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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 PARTIAL FULFILLMENT OF THE  REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY  in THE FACULTY OF GRADUATE STUDIES (Department o f Chemistry)  We accept t h i s t h e s i s as conforming to the r e q u i r e d  standard  THE UNIVERSITY OF BRITISH COLUMBIA June, 1979  (c) David M i c h a e l 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 of the r e q u i r e m e n t s 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 the L i b r a r y  for  that  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 granted by the Head of my Department o r by h i s r e p r e s e n t a t i v e s .  It  i s understood t h a t c o p y i n g o r p u b l i c a t i o n  of 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 written permission.  Department nf  Chemistry  The U n i v e r s i t y of 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  not be a l l o w e d w i t h o u t my  ABSTRACT Muonium (Mu)  i s the atom formed by an e l e c t r o n bound to a  p o s i t i v e muon "nucleus" Since muons are 207 mass of Mu r a d i i and same.  (charge:+l,  s p i n : l / 2 , l i f e t i m e : 2.2  times as massive as e l e c t r o n s , the  i s 0.996 t h a t of the hydrogen atom, and i o n i z a t i o n p o t e n t i a l s of Mu  Therefore,  and  (m  = 1/9  Mu  reduced  the Bohr  H are e s s e n t i a l l y the  the chemical behaviour of the Mu  t h a t of a l i g h t H isotope  ys).  atom i s  ni ) with a g r e a t l y enhanced H  s e n s i t i v i t y to H i s o t o p e " e f f e c t s . Mu  r e a c t i o n r a t e s are measured by a method c a l l e d  Spin R o t a t i o n "  (MSR)  which resembles c o n v e n t i o n a l  techniques such as NMR  or ESR  i s t i c Larmor p r e c e s s i o n or ESR,  the MSR  atom.  absorption,  character-  However, u n l i k e  method does not d e t e c t the Mu  by resonant power  resonance  i n t h a t i t monitors the  of the Mu  "Muonium  Larmor  precession  but r a t h e r through the p e c u l i a r  s p i n dependent r a d i o a c t i v e decay of the muon i t s e l f . t h e o r e t i c a l b a s i s f o r the a p p l i c a t i o n of the MSR  The  technique to  the measurement of muonium r e a c t i o n r a t e s i s d e r i v e d . extensive  d i s c u s s i o n i s given  and  + C l between 3 00 and are r e p o r t e d Mu  + HX ?  and  * * / X = C l , Br, + H  reacts considerably  i s o t o p e s , a t t e n t i o n i s focussed the Mu  + F  0  MuF  and Mu  are r e p o r t e d  + F and  Mu  + C^  room temperature r a t e  f o r the r e a c t i o n s : Mu  MuX  systems Mu  4 0OK,  + F^  I.  the  technique.  a c t i v a t i o n energies  the gas phase r e a c t i o n s : Mu  An  to the p r a c t i c a l aspects of  experimental implementation of the MSR Rate constants  NMR  + Br 2  for  + MuCl  constants  MuBr + Br and  • -  While i n most of these  f a s t e r than the h e a v i e r  H  on hydrogen i s o t o p e e f f e c t s i n  + Cl,, r e a c t i o n s .  T h i s d i s c u s s i o n i s based  on the e x t e n s i v e t h e o r e t i c a l i n v e s t i g a t i o n s of Connor et aJL. , which show the Mu  +  r e a c t i o n to be dominated by quantum  mechanical t u n n e l l i n g at room temperature. quantum t u n n e l l i n g manifests producing two  itself  dramatic i s o t o p e  Experimentally,  in this reaction  by  e f f e c t s a t 30OK: (1) the b i -  molecular r a t e constant f o r the Mu  reaction  i s at l e a s t s i x times t h a t f o r the  analogous H atom r e a c t i o n ,  and  isotope +  Mu  kcal/mole) i s l e s s than h a l f of t h a t f o r H + F  (0.9  In c o n t r a s t ,  the Mu  +  r e a c t i o n does not  show any  such  strong  e f f e c t s a t 300K: (1) the b i m o l e c u l a r r a t e c o n s t a n t f o r (5.1 x l O ^ 1  1/mole-s) i s no more than four times t h a t  of the analogous H r e a c t i o n , and e n e r g i e s f o r both Mu Preliminary  and  (2) the  H reactions  apparent a c t i v a t i o n  are the  same (1.4  c a l c u l a t i o n s o f Connor et a l . on Mu  that c l a s s i c a l  + C^  kcal/mole) suggest  "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  enhancement due  to quantum t u n n e l l i n g .  e f f e c t s cannot be d e f i n e d and  x 10"*"^ 1/mole-s)  (2) the apparent A r r h e n i u s a c t i v a t i o n energy of t h i s  reaction  Mu  (1.4  f o r the Mu  + Br  Quantitative 2  and  Mu  rate isotope  + HX  reactions  t h e i r hydrogen i s o t o p i c analogues because of the absence of  s u f f i c i e n t experimental and are d i s c u s s e d  t h e o r e t i c a l data; these  reactions  i n terms of the g e n e r a l theory of i s o t o p e  effects.  -ivTable of Contents Chapter I - I n t r o d u c t i o n A P o s i t i v e Muons and the y SR Method B Muonium and the MSR Method C Muonium Chemistry - An H i s t o r i c a l Background D O r g a n i z a t i o n o f the D i s s e r t a t i o n  1 1 10 15 20  Chapter I I - Experimental D e t a i l s A TRIUMF and the M2 0 Muon Beam L i n e B The Gas Target, Counters, and Magnetic C Data A c q u i s i t i o n D Data A n a l y s i s  23 23 32 37 49  Field  Chapter I I I - T h e o r e t i c a l Background A Introduction B P o t e n t i a l Energy Surfaces (i) Semi-Empirical P o t e n t i a l Energy Surfaces ( i i ) Contour P l o t s o f the P o t e n t i a l Energy Surface f o r the Reaction A + B C + A B + C ( i i i ) P o t e n t i a l Energy Surfaces f o r the R e a c t i o n s : Y + X„ ->• YX + X, Y = Mu, H, D, T; X = F, C l , Br C Energy (i) C l a s s i c a l T r a j e c t o r i e s ( i i ) Q u a s i c l a s s i c a l and Quantum Mechanical Trajectories ( i i i ) T r a n s i t i o n S t a t e Theory (iv) Reaction Enthalpy (v) Reaction A c t i v a t i o n Energy (vi) P o t e n t i a l Energy Surfaces f o r the R e a c t i o n s :  58 58 60 60 65 73 82 83 87 90 92 94  + TT  YX  Y + HX YH + X ' D Trajectory Calculations E T r a n s i t i o n S t a t e Theory F Tunnelling  Y  =  M  U  ' ' °' H  T  ;  X  =  C 1  '  B r  '  1  9  7  Chapter IV - Experimental R e s u l t s and T h e i r I n t e r p r e t a t i o n A Mu + F -> MuF + F 2  109 118 122 127 12 9  B Mu + C l -> MuCl + C l  157  -y MuBr + Br  D MU + HC1 < JJ^Vc?  171  T-. E Mu +> TTT, HBr < MuBr + H  . __ 185  F Mu + HI <  190  2  C Mu + B r  2  M  M  U  u  I  H  +  +  B  r  H  Chapter V - Summary and C o n c l u s i o n s A Summary B Past P e r s p e c t i v e C Future P e r s p e c t i v e D C l o s i n g Remarks  17 8  194 194 195 198 2 01  -vTable of Contents Literature Chapter Chapter Chapter Chapter Chapter Appendix Appendix Appendix  (Cont'd)  Cited I II III IV V I II III  2 03 0  2  0  6  2  0  7  3  2  1  2  2  1  6  2 2  ^ 18 ^ 7  2 1  Appendix A B C D  2  I - The Time E v o l u t i o n o f 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 State Vectors Time E v o l u t i o n of the Mu^ S t a t e s Time E v o l u t i o n o f the y S p i n p o l a r i z a t i o n i n Mu Experimental I m p l i c a t i o n s o f <P (t)^> +  220 220 223 228 230  (i) Very Weak F i e l d s (<_ 10 gauss) - the Standard MSR S i g n a l ( i i ) Intermediate F i e l d s (10 < B < 150 gauss) Two Frequency Muonium ( i i i ) High F i e l d s (_> 150 gauss) I I - The E f f e c t o f Chemical R e a c t i o n on the Muon P o l a r i z a t i o n A General B Generation o f a Coherent Diamagnetic Muon Background: X ->- 15 y s ~ l , B ^> 10 gauss  231 23 4 236  Appendix  23 9 23 9 242  Appendix  I I I - Data Acquisition w i t h High C u r r e n t Muon Beams: Theory and P r a c t i c e A The Optimal Good Event Rate B S p e c t r a l D i s t o r t i o n s due t o Muon P i l e - u p (i) Pre-y^ Muons and x^: .100% Decay P o s i t r o n Detection E f f i c i e n c y . . ( i i ) Pre-y. Muons and x : e Decay P o s i t r o n I y Detection E f f i c i e n c y ( i i i ) Pre-y^ Muons and the MSR S i g n a l : e Decay l  250 2 50 2 53  .  257 261  Positron Detection E f f i c i e n c y (iv) Post-y^ Muons and x : 100% Decay P o s i t r o n  269  Detection E f f i c i e n c y (v) Post-y. Muons and x : e Decay P o s i t r o n i y Detection E f f i c i e n c y (vi) Post-y^ Muons and the MSR S i g n a l : e Decay  278  H  J  280  ;  Positron' Detection E f f i c i e n c y C The MSR Data A c q u i s i t i o n . System (i) The E l e c t r o n i c L o g i c ( i i ) The Microprogrammed Branch D r i v e r ;  284 2 94 294 2 98  -vi-  List  of  Figures  Chapter I 1 E n e r g y s p e c t r u m o f p o s i t r o n s f r o m muon d e c a y and t h e e n e r g y d e p e n d e n c e o f t h e asymmetry p a r a m e t e r 2 A t y p i c a l y SR t i m e h i s t o g r a m 3 A t y p i c a l MSR t i m e h i s t o g r a m 4 The MSR s i g n a l  5 9 13 16  Chapter I I 5 The TRIUMF C y c l o t r o n and e x p e r i m e n t a l f a c i l i t i e s (1977) 6 The M20 b e a m l i n e ( d e t a i l ) 7 The g a s p h a s e MSR t a r g e t a p p a r a t u s 8 N i t r o g e n v e r s u s argon as moderator gases 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 ) 10 The s p e c t r a l d i s t o r t i o n due t o " e a r l y " s e c o n d y rejection  24 27 33 36 39 45  Chapter I I I 11 A p o t e n t i a l c o n t o u r map f o r t h e e x o t h e r m i c c o l l i n e a r A + BC AB + C r e a c t i o n 12 P o t e n t i a l e n e r g y s u r f a c e f o r t h e Y + reaction 13 P o t e n t i a l e n e r g y s u r f a c e s f o r t h e c o l l i n e a r Y + F r e a c t i o n p l o t t e d i n mass w e i g h t e d c o o r d i n a t e s 14 The b o t t l e n e c k e f f e c t 15 Mass w e i g h t e d p o t e n t i a l e n e r g y s u r f a c e f o r t h e c o l l i n e a r Mu + reaction 16 P o t e n t i a l s u r f a c e s f o r t h e c o l l i n e a r H + HC1 -> H + C l reaction 17 I s o t o p e 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 f r o m mass v a r i a t i o n s o f atom Y f o r t h e r e a c t i o n Y + AB -> YA + B 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 t h e t r u n c a t e d B e l l p a r a b o l a and t h e E c k a r t b a r r i e r  61 71  2  2  C h a p t e r IV 19 The e f f e c t o f t e m p e r a t u r e on t h e Mu + F MSR relaxation rates *" 20 E x p e r i m e n t a l 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 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 (v=0,l) 22 I n t e g r a n d o f t h e quantum c o l l i n e a r r a t e c o n s t a n t a t 300 and 900K f o r Y + F (v=0) 2 3 I n t e g r a n d f o r t h e 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 c o n s t a n t a t 300K f o r Y + F ( v = l ) 24 N o n - r e a c t i v e 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 (v=0) on t h e mass w e i g h t e d LEPS s u r f a c e 25 A r r h e n i u s p l o t s f o r t h e 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 t h e o r y r a t e c o n stants f o r Y + F 2 6 C o m p a r i s o n o f 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 state theory reaction p r o b a b i l i t i e s f o r Y + F (v=0)  74 77 86 100 12 3 12 6  0  132 13 3  2  142  2  144 145  2  2  2  2  147  151 153  -viiL i s t of Figures  (Cont'd)  Chapter IV (Cont'd) 27 Comparison o f quantum and t r a n s i t i o n s t a t e theory 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 Mu + F ( v = 0 , l ) 28 C o l l i n e a r quantum r e l a t i v e p o p u l a t i o n 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 s t a t e s f o r Y + F„(v=0) 2 9 The e f f e c t o f temperature on the Mu + C I MSR relaxation rates 30 Experimental A r r h e n i u s p l o t s f o r Y + C l r e a c t i o n s , Y = Mu, H, D 31 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 A r r h e n i u s p l o t s f o r c o l l i n e a r Y + F„ and Y + Cl„ 3 2 MSR r e l a x a t i o n r a t e s as a f u n c t i o n or Br„ concent r a t i o n i n argon moderator a t 2 95K 33 MSR r e l a x a t i o n r a t e s as a f u n c t i o n o f HCl concent r a t i o n i n N moderator a t 295K 34 The MSR s i g n a l s i n pure N v e r s u s pure HCl a t 295K 3 5 MSR r e l a x a t i o n r a t e s as a f u n c t i o n of HI concent r a t i o n i n argon and N moderator a t 295K 2  2  2  2  2  2  Chapter V 3 6 MSR s i g n a l s i n pure argon and i n B r i n argon a t 1.3 gauss and 295K; data obtained a t LBL  154 156 159 16 0 169 173 180 181 192  ?  Appendix I 1-1 B r e i t - R a b i diagram of the energy e i g e n s t a t e s o f muonium i n a magnetic f i e l d 1-2 The time e v o l u t i o n o f the y s p i n p o l a r i z a t i o n i n a 100 gauss t r a n s v e r s e magnetic f i e l d 1-3 "Two frequency p r e c e s s i o n " o f the muon i n muonium i n fused quartz a t 95 gauss +  Appendix I I I I - l The e f f e c t o f chemical r e a c t i o n on the muonium signal II-2 The l i n e a r dependence of the r e l a x a t i o n r a t e of the muonium s i g n a l on reagent c o n c e n t r a t i o n II-3 The g e n e r a t i o n o f a coherent diamagnetic muon background s i g n a l by f a s t chemical r e a c t i o n s of muonium II-4 The dependence o f the amplitude o f the " r e s i d u a l muon p o l a r i z a t i o n " on muonium r e a c t i o n r a t e a t 7.5 gauss  196  2 26 235 237  241 243 247 248  Appendix I I I I I I - l The n e t good event r a t e as a f u n c t i o n o f beam c u r r e n t f o r v a r i o u s muon decay gates 2 52 III-2 The e f f e c t s o f pre-y. muons on the apparent muon l i f e t i m e with e = 100%. 262 I I I - 3 Logarithmic p l o t of F i g u r e III-2 263 III-4 The e f f e c t s of pre-y. muons on the apparent muon l i f e t i m e w i t h e = 10%" 267 I I I - 5 Logarithmic p l o t s of F i g u r e III-4 268 I I I - 6 The e f f e c t s of pre-y. muons on the MSR s i g n a l w i t h e = 10% 273 1  -viiiL i s t of Figures  (Cont'd)  Appendix I I I (Cont'd) III-7 The e f f e c t s o f pre-y. muons on the MSR s i g n a l w i t h e = 10% (detail}" 274 III-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 t o pre-y. muons 27 6 I I I - 9 A p o s s i b l e example o f the e f f e c t o f pre-y. muons 277 111-10 The e f f e c t s of post-y. muons on the apparent muon l i f e t i m e w i t h e= 100% 281 I I I - l l L o g a r i t h m i c p l o t s of F i g u r e 111-10 282 111-12 The e f f e c t s o f post-y. muons on the apparent muon l i f e t i m e with e = 10% 285 111-13 L o g a r i t h m i c p l o t s of F i g u r e 111-12 286 111-14 The e f f e c t s o f post-y. muons on the MSR s i g n a l w i t h e = 10% 288 111-15 The e f f e c t s o f post-y. muons on the MSR s i g n a l with e = 10% ( d e t a i l ) 28 9 111-16 The o r i g i n s of l i f e t i m e d i s t o r t i o n s due t o post-y. muons 2 92 111-17 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 pile-up 293 111-18 The TRIUMF MSR data acquisition l o g i c ( d e t a i l ) 2 95 111-19 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 F i g u r e 111-18 297 111-2 0 Flow diagram o f the TRIUMF MBD programme 3 07 1  1  1  1  -ix-  List  of  Chapter I II  Tables I Properties Properties  o f p o s i t i v e muons o f muonium  2 11  Chapter II III  Nominal f o r w a r d and backward y momenta 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  and momentum  28  Chapter I I I IV V  Chapter VI VII VIII IX X  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 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  XII XIII XIV XV XIV  Appendix  128 131 13 6 137 138 158 162 17 2 17 9 186  1  ~*YBr XVII  95  IV 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 phase 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 Calculated rate constants f o r the collinear Y + F „ -> Y F + F r e a c t i o n s Calculated rate constant ratios f o r the c o l l i n e a r Y + F« + YF + F r e a c t i o n s Calculated activation energies f o r the collinear Y + F + YF + F r e a c t i o n s 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 + C l „ -> MuCl + C l Experimental rate parameters f o r the reactions Y + C l „ -* Y C 1 + C l , Y = Mu, H, D 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 + B r „ -> MuBr + B r 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 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 B r r e a c t i o n a t 295K E x p e r i m e n t a l r e a c t i o n r a t e parameters f o r Y + Y Br ^YY' + Br 2  XI  93  + Y  1  , Y = Mu,  MSR r e l a x a t i o n a t 295K  rates  H,  and D  f o r the total  187 Mu  + HI  reaction 191  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 e q u a t i o n s 1(8) and 1(12)  variables i n 232  -xAcknowledgement It  i s a p l e a s u r e t o acknowledge the  research director, "colleague" his  than  Dr.  Fleming,  a "director."  characteristic  patient  Don  of  my  i s a l w a y s more o f  a  H i s a d v i c e , always g i v e n  enthusiasm,  r e s p e c t f o r my  who  support  i s u s u a l l y accompanied  with  by  a  independence.  R e s e a r c h 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 efforts  o f an  enormous number o f p e o p l e  o n l y p o s s i b l e t o m e n t i o n a few. experimental  Marshall,  Pifer  t o Ryu  TRIUMF ySR exceedingly  In l e a r n i n g the  A  special  group i s indebted powerful into  data  a new  experience.  to h i s genius  aquisition  keeping  me  t h e s i s work f e l l  into  be  entire  s y s t e m w h i c h has  helped  p a r e n t s , w h i c h I have n e v e r  I t h a n k R o s a Ho  became a muonium c h e m i s t course  UBC  generation.  tingly,  on  Glen  i n c r e a t i n g our  and  supporting  i t became f a r removed f r o m  Finally,  main  acknowledgement s h o u l d  b e f o r e p r o p e r l y done, f o r e n c o u r a g i n g even a f t e r  of  P r o f e s s o r John Warren, of the  I a l s o w i s h t o t h a n k my  she  t h a t my  Hayano o f t h e U n i v e r s i t y o f T o k y o ; t h e  a d v a n c e ySR  education,  "ropes"  of the U n i v e r s i t y of A r i z o n a ,  J e s s B r e w e r , and  P h y s i c s Department. given  - unfortunately i t i s  n u c l e a r p h y s i c s , i t seems t o me  t e a c h e r s were B u r t  the  with  her  (who,  herself,  l o v e and  i s more o f a " d i r e c t o r "  than  a  their  q u i t e unsuspec-  by m a r r i a g e )  support  a state of c r i s i s .  my  As  whenever a muonium  "colleague."  for this chemist  -1CHAPTER I - INTRODUCTION _A  P o s i t i v e Muons and the u SR Method +  The muon i s an u n s t a b l e elementary f i r s t observed Street  as a component of cosmic  rays  (3-7)] and which i s now a r t i f i c i a l l y  energy p a r t i c l e a c c e l e r a t o r s . p o s i t i v e muons  p a r t i c l e t h a t was [Anderson (37),  produced  w i t h high  Some of the p r o p e r t i e s of  ( y ) are summarized i n Table I and i n c l u d e : +  u n i t charge, s p i n H, and a mean l i f e t i m e of 2.2 y s . Muons are decay products of p i o n s , which, i n t u r n , are produced  i n the n u c l e a r i n t e r a c t i o n s t h a t take p l a c e when  a nucleus i s bombarded w i t h h i g h energy protons.  T y p i c a l n u c l e i used f o r p i o n p r o d u c t i o n a t  a c c e l e r a t o r s are copper k i n e t i c energy  and b e r y l l i u m , and the minimum proton  r e q u i r e d f o r p i o n p r o d u c t i o n i n such a nucleus  i s about 145 MeV, the t h r e s h o l d energy. decay w i t h a mean l i f e t i m e o f 26 ns process  p a r t i c l e s such as  [Bjorken TT  ->  P o s i t i v e pions ( T T ) +  i n the p a r i t y  violating  (64)]:  li  +  which i s e x o e r g i c  V  y  (1)  by about 3 4 MeV and produces 4.1 MeV y . +  The muon n e u t r i n o , v , i s a s p i n h, p a r t i c l e w i t h r e s t mass and 100% negative h e l i c i t y .  zero  The h e l i c i t y  operator  i s d e f i n e d as the dot product of the s p i n and momentum y\  -—  d i r e c t i o n h = o_JB, and has eigenvalues of +1  lei  (positive  h e l i c i t y ) i n which the s p i n i s p a r a l l e l t o the momentum, and -1 (negative h e l i c i t y ) i n which the s p i n i s a n t i p a r a l l e l t o the momentum.  In order t o conserve  the muon formed i n p i o n  l i n e a r and angular momentum,  (spin 0) decay comes o f f i n the  -2-  TABLE I : PROPERTIES OF POSITIVE MUONS  CHARGE:  +1  SPIN:  \  MASS:  105.6596 MeV/c  2  =206.7685  m  e  = .0.1126 m P - 0.7570 m + MAGNETIC MOMENT:  4.49048 x 1 0 ~  2 3  =  3.18334 y  =  P 0.004836 y  erg G  _ 1  e g-FACTOR: 2.0023318 = 1.000006 g ^ ^e MEAN LIFETIME: 2.1971 ys Y  13.5544 kHz G "  GYROMAGNETIC RATIO,  1  2TT  COMPTON WAVELENGTH, k: CHARGE RADIUS:  1.86758 fm  <0.01 fm  =- m c y ft  -3d i r e c t i o n o p p o s i t e t o the n e u t r i n o w i t h 100% n e g a t i v e as w e l l .  helicity  T h i s 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  r e s t frame of the p i o n .  That p a r i t y i s v i o l a t e d i n p i o n  decay i s seen from the f a c t t h a t under p a r i t y , the a x i a l v e c t o r a i s unchanged, while the p o l a r v e c t o r 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 c r e a t e d with the p a r t i c l e ' s b i r t h can be t r a n s l a t e d i n t o t h e design of y  +  beams i n which the muons have a  net l o n g i t u d i n a l s p i n p o l a r i z a t i o n .  A more d e t a i l e d d i c u s s i o n  of muon beams i s g i v e n i n Chapter I I . When h i g h energy muons i n t e r a c t w i t h matter,  they —9 '  t h e r m a l i z e p r i m a r i l y by i o n i z a t i o n processes i n about 10 and r e t a i n t h e i r s p i n p o l a r i z a t i o n (67), Brewer  (75)].  In metals  [Hughes (66), Weissenberg  [Brewer  (75), G r e b i n n i k (76)]  and gases such as He w i t h l a r g e i o n i z a t i o n [Stambaugh  (74)], y  other m a t e r i a l s , y environments.  +  +  s  t h e r m a l i z e as " f r e e " y  potentials +  i o n s ; i n many  end up c h e m i c a l l y bound i n diamagnetic  The s p i n s o f such muons w i l l precess i n a  t r a n s v e r s e magnetic f i e l d a t a frequency which i s p r o p o r t i o n a l + y to the y gyro-magnetic r a t i o , ^ = 13.55 kHz/gauss. Because Y  muon beams have a l o n g i t u d i n a l s p i n p o l a r i z a t i o n , a l l o f the muons t h e r m a l i z e w i t h the same i n i t i a l phase w i t h r e s p e c t t o s p i n p r e c e s s i o n i n a t r a n s v e r s e magnetic The u n s t a b l e y  +  field.  decays by another p a r i t y  violating  process: y K  +  -> e  +  + u + u e y  (2)  where u i s an e l e c t r o n n e u t r i n o w i t h n e g a t i v e h e l i c i t y , u e ^ y  is  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  positron with p o s i t i v e predicts  t h a t the three-body  anisotropic with preferentially qualitative by  helicity.  Garwin  helicity,  and e' i s a  Weak i n t e r a c t i o n  decay o f t h e y  theory  i s spatially  +  r e s p e c t t o p o s i t r o n emission, which i s  along  the d i r e c t i o n  b e h a v i o r was f i r s t  (57).  of the y  confirmed  spin.  +  experimentally  The t h e o r e t i c a l p o s i t r o n d e c a y  g i v e n by t h e e x p r e s s i o n  [Sachs  This  spectrum i s  (75)]:  dR(w,6) = w {(3-2w) - P ( l - 2 w ) c o s 6 } dwdfi 2TT 2  = C {1 + DcosG} where w = / E  possible  i-  E M  a  x  energy,  t h e p o s i t r o n energy  s  E M a x  k™-^  =  =  i n units  o f t h e maximal  52.8 MeV, 9 i s t h e a n g l e  between  the  s p i n o f t h e d e c a y i n g 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  muons.  of spin p o l a r i z a t i o n  The p o s i t r o n e n e r g y  spectrum  of the decaying  and t h e asymmetry  p a r a m e t e r , D, f o r P = l a r e shown i n F i g u r e 1. In p r a c t i c e , efficiency The  e (w)  observed  t h e p o s i t r o n s a r e d e t e c t e d w i t h an  which i s n o t constant over  probability  distribution  their  energy  t h e n becomes  range.  [Brewer  (75), Weissenberg (67)] d-R d!T  f ^0 = l_e  1  d-R (w.>80 e(w)dw dwdQ (1 + A c o s 6 )  4TT  I f p o s i t r o n s o f a l l e n e r g i e s were d e t e c t e d w i t h efficiency, IP. T  the observed  In p r a c t i c e ,  average  t h e same  asymmetry, A, w o u l d be  the detection e f f i c i e n c y  o f low e n e r g y  -5-  w= E / E  m a  FIGURE 1: Energy spectrum of p o s i t r o n s from muon decay (upper curve) and energy dependance o f the asymmetry parameter f o r 100% beam p o l a r i z a t i o n (P=l; lower c u r v e ) . The energy i s given as a f r a c t i o n o f t h e maximal p o s s i b l e energy, E =52.8 MeV. ^ max  -6e  +  i s reduced and the lowest energy e  +  are absorbed by  matter b e f o r e r e a c h i n g the d e t e c t o r s r e s u l t i n g i n an observed A l a r g e r than 1 P. 3  T h i s e f f e c t i s o f f s e t , however, by the  r e d u c t i o n i n P due t o k i n e m a t i c d e p o l a r i z a t i o n  ( r e a l muon  beams are not 100% p o l a r i z e d ) and due t o averaging over f i n i t e d e t e c t o r s o l i d angle.  In most muon experiments, the  beam p o l a r i z a t i o n , 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 and s o l i d angle c o r r e c t i o n s are not e x p l i c i t l y known and the r e s u l t a n t e f f e c t i v e muon asymmetry, A , i s t r e a t e d e m p i r i c a l l y i n the expression: R(9) = 1 + A cos9 (3) Since the average decay p o s i t r o n energy i s about 3 5 MeV, _2  corresponding t o a r a d i a t i o n l e n g t h of 15 g. cm  i n Pb,,  most p o s i t r o n s are observable even i f the muon decay occurs deep i n s i d e a s u b s t a n t i a l  target.  The time d i f f e r e n t i a l measurement o f the asymmetric decay of a s p i n p o l a r i z e d ensemble of p o s i t i v e muons p r e c e s s i n g i n a t r a n s v e r s e magnetic technique.  f i e l d forms the b a s i s of the y SR +  The acronym, y SR, stands f o r "muon s p i n +  r o t a t i o n " and was c o i n e d t o draw a t t e n t i o n t o the s t r o n g resemblance  i n i n f o r m a t i o n content t h a t t h i s method bears t o  the f a m i l i a r resonance techniques o f NMR and ESR.  Except i n  s p e c i a l v a r i a t i o n s such as the s t r o b o s c o p i c method  [Schenck  (76)], y SR examines one muon a t a time u s i n g c o u n t i n g +  techniques common t o experimental n u c l e a r p h y s i c s .  The phrase  "muon ensemble" i n the p r e s e n t d i s c u s s i o n , then, r e f e r s t o an ensemble i n time r a t h e r than i n space. In a y SR experiment, a l o n g i t u d i n a l l y s p i n p o l a r i z e d +  -v-  + y  passes  f r o m t h e beam c h a n n e l t h r o u g h a p l a s t i c  counter array The  counters  and t h e r m a l i z e s are arranged  i n a target material  to identify  t a r g e t ; when s u c h an e v e n t o c c u r s , generated which The  s t a r t s some k i n d  muon p r e c e s s e s is  i n the target  i t s gyromagnetic r a t i o  field  e x p e r i e n c e d by t h e y .  decay  spectrum  +  from  muons w h i c h s t o p  precision  phase.  = y^B where  a t a frequency  and B i s t h e t r a n s v e r s e Noting  that  9=OJ t , t h e p o s i t r o n  magnetic  f i e l d of  50 g a u s s t o s e v e r a l k g a u s s i s e x t e r n a l l y a p p l i e d  material,  transverse  f o r ferromagnetic  magnetic  field  may be i n t r i n s i c  i n w h i c h c a s e t h e muon p r e c e s s i o n  frequency  a t the y  material  counter array  (77)].  A positron  p l a n e o f muon p r e c e s s i o n  a t an a n g l e  monitors  JNishida  the y  +  time histogram, t h e c l o c k i s r e s e t , o" 7 t y p i c a l l y 10 - 10 times.  probability  y  i n that  magnetic  placed  < j > to the i n i t i a l  i nthe  muon beam  i s incrementally and t h e p r o c e s s  binned  in a  i s repeated,  decay  i s spatially  asymmetric, t h e  of detecting  the p o s i t r o n  f r o m muon d e c a y r i s e s and  +  as t h e p r e c e s s m g  detectors.  isa  s t a r t e d b y t h e muon e n t e r i n g t h e  The m e a s u r e d t i m e i n t e r v a l  Since  +  to the  d e c a y and g e n e r a t e s an e l e c t r o n i c p u l s e t o  the clock previously  target.  i nthe  targets, a  d i r e c t measure o f t h e i n t e r n a l f i e l d  falls  magnetic  a l l muons have t h e same  A transverse  case o f non-magnetic t a r g e t s ;  stop  clock.  = 1 + A cosw t y y  precession  substantial  i n the  an e l e c t r o n i c p u l s e i s  of high  beam p o l a r i z a t i o n e n s u r e s t h a t  initial  of i n t e r e s t .  ( 3 ) becomes:  R(t) The  scintillator  y  spin  Because t h e s o l i d  swings p a s t  angle  the f i x e d e  s u b t e n d e d by t h e p o s i t r o n  -8counters which period  i s s m a l l , most muon  case  the clock  decays  i s reset  are not detected, i n  after  o f s e v e r a l muon l i f e t i m e s .  some a r b i t r a r y The r e s u l t a n t  "time o u t "  u SR  time  +  h i s t o g r a m has t h e form: N(c(>,t) = N e  - t  ^ y{l  + A  T  where N i s t h e number o f c o u n t s a normalization factor, A  (4)  i n a h i s t o g r a m time b i n , N  T i s the y y  ( t ) i s t h e muon asymmetry w h i c h  to  +  lifetime  i s usually  Q  is  o f 2.2 u s , time  dependent,  i s t h e muon p r e c e s s i o n f r e q u e n c y , and Bg i s a t i m e  independent y S R time features  background  spectrum  +  A  typical  to  upon w h i c h i s  a s y m m e t r i c a l muon d e c a y .  o f t e n decays  phenomena i d e n t i c a l  w i t h t i m e due t o s p i n  relaxation  i  nN  M  R  a n c  The dephasing  ^ commonly h a s  form: A  (t) = A e " y y  where X = l / T ^ • +  events.  i s shown i n F i g u r e 2; i t s most d o m i n a n t  the o s c i l l a t i n g  asymmetry, A ^ ( t ) ,  the  due t o a c c i d e n t a l  a r e t h e e x p o n e n t i a l muon l i f e t i m e  superimposed  y  ( t ) c o s ( w t + d>) j +. Bg  An example o f s u c h a r e l a x a t i o n mechanism i s  spin dephasing  magnetic  field  interstitial  (5)  X t  due t o l o c a l  fluctuations  e x p e r i e n c e d by a y  sites  +  diffusing  i n a ferromagnetic c r y s t a l  y SR i s a p a s s i v e non-resonance +  NMR  i n which  for  signal  i n the i n t e r n a l  the s p e c i a l properties  generation, eliminating  between o f Fe  analogue  different [Nishida  of proton  o f t h e muons a r e r e s p o n s i b l e 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 information  contained  t h e t i m e domain i s ,  in  i n NMR  principle,  spectra transformed into  contained i n y SR +  spectra.  Of c o u r s e , t h e t i m e  LI SR: +  LL  +  IN R L U M I N I U M .  6 9 GAUSS  6000 5000  -  4000  -  CO  LU LU  3000  i I  CL  LU £  2000 1000  h  0 0.0  0.5  1 .0 TIME  1.5  2.0  IN LLSEC  2.5  3.0  3.5  4.0  (20 N S E C / B I N )  FIGURE 2: A t y p i c a l y SR time histogram (data p o i n t s ) and x -minimum f i t to equation (4). The e r r o r bars (on every 10th p o i n t ) are due t o counting s t a t i s t i c s only. The histogram c o n t a i n s about 5 x 1 0 events. The y + asymmetry i s about 35% and X = 0.03 y s . 5  - 1  -10scale  of  phenomena  lifetime. over  detectable  The y + S R method  NMR;  o n e muon  interferences  themselves;  the y+SR  magnetic  materials  penetrate charge y+  without  probe  B_ M u o n i u m  of  a n d t h e MSR most  of  i t s thermalization  Fleming  (79)1.  Since  [Hughes  r e d u c e d mass radius  a light (78)]  isotope  with  difference probe  of  a mass  isotope  external hyperfine  H  form  only  point  often  sites  themselves.  that  of  of  Brewer  Since  behaves Brewer This  Table  electron,  chemically (75),  the  of  mass  sensitive H.  coupled to  to the electron  like  Jean  substantial  spin i s not only  t h e muons  in  H a n d Mu a r e  i n chemical reactions  f i e l d but also  atom,  (75),  as t h e  m a k e s Mu a n e x c e p t i o n a l l y  the y +  stages  H and c o n s e q u e n t l y  of  H,  the  the f i n a l  o f Mu a r e g i v e n  (71),  normal  solids,  the hydrogen-like  as massive  potentials  [Goldanskii  effects  interaction.  bulk  crystals,  and n o n - m e t a l l i c  Mu, t h e r e f o r e ,  1_ t h a t 9  muonium,  magnetic  NMR i s  the properties  i s 207 t i m e s  potentially  In  to  o f Mu i s 0 . 9 9 6  of  and, i n  ( 6 6 ) , Mobley -1-(67 ) ,  t h e same.  with  f o r NMR w i l l  t h e medium d u r i n g  process  and i o n i z a t i o n  essentially  the y +  i s a simple  the l a t t i c e  liquids from  Some o f  t h e muon  of  thereby  Method  an e l e c t r o n  (Mu)  Bohr  the y+  of  +  advantages  from v/ithin  required  the sample;  gases,  captures  muonium  potential  interaction  the r , f ,  by t h e y  i n t h e sample  a complicating structure;  y+  the  due t o  to examination  In  II.  i s present  the i n t e r s t i t i a l region while  constrained  of  s i g n a l i s measurable  while  the skin  i s fixed  h a s a number  at a time  eliminating  by y+SR  an  spin v i a the  are polarized  while  the  -11-  TABLE II: PROPERTIES OF MUONIUM  MASS;  207.8  m  0 . 1 1 3 1 m, H REDUCED MASS:  0.9956 y  H 0.5315  FIRST BOHR RADIUS, (a ) : ' o Mu  10  x  1.0044 ( a 13.54  FIRST IONIZATION POTENTIAL: =  Q  )  -8  cm  H  eV  0 . 9 9 5 6 I.P  "H  2.979  THERMAL DEBROGLIE WAVELENGTH:(300K): =  2.967  10  X X  u  ri  72TTk~Tm  a  HYPERFINE FREQUENCY, co MEAN THERMAL VELOCITY  2.8044 (3 0 0 K )  x 10  0.75 = 2.97  1 0  x 10^ v  H  8^^1/2 um  rad s  _ 1  cm s  -8  cm  -12captured Mu  are  e l e c t r o n s are  5 0%  1  la a ) u e  and  polarization  direction  transverse  magnetic  unpolarized, 50%  1  la 3 ^ y er  i s the  field,  s p i n p o l a r i z a t i o n i n Mu  '  the  i n t h e weak t r a n s v e r s e  GJMU  y  +  at the  ( |a a ) y e  i n the  OJ^  ) precess  , i n the  other  MSR is  oscillation  y  +  field  i n h a l f of  to  ensemble o  "free" y  +  this limits  the  Mu  oscillate  a  rad  frequency, precession;  +  (l y£^)  x 10"^  y  the d e t a i l e d  upshot of  magnetic  O J = 2.8  i s not  the  basis of  observable  totally  and  s  this  . Since  t h e MSR  field method  the  h a l f of  the  depolarized.  time e v o l u t i o n of the  weak t r a n s v e r s e m a g n e t i c  v i a the  y  +  spin i n  asymmetric  for studying  y  Mu  decay  +  muonium.  The  acronym s t a n d s f o r "muonium s p i n r o t a t i o n " and t h e method + + i d e n t i c a l w i t h y SR w i t h t h e e x c e p t i o n s t h a t t h e y  precession and  t h e Mu  ensemble a p p e a r s t o be  forms the  finite  time r e s o l u t i o n i s about 1 nanosecond,  Monitoring in  In  a t t h e muonium Larmor  frequency,  experimental  The  sense o p p o s i t e  h a l f of  hyperfine  hyperfine Mu  I.  g a u s s ) i s , however, s i m p l e :  = 103  the  i n Appendix  of  muon  i s 'quite complicated;  calculation  ensemble  , where t h e  spin states  time e v o l u t i o n of the  i s given  10  initial  quantization axis.  calculation  (<  the  frequency  t h a t the  y  +  asymmetry  shows a t y p i c a l MSR a p p e a r a n c e as stronger.  i n Mu  i s 103 i n Mu  times t h a t  i s r e d u c e d by  s p e c t r u m w h i c h has  for "free" half.  essentially  a y SR spectrum i n a magnetic f i e l d +  I n p r a c t i c e , MSR  N((J,,t)=N e^"  t/T  o  y{l  h i s t o g r a m s have t h e  + A^  the 103  to' a r e y  3  same times  form:  (t) cos ( u ^ t + <J> ) + Mu  (6)  Y  and  +  Figure  A cos (co t -.• <J> ) } + Bg y y y ^ where OJ», Mu  y  t h e muonium and  muon p r e c e s s i o n  frequencies, ^  MSR:  MU I N 780  TORR N  2  RT 6 . 9  GAUSS  12000 10000 CO  LU  8000  O  6000  cm LU CD  -  4000 h 2000 0 0.0  0.5  1.0 TIME  1.5  2.0  IN LLSEC  2.5  (20  3.0  3.5  4.0  NSEC/BIN) 2  FIGURE 3: A t y p i c a l MSR time' histogram (data points) and x -minimum f i t t o equation (6). The histogram c o n t a i n s about 10^ events and the e r r o r bars are due to c o u n t i n g s t a t i s t i c s o n l y . The Mu asymmetry i s about 11% and the background u asymmetry i s about 5.5%. +  4>  Mu  and <f)  and A y  are the i n i t i a l muonium and muon phases, w h i l e  are the muonium and muon asymmetries. -  A M u  (t)  The " f r e e " muon  2  p r e c e s s i o n term may come from s e v e r a l sources: some m a t e r i a l s i s incomplete;  Mu formation i n  i n gas t a r g e t s , muons may  s c a t t e r i n t o the w a l l s of the gas v e s s e l where they may not form Mu; muonium atoms may r e a c t c h e m i c a l l y on the very s h o r t time s c a l e of l e s s than one h y p e r f i n e p e r i o d o f 0.225 ns (probably by epithermal  r e a c t i o n ) i n which case the h y p e r f i n e  i n t e r a c t i o n i s broken, r e s u l t i n g i n coherent polarized y  +  p r e c e s s i o n o f the  ensemble r e s i d i n g i n diamagnetic environments.  muonium experiments, the asymmetry of the f r e e y  +  In  term i s  u s u a l l y independent of time while the muonium asymmetry g e n e r a l l y has  a form s i m i l a r t o equation ( 5 ) : A., (t) = A e ~ Mu Mu  (7)  X t  The muonium r e l a x a t i o n r a t e , A, i s due t o e f f e c t s such as pressure  broadening i n gases, chemical  f i e l d inhomogeneity.  r e a c t i o n and magnetic  The e f f e c t s of chemical  d i s c u s s e d i n some d e t a i l i n Appendix I I . be noted t h a t the phases,  an<  y  +  F i n a l l y , i t should  3 <f>^f have opposite  account f o r the f a c t t h a t the f r e e y opposite d i r e c t i o n s .  r e a c t i o n s are  and y  In condensed media  stopping r e g i o n , the magnitudes o f  however, i n low pressure gases, the y  signs t o  i n Mu precess i n  with a well defined and <f>^' are the same;  stopping r e g i o n i s  smeared out enough t h a t s i g n i f i c a n t d i f f e r e n c e s i n the magnitudes of <j>Mu and <J>  may appear.  Throughout t h i s t h e s i s , r e f e r e n c e w i l l be made t o the MSR " s i g n a l , " S(c|),t), which i s d e f i n e d as:  -15-  S(<f>,t) = A,, e~ cos(w., t + ch ' Mu Mu Mu Xt  Y  v  F i g u r e 4 shows the MSR  f i e l d s used i n  The  time  slow f r e e muon p r e c e s s i o n  appears  l i n e a r background at the weak magnetic  MSR.  _C Muonium Chemistry  - An H i s t o r i c a l Background  Muonium formation was for  (8)  s i g n a l corresponding t o the  histogram of F i g u r e 3. as an approximately  ) + A cos (w t - cf> ) y y y  f i r s t proposed  as an e x p l a n a t i o n  the o b s e r v a t i o n t h a t the " r e s i d u a l muon p o l a r i z a t i o n "  (see Appendix II) [Swanson (58)] . Smilga  i s not the same i n a l l condensed media Nosov and Yakovleva  (63) and  Ivanter and  (68) d e r i v e d a d e t a i l e d model f o r muon d e p o l a r i z a t i o n  phenomena, i n s o l i d s .  F i r s o v and Byakov  the r e s i d u a l p o l a r i z a t i o n to Mu  (65) attempted  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 .misinterpreted.  In 1969 , I v a n t e r and Smilga  formalism to c o r r e c t l y t r e a t Mu simple r e a c t i v e The r e a c t i o n s was  to r e l a t e  (.69)  (66). .were '  extended  the  chemistry i n l i q u i d s f o r  systems.  f i r s t e x t e n s i v e experimental study of thermal by Brewer  (72) who  Mu  a p p l i e d a m o d i f i e d form of the  muonium mechanism of I v a n t e r and Smilga to the measurement of b i m o l e c u l a r r a t e constants of simple Mu These experiments,  conducted  Lawrence Berkeley Laboratory to  a t the 184"  reactions i n liquids. C y c l o t r o n at the  (LBL), grew out of an  experiment  determine" the muon's magnetic-moment p r e c i s e l y • [Hague  which r e q u i r e d s m a l l c o r r e c t i o n s due  to chemical  Brewer a l s o found i t necessary to extend e p i t h e r m a l r e a c t i o n s of Mu  (70)]  effects.  the model t o i n c l u d e  as w e l l as r e a c t i o n s i n which  MSR:  MU I N 7 8 0 TORR N  2  RT 6 . 9 GRUSS  0.15 0.10  >-  0.05  ii  I—  0.00  i  >-  CO d  -0.05 :  -0.10 •0.15 0.0  0.5  1.0 TIME  1 .5  2.0  I N LISEC  (20  2.5  3.0  3.5  4.0  NSEC/BIN)  FIGURE 4: The MSR s i g n a l , S(<J>,t), corresponding t o the histogram shown i n F i g u r e 3. The l i n e i s a x minimum f i t t o equation ( 8 ) . The r a p i d o s c i l l a t i o n s are due t o y+ p r e c e s s i o n i n Mu, while the slowly c u r v i n g d r i f t i s due t o " f r e e " y p r e c e s s i o n . 2-  +  -17t r a n s i e n t muonic r a d i c a l s are formed.  Although  e s t a b l i s h e d the foundations of experimental Mu  Brewer's work chemistry, i t  s u f f e r e d from a number of s e r i o u s shortcomings, notable of which was d i r e c t l y by the MSR  the most  the f a i l u r e to d e t e c t muonium i n l i q u i d s method.  Muonium r e a c t i o n s were measured  by the i n d i r e c t r e s i d u a l p o l a r i z a t i o n method d e s c r i b e d i n Appendix II u s i n g y SR techniques.  With t h i s method, d e t a i l s  +  of the r e a c t i o n mechanisms had to be i n f e r r e d and the r a t e constants obtained were l a r g e l y model dependent.  In  r e a c t i o n s i n v o l v i n g s e v e r a l r a t e p r o c e s s e s , such as those  due  to i n t e r m e d i a t e r a d i c a l f o r m a t i o n , the e x t r a c t e d r a t e constants were h i g h l y c o r r e l a t e d and of q u e s t i o n a b l e accuracy  [ P e r c i v a l 1- (76) , P e r c i v a l  (77)].  absolute  While t h i s  indirect  method has been s u b s t a n t i a l l y • r e p l a c e d w i t h more d i r e c t methods d e s c r i b e d below, Brewer's p i o n e e r i n g work p r o v i d e d a v a l u a b l e p r e l i m i n a r y i n s i g h t i n t o the d e t a i l s of muonium chemistry i n liquids.  For example, i n s p i t e of t e n t a t i v e 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 d e t e c t i o n of muonic r a d i c a l s i n condensed media  [Kent  (77), B u c c i  (78)3  ^ t Brewer's work  p r o v i d e s the most c o n v i n c i n g demonstration  of the  still  importance  of such r a d i c a l s i n l i q u i d phase r e a c t i o n s of Mu  [Brewer (73)].  Although  (78).], the  the s u b j e c t of some c o n t e n t i o n [ P e r c i v a l  r e s i d u a l p o l a r i z a t i o n method might w e l l prove to be the most amenable to the study of e p i t h e r m a l Mu  one  reactions i n  liquids.  During p r e p a r a t i o n of t h i s t h e s i s , the d i r e c t o b s e r v a t i o n of s e v e r a l muonic r a d i c a l s was confirmed a t SIN .{Roduner^l (78)] .  -18The complicated reactions  study of muonium chemistry  by s e v e r a l processes  such as s o l v o l y s i s and  [see, f o r example, Gold  understanding  in liquids is  elementary chemical  (78)].  "spur"  From the viewpoint  r a t e p r o c e s s e s , the gas  of  phase  p r o v i d e s a p h y s i c a l context which i s more t h e o r e t i c a l l y t r a c t a b l e than the l i q u i d phase. chemistry  The  o n l y gas phase  Mu  s t u d i e s p r i o r t o the work i n t h i s t h e s i s were a s e r i e s  of experiments conducted by Mobl-ey .e_t a l . [Mobley ,2-(67)] i n argon gas Nevis Laboratory techniques  at high pressure  (66) , l-(67) ,  (40 atmospheres) a t the  a t Columbia U n i v e r s i t y ,  A v a r i e t y of  i n c l u d i n g d i r e c t o b s e r v a t i o n of Mu  by the  MSR  method were employed t o examine the i n t e r a c t i o n s of Mu C>2, C 2 4 H  a n <  ^  CH-3CI  a n (  3  a number of other reagents.  the c o n v e n t i o n a l muon beam a v a i l a b l e t o Mobley was  with  Unfortunately, of such high  momentum t h a t very h i g h p r e s s u r e gas t a r g e t s were r e q u i r e d to t h e r m a l i z e a u s e f u l f r a c t i o n of the beam. three body  . processes  p l a y an important  r e a c t i o n s ; i t i s p r e f e r a b l e t o use  At 40 atmospheres, r o l e i n the  low p r e s s u r e gas  about 1 atmosphere t o measure b i m o l e c u l a r Mu Motivated muonium Lederman  (y e +  chemical t a r g e t s at  reaction rates.  by a p r o p o s a l to measure the c o n v e r s i o n of  ) to antimuonium  (y e ) +  I see, f o r example,  (77)] which r e q u i r e s the p r o d u c t i o n of thermal Mu  vacuum, a group from the U n i v e r s i t y of A r i z o n a designed k i n d of low momentum muon beam l i n e a t the 184" LBL  IPifer  (76)].  a  in new  C y c l o t r o n at  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 " A r i z o n a " muon beam) are g i v e n i n Chapter I I .  In c o l l a b o r a t i o n with the A r i z o n a group, t h i s  t h e s i s work was  started  at t h e i r s u r f a c e muon f a c i l i t y  at  -19-  LBL d u r i n g 1974-75 . when the f i r s t  low p r e s s u r e gas phase Mu  b i m o l e c u l a r r e a c t i o n r a t e constant was determined t e c h n i q u e , f o r the Mu + B r of  Ar [Fleming  (76)].  2  by the MSR  r e a c t i o n a t 295 K i n 1 atmosphere  The c o l l a b o r a t i o n w i t h the A r i z o n a  group was continued u n t i l J u l y , 1975, when support f o r p h y s i c s experiments a t the 184" C y c l o t r o n ceased machine became a d e d i c a t e d medical  facility.  and the  I t i s , perhaps,  an h i s t o r i c a l f o o t n o t e t o remark t h a t the Mu + C l r e a c t i o n 2  r a t e measurement [Fleming 1-(77)] was the l a s t non-medical experiment executed  on t h a t machine.  A major t e c h n o l o g i c a l advance i n the study of muon p h y s i c s and chemistry of  i n r e c e n t years  i s the development  a new g e n e r a t i o n of a very h i g h c u r r e n t " i n t e r m e d i a t e "  energy p a r t i c l e a c c e l e r a t o r s , the s o - c a l l e d  "meson f a c t o r i e s " .  These machines produce meson beams w i t h i n t e n s i t i e s t h a t are two or more orders of magnitude g r e a t e r than those p r e v i o u s l y available. ional  At p r e s e n t , there are three such f a c i l i t i e s  i n the world:  Nuklearforschung  the Schweizerisches  operat-  Institut fur  (SIN) near Zurich', the C l i n t o n P. Anderson  Meson P h y s i c s F a c i l i t y  (LAMPF) a t Los Alamos, and the  T r i - 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 r e a c t i o n r a t e measurements a t low p r e s s u r e were f i r s t performed a t TRIUMF on the M20 y SR F a c i l i t y o p e r a t i n g +  i n s u r f a c e muon mode. Although for  s u r f a c e muon beam l i n e s r e q u i r e d  low pressure gas phase t a r g e t s are c u r r e n t l y being  commissioned a t LAMPF and under c o n s t r u c t i o n a t SIN, a t p r e s e n t TRIUMF i s the only meson f a c t o r y w i t h an o p e r a t i o n a l f a c i l i t y of  this kind.  Recently, another  s u r f a c e muon f a c i l i t y  (using  the beam components from the o r i g i n a l A r i z o n a beam l i n e at Berkeley) was de-commisioried' a t the 6 00 MeV s y n c h r o c y c l o t r o n the  Space R a d i a t i o n  E f f e c t s Laboratory  of  (SREL) i n V i r g i n a .  With the advent of meson f a c 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 o b s e r v a t i o n  of Mu i n water by the MSR techniqu  at SIN [ P e r c i v a l 2-(76)] .  This discovery,  at TRIUMF [Jean -  ;  recently  confirmed  (78)], has l a r g e l y rendered o b s o l e t e the  r e s i d u a l p o l a r i z a t i o n method used by Brewer i n the study of thermal chemical r e a c t i o n s  of Mu i n the l i q u i d phase and p l a c  l i q u i d Mu chemistry on the f i r m e r experimental  footing  previously  I t should be  enjoyed only by gas phase s t u d i e s .  remarked t h a t l i q u i d phase MSR s i g n a l s are much weaker than gas  phase s i g n a l s .  To date, the SIN group have a p p l i e d  to the study of a number of chemical r e a c t i o n s of l i q u i d media  MSR  i n a variety  [ P e r c i v a l (77), Roduner 2 - ( 7 8 ) ] .  F u r t h e r impetus was g i v e n t o gas phase Mu chemistry when the f i r s t d e t a i l e d 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 r e a c t i o n r a t e was performed by a group i n Europe [Connor 1(77)]  f o r the r e a c t i o n : Mu + F ->MuF + F. 2  Considerable  a t t e n t i o n w i l l be g i v e n t o t h i s and subsequent c a l c u l a t i o n s i n Chapter IV.  ID  Organization  of the D i s s e r t a t i o n  This t h e s i s reports-the  f i r s t , measurements o f Mu  r e a c t i o n r a t e s i n low p r e s s u r e gases  (^1  atmosphere), f o r the  reactions: Mu + .X . •>•• MuX + X, 9  X= F, C l , Br  -21and Mu at 295K. the F  2  and  + HX  ->- MuH  + X,  X = C l , Br,  I  In a d d i t i o n , a c t i v a t i o n e n e r g i e s are r e p o r t e d Cl  2  r e a c t i o n s between 300  and  400K.  As d e t a i l e d i n Chapters I I I and f o r t h i s study i s twofold:  for  IV, the  motivation  (1) as a l i g h t i s o t o p e of H,  Mu  p r o v i d e s a remarkably s e n s i t i v e probe of mass e f f e c t s i n H atom r e a c t i o n s , and chemistry, MSR  (2) u n l i k e the techniques of H atom  i s l i t e r a l l y a one-atom-at-a-time method,  unencumbered by i n t e r a c t i o n s of the Mu  atoms w i t h themselves.  With s u f f i c i e n t understanding o f the f i r s t p o i n t , p o s s i b l e to e x p l o i t the  i t may  be  second p o i n t to o b t a i n accurate v a l u e s  of H atom r e a c t i o n r a t e s f o r systems where they are  not  measurable by other methods. T h i s t h e s i s i s composed of t h r e e main p a r t s . Chapter II d e s c r i b e s  how  the MSR  technique i s a p p l i e d to the measurement  of gas  phase chemical r e a c t i o n : r a t e s .  of the  s u r f a c e muon beam, gas  Included are  descriptions  t a r g e t apparatus, c o u n t i n g  procedures, e l e c t r o n i c l o g i c and  data a c q u i s i t i o n , and  methods  of data a n a l y s i s . Chapter I I I p r e s e n t s a b r i e f general d i s c u s s i o n of gas and  phase r e a c t i o n s of Mu  theoretical  as an H i s o t o p e .  Mu  H are compared i n terms of the k i n e t i c i s o t o p e e f f e c t ;  p o s s i b l e i m p l i c a t i o n s of d i f f e r i n g energy d i s p o s i t i o n s i n the t r a n s i t i o n s t a t e and  among r e a c t i o n products of Mu  r e a c t i o n s are d i s c u s s e d ;  and,  finally,  and  some dynamical  H isotope  -22-  e f f e c t s are examined w i t h p a r t i c u l a r a t t e n t i o n t o quantum mechanical t u n n e l l i n g . In Chapter IV, the experimental gas phase Mu  reaction  r a t e measurements are compared with experimental v a l u e s f o r the analogous H atom r e a c t i o n s  and w i t h t h e o r e t i c a l p r e d i c t i o n s .  CHAPTER I I - EXPERIMENTAL DETAILS I t was mentioned  i n Chapter I t h a t the f i r s t of the  experiments d e s c r i b e d i n t h i s t h e s i s were conducted a t the 184" C y c l o t r o n at Berkeley; d e t a i l s of those experiments are not  g i v e n here but may be found i n s e v e r a l r e f e r e n c e s  (76),  Fleming  (76), Fleming l - ( 7 7 ) ] .  MSR methods are c o n s t a n t l y e v o l v i n g . to  [Pifer  L i k e most t e c h n o l o g i e s , Rather than attempting  p r o v i d e a h i s t o r y of MSR development a t LBL and TRIUMF,  t h i s Chapter w i l l only d e s c r i b e the " s t a t e of the a r t " techniques as p r a c t i c e d a t TRIUMF i n 1978.  Some s p e c i f i c  suggestions f o r f u t u r e improvements, p a r t i c u l a r l y i n the e l e c t r o n i c l o g i c system, are i n c l u d e d . A.  TRIUMF and the M20 Muon Beam L i n e The TRIUMF Annual Reports 1972-76 are a good source f o r  d e t a i l e d i n f o r m a t i o n on the many TRIUMF f a c i l i t i e s ; c e n t r a l p o i n t s are g i v e n here.  The TRIUMF c y c l o t r o n and  experimental areas are shown i n F i g u r e 5. focussed H  only a few  TRIUMF i s a s e c t o r -  c y c l o t r o n t h a t d e l i v e r s protons o f c o n t i n u o u s l y  v a r i a b l e energy ranging from 185-520 MeV a t maximum d e s i g n c u r r e n t s o f 100 yA, a t 500 MeV and 450 yA a t 450 MeV. of  Most  the experiments d e s c r i b e d i n t h i s t h e s i s were conducted  with a 5-10 yA proton beam a t 500 MeV.  One of the most  a t t r a c t i v e f e a t u r e s of the TRIUMF c y c l o t r o n from the viewpoint of  MSR i s i t s 100% macroscopic duty c y c l e : seen on a  macroscopic time s c a l e  (as s h o r t as microseconds), the proton  beam appears t o be a continuous c u r r e n t without a time  EXISTING PROPOSED  FIGURE 5: The TRIUMF C y c l o t r o n  and e x p e r i m e n t a l  facilities  (1977).  structure.  The m i c r o s c o p i c  duty c y c l e i s a 5 nanosecond  b u r s t of protons every 43 ns.  The MSR method r e q u i r e s  at most one muon be i n the t a r g e t a t a time. instantaneous muon stopping  upper l i m i t of the order of 10  of a few muon l i f e t i m e s ) .  Thus, the  r a t e i n the t a r g e t has an 5 "+  absolute  that  A t pulsed  as LAMPF, which has a 6% duty c y c l e 8 ms), the maximum allowable  y  -1 s  (the  inverse  beam f a c i l i t i e s (a 500 ys b u r s t  such every  average counting r a t e i s  decreased by e x a c t l y the duty f a c t o r o f the machine. l i m i t a t i o n o c c a s i o n a l l y , can be side-stepped  This  by the use of  s p e c i a l m u l t i p l e muon techniques such as the s t r o b o s c o p i c method  [Schenck  (76)], but the severe r e s t r i c t i o n s placed on  the experiments t h a t use t h i s method g r e a t l y l i m i t i t s applicability.  Even with i t s 100% duty c y c l e , the i n t e n s e  beams a v a i l a b l e a t TRIUMF are capable of implanting one  more than  muon i n a t a r g e t at a time; a d e t a i l e d d i s c u s s i o n of t h i s  problem of muon " p i l e - u p " i s given  i n S e c t i o n C and Appendix  III. A proton beam, e x t r a c t e d the e l e c t r o n s  from the c y c l o t r o n by s t r i p p i n g  from the H i o n s , passes down beamline-1  i n the "meson h a l l " and s t r i k e s a p i o n p r o d u c t i o n T2.  (BL-1)  target,  The t a r g e t used i n t h i s work c o n s i s t s of a water-cooled  b e r y l l i u m 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 c r o s s reactions  s e c t i o n ; pions are produced here v i a  such as: B e (p, TT ) B e . 9  +  1 0  nuclear  Three secondary beamlines  simultaneously e x t r a c t mesons (TT o r y) produced a t T2: M8, p r i m a r i l y intended f o r use i n IT cancer therapy; M9, a  "stopped" T T or y~ beamline used f o r a v a r i e t y of experiments -  (the modifer "stopped" i n d i c a t e s t h a t the TT o r y beam i s of sufficiently  low energy t o stop i n s m a l l e x p e r i m e n t a l t a r g e t s ,  i n c o n t r a s t t o TT or y beams used f o r s c a t t e r i n g experiments) ; and M20, a stopped to y SR. +  beamline which i s e s s e n t i a l l y  dedicated  The experiments d e s c r i b e d i n t h i s t h e s i s were  performed on M20 which g e n e r a l l y 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 t h e r e i s beam on T2. M2 0 (shown i n F i g u r e 6) t r a n s p o r t s a muon beam i n vacuum t o the e x p e r i m e n t a l t a r g e t i n one of t h r e e o p e r a t i n g modes: " c o n v e n t i o n a l , " "cloud""' o r " s u r f a c e " muon mode.  In  1  c o n v e n t i o n a l mode, p o s i t i v e p i o n s produced from T2 a t 55° t o the p r o t o n beam are c o l l e c t e d i n t o M20 by the quadrupole doublet Q1-Q2 and then momentum s e l e c t e d  ( £E  - 20%,  p <_ 170 MeV/c) by the f i r s t bending magnet, B l ("Patty-Jane"). Of course, B l a l s o charge s e l e c t s p a r t i c l e s s i n c e n e u t r a l s (Y,fT°,n) pass s t r a i g h t through B l and n e g a t i v e 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  decay i n f l i g h t between B l and B2 ("Cal-Tech") Q3-Q7 (the " s t r a i g h t s e c t i o n " ) . TT  +  Thus y  (T  +  +  = 26ns)  i n quadrupoles  Seen i n the p i o n r e s 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  momentum of 29.8 MeV/c.  +  frame,  formed have a  t h a t are formed i n the  momentum d i r e c t i o n of the p i o n beam ("forward" muons) have <_2 9.8MeV/c more momentum than t h a t s e l e c t e d by B l and y  +  that  are formed o p p o s i t e t o the momentum d i r e c t i o n of the p i o n beam ("backward" muons) have >29.8MeV/c l e s s momentum than t h a t s e l e c t e d by B l .  Nominal  forward and backward y  g i v e n as a f u n c t i o n of decaying T T  +  +  momenta are  momentum i n Table I I I .  FIGURE 6: The M2 0 beamline  (detail).  -28TABLE III: NOMINAL FORWARD AND BACKWARD y AS A FUNCTION OF DECAYING T T  TT  /  MOMENTUM  3 +  p^+(MeV/c)  3y+  Backward Muons p^+(MeV/c)  TT  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  20  0 .14  45 . 9  0 .40  -14 .4  -0 .13  10  0 . 07  37 .8  0 .34  -22 .1  -0 .20  0 .00  * 29 .8  0 .27  -29 .8  -0 .27  0 T  MOMENTA AND VELOCITIES  Forward Muons  Pions p +(MeV/c)  +  +  -0 . 07  p and 3 are g i v e n w i t h r e s p e c t t o the TT beam d i r e c t i o n w i t h  * " s u r f a c e " o r "Arizona" muons  N o t i c e t h a t the momentum s e p a r a t i o n between y e x c e l l e n t f o r backward y  +  The  T  and TT" " i s 1  but r a t h e r poor f o r forward y . +  second bending magnet, B2, i s tuned t o momentum  select  e i t h e r forward or backward muons from the other p a r t i c l e s +  +  +  +  i n the beam (e , TT , y , p ) . polarized y  +  The r e s u l t a n t h i g h momentum  beam i s d e l i v e r e d t o the y S R apparatus +  end of the beamline  through the l a s t quadrupole  a t the  doublet,  Q8-Q9. In produced  " c l o u d " muon mode, high momentum  (<170 MeV/c) y  +  from the "cloud" of pions decaying i n f l i g h t between  T2 and B l are c o l l e c t e d and t r a n s p o r t e d through M2 0 w i t h B l and B2 both s e t a t the same momentum-selecting f i e l d s .  The  e s s e n t i a l d i f f e r e n c e between c o n v e n t i o n a l and c l o u d muon modes, then, i s t h a t i n the former case, muons are produced  from pions  decaying i n f l i g h t a f t e r the f i r s t bending magnet, w h i l e 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. higher f l u x e s of y  +  In g e n e r a l , c l o u d muon mode  produces  by a f a c t o r of as much as 4 but w i t h a  lower p o l a r i z a t i o n than c o n v e n t i o n a l forward muon mode (50-60% p o l a r i z a t i o n compared t o 70-80%) and w i t h much worse contamination with protons, pions and p o s i t r o n s .  When  M20 i s tuned f o r backward c o n v e n t i o n a l muons, the f l u x i s lower by about a f a c t o r of 10 compared t o forward muons but beam contamination i s lower by s e v e r a l orders of magnitude (see  Table I I I ) . In p r a c t i c e , w i t h proton c u r r e n t s of <10yA, beam q u a l i t y  i s o f t e n s a c r i f i c e d f o r f l u x by o p e r a t i n g M20 i n c l o u d 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  -30-  yA of p r o t o n s .  The beam d e l i v e r e d i n t h i s mode i s contaminated  with p o s i t r o n s and pions i n a 100:3:1 r a t i o t o muons; there i s a l s o some contamination from forward s c a t t e r e d protons y SR experiments +  utilizing  n e c e s s i t y t o work around  from T2.  c l o u d muons are complicated by the  t h i s contamination.  p a r t i c l e s d e l i v e r e d by the beamline  Since a l l  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 e l i m i n a t e d by d i f f e r e n t i a l a b s o r p t i o n i n degrader upstream o f the y SR t a r g e t . +  placed  P o s i t r o n s are separated from muons  l o g i c a l l y , r a t h e r than p h y s i c a l l y , by p l a c i n g a veto  counter  downstream o f the muon t a r g e t ; the p o s i t r o n s are s u f f i c i e n t l y e n e r g e t i c t o pass through the t a r g e t i n which most of the muons stop.  None o f these methods are completely  successful  i n removing beam contamination r e s u l t i n g i n the appearance o f v a r i o u s background s i g n a l s i n the y SR time spectrum. +  In  a d d i t i o n t o these drawbacks, c l o u d or c o n v e n t i o n a l muons are u n s a t i s f a c t o r y f o r gas phase t a r g e t s s i n c e t h e i r h i g h momentum and concomitant target  long range r e q u i r e a high p r e s s u r e s t o p p i n g  [Mobley 2 - ( 6 7 ) ] . Surface o r A r i z o n a muon mode  [Pifer  (76)] i s s i m i l a r t o  c l o u d muon mode inasmuch as muons are c o l l e c t e d by M20 d i r e c t l y from T2.  The d i f f e r e n c e i s t h a t s u r f a c e muons come from  pions  t h a t decay a t r e s t on the s u r f a c e o f the p i o n p r o d u c t i o n t a r g e t (see Table III).whereas  c l o u d muons come from pions decaying i n  f l i g h t between T2 and B l .  Since c l o u d muons i n c l u d e 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 s u r f a c e muons, however, a r i s e from T T decaying +  i n the forward d i r e c t i o n , g i v i n g them a very h i g h p o l a r i z a t i o n  (>95%) .  Surface muons are n e a r l y monoenergetic  (4.lMeV) w i t h  a nominal momentum of 2 9.8MeV/c corresponding t o a range o f -2 only 148 mg cm atmosphere.  o f CH  2  o r about 130 cm i n argon gas a t one  Contamination of the s u r f a c e muon beam w i t h  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 t o a beamline  transit  time  of 160 ns or 6 p i o n l i f e t i m e s ; thus o n l y about 0.3% of the small number of pions i n i t i a l l y produced to reach the end o f the beamline.  Protons a t 3 0 MeV/c have  i n s u f f i c i e n t range t o p e n e t r a t e the t h i n beamline vacuum window.  a t 3 0 MeV/c s u r v i v e  (0.05 mm  Mylar)  However, as i n other modes of  o p e r a t i o n , t h e r e are about 100 times more p o s i t r o n s than muons i n the s u r f a c e muon beam.  F o r t u n a t e l y , the p o s i t r o n s  can be l o g i c a l l y d i s t i n g u i s h e d from u  +  by p u l s e h e i g h t  d i s c r i m i n a t i o n , as d e s c r i b e d i n S e c t i o n C below [ M a r s h a l l (76)] which e l i m i n a t e s the c o m p l i c a t i o n of a p o s i t r o n veto counter requirement.  Unfortunately, this tidy,  logical  removal of the contamination r e q u i r e s p h y s i c a l placement o f the counters used t o monitor p o s i t r o n s from muon decay a t 9 0° t o the beam t o prevent t h e i r s a t u r a t i o n by beam p o s i t r o n s ; f o r 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 f u t u r e , some of the M20 p o s i t r o n contamination w i l l be removed by adding a very t h i n degrader i n the s t r a i g h t T h i s w i l l reduce the y  +  section.  momentum much more than the p o s i t r o n  momentum, so t h a t by t u n i n g Cal-Tech t o the lower y  +  momentum, the p o s i t r o n contamination w i l l be c o n s i d e r a b l y reduced.  The disadvantage of t h i s simple s e p a r a t i o n technique  i s t h a t i t s a c r i f i c e s both u  +  f l u x and range, the l a t t e r being  at a premium f o r s u r f a c e muons.  However, i t should be p o s s i b l e  to p a r t i a l l y compensate f o r t h i s l o s s by r e d e s i g n i n g the s u r f a c e y SR t a r g e t s to i n c o r p o r a t e fewer and t h i n n e r windows.  At  +  p r e s e n t , M20  d e l i v e r s about 6xl0 ' s u r f a c e \i s 3  10 cm x 10 cm area per uA of 500 MeV  over a  +  protons i n c i d e n t on  the 10 cm Be t a r g e t o f T2.  B_ .  The Gas T a r g e t , Counters and Magnetic F i e l d The gas t a r g e t and counter c o n f i g u r a t i o n are i l l u s t r a t e d  i n F i g u r e 7. The gas t a r g e t v e s s e l i s an aluminum c y l i n d e r 75 cm i n l e n g t h w i t h a 25 cm i n n e r diameter.  One end of the  gas can i s f i t t e d w i t h a t h i n  Mylar) window  (0.13 - 0.25 mm  20 cm i n diameter which i s capable of s u p p o r t i n g a vacuum o f 5 x 10 ^ t o r r or an a b s o l u t e p r e s s u r e of £2.5 The volume of the t a r g e t v e s s e l i s 36,65 ± 0.34  atmospheres. 1 a t an  absolute p r e s s u r e of 800 t o r r , accounting f o r the volume displacement caused by p r e s s u r e d i s t o r t i o n of the Mylar window. The other end of the aluminum c y l i n d e r i s c l o s e d w i t h a f l a n g e housing p r e s s u r e gauges and a s t a i n l e s s s t e e l vacuum r a c k .  The  gas can i s wrapped i n h e a t i n g tape and i n s u l a t i o n p r o v i d i n g the t a r g e t w i t h an o p e r a t i o n a l temperature range from ^300  to^400K.  The temperature i s monitored w i t h a copper-constantan thermocouple which i s p l a c e d t o probe the muon s t o p p i n g r e g i o n ; the temperature variation across the diameter of the v e s s e l i s l e s s than 2K.  C o o l i n g c o i l s are mounted a t both ends of the  v e s s e l to a c c e l e r a t e s t a b i l i z a t i o n of the t a r g e t temperature. T h i s rudimentary t a r g e t system w i l l be r e d e s i g n e d i n the near f u t u r e t o allow a much wider working temperature range and  -33-  •TO GAS RACK  FIGURE 7: The gas phase MSR t a r g e t apparatus (top view). The counter t e l e s c o p e s are designated l e f t and r i g h t according t o the convention: muon's eye view.  -34somewhat l a r g e r pressure The  range.  gas t a r g e t v e s s e l i s mounted on a p o r t a b l e  between dual Helmholtz c o i l s d r i v e n by a c u r r e n t Hewlett-Packard H a r r i s o n 6268A power supply. p r o v i d e a v a r i a b l e magnetic f i e l d  cart  regulated  These c o i l s  from ^ 1 t o 75. gauss which 3  i s homogeneous t o b e t t e r than 0.1% over a volume of 400 cm . A single thin scintillator  (40 mg cm  [Marshall  (76)] serves as a beam d e f i n i n g and  y - stop t i m i n g counter. at  ) 10 cm x 10 cm NE102 p l a s t i c  Two p o s i t r o n t e l e s c o p e s are p l a c e d  ±90° t o both the beam d i r e c t i o n and the t r a n s v e r s e magnetic  field.  Each t e l e s c o p e c o n s i s t s of one 20 cm x 20 cm x 0.6 cm  ( c l o s e s t t o the t a r g e t ) 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 F i g u r e 7, which operate  with  normally  1" of g r a p h i t e degrader between the f i r s t and  second counters.  T h i s degrader serves t o reduce s c a t t e r e d  beam p o s i t r o n background and t o absorb low energy p o s i t r o n s from muon decay, thereby asymmetry  (see F i g u r e 1 ) .  t e l e s c o p e s are designated eye  enhancing the e m p i r i c a l muon The " l e f t "  and " r i g h t " p o s i t r o n  by the mnemonic convention:  muon s 1  view. The  gas phase t a r g e t s c o n s i s t of c h e m i c a l l y  moderator gas c o n t a i n i n g s m a l l c o n c e n t r a t i o n s of i n t e r e s t .  inert  of the reagent  The moderator gas serves not only t o t h e r m a l i z e  the incoming muons, but a l s o t o p r o v i d e the i o n i z a t i o n processes  e t c . f o r the formation  o f 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 t o  be about 1.5 times more e f f i c i e n t than Ar a t 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, A  q  8.  (see Appendix I I ) , as i l l u s t r a t e d i n F i g u r e  Furthermore, N  has a lower muon stopping d e n s i t y than  2  Ar, thereby p r o v i d i n g a longer muon range t h a t a f f o r d s g r e a t e r flexibility  i n the d e s i g n of windows and counters f o r  o p t i m i z i n g the l o c a t i o n of the muon stopping r e g i o n w i t h r e s p e c t t o the p o s i t r o n c o u n t e r s .  The experimental  operating  p r e s s u r e i s u s u a l l y chosen t o be about 780 t o r r i n order t o reduce p o s s i b l e 0  2  leakage  i n t o the r e a c t i o n v e s s e l .  In N  2  at t h i s p r e s s u r e , the r e s i d u a l range of s u r f a c e muons, which have been degraded by p a s s i n g through  the t h i n counter  windows, i s about 30 ± 5 cm a t 300K,  At higher  t h i s o p e r a t i n g p r e s s u r e i s maintained;  and two  temperatures,  the subsequent  lower  d e n s i t y of the gas t a r g e t i s compensated f o r by u s i n g a t h i c k e r window ( r e q u i r e d f o r h i g h temperatures) and a d d i t i o n a l sheets of Mylar  degrader,  A muon range curve i s  taken a t each temperature t o ensure o p t i m a l l o c a t i o n of the muon stopping r e g i o n . Mu w i t h a reagent  O c c a s i o n a l l y , , the r e a c t i o n r a t e of  i s s u f f i c i e n t l y slow  l a r g e c o n c e n t r a t i o n s of t h a t reagent  (eg. HCl) t h a t such  are r e q u i r e d t h a t i t  must then a l s o serve as the moderator.  In r e c e n t  experiments,  the high p u r i t y moderator gases are f u r t h e r p u r i f i e d by p a s s i n g them through  a c t i v a t e d c h a r c o a l o r a Dow Chemical  c a r r i e r p u r i f i e r , reducing 0 1 ppm.  2  contamination  T h i s r e s u l t s i n a reduced  to less  G.C.  than  background Mu r e l a x a t i o n  r a t e , A., although the e f f e c t i s not dramatic. q  Measured c o n c e n t r a t i o n s of r e a c t a n t gas are added t o  -36-  0.15 MU IN 780 TORR N2 AT 6 . 9 GAUSS  0.10 0.05  i ;i:t "  0.00 -0.05 -0.10 -0 . 1 5 0.15  MU IN 670 TORR RR AT 6 . 9 GAUSS  0,10 0.05  ii I  0.00 -0.05 -0.10 -0.15  J 0.0  0.5  1.0  1.5  2.0  TIME IN uSEC FIGURE  8:  _L  2.5  _L  3.0  3.5  (20 NSEC/BIN)  Nitrogen versus argon as moderator gases. Both s p e c t r a were t a k e n under 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 gas p r e s s u r e s were chosen 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 muon 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 muonium s i g n a l amplitude i s ll._£±0.3 % with a relaxation rate of 0.34±0.02 ys . In argon (bottom), the muonium 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 rate of 0.33±0.03 ys  4.0  the r e a c t i o n v e s s e l by f i l l i n g to  a small bulb of known volume  a measured p r e s s u r e , then f l u s h i n g i t i n t o the  t a r g e t can with moderator.  In t h i s way,  evacuated  c o n c e n t r a t i o n s of 15  r e a c t a n t may  be c o n v e n i e n t l y v a r i e d from 10 -3  molecules at  cm  -6 (10  19 - 10  _2 - 10  M).  Reagents which are condensed  S.T.P., such as bromine, are i n t r o d u c e d by f i l l i n g  the  bulb with the e q u i l i b r i u m vapour p r e s s u r e a t a known temperature,  as d e s c r i b e d i n Fleming  (76).  the gas v e s s e l i s " c o n d i t i o n e d " w i t h 300  temperature,  t o r r of r e a c t a n t f o r  about 1 hour before any experiments are run. t h a t s u f f i c i e n t q u a n t i t i e s of reagents  At each  This  ensures  l i k e F^ have enough  time to form i n e r t compounds on the s u r f a c e s of any of  components  the t a r g e t v e s s e l t h a t are c h e m i c a l l y r e a c t i v e w i t h t h a t  reagent.  I n t e r e s t i n g l y , the Mylar window has proven to be  i n e r t even to 300  t o r r of F^ a t 400K,  In order to v e r i f y  the i n e r t n e s s of the t a r g e t v e s s e l s u r f a c e s t o very r e a c t i v e chemicals  l i k e F^, the reagent c o n c e n t r a t i o n s are v a r i e d  randomly from one Mu with metal  r a t e measurement to .the next.  Since r e a c t i o n s  s u r f a c e s tend to f o l l o w -1 order k i n e t i c s  [Frost  (61)], i t should be p o s s i b l e to i d e n t i f y any ongoing i n t e r f e r e n c e r e a c t i o n s from the s y s t e m a t i c s of the Mu  r a t e measurements.  T h i s 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 c o n c e n t r a t i o n of reagent remains constant d u r i n g the experimental C_  runs which t y p i c a l l y take 1 to 2 hours each.  . Data A c q u i s i t i o n Before d e s c r i b i n g the data acquisition system, i t i s  u s e f u l , perhaps, to r e i t e r a t e the e s s e n t i a l f e a t u r e s of an  -38MSR experiment.  Upon l e a v i n g the beamline, a muon passes  through a counter  (designated  'D' i n F i g u r e 7) which  generates -9  a s t a r t p u l s e f o r a high p r e c i s i o n c l o c k .  Within  >10  seconds of r e a c h i n g the stopping r e g i o n o f the t a r g e t , the muon t h e r m a l i z e s as Mu and precesses field.  A t some l a t e r time  i n a weak t r a n s v e r s e magnetic  (up t o s e v e r a l microseconds) the  muon decays, e m i t t i n g a p o s i t r o n p r e f e r e n t i a l l y along i t s s p i n d i r e c t i o n a t the moment of decay.  I f the s p i n v e c t o r of  the muon happens t o p o i n t toward the p o s i t r o n t e l e s c o p e when i t decays, there i s a high p r o b a b i l i t y t h a t the decay p o s i t r o n w i l l be d e t e c t e d , g e n e r a t i n g a stop p u l s e f o r the c l o c k . r e s u l t i n g 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 the c l o c k i s r e s e t and the e n t i r e process times.  Should  The  i n a histogram, 6 7  i s repeated  10  - 10  no decay p o s i t r o n be d e t e c t e d d u r i n g some  a d j u s t a b l e "time-out"  p e r i o d o f s e v e r a l 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 system i s shown i n F i g u r e 9 (taken from Pulses from the counters (corresponding v a r i a b l e delays  t o those (denoted  acquisition  [Marshall (76)]).  a t the top of the diagram i n F i g u r e 7) are time a d j u s t e d by ^ i n the diagram) before being i n p u t t o  d i s c r i m i n a t o r s where " r e a l " s i g n a l s are d i s t i n g u i s h e d from n o i s e . The  thin  'D' counter used f o r d e t e c t i n g s u r f a c e muons a l s o  serves t o d i s c r i m i n a t e muons from p o s i t r o n s : a t 30 MeV/c, positrons travel essentially at c  and are minimum i o n i z i n g ,  d e p o s i t i n g very l i t t l e energy i n the t h i n counter,  i n contrast  to the slower muons t r a v e l l i n g a t < 0.3c which are many more i o n i z i n g .  times  By a d j u s t i n g the v o l t a g e on the D counter  vDISC  .DISC  tLl'L2'L3= e(left) COIN.  y-stop  [Rl-R2'R3 e (right)j COIN. :  COIN.  FAN IN  L route  start stop  reset -P 03  w  c  R route  MBD-11  FIGURE 9:  CAMAC branch highway  MSR data acquisition l o g i c  (simplified)  P  TAC (time-toamplitudel convertgrft  start  TDC 100  PDP-11/40  s t o  I U) I  1  PHA (pulse height analyzer)  UBC computer  -40p h o t o m u l t i p l i e r , p o s i t r o n s i g n a l s can be made t o form a band with a p u l s e h e i g h t of ^50 mV, while muon s i g n a l s form a band with p u l s e h e i g h t s of 300 - 400 mV.  Adjustment of the D  d i s c r i m i n a t o r t h r e s h o l d t o g r e a t e r than 50 mV e f f e c t i v e l y makes the muon " t r i g g e r " t r a n s p a r e n t t o p o s i t r o n s w h i l e r e t a i n i n g a high e f f i c i e n c y  (>95%) f o r muons.  The muon p u l s e s are i n p u t t o  the s t a r t o f an E.G. & G. Model TDC-100 t i m e - t o - d i g i t a l c o n v e r t e r , which has a nominal time r e s o l u t i o n of 0.125 ns and an a d j u s t a b l e range from 4 us t o 34 ms. a c t i v a t e s a f a s t "time-out"  The TDC-100 a l s o  r e s e t i f no stop p u l s e i s accepted  d u r i n g the p r e - s e l e c t a b l e time range. D i s c r i m i n a t e d p u l s e s from the l e f t and r i g h t p o s i t r o n t e l e s c o p e s are i n p u t i n t o separate c o i n c i d e n c e u n i t s "and's") which i d e n t i f y p o s i t r o n s by the Boolean e x p r e s s i o n s : e =Ll*L2«L3 o r e =R1•R2•R3. T  c o i n c i d e n c e requirement to  logical  The t h r e e f o l d  ensures t h a t accepted  events  p o s i t r o n s t h a t pass through a l l three counters  carbon degrader.  (logical  correspond  and the  T h i s d e f i n e s the acceptance s o l i d angle and  e l i m i n a t e s low energy p o s i t r o n s from the muon decay, enhancing the e m p i r i c a l u  +  thereby  asymmetry; more i m p o r t a n t l y , the  degrader absorbs s c a t t e r e d p o s i t r o n s from the beam, thereby reducing background which has the time s t r u c t u r e of the TRIUMF cyclotron "or-ed"  (23.3 MHz), Accepted  p o s i t r o n events  are l o g i c a l l y  from the l e f t and r i g h t t e l e s c o p e s w i t h a " f a n - i n "  u n i t and i n p u t t o the stop of the TDC.  Simultaneous w i t h  stopping the c l o c k , the l e f t or r i g h t p o s i t r o n p u l s e s s e t 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  interface.  -41Upon completion of the d i g i t i z a t i o n , the TDC w r i t e s the measured time i n t e r v a l i n t o a CAMAC i n p u t r e g i s t e r generates a "look-at-me"  which  (LAM) s i g n a l t o a c t i v a t e the Bi-Ra  Microprogrammed Branch D r i v e r  (MBD-11, Model 2) which  s e r v i c e s the d a t a s t o r e d i n CAMAC.  The MBD i s a f a s t  micro computer which i s i n t e r f a c e d v i a UNIBUS t o the main data a q u i s i t i o n computer  (a D i g i t a l Equipment C o r p o r a t i o n  PDP-11/40) and c o n t r o l s the CAMAC c r a t e ( s ) v i a a Branch Highway.  Although under the u l t i m a t e c o n t r o l o f the main  computer, the MBD's o p e r a t i o n i s f u n c t i o n a l l y  independent  of and simultaneous with t h a t 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 a c q u i s i t i o n  tasks,  l i b e r a t i n g i t f o r more s o p h i s t i c a t e d o n - l i n e data a n a l y s i s . The MBD reads the CAMAC data and r e s e t s the e l e c t r o n i c s i n p r e p a r a t i o n f o r acceptance of a new event.  The MBD  identifies  the p o s i t r o n t e l e s c o p e t h a t generated the event, and performs the necessary s h i f t i n g , s u b t r a c t i n g and b a s e - a d d i t i o n f u n c t i o n s r e q u i r e d to increment the address i n the PDP memory r e p r e s e n t i n g the histogram b i n corresponding t o the measured time  interval.  Thus separate l e f t and r i g h t histograms are c o l l e c t e d s i m u l t a n e o u s l y , each normally c o n s i s t i n g of 2000 b i n s of 2 ns each, g i v i n g a t o t a l range o f 4 y s .  The system i s  capable of s u p p o r t i n g almost any number of histograms of any s i z e w i t h a maximum time r e s o l u t i o n of 0.125 ns.  However, a t  present the t i m e - r e s o l u t i o n o f the counters i s about 1.5 ns. The data a c q u i s i t i o n hardware and software i s i n t e r f a c e d to the experimenter through the PDP-11 computer, e x e c u t i n g a s o p h i s t i c a t e d programme w r i t t e n p r i m a r i l y by R.S. Hayano of  -42the U n i v e r s i t y of Tokyo [Hayano 1-(76), Hayano. 2-(76) ] w i t h help from J.H.  Brewer of U.B.C.  T h i s data  programme w i l l support s e v e r a l independent  aquisition experiments  running  simultaneously and i s completely f l e x i b l e with r e s p e c t t o the number, s i z e and time r e s o l u t i o n of histograms each experiment. built-in,  Many experiment-monitoring  required for  f e a t u r e s are  i n c l u d i n g p r o v i s i o n to d i s p l a y a l l or p a r t of  histogram on a g r a p h i c s t e r m i n a l under l i g h t pen  any  control.  The programme p r o v i d e s a h i g h l e v e l of data p r o t e c t i o n by r e g u l a r l y updating histogrammed data on permanent d i s k a powerful  "crash r e c o v e r y " f a c i l i t y minimizes  computer problems.  files;  data l o s s due  Many l e v e l s of redundancy ensure  to  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 n o n - p a t h o l o g i c a l hardware f a i l u r e - f o r example, breakdown of a d i s k d r i v e , the g r a p h i c s t e r m i n a l , or other c o n t r o l t e r m i n a l w i l l c r i p p l e the computer's data t a k i n g f u n c t i o n s .  Several on-line  a n a l y s i s r o u t i n e s such as f a s t f o u r i e r transforms a v a i l a b l e f o r m o n i t o r i n g an experiment.  not  (FFT)  At p r e s e n t , the PDP-11  does not support data a n a l y s i s programmes of s u f f i c i e n t i t y t o perform  Until  data i s w r i t t e n on a 9-track magnetic tape and analyzed  System  (MTS) The  capabil-  " f i n a l " data a n a l y s i s , although implementation  such programmes w i l l be made i n the near f u t u r e .  l i n e on the UBC  are  computer c e n t e r IBM  37 0/16 8 Michigan  of  then, off-  Terminal  as d e s c r i b e d i n the next S e c t i o n .  f o r e g o i n g d e s c r i p t i o n of the MSR  system i s a s i m p l i f i e d overview;  data, a c q u i s i t i o n  the s e r i o u s problem of  "muon p i l e - u p " has been ignored and only s u p e r f i c i a l  treatment  has been g i v e n to the i n t e r t w i n e d data p r o c e s s i n g r e l a t i o n s h i p s  between the e l e c 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 o f muon p i l e - u p and hardware-independent s o l u t i o n s are i d e n t i f i e d below.  A numerical assessment of these problems as  a f u n c t i o n o f muon beam c u r r e n t also provides  i s l e f t t o Appendix I I I , which  a d e t a i l e d d e s c r i p t i o n of the MSR data  acquisition system. In the f o l l o w i n g d i s c u s s i o n , i t i s convenient t o d e f i n e a f i x e d muon decay gate or maximum muon l i f e expectancy time range, T, which i n p r a c t i c e i s s e t t o a few muon l i f e t i m e s . A muon e n t e r i n g the t a r g e t a t the onset of t h i s time i n t e r v a l i s assumed  t o have decayed by the e x p i r a t i o n o f T, correspond-  i n g t o the TDC "time-out" p e r i o d mentioned above. F o r instance,  i f an experimenter sets T = 4^-> , then the u  assumption t h a t the muon has decayed d u r i n g  T i s good t o  b e t t e r than 2%. Following  the e n t r y of a muon, u^, t h a t opens the  T-gate and s t a r t s the TDC, a second " p i l e - u p " muon, y .;__,_-]_> enter the t a r g e t before  Y  the e x p i r a t i o n of T and before any  decay p o s i t r o n i s d e t e c t e d . occurs,  m a  When such an event sequence  an ambiguity i s c r e a t e d  s i n c e there i s no way t o  i d e n t i f y which muon i s a s s o c i a t e d w i t h any subsequently detected  decay p o s i t r o n .  Since  t h a t any decay p o s i t r o n detected y^ ^ +  and  there  i s a high p r o b a b i l i t y  during  T w i l l belong t o  r a t h e r than t o JJN , the time c o r r e l a t i o n between a u i t s decay e i s l o s t i f t h i s p o s i t r o n i s allowed t o stop  the c l o c k .  Acceptance o f these events a t s u f f i c i e n t l y  muon beam c u r r e n t s w i l l r e s u l t i n a time histogram a reduced MSR s i g n a l and a- d i s t o r t e d background.  high  containing To zeroth  -44o r d e r , i t i s necessary t o l o g i c a l l y r e j e c t t h i s event sequence  (called  " e a r l y second y" events) r e p r e s e n t e d  s c h e m a t i c a l l y by:  where time moves from l e f t t o r i g h t and —//—  i n d i c a t e s some  a r b i t r a r y time. R e j e c t i o n o f e a r l y second y events i s n o t , however, a complete  s o l u t i o n t o the problem of m u l t i p l e muons.  a l i n e a r d i s t o r t i o n . •. o f the time spectrum slope) i s generated a t s u f f i c i e n t l y  In f a c t ,  (with a n e g a t i v e  l a r g e y-stop r a t e s when  only e a r l y second y are r e j e c t e d , as i l l u s t r a t e d i n F i g u r e 10. T h i s comes from the f a c t t h a t , g i v e n a c o n s t a n t beam c u r r e n t with muons a r r i v i n g a t times g i v e n by a P o i s s o n 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 t h a t an event w i l l be r e j e c t e d due t o an e a r l y second y i f y^ decays a t l a t e times than i f i t decays a t e a r l y times.  There i s simply  a g r e a t e r o p p o r t u n i t y f o r an e a r l y second muon t o e n t e r the t a r g e t i f the f i r s t muon s u r v i v e s a long time b e f o r e decaying. I t should be noted t h a t the presence of  i n the t a r g e t  merely c r e a t e s an ambiguity i n the a s s o c i a t i o n of any detected e with i t s decaying y; sometimes the d e t e c t e d e does correspond t o y^.  That i s , some e a r l y second y events are  "good" events i n the sense t h a t the decay e t h a t stopped the  -45-  0.15 MU IN 780 TORR N2 RT 6 . 9 GRUSS  0.10  >—  0.05  Ql  0.00 >—  CO 0 1  -0.05 -0.10 I  -0.15 0.25  i  _L  i  _L  i  i  i  1  r  MU IN 1140 TORR N 2 . 7 . 8 G. ERRLY -2ND u 0.15  £  0.05  £ -0.05 ex -0.15 -0.25  h _L  0.0  0.5  1.0  1.5  2.0  TIME IN uSEC  2.5  3.0  3.5  4.0  (20 NSEC/BIN)  FIGURE 10: The s p e c t r a l d i s t o r t i o n due t o " e a r l y " second y r e j e c t i o n : the top spectrum has both " e a r l y " and " l a t e " ( i . e . "post-y.") second y r e j e c t i o n , w h i l e the bottom spectrum has " e a r l y " second y r e j e c t i o n o n l y , g i v i n g r i s e t o a l a r g e background d i s t o r t i o n w i t h a n e g a t i v e s l o p e . I t should be noted t h a t the asymmetry s c a l e s a r e d i f f e r e n t f o r the two s p e c t r a .  -46c l o c k corresponds t o u^, the muon t h a t s t a r t e d the c l o c k , even though they are not i d e n t i f i a b l e as such.  Another way o f  l o o k i n g a t the a r t i f i c i a l d i s t o r t i o n , then, i s t h a t  because  there i s much time a v a i l a b l e f o r an e a r l y second u t o e n t e r the  t a r g e t and cause r e j e c t i o n of an event i f T_K decays a t  l a t e 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 t o the t o t a l number of  good events) i s s m a l l a t l a t e times; c o n v e r s e l y , i f  decays a t e a r l y times, e a r l y second u have  little  o p p o r t u n i t y t o lower the e f f i c i e n c y o f event acceptance.  The  r e s u l t i s t h a t the n o r m a l i z a t i o n o f e q u a t i o n (6) (Chapter I) decreases with time. independence  If fitting  procedures assume time  of the n o r m a l i z a t i o n , i t s a r t i f i c i a l  dependence expresses i t s e l f  time  i n an e r r o n e o u s l y s m a l l apparent  muon l i f e t i m e and an a r t i f i c i a l l y l a r g e apparent muonium  relaxation  rate. It of  i s , t h e r e f o r e , e s s e n t i a l t h a t a constant f r a c t i o n  events per histogram time increment be r e j e c t e d i n order to  a v o i d g e n e r a t i o n o f the a r t i f i c i a l backgrounds d e s c r i b e d above. T h i s i s accomplished by not o n l y r e j e c t i n g e a r l y second u events, but a l s o r e j e c t i n g what are c a l l e d t h a t i s , events i n which  " l a t e second u" events;  enters the t a r g e t b e f o r e the  e x p i r a t i o n of T, but a f t e r a decay p o s i t r o n i s d e t e c t e d :  T  >|  -47-  Thus, an accepted event i s one i n which no second y a r r i v e s d u r i n g T:  -  <  >  T  Higher order c o r r e c t i o n s f o r m u l t i p l e p a r t i c l e events can be made, but the r e j e c t i o n of e a r l y and l a t e second y ely called  (collectiv-  "post-y^ second y") i s the most important, both i n  terms o f a b s o l u t e numbers  (which are muon r a t e dependent) but  a l s o i n terms of the s p e c t r a l d i s t o r t i o n s i n t r o d u c e d by f a i l ure,  t o r e j e c t these events One  (see Appendix I I I ) .  h i g h e r order c o r r e c t i o n comes from c o n s i d e r a t i o n of  the time i n t e r v a l p r e c e d i n g the e n t r y o f y^ i n t o the t a r g e t . In the f o r e g o i n g d i s c u s s i o n , i t was assumed t h a t , upon e n t r y , y^ i s the o n l y muon r e s i d e n t i n the t a r g e t ; but t h i s may not be the case.  Even with p o s t - y ^ second y r e j e c t i o n , t h e r e  are two s i t u a t i o n s i n which a muon may a l r e a d y be r e s i d e n t i n the t a r g e t when y^ e n t e r s . (1) I f y j _ _ T-gate opening muon and U^_-j.  w  a  s  a  n  w  a  s  t  n  e  2  e a r l y or l a t e  previous second  y, then a f t e r the T-gate has c l o s e d i t may be assumed t h a t y^_  2  has decayed, but V-^_-^ may s t i l l be p r e s e n t when y^ opens  the next T-gate.  (2) Whenever t h e r e i s an accepted event,  there i s an i n t r i n s i c e l e c t r o n i c s  "deadtime"  d u r i n g which  t i m e - d i g i t i z a t i o n occurs and the event i s t r a n s f e r r e d t o the histogram.  During t h i s deadtime,  the experiment i s  effectively  "turned o f f " and a l l event m o n i t o r i n g i s suppressed  (reasons f o r t h i s are d e t a i l e d i n Appendix I I I ; t h i s i s a g e n e r a l f e a t u r e of the l o g i c whenever more than one e l e c t r o n i c s module, such as the TDC and p a t t e r n r e c o g n i t i o n u n i t , must be read and r e s e t by the computer.  To preserve the i n t e g r i t y  of the next event, i t i s e s s e n t i a l t h a t a l l such modules be a v a i l a b l e f o r new data a t e x a c t l y the same t i m e ) . Consequently,  when the experiment  i s "turned on" again a t  some a r b i t r a r y l a t e r time, the l o g i c i s unaware o f the presence  of any muons i n the t a r g e t .  To c o r r e c t l y d e a l w i t h  muon p i l e - u p , t h e o n l y a c c e p t a b l e events are those where no muon e n t e r s the t a r g e t d u r i n g a time T e i t h e r before or a f t e r y^ entered the t a r g e t : -^K  ^1  T  However, i t i s shown i n Appendix I I I t h a t "pre-y  multiple  muon events are s e v e r a l orders of magnitude l e s s frequent than "post-y^" m u l t i p l e muon events.  Furthermore,  while such muons  do lower the apparent Mu asymmetry, they d i s t o r t the histogram l e s s - s i g n i f i c a n t l y than po'st'-y-- m u l t i p l e muon events. Another h i g h e r order c o r r e c t i o n can be made f o r events w i t h more than one p o s i t r o n d e t e c t e d d u r i n g T a f t e r y^ e n t e r s the t a r g e t , c r e a t i n g an obvious ambiguity. of " e x t r a " e i n c l u d e a c c i d e n t a l counts beam contamination  P o s s i b l e sources  (possibly r e l a t e d to  and t h e r e f o r e beam c u r r e n t dependent) or e  -49from muons t h a t happen to s u r v i v e T  ( i n the example above  where T = 4 T , 2% of the muons s u r v i v e T ) . telescopes,-  are p r o p e r l y  I f the  s h i e l d e d a g a i n s t a c c i d e n t a l s from  the beam, m u l t i p l e - e events are extremely r a r e and ignored  f o r a l l but  the most p r e c i s e work.  s i g n i f i c a n t d i s t o r t i o n introduced multiple-e  is likely  positron  The  can most  i n the histogram  to be the m i c r o s c o p i c  be  by  time s t r u c t u r e  of the c y c l o t r o n beam. The  experiments d e s c r i b e d  i n t h i s t h e s i s employed  "post-y^" m u l t i p l e muon r e j e c t i o n and m u l t i p l e - e r e j e c t i o n only.  "Pre-y^" m u l t i p l e muon r e j e c t i o n w i l l  incorporated  be  i n t o the data a c q u i s i t i o n l o g i c at higher  beam  currents. A f i n a l high event r a t e c o n s i d e r a t i o n of relevance  to  very high p r e c i s i o n work i s a s s o c i a t e d w i t h counter response characteristics  [Hague (70)].  Counter photomultipliers  have  a minimum recovery  time of about 20 ns. S i g n a l s produced  a photomultiplier  which i s not  i n amplitude and may may  f u l l y recovered  are  be r e j e c t e d by d i s c r i m i n a t o r s .  be prevented by a d d i t i o n a l g a t i n g to ensure t h a t  events of i n t e r e s t were counted by photomultipliers.  fully  This consideration  improving the beam q u a l i t y of M20  by  reduced This the  recovered  i s an argument f o r  to ensure t h a t the  p o s i t r o n contamination does not d i s t o r t y  +  severe  s i g n a l s from  the  D counter.  D_,  Data A n a l y s i s F i g u r e s o f t e n b e g u i l e me, p a r t i c u l a r i l y when I have the arranging of them myself. The remark a t t r i b u t e d  -50to D i s r a e l i would a p p l y - "There a r e t h r e e k i n d s 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. 2 4 6 ) . Most d a t a 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 u s i n g m u l t i p a r a m e t e r  chisquared minimization  performed by a p o w e r f u l , g e n e r a l 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 t h e C o n t r o l Data C o r p o r a t i o n (CDC) 76 00 computer l i b r a r y a t t h e European O r g a n i z a t i o n f o r N u c l e a r Research (CERN), i n Geneva. i s an easy-to-use  programme w i t h enough f l e x i b i l i t y  the u s e r t o d e v i s e a wide v a r i e t y o f f i t t i n g  MINUIT to allow  strategies;  o n l y a few o f i t s c a p a b i l i t i e s a r e mentioned here. Two m i n i m i z a t i o n a l g o r i t h m s a r e n o r m a l l y used: t h e s i m p l e x method o f N e l d e r and Mead [Nelder (67)] and a v a r i a t i o n o f t h e Davidon  (6 8)  v a r i a b l e m e t r i c 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 g i v e n a good set  o f i n i t i a l parameter guesses,  requires f i r s t  d e r i v a t i v e s of the f u n c t i o n being minimized;  partial  t h e s e may be  p r o v i d e d a n a l y t i c a l l y by t h e u s e r o r may be c a l c u l a t e d n u m e r i c a l l y by MINUIT. parameters,  MINUIT. w i l l accomodate up t o 50 v a r i a b l e  any number o f which may be FIXed a t any time and  RESTOREd a t any l a t e r t i m e . any p h y s i c a l l y m e a n i n g f u l  Parameters may be c o n s t r a i n e d t o  n u m e r i c a l 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 a r e c a l c u l a t e d by MINUIT, e i t h e r as an e s t i m a t e generated  by MIGRAD o r from t h e s o - c a l l e d  h e s s i a n m a t r i x , which i s e x a c t f o r a G a u s s i a n distribution.  parent  D e t a i l e d non-symmetric e r r o r e s t i m a t e s o f  parameters f o r n o n - p a r a b o l i c minima may be c a l c u l a t e d by a s e a r c h  -51raethod c a l l e d MINOS.  A number of checks f o r the presence  of  l o c a l minima are a l s o made by MINUIT. The gas phase Mu  data a n a l y s i s i n t h i s t h e s i s has been  performed i n three stages: raw histograms  are analyzed to -  e x t r a c t p s e u d o - f i r s t order r a t e c o n s t a n t s ; the  linear  dependence of these p s e u d o - f i r s t order r a t e constants  on  r e a c t a n t c o n c e n t r a t i o n y i e l d s bimoleeular : r a t e constants at a g i v e n temperature;  and,  f i n a l l y A r r h e n i u s f i t s of the  temperature dependent b i m o l e e u l a r r a t e constants p r o v i d e values of  a c t i v a t i o n energies and p r e - e x p o n e n t i a l Raw  of  histograms  equation  are f i t t e d  (6), Chapter  N (<)>•, t) = N e ~ y  to a model of the b a s i c form  I: t / T K  o  + A  factors.  y  [1 + A., e " c o s ( y Mu Mu A t  1  M  Bt + d>„ ) Mu Y  cos(y Bt - (j) )] + Bg y y' ^ r  1  where t i s the independent v a r i a b l e , and e i g h t unknown parameters are sought: N , A,, , X, B, , A , <$> , and f n o' Mu' ' ' Mu' y' y K  Bg ^  w i t h Y „ B = u as d e f i n e d i n Appendix I, and y B = to . Mu ^ ' 'y y  Of  these, X i s the parameter of c e n t r a l i n t e r e s t , although  A  1  L  M  and A^ p r o v i d e i n f o r m a t i o n about f a s t e p i t h e r m a l Mu r e a c t i o n s (see Appendix I I , S e c t i o n B). experience has c  equal,  shown t h a t <b„ Mu  p o s s i b l y ; because  As e x p l a i n e d i n Chapter and  I,  d>" cannot be assumed to be y  some • f r a c t i o n of the " f r e e "  s i g n a l comes from muons stopped  i n the w a l l s  of the  y  gas  t a r g e t v e s s e l which are g e o m e t r i c a l l y i n e q u i v a l e n t to the ensemble of Mu  stopped  i n the gas.  In a l l cases, the muon  l i f e t i m e i s assumed, to be f i x e d at 2.1971 ys. assumption i s p h y s i c a l l y v a l i d s i n c e y  +  This  lifetimes  are  +  independent of t h e i r environment to at l e a s t a few p a r t s per million  [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 s t r o n g l y upon the i n t e g r i t y of the m u l t i p l e muon r e j e c t i o n l o g i c d e s c r i b e d i n the section. 10%  The  e m p i r i c a l value of A  preceding  normally ranges between  and 15% and A^ i s g e n e r a l l y l e s s than 5%, depending upon  the stopping medium and d e t a i l e d counter c o n f i g u r a t i o n ; c o n v e n i e n t l y measurable values of X range between 0.1 1r 15 ys  ys  and  "I  L e f t and r i g h t histograms y i e l d i n g two tion.  The  are analyzed  independently,  redundant values of X a t each reagent  time b i n corresponding  to "time  concentra-  zero" i s  estimated to a p r e c i s i o n of about 2 ns by performing  a brief  measurement of the time r e q u i r e d f o r beam p o s i t r o n s to s c a t t e r between the accomplish  'D'  counter and p o s i t r o n t e l e s c o p e s .  To  t h i s , the p h o t o m u l t i p l i e r v o l t a g e on the  'D'  counter  i s i n c r e a s e d , thereby i n c r e a s i n g the p o s i t r o n p u l s e h e i g h t above the  'D  1  d i s c r i m i n a t o r t h r e s h o l d , and the p o s i t r o n t e l e s c o p e  t h r e e f o l d c o i n c i d e n c e requirement  i s reduced  to the  single  counter c l o s e s t to the t a r g e t , which always d e f i n e s the t i m i n g of the c o i n c i d e n c e output.  Each 2000 b i n histogram  of 2 ns b i n s  i s normally rebinned to 4 or 8 n s / b i n depending upon the p r e c e s s i o n frequency, of about 1000  r e s u l t i n g i n an e f f e c t i v e histogram  or 500 b i n s c o n t a i n i n g about 10  events  Mu size  (some of  the o r i g i n a l bins, are e l i m i n a t e d because they precede t=0<) . V a l i d data i s normally contained i n the histograms  w i t h i n about  t = 10 ns a f t e r time zero, but c a r e f u l adjustment of the l o g i c t i m i n g can reduce t h i s to about t = 3 ns.  In c o n t r a s t ,  i t may be noted t h a t experiments r e q u i r i n g a p o s i t r o n veto seldom c o n t a i n v a l i d data before before  t = 25 ns, and o f t e n not  t = 100 ns, due t o the width of the a n t i - c o i n c i d e n c e  requirement.  An e i g h t parameter f i t t o a 450 b i n histogram by  MINUIT consumes about 2 t o 2 5 seconds of CPU time, depending on the q u a l i t y of the i n i t i a l guesses t o the parameters. The  f a m i l i a r d e f i n i t i o n of x K  X (x) 2  =  2  -fY. - T . ( x )  _k  k  I k=l  i  s  given by  2 (9)  where x = x^, i = l , n , are the v a r i a b l e parameters, K i s the 2 number of data p o i n t s , Y t h e i r variances model.  fc  and a  fc  are the measured values and  and T ( x ) are the values p r e d i c t e d by the k  Since counting  s t a t i s t i c s g e n e r a l l y f o l l o w a Poisson 2 d i s t r i b u t i o n , the v a r i a n c e i s j u s t Y^ f o r l a r g e Y^, the number of events per f i t t e d time b i n . For histogram a n a l y s i s , 2 the d e f i n i t i o n of x i s modified to o 2  K  / \  X (x) = where  2  weighting  fY, - T (x))2  v  2  ^ k  k — ;  k=l  = Y^ i s r e p l a c e d by  (in)  Tlx)— 2  = T^(x).  This modification i n  i s made t o e l i m i n a t e e x t r a o r d i n a r y weighting  of un-  u s u a l l y low p o i n t s and can be seen as f o l l o w s : c o n s i d e r a s i t u a t i o n i n which one datum i s u n u s u a l l y correspondingly  low; d e f i n i t i o n  high and another i s  (9) p r o v i d e s  the high p o i n t  with  1 2 a s m a l l e r weighting f a c t o r (— ) than the low p o i n t thereby k b i a s i n g the f i t t o the lower p o i n t ; d e f i n i t i o n (10) weights both p o i n t s e q u a l l y .  For most histograms, model (10) p r o v i d e s  -54a x  2  P  degree of freedom of 0.95  e r  < x  2  1.05.  I n d i v i d u a l p s e u d o - f i r s t order r a t e constants from and r i g h t counter t e l e s c o p e s are s i m u l t a n e o u s l y f i t t e d  left to  equation 11(2), Appendix I I : X = k[X] +  x  0  to y i e l d the b i m o l e c u l a r r a t e constant, k. for  X i s determined  a t l e a s t f i v e c o n c e n t r a t i o n s of X i n c l u d i n g 2  [X] =  0.  The t r u e d e f i n i t i o n of x "  from  f i t s , but i n t h i s case  i s g i v e n from the e r r o r s i n X  c a l u l a t e d by MINUIT. histograms  (9) above i s used  f o r these  More experimental data i s accumulated  with f a s t r e l a x a t i o n s i n order t o reduce  in  the  r e l a t i v e u n c e r t a i n t y i n the d e t e r m i n a t i o n of X.  P l o t s of X  versus  X's from  [X] i n t h i s t h e s i s show - weight averaged  and r i g h t t e l e s c o p e s f o r g r a p h i c a l c l a r i t y , but the l i n e s correspond t o s i m u l t a n e o u s l y f i t t e d of  left  fitted  l e f t and r i g h t v a l u e s  X. For temperature  dependent k's, f i t s are made t o the  f a m i l i a r expression: k = Ae" a E  (11)  / R T  where E , the A r r h e n i u s a c t i v a t i o n energy,  and A,  the  cl  p r e - e x p o n e n t i a l factor, are the parameters of i n t e r e s t .  Again,  2 the t r u e d e f i n i t i o n of x l o g a r i t h m i c form of  from  (11): Ink = -E /RT cl  (9) i s used f o r f i t s of the + InA  (12)  Cvetanovic and S i n g l e t o n (77) have p o i n t e d out t h a t the  proper  -55weighting f a c t o r s of the k's i n equation by the i t e r a t i v e  (12) must be obtained  procedure  k.* (k. - k.*) w-l ' = T ,vk / k **)/ - li ln( w  i  where w.  1  = (  ^)  ( 1 3 )  i  i - the exact s t a t i s t i c a l weight f o r an  o. I  perimehtal In k. i n (12), w.  = ( i s the s t a t i s t i c a l a. l an experimental k. i n (11), and k. i s the best f i t 1  of  ex-  s  weight  1  *  p r e d i c t i o n of k^.  Since k_^ are unknown, w^ *  are o b t a i n e d by  1  i t e r a t i o n of c u r r e n t MINUIT v a l u e s of  k^.  Most of the gas phase Mu measurements r e p o r t e d i n t h i s have been taken at a magnetic f i e l d of 7 to 8 gauss.  thesis  Fields  g r e a t e r than 10 gauss are complicated by the beat frequency r 2 o,h o + (see Appendix I) , Q = {(D + j-^) - o~ . - —-.', , _ ' + 4 ^ (i) , At 10 gauss, 5 -1 the envelope of cos fit (ft = 2.8 x 10 s ) reduces A., to Mu w  u  u  2  K  0.4  A  M u  0  a f t e r 4 ys, a t y p i c a l experimental time range.  f i t t e d t o equation r a t e of 0.2  (6), t h i s leads t o an apparent  ys ^ f o r a s t a b l e , l o n g - l i v e d Mu  When  relaxation  signal.  This  bogus " r e l a x a t i o n " r a t e i n c r e a s e s as the square of the appliedf i e l d and has the appearance of a Gaussian r e l a x a t i o n .  In  p r i n c i p l e , a f i t t i n g f u n c t i o n can be d e v i s e d t o i n c l u d e the beat without i n t r o d u c i n g any new  parameters s i n c e 0 depends  only on known constants and B, which i s a parameter anyway. However, i n c l u s i o n of t h i s c o m p l i c a t i o n to the f u n c t i o n i n c r e a s e s computational c o r r e l a t e d to A  and y  fitting  c o s t and the beat i s h i g h l y  X at low f i e l d s .  C l e a r l y , these  factors  are not p r o h i b i t i v e , but they are e a s i l y a v o i d a b l e c o m p l i c a t i o n s .  -56F a i l u r e to account f o r the beat envelope u s i n g model (6) as fitting  f u n c t i o n w i l l generate a systematic  e r r o r reducing  T h i s a r i s e s s i n c e , at 10 gauss f o r example, a s i g n a l w i l l appear to r e l a x at ^ 0.2 s i g n a l (A = 15 ys \  say)  0.  non-relaxing  by the cosSlt envelope  Thus, the systematic  to A decreases with i n c r e a s i n g reagent  For f a s t r e l a x a t i o n s , higher  k.  while a f a s t r e l a x i n g  i s unaffected  which i s almost f l a t near t = introduced  ys  the  error  concentration.  f i e l d s are p r e f e r a b l e , i n  p r i n c i p l e , f o r reasons i l l u s t r a t e d i n the f i g u r e s of Appendix II  and  a l s o because a l a r g e number of o s c i l l a t i o n s i n the  s h o r t - l i v e d Mu and phases.  s i g n a l produce more r e l i a b l e f i t s to the  I t would then seem to be optimal  a f u n c t i o n of A. t h i s proposal, ys FIX  field  to i n c r e a s e B as  There i s a s e r i o u s p r a c t i c a l o b j e c t i o n  however, i n t h a t r e l a x a t i o n r a t e s of 10  are g e n e r a l l y d i f f i c u l t  to f i t .  s e v e r a l parameters i n order  candidates f o r F i x i n g are B and A  A, M u  foreknowledge of these parameters.  The  (6) to a f u n c t i o n  important  which r e q u i r e s  accurate  In p r a c t i c e , 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 i n the Helmholtz c o i l s as a f u n c t i o n of e l e c t r i c because non-reproducable  produced  current  background c o n t r i b u t i o n s to B  f l u c t u a t e over time p e r i o d s r e c a l i b r a t i o n necessary.  - 15  I t i s o f t e n necessary to  to reduce model  t h a t i s s e n s i t i v e to the data and  to  of days, making  constant  These u n r e p r o d u c i b l e  contributions  to B can be t r a c e d to such events as movement of the TRIUMF 50 ton crane over the experimental area, field and  changes i n magnetic  s e t t i n g s of beam l i n e components i n adjacent  constant  r e - s t a c k i n g of s t e e l and  beamlines,  i r o n neutron s h i e l d i n g  -57around adjacent experiments.  Such e f f e c t s c o n s t r u c t i v e l y or  d e s t r u c t i v e l y add s i g n i f i c a n t , though homogeneous, components to  the experimental magnetic f i e l d .  experimental f i e l d  F o r t u n a t e l y , the  i s g e n e r a l l y constant over the time d u r i n g  which a s e r i e s of c o n c e n t r a t i o n s of a r e a c t a n t are examined, Without c a l i b r a t i o n , i t i s i m p o s s i b l e to s e t B reproducably to b e t t e r than a few p e r c e n t .  Experience has  shown t h a t  fitted  values of weak f i e l d s from experimental data taken over 24 - 48 hours are constant w i t h i n a standard d e v i a t i o n of l e s s 1%, g i v i n g a more accurate measure of the f i e l d c a l i b r a t i o n would g i v e .  7-8  i n a feedback  allow r e l i a b l e and c o n s i s t e n t f i e l d The  fitting  than  In the near f u t u r e , the f i e l d w i l l  s t a b i l i z e d by a continuous monitor will  than  procedure  adopted  be  loop, which  settings.  i s to f i r s t  f i tB  ( i n the  G rangey f o r a s e r i e s of runs a t low reagent c o n c e n t r a t i o n  and then t o FIX t h i s value of the f i e l d t o f i t the relaxation  runs.  fast  -58CHAPTER I I I - THEORETICAL BACKGROUND A  Introduction The  i n i t i a l m o t i v a t i o n f o r undertaking the experimental  study of the chemical r e a c t i o n r a t e s of Mu was  t o examine the  behaviour of Mu  (see Chapter  S e c t i o n B). between Mu  as a l i g h t i s o t o p e of hydrogen  I t was  I,  hoped t h a t the s u b s t a n t i a l mass d i f f e r e n c e  and H would p r o v i d e an e x a c t i n g t e s t of modern c a l -  c u l a t i o n s of H atom r e a c t i o n k i n e t i c s , p a r t i c u l a r l y w i t h r e s p e c t t o quantum mechanical [Fleming  (76)].  I t was  e f f e c t s such as t u n n e l l i n g  expected t h a t comparison  of both  t h e o r e t i c a l and experimental r e s u l t s f o r r e a c t i o n s of Mu those of the other H i s o t o p e s might not o n l y l e a d t o  with  improved  t h e o r e t i c a l methods f o r t r e a t i n g H atom r e a c t i o n s , but i t might a l s o p r o v i d e new  i n f o r m a t i o n about  p o t e n t i a l energy  surfaces.  such computational t o o l s as  I t i s shown i n t h i s Chapter  and  the next t h a t many of these o b j e c t i v e s have a l r e a d y reached a high l e v e l of r e a l i z a t i o n .  In the course of s t u d y i n g Mu  r e a c t i o n k i n e t i c s , a second m o t i v a t i o n f o r the became c l e a r ; t h i s i s d i s c u s s e d i n Chapter  experiments  IV.  •The s e l e c t i o n of chemical systems f o r study, namely Mu w i t h the halogen and hydrogen h a l i d e f a m i l i e s , was  based  partly  on the c o n s i d e r a b l e t h e o r e t i c a l and experimental i n t e r e s t i n the H analogue  r e a c t i o n s , and p a r t l y on the experimental .. •.  c o m p a t a b i l i t y of these reagents w i t h the MSR t a r g e t s at about  1 atmosphere may  method: gas phase  be r e a d i l y prepared w i t h a  wide range of r e a c t a n t c o n c e n t r a t i o n s ; and the r e a c t i o n s are s u f f i c i e n t l y f a s t at or near room temperature  t o consume Mu  d u r i n g i t s 2.2  ys l i f e t i m e .  As noted i n Chapter I,  the  experiments have been c o n f i n e d to the measurement of averaged r a t e c o n s t a n t s and state-to-state  a c t i v a t i o n energies.  thermally  Modern  techniques employing l a s e r s , atomic beams,  i n f a r e d chemiluminescence are not I t would be  inappropriate  yet a v a i l a b l e t o Mu  and  studies.  to attempt t o present a  comprehensive review of the t h e o r y of chemical k i n e t i c s i n t h i s t h e s i s 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 r e a c t i o n r a t e theory i s so f r e n e t i c that  such a review would be  other hand, the debut of gas  impossible).  phase Mu  reaction  On  the  r a t e data  has  sparked c o n s i d e r a b l e t h e o r e t i c a l a c t i v i t y , n o t a b l y by Connor, Jakubetz, Manz, and ical  (QMT)  Lagana who  have performed quantum mechan-  [Connor l - ( 7 7 ) , l - ( 7 8 ) , l-(79) ], q u a s i c l a s s i c a l  [Connor 1-(79) ] > and  classical  (CT)  [Jakubetz  ' (79)]  c a l c u l a t i o n s , as w e l l as t r a n s i t i o n s t a t e theory  trajectory  (TST)  [Connor i -  (79)] Jakubetz 1-(78), (79) ] c a l c u l a t i o n s on the r e a c t i o n s H,  D,  and  T w i t h F^  c a l c u l a t i o n s on state-to-state  and  the C l  who 2  are p r e s e n t l y  reactions;  c a l c u l a t i o n s of Mu  performing  (QCT)  of  Mu,  similar  other authors have done reaction  r a t e s which are  of  l e s s d i r e c t r e l e v a n c e to the present experimental work (see [ F i s c h e r 1,2-(77), Korsch  (78)]).  mental r e s u l t s of t h i s t h e s i s are c a l c u l a t i o n s of Connor et_ aJ_.  S i n c e most of the interpreted  experi-  i n terms of  (Chapter IV) , one  of the  eg.  the  aims of  the present Chapter i s to o u t l i n e t h e i r v a r i o u s 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  e x p l o r e some q u a l i t a t i v e p r e d i c t i o n s Mu  versus H based both on the  of the  reaction rates  c a l c u l a t i o n s mentioned above  of and  -60-  on  selected  considerations  from the theory of chemical k i n e t i c s .  B P o t e n t i a l Energy S u r f a c e s Most chemists l i k e l y have a t l e a s t some f a m i l i a r i t y w i t h the n o t i o n s of a p o t e n t i a l energy s u r f a c e r e a c t i o n path. The  and an  These concepts are i l l u s t r a t e d i n F i g u r e 11.  d e t e r m i n a t i o n of a p o t e n t i a l energy s u r f a c e  interatomic point  associated  p o t e n t i a l s of the r e a c t i n g  for a l l trajectory calculations  t o d e s c r i b e the  atoms i s the s t a r t i n g [Johnston(66), L a i d l e r  (65)], and, t o a l e s s e r extent, i t i s a requirement f o r TST c a l c u l a t i o n s as w e l l .  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 r e q u i r e but  the complete p o t e n t i a l  only the minimum energy path f o r the r e a c t i o n .  considering reactions  surface,  Before  p a r t i c u l a r p o t e n t i a l energy s u r f a c e s f o r the  studied  i n t h i s t h e s i s , a few comments should be made  about such s u r f a c e s i n g e n e r a l .  (i) Semi-Empirical P o t e n t i a l Energy S u r f a c e s In p r i n c i p l e , i t should be p o s s i b l e  t o determine p o t e n t i a l  energy s u r f a c e s from ab i n i t i o methods i n v o l v i n g the s o l u t i o n of the Schrodinger equation, perhaps w i t h the a i d of approximat i o n s based on v a r i o u s quantum mechanical c r i t e r i a (65), Jakubetz l - ( 7 8 ) ] .  [Laidler  Unfortunately, i t i s s t i l l  impossible  to perform such c a l c u l a t i o n s w i t h s u f f i c i e n t accuracy t o be o f g e n e r a l use t o r e a c t i o n (78)], w i t h the p o s s i b l e [Liu  (78)]).  kinetics  [Van Hook (70), Jakubetz 1- .  exception of the H +  ' i n the face of t h i s o b s t a c l e ,  system  (see eg.  i t i s customary t o  -61-  FIGURE 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 . The minimum energy path through the saddle p o i n t (+) i s denoted by the dashed line. 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 e x i t v a l f e y depth i s -D (AB). The saddle p o i n t i s above the v a l l e y s , but befow the p l a t e a u .  -62employ s o - c a l l e d " s e m i - e m p i r i c a l " (although,  p o t e n t i a l energy  the degree o f empiricism  surfaces  a c t u a l l y employed o f t e n  b l u r s any d i s t i n c t i o n between " s e m i - e m p i r i c a l "  and "wholly"  e m p i r i c a l methods) which a r e d i s t i n g u i s h e d from ab i n i t i o s u r f a c e s by the f a c t t h a t parameters are l e f t  f o r adjustment  based not on t h e o r e t i c a l grounds, but r a t h e r on a p o s t e r i o r i experimental  results [Laidler  (65)].  The use of a semi-  e m p i r i c a l s u r f a c e n e c e s s a r i l y removes some (but c e r t a i n l y not a l l or even most) o f the p r e d i c t i v e u t i l i t y o f the theory. Indeed, many reviewers  (see eg. [Johnston  (66), L a i d l e r (65),  Thompson (76)]) have p o i n t e d out t h a t while  the accuracy  of a  k i n e t i c c a l c u l a t i o n depends r a t h e r d i r e c t l y on the accuracy o f the p o t e n t i a l energy s u r f a c e , many q u a l i t a t i v e p r e d i c t i o n s can and  have been made from c o n s i d e r a t i o n o f i n a c c u r a t e or even  completely  h y p o t h e t i c a l p o t e n t i a l energy s u r f a c e s  (65), P o l a n y i chemical symbiotic  (69), Mok (69), P o l a n y i  k i n e t i c theory  (78)])..  (eg. [Kuntz  In t h i s way,  and experiments take on an e x p l i c i t  r e l a t i o n s h i p i n a "bootstrap"• procedure whereby  experiments serve not only t o t e s t the accuracy  o f the c a l -  c u l a t i o n s , but a l s o t o a d j u s t the parameters o f the p o t e n t i a l energy s u r f a c e , which, i n t u r n , leads t o improved c a l c u l a t i o n s . Probably the most commonly used s e m i - e m p i r i c a l determining  methods o f  a p o t e n t i a l energy s u r f a c e a r e v a r i a t i o n s o f the  method due t o London, E y r i n g , P o l a n y i and Sato d i s c u s s i o n o f the development o f s e m i - e m p i r i c a l energy s u r f a c e s i s given i n L a i d l e r and P o l a n y i surfaces considered  (LEPS).  A good  potential (65).  The  i n t h i s t h e s i s are a l l e s s e n t i a l l y v a r i a -  t i o n s o f the LEPS s u r f a c e .  The LEPS method i s a m o d i f i c a t i o n  -63of  the H e i t l e r - L o n d o n approximation  of H 2 u s i n g the (unnormalized) $ = ^(1)^(2)  f o r the e l e c t r o n i c  energy  wavefunction  ± ^(1)^(2)  which has H-H i n t e r a c t i o n eigenvalues o f V  (r)  =  =- -  .  T  —  A  where Q . J , and S  are the Coulomb, exchange, and o v e r l a p  i n t e g r a l s which are functions o f r , the i n t e r n u c l e a r s e p a r a t i o n . In  equation  (14), the p l u s s i g n r e f e r s t o the s i n g l e t (bound)  s t a t e and the minus s i g n r e f e r s t o the t r i p l e t ( r e p u l s i v e ) state. r  I t may be noted t h a t J i s n e g a t i v e and | j | > Q near  , the e q u i l i b r i u m i n t e r n u c l e a r s e p a r a t i o n .  and P o l a n y i extended p o t e n t i a l energy V ( r  t o the  system, t o g i v e a  expression  AB' BC' AC r  t h i s treatment  London, E y r i n g ,  r  )  =  Q  AB  +  Q  BC  +  Q  AC  1  4  ±  2  < AB " BcJ J  J  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 s t a t e w i t h the Morse funct i o n s o f the d i a t o m i c p a i r s : Q. . + J . . = ^ ( A r . . ) 1 j  1 3  where ^ ^ j and  of  the atomic  8.. 1 3  = r^_. - r  2 y  = TTV  1  1 3  2 3  ij  A r  ij  - 2e~ ij 3  A r  i j ) (16)  a r e c o n s t a n t s obtained s p e c t r o s c o p i c a l l y and  D  Ar^j  = D, , ( e "  ID  g  with i , j r e f e r r i n g to appropriate  l a b e l s A,B, and C. 1/2  combinations  "*"D. . i s r e l a t e d t o 6 . . by 1 3  where v  1 3  i s a fundamental v i b r a t i o n a l 0  0 1 3  frequency and y i s the reduced mass o f the d i a t o m i c p a i r .  The  2 n e g l e c t s the o v e r l a p 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 b i n d i n g energy (—— ^J -A. A  LEP treatment  +  -64independent o f r . equation V ( r  (16)  The Sato m o d i f i c a t i o n  is:  AB' BC' AC r  r  )  377^2 AB [Q  =  1 I BC  +  where the t r i p l e t  (  ij  ~ i j J  (1 - S )  =  J  +  Q  BC  +  2 -  J  AC>  I  +  Q  (  AC ± 4 1 J  A C  "  (  J  J  B " BC ' J  A  AB'  )  '  2 1/2 1  >  (anti-bonding) s t a t e i s i d e n t i f i e d w i t h a  m o d i f i e d Morse f u n c t i o n Q  [Sato (55)] o f  (anti-Morse o r Sato-Morse):  3„,. , ^D, . , -2 3 . .Ar. . „ - 3 . .Ar. .. V(Ar. .) = 1 3 (e 13 1 3 + 2e " 1 3 1 3 ) (18) 2 3  and the s i n g l e t s t a t e  (equation (16)) becomes:  Q. . + J . . 1 . , 1^ , -23. .Ar. . _ - 6 . .Ar. 13 1 3 - V(Ar..) = D..(e ij 1 3 - 2e ^ 1 3 1 3 ) (19) TT/  (1 +  s) z  The o v e r l a p i n t e g r a l i n the LEPS f o r m u l a t i o n  (equation (17))  is  2 l e f t as an a d j u s t a b l e parameter  (A = S  i s c a l l e d the Sato  parameter) which i s normally found t o be much s m a l l e r than the true o v e r l a p i n t e g r a l . I t should be noted t h a t the Sato 2 m o d i f i c a t i o n s e t S constant over a l l i n t e r n u c l e a r s e p a r a t i o n s and independent o f atomic l a b e l s . Most authors use e m p i r i c a l v a r i a t i o n s o f the LEPS f o r m u l a t i o n such as: replacement o f the 2 constant S Jonathan  2 w i t h S^^ terms f o r each atomic p a i r  (7 2)]  which may or may not be dependent  [Kuntz (66), upon the  i n t e r n u c l e a r d i s t a n c e (Jonathan e t a l . examined both cases, 2 Kuntz e t a l . used S^j independent o f ^ j ) ; e m p i r i c a l adjustment of the t r i p l e t anti-Morse f u n c t i o n (equation (18)) by forms which r e p l a c e D i j by an a d j u s t a b l e 3 D.. ([Jonathan (72)] a l s o r  X  examined t h i s ) ; for  example:  or replacement of the anti-Morse f u n c t i o n by,  -653„ V  i j  /» (  A  r  »  i j  )  3_ =  i j  , -2$; .Ar. . , _ -8. .Ar. . . . (  1 D  1:1  = C ( r . . + A)e 13  3 where  —or  1 D  1 D )  f o r  r  ij  ^  i j-  *  r  for r. . > r  *  -  ID  * D, B, C, A,  a, and  r  are a d j u s t a b l e  parameters  ([White  (73)]; i n f a c t , White used an e m p i r i c a l v a l e n c e bond v a r i a t i o n of the LEP  surface).  C l e a r l y , a l l LEPS f o r m u l a t i o n s mental i n p u t .  LEPS s u r f a c e s  mentioned r e q u i r e  are u s u a l l y "optimized"  experiby  a d j u s t i n g the v a r i a b l e parameters u n t i l some type of t r a j e c t o r y c a l c u l a t i o n s performed on the s u r f a c e reproduce a s e t of experimental r e s u l t s .  For example, Jonathan e t a_l. (72),  t a i l o r e d the LEPS s u r f a c e  f o r the r e a c t i o n H + F  2  -»- HF  + F such  t h a t three dimensional q u a s i c l a s s i c a l t r a j e c t o r y c a l c u l a t i o n s (see S e c t i o n D below) g i v e 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 , r e a c t i o n a c t i v a t i o n energy, r a t e constant enthalpy i n agreement w i t h experiment.  and  reaction  Clearly, this  procedure i s not l i k e l y t o converge to a unique " c o r r e c t " s u r f a c e ; i t i s o n l y hoped t h a t i t produces a u s e f u l ( i i ) Contour P l o t s of the P o t e n t i a l Energy Surface Reaction  A + BC •> AB  11,  provide  computationally process.  f o r the  + C  In t h i s S e c t i o n , i t i s shown how Figure  surface.  contour p l o t s , such as i n  the b a s i s f o r s e t t i n g up a c o n c e p t u a l l y  simple p i c t u r e of the atom-diatom c o l l i s i o n  For the three atom A + BC  system, the  interatomic  p o t e n t i a l energy i s a f u n c t i o n of the p o s i t i o n s of the n u c l e i and it  and  t h e r e f o r e a function of nine c o o r d i n a t e s .  i s only the r e l a t i v e motion of the n u c l e i w i t h i n  three However,  their  -66center of mass frame t h a t i s of r e l e v a n c e  to the  collision  s i n c e t r a n s l a t i o n or r o t a t i o n 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 i n t e r a t o m i c p o t e n t i a l . the i n t e r a t o m i c p o t e n t i a l energy i s a f u n c t i o n of coordinates: r . ^ , r AB  6)ABC' the ABC  , and r BC  three  or, more commonly, r AC  bond angle.  Thus,  , r AB  Consequently, contour  , and BC  p l o t s of  p o t e n t i a l energy f u n c t i o n must be drawn with r e s p e c t to c o o r d i n a t e s , with the t h i r d c o o r d i n a t e contour and  fixed.  two  U s u a l l y , such  p l o t s show the p o t e n t i a l energy as a f u n c t i o n of r  r^^, (as i n F i g u r e 11) with 0 BC AB(  B  f i x e d ; i n f a c t , most d e s c r i b i n g the  c o n f i g u r a t i o n of the three atoms. good reasons why  collinear  There are at l e a s t  two  the c o l l i n e a r p o t e n t i a l s u r f a c e i s the  most o f t e n c o n s i d e r e d :  firstly,  below, the degree of complexity  one  as d i s c u s s e d i n S e c t i o n D of t r a j e c t o r y c a l c u l a t i o n s .  i n c r e a s e s tremendously from the c o l l i n e a r to the coplanar the three dimensional  increased  number of i n t e r n a l degrees of freedom of the system with corresponding  to  cases, not only because of the need to  c o n s i d e r more s u r f a c e s , but a l s o because of the  the  i n c r e a s e i n the number of r e a c t i o n channels  (more product  i n c l u d e d ) ; and, often  A  ABC  commonly, 9 ~. i s f i x e d at 180°  available  the  v i b r a t i o n a l and  secondly,  r o t a t i o n a l s t a t e s are  the simpler c o l l i n e a r c a l c u l a t i o n s  (but by no means always) p r o v i d e a reasonably  accurate  d e s c r i p t i o n of the r e a c t i o n , p a r t l y because i t i s u s u a l l y the case t h a t the c o l l i n e a r r e a c t i o n geometry i s the e n e r g e t i c a l l y favored one.  In f a c t , Jonathan e t a l . (72)  have p o i n t e d  out  t h a t an e n e r g e t i c a l l y favored c o l l i n e a r c o n f i g u r a t i o n seems to be a g e n e r a l  f e a t u r e of LEPS s u r f a c e s ; however, t h i s  general-  -67i z a t i o n does not i n i t s e l f  imply t h a t the c o l l i n e a r  con-  f i g u r a t i o n w i l l n e c e s s a r i l y dominate the r e a c t i o n dynamics because  i t n e g l e c t s other t o p o l o g i c a l f e a t u r e s of the  p o t e n t i a l s u r f a c e as w e l l as the r o l e p l a y e d by m u l t i dimensional i n t e r n a l , energy modes of .the- t a r g e t molecule, which may be a v a i l a b l e f o r promoting  reaction.  The chemical r e a c t i o n A + BC -* AB + C i s e n v i s i o n e d as the movement o f a r e p r e s e n t a t i v e mass p o i n t  (the f e a t u r e s o f  which are d e t a i l e d below) along the p o t e n t i a l energy s u r f a c e through the r e a c t a n t v a l l e y , a c r o s s the saddle p o i n t , and f i n a l l y e x i t i n g along the product v a l l e y . the e l e c t r o n i c p o t e n t i a l energy  This notion that  s u r f a c e mediates  the motion o f  the n u c l e i o f the atoms i s an i m p l i c i t statement o f the BornOppenheimer  (BO) approximation: the e l e c t r o n i c energy o f the  atoms i s separable from t h e i r n u c l e a r energy. imation, sometimes c a l l e d the low k i n e t i c energy [Schatz  (77) , Levine  T h i s approxapproximation  (74)] , i s v a l i d f o r most atoms a t normal  temperatures, where n u c l e a r v e l o c i t i e s are much l e s s than the electron velocities.  Since the mean v e l o c i t y o f an ensemble o f  atoms a t a g i v e n temperature  i s i n v e r s e l y p r o p o r t i o n a l t o the  square r o o t o f t h e i r masses (see S e c t i o n D below), i t i s expected t h a t the BO approximation w i l l break down a t lower temperatures  f o r Mu than f o r H.  To date, the BO approximation  has always been invoked i n c a l c u l a t i o n s of the r e a c t i o n s o f Mu, although i t s v a l i d i t y i n these cases has not y e t been examined [Jakubetz l - ( 7 8 ) ] ; a d i s c u s s i o n of the v a l i d i t y o f the BO approximation w i t h a p p l i c a t i o n t o , among o t h e r s , the H + HD r e a c t i o n i s g i v e n by Bardo and Wolfsberg  (78) who f i n d i t t o be  -68accurate t o w i t h i n a few p e r c e n t i n the cases s t u d i e d . C o r r e l a t e d w i t h the BO approximation i s the assumption of e l e c t r o n i c adiabaticity  [Nikitin  (74)] : the system remains i n the  ground e l e c t r o n i c s t a t e throughout the r e a c t i v e Besides depending on the c o l l i s i o n time, t h i s  collision.  assumption  depends upon the e l e c t r o n angular momentum and the c o r r e l a t i o n of e l e c t r o n i c s t a t e s  [Smith (77)].  The assumption o f  e l e c t r o n i c -adiabaticity i s the standard procedure f o r both t r a j e c t o r y and TST treatments of the r e a c t i o n s c o n s i d e r e d i n t h i s t h e s i s ; the p r a c t i c a l consequence o f t h i s assumption i s t h a t only ground s t a t e p o t e n t i a l energy s u r f a c e s need be considered.  The f i n a l , and, from the p o i n t of view of t h i s  t h e s i s , most important consequence  of the BO approximation i s  t h a t the p o t e n t i a l energy s u r f a c e i s i n v a r i a n t t o i s o t o p i c substitution  [Van Hook (70), N i k i t i n  (74)]; t h a t i s , i d e n t i c a l  p o t e n t i a l energy s u r f a c e s are a p p l i c a b l e t o r e a c t i o n s o f Mu, H, D and T. I t has a l r e a d y been s t a t e d t h a t r e a c t i v e  atom-diatom  c o l l i s i o n s can be p i c t u r e d as the t r a n s m i s s i o n o f some k i n d o f p a r t i c l e across the b a r r i e r o f a p o t e n t i a l energy Levine and B e r n s t e i n  surface.  (72) c a l l t h i s p i c t u r e the "analogue"  f o r m u l a t i o n of the problem and i t i s necessary t o determine the i d e n t i t y of these p a r t i c l e s moving on the p o t e n t i a l To t h i s end, i t i s u s e f u l to c o n s i d e r the freshman  surface. physics  problem of the c o l l i s i o n o f two s t r u c t u r e l e s s b i l l i a r d  balls.  Although t h i s problem i s t r e a t e d c o r r e c t l y by s e p a r a t e l y s o l v i n g the equations o f motion f o r each b a l l ,  i n some ways i t  i s more u s e f u l t o c o n s i d e r the e q u i v a l e n t problem i n which the  motion of the center o f mass i t s e l f  i s p a r t i t i o n e d from the  r e l a t i v e motion w i t h i n the center of mass frame.  This  procedure s i m p l i f i e s both the problem and the i n t e r p r e t a t i o n o f i t s s o l u t i o n by e l i m i n a t i n g the motion o f the center o f mass which i s extraneous t o the c o l l i s i o n  itself.  For example, one  f i n d s t h a t the system k i n e t i c energy i n the c e n t e r of mass frame i s g i v e n by = iyv 2 r  (20)  2  trans where v  M  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  r  t h e i r reduced mass.  Equation  looks j u s t l i k e an equation mass y.  (20) i s remarkable i n t h a t i t  o f motion f o r a s i n g l e p a r t i c l e o f  Given t h i s i n t e r p r e t a t i o n , some of the p r o p e r t i e s o f  the two body c o l l i s i o n  are d e s c r i b e d by the analogue equations  of motion of a s i n g l e r e p r e s e n t a t i v e p o i n t w i t h mass.  some e f f e c t i v e  The use of a p o t e n t i a l energy contour p l o t t o d e s c r i b e  the atom-diatom c o l l i s i o n to three bodies.  i s a g e n e r a l i z a t i o n o f t h i s procedure  P o i n t s along the r e a c t i o n path on the  p o t e n t i a l energy s u r f a c e d e s c r i b e the c o n f i g u r a t i o n o f the three atoms a t v a r i o u s stages  of c o l l i s i o n .  Instead of  s o l v i n g the equations o f motion f o r a l l three atoms (which i s occasionally  done), the r e a c t i v e c o l l i s i o n  i s d e s c r i b e d by  s o l v i n g the equations o f motion o f a s i n g l e r e p r e s e n t a t i v e mass p o i n t moving along the p o t e n t i a l energy s u r f a c e , s i t u a t e d i n the center o f mass frame. The  problem now a r i s e s as t o what e f f e c t i v e mass t o  assign to t h i s representative point. problem, c o n s i d e r  To i l l u s t r a t e  the c o l l i n e a r LEPS p o t e n t i a l energy  this surface  -70due t o Jonathan e t a l . (72) shown i n F i g u r e 12 f o r the r e a c t i o n Y + F ->YF  + F, Y = Mu, H, D, T (adapted from  2  As the r e p r e s e n t a t i v e p o i n t moves along towards the p o t e n t i a l saddle a x i s , i t s motion simply equation  [Connor l - ( 7 8 ) ] ) .  the r e a c t a n t v a l l e y  on a l i n e p a r a l l e l t o the r^p  d e s c r i b e s a two body c o l l i s i o n as i n  (20) with H (say) as one body, and F .  + m )  n^(m  Thus, the e f f e c t i v e mass i s y„ „ = '2 H  a 2  ;  ;  F  m  m  s the o t h e r . - 1 amu, and  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 „, where r E -g^. trans 2 H , F HF' dt M  On the other hand, a f t e r  TT  2  r e a c t i o n , the r e p r e s e n t a t i v e p o i n t moves along  the product  v a l l e y away from the saddle p o i n t on a l i n e p a r a l l e l t o the r„„ a x i s , thereby d e s c r i b i n g another two body system, t h i s time (m + m )m of HF and F. Here the e f f e c t i v e mass i s yHF,F ~ m + m + m R  H  p  F  p  p  - 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 g i v e n 1 * 2 Y  E  trans  =  "^HF^FF"  I  n  9  e n e r a 1  ' i t  c  a  n  b  e  shown t h a t the  - e f f e c t i v e mass •'-."•of- the. r e p r e s e n t a t i v e .point is. a f u n c t i o n o f the d i r e c t i o n o f i t s motion along the p o t e n t i a l energy s u r f a c e (an e x c e l l e n t d i s c u s s i o n of t h i s s u b j e c t i s found i n Johnston varying continuously  from 1 t o 9 amu i n the present  the slope of i t s t r a j e c t o r y v a r i e s from 0 t o . 00  v i b r a t i o n a l motion o f the t a r g e t F  2  (66)),  example as  When the  and product HF are taken  i n t o account, t r a j e c t o r i e s o f the r e p r e s e n t a t i v e p o i n t are o s c i l l a t o r y so t h a t t h e i r e f f e c t i v e masses are a l s o o s c i l l a t o r y , thereby making t h i s analogue p i c t u r e both c o m p u t a t i o n a l l y and conceptually  complicated.  This complication  i s removed by r e p r e s e n t i n g  t i a l energy s u r f a c e with a mass weighted c o o r d i n a t e  the potensystem ( f o r  FIGURE 12:  P o t e n t i a l energy s u r f a c e f o r Y + F due t o Jonathan (7 2) , adapted from Connor 1(78). * S o l i d contours are l a b e l l e d 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 a t ^ ^ = 0.087 eV, the q u a s i c l a s s i c a l t h r e s h o l d . 2  a  n  s  a d e r i v a t i o n , see Johnston  (66)).  F o r the c o l l i n e a r  c o n f i g u r a t i o n o f atoms A, B, and C, i t may be shown t h a t the k i n e t i c energy i n the c e n t e r o f mass frame i s g i v e n by: E  trans  =  ^M A B [m  (m  + m (m c  A  V  ^B .2 + m )r ] +  B  where M = m, + nu + m_,. A B C  +  2 m  A cWBC m  ( 2 1  >  B C  The f i r s t and l a s t terms have been  previewed i n the above d i s c u s s i o n s o f pure A-BC and pure AB-C motion.  The middle  c r o s s - p r o d u c t term p r o v i d e s the continuous  v a r i a t i o n between the two motional extremes and a n t i c i p a t e s t h a t any new c o o r d i n a t e system, q - and q  , that diagonalizes  D  AB  equation  BC  (21) w i l l be skewed w i t h r e s p e c t t o t h e c a r t e s i a n r ^  and r „ .  B  In g e n e r a l , there may be more than one c o o r d i n a t e  D  BC  transform t h a t d i a g o n a l i z e s the k i n e t i c energy for  [Johnston  the c o l l i n e a r case, a common mass weighted c o o r d i n a t e  t r a n s f o r m a t i o n i s [Marcus (77)] : AB AB " BC r  r  =  BC  =  where s -  q  q  C  t  n  a  BC ° in (itv + m_) 1/2  S q  m  and  (66)];  S  C  ( m  A  C  (  2  2  )  a  and cosa =  +  a i s the skewing angle.  mm A C (m, + m_) (m  Equations  1/2 + m )J  (22) g i v e the  k i n e t i c energy e x p r e s s i o n : 1 E  trans  =  •2  2 A,BC AB y  (q  +  •2 q  BC  )  and d e f i n e the constant e f f e c t i v e mass o f the r e p r e s e n t a t i v e p o i n t as V  .  F o r other than the c o l l i n e a r c o n f i g u r a t i o n ,  d i f f e r e n t b u t s i m i l a r e x p r e s s i o n s t o equations r e q u i r e d t o d i a g o n a l i z e the k i n e t i c energy  (22)  are  (see eg. [Gatz  (66)])  With the t r a n s f o r m a t i o n o f the p o t e n t i a l energy s u r f a c e i n t o a  -73mass weighted c o o r d i n a t e be completely  system, the atom-diatom c o l l i s i o n can  understood c l a s s i c a l l y  i n terms o f the t r a j e c t o r y  of a b a l l r o l l i n g along a p h y s i c a l s u r f a c e under the i n f l u e n c e of g r a v i t y .  Although the f o r e g o i n g d i s c u s s i o n  explicitly  r e f e r s t o t r a j e c t o r i e s , i t i s a l s o a p p l i c a b l e t o TST c a l c u l a t i o n s which can be viewed as a s t a t i s t i c a l treatment o f t r a j e c t o r i e s not r e q u i r i n g 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 a l s o s t a t i s t i c a l dynamical t h e o r i e s  (iii)  P o t e n t i a l Energy Surfaces  [Connor l - ( 7 6 ) ] ) .  For the Reactions:  Y +  ->  YX + X, Y = Mu, H, D, T; X = F, C l , Br, I While i n v a r i a n c e of a p o t e n t i a l energy s u r f a c e under i s o t o p i c s u b s t i t u t i o n i s a consequence of the BO approximation, the above d i s c u s s i o n c l e a r l y shows t h a t the e f f e c t i v e p o t e n t i a l energy s u r f a c e  ( i . e . mass weighted) d i s p l a y s no such i n v a r i a n c e .  This i s i l l u s t r a t e d Y + F  2  + Y F + F ,  (adapted from  i n the mass weighted LEPS s u r f a c e f o r the  Y = M u , H, D, T, r e a c t i o n shown i n F i g u r e 13  [Connor l - ( 7 8 ) ] > , corresponding  s u r f a c e o f F i g u r e 12. scheme used i s [Connor a/2 YF  t o the LEPS  In t h i s F i g u r e , the mass  weighting  (75)]: +0.5 (r. YF  r  FF  )  ll/2 (23)  r where R^ i s the d i s t a n c e from Y t o the center of mass of F , T,F and the skewing angle a i s g i v e n by: 2  2  -74-  11  Y=Mu.H.D,T  :  :  L_  0  1  (MY,F /MF/ 2  FIGURE 13:  2  2  R F/A X  P o t e n t i a l energy s u r f a c e s o f Jonathan (72) f o r the c o l l i n e a r Y + F 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 , adapted from Connor l - ( 7 8 ) . The mass weighting scheme i s d e s c r i b e d i n the t e x t . The s i n g l e contours are f o r E = 0.087 eV(2.01 k c a l / 2  t  r  a  n  s  mole) (contour B o f F i g u r e 12). i n d i c a t e d by c r o s s e s .  Saddle p o i n t s are  -752nv, i l1 / 2/-).  fitly +  tana =  F  m Y  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 g i v e n by: E  trans  I ^ F ^ *YF  =  (  +  with an e f f e c t i v e mass o f y^ f o r the r e p r e s e n t a t i v e p o i n t f o r 2 a l l i s o t o p i c forms o f Y.. The s i n g l e contours shown i n F i g u r e F  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  trans  (or the e q u i v a l e n t p o t e n t i a l energy  r e l a t i v e t o zero as d e f i n e d i n S e c t i o n C below), which i s approximately the c l a s s i c a l t h r e s h o l d f o r the r e a c t i o n . skewing angles 77.4,  The  o f the Mu, H, D, and T s u r f a c e s are 86.9, 80.9,  and 74.9 degrees r e s p e c t i v e l y .  Besides  showing t h a t the  skewing angle approaches 90° as the i s o t o p i c mass  decreases,  F i g u r e 13 shows a pronounced c o n t r a c t i o n o f 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 p o i n t f o r the l i g h t e r i s o t o p e .  Another  f e a t u r e of t h i s s u r f a c e to note i s t h a t the p o t e n t i a l energy b a r r i e r or saddle p o i n t i s " e a r l y " ; t h a t i s , i t i s l o c a t e d along the entrance v a l l e y f o r a l l i s o t o p i c v a r i a t i o n s of the reaction. Although the LEPS s u r f a c e s f o r the r e a c t i o n s of H i s o topes with C l , B r 2  the F  2  reaction  2  and I  are not as w e l l known as those f o r  2  [Jakubetz l - ( 7 8 ) , Connor 2-(77)],  i t i s expect-  ed t h a t t h e i r t o p o l o g i c a l f e a t u r e s should be s i m i l a r t o the F surface  [Bauer l - ( 7 8 ) , P a t t e n g i l l  notable  exceptions:  s t r o n g l y favored  (1) while  f o r the F  2  2  (76), B l a i s (74)] with two  the c o l l i n e a r r e a c t i v e geometry i s  and C l r e a c t i o n s 2  [Jakubetz l - ( 7 8 ) ,  -76Polanyi  (75)], t h i s i s probably not the case f o r the B r  reactions  [Baybutt  the e x i s t e n c e  2  (78), Bauer l - ( 7 8 ) , B l a i s (74)]; and  of a s t a t i c p o t e n t i a l energy b a r r i e r  p o i n t on the p o t e n t i a l energy surface) e s t a b l i s h e d f o r the B r (78), B l a i s (74)].  and  2  I  i s not  reactions  2  1^  and (2),  (saddle  clearly  [White (73), Baybutt  For the sake of argument, i n the  d i s c u s s i o n , i t i s assumed t h a t the LEPS s u r f a c e s  for  following the  halogen homologous s e r i e s are s i m i l a r , w i t h p o t e n t i a l energy b a r r i e r s decreasing has  i n the order  C l > Br > I (the F  an anomalously smaller b a r r i e r than the C l  [Pattengill  (76), A n l a u f  t h a t the Mok-Polanyi  2  2  reaction  reaction  ( 7 2 ) ] ) ; furthermore, i t i s assumed  [Mok  (69)]  c o r r e l a t i o n h o l d s : the poten-  t i a l energy b a r r i e r moves to c o n s e c u t i v e l y  e a r l i e r p o s i t i o n s as  the b a r r i e r height decreases i n a s e r i e s of exothermic  reac-  tions . Within  the assumptions made above, i t seems c l e a r t h a t  mass weighted LEPS s u r f a c e s the halogen s e r i e s should as t h a t shown i n F i g u r e  Figure (76)  14  f o r the H i s o t o p e r e a c t i o n s w i t h  d i s p l a y the same e s s e n t i a l behaviour  13,  t i o n s can be made concerning s u b s t a n t i a l "bottleneck  the  from which a number of the r e a c t i o n dynamics.  e f f e c t , " whimsically  (adapted from Connor 2-(77)),  on the b a s i s of F i g u r e  ed t r a j e c t o r y c a l c u l a t i o n s .  13 before  was  being  generalizaA  illustrated  in  p r e d i c t e d by Manz v e r i f i e d by  detail-  T h i s e f f e c t suggests t h a t  classi-  c a l c o n t r i b u t i o n s to the r e a c t i o n p r o b a b i l i t y favor the heavy H isotopes  f o r which the notion of the r e p r e s e n t a t i v e  through the b o t t l e n e c k  points  d i s p l a y s e s s e n t i a l l y laminar flow i n  c o n t r a s t to the t u r b u l e n t  flow e x h i b i t e d i n the Mu  reaction.  F  +  YF  Y+ F The  FIGURE 14:  2  bottle-neck  Adapted from Connor 2-(77).  effect  -7 8On t h i s b a s i s , i t i s p r e d i c t e d t h a t c l a s s i c a l l y the r e a c t i o n p r o b a b i l i t y w i l l f o l l o w the order  T > D > H > Mu.  Similarily,  the c o n t r a c t i o n o f the product v a l l e y i n the Mu case a greater p r o b a b i l i t y f o r c l a s s i c a l non-reactive  presents  "back-  r e f l e c t i o n " o f the r e p r e s e n t a t i v e p o i n t s o f f the s t r o n g l y r e p u l s i v e w a l l i n the product v a l l e y d i r e c t l y o p p o s i t e the saddle  p o i n t , c o r r e s p o n d i n g t o h i g h energy c o l l i s i o n s ;  the c o n t r a c t e d  product v a l l e y makes i t much more d i f f i c u l t f o r  the r e a c t i o n t o " t u r n the corner" other H i s o t o p e s . verified  that i s ,  i n the Mu case than f o r the  T h i s e f f e c t , which has been t h e o r e t i c a l l y  [Connor 2-(78)],  again  suggests a r e a c t i o n p r o b a b i l i t y  •order o f T > D > H > Mu, s i n c e the c l a s s i c a l Mu r e a c t i o n probability  (or c r o s s section) w i l l f a l l o f f a t lower  energies  i n the high temperature regime than f o r the other H i s o t o p e s . A much l e s s important but c o r r e l a t e d e f f e c t i s t h a t due t o the skewing angle of the p o t e n t i a l s u r f a c e : the s m a l l e r the skewing angle,  the l e s s prone t o b a c k - r e f l e c t i o n i s the r e a c t i o n .  Again, t h i s f a v o r s the order The  existence  T > D > H > Mu.  o f an " e a r l y " p o t e n t i a l energy b a r r i e r has  a number o f i m p l i c a t i o n s .  From the p o i n t o f view o f t h i s  t h e s i s , perhaps the most important i m p l i c a t i o n o f t h i s t o p o l o g i c a l f e a t u r e i s the f a c t t h a t the r e p r e s e n t a t i v e crosses  the p o t e n t i a l b a r r i e r w h i l e i t i s s t i l l  t h a t i s more o r l e s s p a r a l l e l t o the r Part  axis.  point  on a t r a j e c t o r y As shown i n  ( i i ) above, t h i s means t h a t the e f f e c t i v e mass o f the  r e p r e s e n t a t i v e p o i n t , on e i t h e r the unweighted s u r f a c e or on the s u r f a c e  t h a t i s mass weighted a c c o r d i n g  t o equations (22),  is p , which i s e s s e n t i a l l y equal t o the H i s o 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 (23)  scheme of  equation  i n which the e f f e c t i v e mass of the r e p r e s e n t a t i v e p o i n t i s  the same f o r a l l H i s o t o p e s , the width of the energy b a r r i e r i s p r o p o r t i o n a l to  (u  ) Y,x  1 /o  ).  Consequently, the f u l l mass  2  e f f e c t of H i s o t o p i c s u b s t i t u t i o n i s u t i l i z e d penetration  (quantum mechanical t u n n e l l i n g )  in barrier  [Jakubetz  l-(78)].  T h i s would not be the case i f the p o t e n t i a l energy s u r f a c e a symmetrically  p l a c e d or l a t e b a r r i e r .  t i o n s , i t i s expected t h a t the Mu  From these  had  considera-  r e a c t i o n w i l l be s u b j e c t to  much g r e a t e r t u n n e l l i n g than the other H i s o t o p e s , tending  to  order  low  the quantum mechanical r e a c t i o n p r o b a b i l i t i e s i n the  temperature regime Mu s c a l e on lengths  > H > D > T.  Furthermore, shown to  the l e f t of F i g u r e 12 are the thermal de B r o g l i e wave(X -  ,  .  B sentative points u  ,1/2)  of the e f f e c t i v e masses of the  motion, and u 1 (- constant f o r a l l Y ) , corresponding to YF-F motion ; s u b s t i t u t i o n of X = C l , Br, or I f o r F does not a f f e c t the u,Y,X wavelength^ but f u r t h e r c o n t r a c t s the u wavelength. As a l. 1 Y  , corresponding  p  to Y-F  repre-  2  X F  F  2  2  Y X  x  rough r u l e , i f the thermal de B r o g l i e wavelength of a p a r t i c l e I  S t r i c t l y speaking, the r e p r e s e n t a t i v e p o i n t 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 t h a t i t s de B r o g l i e wavelength i s a f u n c t i o n of i t s c o o r d i n a t e s on the p o t e n t i a l energy s u r f a c e . Denoting q^ as a g e n e r a l r e a c t i o n c o o r d i n a t e , the de B r o g l i e wavelength of the r e p r e s e n t a t i v e p o i n t i s g i v e n by X(q ) r  =  ( 2 y  V(q  mass, E. trans energy, and V ( q ) r  1  )]y / ' 1  2  i s i t s effective trans r 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 [ E  w  h  e  r  e  u  i s the h e i g h t of the p o t e n t i a l s u r f a c e at q^  above the asymptotic r e a c t a n t v a l l e y [ N i k i t i n (74)]. Consequently, the thermal de B r o g l i e wavelengths shown i n F i g u r e 12 are minimum thermal averages. T h i s p o i n t does not fundament a l l y a f f e c t the arguments made above.  -80i s shorter  than the width o f a b a r r i e r , t u n n e l l i n g  i s minimal;  c o n v e r s e l y , i f i t s thermal de B r o g l i e wavelength i s longer than the b a r r i e r width, t u n n e l l i n g [Nikitin  (74)].  i s expected t o be important  These c o n s i d e r a t i o n s  suggest t h a t quantum  mechanical r e a c t i o n p r o b a b i l i t i e s may be a c c u r a t e l y applying  some s o r t of one dimensional b a r r i e r  correction  t o c l a s s i c a l or TST c a l c u l a t i o n s  estimated by  penetration  [Jakubetz  (79)]  without the need t o c o n s i d e r such c o m p l i c a t i o n s as a l t e r n a t e t u n n e l l i n g paths ("corner c u t t i n g " ) 2 Johnston ( 6 1 ) ] ) .  (see eg. [Marcus (78),  Another i m p l i c a t i o n o f the e a r l y p o t e n t i a l b a r r i e r concerns the f i n a l s t a t e 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 o f the r e a c t i o n products  ( r o t a t i o n a l energy t r a n s f e r cannot take p l a c e  in collinear collisions).  Although f i n a l s t a t e d i s t r i b u t i o n s  are as y e t e x p e r i m e n t a l l y i n a c c e s s i b l e  to Mu s t u d i e s ,  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 t o warrant a b r i e f sion.  For exothermic r e a c t i o n s ,  e r s are a s s o c i a t e d reaction  with repulsive  exoergicity  i s released  discus-  l a t e p o t e n t i a l energy b a r r i energy r e l e a s e as the r e a c t i n g  e a r l y p o t e n t i a l energy b a r r i e r s are o f t e n energy r e l e a s e ,  they are  i n which the atoms separate;  associated  that is, part of i t i s a t t r a c t i v e  w i t h mixed  ( r e l e a s e d as  the p r o j e c t i l e atom approaches the t a r g e t m o l e c u l e ) , and p a r t o f i t . i s repulsive. F^ and  Although the c o l l i n e a r LEPS s u r f a c e s f o r the  reactions  are known t o be predominantly  repulsive  2 A c t u a l l y the b a r r i e r s t o t u n n e l l i n g c o n s i d e r e d i n most t r a j e c t o r y or TST c a l c u l a t i o n s are. not i d e n t i c a l t o the s t a t i c b a r r i e r s d e s c r i b e d by the p o t e n t i a l energy s u r f a c e . Nevert h e l e s s , the present d i s c u s s i o n i s v a l i d because the l o c a t i o n and shape (but not height) o f 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 d i s c u s s e d i n S e c t i o n C below.  -81-  [Polanyi  (75) , P a t t e n g i l l (76) , W i l k i n s  i f t h i s i s the case f o r the B r Blais  (74), Baybutt  Mok-Polanyi  (78)].  and  2  (75)] , i t i s not c l e a r surfaces  [Polanyi (75),  In any case, a c o r o l l a r y o f the  c o r r e l a t i o n i s expected t o h o l d :  " i n a homologous  s e r i e s 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 i n c r e a s e  i n a t t r a c t i v e energy  r e l e a s e w i l l be accompanied by a decrease i n r e p u l s i v e release  [Mok (69)]."  Roughly  energy  speaking, the a t t r a c t i v e p a r t o f  a mixed energy r e l e a s e i s transformed i n t o v i b r a t i o n a l energy of the p r o d u c t s , w h i l e the r e p u l s i v e p a r t i s transformed i n t o t r a n s l a t i o n a l k i n e t i c energy o f the products Since the skewing surfaces  angles o f the Y +  [Polanyi  (72)].  p o t e n t i a l energy  are approximately the same, i t i s expected t h a t as X  changes from F t o I , the i n c r e a s e  i n a t t r a c t i v e energy  w i l l be accompanied by an i n c r e a s e the products  [Wilkins  (75)] .  release  i n the v i b r a t i o n a l energy o f  On the other hand, as Y v a r i e s  from T t o Mu, the product energy d i s t r i b u t i o n s should d i s p l a y the " l i g h t atom anomaly  [Polanyi  ( 7 5 ) ] : " on r e p u l s i v e  surfaces,  when the mass o f the a t t a c k i n g atom i s much l e s s than those o f the t a r g e t molecule, l e s s r e a c t i o n e x o e r g i c i t y i s c h a n n e l l e d i n t o product v i b r a t i o n a l energy as the mass o f the a t t a c k i n g atom decreases.  T h i s may be p i c t u r e d as an i n e r t i a l e f f e c t i n  which the r a p i d r e l e a s e of the r e a c t i o n e x o e r g i c i t y on the dominant r e p u l s i v e p a r t of the s u r f a c e the s e p a r a t i n g  imparts such momentum t o  heavy atoms (B-C) t h a t the r e l a t i v e l y  insignifi-  cant momentum o f the l i g h t a t t a c k i n g atom (A) i s overwhelmed. On t h e / o t h e r hand, i f A were o f a comparable mass t o the atoms of the t a r g e t molecule, i t would have such i n e r t i a t h a t when  -82the r e p u l s i v e r e a c t i o n e x o e r g i c i t y slammed B i n t o 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 e x c i t e d A-B  molecule.  Since the dynamics of the r e a c t i o n s of H i s o t o p e s the hydrogen h a l i d e s are probably  i n f l u e n c e d more by  d i s p o s i t i o n of energy among i n t e r n a l molecular the topology of H-HX  with  the  modes than by  of the p o t e n t i a l energy s u r f a c e s , the d i s c u s s i o n  LEPS s u r f a c e s i s d e f e r r e d u n t i l the next S e c t i o n where  these energy e f f e c t s are taken i n t o account.  C  Energy To t h i s p o i n t , H i s o t o p e e f f e c t s have been d i s c u s s e d  on  the b a s i s of i n t u i t i v e p r e d i c t i o n s of the behaviour of t r a j e c t o r i e s of a r e p r e s e n t a t i v e p o i n t encountering  character-  i s t i c t o p o l o g i c a l f e a t u r e s of e l e c t r o n i c a l l y a d i a b a t i c p o t e n t i a l energy s u r f a c e s . s u r f a c e and  Besides  the p o t e n t i a l energy  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-  a t i v e p o i n t , r e f e r e n c e has been made to other e n e r g i e s  such as  the r e a c t i o n a c t i v a t i o n energy and  thres-  enthalpy,  classical  h o l d energy, and  i n t e r n a l energy of the t a r g e t and  molecules.  task of t h i s S e c t i o n i s to d e f i n e these forms  of energy and  The  i n t e r p r e t t h e i r r o l e s i n the r e a c t i o n  product  process.  F i n a l l y , some of these ideas are a p p l i e d i n a d i s c u s s i o n of H i s o t o p e - hydrogen h a l i d e r e a c t i o n s . Energy d e f i n i t i o n s used i n c o n j u n c t i o n with p o t e n t i a l energy s u r f a c e s depend upon the choice of an a r b i t r a r y r e f e r e n c e p o i n t of zero energy f o r which, u n f o r t u n a t e l y , i s no s i n g l e convention.  For example, i n F i g u r e 12,  there  a l l of  the contours of the LEPS s u r f a c e are drawn with r e s p e c t  to  -83-  zero d e f i n e d as the d i s s o c i a t i o n  l i m i t o f F^r except f o r the  dashed contour r e p r e s e n t i n g a p o t e n t i a l the  energy e q u i v a l e n t t o  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  t h r e s h o l d , which i s drawn w i t h r e s p e c t t o a d i f f e r e n t zero as d e f i n e d below.  This confusion i s further  aggravated by the  f a c t t h a t v a r i o u s authors o f t e n use the same name t o r e f e r t o d i f f e r e n t energies. reactions of Y + F al.l-(79) c a l l  2  For example, i n a d i s c u s s i o n  of the  + YF + F, Y = Mu, H, D, T, what Connor e t  the " b a r r i e r h e i g h t " i s quoted w i t h v a l u e s o f  both 2.35 and 1.08 kcal/mole, the former r e f e r r i n g t o the h e i g h t o f the saddle p o i n t r e l a t i v e t o the bottom o f the asymptotic r e a c t a n t v a l l e y , w h i l e the l a t t e r r e f e r s t o t h i s value l e s s the zero p o i n t v i b r a t i o n a l energy o f the F^ molecule T h i s 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 g e n e s i s i n the m u l t i t u d e o f approaches t o the c a l c u l a t i o n o f r e a c t i o n f o r example, c l a s s i c a l t r a j e c t o r y  calculations  kinetics  apply a d i f f e r -  ent meaning t o the " b a r r i e r h e i g h t " than q u a s i c l a s s i c a l or quantum mechanical t r a j e c t o r y  calculations.  a need f o r c o n s i d e r a b l e care i n d e f i n i n g  Clearly,  there i s  the v a r i o u s energy  terms.  (i)  Classical The  Trajectories  picture  of a b a l l r o l l i n g along the minimum energy  path o f the p o t e n t i a l classical trajectory  energy s u r f a c e corresponds t o a p u r e l y i n which the 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 e n e r g i e s o f the t a r g e t In t h i s case, i t i s u s e f u l Cl h e i g h t , denoted here as E h  molecule are i n i t i a l l y  t o c o n s i d e r the p o t e n t i a l  zero.  barrier  and d e f i n e d as the e l e v a t i o n  o f the  -84-  saddle p o i n t above the bottom of the  asymptotic r e a c t a n t  valley.  Cl W i t h i n the BO variants  the  t a r g e t molecule a t the  same f o r a l l i s o t o p i c  Since there i s no  internal  onset of c o l l i s i o n i n t h i s  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  r e p r e s e n t a t i v e p o i n t i s the over the p o t e n t i a l the  i s the  of the H atom r e a c t i o n s .  energy i n the picture,  approximation, E^  o n l y energy a v a i l a b l e  the  to p r o p e l  it  b a r r i e r to b r i n g  about r e a c t i o n . Therefore, Cl 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 r e s p e c t to zero taken as the  bottom of  asymptotic r e a c t a n t v a l l e y .  to p i c t u r e  relationship p o i n t and  the  between the potential  It i s useful  k i n e t i c energy of the  the the  representative  energy s u r f a c e i n terms of an  airplane  f l y i n g through the v a l l e y a t a constant a l t i t u d e measured from the  asymptotic minimum of the  picture,  the  h e i g h t of the  corresponds to the and  elevation  w i l l occur.  The  In  plane above the v a l l e y  k i n e t i c energy of the  i t i s c l e a r that  exceed the  reactant v a l l e y .  this  floor  representative point,  i f the  a l t i t u d e of the  airplane  does  not  of the  saddle p o i n t , a n o n - r e a c t i v e c r a s h  u t i l i t y of t h i s p e d a n t i c analogy w i l l become  e v i d e n t i n the l a t e r d i s c u s s i o n 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 . C l o s e l y r e l a t e d to the p o t e n t i a l energy b a r r i e r h e i g h t Cl i s the n o t i o n of a c l a s s i c a l t h r e s h o l d energy, denoted E^, d e f i n e d 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 representative point required for reaction. purely c l a s s i c a l picture  d i s c u s s e d i n the  b a r r i e r h e i g h t , but  the  In terms of  the  p r e v i o u s paragraph,  i t might seem t h a t the threshold energy i s i d e n t i c a l to potential  and  t h i s i s g e n e r a l l y not  the  the case.  -85To understand the d i f f e r e n c e , i t must be noted t h a t although the c l a s s i c a l p i c t u r e under c o n s i d e r a t i o n assumes t h a t ly  the t a r g e t molecule possesses no  initial-  i n t e r n a l energy, i t does  not p r o h i b i t the t r a n s f e r of c o l l i s i o n a l k i n e t i c energy i n t o i n t e r n a l energy of the t a r g e t molecule. discussions  As  i m p l i e d i n the  i n S e c t i o n B on b a c k - r e f l e c t i o n and  the  light  atom anomaly, i n e r t i a w i l l cause the t r a j e c t o r y of the s e n t a t i v e p o i n t to d e v i a t e  from the minimum energy path as i t  attempts to " t u r n the corner" [Nikitin be  (74)].  Not  assumed t h a t the  of the p o t e n t i a l energy  + F2 -> MuF  surface  even i n the case of e a r l y b a r r i e r s can i t  saddle  p o i n t i s c o l l i n e a r w i t h the  minimum energy path, as i l l u s t r a t e d i n F i g u r e Mu  repre-  + F reaction  15  incident  f o r the  (adapted from Connor l - ( 7 7 ) ) .  Consequently, the r e p r e s e n t a t i v e p o i n t g e n e r a l l y attempts to c r o s s the p o t e n t i a l b a r r i e r at a p o i n t other b a r r i e r height.  For  than the minimum  s u r f a c e s w i t h symmetrically  placed,  or  l a t e b a r r i e r s , the r e p r e s e n t a t i v e p o i n t w i l l possess a r e l a t i v e l y l a r g e component of v e l o c i t y p e r p e n d i c u l a r  to  the  minimum energy path as i t attempts to c r o s s the b a r r i e r , corresponding al  to c o n v e r s i o n  of some of the i n i t i a l t r a n s l a t i o n -  k i n e t i c energy i n t o v i b r a t i o n a l energy of the  species.  In s h o r t , t h r e s h o l d e n e r g i e s  t i e s while  energy b a r r i e r s are s t a t i c .  are dynamical From these  t i o n s of the c l a s s i c a l t r a j e c t o r y , i t can be c l a s s i c a l t h r e s h o l d energy must be g r e a t e r p o t e n t i a l energy b a r r i e r h e i g h t . s i c a l t r a j e c t o r i e s , i t may  reacting quanti-  considera-  seen t h a t  the  than or equal to  In the case of q u a s i c l a s -  happen t h a t the r e a c t i o n t h r e s h o l d  Cl energy i s l e s s than  the  , as d i s c u s s e d  later in this  Section.  -86-  1.0  R  MUF/A  30  P  0.2 FIGURE 15:  0.3  x/A  0.4  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 + F r e a c t i o n , adapted from Connor l - ( 7 7 ) ; x i s d e f i n e d i n the t e x t . The dash-dot l i n e i s the minimum energy path. The dashed l i n e s i n d i c a t e cont o u r s where the r e a c t a n t 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 a t ° ^ = 0.08, 0.16, and 0.24 eV. L i n e P i s the " l i n e of no r e t u r n " mentioned i n Chapter IV. 0  E  a  n  s  -87-  For s u r f a c e s p o s s e s s i n g e a r l y b a r r i e r s , i t i s expected  that  t h r e s h o l d e n e r g i e s f o r the v a r i o u s H i s o t o p e s w i l l be s i m i l a r (but not i d e n t i c a l ) t o each other due to the r e l a t i v e l y m i l d d i s t o r t i o n s o f the r e a c t a n t v a l l e y under the t r a n s f o r m a t i o n t o mass weighted c o o r d i n a t e s  (eg. F i g u r e 13).  As d i s c u s s e d i n  the p r e v i o u s S e c t i o n , H i s o t o p e r e a c t i o n s with the halogens Cl d i s p l a y a " b o t t l e n e c k " e f f e c t which tends t o order E  T  : Mu >  H > D > T. I t should be noted  t h a t the commonly used phrase, "the  c l a s s i c a l t h r e s h o l d energy o f the r e a c t i o n , " implying the e x i s t e n c e o f a unique v a l u e , o f t e n r e p r e s e n t s a misuse o f the language.  C e r t a i n l y , i n the case o f p u r e l y  classical  t r a j e c t o r i e s , there i s a unique t h r e s h o l d energy f o r each surface.  However, m u l t i d i m e n s i o n a l  trajectory calculations  must be performed on s e v e r a l s u r f a c e s , each with i t s own t h r e s h o l d energy.  Thus, w h i l e one may speak of the c l a s s i c a l  t h r e s h o l d energy f o r a one dimensional dimensional  trajectory,  three  t r a j e c t o r i e s have a range of t h r e s h o l d e n e r g i e s  over the v a r i o u s ABC bond angles and impact parameters  (that i s ,  the minimum d i s t a n c e between the approach t r a j e c t o r y and the center of mass of the t a r g e t m o l e c u l e ) .  The n o t i o n o f a  s i n g l e r e a c t i o n t h r e s h o l d energy i s even l e s s p r e c i s e 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 , d i s c u s s e d  next.  ( i i ) 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 Although  i t was shown i n the p r e v i o u s S e c t i o n t h a t  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 u s e f u l p i c t u r e of the dynamics o f a r e a c t i o n , i t i s u n r e a l i s t i c t o ignore the  -88initial  i n t e r n a l v i b r a t i o n a l and  t a r g e t molecule.  r o t a t i o n a l energy of  the  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 t a r g e t molecule i s a v a i l a b l e to promote r e a c t i o n the B-C  s t r e t c h i n the r e a c t i o n A + BC  -> AB  + C corresponds to  the r e a c t i o n c o o r d i n a t e along the product v a l l e y . c l a s s i c a l t r a j e c t o r y c a l c u l a t i o n s are b e f o r e any  since  Quasi-  formulated such t h a t  i n t e r a c t i o n of the c o l l i s i o n p a r t n e r s o c c u r s ,  i n t e r n a l s t a t e s of the  t a r g e t molecule are d e s c r i b e d  mechanical p r o b a b i l i t y d e n s i t y functions; but t o r y begins, a l l of the motion i s c l a s s i c a l  by quantum  once the t r a j e c [Thompson  Of course, quantum mechanical t r a j e c t o r y c a l c u l a t i o n s quantum s t a t e d i s t r i b u t i o n s throughout the r e a c t i o n . detailed discussion given i n Section For QCT  and  of the v a r i o u s  the  (76)]. involve A more  types of t r a j e c t o r i e s i s  D. QMT  c a l c u l a t i o n s , i t i s customary to d e f i n e  number of e n e r g i e s r e l a t i v e to zero taken as the h e i g h t of v i b r a t i o n a l energy of the t a r g e t molecule the bottom of the asymptotic r e a c t a n t  (denoted  a  the  above  v a l l e y , thereby assuming  t h a t a l l of t h i s v i b r a t i o n a l energy i s a v a i l a b l e f o r  reaction.  For example, t h i s zero energy i s shown as the dashed contour i n Figure  15  f o r the r e a c t i o n Mu  v = 0 state. height,  h  phys  I t may  2  MuF  Based on t h i s energy zero,  E ^ ^ , i s defined b  + F  s  =  Cl  to  + F with F  2  i n the  the p h y s i c a l b a r r i e r  as  _  (  2  4  )  v  be noted t h a t a l l of these q u a n t i t i e s are  invariant  under i s o t o p i c s u b s t i t u t i o n of the p r o j e c t i l e atom. serves as the boundary t h a t d i f f e r e n t i a t e s dynamical from s t a t i c t u n n e l l i n g as d i s c u s s e d  E^^  S  tunnelling  i n S e c t i o n F below.  Since  -89a l l of the v i b r a t i o n a l energy of the available  to promote r e a c t i o n ,  energy of the  molecule may  translational  kinetic  of an a i r p l a n e  QC i s denoted E. . trans  f l y i n g up  the  In terms of  corresponding to the  dashed contour of F i g u r e 15,  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  p o i n t i s e q u i v a l e n t to the l e v e l , and  a l t i t u d e of the  i f t h i s a l t i t u d e does not  f o r example.  representative  airplane  exceed the  above  the  of Mu  there i s no p h y s i c a l For  QCT,  2  i n the v = 1 s t a t e ,  barrier  to  sea  elevation  b a r r i e r , a n o n - r e a c t i v e c r a s h w i l l occur. Cl with F  new  shoreline  the p h y s i c a l reaction  the  reactant v a l l e y , t h i s  zero energy corresponds to a f l o o d e d v a l l e y with a  The  be  r e p r e s e n t a t i v e p o i n t i s a l s o measured r e l a t i v e  to t h i s zero contour and picture  the  target  E^  of For  > E^  so  reaction.  i t i s a l s o common to d e f i n e a t h r e s h o l d 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. has  While the n o t i o n of a t h r e s h o l d energy  l i t t l e meaning i n terms of QMT  calculations,  the  quasi-  c l a s s i c a l t h r e s h o l d energy i s u s e f u l  for partitioning  r e s u l t s i n t o c l a s s i c a l l y allowed and  p u r e l y quantum mechanical  p r o c e s s e s , as d i s c u s s e d i n S e c t i o n F below. reasons mentioned i n the  b a r r i e r height, E J ^  y S  .  be  available  to promote r e a c t i o n ,  t h r e s h o l d energy may also  l e s s than the  A good d i s c u s s i o n  not  o n l y be  potential of the  same energies,  g r e a t e r than or However, because  amount of the v i b r a t i o n a l energy of the may  the  case of c l a s s i c a l t h r e s h o l d  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 i e s must be to the p h y s i c a l  For  QMT  target  the  any  molecule, E ,  quasiclassical  g r e a t e r than or equal t o , Cl  b a r r i e r h e i g h t , E^  o r i g i n s and  equal  [Nikitin  interpretation  of  but (74)].  quasi-  -90c l a s s i c a l t h r e s h o l d e n e r g i e s i s g i v e n i n P o r t e r e t a l . (73). A major problem i n d e f i n i n g q u a s i c l a s s i c a l t h r e s h o l d s f o r even a s i n g l e s u r f a c e i s the f a c t t h a t the r e a c t i o n p r o b a b i l i t y i s not o n l y energy dependent but i t a l s o depends on the phase of o s c i l l a t i o n of the t a r g e t molecule.  Quasiclassical threshold  d e f i n i t i o n s are obtained by some k i n d of averaging (such as Monte C a r l o averaging) ever, d i f f e r e n t procedures old energies  process  over the o s c i l l a t o r phase; how-  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-  [Connor l-(76)].  As i n the d i s c u s s i o n of c l a s s i c a l  t h r e s h o l d s 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  s u r f a c e s f o r H i s o t o p e - halogen r e a c t i o n s  suggest t h a t the 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 i e s are a l s o ordered Mu  (iii)  > H > D > T.  T r a n s i t i o n S t a t e Theory While  TST c a l c u l a t i o n s may  be based  on the p o t e n t i a l  Cl energy b a r r i e r , E^  [Persky  (77), Jakubetz  common to make the assumption "the r e a c t a n t v i b r a t i o n s  (7.9)], i t i s more  of v i b r a t i o n a l adiabaticity(VA) :  (except f o r the one t h a t becomes the  r e a c t i o n coordinate) evolve smoothly i n t o those of the a c t i v a t e d complex, and f i n a l l y i n t o those of the product, w i t h out any change i n v i b r a t i o n a l quantum numbers Of course, the amount of v i b r a t i o n a l energy because the v i b r a t i o n a l f o r c e constants  [Weston  (72)]."  i s not constant  (or c u r v a t u r e of the  p o t e n t i a l surface) change d u r i n g the progress of the r e a c t i o n . For the r e a c t i o n A + BC -> AB + C, the VA b a r r i e r h e i g h t s are d e f i n e d as E™(A)  [Connor 1-(79), Jakubetz = E^  1  + E+(A)  - E  v  (79)]: (25)  -91where E^(A)  i s the energy of the bound normal mode(s) o f  (ABC)^  Cl and E, and E have been d e f i n e d above. The double dagger b v r e f e r s t o the t r a n s i t i o n s t a t e . The v a l u e s of E^ and hence v VA E  depend on the H i s o t o p i c mass and are o r d e r e d : Mu  > H > D  > T, p r e d i c t i n g an i n v e r s e i s o t o p e e f f e c t f o r a l l i s o t o p i c H atom r e a c t i o n s which o r d e r s the r e a c t i o n r a t e s : T > D > H > Mu ( t h i s i s o f t e n r e f e r r e d to as the "secondary i s o t o p e [Nikitin Y + X  2  (74), Van  (70)]).  Hook  -> YX + X, Y = 'Mu,  H, D,  effect"  In many c a s e s , such as  T; X = F, C l ; the presence of  Y i n the t r a n s i t i o n s t a t e weakens the X-X bond without complete compensation  from the f o r m a t i o n of the Y-X  (79), Connor l-(79)] w i t h  [Jakubetz ±  VA  two r e s u l t s :  C l  bond  (1) E  v  >  E'(Y) and thus E (Y) < E, and (2) due t o v i b r a t i o n a l v v ^ D anharmonicity, h i g h e r energy v i b r a t i o n a l s t a t e s are more 1 X c l o s e l y spaced than lower ones so t h a t E^(Y) VA and thus E  Q  - E  Q  > E|(Y) - E^^  VA (Y) > E^  (Y) . Linear t r a n s i t i o n s t a t e t r i a t o m i c s : have  four normal modes of v i b r a t i o n : two bound bending modes, the bound symmetric  s t r e t c h c o r r e s p o n d i n g to motion along a l i n e  p e r p e n d i c u l a r to the r e a c t i o n path a t the saddle p o i n t , and the unbound asymmetric  s t r e t c h c o r r e s p o n d i n g t o motion a l o n g  the r e a c t i o n path i t s e l f  ( t h i s mode has an imaginary f r e q u e n c y ) .  For the c o l l i n e a r r e a c t i o n , the symmetric  s t r e t c h i s the only  bound normal mode, u n i q u e l y d e f i n i n g E^(A) . In g e n e r a l , the assumption of v i b r a t i o n a l approximately v a l i d a t normal temperatures  adiabaticity i s  [Levine  (74)].  In  the p a r t i c u l a r cases of H i s o t o p e r e a c t i o n s w i t h halogens, the e a r l y b a r r i e r s are expected to f a v o r the VA assumption  because  the t r a n s i t i o n s t a t e corresponds to an only s l i g h t l y p e r t u r b e d  -92-  [Connor l - ( 7 9 ) ,  t a r g e t molecule  Jakubetz  (79)].  q u a s i c l a s s i c a l t r a j e c t o r y c a l c u l a t i o n s do not al  adiaba1d.city (allowing  the v i b r a t i o n a l and  Although  assume v i b r a t i o n -  a continuous energy t r a n s f e r between  t r a n s l a t i o n a l modes i n e i t h e r d i r e c t i o n ) ,  the p r e c e d i n g arguments on the expected v a l i d i t y of VA  provide  the b a s i s  the  f o r making the  " f i r s t guess" p r e d i c t i o n t h a t OC  q u a s i c l a s s i c a l thresholds, v e r y s i m i l a r to the VA  , f o r these r e a c t i o n s w i l l  barrier  be  heights.  Table IV compares the v a l u e s of the energy d e f i n i t i o n s made so f a r f o r the c o l l i n e a r r e a c t i o n Y + F Y = Mu, (72)  H,  based on the LEPS s u r f a c e  due  YF +  2  F,  to Jonathan et a l .  .  (iv) R e a c t i o n Enthalpy Thermodynamic r e a c t i o n e n t h a l p i e s Hess's Law  by  summing the heats of formation of r e a c t a n t s  products under i s o t h e r m a l AH^.  .  The  are c a l c u l a t e d with  standard s t a t e c o n d i t i o n s  at 298K,  r e s u l t s are averaged over Maxwell-Boltzmann  i n t e r n a l energy s t a t e d i s t r i b u t i o n s at 298K and any  and  contributions  vaporization,  due  also  include  to p h y s i c a l s t a t e changes (heats of  solidification, etc.).  From the viewpoint of  c a l c u l a t i o n s of the r a t e s of i s o l a t e d atom-diatom  reactions,  it  as  i s more u s e f u l to c o n s i d e r  reaction enthalpies  d i f f e r e n c e between the bond d i s s o c i a t i o n e n e r g i e s , product and i s defined  reactant by D 2  0  = D  d i s s o c i a t i o n energy the  zero p o i n t  molecules. e  The  the  DQ, of  bond d i s s o c i a t i o n energy  - E~, where D i s the 0 e  equilibrium ^  (depth of the Morse p o t e n t i a l ) , and  energy  (ZPE).  the  Since t h i s d e f i n i t i o n of  EQ is reac-  - 9 3 -  TABLE I V :  ENERGY " DEFINITIONS FOR THE COLLINEAR Y + 1  REACTION  Y = Mu, H, FOR THE LEPS SURFACE OF JONATHAN (72)  Mu C L A S S I C A L  P H Y S I C A L  B A R R I E R  B A R R I E R  H E I G H T ,  H E I G H T ,  E ^  1  E^ yS  H  2 . 3 5  2 . 3 5  1.08  1.08  § Q U A S I C L A S S I C A L  T H R E S H O L D  Z E R O  P O I N T  E N E R G Y  O F  F  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  2  ,  E N E R G Y ,  E  Q  ( F  2  E ^  1.8  C  )  A D I A B A T I C  B A R R I E R  2.06  1.27  1.27  1.20  1.12  2.28  2.20  2.15  1.91  C O M P L E X ,  E J ( Y F F ) V I B R A T I O N A L L Y  0  H E I G H T , , VA V  E' Q 0 , VA V  l  EJ ,  kcal/mole, taken from Connor  (7 9 ) OC  the o r i g i n of the lower v a l u e of E ^ Chapter I V , p. 1 4 8 .  f o r Mu i s e x p l a i n e d i n  -94t i o n enthalpy i s based on ZPE s, i t corresponds t o the Maxwell1  a t O K and i s o f t e n denoted A H Q [Wolfrum  Boltzmann p o p u l a t i o n (77),  Douglas  (76)].  AHQ  i s a l s o independent of p h y s i c a l  s t a t e changes. The  bond d i s s o c i a t i o n e n e r g i e s ,  enthalpies  ZPE's, and r e a c t i o n  of the molecules and r e a c t i o n s  t h e s i s are summarized i n Table v .  studied  i n this  This t a b l e shows t h a t some  of the r e a c t i o n s o f Mu with the hydrogen h a l i d e s are endothermic; some i m p l i c a t i o n s of t h i s are d i s c u s s e d Section.  In g e n e r a l ,  containing  l i g h t e r H isotopes,  below i n t h i s  because of the l a r g e r ZPE of products the e x o t h e r m i c i t y  of H  isotope  r e a c t i o n s based on A H ^ are ordered: T > D > H > Mu.  (v) Reaction A c t i v a t i o n Energy Although a general l e f t t o the next S e c t i o n ,  d i s c u s s i o n of t r a j e c t o r y methods i s i t i s useful at t h i s point to  a n t i c i p a t e one o f the major concepts common t o those methods, i n order t o d e r i v e the Tolman i n t e r p r e t a t i o n o f the a c t i v a t i o n energy  (following  [Levine  (74)]).  A l l trajectory calculations  p r o v i d e v a l u e s of some form o f r e a c t i o n r a t e constant t h a t i s a f u n c t i o n o 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 of the c o l l i d i n g species.  In order t o c a l c u l a t e t h e r m a l l y  averaged  r a t e c o n s t a n t s from these r e s u l t s , i t i s necessary t o compute an i n t e g r a l o f the f o l l o w i n g general k(T-) = where e  E / / k  B  T  form:  °° -E/k_T k(E)dE e  B  (26)  o  i s the Boltzmann weighting f a c t o r and Q i s the  p a r t i t i o n f u n c t i o n which normalizes the r e s u l t .  Partition  TABLE V:  reactant molecule  BOND DISSOCIATION ENERGIES,  D  e 37.59  u 1. 27  57.88  •-  ZERO POINT ENERGIES, AND  product molecule  REACTION  D  E  ENTHAPLIES  0  MuF HF DF TF  141.13 141.13 141.13 141.13  16.61 5.78 4.25 3.56  -88.2 -9 9.0 -100.6 -101.2  0.81  MuCl HCl DC1 TCI  106.43 106.43 106.43 106.43  12.24 4.27 3.07 2.52  -37.1 -45.1 -46.3 -46.8  45.92  0.46  MuBr HBr DBr TBr  90.36 90.36 90.36 90.36  10.92 3.79 2.68 2.20  -34 .0 -41.1 -42.2 -42.8  HCl  106.43  4.27  MuH(MuCl) HH(HCl) DH(DCl) TH(TCI)  109.46(106.43) 109.46(106.43) 109.46(106.43) 109.46(106.43)  13 .53(12. 24) 6 .23 (4. 47) 5 .38 (3. 04) 5 .07 (2. 52)  + 6.2 ( + 7.9) - 1 . 1 ( 0 .0) - 1 . 9 ( - 1 .2) - 2 . 2 ( - 1 .8)  HBr  90.36  3.79  MuH(MuBr) HH(HBr) DH(DBr) TH(TBr)  109.46(90.36) 109.46 (90.36) 109.46(90.36) 109.46(90.36)  13 .53(10. 92) 6 .23 (3. 79) 5 .38 (2. 68) 5 .07 (2. 20)  -9.4(+7 .1) -16.7 (0 .0) - 1 7 . 5 ( - 1 .1) - 1 7 . 8 ( - 1 • 7)  HI  73.66  3.27  MuH(Mul) HH (HI) DH (DI) TH(TI)  109.46(73.66) 109.46(73.66) 109.46(73.66) 109.46(73.66)  13 .53 6 .23 5 .38 5 .07  (9. 52) ( 3 . 27) (2. 33) (1. 91)  -25.5(+6 • 3) -32.8 (0 .0) -33.7(-0 .9) - 3 4 . 0 ( - 1 .4)  F  2  c i  Br  2  2  a l l v a l u e s a r e i n k c a l / m o l e c a l c u l a t e d from s p e c t r o s c o p i c d a t a f r o m G. H e r z b e r g , S p e c t r a o f D i a t o m i c M o l e c u l e s , 2nd e d . , Van N o s t r a n d , P r i n c e t o n , 1950.  -96-  f u n c t i o n s have the g e n e r a l form Q = Ee" i e  where  / k  B  (27)  T  i s the energy of the i t h s t a t e .  On the other hand, —E /k T  the A r r h e n i u s a c t i v a t i o n energy e x p r e s s i o n , k(T) = Ae  a  B ,  i s a measure o f the r a t e o f change o f the r a t e c o e f f i c i e n t as a f u n c t i o n o f i n v e r s e temperature.  Assuming temperature i n -  dependence o f t h e p r e - e x p o n e n t i a l f a c t o r  (which i s , i n f a c t ,  weakly temperature dependent), t h i s may be r e - w r i t t e n : E  =  -k_ d [ l n k(T) ] — ,  (28  S u b s t i t u t i o n o f e q u a t i o n (26) i n t o /bEe- /V  k(E) dE _  E  E  a " r -E/k T e  o  B  k  (  E  )  d  f B k  [  E  (28) y i e l d s :  d  (  d  [  l  l  n  Q  (  T  )  ]  (29)  }  The f i r s t term of t h i s e x p r e s s i o n i s c l e a r l y an average energy and i t i s i n t e r p r e t e d as the average energy o f those c o l l i s i o n s * which r e s u l t i n r e a c t i o n , <E >.  D i f f e r e n t i a t i o n of the second  term of equation (29) f o l l o w e d by s u b s t i t u t i o n o f equation (27) yields: k T B  2  d [Q (T) ] _ l i  e Q  Q(T)  1  Se~ i i e  / k  B  T  which i s j u s t the average energy of the r e a c t a n t s , <E>. equation E  a  (29) i s simply: * = <E > - <E>  Thus,  (30)  t h a t i s , the a c t i v a t i o n energy i s j u s t the d i f f e r e n c e between the average energy o f those c o l l i s i o n s t h a t a c t u a l l y r e s u l t i n r e a c t i o n and the average energy of a l l c o l l i s i o n s .  This  -97-  c o n c e p t u a l l y u s e f u l r e s u l t i s due t o R. C. Tolman (27). The process o f quantum mechanical  t u n n e l l i n g 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 h i g h temperature  regime, the r e l a t i v e c o n t r i b u t i o n of  t u n n e l l i n g t o the r e a c t i o n r a t e i s d i m i n i s h e d from t h a t o f the low temperature  regime s i n c e a h i g h e r f r a c t i o n o f c o l l i s i o n s  are e n e r g e t i c a l l y capable of r e a c t i n g c l a s s i c a l l y . c o n s i d e r a t i o n s p r e d i c t t h a t the t u n n e l l i n g process  These will  m a n i f e s t i t s e l f e x p e r i m e n t a l l y i n terms o f the temperature dependence o f equation  (30): E  w i l l decrease w i t h d e c r e a s i n g a  temperature.  S i m i l a r i l y , i t has a l r e a d y been mentioned t h a t  Mu i s expected  to t u n n e l more e a s i l y than H i n r e a c t i o n s with  halogens  due t o the s m a l l e r e f f e c t i v e mass o f the r e p r e s e n t a -  tive point.  Thus, i n the same temperature  range  one expects t o f i n d experimental v a l u e s o f E  &  ( f i x e d <E>),  t o be reduced f o r  the Mu r e a c t i o n s .  Other dynamical e f f e c t s b e s i d e s t u n n e l l i n g * may c o n t r i b u t e t o <E >; f o r example, a l l o f the c l a s s i c a l dynamical e f f e c t s d i s c u s s e d so f a r i n the H i s o t o p e - halogen * r e a c t i o n s tend t o r a i s e <E > f o r the Mu r e a c t i o n , p o s s i b l y o f f s e t t i n g any t u n n e l l i n g e f f e c t s .  thereby  These c o n s i d e r a -  t i o n s c l e a r l y show t h a t a c t i v a t i o n e n e r g i e s are not j u s t energy  averages  but a l s o dynamical  averages. ^YH  + X  (vi) P o t e n t i a l Energy S u r f a c e s f o r the Reactions Y + HX -> YX + H Y = Mu, H, D, T; X = C l , Br, I. Before c o n s i d e r i n g s p e c i f i c p o t e n t i a l energy  surfaces f o r  the hydrogen - hydrogen h a l i d e (HX) r e a c t i o n s , i t should be noted t h a t t h i s seemingly  simple s u b s t i t u t i o n o f the x „ mole-  -98c u l e by an HX molecule g r e a t l y complicates a l and  both the experiment-  t h e o r e t i c a l s t u d i e s of t h i s s e r i e s of r e a c t i o n s .  imony to t h i s i s the v a s t amount of c o n f l i c t i n g published  debate on the H + HC1 Weston (79)]. The  -> YX  r e a c t i o n may  Y + HX  hydrogen a b s t r a c t i o n (Y + HX  systems have two  (Y + HX  -> YH  experimental  r e a c t i o n channels:  + X) and  hydrogen exchange  p o t e n t i a l energy  surfaces.  t h i s means t h a t r a t e data f o r the i n d i v i d u a l  r e a c t i o n channels must probably be obtained product formation  v i a measurements of  r a t h e r than r e a c t a n t d e p l e t i o n .  method i s of the l a t t e r v a r i e t y , i t has  Since  the  so f a r only been  p o s s i b l e to measure the totaK.Mu r e a c t i o n r a t e s c  and,  an  be found i n [Bauer 2-(78),  + H) , each with i t s own  Experimentally,  MSR  literature  i n the p a s t twenty years on these r e a c t i o n s ; as  example, a good summary of the t h e o r e t i c a l and  Test-  (k , abs  + k  exc  )  i n f a c t , o n l y the room temperature r e a c t i o n r a t e s have  been measured to date. determine the Arrhenius channels of Mu  by  i n p r i n c i p l e , i t may  be p o s s i b l e to  parameters f o r the i n d i v i d u a l r e a c t i o n  simply  measuring the temperature dependence  of the t o t a l r e a c t i o n r a t e s i n the u s u a l way. exchange and  I f both the  a b s t r a c t i o n reactions d i s p l a y Arrhenius  Mu  straight-  l i n e behaviour over a wide temperature range (which, i n view of the r e s u l t s d i s c u s s e d  i n Chapter IV, might not be the  case,  and,  i n f a c t i t i s not c l e a r t h a t even H d i s p l a y s t h i s behav-  iour  [Bauer 2-(78), Clyne  g i e s f o r the two  ( 6 6 ) ] ) , and  r e a c t i o n channels are s u b s t a n t i a l l y d i f f e r e n t  ( t h i s i s probably t r u e ; see eg. Arrhenius  i f the a c t i v a t i o n ener-  [Bauer 2 - ( 7 8 ) ] ) ,  then  the  p l o t f o r the t o t a l r e a c t i o n c o u l d show a break with  the high a c t i v a t i o n energy r e a c t i o n d e s c r i b e d  by the  high  -99-  temperature p a r t of the p l o t and the low a c t i v a t i o n energy r e a c t i o n d e s c r i b e d by the low energy p a r t . Reactions of HF are not c o n s i d e r e d .in t h i s because i t i s expected  thesis  t h a t the r e a c t i o n r a t e s of Mu with  are so slow t h a t they are immeasurable  by the MSR  HF  technique.  Accurate e x p e r i m e n t a l l y optimized LEPS s u r f a c e s do  not  e x i s t f o r e i t h e r the exchange or a b s t r a c t i o n r e a c t i o n s of H i s o t o p e s w i t h the hydrogen h a l i d e s because many of the experimental (75)]."  Two  r e s u l t s are " e q u i v o c a l or c o n t r a d i c t o r y [McDonald t o p o l o g i c a l l y d i f f e r e n t s u r f a c e s have been  r e c e n t l y c o n s i d e r e d f o r the a b s t r a c t i o n r e a c t i o n s (Figure 16 shows the H + HCl -> H  2  + C l examples) : (1) the simple LEPS  s u r f a c e shown a t the top of the F i g u r e  (adapted  (78)], optimized f o r the r e v e r s e r e a c t i o n : H  2  from  + C l -»- HCl + H)  which has an e a r l y b a r r i e r to H a b s t r a c t i o n and e s s e n t i a l f e a t u r e s as the H i s o t o p e - halogen 12), and  (2) the valence bond m o d i f i e d LEP  the same  s u r f a c e s (Figure  s u r f a c e mentioned  i n S e c t i o n B, shown a t the bottom of the F i g u r e [Porter  [Persky  (adapted  from  (73)]; a l s o optimized f o r the r e v e r s e r e a c t i o n ) which,  b e s i d e s having an e a r l y b a r r i e r to H a b s t r a c t i o n , shows shallow p o t e n t i a l w e l l s i n both the r e a c t a n t and product Although Persky  three dimensional QCT  c a l c u l a t i o n s performed on  s u r f a c e are i n very good agreement with  r e s u l t s of C l + H  2  versus C l + D  2  2  the  experimental  i s o t o p e e f f e c t s and  a b s o l u t e r a t e constant f o r the C l + H and,  valleys.  the  r e a c t i o n [Persky  (78)] ,  although a t r u l y a c c u r a t e s u r f a c e w i l l p r o v i d e the b a s i s  f o r a c c u r a t e d e s c r i p t i o n s of a chemical r e a c t i o n i n both d i r e c t i o n s , s u r f a c e s t h a t have been optimized w i t h r e s p e c t to a  -100thermal (300K) de B r o g l i e wavelengths  R,/  FIGURE 16:  «.u.  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„ + C l r e a c t i o n , adapted from Persky (78) (top) and P o r t e r (73) (bottom). Note t h a t the Persky s u r f a c e i s drawn r e v e r s e d from the other contour p l o t s shown i n t h i s t h e s i s ; i t has i t s r e a c t a n t v a l l e y at the top and i t s product v a l l e y a t the r i g h t . The Persky s u r f a c e 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 B r o g l i e wavelengths of the r e p r e s e n t a t i v e p o i n t s on t r a j e c t o r i e s p a r a l l e l t o the axes are shown a t the l e f t of each s u r f a c e .  -101-  r e a c t i o n i n one d i r e c t i o n must be viewed w i t h c a u t i o n when applied to the reverse r e a c t i o n  [Heidner  (76)].  In f a c t , a  g e n e r a l f a i l i n g o f LEPS s u r f a c e s i s " t h a t s e m i - e m p i r i c a l s u r f a c e parameters  o b t a i n e d by c a l i b r a t i o n on one r e a c t i o n a r e  o f t e n n o t t r a n s f e r r a b l e t o another r e a c t i o n i n v o l v i n g the same atoms [White  (73)];"  ( i n h i s paper, White d i s c u s s e s some  successful exceptions t o t h i s g e n e r a l i z a t i o n ) . sometimes appears  T h i s problem  as a g e n e r a l c o n s t r a i n t i n the c o n s i d e r a t i o n  of H i s o t o p e exchange versus a b s t r a t i o n r e a c t i o n s with the hydrogen h a l i d e s , although many authors simply t r e a t r e a c t i o n channel independently o f the other  each  (eg. [ K l e i n  (78)]).  P o r t e r e t al_. p r e d i c t t h a t the a b s t r a c t i o n r e a c t i o n s u r f a c e s f o r the l i n e a r c o n f i g u r a t i o n possess w e l l s c o r r e sponding t o weakly s t a b l e H — X and H—HX, X = C l , Br, I , with 2  depths ranging from 1-4 kcal/mole.  Furthermore,  they have  found t h a t some o f these minima are s u f f i c i e n t l y deep t o accomodate one o r more v i b r a t i o n a l l e v e l s f o r the H, D, o r T i s o t o p i c v a r i a t i o n s o f the complexes and t h a t these complexes should be s t a b l e enough t o permit i s o l a t i o n a t low temperatures. However, there i s not y e t any experimental evidence to support these p r e d i c t i o n s .  On the other hand, experimental  data e x i s t s t o suggest t h a t HX [Noble  2  s p e c i e s have been i s o l a t e d  (68), Bondy&ey (71) , Noble  t h a t by u s i n g t h e i r parameters c o n s t r u c t H-X  2  available  (72)] and P o r t e r e t a l . found  f o r the H ~X s u r f a c e s , they can 2  s u r f a c e s t h a t q u a l i t a t i v e l y agree w i t h these  experimental o b s e r v a t i o n s .  From t h i s f a c t , the c r e d i b i l i t y  of the presence o f w e l l s i n the H ~X s u r f a c e s might be i n f e r r e d . 2  However, a number o f counter-arguments  on t h e q u e s t i o n o f t h e  -102e x i s t e n c e of s t a b l e H-X2  s p e c i e s are g i v e n i n Bauer e t a l .  2-(78) and r e f e r e n c e s t h e r e i n . I f the e x i s t e n c e of p o t e n t i a l w e l l s l i k e those of the P o r t e r s u r f a c e i s assumed, i t i s i n t e r e s t i n g to s p e c u l a t e what e f f e c t they would have on the r e a c t i o n s .  It is  on  expected  t h a t the Mok-Polanyi c o r r e l a t i o n holds f o r t h i s s e r i e s of reactions  [Porter ( 7 3 ) ] : as X changes from C l to I, the b a r r i e r  h e i g h t decreases  and moves to p r o g r e s s i v e l y e a r l i e r p o s i t i o n s .  U n f o r t u n a t e l y , the b a r r i e r h e i g h t s f o r these r e a c t i o n s are w e l l known, but they appear to be about twice as h i g h as f o r the corresponding 5 kcal/mole [Klein  Y +  f o r H + HC1  (78), Persky  those  r e a c t i o n s and range from about  to about 0.5  (77), White  q u e n t l y , f o r the H + HC1  not  kcal/mole  f o r H + HI  (73), Bauer 2-(78)].  and H + HBr  Conse-  reactions at l e a s t ,  reac-  t i v e c o l l i s i o n s r e q u i r e such energy t h a t i t i s u n l i k e l y t h a t e i t h e r the r e a c t a n t s or products w i l l be trapped or even much a f f e c t e d by the w e l l s , u n l e s s an extremely t r a n s f e r mechanism e x i s t s . l o n g - l i v e d complexes may  For the H + HI  e f f i c i e n t energy system, r e l a t i v e l y  e x i s t even f o r r e a c t i v e c o l l i s i o n s .  In t h a t case, the r e a c t i o n would no longer be  "direct"  [Levine  (74)] with a c o l l i s i o n time s h o r t e r than one v i b r a t i o n a l p e r i o d -13 (^10  s ) , but would be  complicated  "compound" or "complex", with  trajectories.  Similarily,  very  low temperature  (low  r e l a t i v e v e l o c i t y ) n o n - r e a c t i v e c o l l i s i o n s of H with HX, C l , Br, I, may  be expected  than d i r e c t .  T h i s has  measurements of Mu MSR  method.  to be of a compound nature,  X =  rather  i n t e r e s t i n g i m p l i c a t i o n s f o r the  r e a c t i o n r a t e s with HX molecules  with  the  D i r e c t n o n - r e a c t i v e c o l l i s i o n s of Mu with t a r g e t  -103molecules are not expected to cause much d e p o l a r i z a t i o n of the -14 muon because the i n t e r a c t i o n times a r e s h o r t , t y p i c a l l y <10 s, compared t o the h y p e r f i n e  f a c t t h a t a l o n g - l i v e d Mu s i g n a l i s observed i n N mental evidence of t h i s .  s " S the  frequency of Mu, <10 2  i s experi-  On the other hand, compound non-  r e a c t i v e c o l l i s i o n s may be s u f f i c i e n t l y i n t i m a t e  t h a t the muon  w i l l be e f f i c i e n t l y d e p o l a r i z e d by the q u a s i - s t a b l e muonic r a d i c a l formed i n the c o l l i s i o n . one  From these  considerations,  might expect Mu r e a c t i o n s t o d i s p l a y apparent  Arrhenius  inverse  behaviour a t low temperatures where the r e l a t i v e  numbers o f t r u l y r e a c t i v e c o l l i s i o n s a r e few; as the temperat u r e i s lowered, the e f f i c i e n c y o f n o n - r e a c t i v e  depolarization  of the muon i n c r e a s e s , thereby i n c r e a s i n g the apparent Mu reaction rate.  In t h i s way, the MSR method may present an  experimental means of t e s t i n g the e x i s t e n c e i n these r e a c t i o n s u r f a c e s . subject  of p o t e n t i a l wells  One f i n a l c o n s i d e r a t i o n on t h i s  i s the l a r g e ZPE o f Mu-containing molecular bonds:  s i n c e the w e l l s are r e l a t i v e l y shallow, they may capable o f supporting  not be  any bound v i b r a t i o n a l s t a t e s of the muon-  i c complex m o l e c u l e s , i n which case the complexes would not be long-lived. i s required  A d e t a i l e d c o n s i d e r a t i o n o f the p o t e n t i a l w e l l s to c l a r i f y t h i s  question.  Besides having w e l l s , the P o r t e r  surfaces d i f f e r  from  Persky's s u r f a c e by the f a c t t h a t t h e i r v a l l e y s possess bottlenecks and  near the saddle  H-HI s u r f a c e s  [Porter  p o i n t , p a r t i c u l a r l y i n the H-HBr  (73)].  Other than these d i f f e r -  ences, the main t o p o l o g i c a l f e a t u r e s o f the two types o f surfaces  are s i m i l a r .  The r e a c t i o n dynamics f o r a b s t r a c t i o n  -104are dominated by the c o l l i n e a r r e a c t i o n f o r both [Klein  surfaces  (78), Thompson (75)], mainly because the b a r r i e r  i n c r e a s e s by a f a c t o r of about s i x as the H-H-Y changes from 180° t o 90°.  bond angle  Although these s u r f a c e s  possess  e a r l y b a r r i e r s , they are not as e a r l y as the corresponding Y-X^ surface b a r r i e r s , so t h a t the saddle d i s p l a c e d from the l i n e along The  p o i n t s are more  the r e a c t a n t approach v a l l e y .  r e l a t i v e " l a t e n e s s " of the b a r r i e r s suggest t h a t the  Cl E^ , w i l l be s u b s t a n t i a l l y g r e a t e r than Cl the p o t e n t i a l b a r r i e r h e i g h t s , E^ , and t h a t the q u a s i c l a s OC  c l a s s i c a l thresholds,  s i c a l t h r e s h o l d s , E ^ , w i l l be s u b s t a n t i a l l y g r e a t e r physical b a r r i e r heights,  EJ^  V S  .  than the  Furthermore, the a c t i v a t e d  complexes of Y-HX are expected to have r e l a t i v e l y strong and  H-X bonds, u n l i k e the Y-X  j u s t s l i g h t l y perturbed  X  2  2  Y-H  a c t i v a t e d complexes which are  molecules.  T h i s e f f e c t , combined  w i t h the r e l a t i v e l y l a r g e ZPE's o f Y-H m o l e c u l a r bonds, suggest t h a t E^(Y) > E v  3  ., so t h a t : ( l ) E ( Y ) V A  a* v  v  ^  > E?  1  and (2) E^ (Y)  b  A  0  VA < E, (Y), o p p o s i t e  t o the Y-X„ case.  p l u s the r e l a t i v e h e i g h t s  These  considerations,  o f the p o t e n t i a l b a r r i e r s , suggest  t h a t the a b s t r a c t i o n r e a c t i o n s , Y + HX, w i l l be slower than the corresponding Y + X^ r e a c t i o n s . vibrational  Although the assumption o f  adiabaticity i s not l i k e l y t o be as v a l i d f o r the  Y + HX a b s t r a c t i o n r e a c t i o n s as f o r the Y-X  2  r e a c t i o n s because  of t h e i r b a r r i e r l o c a t i o n s , i f the VA b a r r i e r h e i g h t s  are taken  as " f i r s t guesses" o f the q u a s i c l a s s i c a l t h r e s h o l d s ,  then i t i s  expected t h a t the i n v e r s e i s o t o p e e f f e c t t h a t orders  the r e a c -  t i o n r a t e s T > D > H > Mu w i l l be much more severe f o r the Y + HX a b s t r a c t i o n r e a c t i o n s .  The VA b a r r i e r s a l s o suggest  -105-  t h a t v i b r a t i o n a l e x c i t a t i o n of the t a r g e t molecules w i l l be l e s s e f f e c t i v e i n promoting the a b s t r a c t i o n r e a c t i o n s than i n the halogen molecule r e a c t i o n s ; t h i s p r e d i c t i o n has been experimentally  confirmed  [Wolfrum ( 7 7 ) ,  Arnoldi  (76)].  It  may a l s o be expected t h a t the r e l a t i v e l y l a r g e v i b r a t i o n a l nonadiabaticity of the a b s t r a c t i o n r e a c t i o n s causes VA- TST. t o overestimate the e f f e c t i v e b a r r i e r h e i g h t , predicted r e a c t i o n rates erroneously The  displacement o f the saddle  thereby making i t s  small. p o i n t s from the r e a c t a n t  approach v a l l e y s a l s o has a number o f e f f e c t s on the t u n n e l l i n g process.  I t i s l e s s l i k e l y t h a t a s i n g l e one dimensional  b a r r i e r penetration  c o r r e c t i o n a p p l i e d t o QCT or TST c a l c u l a -  t i o n s o f Y + HX a b s t r a c t i o n w i l l provide imation  an accurate  o f the QMT r e s u l t s as i n the case o f Y + X^  because: the e f f e c t i v e mass of the r e p r e s e n t a t i v e during  the t u n n e l l i n g process;  there  approxreactions  p o i n t changes  i s no obvious s i n g l e  t u n n e l l i n g path or b a r r i e r due t o the i n e r t i a l e f f e c t s t h a t cause the r e p r e s e n t a t i v e  point to deviate  r e a c t i o n path; and "corner to be important  c u t t i n g " t u n n e l l i n g paths are l i k e l y  [Marcus ( 7 7 ) ,  i s r e a d i l y appreciated  Johnston  and  (61)].  The l a s t  45.8°,  system f o r the l i n e a r Y-HCl  36.4°,  and 3 1 . 4 ° f o r Y = Mu,  T r e s p e c t i v e l y ; skewing angles f o r Y-HBr and Y-HI  are q u i t e s i m i l a r . path lengths  point  when i t i s noted t h a t the skewing angles  of the mass weighted c o o r d i n a t e s u r f a c e s are 7 1 . 6 ° ,  from the minimum  Consequently, corner  D,  H,  surfaces  cutting tunnelling  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 t u n n e l l i n g advantage Mu enjoys due t o the e f f e c t i v e mass of the r e p r e s e n t a t i v e  point, y  H y  (equations  (22)).  Figure  16  -106shows the  thermal de B r o g l i e wavelengths of the  ative points  f o r the  represent-  i s o t o p i c v a r i a n t s o f H c o r r e s p o n d i n g to  motion p a r a l l e l to the unweighted s u r f a c e u l a r mass combinations f o r Y + HX  axes.  The  partic-  a b s t r a c t i o n do not  contract  the product path de B r o g l i e wavelengths as much as i n the of Y + X  reactions  2  (c.f. Figure  12).  Since the  case  represent-  a t i v e p o i n t c r o s s e s the p o t e n t i a l b a r r i e r on a t r a j e c t o r y t h a t is  between the  asymptotic r e a c t a n t  Mu  holds a s m a l l e r  than i n the Y + X  2  and  product t r a j e c t o r i e s ,  t u n n e l l i n g advantage i n these reactions.  reactions  Thus, w h i l e t u n n e l l i n g  still  o r d e r s the r e a c t i o n r a t e s Mu  > H > D > T, t u n n e l l i n g i s not  expected to g r e a t l y f a v o r Mu  over the other H i s o t o p e s  reactions. reaction  A H Q e n d o t h e r m i c i t y of the Mu  The  (Table  5) due  to the  ZPE  of Mu-H  + HC1  i n these  abstraction  also r e s t r i c t s  Mu  t u n n e l l i n g by r e n d e r i n g a s u b s t a n t i a l p a r t of the b a r r i e r i n a c c e s s i b l e as a t u n n e l l i n g path. It  has  already  holds f o r the Y-HX  been s t a t e d t h a t the Mok-Polanyi r e l a t i o n  abstraction reactions  and  thus i t i s expect-  ed t h a t as X changes from C l to I, more r e a c t i o n energy i s transferred  i n t o 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 s t r o n g l y on these  reactions  s i n c e the r e a c t i o n e x o e r g i c i t y slams an H atom i n t o 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 e x c i t e d Thus, i t i s expected t h a t a much g r e a t e r  f r a c t i o n of  the  r e a c t i o n energy appears as product v i b r a t i o n i n the abstraction reactions atom anomaly s t i l l  than i n the Y-X  2  reactions.  product.  Y-HX The  p r e d i c t s t h a t the products of the Mu  a b s t r a c t i o n w i l l have l e s s r e a c t i o n energy c h a n n e l l e d  light +  into  HX  -107v i b r a t i o n than i s the case i n the H + HX The  p o t e n t i a l energy s u r f a c e s  reactions.  f o r the Y + HX  hydrogen  exchange r e a c t i o n s are much more p o o r l y known than those f o r the hydrogen a b s t r a c t i o n r e a c t i o n s . exchange r e a c t i o n , f o r example, those w i t h symmetrically b a r r i e r s ) of 5-9 15-25 The  kcal/mole  HCl  proposed s u r f a c e s range from  placed p o t e n t i a l wells  ( i n s t e a d of  kcal/mole, to those w i t h p o t e n t i a l b a r r i e r s of ([Bauer 2-(78)] and  LEPS f o r m u l a t i o n  has  i s of l i t t l e use  the r e f e r e n c e s  been d e c l a r e d  model these p o t e n t i a l s u r f a c e s it  For the H +  [Valencich  to c o n s i d e r  exchange r e a c t i o n s u r f a c e s .  "too  any  therein).  inflexible"  (77)].  Consequently,  s p e c i f i c examples of  Nonetheless, i t i s p o s s i b l e  comment on some of the gross t o p o l o g i c a l f e a t u r e s of surfaces  f o r these r e a c t i o n s .  t h e o r e t i c a l and t h a t Y + HX  The  the  experimental papers on these r e a c t i o n s agree  exchange r e a c t i o n s possess p o t e n t i a l b a r r i e r s , configuration  (78), Bauer 2-(78), Endo (76), Botschwina  (77), Wolfrum  (77), V a l e n c i c h  (77)], and  (77), Dunning  s e v e r a l of these  authors b e l i e v e t h a t the exchange b a r r i e r s exceed the sponding a b s t r a c t i o n b a r r i e r s [Bauer 2-(78), Endo Botschwina  (77), Dunning  (77), Wolfrum  (77)].  corre-  (76),  Unlike  a b s t r a c t i o n r e a c t i o n s u r f a c e s , the exchange r e a c t i o n do not seem to be very Thompson (75)], and, surface surface.  to  bulk of the post-1970  r a t h e r than w e l l s , i n the symmetrical Y-X-H [Klein  to  s e n s i t i v e to the bond angle  i n f a c t , K l e i n and Veltman's  surfaces  [Klein (78)  s l i g h t l y f a v o r s a bond angle of 90° over the I t has  the  (78),  LEPS  collinear  a l s o been suggested t h a t the exchange r e a c -  t i o n s u r f a c e s possess p o t e n t i a l w e l l s i n the product and  reac-  -108t a n t v a l l e y s s i m i l a r to those proposed f o r the reaction surfaces All  abstraction  [Thompson.'(75) ] .  c o l l i n e a r exchange r e a c t i o n 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 and  saddle  p o i n t at 45°  to each a x i s .  origin  I f i t i s assumed t h a t  the exchange r e a c t i o n s u r f a c e s possess b a r r i e r s r a t h e r  than  w e l l s , then the saddle  p o i n t corresponds to a complex w i t h  e q u a l l y strong Y-X  X-H  t h r e s h o l d and  and  bonds and  a l l of the energy  b a r r i e r r e l a t i o n s predicted  i n the  abstraction  r e a c t i o n s w i l l a l s o h o l d f o r the exchange r e a c t i o n s , except t h a t the  i n e q u a l i t y r e l a t i o n s may  s u r f a c e s w i t h symmetrically prone to "corner Johnston  be even stronger.  p l a c e d b a r r i e r s are expected to  c u t t i n g " t u n n e l l i n g paths  (61)], the strong e f f e c t due  [Marcus  the exchange s u r f a c e s where the  diminished  The  f o r Mu  of the r e a c t i o n s  (78),  i s absent i n  skewing angles f o r Y + C1H,  example, a r e : 89.5°, 88.4°, 87.8°, and  be  to the sharp skewing  angles f o r the mass weighted a b s t r a c t i o n s u r f a c e s  respectively.  Although  87.3° f o r Mu,  for  H, D and  importance of t u n n e l l i n g i s , however,  + XH  exchange because of the  endothermicity  (see Table 5) which r e s t r i c t s t u n n e l l i n g to  the top p a r t of the b a r r i e r s . Without knowledge of the p o t e n t i a l energy b a r r i e r s , i t i s impossible  to p r e d i c t which.reaction  hydrogen,atom a b s t r a c t i o n or exchange.  channel -is f a s t e r : Most experimental  evidence suggests t h a t a b s t r a c t i o n i s f a s t e r than exchange at o r d i n a r y temperatures, but t h a t the r e v e r s e temperatures  i s true a t  (>2000K) [Endo (76), Bauer 2-(78)];  high  these r e s u l t s  have been i n t e r p r e t e d as evidence t h a t hydrogen exchange  has  T  -109an u n u s u a l l y  small s t e r i c f a c t o r  exchange has a much higher [Endo (76), Bauer 2-(78)].  [Thompson (75)]  or t h a t  a c t i v a t i o n energy than a b s t r a c t i o n A dynamical argument has been  proposed t o e x p l a i n these experimental o b s e r v a t i o n s 2-(78), K l e i n (78)].  [Bauer  A t room temperature, the most populated  r o t a t i o n a l s t a t e s o f hydrogen h a l i d e molecules are 2 or 3 12 corresponding t o r o t a t i o n a l f r e q u e n c i e s Since  the HX center  nucleus,  o f about 2 x 10  t o a slowly approaching atom, the HX molecule  a b s t r a c t i o n r e a c t i o n c o n f i g u r a t i o n : Y-H-X. v e l o c i t y o f the c o l l i s i o n p a r t n e r s  Y-X-H.  As the r e l a t i v e  i n c r e a s i n g the  f o r the exchange r e a c t i o n c o n f i g u r a t i o n t o occur:  T h i s e f f e c t has not been p r e d i c t e d  a l QCT c a l c u l a t i o n s [Thompson (75), White little  i n the  increases, t h i s r o t a t i o n a l  o f the halogen atom d i m i n i s h e s ,  opportunity  looks  the l a r g e r X  Consequently, the c o l l i s i o n takes p l a c e  screening  s  of mass i s almost c o i n c i d e n t with the X  l i k e a sphere with an H atom " c r u s t " c o v e r i n g atom.  -1  i n three  dimension-  (73)] which show  s e n s i t i v i t y t o the t a r g e t molecule r o t a t i o n a l s t a t e s ;  statistical  phase space c a 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 w i t h t h i s e f f e c t [Truhlar  (69)].  If i t i s  assumed t h a t t h i s e f f e c t i s important, then i t i s expected t h a t the branching r a t i o s f o r a b s t r a c t i o n t o exchange are smaller  f o r Mu r e a c t i o n s than f o r other H isotope  reactions at  the same temperatures, s i n c e the mean v e l o c i t y o f the l i g h t e r Mu atom i s three times t h a t o f H. D Trajectory Calculations The  f i r s t o b j e c t i v e of a l l t r a j e c t o r y c a l c u l a t i o n s , i s  -110to determine the s t a t e - t o - s t a t e  r e a c t i o n p r o b a b i l i t e s as a  f u n c t i o n o 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 or v e l o c i t y o f the c o l l i d i n g s p e c i e s . denoted P . (E, ) , where s s' «-s t r a n s s t a t e s and s r e f e r s t o r e a c t a n t and  1  r e f e r s t o product molecule ^ molecule s t a t e s .  type o f quantum s t a t e s i n c l u d e d  dimensionality  These p r o b a b i l i t i e s are  i n s and s  1  depends on the  o f the c a l c u l a t i o n s and the l e v e l o f approxima-  t i o n t o which t h e c a l c u l a t i o n s are c a r r i e d out. i n Section  The number  As mentioned  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 a t t a c k i n g  atom approaches the center of  mass o f the t a r g e t molecule end on.  F o r such a c o l l i s i o n , the  r o t a t i o n a l s t a t e s of the t a r g e t 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 t r a n s f e r r e d the impact parameter, b = 0.  since  Furthermore, i t has already  been  noted t h a t the bending v i b r a t i o n s o f the a c t i v a t e d complex are a l s o ignored i n c o l l i n e a r c o l l i s i o n s .  These  considerations,  p l u s the f a c t t h a t t r a j e c t o r i e s need only be performed on the one  p o t e n t i a l energy s u r f a c e  180°,  corresponding t o a bond angle of  indicates that c o l l i n e a r state-to-state reaction  b i l i t i e s may be determined w i t h r e l a t i v e ease. al  (coplanar) t r a j e c t o r y c a l c u l a t i o n s i n c l u d e  case as w e l l as t r a j e c t o r i e s on a l l s u r f a c e s bond angles ranging from 0° t o 180°. and  v i b r a t i o n s are i n c l u d e d  a c t i v a t e d complex molecules. are not constrained may occur. included  to zero,  proba-  Two dimensionthe c o l l i n e a r  corresponding t o  A l l in-plane  rotations  i n the t a r g e t , product, and Because the impact parameters o r b i t a l angular momentum t r a n s f e r  A l l i n t e r n a l s t a t e s and impact parameters are  i n three dimensional c a l c u l a t i o n s , thereby r e q u i r i n g  -111the sampling o f a continuum of p o s s i b l e t r a j e c t o r i e s . S t a t e - t o - s t a t e r e a c t i o n p r o b a b i l i t i e s are g i v e n by [Persky  (77) ] : N ,  (E. ) trans  R  P  (E ) = S'*B trans  5  5  (  }  s where N ,  trans  i s the number of r e a c t i v e s t a t e - t o - s t a t e t r a j e c -  t o r i e s a t a g i v e n energy and N  i s the t o t a l number o f  g  t r a j e c t o r i e s c a l c u l a t e d a t t h a t energy f o r i n i t i a l For three dimensional to define a t o t a l  s t a t e , s.  t r a j e c t o r y c a l c u l a t i o n s , i t i s common  r e a c t i o n c r o s s s e c t i o n [Persky  (77), White  (73)] : a  ,  (E  s -<-s  where b  trans  ) = TTb  2  Max  (E^  trans  )P  ,  s'-<-s  (E  )  (31)  i s the l a r g e s t impact parameter t h a t y i e l d s an  a p p r e c i a b l e r e a c t i o n p r o b a b i l i t y ; i n order c a l c u l a t i o n s with experimental it  trans  to compare the  atomic and molecular  i s u s e f u l to d e f i n e a d i f f e r e n t i a l  cross section  beam  data,  [Persky  (77)] : d a  s^s  ( E  trans  dfi  )  u b =  Max s^s N  ( E  tran '  9 )  (32)  S  2TrsineN (E s  t r a n s  )A9  where N , (E^_ ,6) i s the number of r e a c t i v e c o l l i s i o n s s'<-s trans  with  s c a t t e r i n g angle between 9 and 9 + A9, and dfi i s an increment of s o l i d angle. s e c t i o n s have the  Of course, three dimensional  u n i t s o f area and d i f f e r e n t i a l  s e c t i o n s have the u n i t s o f a r e a / s o l i d angle. so-called tions  (76)] analogous t o equations  cross  Occasionally,  "cross s e c t i o n s " are d e f i n e d f o r coplanar  [Baer 0  t r a j e c t o r y cross  calcula-  (31) and (32):  , (E ) = 2b • (E. )P , (E. ) s'-<-s t r a n s Max trans s'«-s t r a n s  (33)  -112and da  , (E. .), 2b N , (E ,9) s -«-s t r a n s _ Max s'^s t r a n s dfi 2N (E.)A9 R  s  ^4)  trans  These " c r o s s s e c t i o n s " have u n i t s of l e n g t h and l e n g t h / u n i t angle r e s p e c t i v e l y . S t a t e - t o - s t a t e r e a c t i o n p r o b a b i l i t i e s or c r o s s s e c t i o n s are o f t e n summed over  a l l final  s t a t e s , s', t o give a t o t a l  r e a c t i o n c r o s s s e c t i o n o f an i n i t i a l o {Ex_ t  s  Equation ing  •  trans  ) =• E, a  (E^  trans  v  )  (26) ( S e c t i o n C) g i v e s a g e n e r a l i n t e g r a l f o r c o n v e r t -  energy dependent r a t e c o n s t a n t s  constants; equation  equivalent to  (26) are [ E l i a s o n (59), Connor l - ( 7 8 ) , Weston ( 7 2 ) ] :  k (T) = — h ^ ) s ^2Tryk T  (2D)  k  1  /  E  J  B  (T) =  (2 l/2  1  }  s  P'NE. )e t r a n s s trans  2  ;  '  t o t h e r m a l l y averaged r a t e  p a r t i c u l a r i n t e g r a l expressions  (ID) ' v  v  ,  s' s ' - f - s  s t a t e , s [Connor l - ( 7 8 ) ] :  0  r  k T  J  R  a  0  t  s  1/2  trans  e  -E  / k  t  r  B a  dE,. (35) trans  T  n  s  A T B  trans  trans (36)  (3D.  k  . ( T ,  ,  r a  B  p  B  t  ( B  t  r  a  n  . » E  t  r  a  n  . « -  B  t r „ ^ B  I  0  • trans d E  (  3  7  )  where u i s the reduced mass o f the r e a c t a n t s and k (T) i s an s i n i t i a l s t a t e thermal r e a c t i o n r a t e constant w i t h u n i t s of cm 1  molecule  —  1  s  —1  , cm  2  . . -1 -1 , 3 , , -1 -1 , molecule s , and cm molecule s for  the one, two, and three dimensional F i n a l l y , t o t a l thermal averaging  k ( T ) over  s t a t e s , s:  g  cases r e s p e c t i v e l y .  r e a c t i o n r a t e constants  are obtained by  the Boltzmann d i s t r i b u t i o n o f i n i t i a l  -113k(T)  = Ef s  (T) k  s  where f ( T ) g  given  (T)  (38)  s  i s the f r a c t i o n of t a r g e t molecules i n s t a t e s  by: f (T) s  where g s  s  i s a degeneracy f a c t o r and  [Weston (72)].  (37)  An  e  s  i s the energy of s t a t e ,  important f e a t u r e of equations  i s t h a t the thermal r a t e constants  (35)  are f u n c t i o n s of  the  -1/2 temperature independent r e a c t a n t reduced mass term, u r e g a r d l e s s of the t r a j e c t o r y d i m e n s i o n a l i t y . of H i s o t o p e s with r e l a t i v e l y heavy molecules  For  ,  reactions  (M.W.  >_ 35 amu) ,  t h i s mass f a c t o r p r e d i c t s t h a t k„ : k - k :k - 2.9:1.0:0.72:0.59 ^ Mu H D T [Connor l - ( 7 8 ) ] .  T h i s i s o t o p e e f f e c t may  be  simply  inter-  c t i the n g smass p e c i e dependence s, s i ot fyt eof n pthe r e t erde a as of the [Weston mean r e l(72)] a t i v e and v e l io c 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 - ( 7 8 ) ] because i t i s not dependent on the r e a c t i o n dynamics; a l l dynamical i n f o r m a t i o n as the e f f e c t s of the mass weighting s u r f a c e , are contained  about the r e a c t i o n , as w e l l of the p o t e n t i a l energy  i n the r e a c t i o n c r o s s s e c t i o n .  S t a t e - t o - s t a t e r e a c t i o n p r o b a b i l i t i e s are  artificial  c o n s t r u c t s 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 t r a j e c t o r y c a l c u l a t i o n s which, by d e f i n i t i o n , have access continuous r a t h e r than quantal both cases, s t a t e s by  the f i n a l  range of energy t r a n s f e r .  state energies  In  are r e l a t e d to quantum  some a r b i t r a r y b i n n i n g procedure which a s s i g n s  range of f i n a l s t a t e e n e r g i e s  to a  extending  a  above and below a given  -114quantum energy s t a t e t o t h a t s t a t e be r e c a l l e d  [Thompson (76)].  ( S e c t i o n C) t h a t i n the case o f p u r e l y  t r a j e c t o r i e s , the t a r g e t molecules i n i t i a l l y i n t e r n a l energy  I t may classical  possess no  ( i n v i o l a t i o n of the zero p o i n t energy) w h i l e  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 t a r g e t molecule i n i t i a l l y  possesses proper quanta o f i n t 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 o f t e n c a l c u l a t e d by s o l v i n g Lagrange's motion.  Lagrange's  d  r  3  L  dt  ^  o r Hamilton's equations o f  equations are g i v e n by [Messiah (58)]  f | •= 0 r  (r = 1,2,...,R)  q  w i t h the Lagrangian f u n c t i o n g i v e n by L(q ,q ,••.q ,q ,q ,•••,4 ;t) ±  2  R  ±  2  = T(q ,q ,...,q )  R  ±  2  R  - V(q ,q ,...,q ) ±  2  R  where q^ are g e n e r a l i z e d c o o r d i n a t e s , T i s the k i n e t i c and V i s the p o t e n t i a l energy of the system.  energy  The c l a s s i c a l  H a m i l t o n i a n f u n c t i o n which spans the 2R dimensional c o o r d i n a t e and momentum space  (phase space)  H(q , . . . ^ ^ p . ^ . . . ,p ;t) = ±  R  i s g i v e n by [Messiah R 9L I_qr ^  (58)]:  #  - L  = T ( p , . . . ,p ) ±  R  + V ( q , . . . ,q ) 1  R  and the equations o f motion are g i v e n by  q  r  _ 9H 9p  ; r  P  r  8H ~3q  r  (r = 1,2,...,R)  In  p r i n c i p l e , these equations of motion can be s o l v e d e x a c t l y  to  o b t a i n the completely d e t e r m i n i s t i c c l a s s i c a l  trajectories.  S t a t i s t i c a l l y averaged r e a c t i o n p r o b a b i l i t i e s must be o b t a i n e d by m u l t i p l e i n t e g r a t i o n over c o l l i s i o n v a r i a b l e s such as impact  -115parameter, molecular o r i e n t a t i o n and Generally,  v i b r a t i o n a l phase.  i t i s d i f f i c u l t t o determine the  f u n c t i o n a l form of  the r e a c t i o n p r o b a b i l i t y dependence on these c o l l i s i o n v a r i - . , ables which i s required  to perform t h i s i n t e g r a t i o n . Consequent-  l y , s t a t i s t i c a l averaging i s o f t e n accomplished by a procedure such as Monte C a r l o  i n t e g r a t i o n which determines the  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 v a l u e s of the c o l l i s i o n s e l e c t e d randomly from a weighted d i s t r i b u t i o n . dure, which normally r e q u i r e s thousand t r a j e c t o r i e s , has  reaction  the c a l c u l a t i o n of  variables This  proce-  several  the p h y s i c a l l y appealing-  feature  "that i t s i m u l a t e s the random process by which c o l l i s i o n s i n the  laboratory  a c t u a l l y occur  [Thompson (76)]."  QCT  t i o n s are o f t e n c a l l e d "Monte C a r l o " c a l c u l a t i o n s  calcula-  (eg.  [Blais  (74)]) or simply " c l a s s i c a l " c a l c u l a t i o n s to d i s t i n g u i s h them from quantum mechanical treatments. The  c o n v e n t i o n a l quantum mechanical approach to atom-  diatom r e a c t i o n 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 e v a l u a t e the  s c a t t e r i n g matrix  r i x ) elements which l e a d d i r e c t l y to the quantized 2 probabilities,  p  s  i ^ _  s  =  l  s  s  ' + -  s  l  [Manz (75)].  which i s almost always c a r r i e d out by only g i v e s  the net r e s u l t of the  This  (S-mat-  reaction procedure,  approximation methods,  c o l l i s i o n i n terms of a  r e a c t i o n p r o b a b i l i t y , s i n c e the e n t i r e p o t e n t i a l i s i n s e r t e d i n t o Schrddinger's equation and and  only the  asymptotic  product wavefunctions are determined w h i l e the  region  i s treated  l i k e a "black box."  reactant interaction  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-  -116tudes of p r o b a b i l i t y f u n c t i o n s quantum m e c h a n i c a l l y  and may  g i v e r i s e t o such p u r e l y quantum mechanical e f f e c t s as wave interference. Although the quantum mechanical treatment may  represent  an exact f o r m u l a t i o n of the r e a c t i o n problem, i n g e n e r a l ,  only  approximate s o l u t i o n s can be found f o r i t , whereas.the a p p r o x i mate c l a s s i c a l f o r m u l a t i o n of the r e a c t i o n can u s u a l l y be solved exactly.  From the viewpoint  o f understanding  the  r e a c t i o n 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 p o s s i b l e t o d i s -  t i n g u i s h p u r e l y quantum mechanical e f f e c t s such as t u n n e l l i n g from c l a s s i c a l e f f e c t s on the b a s i s of quantum mechanical r e s u l t s alone  (this point i s discussed further i n Section F ) .  More i m p o r t a n t l y , 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 p r o v i d e a d e t a i l e d p i c t u r e of the r e a c t i v e or non-reactive  s c a t t e r i n g processes;  because S-matrix quantum  mechanical r e s u l t s do not r e a l l y d e f i n e 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 t o p o l o g i c a l f e a t u r e s on the p o t e n t i a l cannot be determined d i r e c t l y . A number of methods have been d e v i s e d t o o b t a i n more dynamical i n f o r m a t i o n from quantum mechanical c a l c u l a t i o n s . One approach i s t o monitor the S-matrix as the p o t e n t i a l energy f u n c t i o n i s changed t o i n d i r e c t l y i n f e r the e f f e c t s of v a r i o u s potential features.  A more d i r e c t approach i s t o c a l c u l a t e  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 i n t e r a c t i o n r e g i o n  ( s o - c a l l e d " s t r e a m l i n e s , " 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 f o r m u l a t i n g the r e a c t i o n as a quantum hydrodynamics problem  (see eg. [ H i r s c h -  -117felder  (76)]).  Another approach i s an ingenious  modification  of the c o n v e n t i o n a l S-matrix c a l c u l a t i o n s known as the path sum  method  described  [Manz (74),  here s i n c e i t was  (75)]."  "state  T h i s method i s b r i e f l y  employed by Connor e t a l . l-(78) ,  l-(77) i n t h e i r quantum mechanical c a l c u l a t i o n s of the reaction rates.  Normal S-matrix c a l c u l a t i o n s can be  of as d i v i d i n g the p o t e n t i a l energy s u r f a c e the asymptotic r e a c t a n t i n t e r a c t i o n region the r e a c t a n t  and  and  product r e g i o n s ,  of the p o t e n t i a l .  i n t o three  box"  S-matrix.  action region sectors  and  the  the corresponding quantum s t a t e interaction  the c o n n e c t i o n being made v i a the  With the  the  state  s o l u t i o n of  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 product r e g i o n s ,  regions:  separated by  the quantum  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  and  thought  At the boundary between  i n t e r a c t i o n regions,  Schrodinger equation g i v e s  Mu  s t a t e path sum  "black  method, the i n t e r -  i s s u b d i v i d e d i n t o an a r b i t r a r y number of  an S-matrix i s c a l c u l a t e d  f o r each.  As  a result,  v i r t u a l quantum s t a t e 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 s e c t o r boundary of the surface. s i o n has  i n t e r a c t i o n region  of the  Consequently, i f , f o r example, a p a r t i c u l a r c o l l i a low  net r e a c t i o n p r o b a b i l i t y , i t i s p o s s i b l e  determine which p a r t of the p o t e n t i a l energy s u r f a c e responsible  for this result.  asymptotic product s t a t e , s', i n t e r a c t i o n region.  sectoring  the  s t a t e paths.  to  is  A " s t a t e path" i s a complete  l i n e c o n n e c t i n g an asymptotic r e a c t a n t  the  potential  s t a t e , s, to  through the v a r i o u s  Manz has  i n t e r a c t i o n region  an  sectors  of  devised algorithms for  and  i d e n t i f y i n g the dominant  -118E T r a n s i t i o n State  Theory  No attempt i s made t o d e r i v e TST i n t h i s S e c t i o n ; i t merely examines TST p r e d i c t i o n s f o r H i s o t o p e e f f e c t s . normally formulated (79)],  [Johnston  the TST e x p r e s s i o n  (66) , Weston  As  (72) , Kuppeirmann  f o r the r a t e constant of the  r e a c t i o n Y + AB + YA + B i s : V  k = r t  where  Q  h  Q  +  e  -E  V A  A T ' B  (39)  Y AB Q  i s a t u n n e l l i n g c o r r e c t i o n f a c t o r , h i s Planck's  c o n s t a n t , c T , Q , and Q Y  are the products of the t r a n s l a t i o n a l , AB  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 f u n c t i o n s of VA the a c t i v a t e d complex and r e a c t a n t s  r e s p e c t i v e l y , and E  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 (Section has  (C) , P a r t  (iii),  shown t h a t equation  and e q u a t i o n  Kuppeririann' (7 9)  when the a p p r o p r i a t e  f u n c t i o n s and VA b a r r i e r s are-used.  (implicit  treatment  (39) a p p l i e s -to both the c o l l i n e a r and  three dimensional-reactions  assumes the e x i s t e n c e  (25)).  is  -Equation  partition(39)' o n l y  of- thermal e q u i l i b r i u m among the r e a c t a n t s  i n the d e f i n i t i o n of a thermal r e a c t i o n r a t e  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 e f f e c t s due t o the curvature  of the r e a c t i o n path  Since  [Kuppermann  (7 9 ) ] .  the t a r g e t molecule i s the same f o r H  r e a c t i o n s of the type, Y + AB  isotope  YA + B, the r a t e constant  Cr aL tpiaors t if t i othe n fYu nand c t i oY' n s : i s o t o p i c r e a c t i o n s do not i n c l u d e or AB  Q lQ  v  i  trans  T  TT  Q+  Q  TT 7¥ rot Q vib Q trans  , -[E+(Y) - E+(Y') ]/k T e  B  (40)  -119where the e l e c t r o n i c  p a r t i t i o n f u n c t i o n s are assumed to  c a n c e l due to the BO approximation. molecule the three dimensional  For a n o n - l i n e a r YAB  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 f u n c t i o n s are g i v e n by [Van Hook (70)] : Q.  =  trans  ( 2 T r M k T )  3  /  V/h  2  B  (41)  3  3/2  k T B  Qrot =  i*h)l**\W  1/2  where M i s the molecular I  B  (42)  h J 2  mass, V i s the c o n t a i n e r volume, T_ , A  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 f a c t o r . t i 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  The quanf u n c t i o n s are  g i v e n by [Frost .(61).]'": i n t e r n a l modes Q  n  vib  where  ... -hv./k T -1 (1 - e I B )  i s a normal mode v i b r a t i o n a l  equations  (43)  i  (41) t o (43) i n t o  Substituting  frequency  (40) and expanding the VA b a r r i e r  term y i e l d s : y i  k k'  r  3/2  3/2  I' I • I' A B C  t  3n-7 , n  i  (44)  - i _ -hv|(Y-)A T) e - 2 Bk i TT^4(Y) 1 ? 1 B - h ! _ - hM v T*( Y ) / k T R  e  "  V  i  are the molecular  1  and  masses of the a c t i v a t e d  v f are the bound normal v i b r a t i o n a l e n e r g i e s  activated  complexes  the r e a c t i o n further  Y  '  )  ]  B  where m^ and nu., are the atomic masses of the Y and Y M and M  (  (the unbound v i b r a t i o n  coordinate  i s excluded).  s i m p l i f i e d by s u b s t i t u t i o n  1  isotopes,  complexes  o f the  corresponding  to  T h i s e x p r e s s i o n may be  of the R e d l i c h - T e l l e r  -120product theorem  [Johnston  VB ^ 1  M > 3/2  f—1  m, > ,3/2  n = II. i  1 / 2  I' I' I'  ABC  where  (66) , Van Hook  (70)]  3n-6 n  V -4  l  I  (45)  are the masses of the atoms comprising the  of mass M,  molecule  to y i e l d :  k  T  J . v^(Y)  ^  -  n  k' • e-2kTT  B  where the imaginary  3  n  "  vl(Y')  x  I  i  7  e  -hvt(Y')/k T B  -4—  i H  1 -  )n-7 vvT(Y) t 3n-7  i  _ -hvJ(Y)/k T e  B  ±  ?-. tvJ(y) - v f ( Y ' ) ] l :  (unbound) f r e q u e n c i e s , v , T  corresponding  to the r e a c t i o n c o o r d i n a t e , have been f a c t o r e d out. ut" =  h  i , k T v  the l a s t t h r e e f a c t o r s i n (46) may  Denoting  be combined to g i v e  B  ut(Y) T  v (Y) T  n k  '  r  .sinh  3n-7  t V+(Y')  sinh  uJ(Y)  ut(Y«) l (46) fuJ(Y')l  Noting t h a t the i s o t o p i c frequency r a t i o s are r e l a t e d to the i s o t o p i c masses a c c o r d i n g t o Karplus  [Weston (72), N i k i t i n  (74),  (70)] ,  v(Y) v(Y')  (Y')^  y  1 / 2  (Y)  the r a t e c o n s t a n t r a t i o becomes y (Y') .1/2 y (Y) J  k k'  where u  denotes  3n-7  u+ (Y)  sinh  (Y')  uj  (Y')  (47)  n  i sinh  (irfe(Y) l  the e f f e c t i v e masses possessed by  r e p r e s e n t a t i v e p o i n t on the b a r r i e r - c r o s s i n g Equation  uf l  the  trajectory.  (47) c o n t a i n s a number of i n t e r e s t i n g  terms,  U n l i k e the temperature independent term o f c o l l i s i o n theory ( S e c t i o n D) which only depends on the reduced mass of the 1/2 y* Y' r e a c t a n t s , the TST temperature independent term, u* Y depends on the e f f e c t i v e mass o f the r e p r e s e n t a t i v e p o i n t as i t c r o s s e s the p o t e n t i a l b a r r i e r ; t h a t i s , i t depends on the l o c a t i o n of the p o t e n t i a l b a r r i e r .  In t h i s sense, the TST  temperature independent term c o n t a i n s dynamical i n f o r m a t i o n , i n c o n t r a s t t o the c o r r e s p o n d i n g c o l l i s i o n theory term. the l i m i t i n g case o f a v e r y e a r l y b a r r i e r Y + F  In  (such as i n the  Mu r e a c t i o n s ) , t h i s term does p r e d i c t n — ^ 2.9, i n k  2 0  R  accord w i t h 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 o f t e n c a l l e d the "primary" i s o t o p e effect  [Nikitin  — — 4 — t o smh(u/2)  (74)] .  I t i s customary t o denote T  v  =  i n d i c a t e the quantum nature o f the v i b r a t i o n a l ^  U  :  p a r t i t i o n functions  [Johnston (66)].  In the l i m i t o f low  v i b r a t i o n a l f r e q u e n c i e s and h i g h temperatures  [Johnston (66),  Weston (72)] \  "TTTuT nh ( ^ - J * sir •2 limr = 1  r  and  v  =  u^O  u ( 1  +  24  +  2  u T920  u  4  +  -1  6  77^6 7 ! 2 ••• +  )  v  Conversely, i n the l i m i t of low temperature and h i g h frequencies , -u/2 T - ue ' v  and  limr u->  v  = 0  00  C l e a r l y , the e x p o n e n t i a l dependence of  on u i n d i c a t e s  that  i f the i s o t o p i c s u b s t i t u t i o n o f Mu r e s u l t s i n a s u b s t a n t i a l i n c r e a s e i n u^, there w i l l be a v e r y s t r o n g r e d u c t i o n i n  -122(Mu)/r^(H).  In g e n e r a l , i f the b a r r i e r i s e a r l y , the a c t i -  vated complex corresponds t o a very s l i g h t l y perturbed  target .  molecule with symmetric s t r e t c h v i b r a t i o n s t h a t d i s p l a y a very weak dependence on i s o t o p i c s u b s t i t u t i o n , and thus r^(Mu)/r^(H) 1; c o n v e r s e l y ,  as the b a r r i e r becomes p r o g r e s s i v e l y l a t e r ,  the v a l u e s of u^ i n c r e a s e and take on strong i s o t o p i c depen-i dences, and thus r^(Mu)/r^(H)  0. S t r e t c h i n g v i b r a t i o n s are  u s u a l l y stronger than bending v i b r a t i o n s [Johnston thus they might be expected t o have a stronger  (66)]  and  i n f l u e n c e on  (Mu)/r^(H); F i g u r e 17 shows t h i s t o be the case.  The F i g u r e  p l o t s r ^ / r j , a s a f u n c t i o n o f the percent o f u/u' ( i . e . u c o r responds t o a l i g h t e r i s o t o p e than u ) f o r v a r i o u s values o f 1  u . 1  From the F i g u r e , i t i s c l e a r t h a t a small i n c r e a s e i n u  over u' f o r a strong s t r e t c h i n g v i b r a t i o n ( t y p i c a l l y >300 cm f o r an e a r l y b a r r i e r ) reduces r ^ / r ^ , i n u/u  1  more than a l a r g e i n c r e a s e  f o r a weak bending v i b r a t i o n ( t y p i c a l l y <50 cm  early barrier).  Isotope  f o r an  e f f e c t s due t o r^/T^ are r e f e r r e d t o  as "secondary" i s o t o p e 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 t u n n e l l i n g a p p l i c a b l e t o chemical  reactions  [Connor l - ( 7 6 ) ] .  The f i r s t i s the standard  " s t a t i c " or " e n e r g e t i c " d e f i n i t i o n a s s o c i a t e d with b a r r i e r p e n e t r a t i o n ; the model of n u c l e a r alpha decay i s one o f the more celebrated examples.  In a chemical  reaction, static  t u n n e l l i n g occurs when there i s a non-zero r e a c t i o n p r o b a b i l i t y d e s p i t e the f a c t t h a t the t o t a l energy o f 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  kinetic  -123-  ISOTOPE  EFFECT  DUE  TO  f*/f*.  100 125 150 175 200 225 250 275 U/U'  v  ( C M  "  1 ]  300  {%)  FIGURE 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. r . / r * , i s p l o t t e d as a f u n c t i o n of the per cent i n c r e a s e Y n the i s o t o p e - d e p e n d e n t . v i b r a t i o n a l frequency of v over v f o r the v a r i o u s v a l u e s of v i n d i c a t e d on the r i g h t . I t i s assumed t h a t the f r e q u e n c i e s of v and v correspond t o t r a n s i t i o n s t a t e molecules c o n t a i n i n g l i g h t and heavy i s o t o p e s of atom Y r e s p e c t i v e l y , , at 300K. 1  1  1  energy p l u s 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 l e s s than the p o t e n t i a l b a r r i e r h e i g h t . terms of the energy d e f i n i t i o n s of S e c t i o n  C,  b a r r i e r h e i g h t , E^* ^ , d e f i n e d  (24) , i s the  1  by equation  3  the  In  physical static  t u n n e l l i n g b a r r i e r ; r e a c t i v e c o l l i s i o n s with l e s s 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 t h a t E ^ tunnelling. reactions  occur by  y S  As d i s c u s s e d i n the next Chapter, although  show an a p p r e c i a b l e amount of s t a t i c  t h i s form of t u n n e l l i n g temperatures  static  tunnelling,  i s r e l a t i v e l y unimportant at normal  (>200K) [Connor l - ( 7 7 ) , l-(76)].  of t u n n e l l i n g  Mu  i s "dynamic."  This r e f e r s to  The  second form  reactive  c o l l i s i o n s t h a t are e n e r g e t i c a l l y allowed and  which do  occur  quantum m e c h a n i c a l l y but which are c l a s s i c a l l y f o r b i d d e n , because of energy, but  because of the  reaction  A c c o r d i n g to the d e f i n i t i o n s of S e c t i o n  C,  not  dynamics.  the q u a s i c l a s s i c a l  OC threshold  energy, E^, , i s the dynamic t u n n e l l i n g  barrier;  r e a c t i v e c o l l i s i o n s with l e s s 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 ^ ling.  C  but more than E ^ ^ P  Dynamic t u n n e l l i n g  of t u n n e l l i n g  i s by  C i t was  occur by dynamic  [Connor l-(7.6)].  noted t h a t q u a s i c l a s s i c a l  e n e r g i e s are d i f f i c u l t to d e f i n e  i t i s defined  i n terms of E  noted here t h a t Connor l-(76) has unambiguously d e f i n e d  T  be .  somewhat I t i s simply  shown t h a t t u n n e l l i n g may  i n terms of complex-valued  t r a j e c t o r i e s a r i s i n g from s e m i c l a s s i c a l The  threshold  p r e c i s e l y because of Monte  averaging; thus, dynamic t u n n e l l i n g may QC  ambiguous since  tunnel-  f a r the most dominant form  i n chemical r e a c t i o n s  In S e c t i o n  Carlo  S  t u n n e l l i n g c o r r e c t i o n term, V  classical  scattering  , applied  theory. to  TST  be  -125-  (c.f.  (39))  equation  i s normally c a l c u l a t e d as a quantum  b a r r i e r p e n e t r a t i o n c o e f f i c i e n t f o r a mathematically dimensional b a r r i e r  (66) , Jakubetz  [Johnston  ( 6 6 ) . notes t h a t t h i s approach may  Johnston  (79)]  .  be " b e t t e r  chemical e n g i n e e r i n g than n a t u r a l p h i l o s o p h y ; "  nonetheless,  t h i s a t t r a c t i v e l y simple quantum c o r r e c t i o n t o the r a t e e x p r e s s i o n i s o f t e n remarkably  one  successful.  classical Three  one  dimensional b a r r i e r p e n e t r a t i o n c o r r e c t i o n models are most commonly used [Johnston 2TT  (66), 1/2.  1_  V  for r , .  The  Nikitin  f i r s t order Wigner c o r r e c t i o n 2 (74)] i s Y. = 1 + where | v * | 2  Z  i s the imaginary  r e a c t i o n c o o r d i n a t e and  |F*|  4  J  =  V  frequency corresponding to the  =  d V(g) 2  dq  i s the f o r c e constant  2  (curvature) of the p o t e n t i a l s u r f a c e a t the saddle p o i n t ; t h i s f i r s t order c o r r e c t i o n i s v a l i d f o r Wigner expansion may  h I v* I J _1 B  << 1 .  While  be a p p l i e d to any shape of one  the  dimension-  a l b a r r i e r , exact t u n n e l l i n g c o r r e c t i o n s have been worked out for  two  stylized reaction barriers  [Johnston  ( 6 6 ) ] : the  t r u n c a t e d , i n v e r t e d p a r a b o l a , c a l c u l a t e d by R. P. B e l l ; the b a r r i e r due  to C. E c k a r t .  r e a l i s t i c of the two  and  The E c k a r t b a r r i e r , the most  s i n c e i t has a smooth, continuous base  u n l i k e the t r u n c a t e d p a r a b o l a , may ing  t o a thermoneutral  ing  t o an exothermic  be symmetrical,  correspond-  r e a c t i o n , or unsymmetrical,  correspond-  reaction.  The p a r a m e t e r i z a t i o n of the  B e l l and E c k a r t b a r r i e r s and the a n a l y t i c a l forms of the t r a n s m i s s i o n c o e f f i c i e n t s may Jakubetz  (79).  F i g u r e 18  be found i n Johnston (adapted from Johnston  compares the t r a n s m i s s i o n p r o b a b i l i t i e s f o r the two as a f u n c t i o n of energy  (66)  or  (66)) barriers  a t v a r i o u s v a l u e s of the b a r r i e r h e i g h t .  -126-  FIGURE 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 as a f u n c t i o n of energy normalized to the b a r r i e r height f o r the t r u n c a t e d B e l l parabola (top) and E c k a r t b a r r i e r (bottom), adapted from Johnston (61). The a parameter d e s c r i b e s the shape of the b a r r i e r : l a r g e a -> high, wide b a r r i e r ; small a -> s h o r t , narrow b a r r i e r . Johnston has noted t h a t the B e l l t r a n s m i s s i o n c o e f f i c i e n t s are symmetrical t o i n v e r s i o n about K = 0.5, £ = 1. Furthermore, because of the B e l l t r u n c a t i o n , K does not approach zero a t £ = 0 f o r low values of a. I t i s a l s o noted t h a t quantum mechanical r e f l e c t i o n as w e l l as p e n e t r a t i o n occurs with these b a r r i e r s .  -127CHAPTER IV - EXPERIMENTAL RESULTS AND In t h i s Chapter,  THEIR INTERPRETATION  the experimental r e s u l t s f o r each Mu  r e a c t i o n are r e p o r t e d and compared w i t h r e c e n t experimental r e s u l t s f o r the analogous  H atom r e a c t i o n s and w i t h t h e o r e t i c a l  p r e d i c t i o n s , where a v a i l a b l e ; Table VI  summarizes the r e s u l t s .  For some r e a c t i o n s , s e v e r a l H atom r e a c t i o n r a t e parameters are r e p o r t e d w i t h r a t e constants v a r y i n g by f a c t o r s of three or more and a c t i v a t i o n e n e r g i e s v a r y i n g by 50%.  T h i s underscores  the  f a c t t h a t " the wealth of data on b i m o l e e u l a r r e a c t i o n s t h a t i n v o l v e f r e e r a d i c a l s or atoms i s more testimony to the growing awareness of the importance  of these i n t e r m e d i a t e s i n k i n e t i c  systems and the frequency of t h e i r occurence accuracy of the r e s u l t s  [Benson  (60)]."  than t o the g r e a t  The r e l a t i v e l y poor  knowledge of gas phase H atom r e a c t i o n r a t e s a v a i l a b l e 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 a v a i l a b l e t o measure gas phase H atom r e a c t i o n r a t e s d i r e c t l y , e i t h e r by m o n i t o r i n g r e a c t a n t d e p l e t i o n v i a some observable of H atoms or i t s r e a c t a n t p a r t n e r , or by m o n i t o r i n g product formation; r a t h e r , H atom r e a c t i o n r a t e s were i n d i r e c t l y i n f e r r e d from a p o s t u l a t e d r e a c t i o n mechanism and a s s o c i a t e d "steady s t a t e " approximations, r e s u l t s model dependent;  thereby making the  (2) while the advent  of modern t e c h -  niques such as mass s p e c t r o m e t r i c f a s t flow sampling formation or ESR  of product  d e t e c t i o n of H atoms i n d i l u t e gases has made  r a t e measurements d i r e c t , i t has not completely removed the systematic e r r o r s due  to c o m p e t i t i v e  r e a c t i o n s among the  t i v e l y l a r g e c o n c e n t r a t i o n s of h i g h l y r e a c t i v e atomic molecular  s p e c i e s simultaneously present i n the  and  experimental  rela-  TABLE V I :  SUMMARY OF THE REACTION RATE PARAMETERS FOR Mu AND H IN THE GAS PHASE  Reaction Y + F  Y + C l  YF+F  2  2  T  0. 92 + 0.23  1.4 + 0. 1  Hydrogen , E (kcal/mole) k(295K) a 0.20 + 0. 05 2.4 + 0. 2 T  2.2 + 0. 1 YC1+C1 1. 36 + 0.21  2  Y+Br  Muonium , E (kcal/mole) k(295K) a  ->  YBr+Br  5.1 + 0. 2  products  Y+HC1  YH + C l  Y+HBr  products products  Y+ HI  *  ->-  10  1/mole-s  estimates only  ref.  K  6.8 + 1.5  0. 09 + 0. 01 14.6 + 1.6  Dodonov(7 0) Homann(77)  2.9 + 1.0  Dodonov(7 0)  13 + 1.2  Ambidge (7 6)  1.8 + 0. 6  1.7 + 0. 6  1.4 + 0. 2  0.41 + 0. 04  1.20 + 0. 14  1.2 + 0. 1  4.4 + 0.4  Wagner(7 6)  1.14 + 0. 17  1.3 + 0. 1 2.2 + 1. * 5ic 5.1 + 0. 6  4.1 + 0.3 * 11 + 8 k 4.7 + 0.8  Bemand(77)  0. 009 + 0. 004  <0.004 + 0.002  Bott (7 6)  <0.016* + 0.003  Weston (79)  24 + 3 1.8 + 0. 4  Y+HC1  k r-—(295)  <0.000034 + 0.000005 3.18 + 0. 17  0. 0021 + 0. 0002  0.91 + 0. 10  2. 57 + 0. 11  0.21 + 0. 02  2.53 + 0. 13  0.00 + 0. 25  0.11 + 0. 02  0.70 + 0. 25  1.5 + 0. 5  4.4 + 0.6 23 + 4  Fleming(7 6) Fass (70) + Endo(7 6)  Endo  (7 6)  Sullivan(62  1.7 + 0.6* Jones(73)  -129-  apparatus. * HCl  +C1,  For example, the very  f a s t r e a c t i o n of H + C l  2  where * denotes a v i b r a t i o n a l l y e x c i t e d molecule,  be accompanied by the f o l l o w i n g s i d e r e a c t i o n s H + HCl* + H Cl + wall +  which consume a d d i t i o n a l H and  [Wagner (76)] :  + Cl  2  1  /  2  C  1 2  C l atoms and  regenerate C l , 2  thereby a l t e r i n g the r e a c t i o n s t o i c h i o m e t r y .  To reduce these  i n t e r f e r e n c e s , H atom k i n e t i c i s t s are c o n s t a n t l y considerable  may  striving  (with  success) to perform r a t e measurements under more  d i l u t e c o n d i t i o n s , but an i m p a i r i n g r e d u c t i o n  i n the  observable  s i g n a l i n e v i t a b l y accompanies these e f f o r t s . In Chapter I I I , i t was  noted t h a t the f i r s t m o t i v a t i o n  undertaking the k i n e t i c study of the r e a c t i o n s of Mu i n v e s t i g a t e i s o t o p e e f f e c t s i n H atom r e a c t i o n s . motivation  was  The  f o r the study a r i s e s from the f a c t t h a t MSR  for  to  second measure-  ments are l i t e r a l l y one-atom-at-a-time experiments which are  not  s u s c e p t i b l e to the kinds of i n t e r f e r e n c e s t h a t plague H atom measurements as o u t l i n e d above. might w e l l provide  As a r e s u l t , the MSR  the most accurate  atom r e a c t i o n r a t e s .  ( i s o t o p i c ) values  T h i s i s not to say t h a t MSR  fying  A Mu  + F  measures  s i g n a l , care must be taken i n i d e n t i -  the source of t h i s r e l a x a t i o n which need not be  reaction  of H  measurements  are n e c e s s a r i l y unambiguous - s i n c e the method simply the r e l a x a t i o n of the MSR  method  chemical  (see Appendix I I ) . 2  The  ->• MuF MSR  measured i n N  + F  r e l a x a t i o n r a t e s at v a r i o u s F 0  moderator between 295  and  2  concentrations,  383K, are l i s t e d i n  -130-  Table V I I .  The  i n f l u e n c e o f temperature  i s i l l u s t r a t e d i n F i g u r e 19 p l o t s t h e MSR  on the r e a c t i o n r a t e s  [adapted from Garner (78)]  r e l a x a t i o n r a t e d a t a a t 295 and 2  m o l e c u l a r r a t e c o n s t a n t s determined  by x  383K.  which The b i -  minimum f i t s of the  r e l a x a t i o n r a t e d a t a t o e q u a t i o n II (3) are a l s o g i v e n i n the Table and i l l u s t r a t e d i n t h e A r r h e n i u s p l o t of F i g u r e 2 [Garner  (78)].  The x  minimum f i t o f t h e s e d a t a t o the l o g a -  rithmic Arrhenius expression log k(1/mole-s)  + 0.23)  ( e q u a t i o n (12))  = (10.83 + 0.20)  1Q  w i t h k(300K) = (1.46  20.  + 0.11)  x 10  1 0  yields:  - (200 + 50/T)  1/mole-s and E  =  , (la) (0.92  kcal/mole. The  e x p e r i m e n t a l r a t e parameters o f the r e a c t i o n : H + F^ ~+  HF + F have been r e v i e w e d Kaufman (75)..  by Jones e t a l . (73) and Foon and  These a u t h o r s recommend t h e d i r e c t mass s p e c t o -  m e t r i c probe measurements o f a f a s t f l o w system by A l b r i g h t e t al.  (69) and Dodonov e t a l . (70.) from 294 t o  k(300) = (2.15 + 0.46)  x 10  565K'.which-yielded  1/mole-s w i t h E  9  = 2.4  + 0.2  kcal/  cl  mole and l o g A ( 1 / m o l e - s ) = 1Q  (11.079 + 0.035).  i n good agreement w i t h the more r e c e n t EPR ments o f Rabideau e t a l . (72) who x 10  1/mole-s and e s t i m a t e d E  &  f l o w system measure-  determined  = 2.6  These r e s u l t s are  k(300K) =  (2.5 +  0.2)  k c a l / m o l e , 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 o f Levy and Copeland (68) o b t a i n e d by t h e r m a l , 0^ - i n h i b i t e d E^ - F^ r e a c t i o n , w h i c h gave k(288K) = 9 1.8  x 10  1/mole-s.  However, the most r e c e n t measurement of  t h i s r e a c t i o n r a t e i s the f l o w system mass s p e c t r o m e t r i c d e t e r m i n a t i o n from 224 t o 493K by Homann e t a_l. (77) w h i c h y i e l d e d E, = 2.2 3.  + 0.1  kcal/mole, log. A(1/mole-s) = n  —  k(300K) = (1.00  _|_ U  + 0.08)  x 10  9  1/mole-s.  (10.6 + 0.1)  and  —  While the a c t i v a t i o n  -131TABLE V I I :  MSR RELAXATION RATES FOR THE REACTION Mu + F  2  -» MuF + F  Bimoleeular Temperature (K) 295 + 2  327 + 3  353 + 4  383 + 2  Rate Constant k(10  1 0  M"  1  s" ) 1  [F,,]  Relaxation  ( 1 0 ~ M)  Rate A (us" )  4  1  +  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  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  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  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  1.63 + 0.10  1.84 + 0.13  2.03 + 0.14  R e l a x a t i o n r a t e s r e p o r t e d are weighted averages o f the l e f t and r i g h t p o s i t r o n t e l e s c o p e histograms.  -132-  MU  0.0  0.5 F  FIGURE 19:  IN  2  F  1.0  2  / N  2  : ± =  1.5  295K . • =  2.0  CONCENTRATION  2.5 •  383K  3.0  .(10"  4  3.5  M)  The e f f e c t of temperature on the Mu + F MSR relaxation rates. The l i n e s are x minimum f i t s of the data t o equation 11(3) corresponding to k = ( 1 . 4 2 + 0 . 0 7 ) x 1 0 1/mole-s a t 295 K ( t r i a n g l e s ) and k = (2.03 + 0.14) x 1 0 1/mole-s at 3 83K (squares). Experimental p o i n t s shown are weighted averages from the l e f t and r i g h t t e l e s c o p e histograms. 2  2  1 0  1 0  -133-  0.01 1.5 1  1 2.0  1 2.5  1 I 3.0 3.5 4.0 1000/TEMP (K J I  I 4.5 5.0 I  _ 1  FIGURE 20:  Experimental A r r h e n i u s p l o t f o r the Y + F r e a c t i o n s , Y = Mu, H. The Mu data i s on the top l i n e ( t h i s work). The H data i s due t o Rabideau(72)(diamond), Levy(68)(octagon), Dodonov(70)(squares), and Homann(77)(triangles). The e r r o r bars on the Mu data are s t a t i s t i c a l only; the e r r o r bars on the H data are e s t i m ates g i v e n by the authors which a p p a r e n t l y . i n c l u d e systematic e r r o r s . 2  energy i s i n agreement w i t h the p r e v i o u s d e t e r m i n a t i o n s , k(300K) i s a f a c t o r o f two s m a l l e r . possible  Homann e t aJL. (77) c i t e  reasons f o r t h i s d i s c r e p a n c y .  f o r the H atom r e a c t i o n  several  The experimental  results  from a l l o f these authors a r e a l s o  shown i n the A r r h e n i u s p l o t o f F i g u r e 20. C l e a r l y , the a c t i v a t i o n energy f o r the Mu r e a c t i o n w i t h F^ i s l e s s than h a l f o f t h a t  f o r the analogous H atom reaction a t  300K, i n d i c a t i n g t h a t the average energy o f r e a c t i v e Mu c o l l i s i o n s i s much l e s s than t h a t o f r e a c t i v e  H c o l l i s i o n s , a c c o r d i n g t o the  Tolman i n t e r p r e t a t i o n o f a c t i v a t i o n energy Section  C).  (see Chapter I I I ,  Furthermore, the Mu:H r a t e constant r a t i o i s  M e i t h e r r—^(300K) = 6.8 + 1.5, u s i n g the H atom r e s u l t s o f H k  K  Albright al.  e t a l . , or 14.6 + 1.6, u s i n g the r e s u l t s o f Homann e t  Certainly,  H atom r e a c t i o n ; predicted  i t i s at l e a s t  a t 3 0OK i s much f a s t e r than the (2.3 + 0.5) times f a s t e r than  by the temperature independent mass f a c t o r o f 2.9.  This extra and,  the Mu r e a c t i o n  r a t e enhancement must be due t o dynamical e f f e c t s ,  as d i s c u s s e d throughout Chapter I I I , the o n l y such e f f e c t  l i k e l y t o enhance the r a t e o f Mu r e a c t i o n w i t h F^ i s quantum tunnelling.  The measured r e d u c t i o n i n the Mu a c t i v a t i o n energy,  r e l a t i v e t o the H atom v a l u e s , i s a l s o c o n s i s t e n t tunnelling  with t h i s  interpretation.  The o n l y "stand alone" experimental i n d i c a t o r o f the presence o f t u n n e l l i n g  i n t h e r m a l l y averaged r e a c t i o n s i s  curvature i n Arrhenius p l o t s  (Chapter I I I , S e c t i o n  C, [ L a i d l e r  (65)]), but t h i s t e s t i s not unambiguous s i n c e the p r e e x p o n e n t i a l factor  i s a l s o weakly temperature dependent.  d i f f i c u l t to obtain  Besides, i t i s  s u f f i c i e n t experimental p r e c i s i o n over a wide  -135enough temperature range t o demonstrate s i g n i f i c a n t A r r h e n i u s p l o t curvature, p a r t i c u l a r l y for reactions Jakubetz  (7 9 ) ] .  l i m i t e d Mu of the  Consequently, the  data of F i g u r e  20  of gases  [Laidler  absence of c u r v a t u r e i n  i s more l i k e l y a  manifestation  On  hand, the H atom data of A l b r i g h t e t a_l. does c u r v a t u r e although i t may  not  be  the  other  show a  slight  s i g n i f i c a n t given t h e i r  estimated r a t e constant u n c e r t a i n t i e s of 25 to 3 0%.  In any  these data are s u g g e s t i v e of a t u n n e l l i n g c o n t r i b u t i o n to H + F2 r e a c t i o n and  kcal/mole from 300--  4 0OK  QCT,  performed on the r e a c t i o n s  and  and  3.3  TST  12)  has  kcal/mole from 450  method  - 57OK.  i n c l u d i n g Mu,  with  of Jonathan e t a l .  state  [Connor l - ( 7 7 ) , l - ( 7 8 ) , l-(79)]. c o l l i n e a r q u a s i -  ar v i b r a t i o n a l l y a d i a b a t i c TST  calculations  Jakubetz l-(78),(7 9) a l s o used t h i s s u r f a c e to TST  calculations.  The  to i n v e s t i g a t e  are  listed  i n Tables V I I I ,  IX,  v i b r a t i o n a l p o p u l a t i o n s of F and  900K are  98%,  89%,  and  2  74%  and  H,  D,  tunnel-  collinear reaction apparent  a c t i v a t i o n e n e r g i e s c a l c u l a t e d by these authors f o r the 1) -> YF + F, Y = Mu,  colline-  [Connor 1- (7-9) ] ;  r a t e c o n s t a n t s , i s o t o p i c r a t e constant r a t i o s , and  Y + F2 (v = 0,  2  been used by Connor et aJL. to c a l c u l a t e  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  l i n g corrections  F .  (72)  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 path sum  the  i n v e s t i g a t i o n s have been  of H i s o t o p e s ,  c o l l i n e a r m o d i f i e d LEPS s u r f a c e  (shown i n F i g u r e  case,  g i v e apparent a c t i v a t i o n e n e r g i e s of about  A number of QMT,  The  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 t u n n e l l i n g .  2.2  (65),  T between 200  X respectively.  2%,  and  HOOK  The  a t thermal e q u i l i b r i u m f o r v = 0, and  reactions  9%,  at 3 00, and  19%  550, for  TABLE V I I I : (a)  k (Y) n  T/K  CALCULATED RATE CONSTANTS FOR  (cm s Mu  molecule  )  D  H  THE COLLINEAR Y + F_ -»- YF + F REACTIONS (b) k  T  (Y)  T/K  (cm s Mu  molecule 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  t  Quasiclassical  Quasiclassical 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  TST  (no t u n n e l l i n g )  t  (no  tunnelling)  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  tunnelling correction)^  TST  (Eckart  tunnelling correction)^  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, o r T. The number i n p a r e n t h e s i s e n t r y s h o u l d be m u l t i p i e d . f r o m Connor from. J a k u b e t z  l-(79) (79)  i n d i c a t e s t h e power o f 10 by w h i c h t h e  TABLE IX: (a) k ( Y ) / k Q  T/K  CALCULATED RATE CONSTANT RATIOS FOR THE COLLINEAR Y + F„ (H)  (b) k ( Y ) / k ( H ) 1  T/K  T  D  H  Mu  YF + F REACTIONS  1  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  Quasiclassical  t  41  Quasiclassical  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 t u n n e l l i n g )  f  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  TST  correction)^  (Eckart t u n n e l l i n g  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  Independent Factor.  2.9  1  0.72  0.59  *  Temperature Y = Mu, H, D, or T. from Connor  l-(79)  from Jakubetz  (7 9)  TABLE X: (a) E  ( 0 )  a  CALCULATED ACTIVATION ENERGIES FOR THE COLLINEAR Y + F „ - * Y F + F  (Y)  (kcal mole "  (b) E  V  D  H  Mu  T/K  •• -  i.  T  ( 1 )  a  T/K  (Y)  REACTIONS  -1 * (kcal mole )  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  t  Quasiclassical  Quasiclassical  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  TST  tunnelling)  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 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  1.2  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  Y = Mu, H, D, or T. +  from Connor from Jakubetz  l-(79) (7 9)  U) CO I  -139f o r v = 1 r e s p e c t i v e l y ; consequently, b e t t e r than 0.5%  at 300K and  b e t t e r than 2.8%  be noted t h a t the o r i g i n a l QMT reaction  at 900K.  about 12%  to  It  may  r e a c t i o n p r o b a b i l i t i e s f o r the  [Connor l - ( 7 7 ) , Connor l - ( 7 8 ) ] were r e c a l c u l a t e d  found to be the new  (T) approximates k(T)  l a r g e r than f i r s t reported  and  [Connor  1(79)];  r e s u l t s are thought to be a c c u r a t e to b e t t e r than  3%.  I t should a l s o be noted t h a t the quantum c a l c u l a t i o n s f o r the r e a c t i o n s were c a l c u l a t e d by d e f i n i n g a " l i n e of no r e t u r n " the p o t e n t i a l s u r f a c e  Mu  Mu  on  such t h a t the r e a c t i o n i s presumed to  proceed once a g i v e n c o l l i s i o n c r o s s e s t h i s l i n e ; t h i s procedure does not  a l l o w f o r r e f l e c t i o n of the r e p r e s e n t a t i v e  the r e p u l s i v e product v a l l e y w a l l and  therefore  point  from  over-estimates  the r e a c t i o n p r o b a b i l i t y f o r c o l l i s i o n s at very h i g h 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. t h a t t h i s e f f e c t should not  However, QCT  c a l c u l a t i o n s show  i n f l u e n c e the r a t e c o n s t a n t s i n the  temperature range below 1000K [Connor 2-(78), l,-(:79)]. At 300K, the p r e d i c t e d f o r the Mu  and  H reactions,  c o l l i n e a r QMT 1.2  and  2.1  a c t i v a t i o n energies  kcal/mole r e s p e c t i v e l y ,  are i n good agreement w i t h the experimental v a l u e s , and  2.3  + 0.2  kcal/mole r e s p e c t i v e l y .  comparison of the p r e d i c t e d  Mu:H  0.9  but  + 1.5  0.2  Furthermore, at 300K,  r a t e constant r a t i o of 6.6  the experimental v a l u e s shows remarkable agreement w i t h v a l u e of 6.8  +  with  the  o b t a i n e d from k^ measured by A l b r i g h t et a l . ,  c l e a r disagreement w i t h the v a l u e of 14.6  from the r e s u l t s of Homann et al_.  + 1.6  obtained  That these c o l l i n e a r  c a l c u l a t i o n s a p p a r e n t l y agree w e l l w i t h most of the results i s , in i t s e l f , agreement might w e l l be  an i n t e r e s t i n g f a c t . fortuitous since:  (1)  The "the  experimental  quantitative sucessful  -140t h e o r e t i c a l p r e d i c t i o n of an energy o f a c t i v a t i o n does not imply t h a t 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 c o r r e c t [Truhlar predicted  (78)]," and (2) d e s p i t e Mu:H  the agreement between the  r a t e constant r a t i o and the experimental v a l u e  t h a t uses the r e s u l t s o f A l b r i g h t e t a l . , t h i s cannot be i n t e r preted  as removing the ambiguity o f the experimental r a t e con- -  s t a n t r a t i o s s i n c e the Jonathan e t a l . s u r f a c e  used i n the  c a l c u l a t i o n s 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 d i d not i n c o r p o r a t e [Connor 2-(78)].  t u n n e l l i n g , and may, t h e r e f o r e ,  be  inaccurate  On the other hand, there are a number of  reasons t o suppose t h a t the c o l l i n e a r c a l c u l a t i o n s do f a i t h f u l l y describe  the r e a c t i o n , a t l e a s t q u a l i t a t i v e l y [Connor l - ( 7 9 ) ,  Jakubetz  (79)]:  configuration surface,  (1) as mentioned i n Chapter I I I , the c o l l i n e a r  i s e n e r g e t i c a l l y favored  f o r the Jonathan e t a l .  (2) t h r e e dimensional t r a j e c t o r y c a l c u l a t i o n s show t h a t  the r e a c t i o n i s c o l l i n e a r l y dominated due t o the r e a c t i o n dynamics  [Polanyi  (75)], and (3) because the saddle p o i n t i s  very e a r l y , the t r a n s i t i o n s t a t e i s j u s t a s l i g h t p e r t u r b a t i o n of the t a r g e t molecule and thus the three dimensional bending v i b r a t i o n s o f the t r a n s i t i o n t r i a t o m i c the c o l l i n e a r p o t e n t i a l  should not g r e a t l y a l t e r  [Connor 1-(79)].  L i k e the experimental r e s u l t s , the QMT c a l c u l a t i o n s are strongly  s u g g e s t i v e of t u n n e l l i n g i n the Mu r e a c t i o n .  The  quantum k(Y)/k(H) r a t e constant r a t i o s o f Table IX f o r both the F^ (v = 0) and F^ (v •'= 1) r e a c t i o n s ture  approach the l i m i t i n g tempera-  independent mass f a c t o r r a t i o s of 2.9:1.0:0.72:0.59 (see  Chapter I I I , S e c t i o n  D, p. 113) as the temperature approaches  900K, i n d i c a t i n g t h a t the l a r g e dynamical e f f e c t s t h a t enhance  -141the room temperature Mu r e a c t i o n cease to operate i n the h i g h temperature " c l a s s i c a l " regime. Mu r e a c t i o n a c t i v a t i o n energy  The dramatic i n c r e a s e i n the  (Table  X), which approaches  an  i s o t o p e independent v a l u e near 900K, i s a l s o c o n s i s t e n t w i t h t u n n e l l i n g i n the context of the Tolman i n t e r p r e t a t i o n of a c t i v a t i o n energy  (Chapter I I I , p. 97).  A r e v e a l i n g i n d i c a t i o n of the dynamics of the Y + F^ r e a c t i o n s i s i l l u s t r a t e d i n F i g u r e 21  (adapted from  which compares the energy dependence of the t o t a l  [Connor l-(79)])  reaction  p r o b a b i l i t i e s , P" = E P (see Chapter I I I , S e c t i o n D) , f o r S 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 1  t  1  i  OC t , P i s ordered Mu>H>D>T, w h i l e trans s OC t at h i g h e r v a l u e s of E^ , P d i s p l a y s the o p p o s i t e behaviour; trans s  cases, a t lower v a l u e s of  the curves c r o s s near P^ = 0.5. b a r r i e r height, E ^ ^ ,  f o r the F^ (v = 0) r e a c t i o n i s i n d i c a t e d by  S  OC an arrow a t E^ = 1.08 trans to the F^(v  kcal/mole (there i s no p h y s i c a l  = 1) r e a c t i o n ) .  S e c t i o n F, t h a t E ^  y  In the F i g u r e , the p h y s i c a l  I t may  be r e c a l l e d from Chapter I I I ,  r e p r e s e n t s the " s t a t i c " t u n n e l l i n g  S  barrier  barrier:  c o l l i s i o n s w i t h l e s s 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  Phys  no  ^_  deficit.  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 t o t h e i r  energy  From F i g u r e 21, i t i s c l e a r t h a t o n l y the Mu  displays considerable s t a t i c tunnelling.  reaction  I t w i l l a l s o be . OC  r e c a l l e d t h a t the 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 i e s , E~ , r e p r e sent the b a r r i e r s t o "dynamic" t u n n e l l i n g : c o l l i s i o n s w i t h l e s s 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 w i t h more than E ^ ^ , S  are c l a s s i c a l l y f o r b i d d e n , 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 r e a c t i o n dynamics.  The F i g u r e  1  1  1  ;  : !:  Quasiclassical 0.8  1  I  i  i  i —  i  i;1  /  j  / /  /  j  /  1  / 1 /  i '  _  1  /  /  /  /  Quasiclassical 0.8  /  '  / ll1 / A  0.4 '  '  /  i  '  />  /  i  /  0.4  /i  / •• i 1  1  i  tO.OtF lc  Quantum  0.8  0.4  0.0  EirL/kcalmof  ^ a n s A a l mof1 FIGURE 21:  1  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 as a f u n c t i o n of c o l l i s i o n energy f o r Y + F (v=0>l) •*• YF + F, Y = Mu ( ), H( ), D ( - - - ) , or T (• -•-) 2  adapted from  [Connor 1-(-79.)}-.  The s i g n i f i c a n c e of the arrows i s d e s c r i b e d  i n the t e x t  -143c l e a r l y shows t h a t a l l i s o t o p i c v a r i a n t s of the r e a c t i o n considerable  display  dynamic t u n n e l l i n g which i s much more dominant than  s t a t i c t u n n e l l i n g , even i n the case of muonium. The  Boltzmann d i s t r i b u t i o n must be c o n s i d e r e d i n order to  appreciate  the  importance of the tunnelling-enhanced  p r o b a b i l i t y on the ensemble r e a c t i o n process. the  integrand  f u n c t i o n of t i e s at 300  of equation trans  and  (35)  In the F i g u r e ,  kcal/mole),  (a) 300K and  -E  trans  plots  /k_T B ) as a  (1.08  (from  arrow A i n d i c a t e s the  physical  k c a l / m o l e ) , arrow B, at 0.087  [Jonathan  (b) 900K.  the average  (adapted from  i n d i c a t e s the q u a s i c l a s s i c a l t h r e s h o l d  q u a s i c l a s s i c a l thresholds imately  )e  900K f o r the Y + F ^ (v = 0) r e a c t i o n  the H atom r e a c t i o n kgT at  (E  22  c  b a r r i e r h e i g h t at 0.0472 eV (2.01  t  Figure  f o r the quantum mechanical r e a c t i o n p r o b a b i l i ^  [Connor l - ( 7 8 ) ] ) .  eV  (P  reaction  (72)]), and  arrow C  for  indicates  A comparison of arrow B w i t h  of F i g u r e  the  21 r e v e a l s t h a t i t i s approx-  of the q u a s i c l a s s i c a l t h r e s h o l d s  i s o t o p i c v a r i a n t s of t h i s r e a c t i o n .  For  the  sake of  for  the  illustration,  OC arrow B,  the average E^  , d i v i d e s the  from the  " c l a s s i c a l " region.  "tunnelling" reaction  Thus, F i g u r e  22 c l e a r l y demon-  s t a t e s t h a t the room temperature muonium r e a c t i o n by t u n n e l l i n g , which a l s o c o n t r i b u t e s  region  i s dominated  s i g n i f i c a n t l y to the room  temperature H atom r e a c t i o n , whereas, at 900K, c l a s s i c a l processes dominate the r e a c t i o n s from  for a l l H isotopes.  [Connor 1- (79) ]),. a p l o t s i m i l a r to F i g u r e  r a t e constant integrand mechanical r e a c t i o n s  Figure 22,  f o r the q u a s i c l a s s i c a l and  of the F^ OC  (v = 1)  .again, i f the average' E j ^ i s taken as the  23  compares  the  quantum  s t a t e at 3 00K. l i n e that  (adapted  Once  approximately  separates c l a s s i c a l from t u n n e l l i n g p r o c e s s e s , i t i s seen t h a t  H G M M M  integrand  O •  for  rate  constant  rt fD \  -t>t>T-  /[orb.  units}  1  0.04  -0-  Quasiclassical  Quantum  o  §0.03 o  0.02  c cn CD  I  0.01  0  6 E  trans  / k c a L mol  FIGURE 23: 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 k.. (T) a t 3 00K f o r the r e a c t i o n : Y + F, (v=l) YF + F, Y = Mu ( ), H ( ) , D( - • •) , and T( ) , adapted from [Connor l-'(79),]. The arrows i n d i c a t e kgT a t 300K.  -146t u n n e l l i n g completely  dominates the muonium r e a c t i o n a t room  temperature and c o n t r i b u t e s s i g n i f i c a n t l y t o the H atom r e a c t i o n r a t e as w e l l . The  c a l c u l a t i o n s of Connor e t a l . a l s o r e v e a l a g r e a t d e a l  about the high temperature " c l a s s i c a l " behaviour reactions.  of the Y +  F i g u r e 21 shows t h a t P^ f o r high energy c o l l i s i o n s  i s ordered T>D>H>Mu f o r both the QCT and QMT  calculations.  From the d i s c u s s i o n i n Chapter I I I , S e c t i o n B, p.76, t h i s behaviour  may be e x p l a i n e d 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 i n the r e a c t i o n v a l l e y f o r the l i g h t e r H i s o t o p e s . i s given i n F i g u r e 24 (adapted  V e r i f i c a t i o n of t h i s  from [Connor 2-(78)])  effect  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  collision  e n e r g i e s on the mass weighted muonium p o t e n t i a l energy s u r f a c e with F A i n i t i a l l y 0  kcal/mole,  i n the v = 0 s t a t e .  OC t r a n s =1.6  The p l o t a t  which i s g r e a t e r than the p h y s i c a l b a r r i e r  height,  but l e s s than the q u a s i c l a s s i c a l t h r e s h o l d , shows c o l l i s i o n s a t a l l v i b r a t i o n a l phases of F^ t o be not o n l y n o n - r e a c t i v e , but a l s o e l a s t i c s i n c e the v i b r a t i o n a l frequency a l t e r e d by the c o l l i s i o n s .  of F  2  i s not  The other p l o t s are f o r v a l u e s of  OC OC E^ > E „ and show ranges o f v i b r a t i o n a l phase f o r which the trans T ^ ^ c o l l i s i o n s are n o n - r e a c t i v e ; i t may be noted t h a t some o f the n o n - r e a c t i v e 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 a t high OC v a l u e s o f ET;  .  A l l four p l o t s o f F i g u r e 24 show the q u a s i -  t3T3.ll S  c l a s s i c a l n o n - r e a c t i v i t y of the Mu + F  2  c o l l i s i o n s a t moderately  high e n e r g i e s t o be due t o the b o t t l e n e c k e f f e c t .  F i g u r e 21  i n d i c a t e s the importance o f t h i s e f f e c t i n the r e l a t i v e l y r i s e o f the q u a s i c l a s s i c a l P^ curves  slow  from 0.5 t o 1 f o r the Mu  -147-  FIGURE 24:  N o n - r e a c t i v e 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 the r e a c t i o n : Mu + F2(v=0) ->- MuF + F on the mass weighted LEPS p o t e n t i a l energy s u r f a c e of [Jonathan (72)], adapted from [Connor 2 - ( 7 8 ) ] . In the n o t a t i o n of the t e x t , E^ = E § . The q u a s i c l a s s i c a l t h r e s h o l d t trans ^ energy f o r the r e a c t i o n i s 1.8 0 kcal/mole. c  -148r e a c t i o n , p a r t i c u l a r l y f o r the F_(v=l) c o l l i s i o n s f o r which P £ s t  shows some s t r u c t u r e .  The o r i g i n of t h i s s t r u c t u r e i s non-  r e a c t i v e back r e f l e c t i o n of the r e p r e s e n t a t i v e  p o i n t o f f the  s t r o n g l y r e p u l s i v e w a l l o f the product v a l l e y , as d i s c u s s e d i n Chapter I I I , p. 78. confirm energies  The QCT c a l c u l a t i o n s of Connor 2-(79)  t h a t the onset of t h i s e f f e c t o c c u r s a t lower f o r Mu than f o r the other H i s o t o p e s  collision  due t o the extreme  c o n t r a c t i o n of the mass weighted product v a l l e y f o r the l i g h t e r isotope.  F o r the v=0 r e a c t i o n , the onset of w a l l  reflection  OC occurs a t Er; = 7 and 4 0 kcal/mole f o r Mu and H r e s p e c t i v e l y . *cr an s OC For Mu + F _ ( v = l ) t w a l l r e f l e c t i o n begins at = 2 kcal/mole, 2 trans thereby competing w i t h b o t t l e n e c k the observed  s  structure,  Arrows i n F i g u r e barriers 2.16  r e f l e c t i o n and g i v i n g r i s e t o  21 i n d i c a t e the v i b r a t i o n a l l y a d i a b a t i c  VA (see Chapter I I I , S e c t i o n C) E VA Q  kcal/mole and E *  = 2.28, 2.20, 2.17, and  = 2.15, 1.91, 1.84, and 1.80 kcal/mole  f o r Mu, H, D, and T r e s p e c t i v e l y .  In d i s c u s s i n g VA i n the  context of an e a r l y b a r r i e r , i t was p o i n t e d VA VA  assumption holds,  E  v  q u a s i c l a s s i c a l threshold  out t h a t when the  g i v e s good " f i r s t guess" v a l u e s f o r the energies.  Figure  an e x c e l l e n t approximation f o r t h e Y +  21 shows t h i s t o be  r e a c t i o n s , w i t h the  e x c e p t i o n s o f the muonium r e a c t i o n and the f a c t t h a t E  V A  (Y)  have  v OC the opposite  ordering  to E ^ (Y).  assumption, which has g e n e r a l b e t t e r f o r the heavier  In other words, the VA  v a l i d i t y f o r these r e a c t i o n s , i s  H isotopes  than f o r the l i g h t e r ones.  T h i s may a l s o be understood i n terms of the b o t t l e n e c k . r e f l e c t i n g representative bottleneck  points non-reactively  Besides  (the normal  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 p o i n t  region  -149promotes v i b r a t i o n a l non-odiabaticity- by p r e s e n t i n g s u r f a c e geometry t h a t g r e a t l y p e r t u r b s trajectories. the c o n v e r s i o n  a potential  the q u a s i c l a s s i c a l  For a r e s t r i c t e d range o f v i b r a t i o n a l phases, o f v i b r a t i o n a l energy of the F^ molecule t o  t r a n s l a t i o n a l energy o f the r e p r e s e n t a t i v e p o i n t may propel  the system t o r e a c t i o n , thereby reducing  c l a s s i c a l t h r e s h o l d energy.  the q u a s i -  E v i d e n t l y , the sharper  of the l i g h t e r H i s o t o p e s cause g r e a t e r  molecular  bottlenecks  vibrational-translational  energy t r a n s f e r s i n c e they cause a more dramatic i n the q u a s i c l a s s i c a l t r a j e c t o r y .  help  perturbation  Expressed i n the jargon o f  dynamics, the b o t t l e n e c k  encounter f o r the l i g h t  i s o t o p e takes p l a c e i n the "sudden" regime, while the heavy i s o t o p e encounter i s i n the " a d i a b a t i c " regime The  [Levine  (74)].  f o r e g o i n g d i s c u s s i o n of the g e n e r a l v a l i d i t y of the  vibrational  adiabaticity assumption suggests t h a t simple  TST  c a l c u l a t i o n s o f the Y + F^ r e a c t i o n r a t e s u s i n g a 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 should p r o v i d e  f a i r l y accurate  estimates o f  the q u a s i c l a s s i c a l r e a c t i o n r a t e s , with the p o s s i b l e of the muonium r e a c t i o n . t h i s t o be the case. constants,  exception  Indeed, Connor e t a l . l-(79) have found  Tables V I I I - X show the VA-TST r a t e  r a t e constant  r a t i o s , and a c t i v a t i o n energies  f o r the  H, D, and T r e a c t i o n s t o be w i t h i n 5% o f 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;  f o r muonium, the somewhat l e s s s p e c t a c u l a r  agreement i s t y p i c a l l y i n the 10 - 2 0% range, except f o r the case of the F ^ ( v = 1) r a t e constant  a t 300K which d i f f e r s by about 40%.  C l e a r l y , the e a s i l y c a l c u l a t e d VA-TST r a t e constants are s u f f i c i e n t l y accurate,  i n g e n e r a l , t o be used as s u b s t i t u t e s f o r  the much more l a b o r i o u s q u a s i c l a s s i c a l r a t e constant c a l c u l a t i o n s .  -150Thus, VA-TST may be used t o e c o n o m i c a l l y o p t i m i z e p o t e n t i a l energy s u r f a c e s , reactions  not o n l y f o r t h i s r e a c t i o n ,  of the same g e n e r a l type  heavy atom r e a c t i o n s  but a l s o f o r  ( i . e . exothermic,  with e a r l y b a r r i e r s t h a t are dominated by  the c o l l i n e a r r e a c t i o n  geometry).  F i g u r e 25 (adapted from  [Connor 1-(.79),]) d i s p l a y s  QCT, and QMT r a t e constants as A r r h e n i u s p l o t s the TST r e s u l t s are e s s e n t i a l l y c o i n c i d e n t and  therefore  are not shown).  (for H, D, and T,  w i t h the QCT r e s u l t s  c u r v a t u r e due t o t u n n e l l i n g , but  should be noted t h a t the q u a s i c l a s s i c a l p l o t s are a l s o weakly  curved.  Although the c u r v a t u r e i n the quantum A r r h e n i u s p l o t  f o r the Mu r e a c t i o n the  the TST,  As expected, the quantum  A r r h e n i u s p l o t s show n o t i c e a b l e it  light-heavy-  i s s i g n i f i c a n t , i t i s not dramatic; i f  t h e o r e t i c a l p l o t proves t o be p h y s i c a l l y a c c u r a t e , then the  experimental demonstration of the A r r h e n i u s p l o t c u r v a t u r e , even f o r the case of the Mu r e a c t i o n , w i l l r e q u i r e  t h a t the e x p e r i -  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 c a l c u l a t i o n s  i n r e p r o d u c i n g the q u a s i c l a s s i c a l r a t e  c o n s t a n t s f o r the Y + Y^ r e a c t i o n  and having noted the s t r i k i n g  resemblance between the quantum mechanical r e a c t i o n  probability  curves of F i g u r e 21 w i t h the one d i m e n s i o n a l t u n n e l l i n g  trans-  m i s s i o n c o e f f i c i e n t s of F i g u r e 18, Jakubetz l-(78),(79)  investi-  gated the a p p l i c a t i o n of one dimensional t u n n e l l i n g to VA-TST c a l c u l a t i o n s f o r the r e a c t i o n s : Three t u n n e l l i n g c o r r e c t i o n s c o r r e c t i o n , and c o r r e c t i o n s  Y +  were i n v e s t i g a t e d :  corrections  and Y + Cl,,. the Wigner  f o r the t r u n c a t e d B e l l p a r a b o l a and  1000 I  T/K  o  o o r O  If) I  I  11  1  1  1  I  Quasiclassical (TST )  I  I  i1  a o  T / K  o o in  000 I  o  r O I  1  i  Quasiclassical  ^ ^ ^ ^  (TST  Mu  )  10  •-—  \>  T^  8-  ^ ^ • ^  """^  ^ ^ ^ * * ^ ^ ^ ^ ^ ^  ^ ^ ^ ^ - - - ^  ^ *  ^  ~  O  -  £  2  6  -  -  3 1000K/T  25:  1000K/T  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 constants f o r the r e a c t i o n : Y + F + Y F + F , Y = M u , H, D, and T, adapted from [Connor l-(79). The TST r e s u l t s f o r the H, D, and T i s o t o p e s are e s s e n t i a l l y c o i n c i d e n t with the q u a s i c l a s s i c a l r e s u l t s , and t h e r e f o r e are not illustrated. 2  -152the unsymmetrical E c k a r t b a r r i e r s (see Chapter I I I , S e c t i o n F ) . F i g u r e 26  (adapted  from  [Jakubetz  (79)]) compares the  (v =  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 r e s u l t i n g from the v a r i o u s  0)  tunnel-  l i n g c o r r e c t i o n s with the exact quantum mechanical r e s u l t s of Connor e t aJL. reactions.  and with u n c o r r e c t e d In a l l cases,  TST  r e s u l t s f o r the Y +  the E c k a r t b a r r i e r - T S T curves  F^  are i n  e x c e l l e n t agreement with the exact quantum r e s u l t s - the agreement i s almost p e r f e c t f o r the D and p l o t s of F i g u r e 27  (adapted  from  T reactions.  [Jakubetz  The  Arrhenius  (79)]) show t h a t the  B e l l and E c k a r t c o r r e c t i o n s p r o v i d e e s s e n t i a l l y the same e x c e l l e n t agreement w i t h the quantum r e s u l t s f o r the H, D, r e a c t i o n s ; indeed,  and  T  even the v e r y much simpler Wigner c o r r e c t i o n  (not shown i n F i g u r e 27)  p r o v i d e s good " f i r s t guess" approx-  imations to the quantum r e s u l t s f o r these r e a c t i o n s .  However,  F i g u r e 27 a l s o shows t h a t t h i s i s not the case f o r the muonium r e a c t i o n which i s o n l y w e l l d e s c r i b e d by the E c k a r t c o r r e c t i o n . Jakubetz has p o i n t e d out t h a t the f a i l u r e of the B e l l c o r r e c t i o n for  Mu  i s due  to the u n r e a l i s t i c t r u n c a t i o n of the p a r a b o l i c  b a r r i e r which r e s u l t s i n an over estimate energy r e a c t i o n p r o b a b i l i t i e s  of the low  ( c . f . F i g u r e 26).  collision  The  general  success of these t u n n e l l i n g c o r r e c t i o n s can be l a r g e l y  attri-  buted to the e a r l y b a r r i e r l o c a t i o n i n these r e a c t i o n s as d i s c u s s e d i n Chapter I I I , p. 78. E c k a r t f i t to the  F i g u r e 26(d)  a l s o shows an  " c o n s e r v a t i o n 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 j u s t the c l a s s i c a l b a r r i e r h e i g h t , E^ d e f i n e d on page 83;  the f a i l u r e of t h i s b a r r i e r supports  assumption of v i b r a t i o n a l The  , as the  adiabaticity.  l a s t p o i n t of d i s c u s s i o n on the Y + F  9  reaction i s  -1  FIGURE 26: Comparison o f quantum and 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 the r e a c t i o n : Y + F -> YF + F, Y = Mu, H, D, or T, adapted from [Jakubetz (79)]; quantum ( -) [Connor 1- (78)]; tunnelling corrected E c k a r t b a r r i e r VA-TST ( ), t u n n e l l i n g corrected B e l l barrier • VA-TST (•••), uncorrected VA-TST' (•-•-); p l o t (d) a l s o shows t u n n e l l i n g corrected Eckart b a r r i e r CVE-TST (•••)• n o t a t i o n o f the t e x t , E^ = E g C t trans.  r"  1.0 (a)  M u + F (v=0) 2  — MuF+F  0  I  n  t  n  e  E / k J mot t  12 E|/kJmor  1  Mu+F2(v)-*MuF + F v--0  2  3  1000K/T 27: Comparison o f quantum and t r a n s i t i o n s t a t e theory A r r h e n i u s p l o t s f o r c o l l i n e a r Mu + F ( v = 0 , l ) ( l e f t ) and II + F (v=0) and T + F (v=0) ( r i g h t ) , adapted from [Jakubetz (79)]. R e s u l t s shown a r e exact quantum (Q) [Connor l - ( 7 8 ) ] , E c k a r t VA-TST (E), B e l l VA-TST (B), and u n c o r r e c t e d VA-TST (CL). F o r the Mu r e a c t i o n , kQ and k^ c o i n c i d e over the whole temperature range, w h i l e f o r the H and T r e a c t i o n s , k°j and k^ c o i n c i d e over the whole range. 2  2  2  -155the  product v i b r a t i o n a l s t a t e d i s t r i b u t i o n .  reaction,  F i g u r e 28 (adapted from  calculated  F o r the Y + F  [Connor l - ( 7 8 ) ] )  s , as a f u n c t i o n 1  of the f r a c t i o n o f product v i b r a t i o n a l energy,'f  , = E^/Dg,  i s the d i s s o c i a t i o n energy o f YF and E , i s the energy s  u n  of the s  p l o t s the  r e l a t i v e p o p u l a t i o n d i s t r i b u t i o n o f product v i b r a t i o n a l  s t a t e s , normalized t o t h e most populated s t a t e ,  where D  2  1  l e v e l , both measured r e l a t i v e t o s' = 0.  The F i g u r e  OC shows t h e r e s u l t s a t 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 t o t h e c o l l i s i o n [Connor s  1  l-(78)].  energy  The most populated l e v e l has the v a l u e s  = 1, 6, 9, and 12 f o r Mu, H, D, and T r e s p e c t i v e l y ;  r e s u l t i s i n agreement w i t h the i n f a r e d r e s u l t s of Jonathan e t a l . (72)  the H atom  chemilluminescence  and P o l a n y i e t 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, d e f i n e d by < C f 0> £ s'«-0 ^ s 0 ' s' =  P  l / / P  i s 0.40, 0.58, 0.64, and 0.68 f o r Mu, H, D, and T r e s p e c t i v e l y ; a g a i n , the H atom r e s u l t s a r e i n agreement w i t h the  corrected  [Jakubetz 2-(78)] experimental v a l u e s o f 0.55 due t o Jonathan e t a l . and  0.62 due t o P o l a n y i e t a l .  The order Mu<H<D<T  for  i s i n q u a l i t a t i v e agreement w i t h the l i g h t atom anomaly I I I , p. 81), i n which l e s s r e a c t i o n  exoergicity  and V e n z l  succeeds w e l l  f  s V  >  (Chapter  i s transformed  i n t o product v i b r a t i o n as t h e mass o f the a t t a c k i n g Fischer  <C  atom decreases.  (78) d e r i v e d an a n a l y t i c e x p r e s s i o n  that  i n c a l c u l a t i n g t h e 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 f o r exothermic light-heavy-heavy atom r e a c t i o n s and which i s s e n s i t i v e t o the i n t e r a c t i o n l e n g t h location) surface  and the r e l a t i v e a t t r a c t i v e n e s s (see Chapter I I I , p. 80).  (saddle  point  o f the p o t e n t i a l  energy  T h i s e x p r e s s i o n may be used  -156-  FIGURE 28:  C o l l i n e a r quantum mechanical r e l a t i v e p o p u l a t i o n 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 s t a t e s normalized to the most populated s t a t e s a t E =2.45 k c a l / 1  mole f o r the Y + F (v=0) ^ YF + F r e a c t i o n s , Y = Mu, H, D, and T, adapted from [Connor l - ( 7 8 ) ] . 2  -157to  e c o n o m i c a l l y narrow the parameter range f o r LEPS s u r f a c e s by  f i t t i n g the experimental r e s u l t s f o r the v i b r a t i o n a l Korsch  (7 8)  distribution.  d e r i v e d a s i m i l a r , but s i m p l e r , e x p r e s s i o n which o n l y  r e q u i r e s a hand c a l c u l a t o r t o compute.  Although  i t i s not yet  p o s s i b l e t o e x p e r i m e n t a l l y measure product v i b r a t i o n a l d i s t r i b u t i o n s f o r Mu  r e a c t i o n s , the present work has  aided i n the development of the computational above  energy  indirectly  tools described  s i n c e i t prompted the exact quantum mechanical  calcula-  t i o n s 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 e s t of the a n a l y t i c e x p r e s s i o n s subsequently [Jakubetz B Mu  + Cl  developed  1-(78)]. -> MuCl + C l  2  The MSR  r e l a x a t i o n rates at various C l  measured between 29 5 and  concentrations,  38IK, are l i s t e d i n Table XI.  i l l u s t r a t e the influence of temperature F i g u r e 2 9 p l o t s the MSR  2  To  on the r e a c t i o n r a t e ,  r e l a x a t i o n data obtained a t 2 95  and 2  3 84K.  The b i m o l e c u l a r r a t e c o n s t a n t s , 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 11(3), are a l s o listed  i n the Table and 2  F i g u r e 30.  The  x  i l l u s t r a t e d i n the A r r h e n i u s p l o t of  minimum f i t of these data to the l o g a r i t h m i c  Arrhenius expression  (equation (12))  yields:  l o g k ( 1 / m o l e - s ) = (11.72 + 0.14) - (300 + 50/T), (la) w i t h k(300K) = (5.29 + 0.14) x 1 0 1/mole-s and E = (1.36 + — a — 1Q  1 0  0.21)  kcal/mole.  As i n d i c a t e d i n the Table, the moderator f o r  three of the r a t e constant measurements i s argon, while N the moderator f o r the 37OK measurements. s i g n a l enhancement due t o N  9  moderator  Other than the  (see F i g u r e 8), no  2  is MSR  -158-  TABLE XI:  MSR RELAXATION RATES FOR THE REACTION: Mu + C l  2  MuCl + C l Bimolecular Temperature  (K)  [Moderator g a s ] 295  + 2  [Argon]  Rate Constant k(10  1 0  5.17  M  - 1  S "  Relaxation 1  + 0.24 *  (5.45 + 0.19)  (4.73 + 0.56)  336 + 2 [Argon]  6.83 + 0.59  370 + 3 [Nitrogen]  7.27 + 0.72  381 + 2 [Argon]  9.22 + 0.65  )  ( 1 0 ~ M) 5  0 0 .. 0 0  0. 1. 1. 2. 3. 3. 4. 6.  98 58 80 45 12 71 90 85  Rate  XCys" ) " 1  1  *  + + + + + + + +  0. 0. 0. 0. 0. 0. 0. 0.  04 05 07 08 07 11 11 15  0.13 + 0.02*  0.66 1.45  + 0.12* + 0.22  1.09 + 0.03* '* 1 . 6 5 + 0 . 1 7r * 1 . 90 + 0 . 0 3 !  2.36 + 0.17* 2.49 + 0.27* 3. 07 + 0.17  0.0 0. 0 1 . 89 3 . 92 7 . 40  + 0 . 05 + 0 . 10 + o . 17  0.29 1.09 2.32 3.03  + + + +  0. 0 2 . 23 3 . 84  + +  0 . 07 0 . 11  0 . 42 1 . 82 3 . 40  + 0 . 19 + 0 . 40  0. 2. 3. 4. 5. 5. 6.  0 19 40 47 81 94 79  + + + + + +  0. 0. 0. 0. 0. 0.  07 10 13 16 16 19  0. 1. 3. 3. 5. 3. 4.  00 58 62 64 23 83 09  + + + + + + +  0. 0. 0. 0. 1. 1. 0.  23 43 77 92 05 75 91  0. 0. 1. 6.  0 74 94 69  + + +  0 . 05 0 . 07 0 . 18  0. 1. 2. 7 .  47 07 21 25  + + + +  0. 0. 0. 0.  03 09 27 64  0.07** 0.15** 0.25** 1.11 0 . 02  R e l a x a t i o n r a t e s r e p o r t e d are weighted averages of the l e f t and r i g h t p o s i t r o n t e l e s c o p e histograms. Room temperature data o b t a i n e d a t LBL (1975). Room temperature data o b t a i n e d a t TRIUMF (1976). MSR r e l a x a t i o n r a t e s i n l e f t and r i g h t histograms d i f f e r syst e m a t i c a l l y because o f the use o f a d i f f e r e n t geometry f o r each t e l e s c o p e r e s u l t i n g i n d i f f e r e n t X ' s f o r l e f t and r i g h t . R e l a x a t i o n r a t e s r e p o r t e d a r e weighted averages o f (X - X ) . n  n  -159-  0  10  20  CL 2 FIGURE 29:  30  40  50  60  70  80  CONCENTRATION (uM)  The e f f e c t o f temperature on t h e Mu + C I 2 MSR relaxation rates. The l i n e s a r e x minimum f i t s t o the p s e u d o - f i r s t order k i n e t i c expression of e q u a t i o n I I (3) c o r r e s p o n d i n g t o k = (5.17 + 0.24) x 1 0 1/mole-s a t 295K (squares and t r i angles) and k = (9.22 + 0.65) x 1 0 1/mole-s a t 381 K (diamonds). The t r i a n g l e s r e p r e s e n t d a t a t a k e n a t LBL d u r i n g 197 5, w h i l e t h e squares and diamonds r e p r e s e n t d a t a taken a t TRIUMF. The 295 K d a t a r e p r e s e n t (X - X ) o r d e r t o account f o r t h e d i f f e r e n t . . X ' s o b t a i n e d a t LBL and TRIUMF. 1 0  1 0  i  n  -160-  1.0  1.5  2.0  2.5  1000/TEMP  FIGURE 30:  3.0  3.5  4.0  (K" ] 1  Experimental A r r h e n i u s p l o t f o r the Y + CI2 r e a c t i o n s , Y = Mu, H, D. The Mu data i s on the top l i n e ( t h i s work). The H data i s due t o Stedman(70)(open octagon), Dodonov(70)(squares), Ambidge(76) ( t r i a n g l e s ) , Wagner(76) ( + ) , and Bemand(77)(x). The D datum i s due t o Stedman (7 0)(diamond). The e r r o r bars on the Mu data are s t a t i s t i c a l only.  -161moderator e f f e c t s are d e t e c t a b l e  i n the r a t e constant  measure-  2 ments.  A x  minimum f i t to equation  (12) u s i n g o n l y the  moderator data y i e l d s l o g ^ k ( 1 / m o l e - s ) =  (11.78 +0.14) -  (320 + 50/T), c o n s i s t e n t with the r e s u l t r e p o r t e d discussed  i n [Fleming  measured both at LBL completely (5.4  MSR  and  above.  K reaction rate  and  As  was  TRIUMF u s i n g the same method but  d i f f e r e n t equipment, y i e l d i n g r a t e constants  + 0.2)  date,  l - ( 7 7 ) ] , the 295  (4.7 + 0.6)  x 10  1 0  of  1/mole-s r e s p e c t i v e l y .  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  method i n determining The  H +  + HC1  gas phase Mu  + C l r e a c t i o n r a t e has  are compared with the Mu  r a t e s i n Table XII and  been  constants  and  preexponential  ancies.  The  two  F i g u r e 30.  rate  f a c t o r s show some s e r i o u s d i s c r e p to Bemand et a i .  The  r e c e n t ESR  a value of A a f a c t o r of two  measurements of Ambidge  Bemand e t a i . and Wagner e t a l . d i s c u s s  and  suggest t h a t the ESR  due  to t h e i r use of a n e a r l y equal H/C1  l e a d to i n t e r f e r e n c e s due  discrepancies  r e s u l t s of Ambidge e t a l . may 2  be i n e r r o r  s t o i c h i o m e t r y which  to C l atom-wall r e a c t i o n s .  the Lyman-a f l u o r e s c e n c e experiments used a C l / H r a t i o 2  The  three  s m a l l e r t h a t the Lyman-a  the p o s s i b l e o r i g i n s of the v a r i o u s experimental  from 5 - 1 5 .  (77)  essentially  e t a l . (76), however, g i v e a v a l u e of k(300K) a f a c t o r of  fluorescence r e s u l t s .  From  various  and Wagner e t a l . (76), u s i n g Lyman-a f l u o r e s c e n c e , are  s m a l l e r and  the  measured  agreement; however, the  most r e c e n t r e s u l t s , due  i d e n t i c a l to each other.  of  their results  the a c t i v a t i o n e n e r g i e s determined by the  i n v e s t i g a t o r s are i n reasonable  To  reaction rates.  d i r e c t l y by s e v e r a l authors i n the past 10 years;  the Table,  Ar  l e s s s e n s i t i v e , e a r l i e r mass  may  In c o n t r a s t , varying  spectrometrie  TABLE X I I :  EXPERIMENTAL RATE PARAMETERS FOR THE REACTION: Y + C]_ -> YC1 + C l , Y = Mu, 2  log  H Isotope  Method  Temp(K)  (kcal/mole)  Mu  MSR  295 - 384  1.36 + 0.21  H  t  294 - 565  1.8 + 0.3  t  300  H H H H  ms ms ESR Lf  §  Lff ms §  D  252 - 458  1.20 +  300 - 750  1.14 + 0.17  0.14  A s  k(300K) - 1  )  ,, 10 -1 -1. (10 M s ) A  fast  fluorescence  flow  Mu k  (300K) reference  H(D)  present  work  11.72 + 0.14  5.3 +  0.1  11.57  1.8 +  0.6  2.9  +  1.0  Albright(69)  2.1 +  0.7  2.5 +  0.8  Stedman (7 0)  13 +  1.2  Ambidge(7 6)  + 0.04  10.66 + 0.11  0.42 +  10.94 + 0.08  ;i.2 +  0.1  4.4 +  0.4  Wagner(7 6)  1.3 +  0.1  4.1 + 0.3  Bemand(77)  0.30  7.4 + 3.1  Stedman(7 0)  10.93  + 0.07  0.72 +  300  mass s p e c t r o m e t r i c Lyman-a  1.4 + 0.2  292 - 434  (M  _ 1  1 Q  H, D  0.04  i  I  -163measurements agree w i t h the Lyman-a f l u o r e s c e n c e the experimental e r r o r .  results within  I t may a l s o be noted from Table XII  t h a t Stedman e t a_l. (70) measured the D + C l r e a c t i o n r a t e a t 2  300K mass s p e c t r o m e t r i c a l l y . D e s p i t e the v a r i a t i o n i n the measured H atom r e a c t i o n r a t e data, two p o i n t s c l e a r l y emerge from t h e i r comparison w i t h the Mu r e a c t i o n r a t e parameters: t i o n energies and  (2)  (1) i n a l l cases, the apparent a c t i v a -  are the same w i t h i n the experimental u n c e r t a i n t i e s ,  compared w i t h the mass s p e c t r o m e t r i c  c o n s t a n t s a t 300K, there  H atom r e a c t i o n r a t e  i s no r a t e enhancement f o r the Mu  r e a c t i o n beyond the temperature independent mass f a c t o r o f 2.9 (Chapter I I I , p 113), o r , i f compared w i t h the Lyman-a f l u o r e s cence r e s u l t s a t 300K, the r a t e constant i s enhanced by a f a c t o r of o n l y  1.48 + 0.14 beyond the f a c t o r o f 2.9.  (As p r e v i o u s l y  mentioned, the anomalously l a r g e Mu:H r a t e constant r a t i o due t o Ambidge e t a l . (76) appears t o be i n e r r o r ) . r a t e constant r a t i o s , Mu:H:D a t 3 00K, 0.18 u s i n g  the mass s p e c t r o m e t r i c  + 0.4:1.0:0.58 + 0.24 u s i n g constants;  Furthermore, the  a r e 2.7 + 0.9:1.0:0.34 +  H atom r a t e c o n s t a n t s ,  the Lyman-a f l u o r e s c e n c e  o r 4.3  H atom r a t e  these may be compared w i t h the temperature independent  mass f a c t o r s o f 2.9:1.0:0.72. C l e a r l y , there  i s no experimental evidence t o i n d i c a t e  t h a t Mu e x h i b i t s a s u b s t a n t i a l t u n n e l l i n g advantage over the other H i s o t o p e s the Y + F  2  when r e a c t i n g w i t h C l , i n sharp c o n t r a s t t o  reaction.  2  One p o s s i b l e e x p l a n a t i o n  i s t h a t none o f the i s o t o p i c v e r s i o n s s i g n i f i c a n t t u n n e l l i n g a t 300K. by the Lyman-a f l u o r e s c e n c e  for this result  o f the r e a c t i o n d i s p l a y  T h i s p o s s i b i l i t y i s supported  Mu:H:D r a t e constant r a t i o s which  -164are very c l o s e t o the c l a s s i c a l mass f a c t o r r a t i o s . apparent  c l a s s i c a l behaviour  experimental  can be understood  i n d i c a t i o n s of a r e l a t i v e l y low  t h r e s h o l d energy.  i n terms of the  quasiclassical  Although there i s no simple  r e l a t i o n s h i p between a c t i v a t i o n e n e r g i e s and  This  analytic  quasiclassical  t h r e s h o l d e n e r g i e s , i n the absence of s t r o n g dynamical butions, such as t u n n e l l i n g or the b o t t l e n e c k e f f e c t , a c t i v a t i o n energy t h r e s h o l d energy  [Levine (74)].  Thus, a t h r e s h o l d energy  kcal/mole  which may  be compared w i t h the value of 2 kcal/mole  2  f o r the  kcal/mole.  Clearly,  r e a c t i o n s are much more capable of r e a c t i n g  a l l y at 300K than the Y + F importance  of  i s i n d i c a t e d f o r the Y + C l r e a c t i o n s ,  Y + F„ r e a c t i o n s and w i t h k_T(300K) = 0.6 z a 2  the  i s very n e a r l y equal t o the q u a s i c l a s s i c a l  about 1.4  the Y + C l  contri-  2  classic-  r e a c t i o n s , thereby m i n i m i z i n g the  of t u n n e l l i n g .  Indeed, the above hypothesis i s confirmed by r e c e n t p r e l i m i n a r y QCT  c a l c u l a t i o n s , performed  on a newly o p t i m i z e d  LEPS s u r f a c e , which g i v e s 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 i e s f o r the c o l l i n e a r Mu  and H r e a c t i o n s w i t h C l ( v = 0 ) of 1.2 2  kcal/mole r e s p e c t i v e l y  [Lagana (79)].  and  1.4  P r e l i m i n a r y quantum  c a l c u l a t i o n s i n d i c a t e a l a r g e r e d u c t i o n i n t u n n e l l i n g f o r the Y + Cl  2  [Lagana  r e a c t i o n s compared w i t h the Y + F (7 9 ) ] .  T o p o l o g i c a l l y , the Y + C l  i n these c a l c u l a t i o n s c l o s e l y resembles of Jonathan  2  (72) w i t h t h e i r  r e a c t i o n s at 300K 2  LEPS s u r f a c e  the Y + F  2  used  LEPS s u r f a c e  saddle p o i n t s p l a c e d i n almost  e x a c t l y the same r e l a t i v e p o s i t i o n s .  Of course, the Y + C l  s u r f a c e has a deeper r e a c t a n t v a l l e y and a shallower product valley  (see Table V) than the Y + F„ r e a c t i o n .  The most  2  -165s i g n i f i c a n t d i f f e r e n c e between these Y + C l i s t h e i r c l a s s i c a l b a r r i e r heights: Cl  2  surface  [Jakubetz  compared w i t h 2.35 l-(78).  f o r the Y + C l  2  Again, the  reactions  ^1.5  2  and  Y +  surfaces  kcal/mole f o r the Y +  kcal/mole f o r the Y + F  surface  2  reduced t u n n e l l i n g enhancement  at 300K i s c o n s i s t e n t  with t h i s  low  r e a c t i o n b a r r i e r which allows a l a r g e f r a c t i o n of t r a j e c t o r i e s t o proceed to r e a c t i o n c l a s s i c a l l y . t h i s smaller the Y + C l Y + F  2  Y + Cl  I t should be noted  b a r r i e r also explains  2  reactions  that  the o b s e r v a t i o n  proceed f a s t e r than the  that  corresponding  reactions.  2  The  preliminary  c a l c u l a t i o n s of Lagana  (79)  also  t h a t the onset of n o n - r e a c t i v e back r e f l e c t i o n of the  indicate represen-  OC t a t i v e points  o f f the product v a l l e y w a l l occurs at  kcal/mole f o r the Mu H isotopes c  phenomenon does not Mu  and  + C l ( v = 0 ) r e a c t i o n , whereas f o r the 2  i t occurs at  interesting contrast  OC >> trans  3.5  kcal/mole.  i s an  QC occur u n t i l E7_ -7 trans  and  40 kcal/mole f o r  H r e s p e c t i v e l y - c o l l i s i o n e n e r g i e s which are c e r t a i n l y  t i o n s are p r e s e n t l y  [Lagana  other  2  The  e f f e c t of w a l l r e f l e c t i o n i s  to reduce the r e a c t i o n p r o b a b i l i t y , P*",  + Cl  This  2.0  to the Y + F (v=0) r e a c t i o n where t h i s  unimportant even at 1000K.  Mu  <  being undertaken to determine how  r e a c t i o n r a t e i s reduced due  2  (79)].  from u n i t y .  Calculamuch the  to w a l l r e f l e c t i o n  Q u a l i t a t i v e l y , i t i s c l e a r that for t h i s  r e a c t i o n , w a l l r e f l e c t i o n w i l l tend t o o f f s e t t u n n e l l i n g more and  more w i t h i n c r e a s i n g temperature.  Thus, i t appears t h a t  the v a l u e of k„„ /k at 300K i s l e s s f o r the Cl„ r e a c t i o n Mu H 2 TT  f o r the F  2  r e a c t i o n , not  of t u n n e l l i n g due  to the  than  only because of the reduced importance smaller  r e a c t i o n b a r r i e r , but  also  because some o f the t u n n e l l i n g t h a t does occur i s c a n c e l l e d due t o c l a s s i c a l w a l l r e f l e c t i o n . consequences  There are other i n t e r e s t i n g  o f t h i s w a l l r e f l e c t i o n phenomenon.  As the  temperature i s r a i s e d and the c e n t r o i d of the Boltzmann  dis-  t r i b u t i o n s h i f t s toward e n e r g i e s where w a l l r e f l e c t i o n domina t e s , the r a t e of i n c r e a s e i n k (Mu + C l ) w i l l  f a l l and even-  2  t u a l l y , a t s u f f i c i e n t l y h i g h temperatures  (perhaps 1000K),  k(Mu + C l ^ ) i t s e l f w i l l a c t u a l l y decrease.  One might  there-  f o r e expect the a c t i v a t i o n energy o f t h i s r e a c t i o n t o pass through a maximum as i t passes from the low temperature ling-dominated r e g i o n t o the h i g h temperature w a l l dominated  tunnel-  reflection-  region.  I t i s i n t e r e s t i n g t o s p e c u l a t e on the reason f o r the dramatic r e d u c t i o n i n the minimum c o l l i s i o n energy f o r the onset of w a l l r e f l e c t i o n i n going from  t o Cl,,.  Connor mass weighting scheme (p 73), X  2  from F  2  to C l  2  A c c o r d i n g t o the  changing the mass o f  f u r t h e r c o n t r a c t s the product v a l l e y by about  26% f o r each H i s o t o p e .  While t h i s g r e a t e r c o n t r a c t i o n i n the  Y + C l product v a l l e y undoubtably enhances w a l l 2  reflection  q u i t e s i g n i f i c a n t l y , i t seems u n l i k e l y t h a t t h i s alone accounts f o r the r e d u c t i o n i n the w a l l r e f l e c t i o n from 7 kcal/mole f o r Mu + example.  "threshold",  t o 2 kcal/mole f o r Mu + C l , f o r 2  I t i s l i k e l y t h a t the e x o t h e r m i c i t y of the r e a c t i o n s  a l s o p l a y s an important r o l e .  For the Y + F  r e a c t i o n , the  2  bottom o f 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 f o r the Y + C l r e a c t i o n the 2  product v a l l e y i s o n l y about 50 kcal/mole below the c l a s s i c a l barrier.  Since the saddle p o i n t s f o r the two r e a c t i o n s l i e  -167at about the same p o s i t i o n s r e l a t i v e t o the r e a c t a n t  and  product v a l l e y s , i t i s c l e a r t h a t the f o r c e t e n d i n g t o make the representative  point  "round the corner" and "bobsled" down the  product v a l l e y i s g r e a t e r  on the steeper Y +  on the more g e n t l y  Y + Cl^  sloped  surface.  surface  than  F i n a l l y , s i n c e the  angle of r e f l e c t i o n i s equal t o the angle of i n c i d e n c e ,  i t is  l i k e l y t h a t the f a c t , t h a t the skewing angle f o r the mass weighted Mu + C l surface CI2  2  surface  i s about 1 degree more than f o r the Y + F^  a l s o makes a minor c o n t r i b u t i o n  i n reducing the Mu +  wall r e f l e c t i o n threshold. Before c l o s i n g t h i s S e c t i o n ,  i t should be remarked  that  the r o l e played by the experimental Mu r e a c t i o n r a t e measurements i n the development of the t h e o r e t i c a l c a l c u l a t i o n s of Connor e t a l . i s q u i t e d i f f e r e n t f o r the F2 and C l  2  reactions.  In the former case, the reasonably a c c u r a t e p o t e n t i a l energy surface  due t o Jonathan  (72) , optimized f o r the H + F2 r e a c t i o n ,  e x i s t e d b e f o r e c a l c u l a t i o n s were performed f o r the Mu + F2 reaction.  Thus, i n t h i s case, the experimental Mu  reaction  r a t e data p r o v i d e d a t e s t of the q u a l i t y of the t h e o r e t i c a l p r e d i c t i o n s based on t h i s s u r f a c e .  The f i r s t  calculations,  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 r e a c t i o n i s dominated by quantum t u n n e l l i n g , and  (2) d e s p i t e  c a l c u l a t e d using  the f a c t s t h a t the Mu + F^  r e a c t i o n r a t e s were  a " l i n e of no r e t u r n " method  c a l c u l a t i o n s were only  (p 139) and the  c o l l i n e a r , the QMT r e s u l t s seemed t o  p r e d i c t the r a t i o k / k ^ and the a c t i v a t i o n e n e r g i e s M u  accurately. classical  Next, QCT c a l c u l a t i o n s showed t h a t  quite  (1) the  (high temperature) Y + F„ r e a c t i o n r a t e s are governed  -168by the b o t t l e n e c k  e f f e c t and  (2) w a l l r e f l e c t i o n i s unimportant  up to 1000K, thereby e x p l a i n i n g the success of the r e t u r n " method.  F i n a l l y , i t was  discovered  that  " l i n e of simple  E c k a r t t u n n e l l i n g c o r r e c t e d VA-TST worked w e l l f o r the Y + r e a c t i o n s , and,  more i m p o r t a n t l y ,  no  F  2  the success of t h i s method  c o u l d be understood i n terms of the very e a r l y r e a c t i o n b a r r i e r and  the favored The  c o l l i n e a r geometry.  n a t u r a l next t h e o r e t i c a l step was  c a l c u l a t i o n s f o r the Y + C l conclusions  to make s i m i l a r  r e a c t i o n s to determine i f the  2  p r e v i o u s l y drawn c o u l d be g e n e r a l i z e d  heavy-heavy atom r e a c t i o n s w i t h e a r l y b a r r i e r s . u n l i k e the Jonathan s u r f a c e f o r the Y + F experimentally the Y + C l  2  optimized  and  Two Baer  very (74)  i n v e s t i g a t i o n s of the H + C l [Essen  (76)],  [Truhlar  light-  Unfortunately,  r e a c t i o n s , no  2  accurate  p o t e n t i a l energy s u r f a c e e x i s t e d f o r  reactions.  Kuntz et a l . (66)  for  2  s i m i l a r LEPS s u r f a c e s due  have been used i n s e v e r a l  reaction  (eg.  [Wilkins  (75),  ( 7 8 ) ] ) , but the primary aim of these  s t u d i e s has  been t o compare computational methods [Truhlar  such as TST  v e r s u s QMT,  results.  Jakubetz  to  (79)],  r a t h e r than t o model experimental  (79)  used the Kuntz s u r f a c e t o t e s t i f  t u n n e l l i n g c o r r e c t e d VA-TST g i v e s comparable estimates for. the QMT  c a l c u l a t i o n s f o r the H + C l  2  r e a c t i o n and for the H +  F  2  r e a c t i o n , or i f t h e r e 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  of the t u n n e l l i n g c o r r e c t e d VA-TST r a t e constants Cl  2  plots  f o r the Y +  r e a c t i o n , c a l c u l a t e d on the Kuntz s u r f a c e , are compared  w i t h those f o r the Y + F surface, i n F i g u r e  31.  2  r e a c t i o n , c a l c u l a t e d on the Jonathan  As expected  f o r the H + Cl„  reaction,  i  FIGURE 31:  %  1  r  A r r h e n i u s p l o t s f o r c o l l i n e a r Y + F and Y + C l , f o r Boltzmann d i s t r i b u t e d r e a c t a n t s , c a l c u l a t e d by t u n n e l l i n g c o r r e c t e d E c k a r t b a r r i e r VA-TST, adapted from Jakubetz (79). LEPS surface due t o Jonathan (72) used f o r Y + F , LEPS s u r f a c e due t o Kuntz (66) used f o r Y + Cl„. 2  2  2  -170t u n n e l l i n g c o r r e c t e d VA-TST i s as good as QMT.  However,  these c a l c u l a t i o n s do r a t h e r p o o r l y i n reproducing perimental ratio, value  k  results.  For example, the p r e d i c t e d r a t e  Mu+Cl ^ H + C l  of 4.3  + 0.4  '  l s  ^'^  at 3 0OK.  c o m  P  a r e  < i w i t h the  H + Cl  2.4  kcal/mole r e s p e c t i v e l y , compared w i t h the  values  a c t i v a t i o n energies  of about 1.4  exconstant  experimental  Furthermore, at 30OK the Mu  and  2  the  are c a l c u l a t e d t o be 1.4  kcal/mole f o r both.  C±2  + and  experimental  In  fact/these  p r e d i c t i o n s are q u i t e s i m i l a r t o those f o r the Y +  reaction.  T h i s r e s u l t i s not unexpected s i n c e the Kuntz s u r f a c e , which i s known t o be i n a c c u r a t e ,  i s very  w i t h c l a s s i c a l b a r r i e r s of 2.42  s i m i l a r t o the Jonathan and  2.35  Kuntz and Jonathan s u r f a c e s r e s p e c t i v e l y .  surface,  kcal/mole f o r the Besides u n d e r l i n i n g  the d e f i c i e n c y of the Kuntz s u r f a c e , the c a l c u l a t i o n s of Jakubetz suggest t h a t t u n n e l l i n g c o r r e c t e d VA-TST i s a l s o e s s e n t i a l l y a p p l i c a b l e t o the Y + C l ^ r e a c t i o n s , where Y = D, and  T;  f o r Y = Mu,  t h i s inference  H,  c o u l d not be made s i n c e  no quantum c a l c u l a t i o n s were a v a i l a b l e f o r comparison. From t h i s p o i n t , the c a l c u l a t i o n s on the Y + C l ^ r e a c t i o n s have been proceeding i n the r e v e r s e reactions.  order  Jakubetz found t h a t by reducing kcal/mole, the r a t e constant  became 4.1, I t has  F^  F i r s t , Jakubetz used t u n n e l l i n g c o r r e c t e d VA-TST  t o "tune" the c a l c u l a t i o n of an optimized  1.5  to the Y +  Y + Cl  2  surface.  the b a r r i e r height  from 2.4  r a t i o , k,„ /k , at 3 00K Mu H TT  i n good agreement w i t h the experimental  been noted by Connor l - ( 7 8 ) , t h a t a much  result.  stronger  c o n s t r a i n t i s p l a c e d on the c h o i c e of a p o t e n t i a l energy by the k- /k Mu  H  to  r a t i o than by the a c t i v a t i o n e n e r g i e s .  surface  The  d e t a i l s of the newly optimized  i n press constant  [Connor 3-(79)].  In t h i s way,  measurements of the Mu  to optimize  the Y + C l  Y + Cl  + Cl  2  the experimental r a t e  r e a c t i o n have been used  s u r f a c e , r a t h e r than to t e s t the  2  q u a l i t y of the t h e o r e t i c a l p r e d i c t i o n s . inappropriate to c a l l of the Mu  + Cl  any  of the  Consequently, i t i s  subsequently c a l c u l a t e d  reaction rates "predictions."  2  t h i s S e c t i o n has new  s u r f a c e are able to e x p l a i n the o r i g i n s of the  increase  in wall r e f l e c t i o n .  preliminary  r e s u l t s discussed  t i o n suggest t h a t the a p p l i c a b l e to the Mu makes use  I t may  2  may  also f a i l  Since TST  and  the  i n t h i s S e c t i o n on w a l l  of a " l i n e of no r e t u r n "  s u r f a c e " ) , TST  in tunnelling  be noted t h a t  reaction.  this  experimental  " l i n e of no r e t u r n " method may + Cl  values  However, as  shown, t h e o r e t i c a l c a l c u l a t i o n s u s i n g  r e s u l t s by such e f f e c t s as the r e d u c t i o n  C Mu  s u r f a c e are c u r r e n t l y  2  reflecnot  be  implicitly  also  (the s o - c a l l e d " d i v i d i n g  f o r muonium at high  temperatures.  + Br^ -»• MuBr + Br The  MSR  r e l a x a t i o n r a t e s at v a r i o u s B r  2  concentrations,  measured i n argon moderator at 295K, are l i s t e d i n Table and p l o t t e d i n F i g u r e 32. 2 295K, determined by x  The  bimolecular  (2.4  D e t a i l s of t h i s Mu  + 0.3)  x 10  1 1  2-(77)] 1/mole-s.  (la)  r e a c t i o n r a t e measurement, which  conducted at LBL,  are given  F i g u r e and  two  Table,  at  minimum f i t s of the r e l a x a t i o n r a t e data  t o equation 11(3), i s [Fleming (76), k(295K) =  r a t e constant  XIII  i n Fleming  of the MSR  (76).  As  was  shown i n the  r e l a x a t i o n r a t e s are anomal-  o u s l y l a r g e ; these p o i n t s were taken d u r i n g  a time of known  -172-  TABLE X I I I :  MSR RELAXATION RATES FOR THE REACTION Mu + B r ->• 2  MuBr + B r  §  [Br ]  R e l a x a t i o n Rate  (10~ M)  A( u s ' V  2  6  0.0  0.19 + 0.03  0.0  0.17  + 0. 04  0.72  + 0.12  1.74  + 0.23  2.19 + 0.24  0.71 + 0.06  2. 95 + 0.32  0. 98 + 0.10  3.45 + 0.44  1.21 + 0.15  + 0.45  1.11 + 0.10  4.13  5.75 + 0. 61  1.43  + 0.16  + 0.81  2.22  + 0.25  7.73  1. 02 + 0. 09  1. 46 + 0.20  4.25 + 0.55  3. 47 + 0.58  d a t a from [Fleming  (76)].  r e l a x a t i o n r a t e s r e p o r t e d a r e weighted averages o f t h e l e f t and r i g h t p o s i t r o n t e l e s c o p e h i s t o g r a m s . p o i n t s t a k e n w i t h poor magnetic f i e l d r e g u l a t i o n (see t e x t )  -173-  5i  i  1  i  r  o 4-  B R O M I N E {fx. FIGURE 32:  C O N C E N T R A T I O N M O L E S / L I T E R )  MSR r e l a x a t i o n r a t e s as a f u n c t i o n o f Br~ concentrat i o n i n argon moderator a t 295K; data taken a t LBL d u r i n g 1975 [Fleming (76)]. The high p o i n t s are d i s c u s s e d i n the t e x t .  - 1 7 4 -  poor magnetic f i e l d supply. field  r e g u l a t i o n due t o an u n s t a b l e power  T h i s serves t o i l l u s t r a t e the f a c t t h a t magnetic  inhomogeneities  c o n t r i b u t e t o the background  r a t e , A Q , i n pure i n e r t moderator gas.  relaxation  To ensure  that  such  e f f e c t s do not i n t e r f e r e w i t h the Mu r e a c t i o n r a t e measurements, A Q i s p e r i o d i c a l l y checked d u r i n g the experiments. i s a l s o noted t h a t the B r determined  c o n c e n t r a t i o n s a r e not as p r e c i s e l y  2  as the c o n c e n t r a t i o n s o f the other gases s t u d i e d i n from B r ( 1 )  t h i s t h e s i s ; these were determined for  It  vapour p r e s s u r e s ,  2  which r e p o r t e d v a l u e s vary by up t o 3 0 % [Nesmeinov  (63)] .  I n s o f a r as i t i s known, the b i m o l e e u l a r thermal r a t e constant f o r the H + B r  2  r e a c t i o n has never been d i r e c t l y  measured, although i t i s c u r r e n t l y being i n v e s t i g a t e d w i t h Lyman-a f l u o r e s c e n c e [Clyne [Fleming x  1.5)  1 0  ( 7 6 ) ] , 1  0  From a l i t e r a t u r e  ( 7 9 ) ] .  i t has been estimated t h a t  1/mole-s and  k  D  ( 2 9 5 )  =  ( 6 . 1  which g i v e s r a t e constant r a t i o s a t 1 . 0 :  0 . 3 + 0 . 2 .  +  k  U  ( 2 9 5 )  x  3 . 2 )  1 0  =  ( 2 . 2  +  1/mole-s,  9  o f Mu:H:D =  295K  survey  11  + 8 :  Perhaps more r e l i a b l e e s t i m a t e s o f these  r e a c t i o n r a t e c o n s t a n t s can be o b t a i n e d by combining r e c e n t d i r e c t ESR measurements o f the r a t e constants f o r 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 p h o t o l y s i s measurements of the r a t i o s of these r a t e c o n s t a n t s t o those f o r the H + B r [Fass x 1 0 2.3  ( 7 0 ) , 9  ( 7 2 ) ] .  1/mole-s  [Fass  1/mole-s. 1/mole-s  ( 7 0 ) ] ,  [Endo  For H a t (76)]  2  2 9 5 K ,  and D + B r r e a c t i o n s 2  k(H + HBr) =  0 . 2 )  2  2  2 9 5 K ,  +  and k (II + Br )/k (H + HBr) = 2 2 . 7 +  which g i v e s k(H + B r ) =  For D a t  ( 2 . 2  k(D + DBr) =  [Endo ( 7 6 ) ] and k ( D + B r ) / k ( D 2  ( 5 . 1 ( 8 . 0  + +  0 . 6 ) 1 . 0 )  x x  1 0 1 0  8  + DBr) = 5 8 + 1 . 7  1  0  [Fass  (72)] which g i v e s k(D + B r )  =  2  1/mole-s.  (4.6 + 0.6)  x 10  Using these r e s u l t s , the r a t e constant r a t i o s at  2 95K are Mu:H:D = 4.7 temperature  + 0.8:1.0:0.9 + 0.2  compared with the  independent v a l u e s of 2.9:1.0:0.72.  L i k e the r a t e c o n s t a n t s , the a c t i v a t i o n e n e r g i e s f o r the H + Br2 and D + B r  r e a c t i o n s are not well-known, though  2  are known t o be small [ B l a i s measurements of Endo may  (74)] .  Again, the d i r e c t  For H, E  ESR  be combined w i t h the p h o t o l y s i s r e s u l t s  of Fass t o g i v e estimates of the H + Br 2 and D + B r energies.  they  (H + HBr)  = 2.6 + 0.1  2  kcal/mole  activation [Endo (7 6)]  cl  and E (H4 HBr) - E (H + Br„) = 0.8 a a z. which g i v e s E  (H + Br„) = 1.8 A  cl  f o r D, E DBr)  a  - E  (D + DBr)  + 0.4  kcal/mole  kcal/mole.  [Fass (70)] Similarily  —  = 1.7  + 0.1 —  kcal/mole  [Endo (76)] and E (D + a  (D + Br„) = 0.9  + 0.2  kcal/mole  [Fass (72)], which  Z  3.  gives E  + 0.3  —  —  (D + Br„) = 0.8  + 0.3  kcal/mole.  However, r e c e n t  molecular beam r e s u l t s of Hepburn et a l . (78) suggest  E  &  1 kcal/mole f o r both the H and D r e a c t i o n s with B r . 2  Without d i r e c t l y determined r a t e parameters Br2 and D + B r  2  r e a c t i o n s and an a c t i v a t i o n energy measure-  ment f o r the Mu + B r  2  r e a c t i o n , i t i s d i f f i c u l t to s p e c u l a t e  on i s o t o p e e f f e c t s i n t h i s r e a c t i o n f a m i l y . ed' D:H  f o r the H +  r a t e constant r a t i o of 0.9  times the temperature  From the e s t i m a t -  + 0.2, which i s 1.3 +  independent mass f a c t o r of 0.72,  0.3 there  appears t o be an " i n v e r s e " i s o t o p e e f f e c t at room  temperature,  if,  collision  i n f a c t , t h e r e i s any d i f f e r e n c e at a l l .  At  e n e r g i e s g r e a t e r than the c l a s s i c a l b a r r i e r h e i g h t (estimated to be about 1 kcal/mole  [Hepburn  (78)]), t h i s e f f e c t seems t o  be w e l l - e s t a b l i s h e d i n the r e a c t i o n c r o s s s e c t i o n measurements  -176of Hepburn e t a l . (78) and i n the t r a j e c t o r y c a l c u l a t i o n s of Malcolme-Lawes (78) and White classically  [White  (73).  (73), Hepburn  back r e f l e c t i o n o f the  T h i s has been  explained  (78)] i n terms of n o n - r e a c t i v e  r e p r e s e n t a t i v e p o i n t s o f f the r e p u l s i v e  w a l l of the c o n t r a c t e d product v a l l e y , which has been d i s c u s s e d f o r the Mu r e a c t i o n s with F  2  and Cl,,.  An e q u i v a l e n t way of  p i c 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 t o p o t e n t i a l s u r f a c e s , i s t o note t h a t chemical  r e a c t i o n r e q u i r e s momentum  t r a n s f e r between the l i g h t a t t a c k i n g atom and the heavy p a r t i n g product molecule; but i n the case of H + Br,, t h i s has l i t t l e time t o occur  because the H atom moves much f a s t e r than the  heavy Br atoms.  A t a g i v e n value o f E , the H - Br„ trans 2. c o l l i s i o n i s about /2~ f a s t e r than the D - B r c o l l i s i o n , and 2  thus H + B r  2  has a lower r e a c t i o n c r o s s s e c t i o n .  Indeed,  Hepburn e t a l . (78) found t h a t f o r E^_ > 1 kcal/mole, the H + * trans — ' Br„ and D + Br c r o s s s e c t i o n s a r e c o i n c i d e n t when p l o t t e d as a ^ 2 f u n c t i o n 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 r a t h e r than energy. As d i s c u s s e d  i n the preceding  S e c t i o n , the r e d u c t i o n i n  the w a l l r e f l e c t i o n t h r e s h o l d f o r the H - halogen r e a c t i o n , from 40 t o <1 kcal/mole as X^ changes from F probably  2  t o B r , can 2  be a t t r i b u t e d t o t h r e e f a c t o r s : (1) according  Connor mass weighting Br  2  t o the  scheme, the product v a l l e y s f o r the Y +  r e a c t i o n s are about 50% narrower than f o r the Y + F  reactions,  (2) the "down h i l l "  2  p a r t of the p o t e n t i a l s u r f a c e  i s about 106 kcal/mole f o r Y + F  2  compared with about 45 k c a l /  mole f o r Y + B r , and (3) the skewing angle f o r H + B r i s 2  2  85° 'compared with.81° f o r _ t h e H + F Mok-Polanyi r e l a t i o n s h i p holds  2  reaction.  A l s o , i f the  (p 76), then the H + B r b a r r i e r 2  -177i s e a r l i e r than the Y + F^  Y + Cl^  and  slope of the down h i l l p a r t of the probably l e s s than i t would be the H + B r  surface  2  b a r r i e r s , and  surface  at the  i s more a t t r a c t i v e , p 81);  thus, t h e r e i s  2  and  2  f o r Mu  + F^  and  Mu  to  + Cl  p r e d i c t s t h a t t h i s e f f e c t w i l l dominate the  r a t e f o r the + Br  point  of the t h e o r e t i c a l p r e d i c t i o n s  of the w a l l r e f l e c t i o n t h r e s h o l d s + Br  corner i s  corner."  Certainly, extrapolation  Mu  the  i f the b a r r i e r were l a t e r ( i . e .  even l e s s of a tendancy f o r the r e p r e s e n t a t i v e "round the  thus  l a t t e r system.  r e a c t i o n t o be  In f a c t , one  to  2  reaction  might expect the  slower than the H + B r  2  Mu  r e a c t i o n a t 300K  t h a t the estimated r a t i o : k__ /k (300K) i s i n e r r o r . Mu H  How-  TT  ever, i t must be c a u t i o n e d t h a t when the Boltzmann d i s t r i b u t i o n i s taken i n t o account, the r e a c t i o n c r o s s e n e r g i e s l e s s than 1 kcal/mole r e f l e c t i o n ) have a strong  (where t h e r e may  sections  for Y + Br  2  much l a r g e r than f o r the F^ very low  be no  wall averaged  While these low  or C l  2  the  reactions  energy  because of  On  the  [Baybutt  the other hand, i t cannot  expected t h a t quantum t u n n e l l i n g g r e a t l y enhances the Mu r e a c t i o n r a t e at 300K s i n c e the b a r r i e r i s so low. f o r the Mu  + Cl  2  r e a c t i o n , one  be  f a c t t h a t the c o l l i n e a r  r e a c t i o n geometry does not dominate t h i s r e a c t i o n (74)].  collision  are as yet unknown, they should  c l a s s i c a l b a r r i e r and  Bauer l - ( 7 8 ) , B l a i s  at  i n f l u e n c e on the t h e r m a l l y  r e a c t i o n r a t e s at room temperature. cross  sections  As  (78) be  +  Br  2  discuss  might p r e d i c t t h a t the Y +  Br  2  r e a c t i o n apparent a c t i v a t i o n e n e r g i e s pass through a maximum as they change from the ( i f one  e x i s t s ) t o the  low  temperature t u n n e l l i n g  region  high temperature w a l l r e f l e c t i o n r e g i o n .  -178I f i t t u r n s out t h a t the apparent a c t i v a t i o n energy f o r the Mu  + Br  2  r e a c t i o n i s l e s s than t h a t f o r H + B r  2  at 300K, then  t h i s would not n e c e s s a r i l y i n d i c a t e t u n n e l l i n g , u n l i k e  the  case of the F^ r e a c t i o n .  D Mu  ^ MuCl + H + HCl •> MuH + C l In Chapter I I I , p 98,  i t i s noted t h a t i t i s not  p o s s i b l e to determine the i n d i v i d u a l r e a c t i o n r a t e f o r the exchange and Mu  + HX  method - only the t o t a l r e a c t i o n  are measured.  The  MSR  r e l a x a t i o n r a t e s at  v a r i o u s HCl c o n c e n t r a t i o n s ,  measured i n N  are l i s t e d i n Table XIV  i l l u s t r a t e d i n Figure  minimum f i t : yields a  and  2  moderator at 295K, 2 33.  Ax  of the r e l a x a t i o n r a t e data to equation  bimolecular  k(295K) < (3.41 The  constants  a b s t r a c t i o n r e a c t i o n channels f o r the  r e a c t i o n s by the MSR  r a t e constants  r a t e constant  yet  rate  11(3)  constant  + 0.46)  x 10  f o r t h i s very  5  1/mole-s  (la)  slow r e a c t i o n i s w r i t t e n as  an  i n e q u a l i t y to emphasize the f a c t t h a t i t o n l y r e p r e s e n t s  an  upper l i m i t .  only  There are two  reasons why  t h i s experiment  g i v e s an upper l i m i t to the r a t e constant. r a t e s are known f o r only two i s pure HCl at one reduced MSR  HCl c o n c e n t r a t i o n s ,  atmosphere.  s i g n a l amplitude  these data and  t h e i r small MSR  errors introduced  (Table XIV  one  and F i g u r e  of which  34)  r e l a x a t i o n r a t e s , any  the for systematic  to the measurements c o u l d e a s i l y exceed  inherently unreliable.  a known, estimable  relaxation  Moreover, because of  s t a t i s t i c a l e r r o r i n the r a t e c o n s t a n t s . both sparse and  (1) MSR  systematic  the  Thus, the data i s (2) More  importantly,  e r r o r can account f o r h a l f of  -179-  TABLE XIV:  MSR RELAXATION RATES FOR THE TOTAL Mu + HC1 REACTION AT 2 95K  [HC1] (10" M) 2  R e l a x a t i o n Rate A (ys )  Total' A (%) -Mu-—•  _ 1  1  A+(%) 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  r e l a x a t i o n r a t e s r e p o r t e d are weighted averages of the l e f t and r i g h t p o s i t r o n t e l e s c o p e histograms. t o t a l asymmetry = 2 A  M u  + A^+  (see Appendix II)  -180-  MU IN HCL WITH N  0.2  2  MODERATOR. 295 K  h  0.0 0.00 1  1  0.01 HCL  FIGURE 33:  1 0.02  1 0.03  I  0-.04  I  0.05  CONCENTRATION (M)  MSR r e l a x a t i o n r a t e s as a f u n c t i o n o f HCl c o n c e n t r a t i o n i n N a t 2 95K. The high c o n c e n t r a t i o n p o i n t r e p r e s e n t s / 1 atmosphere of pure HCl. The l i n e i s a x minimum f i t o f the data y i e l d i n g k(295K) £ (3.41 + 0.46) x 10 M s . These data r e p r e s e n t o n l y an upper l i m i t , as d e s c r i b e d i n the t e x t . 2  2  -181-  CO 0 1  -0.05  >— t—  CO  cr  -0.15 0-0  0-5  1.0  1.5  TIME IN FIGURE 34  M  2.0 SEC  2.5  3.0  3.5  4.0  (20 NSEC/BIN)  The MSR s i g n a l s i n pure N versus pure H C l . Both s p e c t r a r e p r e s e n t about 10 events. The l i n e s a r e x minimum f i t s t o equation (8). For N , A = 12.2 %, A = 4.0%, X = 0.4 y s ; f o r HCl, A = 5.9%, A = 9.7%, and X = 1.8 y s . 2  2  - 1  2  M u  - 1  M u  -182the observed MSR  relaxation.  i n t h i s experiment,  of 10 0 ppm  [Fleming  ppm.  Mu undergoes s p i n exchange  oxygen which r e l a x e s the MSR  b i m o l e c u l a r r a t e constant of gas phase  used  obtained from Canadian L i q u i d A i r L t d . , has  a t y p i c a l 0^ i m p u r i t y of <100 with paramagnetic  The anhydrous HCl reagent  signal with a  (1.6 + 0.1) x 10"'""'" 1/mole-s i n the  (79), M a r s h a l l (78)].  An 0^ c o n c e n t r a t i o n  i n one atmosphere of HCl g i v e s a s p i n exchange  r e l a x a t i o n r a t e of about 0.7 The experiment  ys "*", h a l f of the observed  MSR  effect.  must be repeated with e l e c t r o n i c grade HCl which  has an 0^ c o n c e n t r a t i o n of <4 ppm.  I t i s important  emphasize t h a t a s i m i l a r systematic e r r o r due t o  to -contaminated  reagents does not a r i s e i n the other systems s t u d i e d i n t h i s thesis.  In the f i r s t p l a c e , the 0^ contamination of the other  gases used i s <10  ppm  and,  secondly, a l l of the other r e a c t i o n  r a t e constants are w i t h i n two s p i n exchange r a t e constant. e r r o r due  to  than HCl)  i s l e s s than  I t may  contamination  o r d e r s of magnitude of the Consequently, i n the X^  the systematic  and HX gases (other  0.1%.  be noted t h a t i n a d d i t i o n t o p r o v i d i n g an  of the thermal Mu + HCl r e a c t i o n r a t e constant, t h i s  estimate  experiment  p o s s i b l y g i v e s some information, about the muonium formation process or the r o l e played by "hot" atom r e a c t i o n s of Mu. i l l u s t r a t e d i n F i g u r e 34, the muonium s i g n a l amplitude HCl i s about h a l f of t h a t i n pure As shown i n the data of Table XIV,  As  i n pure  under i d e n t i c a l c o n d i t i o n s . the r e d u c t i o n i n the muonium  s i g n a l w i t h i n c r e a s i n g HCl c o n c e n t r a t i o n i s accompanied by an i n c r e a s e i n the background " f r e e " y data may  +  s i g n a l amplitude.  be e x p l a i n e d i n two ways: (1) Mu may  These  undergo f a s t  -183hot atom r e a c t i o n s w i t h HCl before  the muon s p i n has time t o  precess s i g n i f i c a n t l y , thereby p l a c i n g muons i n t o diamagnetic product molecules where they precess c o h e r e n t l y y  +  ions  like  "free"  (see Appendix I I ) , or (2) the high energy charge  exchange c r o s s s e c t i o n s of Mu w i t h HCl may be such t h a t a l a r g e f r a c t i o n o f the muons t h e r m a l i z e than as Mu atoms.  as f r e e y  ions,  +  rather  The t o t a l s i g n a l amplitude, given by 2 A  M u  + A^-, appears t o decrease w i t h i n c r e a s i n g HCl c o n c e n t r a t i o n ; unfortunately, "free" y  +  these data cannot be t r e a t e d  quantitatively, since the  s i g n a l amplitudes r e s u l t from f i t s of data  only about one p e r i o d o f the slow y  precession  +  covering  a t 6.9 gauss.  Again, f u r t h e r experiments a r e r e q u i r e d t o c l e a r l y i n t e r p r e t these e f f e c t s .  In p r i n c i p l e , i t i s p o s s i b l e t o d i s t i n g u i s h  the hot atom process from the charge exchange process by the use o f the r e s i d u a l p o l a r i z a t i o n method [Brewer  (72)]).  r a t e constant  (Appendix I I , and  I t must be emphasized t h a t the l i m i t i n g  f o r Mu + HCl r e p o r t e d  i n t h i s S e c t i o n i s f o r the  thermal r e a c t i o n , not the hot atom r e a c t i o n . processes take p l a c e during  the f i r s t  Hot atom r e a c t i o n  s e v e r a l nanoseconds o f  the muon's e n t r y i n t o the t a r g e t , whereas the MSR s i g n a l r e l a x a t i o n i s measured over s e v e r a l microseconds. As d i s c u s s e d  i n Chapter I I I , the experimental and  t h e o r e t i c a l s i t u a t i o n with respect and  t o the H + HCl a b s t r a c t i o n  exchange r e a c t i o n s i s r a t h e r confused.  An e x c e l l e n t  review of these r e a c t i o n s has r e c e n t l y been prepared by Weston  (79).  F o r the H + HCl + H  2  + C l r e a c t i o n , Weston (79)  recommends values  o f E = 3 . 1 8 + 0 . 1 7 kcal/mole, log. A(1/mole-s) a — iu = 9.87 + 0.11 and k(295K) = (2.1 + 0.2) x 1 0 1/mole-s. n  7  -184Although most experimental evidence i n d i c a t e s t h a t  the  a b s t r a c t i o n r e a c t i o n i s f a s t e r than the exchange r e a c t i o n , t h i s question  remains unresolved  [Weston (79)].  Bott and  Heidner  (76) measured the t o t a l r e a c t i o n r a t e f o r H + HCl d i r e c t l y l a s e r induced f l u o r e s c e n c e 1/mole-s.  and  found k(295K) =  Since the r a t e constant r e p o r t e d  (9+4)  x  by 7  10  here f o r the  Mu  + HCl r e a c t i o n i s a l s o a t o t a l r a t e c o n s t a n t , the Bott and . Heidner H atom r a t e constant p r o v i d e s the most u s e f u l comparison g i v i n g k  M u  /k  at 2 95K  H  + 0.002.  <_Q"'.004  the v e r y l a r g e u n c e r t a i n t i e s i n the H and Mu c e r t a i n t h a t the t o t a l Mu 100  Even w i t h  data, i t i s  r e a c t i o n r a t e w i t h HCl  i s at l e a s t  times slower than the corresponding H atom r a t e !  It  i s unnecessary to t u r n t o fancy d e t a i l e d t h e o r e t i c a l c a l c u l a ions to e x p l a i n t h i s r e s u l t .  From Table V,  both r e a c t i o n channels f o r Mu A H Q = +6.2  and  respectively. -1.1 The  and  +7.9  with HCl  are  i t i s seen t h a t  endothermic:  kcal/mole f o r a b s t r a c t i o n and  In c o n t r a s t , the H + HCl r e a c t i o n s g i v e A H Q =  0 kcal/mole f o r a b s t r a c t i o n and  exchange r e s p e c t i v e l y .  c l a s s i c a l b a r r i e r s f o r these r e a c t i o n s , though  known, are thought t o be ^4 reaction  [Thompson  reaction  [Weston (79)].  were zero, Mu  exchange  (75)]  kcal/mole f o r the  and  abstraction  even more f o r the exchange  However, even i f the c l a s s i c a l b a r r i e r s  must overcome an enormous zero p o i n t  b a r r i e r of at l e a s t 6.2  poorly  energy  kcal/mole i n order to r e a c t with  HCl;  a b a r r i e r which i s at l e a s t 6.2  kcal/mole g r e a t e r  than  the  r e a c t i o n b a r r i e r f o r H + HCl.  In the absence of  reliable  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 g e n e r a l  p r e d i c t i o n s on the Y +  Cl„  -185reactions given  i n Chapter I I I , p 97-109.  ^rMuBr + H E Mu  + HBr The  -> MuH MSR  + Br  .  relaxation r a t e s at v a r i o u s HBr  concentrations,  measured i n Ar moderator at 295K, are l i s t e d illustrated  i n Figure  II-2  (Appendix I I ) .  i n Table XV 2 Ax  minimum f i t  of these r e l a x a t i o n r a t e data to equation 11(3) t o t a l bimoleeular tion reactions k(295K) =  r a t e constant  and  yields a  f o r the exchange p l u s  abstrac-  of (9.09  + 0.97)  x 10  9  1/mole-s  (la)  I n s o f a r as. i t i s known, Endo et a l . (76) and Takacs e t a l  v  (73)  have made the only d i r e c t measurements of the analogous H r e a c t i o n r a t e using ESR  d e t e c t i o n of H atoms i n a flow  T h e i r measurements, summarized i n Table XVI, parameters f o r the  sum  of the exchange p l u s  channels f o r the four H and r e a c t i o n between 230 f o r the H + DBr  and  and  as w e l l as the r a t e  r e a c t i o n s at 2 95K  D,  Y'Br constants  from which the  abstraction:exchange branching r a t i o s are obtained. measurements c l e a r l y show t h a t f o r H and  rate  abstraction  D v a r i a t i o n s of the Y +  318K,  D + HBr  gave the  system.  the  These  abstraction  r e a c t i o n channels are much f a s t e r than the exchange r e a c t i o n channels at room temperature. r e a c t i o n a c t i v a t i o n energies  From the measured a b s t r a c t i o n and  the estimated exchange r e a c -  t i o n a c t i v a t i o n energy, i t can be i n f e r r e d t h a t the  classical  b a r r i e r to a b s t r a c t i o n i s about l.,5-3 kcal/mole whereas the b a r r i e r to exchange i s about 5 kcal/mole. inference-  i t must be cautioned  In drawing t h i s  t h a t a c t i v a t i o n energies  approximately equal t o the c l a s s i c a l b a r r i e r heights  are  only i n  -186-  TABLE XV:  MSR RELAXATION RATES FOR THE TOTAL Mu + HBr REACTION AT 2 95K  [HBr] (10~ M) 4  Relaxation  Rate  A (ys )  0.00  0.27 + 0.04  0.77 + 0.07  0.92 + 0.09  1.50 + 0.03  1.60 + 0.26  2.85 + 0.08  2.56 + 0.71  4.49 + 0.10  4.76 + 0.94  5.92 + 0.13  6.66 + 1.20  r e l a x a t i o n r a t e s r e p o r t e d are weighted averages of the l e f t and r i g h t p o s i t r o n t e l e s c o p e histograms.  TABLE XVI:  EXPERIMENTAL RATE PARAMETERS FOR THE REACTIONS: Y + Y'Br^T C-r> YBr I + vYi H, D; Y' = H,D  Reaction  E ' a (kcal/mole)  log A(M s ~ ) 1  1  1Q  (10  Mu+HBr -»• products H+HBr  k(295K) 9  1/mole-s)  Y+HBr k  , Y = Mu,  (295K)  H+HBr  AH°  [type]  (kcal/mole)  9. 09 + 0.97  4.4 + 0.6  products  2.57 + 0.11  11.22 + 0.05  2.08 + 0.16  1.0  D+HBr •> products  2.13 + 0.08  10.59 + 0.03  1.02 + 0.05  0.49 + 0.04  -9.4[abs] +7.1[exc] -16.7 [abs] 0 [exc] -17.5[abs] -1.1 [exc]  Y+DBr (295K) H+DBr H+DBr ->• products  2.19+0.11  10.82+0.04  1.57 + 0.18  1.0  D+DBr  1.69 + 0.13  10.14 + 0.03  0.78 + 0.11  0.50 + 0.09  products  "abs  (295K)  exc D+HBr -> DBr+H  5.2  H+DBr -* HBr+D  10  0.0078 + 0.0024 <0.023  -16.4[abs] +1.1 [exc] -17.4[abs] 0 [exc]  137  + 60 -32  >69  between 23 0 and 318K estimates based on the measured r a t e constant a t 2 95K and high temperature beam and p h o t o l y s i s data  [Endo (7 6) and r e f e r e n c e s t h e r e i n ] .  molecular  -188the absence of strong dynamical p 109,  e f f e c t s ; as d i s c u s s e d on  i t i s p o s s i b l e t h a t the a c t i v a t i o n e n e r g i e s are  ed by r o t a t i o n a l s c r e e n i n g of the H i s o t o p e . the Table i t appears  t h a t a t 295K, the  branching r a t i o f o r D + HBr  govern-  In f a c t ,  from  abstraction:exchange  i s g r e a t e r than f o r H +  DBr.  T h i s i s c o n s i s t e n t w i t h the r o t a t i o n a l s c r e e n i n g hypothesis s i n c e the D atom approaches the more q u i c k l y r o t a t i n g  HBr  about /2 times slower than the H atom approaches the more s l o w l y r o t a t i n g DBr.  As remarked on p 109,  s c r e e n i n g i s important, abstraction:exchange  i t i s expected  i f rotational  t h a t f o r Mu,  the  branching r a t i o w i l l be s m a l l e r than f o r  the o t h e r H i s o t o p e s a t 2 95K due t o the g r e a t e r mean v e l o c i t y of the l i g h t e r muonium atoms. argued  On the other hand, i t can  t h a t the exchange channel  much suppressed  because AHg  f o r Mu + HBr  = +7.1  minimum exchange b a r r i e r f o r Mu  should be very  kcal/mole  + HBr.  ; t h i s i s the  I f the b a r r i e r h e i g h t s  estimated from the a c t i v a t i o n e n e r g i e s are c o r r e c t , then e f f e c t i v e Mu.+  HBr  be  the  exchange b a r r i e r i s at l e a s t twice as l a r g e  as the a b s t r a c t i o n b a r r i e r ; t h e r e f o r e , i t seems reasonable t o assume t h a t the measured v a l u e of k f o r Mu + HBr corresponds  essentially  t o the a b s t r a c t i o n r e a c t i o n channel o n l y .  Table XVI g i v e s the r a t e constant r a t i o s Mu:H:D a t 300K of 4.4  + 0.6:1.0:0.49 + 0.04,  independent Mu:H  and H:D  which exceeds the  mass r a t i o s by 1.52 respectively.  t h a t t u n n e l l i n g i s expected  + 0.21  and 1.47  temperature + 0.12  for  T r u h l a r (79) has p o i n t e d out to be more important  f o r a given  H + HX a b s t r a c t i o n r e a c t i o n than f o r the corresponding H + r e a c t i o n s , because the imaginary  frequency of the unbound normal  -189-  mode v i b r a t i o n  of the t r a n s i t i o n s t a t e tends t o be much  l a r g e r f o r the H + HX systems than f o r the H + X^  systems  (see F i g u r e 18; i n t h i s F i g u r e , the a parameter i s i n v e r s e l y p r o p o r t i o n a l t o the imaginary  frequency  [Johnston  (61)]).  T h i s i s e q u i v a l e n t t o s a y i n g t h a t H + HX b a r r i e r s tend t o be narrower than the H + X^ b a r r i e r s .  Assuming t h a t the r a t e  constant r a t i o s r e p o r t e d above do correspond t o a b s t r a c t i o n , then they may be i n t e r p r e t e d as an i n d i c a t i o n o f t u n n e l l i n g i n this reaction. appears  Since the b a r r i e r h e i g h t f o r a b s t r a c t i o n  t o be comparable t o the H + F^  by analogy  i t may a l s o be expected  important a t 3 00K.  barrier  (^2 k c a l / m o l e ) ,  t h a t t u n n e l l i n g w i l l be  Although the degree of t u n n e l l i n g  cannot  be i n f e r r e d from a s e t of r a t e constant r a t i o s a t one temperat u r e , i t appears  t h a t Mu and H t u n n e l comparable amounts when  r e a c t i n g with HBr a t 295K.  T h i s i s c o n s i s t e n t w i t h the  d i s c u s s i o n on p 105 where i t i s suggested  t h a t "corner c u t t i n g "  might e q u a l i z e the t u n n e l l i n g advantage o f the v a r i o u s i s o t o p e s due t o the mass weighted  c o o r d i n a t e skewing angles o f 7 2 ° ,  4 5 ° , 3 6 ° , and 31° f o r Mu, H, D and T r e s p e c t i v e l y . Table XVI shows the a c t i v a t i o n e n e r g i e s f o r the H r e a c t i o n s w i t h HBr and DBr t o be l a r g e r than those f o r D by about 25-30%.  T h i s r e s u l t may be e x p l a i n e d i n terms o f the  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 s which should be c o n s i d e r a b l y l a r g e r f o r the l i g h t e r H i s o t o p e s (see p 104 and Table V ) . On the other hand, t h i s seems t o c o n t r a d i c t the t u n n e l l i n g hypothesis j u s t d i s c u s s e d , s i n c e , d e s p i t e corner c u t t i n g , H i s expected  t o t u n n e l more than D, thereby p r e d i c t i n g s m a l l e r  a c t i v a t i o n e n e r g i e s f o r H + HBr than f o r D + HBr.  Of course,  -190the p o s s i b i l i t y always e x i s t s t h a t the experimental data in error. i n order  are  I f i t i s assumed t h a t these data are c o r r e c t , then, to r a t i o n a l i z e the two  t i o n s t h a t even though H + HBr than D + HBr,  H + HBr  still  seemingly c o n t r a d i c t o r y observahas  a l a r g e r a c t i v a t i o n energy  r e a c t s f a s t e r at 295K, i t seems  necessary to p o s t u l a t e t h a t these systems have unusual e x c i t a tion  functions  curves).  ( i . e . c r o s s s e c t i o n v e r s u s c o l l i s i o n energy  I t w i l l be i n t e r e s t i n g t o see i f Mu  + HBr  also  f o l l o w s t h i s t r e n d by having a larger a c t i v a t i o n energy than H +  HBr.  F Mu  + HI The  ^ Mul MuH MSR  + H + I  r e l a x a t i o n r a t e s at v a r i o u s HI  measured i n both argon and listed  i n Table XVII and  N  2  concentrations,  moderator gases at 295K, are 2  i l l u s t r a t e d i n Figure  35.  Ax  minimum f i t of these r e l a x a t i o n r a t e data to equation y i e l d s a t o t a l bimoleeular  r a t e constant  11(3)  f o r the exchange  p l u s a b s t r a c t i o n r e a c t i o n s of k(295K) =  (2.53  + 0.12)  x 10  1/mole-s  1 0  (la)  I n s o f a r as i t i s known, the analogous H atom r e a c t i o n r a t e has  never been d i r e c t l y measured, and  the  indirect  measurements t h a t have been made are sparse and [Bauer 2-(78)]. (73)  report E  k(295K) =  &  unreliable  For the a b s t r a c t i o n r e a c t i o n , Jones et a l . = 0.7  + 0.25  (1.5+0.5) x 1 0  kcal/mole, l o g A (1/mole-s) = 10.7, 1 Q  1 0  1/mole-s, based on the  thermal r e a c t i o n experiments of S u l l i v a n (62) 800K, which gives k  M u  /k  H  = 1.7  + 0.6  H /I  2  between 6 67  when e x t r a p o l a t e d  However, t h i s estimate must r e p r e s e n t  2  and  and  to 295K.  some type of r e - a n a l y s i s of  TABLE XVII:  MSR RELAXATION RATES FOR THE TOTAL Mu + HI REACTION AT 2 95K  [HI] (10 M) _4  0 . 00 0.36  Relaxation A - A 0 . 00  (us  Rate - 1  )+  *  + 0 . 01  0.96  + 0.13*  1 . 03 + 0 . 03  2.35  + 0.20*  1.39  + 0.04  4.19  2.03  + 0 . 06  5.11  +• 0 . 5 5 * * + 0 . 67  0.00 0.46  0. 00§  + 0.02  §  0 . 98 + 0 . 03  3.58  + 0 . 42 + 0. 6 3 §  1.25  3.84  + 0.52§  + 0.03  1.29  r e l a x a t i o n r a t e s r e p o r t e d are weighted averages of the l e f t and r i g h t  t e l e s c o p e histograms, g i v e n as A -  f o r the s l i g h t l y d i f f e r e n t background argon and N . 2  j argon  moderator  t  N~  moderator  A  q  t o account  relaxation rates i n  -192-  M U / H I . •=  •  50  N . = 2  A  RR MODERATOR.  100  150  200  295 K  250  HI CONCENTRATION (uM) FIGURE 35:  MSR r e l a x a t i o n r a t e s as a f u n c t i o n of HI c o n c e n t r a t i o n i n argon (diamonds) and N2 (squares). The data are p l o t t e d as A - A Q to account f o r the small d i f f e r e n c e s i n A Q f o r 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 M" ,'s' . 1 0  1  1  -193S u l l i v a n ' s data s i n c e the paper r e f e r e n c e d r e p o r t s E + 0.25 =  kcal/mole,  (1.1 + 0.2) —  x 10  l o g A ( 1 / m o l e - s ) = 9.05 1Q  9  and  0.0  and k(295K)  TT  P h o t o l y s i s experiments  Kuppermann (74) g i v e a b s t r a c t i o n f r a c t i o n s + 0.04  =  1/mole-s which g i v e s k.. /k = 23 + 4 when ^ Mu H —  e x t r a p o l a t e d to 2 95K.  = 0.95  + 0.07  &  0.88  + 0.08  of Persky  (k , /[k ,  f o r H + DI and D + HI  and  + k  ].)  respec-  t i v e l y , which again, i n d i c a t e s t h a t a b s t r a c t i o n i s much f a s t e r than exchange f o r these H-HX  reactions.  be c a u t i o n e d t h a t i n an analogous  However, i t should  experiment  w i t h HBr,  a b s t r a c t i o n f r a c t i o n s i n d i r e c t l y obtained by Persky  the  and  Kuppermann (74) have the o p p o s i t e o r d e r i n g t o 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 w i t h these  r e a c t i o n s , l i t t l e can be s a i d about the Mu + HI r e a c t i o n isotope e f f e c t s .  I t appears  t h a t the r e a c t i o n b a r r i e r f o r  a b s t r a c t i o n i s very s m a l l , which e x p l a i n s why r a t e constant i s l a r g e r than the Mu + HBr a l s o appears c  c  and  the Mu + HI  rate constant.  It  t h a t k„„ /k a t 295K f o r t h i s r e a c t i o n i s g r e a t e r Mu .H TT  than one, as expected,  though t h i s estimate i s based on  e x t r a p o l a t i o n of very q u e s t i o n a b l e data.  an  Finally,, based  on  the experimental r e s u l t s of Persky and Kupperman and i n v o k i n g the e n d o t h e r m i c i t y arguments of the p r e v i o u s S e c t i o n s , i t seems reasonable t o again suggest t h a t the a b s t r a c t i o n r e a c t i o n channel dominates, the Mu  + HI r e a c t i o n a t 3 0OK.  ment of the a c t i v a t i o n energy  of Mu + HI a t 3 0OK  A measurewould  c e r t a i n l y r e p r e s e n t a s u b s t a n t i a l i n c r e a s e i n the a v a i l a b l e data on the Y + HI  system.  -194CHAPTER V - SUMMARY AND  CONCLUSIONS  A Summary This t h e s i s describes,  i n c o n s i d e r a b l e d e t a i l , the present  experimental and t h e o r e t i c a l s t a t e o f the study of gas phase muonium r e a c t i o n k i n e t i c s .  On the experimental s i d e , i t out-  l i n e s most o f 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 c u r r e n t l y implemented s o l u t i o n s t o these problems,  and makes some s p e c i f i c suggestions f o r f u r t h e r  improvements.  P a r t i c u l a r a t t e n t i o n i s p a i d t o coping w i t h the  data acquisition problems t h a t a r i s e i n h a n d l i n g 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 s i d e ,  the remarkably  l a r g e body of c a l c u l a t i o n s , mainly due t o Connor,  Jakubetz, Manz, and Lagana, p r o v i d e d e t a i l e d i n t e r p r e t a t i o n s o f the experimental r e s u l t s and e s t a b l i s h the r e l e v a n c e o f gas phase muonium r e a c t i o n k i n e t i c s t o the more c o n v e n t i o n a l and more g e n e r a l f i e l d s o f chemical k i n e t i c s and molecular dynamics.  Two main c o n t e n t i o n s are made:  reaction  (1) t h a t muonium  p r o v i d e s an u n u s u a l l y u s e f u l t o o l w i t h which t o i n v e s t i g a t e hydrogen i s o t o p e of the Y + X  2  e f f e c t s , s p e c i f i c a l l y i n terms o f the dynamics  and Y + HX r e a c t i o n s ,  and (2) t h a t the p e c u l i a r  p r o p e r t y o f the MSR technique - t h a t i t l i t e r a l l y examines one atom a t a time - b l e s s e s  i t w i t h some d i s t i n c t advantages over  c o n v e n t i o n a l H atom s t u d i e s .  Thanks t o the t h e o r e t i c a l work  of Connor e t a_l. , the experimental study o f gas phase muonium reactions  seems t o have sparked progress i n both the understand-  i n g o f the Y + F^  and Y + C l r e a c t i o n s 2  and i n the development  of u s e f u l computational t o o l s f o r d e a l i n g w i t h them. more experimental data on the r e a c t i o n s  However,  o f Mu and H are r e -  -195q u i r e d i n order t o f i r m l y e s t a b l i s h the MSR method.  B Past P e r s p e c t i v e  Chapter  I i n c l u d e s a b r i e f h i s t o r i c a l summary t h a t sketches  the development of muonium chemistry i n gases up t o 197 5; the d i s c u s s i o n then takes a quantum leap by d e s c r i b i n g the present s t a t u s o f the s u b j e c t .  In many ways, t h i s c r e a t e s a d i s t o r t e d  p e r s p e c t i v e which t h i s S e c t i o n s h a l l attempt  to correct.  The  work i n t h i s t h e s i s took p l a c e d u r i n g a time when the use o f muons as probes  of p h y s i c a l phenomena matured from seed t o  s e e d l i n g - 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, i n 1975, was r a t h e r n e g l e c t e d compared w i t h the a p p l i c a t i o n o f u SR t o s o l i d +  physics.  state  I n d i c a t i o n s o f the maturation o f gas phase muonium  chemistry from 1975 t o the present a r e many; F i g u r e 36 p r o v i d e s a g r a p h i c example o f i t s experimental development.  The F i g u r e  i s a r e p r o d u c t i o n of t y p i c a l MSR s i g n a l s obtained d u r i n g the study o f the Mu + Br^ r e a c t i o n a t the Lawrence Berkeley a t o r y i n 1975 [Fleming r a t e o f 2 x 10  (76)].  Labor-  With an average muon stopping  3-1 . . . s , these were n e c e s s a r i l y low s t a t i s t i c s 5  of t y p i c a l l y 1.5 x 10  events.  the TRIUMF s p e c t r a i l l u s t r a t e d less primitive conditions.  runs  F i g u r e 36 may be compared w i t h i n Figure I I - l ,  taken under much  The e a r l y MSR experiments were  c h a r a c t e r i z e d by an almost t o t a l p r e o c c u p a t i o n w i t h gadgets and gizmos r e q u i r e d t o o b t a i n muons and u l t i m a t e l y t o o b t a i n data  -196-  0.20  0.I0H LxJ O O.OOH CL  <  -0.I0H  -0.20H 0  1 25  1 50  1 75  1 100  1 125  1 150  - r — - r 175 2 0 0  _ 225  BIN NUMBER (20 NANOSECONDS / BIN) 0.20-  O.IOH LU Q Z>  • O.OOH -o.ioH  -0.20 0  25  50  75  100  125  150  175  200  225  BIN NUMBER (20 NANOSECONDS / BIN) FIGURE 36:  Gas phase MSR s i g n a l s i n a magnetic f i e l d of 1.8 gauss obtained a t LBL i n 1975. The t a r g e t cont a i n e d pure Ar (top) and Ar w i t h ^10 ppm Br,, (bottom) a t 295K and one atmosphere p r e s s u r e . The e r r o r bars are due t o counting s t a t i s t i c s ^ only. Each histogram c o n t a i n s about 150 x 10 events. The l i n e i s a x minimum f i t t o an approximation of equation (.8) • [Fleming (7 6) ] .  -197-  from them; "doing p h y s i c s " seemed t o p l a y a s u b o r d i n a t e r o l e . A t t i m e s i t appeared t h a t the p o s s i b i l i t y o f h a v i n g t h e c y c l o t r o n , beam'.lines, c o u n t e r s , e l e c t r o n i c s , and d a t a 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 s i m u l t a n e o u s l y was f a n c i f u l dream.  l i t t l e more than a  Today, t h e p r i o r i t i e s a r e u s u a l l y r e v e r s e d .  G e t t i n g muons and t a k i n g d a t a are m o r e - o r - l e s s r o u t i n e ; e q u i p ment breakdowns are l e s s f r e q u e n t and tend t o be r a t h e r than c a t a s t r o p h e s . now to  irritations  The b u l k of the e x p e r i m e n t a l e f f o r t  goes i n t o d e s i g n i n g 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 e x p l o r e new  and o f t e n more s u b t l e phenomena.  T h i s t h e s i s work l e a n e d h e a v i l y on t h e t h e o r e t i c a l work o f Connor e t al_. , not o n l y i n o r d e r t o e x p l a i n t h e r e s u l t s , but a l s o as a g u i d e w i t h which t o f o r m u l a t e an e x p e r i m e n t a l strategy.  T h i s happy s y m b i o s i s o f t h e o r y and experiment  about by a f o r t u n a t e c h a i n o f c i r c u m s t a n c e .  The  came  f i r s t pub-  l i c a t i o n o f a low p r e s s u r e gas phase r e a c t i o n r a t e c o n s t a n t , f o r the Mu + B r MSR  2  reaction  [Fleming  ( 7 6 ) ] , s e t f o r t h t h e b a s i s of t h e  t e c h n i q u e 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 o f i t s p r o s p e c t u s . a t h e o r e t i c a l QMT  Connor e t al_. , who  had j u s t  study o f the c o l l i n e a r H + F^  reaction  ( 7 6 ) ] , p i c k e d up on t h i s Mu paper, d e c i d e d t o extend c a l c u l a t i o n s t o i n c l u d e t h e Mu, suggested  A l t h o u g h e x p e r i m e n t a l work was  and  r e a c t i o n be  i n p r o g r e s s on the  o t h e r r e a c t i o n s r e p o r t e d i n t h i s t h e s i s , t h e r e was ^  [Connor 2-  their  H, D, and T r e a c t i o n s ,  t h a t an e x p e r i m e n t a l s t u d y o f the Mu + F^  c a r r i e d out.  completed  reason to  Around c y c l o t r o n f a c i l i t i e s , one speaks o f "doing p h y s i c s " s a y i n g "doing c h e m i s t r y " 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 o d i o u s manners, except when you prop up t h e i r t h e o r y ; then you can borrow money from them." Mark Twain, What i s a Man and Other E s s a y s , 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  and, b e s i d e s , the use o f F^  w i t h t h e thin-windowed  slow,  target  a p p a r a t u s would c r e a t e f o r m i d a b l e ( p r o b a b l y i n s u r m o u n t a b l e ) problems. the  F o r t u n a t e l y , t h e t h e o r i s t s ' judgement p r e v a i l e d , and  experiment proved t o be f e a s i b l e .  i c a l c a l c u l a t i o n s on t h e Mu + F^  The p r e l i m i n a r y t h e o r e t -  reaction  [Connor 1-(77)] were  completed a few months b e f o r e t h e experiment [Garner  (78)].  As d i s c u s s e d i n Chapter IV, t h e t h e o r e t i c a l c a l c u l a t i o n s have s i n c e been r e v i s e d and supplemented.  S u b s e q u e n t l y , the  e x p e r i m e n t a l 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 . E x p e r i m e n t a l r e s u l t s on t h e Mu + Cl,, r e a c t i o n  [Fleming (79) ] have  been f o l l o w e d by TST c a l c u l a t i o n s of J a k u b e t z (7 9),  while  and QCT  2-(78)].  c a l c u l a t i o n s a r e p r e s e n t l y underway [Connor  QMT  C Future Perspective The. .experimental i n t e r e s t i n gas phase muonium i s by no means c o n f i n e d t o t h e s t u d y o f i t s t h e r m a l c h e m i c a l r e a c t i o n s . A t TRIUMF, programmes a r e i n , p r o g r e s s t o examine t h e muonium formation process ( u M i k u l a and D.G. species  +  charge exchange) i n v a r i o u s gases [ R . J .  F l e m i n g ] , muonium s p i n exchange w i t h paramagnetic  [D.G. F l e m i n g , R.J. M i k u l a , and D.M.  G a r n e r ] , and t h e  p r o d u c t i o n o f t h e r m a l muonium i n vacuum t h r o u g h the use o f f i n e powdered i n s u l a t o r s as a s t o p p i n g medium [G.M. M a r s h a l l , R. and J.B. Warren].  I t i s highly l i k e l y  Kiefel,  t h a t b o t h t h e under-  s t a n d i n g of t h e s e phenomena and t h e development  of experimental  t e c h n i q u e s w i t h which t o s t u d y them w i l l have a mutual impact on the  f u t u r e s t u d i e s o f Mu c h e m i c a l r e a c t i o n r a t e s . I t seems t h a t t h e immediate o b j e c t i v e , o f f u t u r e gas  phase  -199Mu r e a c t i o n r a t e experiments ought t o be t o complete the present study.  In p a r t i c u l a r ,  i t i s d e s i r a b l e t o develop a new t a r g e t  r e a c t i o n v e s s e l that provides  an o p e r a t i o n a l temperature range  of from about 200 - 600K, i f p o s s i b l e , and t o extend the r a t e measurements o f the r e a c t i o n s s t u d i e d i n t h i s t h e s i s t o span t h a t temperature range. atom r e a c t i o n r a t e data, of i s o t o p i c  In c o n j u n c t i o n  w i t h the analogous H  t h i s would provide  r a t e parameters f o r two dynamically  c l a s s e s o f elementary chemical r e a c t i o n s . reactions  ( i f p o s s i b l e , X should  temperature range should Arrhenius  a very complete s e t different  For the Y + X^  be extended t o i n c l u d e I ) , t h i s  be s u f f i c i e n t t o check the p r e d i c t e d  p l o t curvature,  thereby p l a c i n g a f i r m e r  handle on the r e a c t i o n dynamics. the r e a c t i o n dynamics should  experimental  In changing X from F t o I,  g r a d u a l l y s h i f t from c o l l i n e a r  domination t o the f u l l three dimensional r e a c t i o n which may d r a m a t i c a l l y a f f e c t the H i s o t o p e e f f e c t s . the Y + HX r e a c t i o n s are dynamically  As a l r e a d y  described,  much more complicated  than  the Y + X^ r e a c t i o n s , and, i n f a c t , the Mu + HX r e a c t i o n s may be quite different p.98,  from the H + HX r e a c t i o n s .  i t i s described  In Chapter I I I ,  how i t may be p o s s i b l e t o determine the  e x c h a n g e : a b s t r a c t i o n branching r a t i o s f o r Mu + HX r e a c t i o n s by simply  obtaining Arrhenius  data over a wide temperature range.  Chapter I I I , p. 102, a l s o d e s c r i b e s  how low temperature r a t e  data on these r e a c t i o n s might provide  information  about the  presence or absence o f w e l l s i n the Mu + HX p o t e n t i a l s u r f a c e s . I t seems probable t h a t t h i s temperature range extension present  experiments w i l l provide  o f the  new dynamical i n f o r m a t i o n on  these important elementary r e a c t i o n s .  -200For many years, the u l t i m a t e r e a c t i o n s f o r experimental k i n e t i c s study have been the i s o t o p i c v a r i a t i o n s of the H + E^ r e a c t i o n s , mainly because they have been the s u b j e c t of exhaustive t h e o r e t i c a l i n v e s t i g a t i o n . Mu + E^ r e a c t i o n  P r e l i m i n a r y data on the  [Mikula (7 9)] i n d i c a t e s t h a t a t room  temperature, 5  the r a t e i s a t or near the lower l i m i t of the MSR 1 mole ^ s "S .  (-10  However, t h i s r e a c t i o n , which has a high  a c t i v a t i o n energy between 400  method  (_> 7 kcal/mole  [Jones  (73)]), may  be measurable  - 600K where i t i s a t l e a s t from 20 - 350  f a s t e r than at 3 00K.  C e r t a i n l y , the experimental  times  investigation  of t h i s r e a c t i o n should have a h i g h p r i o r i t y i n the immediate future. Looking deeper i n t o the c r y s t a l b a l l , where should phase Mu  gas  r e a c t i o n k i n e t i c s go a f t e r the programme o u t l i n e d above  i s completed?  T h i s may  depend s t r o n g l y on the p r e v a i l i n g  technology and theory a t 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 t o measure the r e a c t i o n r a t e s of muonium w i t h other molecules.  The d e s i r a b i l i t y of such a  programme w i l l depend l a r g e l y on the understanding effects.  I f the d i f f e r e n c e s and  of Mu  isotope  s i m i l a r i t i e s between Mu  and  H r e a c t i o n r a t e s c o u l d be p r e d i c t e d w i t h c o n f i d e n c e , then the MSR  method may  p r o v i d e a more a c c u r a t e means of measuring H  atom r e a c t i o n r a t e s .  More i m p o r t a n t l y , MSR  may  a l s o be a p p l i e d  to chemical systems where d i r e c t measurements of the H atom r e a c t i o n r a t e s are not e x p e r i m e n t a l l y p o s s i b l e . One  v e r y i n t e r e s t i n g d i r e c t i o n gas phase muonium k i n e t i c s  might take i s to venture i n t o the realm of chemistry.  state-to-state  U n f o r t u n a t e l y , the experimental o b s t a c l e s t o such  -201-  a study p r e s e n t l y appear t o 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 e x i s t ; o n l y v e r y h i g h energy muon ion  beams do.  Some p r o g r e s s has been made i n producing thermal  muonium i n vacuum  [Marshall  (78)]; however, even i f an  ideal  "muon t o muonium c o n v e r t e r " e x i s t e d , 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 t r a v e l s about one i n c h d u r i n g i t s  lifetime. intrinsic  A s s o c i a t e d w i t h the l i f e t i m e problem beam i n t e n s i t y problem.  would l i k e l y not be MSR  i s the .  State-to-state  experiments  experiments, but would employ some other  d e t e c t i o n technique such as i n f a r e d chemiluminescence.  Even  at meson f a c t o r i e s , the most i n t e n s e muon beam i d e a l l y would d e l i v e r no more than 10 A c c o r d i n g t o Appendix  8  y  +  s  —1  over a 1 cm  2  available  area.  I I I , the average number of muons i n a  t a r g e t a t any time, g i v e n a 100% duty c y c l e beam, i s ^ 8  T u  »  where  —1  71 i s the beam c u r r e n t ; w i t h 91 = 10 s more than 2 00 muons i n i t a t a time.  , the t a r g e t would have no T h i s may be improved by  u s i n g a low duty c y c l e a c c e l e r a t o r which c o u l d d e l i v e r b u r s t s of 4 10  5 - 10  muons a t any i n s t a n t .  formidable problem of measuring  Still,  an o b s e r v a b l e a t such  on the time s c a l e of the muon l i f e t i m e . t h a t Mu  infared  one i s faced wxth the  chemiluminescence  intensities  For example, i t appears  experiments would r e q u i r e  i n f a r e d , energy s e l e c t i v e , s i n g l e photon counters - the i n f a r e d analogue of such gamma r a y d e t e c t o r s as sodium  iodide  crystals.  D Concluding Remarks "He Two  i s not a l i a r , but he w i l l become one i f he keeps on." Mark Twain, F o l l o w i n g the Equator, p. 2 91  a s p e c t s o f t h i s t h e s i s work have, I t h i n k , g i v e n me  an  -202-  u n u s u a l and unique v i e w o f gas phase c h e m i c a l k i n e t i c s i n g e n e r a l , and muonium c h e m i s t r y i n p a r t i c u l a r :  (1) t h e i n h e r e n t  i n t e r d i s c i p l i n a r y n a t u r e o f a s u b j e c t which e x p l o i t s p a r t i c l e p h y s i c s as a c h e m i c a l t o o l , and (2) t h e t i m i n g o f my  involvement  i n t h e programme, which spanned t h e t w i l i g h t days o f t h e 18 4" C y c l o t r o n a t Berkeley t o the e a r l y stages of the o p e r a t i o n of the TRIUMF meson f a c t o r y .  On t h e 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 o f innocence - as I r e c a l l , t h e f i r s t q u e s t i o n I ever asked my r e s e a r c h s u p e r v i s o r was, " What i s a muon?" questions  To t h i s day, my w i f e , an a r t h i s t o r i a n ,  still  ( q u i t e s e n s i b l y , I t h i n k ) t h e s a n i t y o f people who  c l a i m t o 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 t h e i n f a n t r e s e a r c h p r o j e c t ,  t h i s gave me a range o f e x p e r i e n c e i n MSR t h a t w i l l n o t o r d i n a r i l y be a v a i l a b l e t o f u t u r e MSR w o r k e r s .  As an o l d t i m e r ,  I have h e l p e d b u i l d t h e c y c l o t r o n and beamlines  (which a r e now  b u r i e d 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 i n v o l v e d i n beam l i n e t u n i n g and t h e d e s i g n and i m p l e m e n t a t i o n of  d a t a a c q u i s i t i o n and a n a l y s i s systems.  Of c o u r s e , I d i d n o t  always c h e r i s h t h i s e x p e r i e n c e which seemed a t many t i m e s t o be f r u s t r a t i n g drudgery.  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Fleming, and J.H. Brewer, Chem. Phys. L e t t e r s 55 (1978) 163.  (7 8)  Jakubetz (79)  W. Jakubetz, J . Am. Chem. S o c , 101 (1979) 29i  Jones (73)  W.E. Jones, S.D. MacKnight, and L. Teng, Chem. Rev. J! (1973) 407.  M a r s h a l l (78)  G.M. M a r s h a l l , J.B. Warren, D.M. Garner, G.S. C l a r k , J.H. Brewer, and D.G. Fleming, Phys. L e t t e r s 65A (1978) 351.  M i k u l a (79)  R.J. M i k u l a , p r i v a t e cummunication,  (1979)  -217-  L i t e r a t u r e C i t e d - Appendix I B r e i t (31)  G. B r e i t and I, Rabi, Phys. Rev. 38 (1931) 2082.  Brewer  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.  (7 5)  C a r r i n g t o n (67)  A. C a r r i n g t o n and A.D. McLachlan, I n t r o d u c t i o n to Magnetic Resonance, Harper and Row, New York, (1967), 14.  Fleming  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, H.J. Ache, Ed., 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 .  (7 9)  Gurevich (71)  I.I. B. A. V.P. Sov.  G u r e v i c h , I.G. I v a n t e r , E.A. Meleshko, N i k o l ' s k i i , V.S. Roganov, V . I . S e l i v a n o v , Smilga, B.V. Sokolov, and V.D. Shestakov, Phys. JETP 33 (1971) 253.  P e r c i v a l l-(76)  P.W. P e r c i v a l and H. F i s c h e r , Chem. Phys. 1(5 (1976) 89.  Schenck (76)  A. Schenck, i n Nuclear and P a r t i c l e P h y s i c s at Intermediate E n e r g i e s , 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-  L i t e r a t u r e C i t e d - Appendix I I Arnold  (68)  V . I . A r n o l d and A. Avez, E r g o d i c Problems i n Quantum Mechanics, Benjamin, New York, (19 68) 16.  Brewer  (7 2)  J.H. Brewer, Ph.D. T h e s i s , Lawrence Berkeley Laboratory Report, LBL-950, (1972).  Brewer  (75)  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.  Fleming  (7 9)  D. G. Fleming, D.M. Garner, L.C. v a z , 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 .  Schenck  (71)  A. Schenck and K.M. 26 (1971) 57.  Crowe, Phys. Rev. L e t t e r s  -219-  L i t e r a t u r e C i t e d - Appendix I I I Biswell  (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 L a b o r a t o r y Report LA-5144, (1973) .  Feller  (50)  W. F e l l e r , An I n t r o d u c t i o n t o 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 , W i l e y , London, (1950), 337.  Shlaer  (74)  S. S h l a e r , Los Alamos S c i e n t i f i c Report LA-511-MS, (1974).  Thomas (73)  Laboratory  R.F. Thomas, Los Alamos S c i e n t i f i c Report LA-5404-MS, (1973).  Laboratory  -220-  Appendix  I - The 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 Muonium i n a Transverse Magnetic F i e l d . S o l u t i o n s to the problem t r e a t e d i n t h i s  may  be found i n s e v e r a l r e f e r e n c e s  matrix formalism [Gurevich Percival l  -  (76)]  .  Appendix  which use the d e n s i t y  ( 7 1 ) , Brewer  ( 7 5 ) , Schenck ( 7 6 ) ,  In t h i s Appendix, the approach to the  problem f o l l o w s t h a t of Fleming (79) which i s , perhaps, more p h y s i c a l l y transparent  A. State  than the d e n s i t y matrix approach.  Vectors The i n i t i a l  described  using  quantization  the u  axis.  s t a t e s of the system are most e a s i l y +  s p i n p o l a r i z a t i o n d i r e c t i o n as the  Since  a l l of the muons are p o l a r i z e d  w h i l e the e l e c t r o n s captured initial  = \a^>  s t a t e s are  standard  t o form Mu are u n p o l a r i z e d , ±  B  convention i n which the f i r s t  s p i n and the second r e f e r s to e i n d i c a t e t h a t the q u a n t i z a t i o n a p p l i e d magnetic f i e l d . transverse a new  and ^ ( 0 )  spin.  = |a$>j_  the  u s i n g the  a or 3 r e f e r s to u  +  The s u b s c r i p t s j_  axis i s perpendicular  to the  The a p p l i c a t i o n of a magnetic  field  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 d e f i n e s  q u a n t i z a t i o n a x i s and i t i s the task of t h i s s e c t i o n t o  show the a p p r o p r i a t e functions  transformation  i n t o t h i s new  coordinate  of the i n i t i a l Mu system.  state  -221-  The illustrated  required  transformation  of s t a t e v e c t o r s i s  below:  Z  + y spin polarization  y/^ B  1  + y ^ P o l spin arization  -77 rotation 2  >  about Y - a x i s  old The  system  new  r o t a t i o n i s most e a s i l y a p p l i e d  respect  system  to spin states  labled with  t o the t o t a l Mu s p i n angular momenta, | JM>j_  , where J  i s the t o t a l Mu s p i n and M i s i t s p r o j e c t i o n on the o r i g i n a l quantization  a x i s , as u s u a l .  The i n i t i a l  expressed i n t h i s way by the a p p r o p r i a t e Wigner or Clebsch-Gordon r  M  m  from which one  |1,-t>  m a n i p u l a t i o n of  c o e f f i c i e n t s [Tinkham  mM-mM m v  s p i n s t a t e s can be  (64)] ,  M-m  Y  K  obtains:  =  ±  |1,0>.  =  I 0,0>,  |6B>L JL( | a B > L  + I Ba>, )  = 1 ( |aB>, /2  - I Ba>.  KD )  X  Thus, iK(0)  = I aa).  I{J (0) = b  I a3>,_  1,1> =1(11,0^  +  |0,0>L  )  /2* A r o t a t i o n of a s t a t e f u n c t i o n which i s a l i n e a r combination of b a s i s  vectors,  - 2 2 2 -  m m  m  —  —  about E u l e r i a n angles a,B,  and  y, i s accomplished w i t h  a p p l i c a t i o n of the r o t a t i o n operator [D (j)  R ( a , B , y ) ^ = Z.<f>  J  m Following  [Tinkham  (64)]  m'  m  the Condon and  the  S h o r t l e y phase convention g i v e n  in  IT ••'  Tinkham, the r e q u i r e d r o t a t i o n i s a=0,  3 = — y = 0 , where 3 i s  negative  s i n c e i t would advance a r i g h t handed screw i n the  negative  y-direction,  n  (j) „ f  U  I  A  R  v  ^  ' ^ '  V M  Since we  v  E  •  B > 2 j - 2 K-m +m ,  ! (J+.m') 1 (J-m' ) ,1  . B\2K+m'-m  1  (cos|)  (-sin|)  J  are r o t a t i n g the c o o r d i n a t e  system  (basis  vectors)  r a t h e r than f u n c t i o n s , f o r which the above e x p r e s s i o n was  c a l c u l a t e d , the r e q u i r e d r o t a t i o n corresponds to  D  (-Y,-B,-cO  ( j )  The  as  KI(j+m-K)I(j-m'-K)!(K+m'-m)I  K  E  .  m a t r i x elements are d e f i n e d  -imy ( - l ) V ( J + m ) i (J-m)  _ -im'g Y  The  f o r E) (j)  r e q u i r e d m a t r i x i s , then, -1  -1_ /2  1 /2  Thus , (0,|,0) 0) = i | l , ^ R  and  V  + /  J|l,0j>  +  R(0,|,0)^ (0) = ~|.|1,1^> + 1|1,-1^ B  where the s u b s c r i p t s || i n d i c a t e t h a t the respect  to the magnetic f i e l d  totally  symmetric b a s i s f u n c t i o n  rotation.  Using 1(1)  and  to transform  || !,-]>> +/  ||0,0^  s t a t e s are quantized  where i t i s noted t h a t | 0, (£>  i s unaffected  with  the by  back i n t o uncoupled s p i n  the  -223-  s t a t e s , [ro m ) , we o b t a i n : ' ' y e ' + |33^j>  ij^(0) = j(|aoCj[ ^ (0)  = |.(-|acxj>  B  B  |3fcCj|  +  + |a3^j> +  + | 3aj>)  |aB>.  K2)  - |Bc^)  Time E v o l u t i o n of the Mu States In order t o f i n d the e x p e c t a t i o n value of the y  p o l a r i z a t i o n i n the Schrodinger  p i c t u r e , i t i s necessary t o  determine the time e v o l u t i o n of the Mu s p i n s t a t e s . simply g i v e n by Tp (t)  +  These are  = U ( t ) ^ ( 0 ) where U(t) i s the time  e v o l u t i o n operator: U(t) = e  -i#t/n * ' where- 3X = 0 and  (0)  are the  9t eigenvectors of X . e  The  Hamiltonian  Zeeman terms and the y - e  hyperfine  +  s\  /\  interaction:  /\  X = g 3 BS ^ e  i s the sum o f the y and  - g3 Bl z  ^ y  + aS«I  z  where g i s the e l e c t r o n or muon g - f a c t o r  (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 s p i n o p e r a t o r s , and 'a' i s the Fermi c o n t a c t constant.The e l e c t r o n and muon magnetons are equal t o eh where m i s the 2mc corresponding  e l e c t r o n or muon mass.  c a l c u l a t e d t o be  The 'a' term has been  [see, f o r example, C a r r i n g t o n (67)]  a = 1p g 3 e g 3 k ( 0 ) |  2  y  2  i  where muon.  |^(0)|  i s the p r o b a b i l i t y d e n s i t y of the e l e c t r o n a t the  F o r the Is e l e c t r o n o r b i t a l ^  - -I' e- /% r  ( r )  -224-  with *  a  Q  2  /  2  = ft /ye  which i s the f i r s t Bohr r a d i u s ; here y r e p r e s e n t s the  reduced  mass of the e l e c t r o n and muon. The  eigenvalues  of t h i s Hamiltonian  are most e a s i l y  determined by r e - e x p r e s s i n g the s p i n v e c t o r operators of  the  h y p e r f i n e term i n t h e i r v e c t o r components: X-  = g3  BS e z  - g$ B l + a (S I ^ y z xx  D e f i n i n g the r a i s i n g and  and it  S  =  S  = S  i s readily  s  x x  +  is  - iS  + S I yy  + S I ) zz  K 3 )  lowering o p e r a t o r s  i n the u s u a l  way  y y  shown t h a t  S I„ + S I = h (S'I x y y y  + S I' )  S u b s t i t u t i o n of t h i s e x p r e s s i o n i n t o 1(3) g i v e s y\  7t  /\  = g3  ^  BS,, - g 3 BI e z ^ y z  /\  + a{%(S  (  S\  /\  s\  I ~ + S~I  /\  The matrix r e p r e s e n t a t i o n of ^ given by H^^ r e s p e c t to the b a s i s s e t Ira m )  is:  CO  H = ft  CO  /\  ) + SI } zz  03  CO  =  <^JK  |^| cf>^> with  -225-  where  = k ( a B B/ft ± g3 B/ft) and to - e ^ y o  ±  2  frequency.  03  +  V  E /h 3 0  —I'  =  /  2  o  03 t  ft  The eigenvalues of the Harailtonian a r e :  E./fi = to  f  = — i s the h y p e r f i n e  +  (w  +  ,  03  4^  +  I  (  4  )  03  =  + -rE 4  -oo  03 • O  2 .  0  ,  2  03. ,  0 *5 X  The e i g e n v e c t o r s of H may be obtained by i n s p e c t i o n f o r the l x l submatrices  and by a t e d i o u s but s t r a i g h t f o r w a r d  c a l c u l a t i o n f o r the 2x2 |,1>  - |ac£|  I 2>  = s I a3>  |3>  =  | 4>  = c|aB>  submatrix:  + c I f?a>  II  f  1(5)  |BB^  - s| 3a>  *  II  where ~  -  Q  ~ /2  _L (  1  " •  2  /oo  and 1  ( 1  -  •_.  03  with x =  A  J  +  s  h  1,,  / T — 2 >  +  2.to.+ 03  o  2  0  .  +  4OJ  1 ± +  /2  =  ,  , ( 1  +  ,h  _  l  x  ,  / T — 1 / l+ X  +  n  _  h  }  x  %  403 , ' +  I t should be noted t h a t c  2  2 + s = 1.  0  The above r e s u l t s are i l l u s t r a t e d i n the f a m i l i a r B r e i t - R a b i diagram i n F i g u r e 1-1 [ B r e i t  (31)] .  The zero  field  -226-  FIELD  FIGURE: 1-1: B r e i t - R a b i diagram of the energy e i g e n s t a t e s o f muonium i n an e x t e r n a l magnetic f i e l d . The four allowed t r a n s i t i o n s (Am=+1) are i n d i c a t e d . In weak f i e l d s (B<L0 g a u s s ) , the t r a n s i t i o n s , CO12 and oo 2 3 / are degenerate and p r o v i d e the c h a r a c t e r i s t i c muonium frequency, =w = 8.76 x 10 *B r a d s  -227-  s p i t t i n g or h y p e r f i n e frequency, to s ^.  The c r o s s - o v e r o f u  and u  1  f o r Mu i s 2.8 x 1 0 ^ r a d  f  may be c a l c u l a t e d bv s e t t i n g  2  2  Denoting A of u  = g3 /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  g  e  and w_, t h i s c r o s s - o v e r i s found t o be a t A  - A e y 2A - A e y  -i  1 2A y  o  o  m- ft • :y 2gm 3 ^ e e  o  which corresponds t o about 160 kgauss. I t i s now p o s s i b l e t o express the i n i t i a l  state  v e c t o r s 1(2) i n terms of the e i g e n v e c t o r s of the Hamiltonian I (5) : iD (0) = ||1> + p|2> + |;|3> - q|4> A  and  ^ ( 0 ) . = -||1> + q|2> + || 3> + p | 4>  where p =  s + c s - c 2 and q = 2  B  The time dependent Mu s t a t e s  are computed from the e i g e n v a l u e s 1 ( 4 ) : • ,. 1.-icoiti.v . -io), 11 „\ , 1 —iu), 11 _\ --iwut i .\ ^ ( t ) = -je 11/ + pe | 2/ + -e | 3> - pe * | 4/ x  1  2  3  A  K6)  and *B  ( t )  =  ~\ e "  i a 3 l t  U>  +qe-  i a , 2 t  |2>  + \ ^ ^ \ ^  + p e ^ ^ l 4>  where OJ . = E . /ft Since the c a l c u l a t i o n of 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 w i l l r e q u i r e computation of the e x p e c t a t i o n value of the muon s p i n o p e r a t o r , 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 b a s i s become:  |m in ) . 'ye'  In t h i s b a s i s , equations 1(6)  -228-  y  A  ) = l -  i a 3 l t  2  ( t  e  + |e-  and  ij; (t)  H  i a , 3 t  le"  IA>3T  +  |33>  = -|e~  B  +  |aa>  i U l t  |B3>  (pse-  +  - qce  i a j 2 t  ( ce-  i w 2 t  P  |aa>  +  +  Time E v o l u t i o n o f t h e u  + qse"  (qse~  (qce-  +  _ i u 4 fc  ) | a3>  i W l t t  ) |6a>  + pee" "" ) | a3>  i u 2 t  1  - pse" "*  i a ) 2 t  this,  = a  y  Thus P  i n transverse magnetic  + i a  x  y  i n t h e x-y p l a n e .  =  y  operator  y  Pauli  spin  i s a f o r m o f t h e muon s p i n r a i s i n g p  field,  Y  and cr a r e t h e f a m i l i a r  y  i n Mu  we d e f i n e t h e 'complex muon p o l a r i z a t i o n " P  where a  | ga>  1  we a r e i n t e r e s t e d i n t h e muon p o l a r i z a t i o n For  t  Spin P o l a r i z a t i o n  In an MSR e x p e r i m e n t  I (7)  [o  A= 2i  \o  oj  matrices:  operator:  +  o p e r a t o r o n l y a c t s on t h e muon s p i n p a r t o f t h e I M ^ m ^ s t a t e s i n t h e u s u a l way:  This spin  Py|a3>  eg.  = 0, P  | 3a>  = 21 aa)  ,  etc. A,  The  object of this  <P (t)> y  s e c t i o n , then,  = f <> (t) | P ! ^ ( t ) > A  u  A  +  i s t o c a l c u l a t e <^P (t))> : y  d - f ) <^ (t)|P B  where  0£ f<_ 1 i s t h e f r a c t i o n  state  ^,(0) and ( 1 - f ) i s t h e f r a c t i o n  is  assumed  f= %  unpolarized.  [Fleming  of y  +  i j  |^ (t)> B  that i n i t i a l l y i n \0 (0) .  f o r m Mu  Normally,  (79)] s i n c e t h e e l e c t r o n s a r e  However, t h e more g e n e r a l  case  i s derived  here  -229-  s i n c e i t does not g r e a t l y may  complicate the c a l c u l a t i o n  and i t  have p h y s i c a l  a p p l i c a b i l i t y i n c e r t a i n ordered systems such s +c as c h i r a l molecules. A f t e r s u b s t i t u t i n g p = 2 and s -c q = 2 and r e a r r a n g i n g , the c a l c u l a t i o n y i e l d s :  ( t ) ^ = hi [ ( 2 f - l ) s c + c ]  e i^ i 2  2  - [(2f-l)sc-s ] e ^ u 2  t  1  1(8) +  [ ( 2 f - l ) sc+s ] e 2  l  u  where CO^_. = CO^ - W J which are eauations  OJ.  =  1  UO  OO  +  %  -  W_ +  2  l k  .  0)  2  00  (CO  =  W  -  %  +  2  W  -CO  +  2*  ( W  ( O J  2  =  e ~  I  A  ) 3  4  T  }  e x p l i c i t l y g i v e n below from  <?o' +  h  O)  4  +  CO2, h O)  , +  '  +  3  -  +  2  +  =  2  3  1(4):  2  ..  - [ (2f-l) sc-c ]  t 2  4  +  2  +  (03  2  %  1(9)  )  +' 0 W  %  )  k  2 Defining n =  ^ ( O O , - OJ - ) = 3  .2  . 1 2  where i t i s r e c a l l e d that  (CO  +  2  x = +  OO  = 1  2  OO  - n  O )^ '- J V  4  2  = % ['(1+x )^ - 1] 2  2  , equations 1(9)  2ai  CO  W  become:  0 A  =  -W_ +  -  W Q  + .  1(10) CO  =  CO  +  CO  +  Q  00  = 0 0 + ^  -230-  S u b s t i t u t i n g equations e  l k t  <(p  1(10) i n t o 1(8) and expanding exponentials,  = coskt + i s i n k t ,  (t£> = e  l w  -  yields:  +-%):t - i v sin(fi + %) t]  {cdsV[cos(fi  t  + 2(2f - l ) s c . . s i h o t sin(ti  where i t i s noted  c  2  + s  2  =1  1(11)  + o).t}  u  u  and c  2  2 - s = v =  X  vl  Noting  1  t h a t sc = %  and m a n i p u l a t i n g  Al + x' i d e n t i t i e s g i v e s the g e n e r a l  <P (t)> = J e i u  y  x .••  trignometric  expression:  Is / (2f - l ) ] c o s f i t  {[1 +  t  5  '+  1 + x' %'-( f  + [1  -  2  \1  y + x^  a  D„  1)]COS(C00+  fi)t  [ s i n (to  + ti) t + sinftt] }  .- o  i s g e n e r a l f o r a l l magnitudes of magnetic  Experimental  K12)  X  X  X  T h i s equation  +  -  fields.  I m p l i c a t i o n s of <^P (t)^> y  I t i s the task of t h i s s e c t i o n t o s i m p l i f y the complex g e n e r a l e x p r e s s i o n f o r <(p^(t.)^> p r a c t i c a l experimental  (equation 1(12)) i n terms o f  considerations.  Two  experimental  c o n s t r a i n t s must be borne i n mind throughout t h i s s e c t i o n : (1) the p r a c t i c a l t i m i n g r e s o l u t i o n of c o n v e n t i o n a l counting and t i m i n g 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 t o , a t  most, about 10. ys..  to i s 2.8 x 10^~®  The h y p e r f i n e frequency,  -231-  rad not,  s ^ which corresponds t o a p e r i o d of about 0.225 ns and i s therefore, experimentally observable. 5  A t the o t h e r l i m i t ,  -1  f r e q u e n c i e s slower than 1.2 x 10  rad s  which have p e r i o d s of  g r e a t e r than 50 y s a r e not observable w i t h the y 10 ystime range. <3?^ (t)> i s r e l a t e d t o the experimental MSR " s i g n a l " +  S(<f>,t) = A ( t ) <P  (<(>,t)>  where A ( t ) i s the time dependent the  e m p i r i c a l asymmetry, and <j) i s  angle between the p o s i t r o n counter and i n i t i a l muon  polarization.  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 s i g n a l .  The s i g n a l appears i n the e x p o n e n t i a l  histogram: N(c|),t) = N e  _ t  / y T  .[1 + S((b,t)]  which i s analogous t o equations  + Bg  1(13)  (4) and (6) i n Chapter I .  Hence,  the  s i g n a l i s the o s c i l l a t o r y muon f u n c t i o n which remains when  the  e x p o n e n t i a l muon l i f e t i m e and background  time histogram. are  are removed from the  Chemical r e a c t i o n s and other r e l a x a t i o n  effects  not i n c l u d e d i n S(cf>,t) here; these e f f e c t s are i n c o r p o r a t e d  i n t o the formalism i n Appendix I I . Table 1-1 l i s t s v a l u e s of the magnetic v a r i a b l e s of equation 1(12) magnetic  field  f o r a number of p r a c t i c a l l y  s t r e n g t h s r a n g i n g from 1 gauss  s e c t i o n examines ^P ( t ) ^ f o r t h r e e magnetic (i) Very TA7eak F i e l d s  field  (^lOgauss)  dependent available  t o 10 kgauss. field  This  regimes.  - the Standard MSR S i g n a l  From Table 1-1, i s i s seen t h a t f o r B. £10-gauss,  x = 0, /  1  - \ -y = 1 , and v = 0 t o b e t t e r than 1%. \1 + x / 2  Furthermore,  VALUES OF MAGNETIC FIELD DEPENDENT VARIABLES IN EQUATIONS 1(8) AND I ( 1 2 ) t  TABLE I -1: Field  2 X  V  c  2  2sc  (gauss)  CO!  CO  (JO  (10 )  (10 )  7 (It)')  6  6  2  C0  2  3  CO!  CO  4  3  10 (10 ) ( 1 0 )  (10 )  10  10  h  (10 ) 10  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 f r e q u e n c i e s are given i n u n i t s of rad s  -233-  2 ^ 2 f o r s m a l l x, (1 + x ) - 1 + %x ; thus 2  n  =  % [ ( i  +  *  2  )  -  h  -  i]  %  2.  + %x 2  i)  =  2  In t h i s l i m i t , equation 1(12)  %  u  o  becomes:  ' 4 -  ^  /p  (i  y  (t£>  = e B<10gauss  1 W  - [f, cosflt + ( l - f ) c o s ( w  0  + Q)t]  1(14)  The r e a l p a r t o f the muon p o l a r i z a t i o n i s ( i n c l u d i n g the counter phase  dependence):  Re <^P (cj>,t£> y  = f cos(io_t + cf))cosfit B<10gauss 1(15) + (1-f) cos(to_t + (J)) cos(co  where w_ = h(g$ B/fi-g$^B/fi) =  1 0 3 t 0  e  y  + ft)t  o  corresponds t o the  c h a r a c t e r i s t i c muonium p r e c e s s i o n frequency.  N o t i c e that the  counter phase dependence i s added only t o the Larmor p r e c e s s i o n p a r t s of each term i n 1(15). ^P  By c o n s t r u c t i o n , the r e a l p a r t of  ( 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 p a r t corresponds t o the u y_.direction.  +  p o l a r i z a t i o n i n the  I n t r o d u c t i o n of the counter phase t o e i t h e r the r e a l  or imaginary p a r t s g e n e r a l i z e s d i r e c t i o n i n the x-y p l a n e . Im <(^P^ (^-', t)^>  .  <^P^(t)y  t o correspond t o any  F o r example, Re<^P (0,t)^>  =  Since the h y p e r f i n e frequency i s too f a s t t o  be e x p e r i m e n t a l l y r e s o l v a b l e , t h e second term i n 1(15) to zero and t h i s f r a c t i o n  (1-f) of the u  +  averages  appears to be u n p o l a r i z e d .  The remaining term shows the muonium p r e c e s s i o n , CO_, modulated by 5 -1 the slower beat frequency Q<2.8 x 10 r a d s to a period->22.5 ys f o r B<10 gauss.  which corresponds  T h i s beat frequency i s slow  enough t h a t i t may be i g n o r e d i n f i e l d s of l e s s than 10 gauss  -234-  except, p o s s i b l y , i n experiments which attempt h i g h l y measurements of very slow Mu r e l a x a t i o n r a t e s  precise  (X^0.2ys "*") .  The  net  observable s i g n a l i n t h i s weak f i e l d regime thus reduces t o  the  very simple e x p r e s s i o n : (A,t)>  =-f .cos(w_t + <M  1(16)  Experiments of t h i s type which monitor a s i n g l e Mu frequency,  IO_  =  COMU  d e f i n e d i n Chapter I .  = 103  TO  characteristic  , are c a l l e d MSR experiments as  In most cases, f i s assumed t o be h .  G e n e r a l l y , f o r reasons c i t e d i n Chapter I, the experimental asymmetry i s t r e a t e d e m p i r i c a l l y so t h a t t h i s f r a c t i o n i s indeterminable. F i g u r e 1-2 i l l u s t r a t e s the time e v o l u t i o n of y  +  given  by 1(15) assuming f = h and p e r f e c t experimental time r e s o l u t i o n . Although F i g u r e 1-2 was c a l c u l a t e d f o r a lOOg f i e l d , equation / l \ 35 1(15) i s v a l i d a t e a r l y times (a few ns) s i n c e • •—— =d = 1  U + x| z  at:--100 .gauss t o b e t t e r than 1%.  The e f f e c t s of 9, are  unobservable a t e a r l y times a t t h i s f i e l d .  The f i g u r e shows the  f a s t h y p e r f i n e o s c i l l a t i o n i n the envelope of the slower muonium p r e c e s s i o n .  In p r a c t i c e , the f a s t o s c i l l a t i o n i s averaged  over, l e a v i n g an observable muonium envelope of reduced amplitude. (ii)  Intermediate F i e l d s  At  magnetic  Muonium  f i e l d s l e s s than 150 gauss, Table 1-1 shows  = 1 and v<0.1. \1 + x equation I(T2). i s 2  2  (1 0<_B<_15 0gauss) - Two Frequency  In these f i e l d s , the. r e a l p a r t o f  -235-  FIGURE 1-2:  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 a 100 gauss t r a n s v e r s e magnetic f i e l d from equation 1(15), assuming f=%,-- $=0. The f a s t o s c i l l a t i o n s are e s s e n t i a l l y a t ^ t h e h y p e r f i n e frequency, co = 2.8 0 .x 10 rad s and the envelope i s a t thg mujnium frequency, c o (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 t o be observed on this'_time s c a l e . E x p e r i m e n t a l l y , the f a s t h y p e r f i n e o s c i l l a t i o n i s averaged over, l e a v i n g and observable co envelope of reduced amplitude, i n d i c a t e d by the broken l i n e . T  Mu  M  -236-  Re  <(P (<f>,t£> = f'cos (00 t + <)>) c'osflt 10<B<150gauss y  + (1-f) • cos  (CO_t  + 4) cos  (CO  q  + fi)t  + |-sin(a)_t + (J)). [ s i n (CO + fi)t +• sinftt] q  As b e f o r e , the h y p e r f i n e frequency i s unobservably the e f f e c t i v e Re  e x p r e s s i o n f or Re<^P^ (<p , t)y  (s>,t)>  -=  f ,cos(CO_t  f a s t so t h a t  becomes:  + *)'cos«t  10£B<150gauss + ^ sin(TO_t  + p)  sinnt  G e n e r a l l y t h i s e x p r e s s i o n i s r e - w r i t t e n by m a n i p u l a t i o n of trignometric i d e n t i t i e s :  Re  <(p (c?>,t)>  =  y  hi (f-|) c o s [  +. nyt + <?]  10<B<150gauss - + ( f + J . ) . c o s [ ( C O _ - fi)t + •<!>]}• 9 -1''  1(17)  At 150 gauss, co_ = 1.3 x 10 rad s - corresponding t o a p e r i o d of ^5ns which i s j u s t observable w i t h a time r e s o l u t i o n resultant  signal  i s a f a s t muonium o s c i l l a t i o n b e a t i n g a t the  slower frequency fi. T h i s was f i r s t observed Gurevich 1-3.  of 1 ns. The  e x p e r i m e n t a l l y by  (71) i n q u a r t z a t 95 gauss; the data are shown i n F i g u r e  This i s referred  t o as the "two frequency p r e c e s s i o n " o f  the muon i n muonium. (iii)  High F i e l d s  (>150 gauss)  At these f i e l d s , frequency, CO  = OJ  the c h a r a c t e r i s t i c  muonium  , becomes immeasurably l a r g e .  experimental, time r e s o l u t i o n  With an  o f about 1 ns, an observable  ~ 4400 § 4200 3 4000 38003600FIGURE 1-3  60  80  t(nsec) "Two frequency p r e c e s s i o n " of the muon i n muonium i n fused quartz a t 95 gauss [from G u r e v i c h ( 7 1 ) ] . The smooth l i n e i s a t h e o r e t i c a l f i t t o t h e data. The f a s t o s c i l l a t i o n s a t the c h a r a c t e r i s t i c muonium frequency, • ~ — x 10 - i ^ O r ajd s _—\ -L 3. _ T „ j J 4^T.IA i ~ (95 gauss)- = 8.7 a r e modulated by t h e slower beat4frequency, ft (95 gauss) = 2.7 x 10' r a d s T  -238-  s i g n a l i n the time domain must have a p e r i o d o f 5 ns or more, c o r r e s p o n d i n g t o a frequency o f <_ 0.13  x 10^  rad s ^. Table  1-1  shows t h a t none o f the B r e i t - R a b i t r a n s i t i o n s have such a frequency between 150 gauss and 10 kgauss. fields but  (_> 100 kgauss), O J  1 2  At much higher  w i l l once more become r e s o l v a b l e ,  such f i e l d s are not e x p e r i m e n t a l l y a v a i l a b l e .  I f the  experimental time r e s o l u t i o n c o u l d be improved by a f a c t o r of 10, i t would be p o s s i b l e t o observe co  x2  up t o 1 kgauss.  This i l l u s t r a t e s  between y SR and MSR: +  magnetic  up to 10 kgauss and to 3 2  an e s s e n t i a l  w h i l e the former may  be performed i n  f i e l d s of up t o 15 kgauss, the l a t t e r  to f i e l d s of l e s s than a:few hundred  gauss.  difference  i s constrained  -239-  Appendix I I - The E f f e c t o f Chemical R e a c t i o n on the Muon Polarization _A  General T h i s Appendix w i l l  examine the muon s p i n p o l a r i z a t i o n  i n a muonium ensemble which i s undergoing weak magnetic  field  (B £ 1 0 gauss).  chemical r e a c t i o n i n  E x c l u d i n g the e f f e c t s of  chemical r e a c t i o n , the muon p o l a r i z a t i o n i s c h a r a c t e r i z e d by equation 1(16) of Appendix I : P  (cj),t) =-f cos (to_t + *)  When a muonium atom r e a c t s c h e m i c a l l y , i t s e l e c t r o n forms a chemical bond and becomes p a i r e d w i t h another e l e c t r o n breaking the h y p e r f i n e i n t e r a c t i o n between the Mu e l e c t r o n and muon (in g e n e r a l , i n t e r m e d i a t e muonic r a d i c a l s are expected t o be formed  - t h i s s i t u a t i o n i s not c o n s i d e r e d here s i n c e the  l i f e t i m e s of any such r a d i c a l s formed  i n the simple gas phase  r e a c t i o n s s t u d i e d i n t h i s t h e s i s are surely '.shorter than one h y p e r f i n e p e r i o d of 0.225 ns) . diamagnetic environment  Such a muon f i n d s i t s e l f  ina  where i t precesses a t e s s e n t i a l l y the  " f r e e " muon frequency to ='^ir-to . Since to = U (g3 B/fi - gB B/ft) u 103 e y i s dominated by the e l e c t r o n magnetic moment, the sense of to_ p r e c e s s i o n i s o p p o s i t e t o t h a t of to . y  The c o r r e c t i o n between the  f r e e muon frequency and the diamagnetic muon frequency due t o electron shielding  (the s o - c a l l e d  "chemical s h i f t " ) i s a t the  p a r t per m i l l i o n l e v e l which i s not r e s o l v a b l e w i t h p r e s e n t MSR technology.  The s t r o n g e r d i p o l a r c o u p l i n g between diamagnetic  muons and protons i n water molecules c r y s t a l l i n e gypsum  (MuHO) has been r e s o l v e d f o r  (CaSO. •2H„0) [Schenck  (71)]  ,  In f l u i d s ,  -240-  however, t h i s e f f e c t appears fundamentally u  +  frequency  as a broadening which does not  e f f e c t the approximation  t h a t the  diamagnetic  i s to .  To z e r o t h o r d e r , because muonium atoms r e a c t 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 c o h e r e n t l y p r e c e s s i n g Mu ensemble becomes an i n c o h e r e n t l y p r e c e s s i n g y the r e a c t i o n procedes. Mu  signal.  The Mu  ensemble as  +  The net r e s u l t i s a r e l a x a t i o n of the  s i g n a l , then, becomes a measure of the  time-dependent p r o b a b i l i t y of a Mu chemical r e a c t i o n .  atom s u r v i v i n g  without  Hence, the r e l a x a t i o n of the Mu  a simple e x p o n e n t i a l decay as g i v e n by equation  S(ct),t) = A  M u  e"  X t  s i g n a l has  (8) i n Chapter I  c o s ( w t + d»)-.+ -A cos.(u t * A)  1 1  M u  d>  where i t i s noted t h a t to Eoo_ (see F i g u r e I I - l ) . M  v  From the viewpoint of chemical k i n e t i c s , the r a t e equation f o r a b i m o l e c u l a r r e a c t i o n of Mu with reagent X i s g i v e n by the standard second d[Mu] dt  order e x p r e s s i o n :  = -k[X][Mu]  11(2)  where k i s the b i m o l e c u l a r r a t e c o n s t a n t , c o n c e n t r a t i o n of reagent X, and "concentration."  [Mu]  [X] i s the  i s the muonium  Here, of course, the concept of a Mu  c o n c e n t r a t i o n invokes the e r g o d i c p r i n c i p l e :  an ensemble i n  time i s f o r m a l l y the same as an ensemble i n space (68)] .  Since the t o t a l number of Mu  [Arnold  atoms i n v o l v e d i n a  r e a c t i o n i s minute compared w i t h the number of reagent molecules  (-10  7  compared w i t h -10  19  ), equation 11(2) may  be  -241-  0.15  MU  0.10  >~  LaJ >CO  cr.  I N 780 T O R R N  2  A T 6.9 GAUSS  0.05  iiii  0.00 -0.05 -0 .10 -0.15 0.15 0.10  130 LIM H I I N 780 T O R R N . 6 . 9 G R U S S  h  0.05 cr i—  UJ  0.00  >~ CO  ^  -0.05 - 0 .10 - 0 .15 0.0  0.5  1 .0  TIME  1 .5  2.0  I N LASEC  2.5  3.0  3.5  4.0  (20 N S E C / B I N ;  FIGURE I I - 1 : The e f f e c t o f c h e m i c a l r e a c t i o n on the muonium s i g n a l , S(cj),t) (equation 1 1 ( 1 ) ) . The upper f i g u r e shows the slow background r e l a x a t i o n r a t e of _^ muonium i n pure N g i v i n g X .= 0.34 ± 0.02 ys (equation 1 1 ( 3 ) ) . In the presence of HI reagent (lower f i g u r e ) , the e x p o n e n t i a l decay o f the muonium s i g n a l due t o removal o f muonium atoms by chemical r e a c t i o n i s pronounced, g i v i n g a r e l a x a t i o n r a t e , X = 3.75 ± 0.38 ys . Each histogram c o n t a i n s about 10 events and the l i n e s a r e x -minimum f i t s t o equation 11(1). 2  -242-  rigorously  r e - w r i t t e n as a p s e u d o - f i r s t o r d e r r a t e , f o l l o w i n g  the c o n v e n t i o n s o f c h e m i c a l k i n e t i c s : d  [Mu] = -X[Mu] dt  which i d e n t i f i e s t h e Mu r e l a x a t i o n r a t e ( e q u a t i o n 11(1)) as t h e p s e u d o - f i r s t o r d e r r a t e c o n s t a n t , X, as g i v e n by t h e l i n e a r relation  (see F i g u r e I I - 2 ) : X = k[X] + X o  11(3)  The i n t e r c e p t , X , i s i n t r o d u c e d t o account f o r "background" r e l a x a t i o n o f t h e Mu s i g n a l due t o e f f e c t s o t h e r than t h e c h e m i c a l r e a c t i o n o f i n t e r e s t , such as magnetic inhomogeneity, or  pressure broadening  field  from t h e r e a c t i o n medium,  background r e a c t i o n s w i t h c h e m i c a l i m p u r i t i e s i n t h e  moderator gas. Thus, a b i m o l e c u l a r r a t e c o n s t a n t i s s i m p l y determined  from e q u a t i o n 11(3) by t h e d i r e c t o b s e r v a t i o n o f  the r e l a x a t i o n r a t e o f a Mu; s i g n a l as a f u n c t i o n o f concentration of the r e a c t i n g molecules, [ X ] . B_ G e n e r a t i o n o f a Coherent Diamagnetic X->15us  Muon Background:  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 a r e e x t r a c t e d from  time h i s t o g r a m s by t h e s o r t o f f i t t i n g procedures d e s c r i b e d 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 o f t h e muonium s i g n a l be p r o p e r l y d e s c r i b e d . to  Complications  e q u a t i o n 11(1) a r i s e under t h e c o n d i t i o n s : w,, < X << to Mu ~ o  11(4)  because a s i g n i f i c a n t f r a c t i o n o f muons i n Mu a r e p l a c e d i n t o d i a m a g n e t i c environments  by f a s t t h e r m a l c h e m i c a l r e a c t i o n s  -243-  MU IN HBR WITH AR MODERATOR AT 2 9 5 K  0 , 0  1  1  1  1  0.0  1.0  2.0  3.0  1  4.0  HBR C O N C E N T R A T I O N  1  1  5.0 :(10  6.0 - 4  1  7.0  M)  FIGURE I I - 2 : The l i n e a r dependence o f the r e l a x a t i o n r a t e of the muonium s i g n a l on reagent c o n c e n t r a t i o n (equation 1 1 ( 3 ) ) . The data p o i n t s are weighted averages of X's e x t r a c t e d from l e f t and r i g h t h i s t o g r a n ^ a n d the e r r o r bars r e p r e s e n t l c r . The l i n e i s a x ~ minimum f i t g i v i n g a b i m o l e e u l a r r a t e c o n s t a n t , k(295K) = (9.1 ± 1.01 x 10 1 mole s and X^ = 0.26 ± 0.06 u s  -244-  b e f o r e t h e phase coherence i s l o s t due t o t h e r a p i d to,, Mu r  oscillation.  That such a r e a c t i o n i s t h e r m a l i s c l e a r when  i t i s r e c a l l e d from Chapter I t h a t t h e r m a l i z a t i o n o f y place i n  1. ns-•• and c o n d i t i o n 11(4) c o r r e s p o n d s t o r e a c t i o n  times much l o n g e r than 0.2 ns, for  takes  +  the hyperfine period.  Indeed,  e x p e r i m e n t a l reasons c i t e d i n Chapter I I , a d e t e c t a b l e  MSR s i g n a l i s n o t observed  f o r times l e s s than 10ns  and, as  d e s c r i b e d below, a s i g n a l must l a s t f o r a t l e a s t 300 ns measurable.:  The d i a m a g n e t i c y  +  t o be  ensemble g e n e r a t e d under  c o n d i t i o n 11(4) does n o t p r e c e s s i n c o h e r e n t l y as assumed i n S e c t i o n A above, b u t , i n f a c t , may g i v e r i s e t o a s i g n i f i c a n t diamagnetic y  +  signal.  I t i s the task of t h i s s e c t i o n t o  e v a l u a t e t h e 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) t h a t X << to ensures t h a t t h e f a s t o h y p e r f i n e term i n e q u a t i o n 1(15) o f 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 t o 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 c o r r e s p o n d s t o  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 c o u n t i n g statistics). to  9,.ys  1  < to  The magnetic f i e l d range, 1<B<10 gauss, c o r r e s p o n d s < 89 ys" '; t h u s , f o r X - 15 ys , c o n d i t i o n 11(4) 1  M u  i s reasonably w e l l  1  satisfied.  The c a l c u l a t i o n below i n v o k e s t h e f o l l o w i n g assumptions: (1) t h e muon l i f e t i m e i s independent o f t h e y environment; ignored  +  (2) t h e f i e l d i s weak enough t h a t b e a t i n g may be  (0, i n e q u a t i o n 1(15) o f Appendix I ) and X i s s m a l l  enough t h a t t h e h y p e r f i n e o s c i l l a t i o n may be i g n o r e d (terms  -245-  OJ  containing  i n equation 1(15)  process t h a t p l a c e s  a single y  of Appendix I ) , (3) the from Mu  T  environment occurs i n s t a n t a n e o u s l y d i s t r i b u t e d times,  frequency as " f r e e " y all  y  entering  +  the t a r g e t  t h e r m a l i z e as f r e e y environments by hyperfine  ions,  +  say, and  +  initially  form Mu  f a s t epithermal reactions  are the  forming Mu  that  for y  to  +  i n t o diamagnetic t h a t occur before  (6) the  i n i t i a l phases of y  same, (7) thermal f r e e y For  +  do not  one  in  +  relax  convenience, both f r e e y  A geometrical formulation given  required  (allowing  w i l l be r e f e r r e d t o as " f r e e "  +  same  i t i s assumed t h a t these f a t e s are .  a f t e r t=0.  diamagnetic y  schematically  p r e c e s s at the  +  ions or t o be p l a c e d  +  oscillation),  free y  statistically  (5) w h i l e i t i s not  a r b i t r a t e d a t e s s e n t i a l l y t=0, and  i n t o a diamagnetic  over  (4) diamagnetic y  chemical  Mu  by,  ions  +  y . +  of the problem i s  by:  \ Mu  y  y  Mu  +  t=t'  t=0 Here, C i s a counter p l a c e d  at an angle § to the  p o l a r i z a t i o n , \ r e p r e s e n t s the Mu r e p r e s e n t s the  s p i n p o l a r i z a t i o n and  free y+ p o l a r i z a t i o n .  r e p r e s e n t the magnitudes of the Mu arbitrary units. S.  i (<() ft) = A  total t  By  Y  Mu  inspection, (t)  cosU  Mu  initial  The  and  l e n g t h s of the  free y  +  y  +  t arrows  polarizations in  the t o t a l y + s i g n a l i s given t + (j)) + A Y  y  (t)  COS(OJ t - (j>)  y  by  -246-  where  A M u  (t)  and y (t) A  a  r  e  t  n  asymmetries o f muons i n Mu and  e  f r e e y ensembles r e s p e c t i v e l y . +  i n Chapter I and A^ (t) A (t)  = A  y  y  A M u  (t)  i s g i v e n by e q u a t i o n (_7)  is:  + A^jl  e-^'coscopt' d t '  11(5)  where A and A„ a r e the amplitudes of the f r e e y y Mu *  and Mu  +  K  ensembles a t t=0, and O J = (to,, + oo ) i s the r e l a t i v e angular p Mu y ^ v e l o c i t y o f the two ensemble s p i n v e c t o r s p r e c e s s i n g i n opposite d i r e c t i o n s .  Performing the i n t e g r a t i o n  i n 11(5), the  t o t a l s i g n a l becomes: S  total  ( 4 )  '  t )  A X Mu + X + n 2  =  A  Mu " e  X t c o s ( u  Mu  t  +  *  } +  V ° V ~ *) S (  • -Xt .•. • • " -Xt , , , , " x- [oo e sxnoo t - Xe COSOJ t + X ] C O S ( O J t - <p) -OJ P P P I  I  (  6  )  r  y  2  The c o r r e c t i o n  calculated  above i s i l l u s t r a t e d i n  F i g u r e II-3 which p l o t s equation 11(1) (lower l i n e s a s y m p t o t i c a l l y approaching an asymmetry o f zero a t long times) and e q u a t i o n 11(6) (upper l i n e s ) f o r a s e r i e s o f f i e l d s ranging from 2 t o 10 gauss w i t h i> - 1 r a d i a n , 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  obvious because  i n the upper  curves i s not  o f i t s low frequency a t these f i e l d s .  Clearly,  at 10 gauss, equation 11(1) i s a very good approximation t o + S+. > -i ((f),t) w h i l e a t 2 gauss, the coherent diamagnetic y s i g n a l r e q u i r e s d e s c r i p t i o n by equation 11(6). In F i g u r e I I - 4 , equations 11(1) and 11(6) are again plotted  f o r a s e r i e s o f X ' s r a n g i n g from 15 ys 1  t o 300 ys.  at  a f i x e d f i e l d o f 7.5 gauss w i t h A =0.1, A^=0.0, and <j>=l radian, Mu  Although the Mu s i g n a l i n the.. . p l o t s where X>15 ys  i s not  -247-  0.15 0.10 10  cc  GRUSS.  R(MU)=0.1.  x=15u5"  CO cr  5.0  cu  GRUSS.  fl(MU)=0.1.  X-15LIS"'  CO cr  3.0  0.05  GRUSS.  fi(MU)=0.1.  X=15LIS"  0.00 to en  -0.05 0.10 2.0 GRUSS. RIMU)=0.1 . x=15S-'  0.05  M  or  0.00 CO  ^ -0.05 -0.10 -0.15  _L  0.0  0.5  1.0  2.0  1.5 TIME  I N  2.5  3.0  nS  FIGURE I I - 3 : The g e n e r a t i o n o f a coherent diamagnetic muon background s i g n a l by f a s t chemical r e a c t i o n s of muonium. The l i n e s a r e t h e o r e t i c a l muonium s i g n a l s w i t h pseudo f i r s t order r a t e constants, X = 15 ys , i n v a r i o u s weak magnetic f i e l d s f o r counters p l a c e d a t 1 r a d i a n t o 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 f r e e muon amplitude i s 0%. The lower curves correspond t o equation 11(1) and assume complete l o s s o f muon phase coherence d u r i n g chemical r e a c t i o n ; the upper curves correspond t o equation 11(6) and show t h a t i f X >_to , t h e muon phase coherence i s not l o s t d u r i n g chemical r e a c t i o n , but may g i v e r i s e t o a s i g n i f i c a n t " r e s i d u a l muon polarization" signal.  -248-  0.15 0.10  h  0.05  7.5  GRUSS.  R(MU)=0.1 . K=15u5"'  7.5  GRUSS.  R(MU)=0.1.  0.00 to CL  -0.05 0.10 x=50u5"'  0.05 or i— UJ  2: CO  cr  0.00 -0.05 0.10  7.5  GRUSS.  R(MU)=0.1.  x=99 S~ u  0.05 QL  UJ  CO  cr  0.00 -0.05 0.10  h  0.05  7.5  GRUSS. R(MU)=0.1. x=300uS~  0.00 CO a  -0.05 -0.10 -0.15  0.0  0.5  1.0  1.5 TIME IN  2.0  2.5  3.0  uS  FIGURE I I - 4 : The dependence of the amplitude of the " r e s i d u a l muon p o l a r i z a t i o n " on muonium r e a c t i o n r a t e a t 7.5 gauss. As i n F i g u r e I I - 3 , the lower curves correspond to equation 11(1) and the upper curves t o equation 11(6). E a r l y d e t e r m i n a t i o n s of muonium r e a c t i o n r a t e s [Brewer (72)] were made by the r e s i d u a l p o l a r i z a t i o n method by measuring the amplitude and phase of the diamagnetic muon s i g n a l by y SR and r e l a t i n g these t o the f a s t muonium r e l a x a t i o n r a t e s . In the p r e s e n t work, muonium r e a c t i o n r a t e s are measured d i r e c t l y by MSR.  -249-  of long enough d u r a t i o n  t o allow u s e f u l f i t t i n g  of the data,  these p l o t s do i l l u s t r a t e the e f f e c t known as the " r e s i d u a l muon p o l a r i z a t i o n "  [Brewer  (75), Fleming  (79)]  i n which  large  v a l u e s o f X m a n i f e s t themselves as l a r g e r v a l u e s of the " f r e e " 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  l a r g e t h a t the  condition oo  << co  11(7)  < X  o ~  is f u l f i l l e d ,  the e x p r e s s i o n f o r  s  j( ! (  t o t a  )  »h) must i n c l u d e an  i n t e g r a t i o n analogous t o 11(5)  over the h y p e r f i n e terms of  equation 1(15)  The upshot of the c a l c u l a t i o n  i n Appendix I .  i s t h a t f a s t Mu r e l a x a t i o n s  not only express themselves as  l a r g e v a l u e s of the r e s i d u a l p o l a r i z a t i o n , but a l s o with  rate  dependent v a l u e s of the phase of the r e s i d u a l p o l a r i z a t i o n . Previously,  these f a c t s have been e x p l o i t e d  of f a s t r e a c t i o n s  t o measure the r a t e s  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  I I I - Data Acquisition w i t h High C u r r e n t Muon Beams: Theory and P r a c t i c e In  of  Chapter I I , S e c t i o n C, a q u a l i t a t i v e  assessment  the problem o f muon p i l e - u p i s presented; i n the f i r s t two  S e c t i o n s o f t h i s Appendix,  the a b s o l u t e magnitude of v a r i o u s  m u l t i p l e muon events and t h e i r e f f e c t s on the r e s u l t a n t MSR time s p e c t r a are c a l c u l a t e d .  The l a s t S e c t i o n of t h i s  Appendix  d e t a i l s the e x i s t i n g MSR data acquisition system a t TRIUMF which d e a l s w i t h h i g h muon beam c u r r e n t s . A The Optimal Good Event Rate It  i s the task o f t h i s S e c t i o n t o c a l c u l a t e the  o p t i m a l "good"  (see Chapter I I , S e c t i o n C) event r a t e f o r an  experiment w i t h a data a c q u i s i t i o n system t h a t d i s c a r d s the ambiguous m u l t i p l e muon events.  The f i r s t  calculation i s for  " p o s t - i K " second muons a r r i v i n g d u r i n g the o b s e r v a t i o n time T a f t e r the e n t r y o f the i n i t i a l the  target.  c l o c k - s t a r t i n g muon, VK , i n t o  F o r t h i s and subsequent c a l c u l a t i o n s , i t i s  assumed t h a t the a r r i v a l o f beam muons obeys a P o i s s o n time distribution;  t h i s assumption i s v a l i d over time  intervals  much l a r g e r than the m i c r o s c o p i c beam s t r u c t u r e a t the c y c l o t r o n r a d i o frequency - 23 MHz a t TRIUMF.  The P o i s s o n d i s t r i b u -  tion function i s : P (n.*,t) p  = m i  n  e -  n  t  iii,i,  where P i s the p r o b a b i l i t y o f n events o c c u r r i n g i n a time t , g i v e n an average event r a t e 71.  I f a v a l i d event i s d e f i n e d as  one where no other muons appear d u r i n g a time T a f t e r the f i r s t  -251-  muon, then the p r o b a b i l i t y o f an event being v a l i d i s given by P (0,72,T) = e ~ *  T  p  and  the average v a l i d event r a t e , 71 i i s 7Z  = 7lsT  111(2)  m  g  Since 71 = 0 when 71 = 0 or 9l = °°, i t i s c l e a r t h a t g p o s i t i v e f u n c t i o n of ^ has a maximum f o r constant Figure  III-l;  T as shown i n  t h i s p o i n t i s a l s o i n t u i t i v e l y obvious s i n c e a t  low muon beam c u r r e n t s beam c u r r e n t s  this  p i l e - u p i s n e g l i g i b l e , w h i l e a t high  v a l i d events are r a r e .  The optimum beam r a t e  occurs under the c o n d i t i o n  an 7 l  R a x  = k  1  T h i s important r e s u l t i m p l i e s t h a t under optimal  1  1  ( 3 )  conditions,-37%  of the muons are f r e e o f p i l e - u p and 63% must be r e j e c t e d . For T = 4x^ i n the example of Chapter I t , the optimal 5 current  i s 1.1 x 10  i s asymmetrically  -1 s  .  I t may be noted t h a t f u n c t i o n 111(2)  peaked w i t h r e s p e c t  t o 71, r i s i n g r a p i d l y t o  a maximum and t a p e r i n g o f f slowly a t l a r g e 71. p r a c t i c a l consideration lowered i n c r e m e n t a l l y  T h i s becomes a  s i n c e muon beam c u r r e n t s a r e g e n e r a l l y  by c o l l i m a t i o n r a t h e r than by f i n e  adjustment of the proton beam i t s e l f . t h a t any c o l l i m a t o r w i l l p r o v i d e beam c u r r e n t ,  beam  Since  i t i s unlikely  e x a c t l y the optimal e f f e c t i v e  i t i s advantageous t o b i a s i t toward a l a r g e r -  than-optimal value  r a t h e r than a s m a l l e r one.  How can the above c a l c u l a t i o n be extended t o i n c l u d e  FIGURE I I I - l :  The net good event r a t e  (without p i l e - u p ) as a f u n c t i o n of beam c u r r e n t  f o r v a r i o u s muon decay gates. The good event r a t e i s g i v e n by equation  111(2) .  -253-  r e j e c t i o n o f "pre-y^"  second muons a r r i v i n g d u r i n g a time: T  before the e n t r y of y^, the c l o c k - s t a r t i n g muon?  The answer  i s simply t o apply the above arguments backwards i n time. While  i n t u i t i v e l y c o r r e c t , t h i s i s a l s o a r e c o g n i t i o n o f the  f a c t t h a t the a r r i v a l o f muons i s a Markov process  (a random  process i n which the f u t u r e i s completely determined present and independent  by the  of the way i n which the present  and t h a t a Markov process i s a l s o Markov i n r e v e r s e example  [Feller  (50)]).  evolved)  (see f o r  The net r e s u l t f o r both p o s t - y ^ and  pre-y^ event r e j e c t i o n i s t h a t the r a t e of v a l i d events i s given by: _  ^.  -PIT  7? = 7l(e y  -#T, ~ -271T • e ) = Tie.  which i s o p t i m a l when *Max = 2T-  1 1 1  (4)  t h a t i s , when the beam r a t e i s the i n v e r s e o f twice the muon gate width.  Thus, f o r T = 4T , ??  again, corresponds  M  = 5.7 x 1 0  y jyiax  t o an event acceptance  4  s  _  1  which,  r a t e of 37%.  B S p e c t r a l D i s t o r t i o n s due to. Muon P i l e - u p In t h e preceding/- S e c t i o n , i t was shown t h a t p r e - and post-y^ m u l t i p l e muon events reduce the good event r a t e by the same amount, g i v e n p r e - and post-y^ T-gates  of the same width.  However, t h i s does not imply t h a t the s p e c t r a l d i s t o r t i o n s due t o p r e - and p o s t - y ^ m u l t i p l e muon events a r e o f e i t h e r the same magnitude or c h a r a c t e r ; r a t h e r , i t i s shown i n t h i s S e c t i o n  -254-  that post-  y^ events are much more d e v a s t a t i n g  than pre-LK  events. The  e f f e c t s o f muon p i l e - u p upon the time histogram  are c a l c u l a t e d s e p a r a t e l y three  stages:  muon l i f e t i m e  firstly,  f o r t h e pre- and p o s t - y ^ cases i n  the p i l e - u p e f f e c t s on the apparent  ( i g n o r i n g the ySR or MSR s i g n a l ) are c a l c u l a t e d  assuming 100% e f f i c i e n c y f o r decay p o s i t r o n d e t e c t i o n steradians  solid  angle,  100% counter e f f i c i e n c y ) ;  t h i s c a l c u l a t i o n i s extended s i g n a l ) t o a l l o w f o r imperfect  secondly,  (again, i g n o r i n g the ySR or MSR decay p o s i t r o n  e f f i c i e n c y , 0 _< e £ 1, e = counter s o l i d efficiency;  ( i . e . 4TT  detection  angle x counter  f i n a l l y , the muon p i l e - u p e f f e c t s on the ySR or MSR  s i g n a l a r e c a l c u l a t e d f o r the case o f imperfect  decay p o s i t r o n  detection. In t h e f o l l o w i n g d i s c u s s i o n , two concepts must not -t/x be confused:  (1) The f u n c t i o n , 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 g i v i n g the proba b i l i t y t h a t a muon w i l l  survive u n t i l  t h a t a muon w i l l decay before  time t . The p r o b a b i l i t y -t/x  time t i s (1 - e  y).  (2) Given t h a t a muon has s u r v i v e d u n t i l t , the p r o b a b i l i t y t h a t i t w i l l decay d u r i n g for  the next i n t e r v a l d t i s t h e same  a l l muons; t h a t i s , a 10 ys o l d muon has the same prob-  a b i l i t y of decaying d u r i n g  the next ps as a 1 ns o l d muon.  What i s the p r o b a b i l i t y t h a t a muon w i l l decay d u r i n g the i n t e r v a l t and t + dt?  T h i s i s j u s t the product of the -t/x p r o b a b i l i t y t h a t i t has s u r v i v e d 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 o f i t decaying d u r i n g the.next i s a constant independent of t . of e  . d t , which  T h e r e f o r e , the p r o b a b i l i t y  a muon decaying between t and t + d t i s p r o p o r t i o n a l t o  -t/x '  and i n t h i s sense e  "decay" curve.  -t/T  y may be thought o f as a  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  w i t h an experimental time histogram. At  f i r s t g l a n c e , i t might appear t h a t p i l e - u p events  cannot i n t r o d u c e 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 s i n c e the p r o b a b i l i t y of decay per u n i t time i s the same for  a l l muons.  L i f e t i m e d i s t o r t i o n s a r e i n t r o d u c e d as  experimental a r t i f a c t s , however,  because i n a p i l e - u p  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 d e t e c t e d decay e t h a t stops the clock.  T h i s e f f e c t may be understood by c o n s i d e r i n g the  f o l l o w i n g gedanken experiment: imagine a magic beamline t h a t d e l i v e r s e x a c t l y two muons a t i n t e r v a l s o f T, the muon gate width, and imagine 100% decay e d e t e c t i o n  efficiency.  Obviously, one muon w i l l g e n e r a l l y decay b e f o r e the o t h e r . Since i t i s the f i r s t muon decay t h a t stops the c l o c k , an . accumulated histogram w i l l be s t r o n g l y b i a s e d toward times, thereby r e d u c i n g the apparent muon l i f e t i m e .  early For a  g i v e n p a i r o f muons e n t e r i n g the t a r g e t a t t = 0, what 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 b e f o r e some l a t e r time t ?  Denoting t h e p r o b a b i l i t y t h a t the n t h muon w i l l  survive u n t i l  t as -t/x = e ' y  I I I (5)  -256-  and r e c o g n i z i n g t h a t t h e decay o r s u r v i v a l o f 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 i s  p  ( l * s  s  independent e v e n t s , t h e r e q u i r e d )  2  =  P(s )P(s ) 1  = e"  2  probability 111(6)  2 t / x y  That i s , t h e p r o b a b i l i t y t h a t t h e c l o c k w i l l n o t be stopped by t i s j u s t t h e p r o b a b i l i t y t h a t b o t h muons s u r v i v e until  t.  (at least)  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 b e f o r e t as P(d ) = 1 - e "  t / T  y  I I I (7)  i t i s seen t h a t t h e p r o b a b i l i t y t h a t t h e c l o c k w i l l be stopped before t : P(d +d ) = p(d )  + P(d )  - P(d -d )  = P(d )  + P(d )  - P(d )P(d )  1  2  1  2  x  2  1  2  1  2  -t/x . - t / x ,2 (1- -e e y) ) y )u)" - (1  = 2 (11 - e  w  L  n  1 1 1  w  (8)  u  = ,1 - e~ 2 t / x y I t may be noted t h a t P ( s - s ) ,  L  As d i s c u s s e d  2  i n the previous  +  P (dj+d^)  = 1, as i t s h o u l d .  p a r a g r a p h , an e x p e r i m e n t a l  time  histogram corresponds t o the p r o b a b i l i t y that the c l o c k i s not stopped b e f o r e  t , b u t does stop between t and t + d t and t h a t  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 that the clock i s not stopped b e f o r e  time t .  S i n c e t h e r e a r e two muons r e s i d e n t i n  the t a r g e t a t t i n t h i s example, t h e p r o b a b i l i t y o f some muon d e c a y i n g between t and t + d t i s d o u b l e d , and t h e e x p e r i m e n t a l h i s t o g r a m has t h e form N(t)  = 2e"  2 t / T  y  I n t h i s gedanken e x p e r i m e n t , t h e n , t h e measured muon l i f e t i m e  -257-  i s hi  .  T  n  e  last  e x p r e s s i o n must be normalized  2 making i t correspond  by d i v i d i n g by  t o one muon a t t = 0 so t h a t i t may be  compared with the t h e o r e t i c a l n o n - d i s t o r t e d histogram e  -t/x  ,  (N(t) =  . .  y) , g i v i n g : N (t) XT  q  -2t/x = e y  In t h i s example, the n o r m a l i z a t i o n i s t r i v i a l  and i t makes the  argument t h a t l e d t o the e x t r a f a c t o r of 2 i n the f i r s t seem s u p e r f l u o u s .  place  However, when t h e procedure used i n t h i s  example i s a p p l i e d t o more complex cases below, the n o r m a l i z a t i o n s t h a t r e s u l t are n o n - t r i v i a l .  (i) Pre-y^ Muons and x : 100% Decay P o s i t r o n D e t e c t i o n y  Equation  Efficiency  111(6) may be extended t o g i v e the prob-  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 n muons enter t h e t a r g e t a t t = 0: P(s,-s -...-s ) = e ~ 1 2 n 0  n t / x  y  111(9)  K  where P(s )=P(s,,)-•••=P(s ); and equation 1  111(8) may be  extended t o g i v e 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 i f n muons enter the t a r g e t a t t = 0: P(d +d +. . ,+d ) = 1  2  I  n  k  =  (-l)  k-1  (£) P ( d )  1  , -nt/x = 1 - e y  k  k  111(10)  where P. (d, ) =P (d„) =. . . =P (d ). 1 2 n Pre-y^ muons a r r i v e i n the t a r g e t before t = 0 (when y ^ a r r i v e s ) but may not be r e s i d e n t i n the t a r g e t a t t = 0 because they have a l r e a d y 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 t a r g e t counting the y muon)? i  Consider an a r b i t r a r y time i n t e r v a l T  before t = 0 ( i t w i l l be shown e v e n t u a l l y t h a t c a l c u l a t i o n i s independent of ? f o r  the f o l l o w i n g  s u f f i c i e n t l y l a r g e Y and  that T - T, the muon gate width, f u l f i l l s t h i s The  a t t = 0 (not  p r o b a b i l i t y of n muons e n t e r i n g the t a r g e t  condition). d u r i n g 7 f o r an  average beam r a t e of 71 muons per u n i t time i s given by the P o i s s o n 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 o f a muon a r r i v i n g i n the t a r g e t subinterval  d u r i n g any  A ? " of T i s the same f o r a l l AT, the average a muon a r r i v i n g d u r i n g T s u r v i v e s u n t i l t = 0  p r o b a b i l i t y that is:  where P denotes the average p r o b a b i l i t y .  The  used i n equation I I I (5) have been dropped s i n c e  this  probability  a l s o , Y has been i n c l u d e d as an  i s the same f o r a l l muons; argument o f P ( s ) .  subscripts  I t may be noted t h a t  equation 111(11) goes  to the proper l i m i t s of T: lim  y..  r-o and  r  ( 1  l i m y.-.  - 7T 7  "  e  y )  T  r+co  r"  ( 1  "  e  ~Vi  ,  =  ,  1  v  U)  _  =  n  0  S i m i l a r i l y , the average p r o b a b i l i t y t h a t target  a muon e n t e r i n g the  d u r i n g Y has decayed by t = 0 i s : P(d(r) ) = 1 - P(s(r) )  I I I (12)  -259-  Assuming t h a t there are no muons i n the t a r g e t a t t = 0 -V  , the p r o b a b i l i t y of there being n muons i n the t a r g e t  at t = 0 i s the p r o b a b i l i t y t h a t : n a r r i v e during T'x the prob. t h a t a l l  l a s t u n t i l t=0  + n+1 a r r i v e during7'x the prob. t h a t a l l but 1 l a s t u n t i l t=0 + n+2 a r r i v e during T x the prob. t h a t a l l but 2 l a s t u n t i l t=0 + .. . T h i s may be expressed s y m b o l i c a l l y ; by combining equations and  III(11)  111(12) with the Poisson d i s t r i b u t i o n : t k=n  ( ) p (k,^,r)p(s(r)) p(d(r)) " k  n  n  k  H K B )  n  p  Combining t h i s e x p r e s s i o n  with equation I I I (9) m u l t i p l i e d by the  number-of. muons i n the t a r g e t - g i v e s the -unnormalized histogram: oo  N(t,#,r) =  £ n=0  E k=n  ( ) p^(k,^,r)p(s(r)) [i-p(s(r))] k  n  n  k  _  n  p  III(14)  , , -(n+1) t/x • (n+1) e y where n+1 r e f e r s t o n pre-y^ muons p l u s the y^ muon. I t must now be v e r i f i e d a function of Y f o r s u f f i c i e n t l y  t h a t equation 111(14) i s not l a r g e T.  Since the l a s t term  of equation 111(14) i s not a f u n c t i o n o f Y (but only a f u n c t i o n of t > 0), i t may be s e t t o 1 ( i . e . t = 0) and the equation r e - w r i t t e n by expanding the Poisson N(0,?2,7) =  £ (n+l)P(s(r)) e" n=0 n  ? ? ?  term: '  T, ( ) k=n k  [l-p"(s (71 ). ]  Changing the index of the second summation y i e l d s :  k  _  n  -260-  n  in i g p - [ 1-p (s (T) ) ] _ co n equation 111(11) f o r P ( s ( T ) ) and e = £ - h=0 * co  N(0,ft/n  =  00  £ (n+l)P(s(7)) e~^ -^pE n=0 ' m=0 n  7  n  Substituting  m  x  x  j  n  yields:  sO  N o,*,r> = (  (n i)P(s r)) +  ™  i222L n•  n  n  (  (n+l) <»» ,, (1-e y n=0 n! 2  e  n  - ^ - j — -r i n x  y)]  - ^ V ^ e  - H T  (  m  y e  Taking the l i m i t of t h i s e x p r e s s i o n as T •+  s  y  (  r  (e v  )  )  T / x  y) H ;  gives:  00  oo  l i m N(0 *,r) =  or  I  f  r^oo  N(0 7l) = e  n=0  - / ? T  f  ^ r - W - t , , ) n  e"^ y T  n  y  {Tlx )  y y  = i + nx  -  iri=0  m  m!  °°  (fit )  n=0  ri!  n  I l l (15)  y  As a check, one may a r r i v e a t t h i s r e s u l t by answering the q u e s t i o n , "What i s the average number of muons i n the t a r g e t at any time?"  The answer i s simply the i n t e g r a l of the  product o f the beam c u r r e n t / ^ e "  t  /  T  y  and the muon s u r v i v a l p r o b a b i l i t y :  dt = ^ T ( e ° - e ^ y ) y  = Tlx y  where the lower i n t e g r a t i o n l i m i t r e f e r s t o the time the beam i s turned on and the upper i n t e g r a t i o n l i m i t r e f e r s t o some very much l a t e r time.  Thus, when the y ^ muon e n t e r s the  t a r g e t a t t = 0, t h e r e are, on average, a l r e a d y Tlx muons i n the t a r g e t  f o r a t o t a l of l+7lx muons.  pre-y^  It i s easily  v e r i f i e d t h a t equation 111(14) i s reasonably independent of T forT^T,  a t y p i c a l muon decay gate.  For  example, i f the  m  -261-  upper i n t e g r a t i o n l i m i t o f t h e l a s t e x p r e s s i o n i s s e t t o T-  T - 4T / say, the r e s u l t i s accurate t o b e t t e r y 1  than 2%  E q u a t i o n 111(15) p r o v i d e s t h e n o r m a l i z a t i o n f o r e q u a t i o n I I I (14) •t  N  (t,7l) = n=0  k=n  0  P (k,?Z,T)P(s(T) ) [ l - P ( s ( T ) )] n  k  _  n  D  o  1 + 7cT  I I I (16)  y  , ,,, -(n+1)t/x • (n+l)e y -t/T  The  t r u e muon s u r v i v a l c u r v e , e  y, i s compared w i t h  e q u a t i o n 111(16) f o r v a r i o u s beam c u r r e n t s i n F i g u r e s I I I - 2 and I I I - 3 .  As e x p e c t e d ,  t h e e f f e c t o f p r e - y ^ muons i s  pronounced a t e a r l y t i m e s b u t d i m i n i s h e s t o i n s i g n i f i c a n c e a t l a t e t i m e s , as e v i d e n c e d  by t h e f a c t t h a t t h e l o g a r i t h m i c  curves are p a r a l l e l a t l a t e times.  T h i s i s because, by  d e f i n i t i o n , a p r e - y ^ muon i s o l d e r than t h e y^ muon, and so i t s chance o f s u r v i v i n g u n t i l t = 4ys, s a y , i s much l e s s t h a t o f t h e y^ muon.  than  The apparent muon l i f e t i m e s o b t a i n e d by -t/T  f i t t i n g the histogram t o e y over a 4ys time range would be (from F i g u r e I I I - 3 ) 2.0, 1.85, and 1.7 ys f o r beam c u r r e n t s o f 3 -1 50, 100, and 150 x 10 s respectively. ( i i ) P r e - y ^ Muons and T The  y  ; e 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  p r o v i s i o n o f 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 t h a t t h e r e a r e two p o s s i b l e outcomes o f 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 p r e v i o u s S e c t i o n , t h e 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 I I I - 2 :  The e f f e c t o f pre-LK muons on the apparent  muon l i f e t i m e , w i t h e = 100% p o s i t r o n counting e f f i c i e n c y . The upper curves i n each p l o t show the " t r u e " histogram, -t/x e u, w h i l e the lower curves show equation 111(16) f o r 3 -1 beam c u r r e n t s o f 50, 100, and 150 x 10 decay gate T = 20 ys, t h i s c a l c u l a t i o n 1 ppm.  s  . With the muon  i s a c c u r a t e t o about  -263-  0.001  FIGURE I I I - 3 : L o g a r i t h m i c p l o t s o f F i g u r e I I I - 2 . A t l a t e times, the lower, pre-y^ curves are p a r a l l e l t o the t r u e :  muon l i f e t i m e c u r v e s , showing t h a t t h e e f f e c t o f p r e - y ^ muons on t h e measured muon l i f e t i m e i s o n l y i m p o r t a n t a t early  times.  -264-  P(d) = ( l - e  t / T  E  y)  S i m i l a r i l y , t h e p r o b a b i l i t y t h a t a muon decays and i s n o t detected i s P(c()  = (1-e) ( l - e ~  t / x  y)  P r o c e e d i n g as i n t h e p r e v i o u s S e c t i o n , an e q u a t i o n analogous to  e q u a t i o n 111(6) may be w r i t t e n t o g i v e t h e p r o b a b i l i t y  t h a t t h e c l o c k w i l l n o t be stopped b e f o r e t i f two muons e n t e r the t a r g e t a t t = 0: P(s,-s ) + P(s ) + P(di -s ) + P ( % x z . x z x z . x = (e"  t / T  y)  2  + 2(l-e)e~  t / x  y(l-e~  t / x  ) z  111(17)  y ) - + (1-e) ( l - e ~ 2  t / x  y)  2  The f i r s t term i s i d e n t i c a l t o e q u a t i o n 1 1 1 ( 6 ) , t h e second term c o r r e s p o n d s t o one muon s u r v i v i n g and t h e o t h e r d e c a y i n g u n d e t e c t e d , and t h e l a s t term c o r r e s p o n d s t o b o t h muons d e c a y i n g without detection.  S i m i l a r i l y , the p r o b a b i l i t y that the clock  w i l l be stopped b e f o r e t may be w r i t t e n  (analogous t o e q u a t i o n  IIK8) ) : P(d +d ) = 2 e ( l - e ~ x  t / x  2  y) - e (l-e~ 2  t / x  y)  111(18)  2  t h a t i s , i t c o r r e s p o n d s t o t h e p r o b a b i l i t y o f e i t h e r muon decaying w i t h d e t e c t i o n .  A g a i n , i t i s r e a d i l y checked  the sum o f e q u a t i o n s 111(17) and 111(18) i s one.  that  Equation  111(17) may be g e n e r a l i z e d t o c o r r e s p o n d t o t h e case o f n muons e n t e r i n g t h e t a r g e t a t t = 0: n P noti stopped • = ^Z Q  . k (e . - t / xy). n-k, / T .k (rn-\ J (1-e) (1-e -' t y) /  n  K  I I I (19)  -265-  and e q u a t i o n 111(18) may be g e n e r a l i z e d i n l i k e manner P ^stopped nrsn^ =  S  * k=l  ^  111(20)  The e x p e r i m e n t a l h i s t o g r a m c o r r e s p o n d s  t o the p r o b a b i l i t y that  the c l o c k has n o t stopped b e f o r e t b u t does s t o p and t + d t .  between t  When each term i n e q u a t i o n 111(19) i s m u l t i p l i e d  by t h e number