t t ) | 6 a > and ij; B(t) = - | e ~ i U l t | a a > + ( q s e ~ i u 2 t + pee" 1"" t ) | a 3 > + le" I A > 3 T|B3> + ( q c e - i a ) 2 t - pse" 1"* | ga> H Time E v o l u t i o n of the u + Spin P o l a r i z a t i o n i n Mu In an MSR experiment i n t r a n s v e r s e magnetic f i e l d , we are i n t e r e s t e d i n the muon p o l a r i z a t i o n i n the x-y plan e . For t h i s , we d e f i n e the 'complex muon p o l a r i z a t i o n " o perator P = a x + i a Y y y y where a and cr are the f a m i l i a r P a u l i s p i n m a t r i c e s : Thus P y i s a form of the muon s p i n r a i s i n g o p e r a t o r : p = [o A = 2 i + y \o o j T h i s operator only a c t s on the muon s p i n p a r t of the I M ^ m ^ s p i n s t a t e s i n the u s u a l way: eg. P y | a 3 > = 0, P | 3a> = 21 aa) , e t c . A, The o b j e c t of t h i s s e c t i o n , then, i s to c a l c u l a t e <^Py (t))> : < P y ( t ) > = f <> A(t) | P u ! ^ A ( t ) > + d - f ) < ^ B ( t ) | P i j | ^ B ( t ) > where 0£ f<_ 1 i s the f r a c t i o n of y + t h a t i n i t i a l l y form Mu s t a t e ^,(0) and (1-f) i s the f r a c t i o n i n \0 (0) . Normally, i s assumed f= % [Fleming (79)] s i n c e the e l e c t r o n s are u n p o l a r i z e d . However, the more g e n e r a l case i s d e r i v e d here -229-since i t does not greatly complicate the cal c u l a t i o n and i t may have physical a p p l i c a b i l i t y i n certain ordered systems such s + c as c h i r a l molecules. After substituting p = 2 and s - c q = 2 and rearranging, the ca l c u l a t i o n y i e l d s : ( t ) ^ = hi [ (2f-l)sc+c 2] e i ^ i 2 t - [ ( 2 f - l ) s c - s 2 ] e ^ u 1 + [ (2f-l) sc+s 2] e l u 2 3 t - [ (2f-l) sc-c 2] e ~ I A ) 3 4 T } where CO^_. = CO^ - W J which are e x p l i c i t l y given below from eauations 1(4): 1(8) OJ. = OO + % - (CO 2 + W O ) 2 h 1 2 + 4 2 , C O 2 , h l k 2 ' 4 1(9) UO .. = W _ + + ( W + * + O ) 0 ) . = W - % + ( O J 2 + % ) % 2 3 2 + 00 - = -CO + <?o' + ( 0 3 2 +' W 0 ) k 2 Defining n = ^ ( O O , - OJ - ) = ( C O 2 + W O ) ^ ' - J V = % ['(1+x2 )^ - 1] . 2 3 . 1 2 4 2 2 where i t i s r e c a l l e d that x = 2ai+ , equations 1(9) become: CO 0 OO = OO - n A = - W _ + W - + 1 2 - Q . 1(10) CO = CO + CO + Q 00 = 0 0 + ^ -230-Substituting equations 1(10) into 1(8) and expanding exponentials, e l k t = coskt + i sinkt, y i e l d s : <(p (t£> = e l w - t { c d s V [ c o s ( f i +-%):t - i v sin(fi + %) t] + 2(2f - l)sc..sih uot sin(ti + uo).t} 1(11) 2 2 2 2 where i t i s noted c + s = 1 and c - s = v = X vl '+ x .• Noting that sc = % 1 and manipulating trignometric Al + x' i d e n t i t i e s gives the general expression: <P y(t)> = J 5 e i u - t {[1 + Is / (2f - l)]cosfit 1 + x' + [1 - X % ' - ( 2 f - 1 ) ] C O S ( C 0 0 + fi)t \1 + X X a + x y [sin (to + ti) t + sinftt] } ^ . - o K 1 2 ) This equation i s general for a l l magnitudes of magnetic f i e l d s . D„ Experimental Implications of <^ Py (t)^> It i s the task of thi s section to simplify the complex general expression for <(p^(t.)^> (equation 1(12)) i n terms of p r a c t i c a l experimental considerations. Two experimental constraints must be borne i n mind throughout t h i s section: (1) the p r a c t i c a l timing resolution of conventional counting and timing technology i s about I n s , and (2) the l i f e t i m e of y + i s 2,2 us which l i m i t s the maximum experimental time range to, at most, about 10. ys.. The hyperfine frequency, to i s 2.8 x 10^~® -231-rad s ^ which corresponds to a period of about 0.225 ns and i s not, therefore, experimentally observable. At the other l i m i t , 5 -1 frequencies slower than 1.2 x 10 rad s which have periods of greater than 50 ysare not observable with the y + 10 ystime range. <3?^  (t)> i s related to the experimental MSR "signal" S(<f>,t) = A(t) <P (<(>,t)> where A(t) i s the time dependent empirical asymmetry, and <j) i s the angle between the positron counter and i n i t i a l muon po l a r i z a t i o n . S(t) i n equation(8) of Chapter I i s a p a r t i c u l a r example of an MSR si g n a l . The signal appears i n the exponential histogram: N(c|),t) = N e _ t / T y .[1 + S((b,t)] + Bg 1(13) which i s analogous to equations (4) and (6) i n Chapter I. Hence, the signal i s the o s c i l l a t o r y muon function which remains when the exponential muon l i f e t i m e and background are removed from the time histogram. Chemical reactions and other relaxation e f f e c t s are not included i n S(cf>,t) here; these effects are incorporated into the formalism i n Appendix I I . Table 1-1 l i s t s values of the magnetic f i e l d dependent variables of equation 1(12) for a number of p r a c t i c a l l y available magnetic f i e l d strengths ranging from 1 gauss to 10 kgauss. This section examines ^P (t)^ for three magnetic f i e l d regimes. (i) Very TA7eak Fi e l d s (^lOgauss) - the Standard MSR Signal From Table 1-1, i s i s seen that for B. £10-gauss, x = 0, / 1 - \ -y 2 =1, and v = 0 to better than 1%. Furthermore, \1 + x / TABLE I F i e l d (gauss) -1: VALUES OF MAGNETIC 2 X V c FIELD DEPENDENT VARIABLES 2 IN EQUATIONS 1(8) AND I(12)t 2sc (JO CO CO! 2 C 0 2 3 CO! 4 CO 3 h (106) (10 6) 7 10 (It)') (10 ) (10 1 0) (10 1 0) (10 1 0) 1 0. 001 0. 001 0. 500 0. 500 1. 000 8.847 8.761 0. 000 0. 001 0. 001 2. 805 2. 804 3 0. 002 0. 002 0. 501 0. 499 1. 000 26.54 2 6.28 0.003 0. 003 0. 003 2 . 807 2. 802 5 0. 003 0. 003 0. 502 0. 498 1. 000 44.24 43.81 0. 007 0. 0 04 0. 004 2. 809 2. 800 7 0. 004 0. 004 0. 5 02 0. 498 1. 000 61.93 61.33 0.013 0. 006 0. 006 2 . 811 2. 798 10 0. 006 0. 006 0. 5 03 0. 497 1. 000 88.47 87.61 0.028 0. 009 0. 009 2. 813 2. 796 20 0. 013 0. 013 0. 506 0. 494 1. 000 177.0 175.2 0.111 0. 017 0. 018 2. 822 2. 787 30 0. 019 0. 019 0. 509 0. 491 1. 000 265.4 262.8 0.250 0. 026 0. 027 2. 831 2. 778 50 0. 032 0. 032 0. 516 0. 484 1. 000 442.4 438.1 0.697 0. 043 0. 045 2. 849 2. 761 75 0. 047 0. 047 0. 524 0. 476 0. 999 663 .6 657.1 1.569 0. 064 0. 0 67 2. 872 2. 740 100 0. 063 0. 063 0. 531 0. 469 0. 998 884.7 876.1 2.788 0. 085 0. 090 2. 895 2. 720 150 0. 095 0. 094 0. 547 0. 453 0. 996 1327. 1314 . 6.266 0. 125 0. 138 2. 942 2. 679 200 0. 126 0. 125 0. 563 0. 437 0. 992 1770. 1752. 11.12 0. 164 0. 186 2. 991 2. 640 300 0. 189 0. 186 0. 593 0. 407 0. 983 2654. 2628. 24. 90 0. 238 0. 288 3. 092 2. 566 500 0. 315 0. 301 0. 650 0. 350 0. 954 4424. 4381. 68.13 0. 370 0. 506 3. 310 2. 435 1000 0. 631 0. 534 0. 767 0. 233 0. 846 8847. 8761. 255.8 0. 620 1. 132 3. 936 2. 184 2000 1. 262 0. 784 0. 892 0. 108 0. 621 17695 17522 855.5 0. 897 2. 608 5. 412 1. 9 08 3000 1. 893 0. 884 0. 942 0. 058 0. 467 26542 26283 1600. 1. 029 4. 228 7 . 032 1. 77 6 5000 3 . 155 0. 953 0. 977 0. 023 0. 302 44237 43805 3238. 1. 142 7 . 619 10 .42 1. 662 10000 6. 310 0. 988 0. 994 0. 006 0. 153 88474 87610 7556. 1. 205 16 .32 19 .12 1. 599 t frequencies are given i n units of rad s -233-2 ^ 2 for small x, (1 + x ) 2 - 1 + %x ; thus n = % [ ( i + * 2 ) h - i ] - % ( i + % x 2 - i ) = % 2. 2 u o In t h i s l i m i t , equation 1(12) becomes: ^ ' 4 -/p (t£> = e 1 W - [f, cosflt + (l-f)cos(w 0 + Q)t] 1(14) y B<10gauss The r e a l part of the muon p o l a r i z a t i o n i s (including the counter phase dependence): Re <^ P (cj>,t£> = f cos(io_t + cf))cosfit y B<10gauss 1(15) + (1-f) cos(to_t + (J)) cos(coo + ft)t where w_ = h(g$e B/fi-g$^B/fi) = 1 0 3 t 0 y corresponds to the ch a r a c t e r i s t i c muonium precession frequency. Notice that the counter phase dependence i s added only to the Larmor precession parts of each term i n 1(15). By construction, the r e a l part of ^ P ( t ) ^ corresponds to the u + p o l a r i z a t i o n i n the x d i r e c t i o n and the imaginary part corresponds to the u + p o l a r i z a t i o n i n the y_.direction. Introduction of the counter phase to either the re a l or imaginary parts generalizes <^P^(t)y to correspond to any di r e c t i o n i n the x-y plane. For example, Re<^P (0,t)^> = Im <(^ P^  (^-', t)^ > . Since the hyperfine frequency i s too fast to be experimentally resolvable, the second term i n 1(15) averages to zero and t h i s f r a c t i o n (1-f) of the u + appears to be unpolarized. The remaining term shows the muonium precession, CO_, modulated by 5 -1 the slower beat frequency Q<2.8 x 10 rad s which corresponds to a period->22.5 ys for B<10 gauss. This beat frequency i s slow enough that i t may be ignored i n f i e l d s of less than 10 gauss -234-except, possibly, i n experiments which attempt highly precise measurements of very slow Mu relaxation rates (X^0.2ys "*") . The net observable signal i n th i s weak f i e l d regime thus reduces to the very simple expression: ( A,t)> =-f .cos(w_t + <M 1(16) Experiments of th i s type which monitor a single c h a r a c t e r i s t i c Mu frequency, IO_ = C O M U = 103 TO , are c a l l e d MSR experiments as defined i n Chapter I. In most cases, f i s assumed to be h . Generally, for reasons c i t e d i n Chapter I, the experimental asymmetry i s treated empirically so that t h i s f r a c t i o n i s indeterminable. Figure 1-2 i l l u s t r a t e s the time evolution of y + given by 1(15) assuming f = h and perfect experimental time resolution. Although Figure 1-2 was calculated for a lOOg f i e l d , equation / l \ 35 1(15) i s v a l i d at early times (a few ns) since • •—— =d = 1 U + xz| at:--100 .gauss to better than 1%. The effects of 9, are unobservable at early times at th i s f i e l d . The figure shows the fast hyperfine o s c i l l a t i o n i n the envelope of the slower muonium precession. In pra c t i c e , the fast o s c i l l a t i o n i s averaged over, leaving an observable muonium envelope of reduced amplitude. ( i i ) Intermediate F i e l d s (1 0<_B<_15 0gauss) - Two Frequency Muonium At magnetic f i e l d s less than 150 gauss, Table 1-1 shows 2= 1 and v<0.1. In these f i e l d s , the. r e a l part of \1 + x 2 equation I(T2). i s -235-FIGURE 1-2: The time evolution of the y T spin p o l a r i z a t i o n i n a 100 gauss transverse magnetic f i e l d from equation 1(15), assuming f=%,-- $=0. The fast o s c i l l a t i o n s are e s s e n t i a l l y at^the hyperfine frequency, co = 2.8 0 .x 10 rad s and the envelope i s at thg mujnium frequency, co M u (10 0 gauss) = 8^76 x 10^ rad s . The beat frequency, 9, = 2.8 x 10 rad s at 100 gauss, i s too slow to be observed on this'_time scale. Experimentally, the fast hyperfine o s c i l l a t i o n i s averaged over, leaving and observable coM envelope of reduced amplitude, indicated by the broken l i n e . -236-Re <(P (<f>,t£> = f'cos (00 t + <)>) c'osflt y 10<B<150gauss + (1-f) • cos (CO_t + 4) cos (CO q + fi)t + |-sin(a)_t + (J)). [sin (CO q + fi)t +• sinftt] As before, the hyperfine frequency i s unobservably fast so that the e f f e c t i v e expression f or Re<^ P^  (<p , t)y becomes: Re (s>,t)> -= f ,cos(CO_t + *)'cos«t 10£B<150gauss + ^ sin ( T O_t + p ) sinnt Generally t h i s expression i s re-written by manipulation of trignometric i d e n t i t i e s : Re <(py(c?>,t)> = hi (f-|) cos[ +. nyt + <?] 10<B<150gauss - - 1(17) + (f + J . ) . cos [ ( C O _ - fi)t + •<!>]}• 9 -1'' At 150 gauss, co_ = 1.3 x 10 rad s - corresponding to a period of ^5ns which i s just observable with a time resolution of 1 ns. The resultant signal i s a fast muonium o s c i l l a t i o n beating at the slower frequency fi. This was f i r s t observed experimentally by Gurevich (71) i n quartz at 95 gauss; the data are shown i n Figure 1-3. This i s referred to as the "two frequency precession" of the muon i n muonium. ( i i i ) High F i e l d s (>150 gauss) At these f i e l d s , the c h a r a c t e r i s t i c muonium frequency, CO = OJ , becomes immeasurably large. With an experimental, time resolution of about 1 ns, an observable ~ 4400 § 4200 3 4000 3800-3600-FIGURE 1-3 60 80 t ( n s e c ) "Two frequency precession" of the muon i n muonium i n fused quartz at 95 gauss [from Gurevich(71)]. The smooth l i n e i s a th e o r e t i c a l f i t to the data. The fast o s c i l l a t i o n s at the c h a r a c t e r i s t i c muonium frequency, • - ~ — -i^O j _ — -L 3. _ T „ j J T 4- ^T.IA i ~ 4-frequency, ft (95 gauss) = 2.7 x 10' rad s (95 gauss) = 8.7 x 10 rad s \ are modulated by the slower beat -238-signal i n the time domain must have a period of 5 ns or more, corresponding to a frequency of <_ 0.13 x 1 0 ^ rad s ^. Table 1-1 shows that none of the Breit-Rabi t r a n s i t i o n s have such a frequency between 150 gauss and 10 kgauss. At much higher f i e l d s (_> 100 kgauss), O J 1 2 w i l l once more become resolvable, but such f i e l d s are not experimentally available. If the experimental time resolution could be improved by a factor of 10, i t would be possible to observe co x2 up to 10 kgauss and to2 3 up to 1 kgauss. This i l l u s t r a t e s an ess e n t i a l difference between y +SR and MSR: while the former may be performed in magnetic f i e l d s of up to 15 kgauss, the l a t t e r i s constrained to f i e l d s of less than a:few hundred gauss. -239-Appendix II - The E f f e c t of Chemical Reaction on the Muon  Pola r i z a t i o n _A General This Appendix w i l l examine the muon spin p o l a r i z a t i o n i n a muonium ensemble which i s undergoing chemical reaction i n weak magnetic f i e l d (B £10 gauss). Excluding the effects of chemical reaction, the muon p o l a r i z a t i o n i s characterized by equation 1(16) of Appendix I: P (cj),t) =-f cos (to_t + *) When a muonium atom reacts chemically, i t s electron forms a chemical bond and becomes paired with another electron breaking the hyperfine int e r a c t i o n between the Mu electron and muon (in general, intermediate muonic radi c a l s are expected to be formed - t h i s s i t u a t i o n i s not considered here since the life t i m e s of any such r a d i c a l s formed i n the simple gas phase reactions studied i n thi s thesis are surely '.shorter than one hyperfine period of 0.225 ns) . Such a muon finds i t s e l f i n a diamagnetic environment where i t precesses at e s s e n t i a l l y the "free" muon frequency to ='^ir-to . Since to = U (g3 B/fi - gB B/ft) u 103 - - e y i s dominated by the electron magnetic moment, the sense of to_ precession i s opposite to that of to . The correction between the y free muon frequency and the diamagnetic muon frequency due to electron shielding (the so-called "chemical s h i f t " ) i s at the part per m i l l i o n l e v e l which i s not resolvable with present MSR technology. The stronger dipolar coupling between diamagnetic muons and protons i n water molecules (MuHO) has been resolved for c r y s t a l l i n e gypsum (CaSO. •2H„0) [Schenck (71)] , In f l u i d s , -240-however, th i s e f f e c t appears as a broadening which does not fundamentally e f f e c t the approximation that the diamagnetic u + frequency i s to . To zeroth order, because muonium atoms react at s t a t i s t i c a l l y d i s t r i b u t e d times, the coherently precessing Mu ensemble becomes an incoherently precessing y + ensemble as the reaction procedes. The net r e s u l t i s a relaxation of the Mu s i g n a l . The Mu signal, then, becomes a measure of the time-dependent p r o b a b i l i t y of a Mu atom surviving without chemical reaction. Hence, the relaxation of the Mu signal has a simple exponential decay as given by equation (8) i n Chapter I S(ct),t) = A M u e " X t c o s (w M ut + d»)-.+ -A cos.(u t * A) 1 1 d> where i t i s noted that toM Eoo_ v (see Figure I I - l ) . From the viewpoint of chemical k i n e t i c s , the rate equation for a bimolecular reaction of Mu with reagent X i s given by the standard second order expression: d[Mu] = -k[X][Mu] 11(2) dt where k i s the bimolecular rate constant, [X] i s the concentration of reagent X, and [Mu] i s the muonium "concentration." Here, of course, the concept of a Mu concentration invokes the ergodic p r i n c i p l e : an ensemble i n time i s formally the same as an ensemble i n space [Arnold (68)] . Since the t o t a l number of Mu atoms involved i n a reaction i s minute compared with the number of reagent 7 19 molecules (-10 compared with -10 ), equation 11(2) may be -241->~ LaJ >-CO cr. cr i — UJ 0 . 1 5 0 . 1 0 0 . 0 5 0 . 0 0 - 0 . 0 5 -0 .10 - 0 . 1 5 0 . 1 5 0 . 1 0 h 0 . 0 5 0 . 0 0 >~ CO ^ - 0 . 0 5 - 0 .10 - 0 . 1 5 MU I N 780 T O R R N 2 A T 6 . 9 G A U S S iiii 130 LIM H I I N 780 T O R R N. 6 . 9 G R U S S 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 3 . 0 3 . 5 4 . 0 T I M E I N LASEC (20 N S E C / B I N ; FIGURE II-1: The e f f e c t of chemical reaction on the muonium sign a l , S(cj),t) (equation 11(1)). The upper figure shows the slow background relaxation rate of _^ muonium i n pure N 2 giving X .= 0.34 ± 0.02 ys (equation 11(3)). In the presence of HI reagent (lower f i g u r e ) , the exponential decay of the muonium signal due to removal of muonium atoms by chemical reaction i s pronounced, giving a relaxation rate, X= 3.75 ± 0.38 ys . Each histogram contains about 10 events and the l i n e s are x -minimum f i t s to equation 11(1). -242-r i g o r o u s l y r e - w r i t t e n as a p s e u d o - f i r s t order r a t e , f o l l o w i n g the conventions of chemical k i n e t i c s : d [Mu] = -X[Mu] dt which i d e n t i f i e s the Mu r e l a x a t i o n r a t e (equation 11(1)) as the p s e u d o - f i r s t order r a t e constant, X, as given by the l i n e a r r e l a t i o n (see Figure I I - 2 ) : X = k[X] + X 11(3) o The i n t e r c e p t , X , i s introduced to account f o r "background" r e l a x a t i o n of the Mu s i g n a l due to e f f e c t s other than the chemical r e a c t i o n of i n t e r e s t , such as magnetic f i e l d inhomogeneity, pressure broadening from the r e a c t i o n medium, or background r e a c t i o n s w i t h chemical i m p u r i t i e s i n the moderator gas. Thus, a bimolecular r a t e constant i s simply determined from equation 11(3) by the d i r e c t observation of the r e l a x a t i o n r a t e of a Mu; s i g n a l as a f u n c t i o n of concentr a t i o n of the r e a c t i n g molecules, [X]. B_ Generation of a Coherent Diamagnetic Muon Background: X->15us B<_10 gauss In p r a c t i c e , Mu r e l a x a t i o n r a t e s are e x t r a c t e d from time histograms by the s o r t of f i t t i n g procedures described i n Chapter I I ; consequently, i t i s important t h a t the f u n c t i o n a l form of the muonium s i g n a l be p r o p e r l y described. Complications to equation 11(1) a r i s e under the c o n d i t i o n s : w,, < X << to 11(4) Mu ~ o because a s i g n i f i c a n t f r a c t i o n of muons i n Mu are placed i n t o diamagnetic environments by f a s t thermal chemical r e a c t i o n s -243-MU IN HBR WITH AR MODERATOR AT 295K 0 , 0 1 1 1 1 1 1 1 1 0 . 0 1 .0 2 . 0 3 . 0 4 . 0 5 . 0 6 . 0 7 . 0 HBR CONCENTRATION : ( 1 0 - 4 M) FIGURE II-2: The lin e a r dependence of the relaxation rate of the muonium signal on reagent concentration (equation 11(3)). The data points are weighted averages of X's extracted from l e f t and ri g h t histogran^and the error bars represent l c r . The l i n e i s a x ~ minimum f i t giving a bimoleeular rate constant, k(295K) = (9.1 ± 1.01 x 10 1 mole s and X^  = 0.26 ± 0.06 us -244-before the phase coherence i s l o s t due to the r a p i d to,, r Mu o s c i l l a t i o n . That such a r e a c t i o n i s thermal i s c l e a r when i t i s r e c a l l e d from Chapter I tha t t h e r m a l i z a t i o n of y + takes place i n 1. ns-•• and c o n d i t i o n 11(4) corresponds to r e a c t i o n times much longer than 0.2 ns, the hyperfine p e r i o d . Indeed, f o r experimental reasons c i t e d i n Chapter I I , a detectable MSR s i g n a l i s not observed f o r times l e s s than 10ns and, as described below, a s i g n a l must l a s t f o r at l e a s t 300 ns to be measurable.: The diamagnetic y + ensemble generated under c o n d i t i o n 11(4) does not precess i n c o h e r e n t l y as assumed i n Se c t i o n A above, but, i n f a c t , may give r i s e to a s i g n i f i c a n t diamagnetic y + s i g n a l . I t i s the task of t h i s s e c t i o n to evaluate the f u n c t i o n a l form f o r S(cf>,t) under such c o n d i t i o n s . The c o n d i t i o n i n 11(4) tha t X << to ensures that the f a s t o hyperfine term i n equation 1(15) of Appendix I may s t i l l be ignored. The maximum r e l a x a t i o n r a t e which i s p r a c t i c a l to measure d i r e c t l y i s t y p i c a l l y about 15 ys 1 which corresponds to a Mu s i g n a l l a s t i n g f o r about 300-400 ns- ( l i m i t e d by counting s t a t i s t i c s ) . The magnetic f i e l d range, 1<B<10 gauss, corresponds to 9,.ys 1 < t o M u < 89 ys" 1'; thus, f o r X - 15 ys 1 , c o n d i t i o n 11(4) i s reasonably w e l l s a t i s f i e d . The c a l c u l a t i o n below invokes the f o l l o w i n g assumptions: (1) the muon l i f e t i m e i s independent of the y + environment; (2) the f i e l d i s weak enough th a t beating may be ignored (0, i n equation 1(15) of Appendix I) and X i s small enough th a t the hyperfine o s c i l l a t i o n may be ignored (terms -245-containing OJ i n equation 1(15) of Appendix I ) , (3) the chemical process that places a single y T from Mu into a diamagnetic environment occurs instantaneously over s t a t i s t i c a l l y d i s t r i b u t e d times, (4) diamagnetic y + precess at the same frequency as "free" y + ions, (5) while i t i s not required that a l l y + entering the target i n i t i a l l y form Mu (allowing for y + to thermalize as free y + ions or to be placed into diamagnetic environments by fast epithermal reactions that occur before one hyperfine o s c i l l a t i o n ) , i t i s assumed that these fates are . arbitrated at e s s e n t i a l l y t=0, (6) the i n i t i a l phases of y + i n Mu and free y + are the same, (7) thermal free y + do not relax by, say, forming Mu after t=0. For convenience, both free y + ions and diamagnetic y + w i l l be referred to as "free" y + . A geometrical formulation of the problem i s schematically given by: y Mu t=0 \ y + Mu t=t' Here, C i s a counter placed at an angle § to the i n i t i a l y p o l a r i z a t i o n , \ represents the Mu spin p o l a r i z a t i o n and t + + represents the free y p o l a r i z a t i o n . The lengths of the arrows + represent the magnitudes of the Mu and free y polarizations i n + a r b i t r a r y units. By inspection, the t o t a l y signal i s given by S. t i (<() ft) = A (t) c o s U t + (j)) + A (t) COS(OJ t - (j>) t o t a l Y Mu Mu Y y y -246-where A M u ( t ) and A y (t) a r e t n e asymmetries of muons i n Mu and free y + ensembles respectively. A M u ( t ) i s given by equation (_7) in Chapter I and A^ (t) i s : A y(t) = A y + A^jl e-^'coscopt' dt' 11(5) where A and A„ are the amplitudes of the free y + and Mu y Mu * K ensembles at t=0, and O J = (to,, + oo ) i s the r e l a t i v e angular p Mu y ^ ve l o c i t y of the two ensemble spin vectors precessing i n opposite d i r e c t i o n s . Performing the integration i n 11(5), the t o t a l signal becomes: S t o t a l ( 4 ) ' t ) = A M u e " X t c o s ( u M u t + *} + V ° S ( V ~ *) A X • I I ( 6 ) Mu r -Xt .•. • • " -Xt , , , , " + - n x- [oo e sxnoo t - Xe COSOJ t + X]COS(OJ t - <p) X 2 + -OJ2 P P P y The correction calculated above i s i l l u s t r a t e d i n Figure II-3 which plots equation 11(1) (lower l i n e s asymptotically approaching an asymmetry of zero at long times) and equation 11(6) (upper lines) for a series of f i e l d s ranging from 2 to 10 gauss with i> - 1 radian, A.. = 0.1, A =0.0, and ^ ' Mu ' y ' X = 15 ys The to o s c i l l a t i o n i n the upper curves i s not obvious because of i t s low frequency at these f i e l d s . C learly, at 10 gauss, equation 11(1) i s a very good approximation to S+. > -i ((f),t) while at 2 gauss, the coherent diamagnetic y signal requires description by equation 11(6). In Figure II-4, equations 11(1) and 11(6) are again plotted for a series of X ' s ranging from 15 ys 1 to 300 ys. at a fixed f i e l d of 7.5 gauss with A M u=0.1, A^=0.0, and <j>=l radian, Although the Mu signal i n the.. . plots where X>15 ys i s not + -247-cc CO cr c u CO cr to en or 0.15 0.10 0.05 0.00 -0.05 0.10 0.05 0.00 CO ^ -0.05 -0.10 -0.15 10 G R U S S . R(MU)=0.1. x=15u5" 5 . 0 G R U S S . fl(MU)=0.1. X - 1 5 L I S " ' 3.0 G R U S S . fi(MU)=0.1. X=15LIS" 2.0 GRUSS. RIMU)=0.1 . x=15MS-' _L 0.0 0.5 1.0 1.5 2.0 T I M E I N nS 2.5 3.0 FIGURE II-3: The generation of a coherent diamagnetic muon background signal by fast chemical reactions of muonium. The li n e s are t h e o r e t i c a l muonium signals with pseudo f i r s t order rate constants,X = 15 ys , i n various weak magnetic f i e l d s for counters placed at 1 radian to the muon beam. The i n i t i a l muonium amplitude i s 10% and the i n i t i a l free muon amplitude i s 0%. The lower curves correspond to equation 11(1) and assume complete loss of muon phase coherence during chemical reaction; the upper curves correspond to equation 11(6) and show that i f X >_to , the muon phase coherence i s not l o s t during chemical reaction, but may give r i s e to a s i g n i f i c a n t "residual muon po l a r i z a t i o n " s i g n a l . -248-to CL or i — U J 2: CO cr QL U J CO cr 0.15 0.10 h 0.05 0.00 -0.05 0.10 0.05 0.00 -0.05 0.10 0.05 0.00 -0.05 h 0.10 0.05 0.00 CO a -0.05 -0.10 -0.15 7 . 5 GRUSS. R(MU)=0.1 . K=15u5"' 7 . 5 GRUSS. R ( M U ) = 0 . 1 . x=50u5"' 7.5 GRUSS. R (MU)=0 . 1 . x=99 uS~ 7 . 5 GRUSS. R(MU)=0.1. x=300uS~ 0.0 0.5 1.0 1.5 2.0 TIME IN uS 2.5 3.0 FIGURE II-4: The dependence of the amplitude of the "residual muon po l a r i z a t i o n " on muonium reaction rate at 7.5 gauss. As i n Figure II-3, the lower curves correspond to equation 11(1) and the upper curves to equation 11(6). Early determinations of muonium reaction rates [Brewer (72)] were made by the residual p o l a r i z a t i o n method by measuring the amplitude and phase of the diamagnetic muon signal by y SR and r e l a t i n g these to the fa s t muonium relaxation rates. In the present work, muonium reaction rates are measured d i r e c t l y by MSR. -249-of long enough duration to allow useful f i t t i n g of the data, these plots do i l l u s t r a t e the e f f e c t known as the "residual muon p o l a r i z a t i o n " [Brewer (75), Fleming (79)] i n which large values of X manifest themselves as larger values of the "free" muon p o l a r i z a t i o n . When X becomes s u f f i c i e n t l y large that the condition oo << co < X 11(7) o ~ i s f u l f i l l e d , the expression for s t o t a j ( ( ! ) »h) must include an integration analogous to 11(5) over the hyperfine terms of equation 1(15) i n Appendix I. The upshot of the ca l c u l a t i o n i s that f a s t Mu relaxations not only express themselves as large values of the residual p o l a r i z a t i o n , but also with rate dependent values of the phase of the residual p o l a r i z a t i o n . Previously, these facts have been exploited to measure the rates of f a s t reactions of Mu i n l i q u i d s by t h i s i n d i r e c t method mentioned i n Chapter I [Brewer (7 2)]..--250-Appendix III - Data Acquisition with High Current Muon Beams: Theory and Practice In Chapter I I , Section C, a q u a l i t a t i v e assessment of the problem of muon pile-up i s presented; i n the f i r s t two Sections of t h i s Appendix, the absolute magnitude of various multiple muon events and th e i r e f f e c t s on the resultant MSR time spectra are calculated. The l a s t Section of t h i s Appendix d e t a i l s the exis t i n g MSR data acquisition system at TRIUMF which deals with high muon beam currents. A The Optimal Good Event Rate It i s the task of t h i s Section to calculate the optimal "good" (see Chapter I I , Section C) event rate for an experiment with a data acquisition system that discards the ambiguous multiple muon events. The f i r s t c a l c u l a t i o n i s for "post- i K " second muons a r r i v i n g during the observation time T after the entry of the i n i t i a l c l o c k - s t a r t i n g muon, VK , into the target. For t h i s and subsequent calculations, i t i s assumed that the a r r i v a l of beam muons obeys a Poisson time d i s t r i b u t i o n ; t h i s assumption i s v a l i d over time in t e r v a l s much larger than the microscopic beam structure at the cyclo-tron radio frequency - 23 MHz at TRIUMF. The Poisson d i s t r i b u -t i o n function i s : P p ( n . * , t ) = m i n e - n t i i i , i , where P i s the p r o b a b i l i t y of n events occurring i n a time t, given an average event rate 71. If a v a l i d event i s defined as one where no other muons appear during a time T afte r the f i r s t -251-muon, then the p r o b a b i l i t y of an event being v a l i d i s given by Pp(0,72,T) = e~* T and the average v a l i d event rate, 71 i i s 7Zg = 7lsTm 111(2) Since 71 = 0 when 71 = 0 or 9l = °°, i t i s clear that t h i s g p o s i t i v e function of ^  has a maximum for constant T as shown i n Figure I I I - l ; t h i s point i s also i n t u i t i v e l y obvious since at low muon beam currents pile-up i s negligible, while at high beam currents v a l i d events are rare. The optimum beam rate occurs under the condition an 7 lR a x = k 1 1 1 ( 3 ) This important r e s u l t implies that under optimal conditions,-37% of the muons are free of pile-up and 63% must be rejected. For T = 4x^ i n the example of Chapter I t , the optimal beam 5 -1 current i s 1.1 x 10 s . It may be noted that function 111(2) i s asymmetrically peaked with respect to 71, r i s i n g rapidly to a maximum and tapering o f f slowly at large 71. This becomes a p r a c t i c a l consideration since muon beam currents are generally lowered incrementally by collimation rather than by fine adjustment of the proton beam i t s e l f . Since i t i s un l i k e l y that any collimator w i l l provide exactly the optimal e f f e c t i v e beam current, i t i s advantageous to bias i t toward a larger-than-optimal value rather than a smaller one. How can the above c a l c u l a t i o n be extended to include FIGURE I I I - l : The net good event rate (without pile-up) as a function of beam current for various muon decay gates. The good event rate i s given by equation 111(2) . -253-re j e c t i o n of "pre-y^" second muons a r r i v i n g during a time: T before the entry of y^, the cl o c k - s t a r t i n g muon? The answer i s simply to apply the above arguments backwards i n time. While i n t u i t i v e l y correct, t h i s i s also a recognition of the fact that the a r r i v a l of muons i s a Markov process (a random process i n which the future i s completely determined by the present and independent of the way i n which the present evolved) and that a Markov process i s also Markov i n reverse (see for example [Feller (50)]). The net r e s u l t for both post-y^ and pre-y^ event r e j e c t i o n i s that the rate of v a l i d events i s given by: _ ^. -PIT -#T, ~ -271T 7? = 7l(e • e ) = Tie. y which i s optimal when *Max = 2T- 1 1 1 ( 4 ) that i s , when the beam rate i s the inverse of twice the muon gate width. Thus, for T = 4T , ??M = 5.7 x 10 4 s _ 1 which, y jyiax again, corresponds to an event acceptance rate of 37%. B Spectral Distortions due to. Muon Pile-up In the preceding/- Section, i t was shown that pre- and post-y^ multiple muon events reduce the good event rate by the same amount, given pre- and post-y^ T-gates of the same width. However, t h i s does not imply that the spectral d i s t o r t i o n s due to pre- and post-y^ multiple muon events are of either the same magnitude or character; rather, i t i s shown in t h i s Section -254-that post- y^ events are much more devastating than pre-LK events. The e f f e c t s of muon pile-up upon the time histogram are calculated separately for the pre- and post-y^ cases i n three stages: f i r s t l y , the pile-up e f f e c t s on the apparent muon l i f e t i m e (ignoring the ySR or MSR signal) are calculated assuming 100% e f f i c i e n c y for decay positron detection ( i . e . 4TT steradians s o l i d angle, 100% counter e f f i c i e n c y ) ; secondly, t h i s c a l c u l a t i o n i s extended (again, ignoring the ySR or MSR signal) to allow for imperfect decay positron detection e f f i c i e n c y , 0 _< e £ 1, e = counter s o l i d angle x counter e f f i c i e n c y ; f i n a l l y , the muon pile-up e f f e c t s on the ySR or MSR signal are calculated for the case of imperfect decay positron detection. In the following discussion, two concepts must not -t/x be confused: (1) The function, e y, sometimes c a l l e d a "decay" curve, i s r e a l l y a "s u r v i v a l " curve giving the prob-a b i l i t y that a muon w i l l survive u n t i l time t. The p r o b a b i l i t y -t/x that a muon w i l l decay before time t i s (1 - e y). (2) Given that a muon has survived u n t i l t , the p r o b a b i l i t y that i t w i l l decay during the next i n t e r v a l dt i s the same for a l l muons; that i s , a 10 ys old muon has the same prob-a b i l i t y of decaying during the next ps as a 1 ns old muon. What i s the pr o b a b i l i t y that a muon w i l l decay during the in t e r v a l t and t + dt? This i s just the product of the -t/x p r o b a b i l i t y that i t has survived u n t i l t (that i s , e y) -255-and the p r o b a b i l i t y of i t decaying during the.next . dt, which i s a constant independent of t. Therefore, the pr o b a b i l i t y of a muon decaying between t and t + dt i s proportional to "decay" curve. I t i s t h i s p r o b a b i l i t y that i s i d e n t i f i e d with an experimental time histogram. cannot introduce d i s t o r t i o n s i n the measured l i f e t i m e of the muon since the p r o b a b i l i t y of decay per unit time i s the same for a l l muons. Lifetime d i s t o r t i o n s are introduced as experimental a r t i f a c t s , however, because i n a pile-up s i t u a t i o n the experiment cannot i d e n t i f y which muon decays; consequently, i t i s the f i r s t detected decay e that stops the clock. This e f f e c t may be understood by considering the following gedanken experiment: imagine a magic beamline that delivers exactly two muons at in t e r v a l s of T, the muon gate width, and imagine 100% decay e detection e f f i c i e n c y . Obviously, one muon w i l l generally decay before the other. Since i t i s the f i r s t muon decay that stops the clock, an . accumulated histogram w i l l be strongly biased toward early times, thereby reducing the apparent muon l i f e t i m e . For a given pair of muons entering the target at t = 0, what i s the pro b a b i l i t y that the clock w i l l not be stopped before some l a t e r time t? Denoting the p r o b a b i l i t y that the nth muon w i l l survive u n t i l t as -t/x e ' and i n t h i s sense e - t / T y may be thought of as a At f i r s t glance, i t might appear that pile-up events -t/x = e ' y III (5) -256-and r e c o g n i z i n g t h a t the decay or s u r v i v a l of i n d i v i d u a l muons are s t a t i s t i c a l l y independent events, the re q u i r e d p r o b a b i l i t y i s p ( s l * s 2 ) = P ( s 1 ) P ( s 2 ) = e " 2 t / x y 111(6) That i s , the p r o b a b i l i t y t h a t the c l o c k w i l l not be stopped by t i s j u s t the p r o b a b i l i t y t h a t both muons su r v i v e (at l e a s t ) u n t i l t . S i m i l a r i l y , denoting the p r o b a b i l i t y that the nth muon w i l l decay before t as P(d ) = 1 - e " t / T y I I I (7) i t i s seen th a t the p r o b a b i l i t y t h a t the c l o c k w i l l be stopped before t: P(d 1+d 2) = p(d 1) + P(d 2) - P(d 1-d 2) = P(d x) + P(d 2) - P ( d 1 ) P ( d 2 ) - t / x . n - t / x ,2 1 1 1 (8) 1 - e u) - (1 - e u) = 2(1  w Ly) " (1 - e w y) , ~2t/x = 1 - e y I t may be noted t h a t P ( s - L , s 2 ) + P (dj+d^) = 1, as i t should. As discussed i n the previous paragraph, an experimental time histogram corresponds to the p r o b a b i l i t y t h a t the c l o c k i s not stopped before t , but does stop between t and t + dt and that t h i s i s p r o p o r t i o n a l t o the p r o b a b i l i t y t h a t the c l o c k i s not stopped before time t . Since there are two muons r e s i d e n t i n the t a r g e t at t i n t h i s example, the p r o b a b i l i t y of some muon decaying between t and t + dt i s doubled, and the experimental histogram has the form N(t) = 2 e " 2 t / T y In t h i s gedanken experiment, then, the measured muon l i f e t i m e -257-i s hi . T n e l a s t expression must be normalized by di v i d i n g by 2 making i t correspond to one muon at t = 0 so that i t may be compared with the th e o r e t i c a l non-distorted histogram (N(t) = -t/x , . . e y) , giving: X T -2t/x N q ( t ) = e y In t h i s example, the normalization i s t r i v i a l and i t makes the argument that led to the extra factor of 2 i n the f i r s t place seem superfluous. However, when the procedure used i n t h i s example i s applied to more complex cases below, the normaliza-tions that r e s u l t are n o n - t r i v i a l . (i) Pre-y^ Muons and x y : 100% Decay Positron Detection E f f i c i e n c y Equation 111(6) may be extended to give the prob-a b i l i t y that the clock w i l l not be stopped before t i f n muons enter the target at t = 0: P(s,-s 0-...-s ) = e ~ n t / x y 111(9) 1 2 n K where P(s 1)=P(s,,)-•••=P(s ); and equation 111(8) may be extended to give the p r o b a b i l i t y that the clock w i l l be stopped before t i f n muons enter the target at t = 0: P(d 1+d 2+. . ,+dn) = I (-l) k - 1(£) P ( d k ) k k = 1 111(10) , -nt/x = 1 - e y where P. (d, ) =P (d„) =. . . =P (d ). 1 2 n Pre-y^ muons arrive i n the target before t = 0 (when y^ arrives) but may not be resident in the target at t = 0 because they have already decayed. What, then, i s the -258-p r o b a b i l i t y of there being n muons i n the target at t = 0 (not counting the y muon)? Consider an a r b i t r a r y time i n t e r v a l T i before t = 0 ( i t w i l l be shown eventually that the following c a l c u l a t i o n i s independent of ? f o r s u f f i c i e n t l y large Y and that T - T, the muon gate width, f u l f i l l s t h i s condition). The p r o b a b i l i t y of n muons entering the target during 7 for an average beam rate of 71 muons per unit time i s given by the Poisson d i s t r i b u t i o n , equation I I I ( l ) . Since, on average, the p r o b a b i l i t y of a muon a r r i v i n g i n the target during any subinterval A ? " of T i s the same for a l l AT, the average p r o b a b i l i t y that a muon a r r i v i n g during T survives u n t i l t = 0 i s : where P denotes the average p r o b a b i l i t y . The subscripts used i n equation III (5) have been dropped since t h i s p r o b a b i l i t y i s the same for a l l muons; also, Y has been included as an argument of P(s). I t may be noted that equation 111(11) goes to the proper l i m i t s of T: lim y.. - 7 7 T , , r-o r ( 1 " e y ) = 1 and lim Ty.-. ~Vi v _ n r+co r" ( 1 " e U) = 0 S i m i l a r i l y , the average p r o b a b i l i t y that a muon entering the target during Y has decayed by t = 0 i s : P(d(r) ) = 1 - P(s(r) ) III (12) -259-Assuming that there are no muons i n the target at t = 0 - V , the p r o b a b i l i t y of there being n muons i n the target at t = 0 i s the pr o b a b i l i t y that: n a r r i v e during T'x the prob. that a l l l a s t u n t i l t=0 + n+1 arr i v e during7'x the prob. that a l l but 1 l a s t u n t i l t=0 + n+2 arr i v e during T x the prob. that a l l but 2 l a s t u n t i l t=0 + . . . This may be expressed symbolically; by combining equations III(11) and 111(12) with the Poisson d i s t r i b u t i o n : t ( k) p ( k , ^ , r ) p ( s ( r ) ) n p ( d ( r ) ) k " n H K B ) k=n n p Combining t h i s expression with equation III (9) m u l t i p l i e d by the number-of. muons i n the target-gives the -unnormalized histogram: N(t,#,r) = £ n=0 oo E ( k) p ^ ( k , ^ , r ) p ( s ( r ) ) n [ i - p ( s ( r ) ) ] k _ n k=n n p III(14) , , -(n+1) t/x • (n+1) e y where n+1 refers to n pre-y^ muons plus the y^ muon. It must now be v e r i f i e d that equation 111(14) i s not a function of Y for s u f f i c i e n t l y large T. Since the l a s t term of equation 111(14) i s not a function of Y (but only a function of t > 0), i t may be set to 1 ( i . e . t = 0) and the equation re-written by expanding the Poisson term: N(0,?2,7) = £ (n+l)P(s(r)) ne" ? ? ?' T, (k) [l-p"(s (71 ). ] k _ n n=0 k=n Changing the index of the second summation y i e l d s : -260-co n 0 0 in N(0,ft/n = £ ( n + l ) P ( s ( 7 ) ) n e ~ ^ 7 - ^ p - E i g p - [ 1-p (s (T) ) ] m n=0 n ' m=0 m-_ co n Substituting equation 111(11) for P(s(T)) and e x = £ x-j-h=0 n* yi e l d s : N(o,*,r> = s ( n + i ) P ( s ( r ) ) n i222Ln e - ^ V ^ ( s ( r ) ) O n • ™ (n+l)r<»» ,, n - H T m (e T / x y ) 2 - ^ - j — - i n x (1-e y ) ] e y e y v H ; n=0 n! y or N(0f7l) = e - / ? T y Taking the l i m i t of t h i s expression as T •+ 0 0 gives: oo lim N(0 f*,r) = I ^ r - W - t , , ) n e " ^ T y r^oo n=0 n- y {Tlx )m °° (fit ) n y iri=0 m! n=0 ri! I l l (15) = i + nx y As a check, one may arrive at t h i s r e s u l t by answering the question, "What i s the average number of muons i n the target at any time?" The answer i s simply the in t e g r a l of the product of the beam current and the muon survival probability: / ^ e " t / T y dt = ^ T y ( e ° - e ^ y ) = Tlx y where the lower integration l i m i t r e f e r s to the time the beam i s turned on and the upper integration l i m i t refers to some very much l a t e r time. Thus, when the y ^ muon enters the target at t = 0, there are, on average, already Tlx pre-y^ muons i n the target for a t o t a l of l+7lx muons. It i s e a s i l y v e r i f i e d that equation 111(14) i s reasonably independent of T f o r T ^ T , a t y p i c a l muon decay gate. For example, i f the -261-upper i n t e g r a t i o n l i m i t of the l a s t expression i s set to T- T - 4T / say, the r e s u l t i s accurate to b e t t e r than 2% y 1 Equation 111(15) provides the n o r m a l i z a t i o n f o r equation I I I (14) N (t,7l) = n=0 o •t k=n 0 P D(k,?Z,T)P(s(T) ) n [ l - P ( s ( T ) )] k _ n 1 + 7cT , ,,, -(n+1)t/x • (n+l)e y y I I I (16) - t / T The true muon s u r v i v a l curve, e y, i s compared wi t h equation 111(16) f o r v a r i o u s beam currents i n Figures I I I - 2 and I I I - 3 . As expected, the e f f e c t of pre-y^ muons i s pronounced at e a r l y times but diminishes to i n s i g n i f i c a n c e at l a t e times, as evidenced by the f a c t t h a t the l o g a r i t h m i c curves are p a r a l l e l at l a t e times. This i s because, by d e f i n i t i o n , a pre-y^ muon i s ol d e r than the y^ muon, and so i t s chance of s u r v i v i n g u n t i l t = 4ys, say, i s much l e s s than that of the y^ muon. The apparent muon l i f e t i m e s obtained by - t / T f i t t i n g the histogram to e y over a 4ys time range would be (from Figure I I I - 3 ) 2.0, 1.85, and 1.7 ys f o r beam currents of 3 -1 50, 100, and 150 x 10 s r e s p e c t i v e l y . ( i i ) Pre-y^ Muons and T y ; e Decay P o s i t r o n Detection E f f i c i e n c y The p r o v i s i o n of a decay p o s i t r o n d e t e c t i o n e f f i c i e n c y means tha t there are two p o s s i b l e outcomes of a muon decay: e i t h e r i t i s detected or i t i s not. Using the n o t a t i o n of the previous S e c t i o n , the p r o b a b i l i t y t h a t a muon decays and i s detected i s FIGURE III-2: The e f f e c t of pre-LK muons on the apparent muon l i f e t i m e , with e = 100% positron counting e f f i c i e n c y . The upper curves i n each plot show the "true" histogram, -t/x e u, while the lower curves show equation 111(16) for 3 -1 beam currents of 50, 100, and 150 x 10 s . With the muon decay gate T = 20 ys, t h i s c a l c u l a t i o n i s accurate to about 1 ppm. -263-0.001 FIGURE I I I - 3 : Logarithmic p l o t s of Figure I I I - 2 . At l a t e times, the lower, : pre-y^ curves are p a r a l l e l to the true muon l i f e t i m e curves, showing t h a t the e f f e c t of pre-y^ muons on the measured muon l i f e t i m e i s only important at e a r l y times. -264-P(d) = E ( l - e t / T y ) S i m i l a r i l y , the p r o b a b i l i t y that a muon decays and i s not detected i s P(c() = (1-e) ( l - e ~ t / x y ) Proceeding as i n the previous S e c t i o n , an equation analogous to equation 111(6) may be w r i t t e n to give the p r o b a b i l i t y t h a t the c l o c k w i l l not be stopped before t i f two muons enter the t a r g e t at t = 0: P(s,-s ) + P(s ) + P(di -s ) + P ( % ) x z . x z x z . x z 111(17) = ( e " t / T y ) 2 + 2 ( l - e ) e ~ t / x y ( l - e ~ t / x y ) - + (1-e) 2 ( l - e ~ t / x y ) 2 The f i r s t term i s i d e n t i c a l t o equation 111(6), the second term corresponds to one muon s u r v i v i n g and the other decaying undetected, and the l a s t term corresponds to both muons decaying without d e t e c t i o n . S i m i l a r i l y , the p r o b a b i l i t y t h a t the c l o c k w i l l be stopped before t may be w r i t t e n (analogous to equation I I K 8 ) ) : P(d x+d 2) = 2 e ( l - e ~ t / x y ) - e 2 ( l - e ~ t / x y ) 2 111(18) that i s , i t corresponds t o the p r o b a b i l i t y of e i t h e r muon decaying w i t h d e t e c t i o n . Again, i t i s r e a d i l y checked that the sum of equations 111(17) and 111(18) i s one. Equation 111(17) may be g e n e r a l i z e d to correspond to the case of n muons en t e r i n g the t a r g e t at t = 0: n P i • = Z rn-\ . k . - t / x . n-k,n - t / T .k not stopped ^ Q ( K J (1-e) (e / y) (1-e ' y) I I I (19) -265-and equation 111(18) may be g e n e r a l i z e d i n l i k e manner stopped P ^ n r s n ^ = * S ^ 111(20) k=l The experimental histogram corresponds to the p r o b a b i l i t y t h a t the c l o c k has not stopped before t but does stop between t and t + d t . When each term i n equation 111(19) i s m u l t i p l i e d by the number of muons s t i l l i n the t a r g e t at t , the d e s i r e d r e s u l t i s obtained: N(t) = I (?) ( l - e ) k ( e " t / T y ) n " k ( l - e " t / T U ) k ( n - k ) 111(21) k=0 K Notice t h a t the f a c t o r of (n-k) e l i m i n a t e s the l a s t term i n equation I I I (21) corresponding to the s i t u a t i o n i n which a l l muons decay undetected before t . One can now w r i t e the experimental histogram analogous to equation 111(14) corresponding to pre-y^ muons w i t h a counting e f f i c i e n c y e by combining equation 111(21) w i t h expression 111(13) y i e l d i n g : °° °° — — k- 1 N(t,?l) - I [ E (J) P (k,?l,T)P(s(T) ) n [ l - P ( s ( T ) )] K n n=0 [k=n P J -m=n+l 111(22) „ , „. ria\ . £. - t / x ,m-£,T -t/x . £> . Z (m-£) [ J (1-e) (e y) (1-e y) • 1=0 x J where m = n+1 corresponds to n pre-y^ muons plus the y^ muon. Since the i n t r o d u c t i o n of a p o s i t r o n counting e f f i c i e n c y does not a l t e r the d e r i v a t i o n of expression 111(13), equation I I I (21) i s w r i t t e n as being independent of T f o r T = T as discussed p r e v i o u s l y . Noting that m=n+l „ , , n , / „ v / n\ r^h ri \ & t - t / x xm-£,. -t/x . £ T. (m-£) [ J (1-e) (e y) (1-e y) £=0 * -266-- " v 1 f n i i o \ (n+1) 1 , - t / i ,n+l-£ M - t / x . - t / x v £ - ^ (n+l-£) £, ( n + 1 _ & ) , (e y) (1-e-e y+ee y) , - t / x ^ rn-N , - t / T , n-£... - t / T , - t / T . £ = (n+l)e y E (.J (e y) (1-e-e y+ee ' y) i? = 0 111(23) = ( n + l ) e " t / T y [ ( 1 - e ) + e e " t / T y ] n where the l a s t step a p p l i e s the binomial theorem, (a+b) n = n ,-n-i n-£ £ Z ( J a b., equation 111(22) becomes: £=0 N 0(t,7l) = n=0_ Z ( k] P ( k , 7 l , T ) P ( s ( T ) ) n [ l - P ( s ( T ) ) ] k n k=n n p i + mv (n+l)e t / T y [ ( l - e ) + ee t / T y ] n I I I (24) This equation i s i d e n t i c a l w i t h equation 111(14) except f o r the f a c t o r of [(1-e) + ee 1 " / / x y ] n which i s simply the binomial d i s t r i b u t i o n of success or f a i l u r e i n p i l e - u p muon decay d e t e c t i o n . Notice t h a t as e •> 1, equation 111(24) becomes i d e n t i c a l to equation 111(14), as i t should. Furthermore, i n the l i m i t of 71 -> 0 (no pile-up) , the histogram becomes - t / T simply N(t) = e y as i t should since l i m ( 7 l t ) k = 1 i f k = 0 = 0 i f k > 1 - t / T Equation 111(24) i s compared w i t h e y f o r v a r i o u s muon beam cur r e n t s i n Figures I I I - 4 and I I I - 5 w i t h a p o s i t r o n d e t e c t i o n e f f i c i e n c y of 10%. C l e a r l y , the low decay p o s i t r o n d e t e c t i o n e f f i c i e n c y d r a m a t i c a l l y decreases the e f f e c t of pre-y^ muons by decreasing t h e i r opportunity f o r i n t e r f e r e n c e . -267-0 1 2 3 4 5 6 7 8 9 10 TIME IN uSEC FIGURE I I I - 4 : The e f f e c t of pre-y^ muons on the apparent muon l i f e t i m e w i t h e = 10%. Comparison w i t h Figure I I I - 2 shows th a t the i n t r o d u c t i o n of a p o s i t r o n counting e f f i c i e n c y d r a m a t i c a l l y reduces the d i s t o r t i o n due to p i l e - u p since the pre-y^ muons have l e s s chance to i n t e r f e r e w i t h the measurement. -268-FIGURE I I I - 5 : Logarithmic p l o t s of Figure I I I - 4 . In the top p l o t , the e f f e c t of pre-y^ muons i s so small t h a t the d i s t o r t e d curve i s almost c o i n c i d e n t w i t h the t r u e curve on t h i s s c a l e . -269-The dependence of the apparent muon l i f e t i m e on the decay p o s i t r o n d e t e c t i o n e f f i c i e n c y i n d i c a t e s t h a t any muon l i f e -time measurements using more than one p o s i t r o n telescope may not give the same r e s u l t , even though the telescopes look at the same t a r g e t . The apparent muon l i f e t i m e s obtained by -t/x f i t t i n g the histogram to e y over a 4ys time range would be (from Figure I I I - 5 ) 2.16 and 2.13 ys f o r beam cur r e n t s of 100 3 -1 and 150 x 10 s r e s p e c t i v e l y . ( i i i ) Pre-y^ Muons and the MSR S i g n a l ; e Decay P o s i t r o n  Detection E f f i c i e n c y In the f o l l o w i n g c a l c u l a t i o n , i t i s assumed that the muon precession frequency ( e i t h e r i n muonium or as " f r e e " muons) i s s u f f i c i e n t l y l a r g e t h a t a l l pre-y_^ muons are out of phase w i t h the precession of the y^ muon and th e r e f o r e do not co n t r i b u t e t o the precession s i g n a l . This assumption, v a l i d i n almost a l l experimental s i t u a t i o n s , n a t u r a l l y p a r t i t i o n s equation 111(21) i n t o two sets of terms: those due to d e t e c t i o n of y^ decay which manifest a muon precession s i g n a l ; and those due to d e t e c t i o n of pre-y^ decay without a muon precession s i g n a l . Consider the general s i t u a t i o n when n muons are i n the ta r g e t at t = 0, not counting the y_^  muon. The p r o b a b i l i t y t h a t the c l o c k has not been stopped before t , but w i l l be stopped by the ]i. muon between t and t + dt i s -270--(n+1)t/x , -nt/T = e y + ne ... . ... - t / T . , rri\ - ( n - l ) t / T y(1-e)(1-e u) + [ 2J e y ... ,2... - t / T ,2 , , - t / T ,, - t / T .n,, . n • (1-e) (1-e y) + ... + e y (1-e y) (1-e) where the f i r s t term represents the case where a l l (n+1) muons sur v i v e u n t i l t , the second term corresponds to the n permuta-t i o n s i n which one of the pre-y^ muons decay undetected before t , and so on, u n t i l the l a s t term representing the case where a l l n pre-y^ muons decay before t . Re-wr i t i n g t h i s expression and applying the binomial theorem as i n the d e r i v a t i o n of equation 111(23) above, one o b t a i n s : " rm ., , £, - t / T ,n+l-£., - t / T . £ P y = £ ( J (1-e) (e y) (1-e y) £=0 • £ - t / T „ fn>, . , = e y E ( J ( £=0 * - t / T >n-£.1 - t / T ^  - t / T ,£ e y) (1-e-e y+ee y) I I I (25) - t / T r ., , , - t / T , n = e y [ (1-e) + ee y] S i m i l a r i l y , the p r o b a b i l i t y t h a t the c l o c k has not stopped before t , but w i l l be stopped by a pre-y^ muon decay during t and t + dt i s : P = n e - n t / T y ( l - e ) ( l - e - t / T y ) + ( n - 1 ) n e " ( n " 1 J t / x y pre-y_^ (1 ,2,, - t / T ,2 , , „. rn^ - ( n - 2 ) t / T -e) (1-e ' y) + (n-2) [„J e i (1-e)" (1-e >-> - t / T . n , (1-e y) + ne 2 y ) " + ... + ne -(n+1)t/T 3 / n _ - t / T , . , 3 , ^ „^-t/x y d - e ) n I I I (26) , ,. -nt/T y + n ( n - l ) e y -e) ( l - e - ^ y ) + (?) (n-2) e" ^  t / T y (1-e) (1 - t / T .2 , (1-e y) + ,, - t / T , n - l (1-e y) -2t/T , n - l + ne ' y (1-e) -271-where the f i r s t n terms correspond to the cases where the y^ muon i s among the muons that decayed undetected before t and the l a s t n terms correspond to the cases where the y^ muon i s among the s u r v i v o r s at t . In p a r t i c u l a r , the f i r s t term corresponds to the case where the y^ muon has decayed undetected before t , ? l e a v i n g n s u r v i v i n g pre-y^ muons which may decay between t and t + d t ; the second term corresponds to the case where the y^ muon and any one of the n pre-y^ muons have decayed undetected before t l e a v i n g any of n-1 s u r v i v i n g pre-y^ muons which may decay between t and t + d t ; and so on u n t i l the nth term corresponding to the case where the y^ muon and a l l but one of the n pre-y^ muons have decayed undetected before t , l e a v i n g one s u r v i v i n g pre-y^ muon tha t may decay during t and t + d t . The (n+l)th term corresponds to the case where a l l muons have survived u n t i l t l e a v i n g n pre-y^ muons tha t may decay between t and t + dt (decay of the y^ muon belongs to equation 111(25)); the (n+2) term corresponds to the case where any one of n pre-y^ muons has decayed before t l e a v i n g (n-1) pre-y muons tha t may decay between t and t + d t ; i and so on. In the l a s t n terms, y. sur v i v e s both t and 1 t + d t . Proceeding as before, equation 111(26) becomes _ n rn-\ ., . £, -t/x . n+l-£... - t / T . £ pD r e . „ = £ n L j ( l - - e ) (e y) (1-e y) p y i £=0 - t / T n rn-i . - t / T ,n-£,, - t / T , - t / T . i y I l0) (e y) (1-e-e y+ee y) n e . u z ........... _ . . . _ . . . £ £=0 4 . / 4 . / 111(27) - t / T r ., . , - t / T ,n ne y [ (1-e) + ee y] -272-C l e a r l y , the sum of equations 111(25) and 111(27) i s equal to equation 111(23) as expected. Denoting the MSR s i g n a l as S(t) as defined i n Appendix I and Chapter I , and combining equations 111(25) and 111(27) w i t h 111(24), the normalized histogram becomes: N Q (t,7l) = n=0 Z ( k j P (k,7Z,T) P ( s ( T ) ) n [ l - P ( s ( T ) ) ] k n k=n n p 1 + Tlx U - t / x r ... . , - t / x , n , - t / x T T T , „ 0 . ne y [ ( l - e ) + ee y] + e y 111(28) • [(1-e) + e e " t / T y ] n ( l + S ( t ) ) I t may be noted t h a t when 91 = o (no p i l e - u p ) , t h i s reduces to - t / x e y (1 + S ( t ) ) as expected. This c a l c u l a t i o n shows t h a t pre-y^ muons not only d i s t o r t the apparent muon l i f e t i m e upon which the MSR s i g n a l i s superimposed, but they a l s o generate an exponential background c o n t a i n i n g no MSR s i g n a l . Figures I I I - 6 and I I I - 7 i l l u s t r a t e the r e s u l t of t h i s c a l c u l a t i o n at va r i o u s beam cu r r e n t s and a 10% decay p o s i t r o n counting e f f i c i e n c y , f o r the very simple MSR s i g n a l : S (t) = A^ u cosco^t at 5 gauss. Four f u n c t i o n s are p l o t t e d i n each F i g u r e : (1) the upper muon precession curve i s the "true" histogram: N(t) = e ~ t / / T y ( l + A.. cosw.. t) Mu Mu (2) the apparent histogram given by equation 111(28) i s almost superimposed on the true histogram (see the d e t a i l e d p l o t i n Figure I I I - 7 ) . -273-PRE-n, 2ND p. e=lOX. N=100K. f\m=0.\ 0 1 2 3 4 5 6 7 8 9 10 TIME IN uSEC FIGURE I I I - 6 : The e f f e c t of pre-yu muons on the MSR s i g n a l w i t h e = 10%. The top two curves i n each p l o t are almost c o i n c i d e n t , but can be d i s t i n g u i s h e d i n Figure I I I - 7 . The measured s i g n a l , which i s almost i d e n t i c a l w i t h the true s i g n a l i n each p l o t , i s made up of a s i g n a l - b e a r i n g curve (the lowest s i n u s o i d a l curve i n each p l o t ) , and an approximately exponential curve without a s i g n a l which i s due to pre-y. muon decay d e t e c t i o n . .-• P R E - L i j 2ND | i . e=10#. N=150K. RMU=0.1 0 : 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 3 . 0 3 . 5 4 . 0 4 . 5 5 . TIME IN uSEC FIGURE I I I - 7 : The e f f e c t of p r e - i K muons on the MSR s i g n a l . ( d e t a i l ) . Of the top two curve the one wit h the l a r g e s t amplitude i s the t r u e s i g n a l , w h i l e the apparent s i g n a l has a d i s t o r t e d exponential and an amplitude reduced by 1 + ^ T y . -275-(3) the lower s i n u s o i d a l exponential corresponds t o £ n=0 1 £ ( k) P (k,7i,T)P(s(T) ) n [ l - P ( s ( T ) ) ] K _ n k=n n P NQ(t ,n) = 1 + 7lT\x I I I (29) • e ~ t / T y [ ( l - e ) + e e ~ t / T y ] n (1 + S (t) ) (4) the lower exponential corresponds to N 0(t,7l) = £ n=0 OO -E (K) P (k,?Z,T)P(s(t) ) n [ l - P ( s ( T ) ) ] K ~ n k=n p 1 + Tlx y I I I (30) -t/x ,,, . , - t / r ,n • ne y [ (1-e) + ee y] Curve (2) i s the sum of curves (3) and (4). As i l l u s t r a t e d i n Figure I I I - 7 , pre-y^ muons reduce the e f f e c t i v e MSR asymmetry, A.. , by e x a c t l y the f a c t o r (1 + Tlx ) . In order to i l l u s t r a t e Mu' 2 2 y' the o r i g i n s of the muon l i f e t i m e d i s t o r t i o n s i n the histogram, Figure I I I - 8 shows the four curves i n l o g a r i t h m i c p l o t s i n which the asymmetry has been set t o zero (no MSR s i g n a l ) . The upper two curves i n each p l o t are e x a c t l y those i l l u s t r a t e d i n F igure I I I - 5 . F igure I I I - 9 shows an experimental y +SR histogram taken by G.M. M a r s h a l l of U.B.C. using the e l e c t r o n i c l o g i c described i n S e c t i o n C of t h i s Appendix at a beam current of 3 -1 about 50 x 10 s . The l o g i c d i s c a r d s post-y^ muon events, but not pre-y events. Both p l o t s are of the "normalized r e s i d u a l s " of the experimental data; t h a t i s , the experimental data has been d i v i d e d by the best f i t t o a model i n order to expose any d e f i c i e n c i e s i n the model - i f the model f a i t h --276-0 . 0 0 1 1 1 1 1 1 I I I I I 1 0 - 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 3 . 0 3 . 5 4 . 0 4 . 5 5 . 0 TIME IN uSEC FIGURE I I I - 8 : The o r i g i n s of the l i f e t i m e d i s t o r t i o n s due to pre-y^ muons. The top two curves of each p l o t are the same as. the lower p l o t s of Figures I I I - 3 and I I I - 5 . The top curve i n each p l o t i s the true l i f e t i m e and the second curve, showing the net e f f e c t of pre-y^ muons, i s the sum of the t h i r d curve due to decay d e t e c t i o n of the y^ muon, and the f o u r t h curve due t o decay d e t e c t i o n of pre-y. muons. -277-co _ i CL ZD Q I 1 to LU or Q UJ 5 . 0 2 . 5 0 . 0 c r c l - 2 . 5 o - 5 . 0 5 . 0 CO I cx ZD Q CO UJ or CD UJ rvj _j cx c l - 2 . 5 o 2 . 5 0 . 0 - 5 . 0 i i i 1 r S I 0 , IN V A C U U M . 6 9 G A U S S S I 0 2 I N V A C . . 6 9 G . T I M E D E P . B K G D . 0 . 0 0 . 5 1.0 1 . 5 2 . 0 2 . 5 3 . 0 T I M E I N LI S E C (20 N S E C / B I N ) 3 . 5 4 . 0 FIGURE I I I - 9 : A p o s s i b l e experimental example of the e f f e c t of p r e - L K muons (due to G.M. M a r s h a l l ) . The normalized r e s i d u a l s (see text) should be randomly s c a t t e r e d about zero i f the model used to describe the data i s c o r r e c t . The top p l o t c l e a r l y r e v e a l s the inadequacy of the standard ySR model; i n c l u s i o n of an exponential background w i t h a l i f e t i m e shorter than T provides a more c r e d i b l e d e s c r i p t i o n of the data (bottom). This i s c o n s i s t e n t w i t h the e f f e c t s of pre-y^ muons. -278-f u l l y d e s c r i b e s the data, the p l o t of r e s i d u a l s should be randomly s c a t t e r e d about zero. In the top p l o t of the F i g u r e , the data were f i t t e d to the standard y +SR model (equation (4), Chapter I ) : N(t) = N e ~ t / X y ( l + A e ~ A t cos (oo t + (j)) ) + Bg. o p y y T ^ The r e s u l t i s a poor f i t as evidenced by the obvious d r i f t i n 2 the F i g u r e and by the x P e r degree of freedom of 2.8. Modifying the model to N(t) = N e ~ t / x y ( l + A e ~ X t cos (oo t + cj>) ) + Bg + N, e " t / T l o M y v y T ^ 1 r e s u l t s i n a much b e t t e r f i t as evidenced by the lower p l o t of 2 the Figure and by the x P e r degree of freedom of 1.1. The second f i t gives T 1 = 1.42 ys and N = 31^. The q u a l i t a t i v e behavior of these data i s c o n s i s t e n t w i t h the foregoing c a l c u l a t i o n s of the e f f e c t s of p r e - y ^ muons. Un f o r t u n a t e l y , i t i s not p o s s i b l e to make a q u a n t i t a t i v e comparison of these data w i t h the c a l c u l a t i o n s because of the r e j e c t i o n of p o s t - y ^ muons and the deadtime c h a r a c t e r i s t i c s of the e l e c t r o n i c l o g i c system. Thus, i t cannot be s t a t e d u n e q u i v o c a l l y t h a t the apparent d i s t o r t i o n of t h i s spectrum i s due to p r e - y ^ muons. (iv) P o s t - y ^ Muons and x y ; 100% Decay P o s i t r o n Detection  E f f i c i e n c y The p r o b a b i l i t y of n p o s t - y ^ muons a r r i v i n g i n the t a r g e t between t = 0 and t i s again given by the Poisson d i s t r i b u t i o n , Pp(n,7?,t). As i n the c a l c u l a t i o n of equation 111(11), the p r o b a b i l i t y of a p o s t - y . muon a r r i v i n g during any -279-s u b i n t e r v a l At of t i s the same f o r a l l At. Consequently, the'average p r o b a b i l i t y t h a t a muon a r r i v i n g between t = 0 and t s u r v i v e s u n t i l t i s : P ( s ( t ) ) = ^ (1 - e " t / T y ) 111(31) The unnormalized experimental histogram has the very simple form: 00 N(t,7l) = Z P (n,?Z,t)P (s (t) ) n ( n + l ) e " t / T y 111(32) n=0 P where (n+1) r e f e r s to n p o s t - y ^ muons plus the y ^ muon, any of -t/ x which may decay between t and t + d t , e y i s the s u r v i v a l p r o b a b i l i t y of the y . muon, and P (n,71, t) P (s (t) ) n i s the entry and s u r v i v a l p r o b a b i l i t y of n p o s t - y ^ muons. The n o r m a l i z a t i o n of p o s t - y ^ histograms i s somewhat more complicated than f o r p r e - y ^ histograms. In the present case, there i s only one muon i n the t a r g e t at t = 0. However, by t , another muon may have entered the t a r g e t such t h a t i f the t o t a l muon s u r v i v a l p r o b a b i l i t y at t i s ex t r a p o l a t e d back to t = 0, i t w i l l not correspond to one. For example, i f a p o s t - y ^ muon enters the t a r g e t at t , t h i s corresponds to a muon population of —W+J— + 1 at t = 0, where the f a c t o r of one i s e T y due to y ^ . Consequently, t h i s time-dependent n o r m a l i z a t i o n must be a p p l i e d to ensure t h a t the t o t a l muon s u r v i v a l p r o b a b i l i t y at any time e x t r a p o l a t e d back t o t = 0 corresponds to one. The re q u i r e d n o r m a l i z a t i o n i s : oo Z P ( n , ? c , t ) n P ( s ( t ) ) n=0 p 1 + 5-^ : I I I (33) -t/x e y -280-t h a t i s , nP (n,??, t) P (s (t) ) gives the post-yu muon pop u l a t i o n at - t / x t and the f a c t o r e y e x t r a p o l a t e s t h i s p o p ulation back to t = 0. The normalized histogram i s : CO - t / x N o(t f7Z) •= Z P (n,fl,t) P ( s ( t ) )"(n+l)e n=0 p y I I I (34) 00 n P ( s ( t ) ) 1 + Z n=0 P (n,tt,t) F i gures 111-10 and I I I - l l compare equation 111(34) -t/x w i t h the true histogram, e ' y,for v a r i o u s beam c u r r e n t s . As expected, the e f f e c t s of post-y^ muons are f e l t at l a t e times because of the increased opportunity f o r such a muon t o enter the t a r g e t (see Figure I I I - l l ) . The apparent muon 4 ys time range would be (from Figure I I I - l l ) 1.9, 1.6 and 1.5 3 -1 ys f o r beam currents of 50, 100, and 150 x 10 s r e s p e c t i v e l y . (v) Post-y^ Muons and x y; £ Decay P o s i t r o n Detection E f f i c i e n c y The i n t r o d u c t i o n of a p o s i t r o n counting e f f i c i e n c y g r e a t l y complicates the c a l c u l a t i o n of the post-y^ histogram. As before, we must compute the p r o b a b i l i t y t h a t the c l o c k i s not stopped before t but w i l l be stopped between t and t + dt. The p o s s i b i l i t y of an undetected muon decay leads t o the generation of two kinds of terms: those i n which y^ i s a su r v i v o r at t and those i n which y^ has decayed before t . As i n the d e r i v a t i o n of equation I I I (13) , the p r o b a b i l i t y of there being n muons (not counting the y^ muon) i n the t a r g e t at time t without the c l o c k stopping i s : l i f e t i m e s obtained by f i t t i n g the histogram t o e -t/x y over a -281--282-Q L U Ql o ± 0.01 < X to tD O 0.001 fe-ll.0001 L U r-si o 0.01 t-tx L 3 to o 0.001 D 0.0001 1 5 L U rvi o cr z to o o 0.01 fe-0.001 t-0.0001 4 5 6 TIME IN u S E C ' Logarithmic p l o t s of Figure 111-10. C l e a r l y , the e f f e c t s of post-LK muons are important at l a t e times, i n co n t r a s t to the case of p r e - y ^ muons. FIGURE I I I - l l : -283-the p r o b a b i l i t y of n muons a r r i v i n g during t x the p r o b a b i l i t y t h a t a l l s u r v i v e u n t i l t + the p r o b a b i l i t y of n+1 muons a r r i v i n g during t x the p r o b a b i l i t y t h a t one decays undetected before t + the p r o b a b i l i t y of n+2 muons a r r i v e during t x the p r o b a b i l i t y t h a t two decay undetected before t + . . . This may be w r i t t e n s y m b o l i c a l l y as: P (n,7?,t)P(s(t) ) n + P {n+1,71,t) (1-e) [ {n+^) [1-P(s(t) ) ] / /1 \ \ n - t / T r x \ cn-fi^ —, ,. . . n-rx . ., P ( s ( t ) ) e y + l^ +xJ (. n J P ( s ( t ) ) (1-1 ^ rn+1 - n+1 " t / T y ) + P (n+2,^,t)(1-e) ( n+ 2)P(s (t) ) n + L ( l - e - ^ T y . ) [1-P (s (t) ) ] ( n ; 2 ) [ l - P ( s ( t ) ) ] 2 P ( s ( t ) ) n e - t / T 1 n+1 - t / T + P p(n+3,^,t) (1-e) + (nTl) [ n n 2 ) P ( s ( t ) ) n + X ( 1 - e *" l y ) [ l " P ( s ( t ) ) ] ( n+ 3) [1-P(s(t) ) ] J P ( s ( t ) ) n - t / x n+1 - t / T 2) + where the f i r s t term i n the l a r g e brackets corresponds to the case where n p o s t - y ^ muons plus the y ^ muon surv i v e u n t i l t and a l l other p o s t - y ^ muons decay undetected before t ; the second term i n larg e brackets corresponds to the case where (n+1) p o s t - y ^ muons surv i v e u n t i l t and the y ^ muon plus a l l other p o s t - y ^ muons decay undetected before t . Since the muon popu l a t i o n at any time i s independent of the features of the p o s i t r o n counters, the n o r m a l i z a t i o n c a l c u l a t e d i n the preceeding Section remains v a l i d and the normalized histogram i s given by: -284-E (n+l)P(s(t) ) n E P ( k , n , t ) ( l - e ) K _ n (*) N (t,W 1 0 ^ ' ' 00 E P_ (n,??,t) nP (s (t) ) T 1 + n=0 P [1-P(s(t) ) ] k n _ 1 -t/x e ' y [1-P(s(t) ) ] e ~ t / T y 111(35) + P ( s ( t ) ) (1-e t / T y ) Noting t h a t l i m (1-e) = 1 i f k-n = 0 e+1 = 0 i f k-n > 1 i t i s seen th a t equation 111(35) reduces to equation 111(34) as -t/x i t should. Equation I I I (35) i s compared w i t h e y f o r var i o u s beam cur r e n t s i n Figures 111-12 and 111-13 w i t h a p o s i t r o n d e t e c t i o n e f f i c i e n c y of 10%. As i n the case of pre-y^ muons, the i n t r o d u c t i o n of a p o s i t r o n counter e f f i c i e n c y d r a m a t i c a l l y decreases the e f f e c t of the p i l e - u p . The apparent muon l i f e t i m e s obtained by f i t t i n g the histogram to - t / x e y over a time range of 4ys would be (from Figure 111-13) 2.15, 2.12, and 2.07 ys f o r beam cur r e n t s of 50, 100, and 150 3 -1 x 10 s r e s p e c t i v e l y . (vi) Post-y^ Muons and the MSR S i g n a l ; e Decay P o s i t r o n  Detection E f f i c i e n c y As i n the case of pre-y^ muons, the f o l l o w i n g c a l c u l a t i o n assumes t h a t the muon precession frequency ( i n both muonium and as " f r e e " muons) i s s u f f i c i e n t l y l a r g e t h a t a l l -285-1 . 0 Figure 111-12: The e f f e c t s of p o s t - i K muons on the apparent muon l i f e t i m e w i t h e = 10%. As i n the case of pre-LK muons, the i n t r o d u c t i o n of a p o s i t r o n counting e f f i c i e n c y g r e a t l y reduces the d i s t o r t i o n due to p i l e - u p , though not as much. -286-FIGUPvE 111-13: Logarithmic p l o t s of Figure 111-12 -287-post-]_u muons are out of phase wi t h the precession of the L K muon and th e r e f o r e do not c o n t r i b u t e t o the precession s i g n a l . The c a l c u l a t i o n of the preceeding S e c t i o n has already p a r t i -t i o n e d the histogram i n t o terms i n which the y^ muon has survived u n t i l t and those i n which i t has decayed undetected before t . The only terms which c a r r y a MSR s i g n a l are those where y^ s u r v i v e s u n t i l t but decays between t and t + d t . Follow i n g the arguments leading t o the d e r i v a t i o n of equation I I I (28) , i t i s e a s i l y seen t h a t of the terms corresponding to y. s u r v i v a l u n t i l t , 1 (n+1) [1-P(s(t) ) ] e " t / T y the MSR s i g n a l - b e a r i n g terms are [1-P(s(t) ) ] e " t / T y and the remaining terms: n [ l - P ( s ( t ) ) ] e " t / T y correspond to y^ s u r v i v a l u n t i l t + dt ( a t " l e a s t ) . Denoting the MSR s i g n a l as S ( t ) , the histogram has the form: 00 CO E P ( s ( t ) ) n E P (k,7l,t) ( l - e ) K ~ n [ l - P ( s ( t ) ) ] K _ n 1 N 0 ( t f * ) - S=5 * 3 L J ? E P (n,??,t)nP(s(t) ) ! + n=0 ? -t/x e y _ , 111(36) [ 1 - P ( s ( t ) ) ] e ' y [ l + S ( t ) ] + n [ l - P ( s ( t ) ) ]e t / T y + (k-n) P (s (t)) (1-e t / x y ) The e f f e c t s of post-y^ muons are i l l u s t r a t e d i n Figures 111-14 and 111-15 at va r i o u s beam cur r e n t s and a 10% -288-0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 T I M E I N M S E C FIGURE 111-14: The e f f e c t of post-y muons on the MSR I s i g n a l w i t h e = 10%. The top two curves i n each p l o t are more d i s t i n g u i s h a b l e i n Figure 111-15. The equations of each curve are given i n the t e x t . P Q S T - L i , 2ND L i . e = 10X ,N = 1 5 0 K . fl =0 .1 0 . 0 1.0 2 . 0 3 . 0 4 . 0 5 . 0 6 . 0 7 . 0 8 . 0 9 . 0 1 0 . 0 T I M E IN LISEC FIGURE 111-15: The e f f e c t of post-y.^ muons on the MSR s i g n a l ( d e t a i l ) . The top curve i s the tru e s i g n a l and the second curve i s the e f f e c t i v e p i l e - u p s i g n a l . At l a t e times, the damping of the p i l e - u p s i g n a l i s c l e a r l y evident. -290-p o s i t r o n d e t e c t i o n e f f i c i e n c y f o r the very simple MSR s i g n a l : S (t) = A,, cosai,, t at 5 gauss. Four f u n c t i o n s are p l o t t e d i n v Mu Mu ^ c each F i g u r e : -t/x (1) The upper curve i s the "true" histogram: N(t) = e ' y • (1 + A coseo,, t) . This i s the curve w i t h the longest-Mu Mu l i v e d MSR s i g n a l . (2) The apparent histogram given by equation 111(36) i s almost superimposed on the tr u e histogram (see the d e t a i l e d p l o t of Figure 111-15). The MSR s i g n a l i n t h i s function i s much s h o r t e r - l i v e d than f o r the "true" histogram. (3) The lower s i n u s o i d a l exponential corresponds t o : E P ( s ( t ) ) n E P (k,^,t) ( l - e ) k n [ l - P ( s ( t ) ) ] k n 1 * 0 ( t , * > = ^ ^ ^ - ^ =  E P ( n , ^ , t ) n P ( s ( t ) ) 1 + ^ 1 t/x e y I I I (37) • ( k) [ 1 - P ( s ( t ) ) ] e " t / T y [ l + S ( t ) ] (4) The bottom, n o n - s i n u s o i d a l curve corresponds t o : E P ( s ( t ) ) n E P (k,?Z,t) ( l - e ) K _ n [ l - P ( s ( t ) ] K n 1 N It.71) = ^ o °° E P„(n,#,t)nP(s(t) ) y 1 + n=0 P ( k ) - t / x e y n [ l - P ( s ( t ) ) ] e " t / T y + ( k - n ) P ( s ( t ) ) ( l - e " t / T y ) III(38) Curve (2) i s the sum of curves (3) and (4). Besides -291-confirming the f a c t t h a t the muon l i f e t i m e d i s t o r t i o n due to post-y^ muons i s the most dominant at l a t e times, these p l o t s i l l u s t r a t e the very important f a c t t h a t post-y^ muons introduce a "bogus" r e l a x a t i o n i n t o the MSR s i g n a l . The d e t a i l s of the muon l i f e t i m e d i s t o r t i o n s are i l l u s t r a t e d i n Figure 111-16 which p l o t s the four curves on a l o g a r i t h m i c s c a l e w h i l e suppressing the MSR s i g n a l ( A M U = °)• T n e t o P two curves on each p l o t are i d e n t i c a l to those shown i n Figure 111-13. The e f f e c t s of muon p i l e - u p on the r e l a x a t i o n of the MSR s i g n a l are i l l u s t r a t e d i n Figure 111-17 f o r both pre-and post-y^ muons. The pre-y_^ asymmetry p l o t i s of the f u n c t i o n : A ( t ) _ [equation I I I (28)] _ 1 111(39) -t/x e y and the post-y^ asymmetry p l o t i s of the f u n c t i o n : A ( t ) = [equation 111(36)] _ ± 111(40) e / T y wi t h S (t) = A., cosd),, t i n both cases. C l e a r l y , there i s no Mu Mu r e l a x a t i o n of the MSR s i g n a l i n the case of pre-y^ muons. The curving envelope of the precession s i g n a l i s simply due to the muon l i f e t i m e d i s t o r t i o n s d e t a i l e d e a r l i e r . However, the post-y^ curve shows a d i s t i n c t damping of the MSR s i g n a l w i t h an approximately Gaussian shape. Again, the downward d r i f t i n g envelope of the precession curve i s due t o muon l i f e t i m e d i s t o r t i o n s d e t a i l e d above. -292-o . o o o i I 1 1 1 1 1 1 I i i 0.0001 1 1 1 1 1 I i i i I I 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 TIME IN uSEC FIGURE 111-16: The o r i g i n s of the l i f e t i m e d i s t o r t i o n s due to post-yu muons. The top two curves i n each p l o t are i d e n t i c a l to those i n Figure 111-13. The monotonically decreasing curve w i t h the gr e a t e s t curvature i s due t o u. decay d e t e c t i o n , and the lower curve which r i s e s and f a l l s i s due to post-u. decay d e t e c t i o n . -293-0.15 >-CO °= - 0 . 1 5 -0. 2 0 PRE-LI, 2ND y . e - 102,N=150K. flM=0 .1 0.0 1 . 0 2.0 3.0 4.0 5.0 6.0 T I M E IN u S E C 7.0 8.0 9.0 10.0 FIGURE i l l - 1 7 : R e l a x a t i o n e f f e c t s i n the MSR s i g n a l due t o muon p i l e - u p . The top p l o t of pre-y^ muons shows no r e l a x a t i o n i n the MSR s i g n a l , but i t does show t h a t the amplitude of the s i g n a l i s reduced by 1 + Tlx . The bottom p l o t of post-y^ muons shows no such r e d u c t i o n i n the i n i t i a l amplitude of the s i g n a l , but i t does show a strong r e l a x a t i o n of the s i g n a l w i t h a Gaussian shape. The curving envelopes of both s i g n a l s i s are due t o muon l i f e t i m e d i s t o r t i o n s . -294-C The MSR Data Acquisition System This Section presents a d e t a i l e d d e s c r i p t i o n of the data acquisition system t h a t was b r i e f l y sketched i n Chapter I I . The e l e c t r o n i c l o g i c i s designed to d i s c a r d post-y^ and second e + events, but not pre-y^ events; however, m o d i f i c a t i o n s t o the l o g i c t o incor p o r a t e pre-y^ event r e j e c -t i o n are p r e s e n t l y being implemented. The f i r s t p a r t of t h i s S e c t ion d e s c r i b e s the e l e c t r o n i c l o g i c i n c l u d i n g CAMAC modules, while the second part e x p l a i n s the r o l e of the MBD. The main computer, a PDP 11/40, i s discussed only w i t h respect to i t s i n t e r a c t i o n w i t h the MBD. (i) The E l e c t r o n i c Logic A schematic diagram of the pre-CAMAC e l e c t r o n i c l o g i c i s given i n Figure 111-18. The operation of the TDC-100 and CAMAC p a t t e r n r e c o g n i t i o n u n i t (a strobed coincidence u n i t ) . n a t u r a l l y leads to separate r e j e c t i o n of e a r l y and l a t e second muon events (see Chapter I I , S e c t i o n C). E a r l y second muons may be r e j e c t e d i n hardware by i n h i b i t i n g the p o s i t r o n l o g i c thereby preventing >the s e t t i n g of a b i t p a t t e r n i n the CAMAC coincidence u n i t and f o r c i n g the TDC-100 to "time-out" and re s e t . Late second muons, however, are detected a f t e r the CAMAC coincidence u n i t has been set and a f t e r the TDC-100 has begun i t s time d i g i t i z a t i o n ; these events are r e j e c t e d i n software by the MBD. P i l e - u p c o n d i t i o n s are monitored by three LRS 222 Dual Gate Generators designated g l , g2, and g3. A l l three -295-9 0 ° / 2 7 0 ° or "Arizona" data acquisition mode logic diagram s r S \ .1. Comae nim _ . . driver D y-. bit d> LI L2 L3 Rt R2 R3 ryjtyjlii-Uj —J. V i f • .v . -j B C". II1 ~r 0 I 1 II 11 1 R 3 R 2 R l 4 i 11 LI L 2 1 3 Counter configuration u n t e m i i n a t e d TDC 100 | Start pulse height = 8 0 0 1 1 0 0 mV KEY t - logical 'not' - inverted pulse = bridged output a s 33J = gate generators s = discriminator with output pulse shape . logical 'or' tan in /out . logical 'and' coincidence unit E delay dotted lines = denote optional " units s c a l e r e + r i g h t route Comae oomciderce wordaS bit mask Camac coincidence strobe word <f> level . Camac coincidence word 1 bit 12,13.14 or 15 "1 l e ve 7 " . " l \ stop / ' T D C 1 0 0 Stop pulse height = 6 0 0 1 1 0 0 mV C o m a e coincidence strobe word 1 FIGURE 111-18 The TRIUMF MSR data acquisition l o g i c ( d e t a i l ) + This l o g i c r e j e c t s post-y^ muon events and second e events, but not pre-y^ muon events. -296-gates are set t o a width equal to the muon decay gate, T, of se v e r a l muon l i f e t i m e s (T = 4x y i n previous examples). A schematic diagram of the var i o u s accepted and r e j e c t e d event sequences i s given i n Figure 111-19. The p i l e - u p monitor., g l , i s opened by y^, the muon t h a t s t a r t s the clock, under the c o n d i t i o n ^g^Qp*?1' 9^, used t o d i s t i n g u i s h e a r l y from l a t e second muons, i s opened by an accepted decay p o s i t r o n s i g n a l (an accepted TDC stop p u l s e ) . An e a r l y second muon, defined by the c o n d i t i o n y s t o^«gl*g2, opens the t h i r d gate, g3, which serves to i n h i b i t the p o s i t r o n l o g i c . Late second y + , defined as y s t op«gl*g2, or second e + , defined as e + , g l * g 2 , set a veto b i t i n the CAMAC coincidence u n i t , causing subsequent r e j e c t i o n of the event by the MBD. A delayed p u l s e , f i r e d by the c l o s i n g of g l , serves t o c l o s e g2 and g3. Accepted decay p o s i t r o n s , + — defined as e 'gl*g2, serve to stop the TDC-100, open g2 and set the appropriate telescope " r o u t i n g " b i t s of the CAMAC c o i n c i -dence u n i t . The TDC-100 ignores m u l t i p l e s t a r t and stop pulses, Upon completion of time d i g i t i z a t i o n , t a k i n g 2.5 ys from the r e c e i p t of a stop pulse on average, the TDC-100 sends the d i g i t i z e d time to the CAMAC EG&G RI 224 Input R e g i s t e r by the "handshake" method. The CAMAC input r e g i s t e r sends a LAM s i g n a l to the MBD i n d i c a t i n g the presence of data. Two types of i n h i b i t s i g n a l s are used to prevent m u l t i p l e f i r i n g of the CAMAC coincidence u n i t during the slow (20-30 ys) MBD data handling o peration. The primary i n h i b i t i s generated by the MBD through a CAMAC EG&G ND 027 Output R e g i s t e r or NIM d r i v e r which serves t o i n h i b i t the U s t Q p l o g i c , thereby preventing -297-P U L S E T I M I N G accepted event refers to B • T • D -X - e + refers to L 1 L 2 L 3 or R 1 R 2 R 3 -typical g i gate width is 7 / A S or about 3T)I - - / / \ / — denotes pulse at arbitrary time 7/is gi 1  e +-gi g2 e + strobe word 4> V ~ e+route U g 3 is not opened rejected events 'EARLY' 2nd /i+ (2nd /i+ before e + during g i ) "g t -L =// < r early 2ndfi.*=/i.*.g1.g2 v g3 = early 2nd p+ electron veto L g2 is not opened since e + cannot occur 'LATE' 2nd/z+ and 2nd e + (/r* or e+after first e+during gi) Lt+.gTV gi 1ste+=e+-gi-g2 //L~~V g2 e + strobe word oS e + route LJ / / late 2nd ^ + =M + -g1-g2 or / / / / 2nd e + = e K gi -g2 / / V reject strobe word 1 V level bit12,13,14,or15 word 1 g3 is not opened FIGURE 111-19: Pulse t i m i n g and event i d e n t i f i c a t i o n f o r the l o g i c of Figure III-18. -298-re-opening of the g l gate which, i n t u r n , supresses the p o s i t r o n l o g i c . To p r o t e c t any time i n t e r v a l between the c l o s i n g of g l and the generation of the NIM d r i v e r i n h i b i t , a c l o c k "busy" pulse from the TDC a l s o serves to veto the g l s t a r t i n g l o g i c . The TDC busy goes up w i t h the acceptance of a s t a r t pulse and remains on u n t i l the CAMAC input r e g i s t e r i s c l e a r e d by the MBD. Some d e t a i l s i n Figure 111-18, such as pulse height s p e c i f i c a t i o n s and the de s i g n a t i o n of unterminated bridged outputs on some coincidence u n i t s , are hardware-specific f o r . the e l e c t r o n i c modules used at TRIUMF and have no fundamental l o g i c a l f u n c t i o n . A general s i m p l i f i c a t i o n of the e l e c t r o n i c l o g i c and i t s extension to r e j e c t p r e - L u muons are p r e s e n t l y being implemented w i t h the use of p i l e - u p r e j e c t o r s which have r e c e n t l y become a v a i l a b l e . ( i i ) The Microprogrammed Branch D r i v e r The f i r s t p a r t of t h i s S e c tion provides a general o p e r a t i o n a l d e s c r i p t i o n of the MBD and the second part gives a d e t a i l e d d e s c r i p t i o n of the general TRIUMF MSR data a q u i s i t i o n programme. A number of references are a v a i l a b l e on the MBD: B i s w e l l ( 7 3 ) , Thomas(73), and Sh l a e r ( 7 4 ) . The MBD-11 i s an i n t e r f a c e between the PDP-11 computer and CAMAC systems. I t i s a microprocessor c o n t r o l l e d , m u l t i p l e channel, d i r e c t memory access (DMA) branch d r i v e r t h a t looks l i k e a PDP-11 p e r i p h e r a l . In normal op e r a t i o n , the MBD runs l i k e a small stored-programme computer w i t h the programmes -299-contained i n 16-bit'word, 256-word page memory- The MBD i s organized into 8 "channels" which, i n some sense correspond to programmes, each of which has a p r i o r i t y with channel 7 the highest and channel 0 the lowest p r i o r i t y . Each channel has a dedicated set of 14 programmable 16-bit r e g i s t e r s i n which data i s processed. In addition, there are a number of re g i s t e r s common to a l l channels: UNIBUS registers which are used to transfer data to and from PDP memory, CAMAC reg i s t e r s which are used to transfer data to and from CAMAC, and a number of miscellaneous r e g i s t e r s . There are also a group of PDP-11 regi s t e r s accessible to the PDP v i a i t s I/O page through which the PDP exercises ultimate control over the MBD. Some of these non-channel r e g i s t e r s may only be used by the MBD either as sources or sinks. This elaborate r e g i s t e r structure i s designed with the intention that a l l data, whether i t be data to be transfered between PDP and CAMAC or control data, be held in r e g i s t e r s , while the MBD memory i s used to hold programme instru c t i o n s , constants, and buffers. The MBD i n s t r u c t i o n set permits addition, subtraction, masking, s h i f t i n g of 16 b i t integers as well as te s t i n g of r e s u l t s , execution of CAMAC commands, and communication with the PDP. The execution cycle time i s f a s t , t y p i c a l l y 350 ns. Many instructions allow multiple operands thereby eliminating the need for intermediate storage r e g i s t e r s . MBD programmes are assembled with the PDP-11 macroassembler and loaded into MBD memory. MBD communication with the PDP i s car r i e d out usually -300-by NPR's (non-processor requests) through i t s DMA channel. This operation i s asynchronous w i t h the PDP processor and allows t r a n s f e r of data to and from PDP memory without the n e c e s s i t y of i n t e r v e n t i o n by the PDP processor. In a d d i t i o n , each channel of the MBD may have an i n t e r r u p t v e ctor assigned to i t which p o i n t s to the address of a PDP i n t e r r u p t s e r v i c e r o u t i n e . Thus, the MBD has the a b i l i t y to fo r c e the PDP processor to intervene i n data handling. S i m i l a r i l y , each MBD channel has a graded-L (GL) b i t assigned to i t f o r communication w i t h CAMAC. I f the MBD i s t o respond t o LAM's from a CAMAC module, the CAMAC c r a t e c o n t r o l l e r must have the appropriate GL jumper between the c r a t e address of the LAM-generating module and the corresponding MBD channel. Data t a k i n g from s e v e r a l e x p e r i -ments at once could be accomplished by as s i g n i n g d i f f e r e n t MBD channels t o each experiment. Each MBD channel to be used has a programme i n MBD memory ass o c i a t e d w i t h i t . In order t h a t the PDP e x e r c i s e u l t i m a t e c o n t r o l over the MBD, each channel must be s t a r t e d , at l e a s t once, by the PDP. In f a c t , t h i s process assigns a p a r t i c u l a r MBD programmme t o a p a r t i c u l a r channel. At the completion of t h i s channel i n i t i a l i z a t i o n , the channel may " e x i t " or cease execution i n one of four ways, determined by the programme: i t may e x i t such t h a t i t cannot be r e s t a r t e d except by the PDP; i t may e x i t such that i t w i l l be r e s t a r t e d when a LAM as s o c i a t e d w i t h i t becomes a c t i v e ; i t may e x i t from a LAM-started programme such t h a t the PDP must r e s t a r t i t ; and -301-i t may e x i t such that i t w i l l r e s t a r t i t s e l f the next time t h a t channel has the highest p r i o r i t y . To r e t a i n PDP c o n t r o l over the MBD, there i s a s u b - p r i o r i t y h i e r a r c h y which gives the PDP-r e s t a r t e d channels higher p r i o r i t y than channels r e s t a r t e d by LAM's or by themselves. For example, PDP s t a r t e d channel 4 has higher p r i o r i t y than LAM r e s t a r t e d channel 6 which, i n t u r n , has higher p r i o r i t y than s e l f - s t a r t e d channel 5. Since channels cannot i n t e r r u p t each other, channel p r i o r i t y a r b i t r a t i o n only occurs a f t e r an executing channel has e x i t e d . For most a p p l i c a t i o n s , data t a k i n g channels are i n i t i a l i z e d by the PDP, but they e x i t such that they are r e s t a r t e d by LAM's. The PDP i n i t i a l i z a t i o n g e n e r a l l y provides the MBD programme w i t h c o n t r o l i n f o r m a t i o n such as histogram addresses i n PDP memory, histogram s i z e s , data masks etc. This i n f o r m a t i o n i s generated by the PDP data a c q u i s i t i o n programme which s o l i c i t s the i n i t i a l i z a t i o n i n f o r m a t i o n from the experimenter. A f t e r the i n i t i a l i z a t i o n process, the MBD runs i n response to LAM's from CAMAC modules without i n t e r v e n -t i o n from the PDP. A t y p i c a l MBD data handling programme performs such tasks as: reading data from CAMAC modules; checking data masks f o r good/bad event a r b i t r a t i o n or h i s t o -gram d i s p a t c h i n g ; a d j u s t i n g data r e s o l u t i o n by s h i f t i n s t r u c -t i o n s ; c a l c u l a t i n g the word address of the c o r r e c t histogram b i n i n PDP memory; incrementing the histogram b i n i n PDP memory v i a an NPR; and, f i n a l l y , r e s e t i n g the appropriate CAMAC modules. One can e x p l o i t the asynchronous operation of the MBD to minimize both experimental and PDP deadtimes. For -302-example, the MBD can be involved i n a data manipulation operation while data i s being transferred to or from the PDP and to or from CAMAC. The design of an MBD programme for general application to MSR at TRIUMF must have a number of features: i t must allow simultaneous data acquisition for more than one experiment using either the same MBD channel or d i f f e r e n t ones; any experimenter must have the option of st a r t i n g or stopping data acquisition for his experiment at any time without i n t e r f e r i n g with the data acquisition of other experiments simultaneously using the same system; the number, size and time resolution of histograms associated with each experiment must be completely f l e x i b l e (within the physical constraints of computer memory s i z e ) ; i t must be possible to associate a time " o f f s e t " with each histogram (that i s , an experimenter may only want to histogram data corresponding to time ranges greater than some minimum "of f s e t " value); there must be provision for software rejec-t i o n of bad events; and there must not, obviously, be any "cross talk" between one histogram and another. In addition, the MBD programme should be e f f i c i e n t since i t s data processing i s a major contributor to experimental deadtime. While the time range of the TDC-10 0 i s externally adjustable to some extent, i t s time resolution i s not - i t i s fixed at 0.125 ns. The time range may be varied from 8ys to 34 ms i n binary steps. Furthermore, the TDC-100 transfers i t s binary coded data into two words i n the CAMAC input r e g i s t e r . The f i r s t 16 b i t s of data (8* ys) are stored i n word 0 of the -303-input r e g i s t e r and the high order 13 b i t s ( for times > 8 ys) are stored i n word 1 of the input r e g i s t e r . Since both the PDP and MBD are organized i n t o 16 b i t r e g i s t e r s and memory words, TDC-100 time measurements are constrained to 15 b i t s (the highest order b i t i s a sig n b i t ) . The MBD must adjust measured time r e s o l u t i o n s by the execution of s h i f t i n s t r u c t i o n s and concatenate the two words of TDC-100 output i n t o one word. This i s accomplished i n the f o l l o w i n g way: i f the d e s i r e d time 3 r e s o l u t i o n i s 1 ns, say, (0.125 ns x 2 ), the MBD l o g i c a l l y s h i f t s the data i n word 0 three b i t s t o the r i g h t , l o g i c a l l y s h i f t s the data i n word 1 (16-3 =) 13 b i t s t o the l e f t and merges ( v i a an e x c l u s i v e "or" i n s t r u c t i o n ) the data i n t o one 16 b i t word. The d e s i r e to a l l o w each histogram to have i t s own time r e s o l u t i o n presents a problem t o the MBD coding since i t must have a d i f f e r e n t s h i f t f i e l d f o r each histogram. This problem i s solved by programming the MBD w i t h s e l f - m o d i f y i n g code that i n s e r t s the proper s h i f t i n s t r u c t i o n s as r e q u i r e d . The standard TRIUMF MBD data acquisition programme, which i s LAM i n i t i a t e d on channel 6, supports up t o 16 histograms and f u l f i l l s the requirements o u t l i n e d above. When the data acquisition system i s "boot-strapped", the MBD code i s au t o m a t i c a l l y loaded by the PDP. The MBD then executes as b r i e f i n i t i a l i z a t i o n sequence which removes the i n h i b i t on the CAMAC c r a t e 'A' c o n t r o l l e r , enables LAM generation by the TDC-100 CAMAC input r e g i s t e r , enables the "branch demand" on the CAMAC c r a t e c o n t r o l l e r ( t h i s allows the cr a t e c o n t r o l l e r to send the LAM's generated by CAMAC modules along the branch . -3 04-highway t o the MBD), and, f i n a l l y , e x i t s to await r e s t a r t i n g by LAM's. Any LAM received by the MBD at t h i s p o i n t causes the MBD t o execute a dummy code which merely c l e a r s the CAMAC modules but does not communicate w i t h the PDP. I t i s not u n t i l an experimenter a c t u a l l y orders the system to commence data acqu i s i t i o n , thereby modifying the MBD code v i a the PDP, that data a q u i s i t i o n commences. To accomplish the data acquisition o b j e c t i v e s o u t l i n e d above, the MBD uses three sets of t a b l e s contained i n i t s memory, two of which are w r i t t e n by the PDP (these may be modified at any time by the PDP). The f i r s t i s a mask t a b l e , l i s t i n g the v a l i d histogram masks. These masks are simply b i t p a t t e r n s t h a t i d e n t i f y which histogram a p a r t i c u l a r datum belong to . With each event, the MBD reads a mask from the CAMAC coincidence u n i t and i d e n t i f i e s which histogram the event belong to by comparing i t w i t h the masks i n the mask t a b l e . The mask t a b l e i s w r i t t e n by the PDP on advice from the experimenter who s e l e c t s a unique mask p a t t e r n f o r each histogram. Besides p r o v i d i n g histogram i d e n t i f i c a t i o n , t h i s mask t a b l e allows software event r e j e c t i o n s ince the MBD d i s c a r d s any events w i t h masks not contained i n i t s t a b l e . Thus, an experimenter can set a f a l s e mask to r e j e c t s p e c i f i c types of events. By i d e n t i f y i n g a mask i n i t s mask t a b l e , the MBD au t o m a t i c a l l y f i n d s a p o i n t e r i n a "dispatch" t a b l e t h a t gives the l o c a t i o n i n i t s memory of the "histogram" t a b l e t h a t corresponds t o that mask. These t a b l e s , which are w r i t t e n by the PDP, each c o n t a i n s i x words of informat i o n about t h e i r -305-histogram. The f i r s t word t e l l s the MBD i f the histogram i s c u r r e n t l y a c t i v e (that i s , whether data acquisition f o r that experiment i s c u r r e n t l y o f f or on). This word i s modified by the PDP whenever the ass o c i a t e d experimenter decides to s t a r t or stop data acquisition, thereby a l l o w i n g d i f f e r e n t experiments to s t a r t or stop independently. The second word i n the t a b l e gives the address of the f i r s t word of the histogram (the "base" address) stored i n PDP memory. The t h i r d word gives the s i z e (number of bins) of t h a t histogram. The MBD d i s c a r d s any data corresponding to time ranges greater than the histogram s i z e . The f o u r t h and f i f t h words c o n t a i n s h i f t l e f t l o g i c a l and s h i f t r i g h t l o g i c a l i n s t r u c t i o n s r e s p e c t i v e l y , corresponding to the histogram time r e s o l u t i o n . The MBD dynamically i n s e r t s these i n s t r u c t i o n s i n t o i t s data handling code as r e q u i r e d , thereby a l l o w i n g d i f f e r e n t time r e s o l u t i o n s f o r each histogram. The l a s t word i n the t a b l e contains the histogram o f f s e t which the MBD su b t r a c t s from the measured time i n t e r v a l . The MBD c a l c u l a t e s the address of the word i n PDP memory corresponding to the p a r t i c u l a r time b i n of a histogram by a d j u s t i n g the time r e s o l u t i o n of a measurement, s u b t r a c t i n g the o f f s e t , comparing the r e s u l t w i t h the histogram s i z e , and, i f i t i s not greater than the histogram s i z e , adding the base address. The MBD communicates w i t h CAMAC v i a the branch highway and w i t h the PDP v i a the UNIBUS. R e l a t i v e to the 3 50 ns data manipulation operations of the MBD, communication w i t h e x t e r n a l devices i s slow. MBD-CAMAC communication time depends on the p h y s i c a l l e n g t h of the branch highway and takes about -3.06-2.7 \is at TRIUMF f o r each operation. MBD-PDP communication by NPR's depends on the l e v e l of PDP a c t i v i t y since the MBD must compete f o r c o n t r o l of the UNIBUS wit h other devices such as the c e n t r a l processor e t c . Each UNIBUS operation takes an average of 3.5 us when the PDP i s moderately a c t i v e . In order to r e a l i z e the o b j e c t i v e of minimizing MBD data process-ing time, i t i s d e s i r e a b l e to execute data manipulation procedures such as checking masks while simultaneously communicating w i t h CAMAC and the PDP. There i s an inherent c o n t r a d i c t i o n i n t h i s stategy, however, since the MBD has no data to manipulate u n t i l i t completes s e v e r a l CAMAC read operations t o o b t a i n the data. In order to circumvent t h i s problem, the MBD has been coded so t h a t many of i t s data manipulation operations are one programme execution pass behind; t h a t i s , while the MBD i s w a i t i n g f o r r e c e i p t of data from CAMAC corresponding to the present LAM, i t processes the data from the previous LAM which was te m p o r a r i l y stored i n i t s r e g i s t e r s . By the time i t completes execution of the current pass of i t s code, the MBD has f i n i s h e d w i t h the previous data, e i t h e r by w r i t i n g i t i n t o the PDP, or r e j e c t i n g i t , and has stored the current data i n i t s r e g i s t e r s u n t i l i t re-commences execution i n response t o the next LAM. A flow diagram of the standard TRIUMF MBD code i s given i n Figure 111-20. -307-•et previous •vent flag-bad add previous event - offset to h i i t base address •ad PDP hia bin for previous get address of currant event hist table \read t clear \ TDC word 1 \for current \event / bet and store from current levant hist table] base address hist size tread t clear \ TDC word 1 \for current/ \ event / make stored value of curren' event negative |get shift right logical inetrue 'tion for current^ data and insert into code |get shift l e f t logical instrue tion for current data and insert into code shift right logical current data word 0  shift left logical current data word 1 exclusive 'or' shifted words 0*1- merge into one word subtract offset from current data and store result FIGURE 111-20: Flow diagram of TRIUMF MBD programme. 

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