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Atomic capture of negative muons in oxides Stanislaus, Thanthirimudalige Don Shirvel 1983

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ATOMIC CAPTURE OF NEGATIVE MUONS IN OXIDES by THANTHIRIMUDALIGE DON SHIRVEL STANISLAUS B.Sc.(Hons.), University of S r i Lanka, Peradeniya, 1976 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n THE FACULTY OF GRADUATE STUDIES (Department of Physics) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA December 1983 © Thanthirimudalige Don Shirvel Stanislaus, 1983 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements fo r an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I further agree that permission for extensive copying of t h i s thesis for s c h o l a r l y purposes may be granted by the head of my department or by his or her representatives. I t i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of Posies  The University of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date 3o* Dece-m key 138$ DE-6 (3/81) i i ABSTRACT Using the l i f e t i m e technique, atomic capture r a t i o s of negative muons have been measured i n 41 oxides. The experimental method was to detect the decay electrons and to use the unique l i f e t i m e signature to i d e n t i f y the element which captured the muon. Muons were produced v i a the backward decay of pions which were provided by the M20 channel at TRIUMF. The experimental set-up and e l e c t r o n i c l o g i c were tested against the p o s i t i v e muon l i f e t i m e which was measured to be 2197.7 ± 2.6 ns, i n good agreement with the present world average of 2197.138 ± 0.065 ns. The r e s u l t s , though somewhat d i f f e r e n t from the e a r l i e r TRIUMF findi n g s , are consistent with the mesic x-ray measurements. We have also c a r r i e d out new measurements for MnO, FegO^, CoO, NiO, NbO, MoOj, SnO, P r 0 2 , HgO and PbgO^ I t has been noticed that the atomic capture r a t i o s have a l i n e a r dependence on the density, whereas the Z-dependence i s p e r i o d i c . Using our r e s u l t s and data from the l i t e r a t u r e , we have been able to obtain an empirical formula for the atomic capture r a t i o s i n oxides, chlorides and f l u o r i d e s . Our r e s u l t s for oxides, also show evidence for the existence of chemical and/or s o l i d state e f f e c t s i n the atomic capture process. The l i f e t i m e s of negative muons bound i n heavy n u c l e i were also measured during t h i s experiment. Our r e s u l t s are i n good agreement with the e a r l i e r measurements ca r r i e d out at TRIUMF. i i i TABLE OF CONTENTS CHAPTER 1 : Introduction 1 1.1 Discovery of the Muon 1 1.2 Properties of the Muon 2 1.3 Decay of the Muon 3 1.4 Interaction of Negative Muons with Matter 4 1.5 The Nuclear Capture of Muons 7 1.6 Disappearance Rate and Lifetime 8 1.7 Experimental Methods to Measure the Atomic Capture Ratio 8 1.8 Atomic Capture Models 10 CHAPTER 2 : Experimental Method 18 2.1 Muon Beam Line 18 2.2 S c i n t i l l a t i o n Counters 22 2.3 Magnetic Shielding 24 2.4 Collimator and Degrader 25 2.5 Targets 26 2.6 E l e c t r o n i c s 26 2.7 Run Procedure 33 2.8 Run Record 34 CHAPTER 3 : Data Analysis 40 3.1 Data Analysis Procedure 40 3.2 The Carbon Background 45 3.3 Bound Muon Decay 56 3.4 The Atomic Capture Ratio 58 3.5 Correction E 0 for the Lost Electrons 59 iv TABLE OF CONTENTS (Continued) 3.6 Counter Asymmetries 60 3.7 Lifetime Analysis 60 CHAPTER 4 : Experimental Results 62 4.1 Lifetime Measurements 62 4.2 X-ray Analysis 65 4.3 Atomic Capture Ratio Measurements 66 4.4 Comparison with the Past Measurements 69 4.5 Comparison with Theoretical Models 72 4.6 Chemical or S o l i d State E f f e c t s i n Atomic Capture 75 CHAPTER 5 : Dependence of Atomic Capture Ratio on Density 76 5.1 Evidence for a Dependence on Density 76 5.2 Atomic Capture Ratios for Chlorides 84 5.3 Discussion on the Empirical Formulae 86 CHAPTER 6 : Summary 91 REFERENCES 94 V LIST OF TABLES I Counter Geometry 23 II L i s t of Targets (Run #1) 27 II I L i s t of Targets (Run #2) 28 IV Meaning of the Symbols i n Logic Diagram 30 V Run Records (Run #1) 35 VI Run Records (Run #2) 37 VII The values of C o e f f i c i e n t s o m n 50 VIII Target Thicknesses (Run #1) 51 IX Target Thicknesses (Run #2) 53 X Negative Muon Lifetimes 63 XI Atomic Capture Ratios from the 2 Runs 67 XII Atomic Capture Ratios of Negative Muons i n Oxides 70 XIII Chi-squares for Oxides 80 XIV Chi-squares for Chlorides 87 v i LIST OF FIGURES 1 Theoretical decay electron spectrum for free muons 5 2 (a) The dependence of W(Z)/W(0) i n oxides ( Z m 0 n ) on atomic number Z (b) The dependence of e l e c t r o n i c stopping power S e on atomic number Z 16 3 (a) Experimental set up (b) Decay electron counters 19 4 The old M20 beam l i n e 20 5 The new M20 beam l i n e 21 6 E l e c t r o n i c l o g i c 31 7 The decay electron spectrum for negative muons i n Na 20 2 42 8 The decay electron spectrum f o r negative muons i n PbO 43 9 The dependence of carbon normalisation on the target thickness (a) f o r run #1 (b) for run #2 46 10 The o r e t i c a l spectra of decay electrons from negative muons i n lead, antimony, z i n c , i r o n and magnesium 57 11 Experimental atomic capture r a t i o s i n oxides (from this work) and the t h e o r e t i c a l predictions 73 12 The dependence of atomic capture r a t i o s on density (a) Monoxides (b) Dioxides (c) Sesquioxides 77 13 Experimental and t h e o r e t i c a l atomic capture r a t i o s f o r negative muons i n oxides. 85 v i i ACKNOWLEDGEMENTS I would l i k e to extend my sincere gratitude and appreciation to my supervisor Professor David F. Measday for h i s guidance, advice and c r i t i c i s m . His support and patience throughout t h i s work has been a great source of encouragement for me. I am greately indebted to Dr. Farrokh Entezami for h i s invaluable c o l l a b o r a t i o n i n the experiment and his suggestions and discussions during the analysis of the data and the preparation of the the s i s . I would also l i k e to thank Ardeshire Bagheri, Dr. Dave Garner and John Warden for t h e i r invaluable help with the data a q u i s i t i o n . Dr. Dezso Horvath i s thanked for very useful and stimulating discussions. I would l i k e to thank Peter Mulhern of the Physics Department and Stanya Horsky of the Geological Science Department for th e i r help with the x-ray analysis and Visha Saravanabawan f o r her help with the word processor. F i n a l l y , I wish to thank my parents for t h e i r continuous help and encouragement. However, i t i s deepely regretted that my father could not see the completion of my work at UBC. To the loving memory of my father 1 CHAPTER 1  Introduction 1.1 Discovery of the Muon The muon was discovered by Anderson and Neddermeyer 1 i n 1938 during an examination of the processes associated with the development of showers i n cosmic r a d i a t i o n . An apparent anomaly was observed i n the absorption of these showers i n various materials. Some of these p a r t i c l e s penetrated thicknesses of matter which would not be f e a s i b l e i f they were e i t h e r electrons or protons. Furthermore, the p a r t i c l e s appeared to possess e i t h e r p o s i t i v e or negative e l e c t r i c charge. The analysis of the experimental data strongly indicated the existence of a p a r t i c l e of mass i n the region of 100 - 200 electron masses(m e). From a study of a photograph of a p a r t i c l e stopped i n a cloud chamber they estimated the mass to be ~ 240 mg. The existence of a p a r t i c l e with about the same mass had been predicted by Yukawa 2 i n 1935. In his predictions, Yukawa suggested that the p a r t i c l e should undergo strong i n t e r a c t i o n with nucleons and decay spontaneously into an electron and a neutrino with an estimated l i f e t i m e of about 1 0 - 7 s e c . But the measurements of the l i f e t i m e by R a s e t t i 3 and others gave an average value of ~ 2 x 10~ 6 sec. Experiments c a r r i e d out by Conversi et a l . \ who measured the r a t i o of the nuclear absorption of negative muons i n l i g h t elements, showed that the muons did not i n t e r a c t strongly with the nucleus. This evidence suggested that the p a r t i c l e discovered by Anderson and Neddermeyer was not the p a r t i c l e predicted by Yukawa. Later a strongly i n t e r a c t i n g 2 p a r t i c l e with a mass of about 270 me was discovered by Lattes et a l . 5 which was found to decay into a weakly i n t e r a c t i n g p a r t i c l e with a mass ~ 200 me. Consequently, the strongly i n t e r a c t i n g p a r t i c l e was c a l l e d the pion and the product p a r t i c l e the muon. I t was noted that the p a r t i c l e s observed i n cosmic rays were muons, the decay products of pions. 1.2 Properties of the Muon The muons are leptons, thus they are p o i n t l i k e p a r t i c l e s (experimentally t h e i r r a d i i < 0.001 fm. 6). Muons e x i s t i n two charge states, \i~ and u +, the u~ normally being considered the p a r t i c l e and p,+ i t s a n t i - p a r t i c l e . The i n t e r a c t i o n of muons with charged leptons and nucleons consists of an electromagnetic and a weak term with the l a t t e r having a n e g l i g i b l e e f f e c t i n most circumstances. The muon has a mass of 105.65941 + 0.00018 MeV/c2 ( r e f . 7 ) . The most recent measurement of the mass of the muon was car r i e d out by measuring the frequency of the hyperfine Zeeman t r a n s i t i o n s i n the ground state of muonium8. Muonium (p.+e~) i s a hydrogen-like atom consi s t i n g of a p o s i t i v e muon and an el e c t r o n . These measurements give a precise value f o r the muon magnetic moment (or p^/up) which i n turn i s used to obtain the mass of the muon (or m^/mg) from the r e l a t i o n , where g^ refe r s to the gyro magnetic r a t i o . The present world average of the muon l i f e t i m e i s 2.197138 ± 0.000065 microseconds 7. The l a t e s t precise measurement of the muon l i f e t i m e was c a r r i e d out at TRIUMF9, and the l i f e t i m e obtained was 2.19695 ± 0.00006 microseconds, which d i f f e r s s l i g h t l y from the current world average, while increasing i t s p r e c i s i o n . The method used was to detect the muon decay electrons and make use of the exponential character of the decay electrons to determine the l i f e t i m e . The l i f e t i m e of the negative muon ( i n hydrogen) was measured by Bardin et a l . 1 0 to be 2.194903 ± 0.000066 microseconds. This i s consistent with the p + l i f e t i m e when the weak i n t e r a c t i o n absorption i s taken into account (~460 s e c - 1 ) . The anomalous magnetic moments of the muons defined as a u = 1 / 2 ( 8 p " 2> a r e found to b e 1 1 a u+ = (1165911 ± 11) x 10~ 9 and a u - = (1165937 ± 12) x 10~ 9 giving an average value of a^ = (1165924 ± 8.5) x 10~ 9. These r e s u l t s are In good agreement with the predictions of QED c a l c u l a t i o n s . Comparison of the properties of po s i t i v e and negative muons show that they are i d e n t i c a l to a l e v e l of 5 x 10~ 5 with a 95% co n f i d e n c e 1 1 . This serves as a precise v e r i f i c a t i o n of the CPT theorem for muons, which predicts that the properties of a p a r t i c l e and i t s a n t i - p a r t i c l e are either equal or opposite (although there can be minor exceptions to t h i s rule because of CP non-conservation). We s h a l l henceforth assume th i s r e l a t i o n s h i p to hold and use experimental data from one muon to apply to i t s a n t i - p a r t i c l e . 1.3 Decay of the Muon A free muon decays into an electron and two neutrinos as follows; + + -u •*• e + v + v e u 4 The three p a r t i c l e decay scheme was confirmed by the observation of the energy spectrum of the decay e l e c t r o n s 1 2 . Figure 1 shows the energy spectrum of the decay electrons. The maximum energy of the electrons i s found to be 52.8 MeV/c 2. This decay has a branching r a t i o of 98.6%. Other rare decay modes and t h e i r corresponding branching r a t i o s (obtained from reference 7 and updated from references 13 and 14) are: p f * e~ + v + V + Y 1.4 ± 0.4% for E >10 MeV e u y •+ e~ + y + Y < 8 . 4 x 10" 9 -»• e~ + e + + e~ < 1.9 x 10~ 9 •*• e + y < 1.7 x 1 0 _ 1 ° -»• e + v + v <9% e u The decay of the bound muon, which has d i f f e r e n t c h a r a c t e r i s t i c s compared to that of a free muon, w i l l be discussed i n d e t a i l i n section 3.3. 1.4 Inte r a c t i o n of Negative Muons with Matter When a \i~ slows down and stops i n a material, a complex s e r i e s of events take place. The i n t e r a c t i o n can be described i n f i v e steps, corresponding to f i v e d i f f e r e n t energy r e g i o n s 1 5 . 1. High energy to 2 keV: In the f i r s t step, muons of several hundred MeV energy slow down to 2 keV, l o s i n g t h e i r energy mainly through c o l l i s i o n s with atomic electrons. At the end of th i s stage, t h e i r v e l o c i t y i s almost equal to that of the valence electrons. The time needed to slow down, to 2 keV i s about 10""9 to 10~ 1 0 sec. i n condensed matter. 1.25 1.00 0.75 0.50 \ — 0.25 \ — 0.00 10 20 30 40 Energy (MeV) • ? i e u r e 1 : Theoretical decay electron spectrum for free muons. 6 2. 2 keV to capture: The muon of about 2 keV loses more energy by exchanging energy with electrons. The slowing down time i n t h i s region i s around 3 x 10 - 1 1* sec. for metals and about 10" 1 3 sec. f o r i n s u l a t o r s . In gases however, a muon needs about 1 0 - 9 sec. to stop. 3. Atomic capture: Although early c a l c u l a t i o n s 1 6 Indicated that the capture occurred when the muon s t i l l had an energy of hundreds of eV, more recent studies suggest that, at least for hydrogen, the capture occurrs when the muon has an energy of only 15 eV or s o 1 7 . Capture w i l l occur into a muonic o r b i t of the same size as the electron o r b i t s , and i t w i l l have a p r i n c i p a l quantum number n u - (mp,/me)1/2 = 14. 4. Electromagnetic cascade: As a l l muonic states are unoccupied, the muon w i l l cascade down to states of lower energy within about 1 0 - 1 3 sec. The t r a n s i t i o n i s accompanied by the emission of Auger electrons or c h a r a c t e r i s t i c electromagnetic r a d i a t i o n or e x c i t a t i o n of the nucleus. In cascading down, from n ~ 14 to lower states, the i n i t i a l low energy muonic t r a n s i t i o n s w i l l y i e l d a strong Auger e f f e c t . However, as n reaches lower values, the t r a n s i t i o n energy increases r a p i d l y (~n - 3) and the ra d i a t i v e t r a n s i t i o n s dominate. 5. Ultimate f a t e : Once i n the lowest (Is) l e v e l , the muon ei t h e r decays or i s captured by the nucleus v i a the weak i n t e r a c t i o n . The disappearence time i s only s l i g h t l y l e s s than 2.2 x 10~ 6 sec. i n l i g h t elements, but reduces to 80 ns i n heavy elements. 7 1.5 The Nuclear Capture of Muons A muon captured by an atom disappears e i t h e r by decaying or by being captured by the nucleus v i a the weak i n t e r a c t i o n . The basic process of muon capture i n a nucleus i s the reaction: u~ + p n + The capture of the u~ normally leaves the nucleus i n an excited state (mainly a giant resonance state around 20 MeV of excit a t i o n ) which consequently de-excites with the emission of one or more n e u t r o n s 1 8 . The t o t a l capture rate i n the nucleus, R c, was t h e o r e t i c a l l y calculated by Wheeler 1 9 as where Z e f f i s the e f f e c t i v e Z or the reduced charge "seen" by the muon inside the nucleus. This law i s v a l i d approximately for l i g h t elements, but i t overestimates the capture rate for heavy elements. P r i m a k o f f 2 0 derived a formula for the t o t a l muon capture rate, R C(A,Z), i n a complex nucleus with mass number A and atomic number Z by using an e f f e c t i v e hamiltonian and c a l c u l a t i n g the t r a n s i t i o n matrix elements for a l l accessible excited states of the daughter nucleus. He obtained: where R c ( l , l ) i s the muon capture rate i n hydrogen and g i s the nucleon-nucleon c o r r e l a t i o n parameter; the estimated value for g i s 3. K represents the reduction of phase space f o r the neutrino. Goulard and Pr i m a k o f f 2 1 improved on the above formula to obtain (1) R C(A,Z) = (Zeff) 1*. R c ( l , l ) K[ 1 - g(A-Z)/2A] (2) 8 R C(A,Z) = X ( l ) . ( Z e f f ) 1 * [ 1 + (A/2Z).X(2) - (A-Z) .X(3)/2A -((A-Z)/2A + (A-2Z)/8ZA).X(4)] (3) where X ( i ) , i = 1 to 4 are constants. Experiments done i n TRIUMF 2 2 have shown that the two formulae (2) and (3) show almost the same behaviour. 1.6 Disappearance Rate and Lifetime The t o t a l disappearance rate, R r, of a muon i s the sum of the decay rate of the bound muon Rd( -) and the capture rate R c: R t = R c + Rd(-) Rd( -) i s related to the free muon decay rate, R(+) through the Huff factor Qhuff» a s follows R d ( " ) = Qhuff-Rd(+) The muon decay rate w i l l be discussed further i n section 3.3. The l i f e t i m e of the muon i s the inverse of the t o t a l disappearance rate T ^ ( - ) = 1/Rt 1.7 Experimental Methods to Measure the Atomic Capture Ratio When a negative muon i s stopped i n a compound consisting of elements Z^ and Z 2, i t i s captured by eit h e r of the two elements. If W(ZjJ i s the p r o b a b i l i t y of the muon being captured by the atom Z±t then the quantity ACZj/Zj) i s c a l l e d the "atomic capture r a t i o " , where A(Z 1/Z 2) = W(Z 1)/W(Z 2) The atomic capture r a t i o can be measured using two d i f f e r e n t experimental methods. 1. Lifetime Method: This method i s based on the fac t that the l i f e t i m e of a 9 negative muon i n matter depends strongly on the nuclear charge Z. The time spectrum of the py-decay electrons i s a simple exponential with a time constant c h a r a c t e r i s t i c of the element selected. (There are minor modifications of t h i s i n elements such as F and CI due to hyperfine t r a n s i t i o n . ) In a chemical compound, the time spectrum w i l l c o n sist of the sum of as many exponentials as the number of elements i n that compound. Through appropriate analysis of the time spectrum i t i s possible to determine the weight of each exponential i n the t o t a l spectrum and thereby determine the p r o b a b i l i t y of atomic capture of the muon i n a given element i n the chemical compound studied. 2. Mesic X-ray Method: This method makes use of the fact that the t r a n s i t i o n energy of a muon i n a mesic atom depends strongly on the nuclear charge Z; i n l i g h t n u c l e i i t i s proportional to Z 2. The p r i n c i p a l technique i s to sum the i n t e n s i t y of the muonic K x-rays (the Lyman se r i e s ) for a ce r t a i n element i n a compound and to equate t h i s to the number of muons which have been captured by that atom. The i n i t i a l measurements of the capture r a t i o s were c a r r i e d out using the l i f e t i m e method 2 3, but few compounds were studied. The mesic x-ray method was f i r s t used by Zinov et a l . 2 1 * and since then most of the measurements have been c a r r i e d out using t h i s method. In the present work we used the l i f e t i m e method. A de t a i l e d d e s c r i p t i o n of the experimental method and analysis i s given i n chapters 2 and 3 resp e c t i v e l y . 1 0 1.8 Atomic Capture Models The atomic capture of mesons i n chemical compounds was f i r s t studied by Fermi and T e l l e r . In t h e i r p a p e r 2 5 published i n 1947 they estimated the capture p r o b a b i l i t i e s of mesons (muons or hadrons) i n atoms to be roughly proportional to t h e i r atomic numbers and proposed the so c a l l e d Z-law A(Z 1/Z 2) - W(Z 1)/W(Z 2) = Z x/Z 2 (4) In deducing equation (4) i t has been assumed that the capture rates are proportional to the stopping powers of the atoms. This model was f i r s t put to the test by Sens et a l . 2 3 i n 1958 when the f i r s t measurements of atomic capture r a t i o s were c a r r i e d out. They concluded that, at le a s t i n Insulators, the predictions of the Z-law do not hold. Since then the Z-law has undergone numerous experimental tests and modifications. The f i r s t of the modifications was introduced by B a i j a l et a l . 2 6 i n 1963, when they observed that t h e i r experimental capture r a t i o s could be better described by : A(Z 1/Z 2) = ( Z 1 / Z 2 ) n (5) with most of the experimental r e s u l t s f a l l i n g i n the range 0.5 < n < 1.5 ; of course, n = 1 corresponds to the Z-law. In 1966 Zinov et a l . 2 * * were the f i r s t to observe that the atomic capture r a t i o of muons i n oxides varies p e r i o d i c a l l y with increasing nuclear charge i n correspondence with the periods i n the periodic table of elements. They also proposed a modified empirical Z-law for m e t a l l i c halides and a l l o y s as: A(Z X/Z 2) - 0.66 ( Z 1 / Z 2 ) (6) Vogel et a l . 2 7 i n 1975 d i scussed the slowing down and capture of negative muons i n s o l i d s on the basis of c l a s s i c a l equations of 11 motion where the energy d i s s i p a t i o n i s described i n terms of f r i c t i o n a l f o rces. Using a s t a t i s t i c a l model of the atom they deduced a Z-dependence of the capture r a t i o s as: ACZJ/ZJ) = (Zj / Z , , ) 7 ' 6 (7) In a paper published the same year, D a n i e l 2 8 considered the energy loss of the meson i n the atom along a t r a j e c t o r y which i s close enough to the nucleus for the meson to be subsequently captured i n an atomic o r b i t and concluded that the p r o b a b i l i t y of coulomb capture i s proportional to Z 1 / 3 l n ( 0 . 5 7 Z), i . e . 1/3 Z ' ln(0.57 Z ) A(Z /Z ) = — (8) Z 2 i / J ln(0.57 Z 2) This model was tested using the e x i s t i n g data from halides, and produced remarkably good r e s u l t s as compared to the Z-law. For the 27 halides used, the above law gave a reduced chi square ( c h i square per degree of freedom) of 1.61 as compared to 15.98 from the Z-law. By t h i s time the experimental data from atomic capture r a t i o s i n oxides had c l e a r l y shown and confirmed the existence of o s c i l l a t i o n s i n capture r a t i o s with increasing nuclear charge. Four years l a t e r , i n 1979, i n order to describe these o s c i l l a t i o n s D a n i e l 2 9 modified h i s own model by including the atomic r a d i i R(Z), and he showed that a smaller atom ( i . e . denser electron cloud), has a higher p r o b a b i l i t y for capturing the muon v i z : Z. 1 / 3 ln(0.57 Z.).R(Z 9) A(Z /Z ) f r r ± (9) z£ / J ln(0.57 Z 2).R(Z 1) This expression describes the o s c i l l a t i o n s i n the oxides f a i r l y w e l l , 12 although there appears to be an ambiguity i n the d e f i n i t i o n of the atomic r a d i u s 3 0 . In 1976 Petrukhin and Suvorov 3 1, i n an experimental study of pion capture i n mixtures of hydrogen with noble gases, empirically found a Z-dependence somewhat s i m i l a r to (8). A f t e r the removal of the contribution of pion t r a n s f e r , the observed atomic capture r a t i o s A(Z/H) could be well approximated by A(Z/H) a Z 1 / 3 - 1 (10) Later, Vasileyev et a l . 3 2 generalised t h i s expression i n the following form: 1/3 A(Z X/H) zj' - 1 A ( V Z 2 > = ^ H 7 = Z ^ 3 - l < » > and compared i t s predictions with the atomic capture r a t i o s a v a i l a b l e at that time. The equation approximated the experimental values much better than the Z-law (4) and somewhat better than Daniel's formula (8). By t h i s time the e f f e c t of the chemical bond i n atomic capture had been observed. Zinov et a l . 2 1 * and Kessler et a l . 3 3 had shown that the muonic x-ray K-series spectra i n pure metal titanium have more t r a n s i t i o n s from high o r b i t s than those of titanium oxides. In 1978, Schneuwly et a l . 3 1 * reported systematic measurements of capture r a t i o s i n selected compounds of nitrogen, sulphur and selenium, confirming that the chemical structure plays an important r o l e i n the atomic capture of muons. Later, Schneuwly et a l . 3 5 proposed the f i r s t successful theory to take into account the e f f e c t of core and valence electrons. An ess e n t i a l feature of t h e i r approach i s the assumption that the d e c i s i v e 13 role In the capture process i s played by not too strongly bound electrons. They formulated a model where p(E) the e f f i c i e n c y of an electron's p a r t i c i p a t i o n i n atomic capture depend's on the electron's binding energy E, and i s given by (a) a "sharp boundary" p(E) = 1 for E < E Q 0 for E > E Q (12) or (b) a "smooth boundary" p(E) = 1 for E < E Q for E > E Q (13) where E Q and E c are parameters The atomic capture r a t i o i s defined as n 1 + 2V . U 'A(Z 1/ Z 2) - (14) n2 + 2 v 2 ^ 1 ~ w ) where n^ and n 2 are r e s p e c t i v e l y the e f f e c t i v e electron numbers of atoms Zj, and Z 2 and are given by n ± = I p(Ej) n*(Z) where E j * i s the binding energy and n j * the number of electrons i n the j t n subshell of atom Z^. V j and v 2 are the corresponding valencies, u i s the t r a n s i t i o n p r o b a b i l i t y of the meson from the molecular o r b i t to the atomic o r b i t of atom Z^, and i s assumed to have one of the following forms: = ( 1 + p 2 q 2 / p 1 q 1 ) - 1 for a lo n g - l i v e d mesic molecular state, or w2 = p 1B 1 + ( 1 - p ^ - P 2 B 2 ) q 1 for a 14 s h o r t - l i v e d mesic molecular state. The p's are the valence electron d e n s i t i e s expressed i n terms of a, the i o n i c i t y of the Z^- Z 2 bond: P 1 = | ( 1 - C T ) ; P 2 = | ( 1 + C T ) The t r a n s i t i o n from molecular state to atomic state i s described by the quantities q and 8: q x - [ 1 + ( Z 2 / Z 1 ) 2 ] ~ l ; q 2 - 1 - q t a " l 0 "2 P J Y>2 =  1 n + 2v^ n 2 + 2v 2 Schneuwly et a l . showed that the smooth boundary approximation with u ) 2 and parameters E Q = 15 eV E c = 100 eV for Z > 18 70 eV for Z < 18 can describe the atomic capture r a t i o s , measured i n oxides, better than any of the e a r l i e r mentioned models. In 1981 Evseev et a l . 3 6 showed for the f i r s t time that, within the t h e o r e t i c a l concepts of Fermi and T e l l e r , i t i s possible to explain the p e r iodic o s c i l l a t i o n s of the atomic capture r a t i o s . Evseev et a l . assumed the capture p r o b a b i l i t y to be proportional to stopping power of the atom, i . e . A(Z / Z ) (15) se(z2) 1 dE where S = i s the e l e c t r o n i c stopping power of the atom. 6 N dx dE/dx i s the l i n e a r energy loss by charged p a r t i c l e s i n the element 15 target and N Is the number of atoms per cm 3. They showed that, f or low energy p a r t i c l e s stopping i n a target, S e(z) has p e r i o d i c o s c i l l a t i o n s depending on the atomic number Z of the stopping atom of the element target. The pattern of these o s c i l l a t i o n s i s c l o s e l y related to that shown by the atomic capture r a t i o s i n oxides. Figure 2 compares the dependence of S e and A(Z/0) as a funtion of Z. Of course t h i s only transforms the problem to having to explain why there are o s c i l l a t i o n s i n the stopping powers. The e f f e c t of s o l i d state structure on capture r a t i o s was observed i n 1982 by Schneuwly et a l . 3 7 i n an experiment i n which they measured the muonic x-ray i n t e n s i t i e s of the Lyman series i n boron and nitrogen i n the cubic and hexagonal structures of boron n i t r i d e . They observed a difference of 18 ± 3 % i n the boron to nitrogen capture r a t i o i n the cubic and hexagonal structures. This difference has been at t r i b u t e d to the diffe r e n c e i n s o l i d state structures of the two samples. In 1983, Horvath and Entezami of TRIUMF 3 0 tested various models of atomic capture of negative mesons against 321 experimental atomic capture r a t i o s , measured for binary systems, and concluded that the general agreement between theory and experiment i s not s a t i s f a c t o r y . They t r i e d to improve the models v i a introducing adjustable parameters to be estimated by f i t t i n g the experimental data. The model which produced the best agreement with the measured data was the model o r i g i n a l l y proposed by Schneuwly et a l . 3 5 with modified parameters E Q = 0 E c = 86 eV for Z < 15 = 119 eV for Z > 15 Figure 2 : (a) The dependence of W(Z)/W(0) in oxides (Zmon) on atomic number Z. (b) The dependence of electronic stopping power S e on atomic number Z. 17 For mixtures of elements (gases) the empirical formula (8) of Vasileyev et a l . turned out to be the "best" and the o v e r a l l "best" Z-law was shown to have the following form: A(Zj/ Z 2) = 0.69 ( Z j / Z 2)0'86 Recently von Egidy et a l . 3 8 have calculated the muonic coulomb capture p r o b a b i l i t i e s assuming that electrons with binding energies less than a given l i m i t (about 80 eV) contribute with a weight which i s a function of the electron binding energy, the electron quantum numbers n and 1, and Z. Using a quantum mechanical c a l c u l a t i o n they obtained the following parametrized form of the capture p r o b a b i l i t y . P(Z) = £ N ±or 1 a± = C f l - E 1/E 0) 1 / 2.Z^nJ for E±< E Q a± = 0 for E ±> E Q where Is the number of electrons i n o r b i t i , E^ i s the binding energy of these electrons and C, E Q , a and b are f i t t i n g parameters, von Egidy et a l . f i t t e d t h i s function to average experimental p r o b a b i l i t i e s 3 9 and obtained the parameters as follows. E Q = (76 ± 3)eV, a = 0.30 ± 0.03 and C = 0.0864 i f b = 0 or E Q = (79.7 ± 4.0)eV, b = 0.735 ± 0.060 and C = 0.0942 i f a - 0 This model seems to produce a remarkably low chi-square per data point for the average capture p r o b a b i l i t i e s when compared with the models of Schneuwly, Daniel etc. But, apparently to produce t h i s low chi-square value they have normalised t h e i r calculated values i n such a way that i t produces a weighting of 0.87 for capture i n oxygen whereas the data are a l l normalized s p e c i f i c a l l y to give unity for t h i s value. This feature of von Egidy et a l . ' s model has not been understood yet. 18 CHAPTER 2 Experimental Method Atomic capture rate measurements were ca r r i e d out i n 13 days during two d i f f e r e n t scheduled beam times, using a beam of backward muons provided by the stopped muon channel (M20) at TRIUMF. Figure 3 shows the main features of our experimental set up. 2.1 Muon Beam Line A schematic diagram of the stopped muon channel (M20) at TRIUMF i s shown i n figures 4 and 5. The proton beam s t r i k e s a pion production target, T 2, which consists of a water cooled s t r i p of beryllium, 10 cm long i n the beam d i r e c t i o n and 5 mm x 15 mm i n cross section. A beryllium target has the advantages of having a r e l a t i v e l y high production rate for negative pions and a low electron contamination i n the pion beam. Throughout the f i r s t run the proton beam current was f a i r l y stable at 25 pA and during the second run the beam current was steady at 30 pA, except for the l a s t three days when i t was increased to 100 uA. The proton beam energy i n both cases was about 500 MeV. Po s i t i v e muons, depending on t h e i r method of production, are categorized into cloud, conventional and surface muons. On the other hand, negative muons have only the cloud and conventional modes. The conventional mode i s a combinations of forward and backward muons. The muons which decay into the same d i r e c t i o n i n the centre of mass as the pion d i r e c t i o n are c a l l e d forward muons and those decaying into the 19 Figure 3 : (a) The experimental set up (b) Decay electron counters Q - Quadrupole Magnet M20 Experimental Area Beam l i n e 1A Figure 4 : The old M20 beam l i n e . Figure 5 : The new M20 beam l i n e . 22 opposite d i r e c t i o n are c a l l e d backward muons (although they also go forward i n the laboratory system, but with lower energy than the "forward" muons). In t h i s experiment backward muons were employed for the following reasons. 1. They have a very low e l e c t r o n contamination (8%), as compared to the other modes which can be higher than 80%. 2. When negative pions are stopped i n a degrader, they are absorbed i n n u c l e i producing neutrons, protons and gamma rays which contribute s i g n i f i c a n t l y to the background. For backward muons t h i s background i s much lower. o 3. Backward muons have a lower momentum, as compared to the forward muons and can be stopped much more e a s i l y i n targets using only a thin degrader. 2.2 S c i n t i l l a t i o n Counters In t h i s experiment nine counters were used whose dimensions are given i n table I. These counters were made of p l a s t i c s c i n t i l l a t o r s and viewed by RCA 8575 phototubes. S ^ S j ^ g were beam defining counters, S^ was a veto counter and S 5 to S g were decay electron defining counters. Sj : The Sj^ counter was larger than the lead collimator to cover the whole beam spot. S 2 • The S 2 counter was made thicker i n order to make i t possible to d i s t i n g u i s h between electrons, muons, slow muons and double muons, using the method of pulse height a n a l y s i s . Double muons, as well as slow muons which go through the S 2 counter are distinguished by Table I : Counter Geometry Symbol (Name) Size (cm) S l 7.6 (dia) x 0.16 S 2 ( t h i c k counter) 6.4 (dia) x 1.3 S ^ d e f i n i n g counter) 5.0 (dia) x 0.08 S^(veto or cup counter) 5.2 (dia) x 0.3, 2.5 (length) S ^ ( c y l i n d r i c a l counter) 15 (dia) x 0.3, 25 (length) S ^ ( l e f t e. counter) 20 x 20 x 0.6 S^(right e. counter) 20 x 20 x 0.6 Sg(top e. counter) 20 x 20 x 0.6 Sg(bottom e. counter) 20 x 20 x 0.6 24 having a higher pulse height than an ordinary muon. On the other hand electrons lose very l i t t l e energy i n p l a s t i c s c i n t i l l a t o r s and hence have a lower pulse height than ordinary muons. By se t t i n g suitable d i s c r i m i n a t i o n l e v e l s , a l l electrons, slow and double muons were rejected. A t y p i c a l pulse height f o r ordinary muons i s between 100 and 400 mV. S 3 : The S 3 counter was used as a defining counter for muons. I t was made very t h i n i n order to minimise the number of muons stopping i n the counter and causing carbon background events. : The was a cup counter, inside which the targets were housed. This counter acted as a veto for muons which did not stop i n the target. S 5 : This large c y l i n d r i c a l counter was used i n coincidence with S g- S, counters to detect decay electrons. I t was also used as an ad d i t i o n a l veto counter f o r muons which were scattered from the target but missed . Sg- S 9 : These were i d e n t i c a l square - shaped counters surrounding the S 5 counter, forming a cubic box, which i n conjunction with S 5 constituted our electron telescopes. 2.3 Magnetic Shielding Two sources of magnetic f i e l d , which could not be avoided, could r e s u l t i n precessing of muons i n the target which i n turn would a f f e c t the muon l i f e t i m e . These were the magnetic f i e l d associated with the earth and some leakage of magnetic f l u x from the beam l i n e magnets. In order to reduce the magnetic f i e l d at the p o s i t i o n of target, a mu-metal c y l i n d e r of 1 mm thickness was placed i n between 25 Sg and Sg - S g counters. One millimeter was an optimum thickness f o r the magnetic sc h i e l d i n g ; i t was thick enough to avoid the saturation problem and was thi n enough to minimise the number of low energy decay electrons stopped i n the s h i e l d i n g . This reduced the magnetic f i e l d i n the v i c i n i t y of the target from 1 Gauss to about 0.05 Gauss. The mu-metal used was Conetic AA per f e c t i o n annealed a l l o y . Moreover, since our four electron telescopes were placed symmetrically around the target, averaging the four telescopes cancels out the e f f e c t of muon spin r o t a t i o n . I t should also be noted that when i t stops i n most materials the p,+ retains i t s p o l a r i s a t i o n (50% for cloud muons, more for backward muons) whereas a u~ i n the ground state of the mesic atom retains only about 15% of i t s i n i t i a l p o l a r i s a t i o n ( l e s s i f the nucleus has a s p i n ) . 2.4 Collimator and Degrader The M20 beam l i n e formed a broad beam spot of about 10 cm i n diameter. In order to collimate the beam, a lead collimator 5 cm i n length and 5 cm i n diameter was placed before the counter and a brass collimator 5 cm i n length and 3.8 cm (2.54 emf) i n diameter between S 2 and Sg. A CH 2 degrader of variable thickness was placed between Sj and S 2 to reduce the energy of the muons and remove any pions i n the beam. The thickness of the degrader was set at 2 cm (2.8 cm) to obtain an optimum p-stop to p,-incident r a t i o . With the 3.8 cm (2.54 cm) collimator and 2 cm (2.8 cm) of degrader, the incoming negative muon t the numbers within parentheses correspond to the second run. 26 beam rate (defined by 1.2.3) was around 700/sec (1000/sec) and the stopping rate (1.2.3.4.5) was around 400/sec (700/sec) at 25 uA (30 uA) proton beam current. 2.5 Targets The target materials for t h i s experiment consisted of 8 pure metals and 48 oxides. Metal targets were s e l f supporting and the oxides were i n powder form. A d e t a i l e d information on targets i s given i n tables II and I I I . Four of the metal targets (Cu, Fe, A l & Sn) were used i n two (three for Fe) d i f f e r e n t thickneses each. A l l metal targets had a diameter of 4.6 cm. A l l the targets were contained i n t h i n bags of polyethelene. Polyethelene was chosen because i t would produce only a carbon background. Since the carbon background, caused by the containers and the counters, was to be estimated using the metal targets, care was taken to have i d e n t i c a l bags for a l l the targets, i n c l u d i n g the metals. In order to have a l l the targets i n the same p o s i t i o n r e l a t i v e to the detection system, extreme care was taken i n the design of the experimental set up. 2.6 E l e c t r o n i c s The atomic capture rate measurements were c a r r i e d out using the MSR data a q u i s i t i o n system. Figure 6 shows a schematic diagram of the e l e c t r o n i c s . Names of the e l e c t r o n i c modules and meanings of the symbols are l i s t e d i n table IV. The incident muon was defined by a (1.2.3) coincidence, while for a muon stopped i n the target a condition of (1.2.3.4.5) had to be Table II : L i s t of Targets (Run #1) z Target Mass(g) Z Target Mass(g) 13 A l ( l ) 15.0 22 T i 0 2 30 13 Al(2) 29.0 24 Cr0 3 20 26 Fe ( l ) 21.3 25 MnO 30 26 Fe(2) 42.2 25 Mn02 30 29 Cu(l) 25.1 26 F e 2 ° 3 30 29 Cu(2) 55.0 26 F e 3 ° 4 30 42 Mo 35.6 29 CuO 30 79 Au 24.4 29 Cu 20 30 11 Na 20 2 30 30 ZnO 30 12 MgO 20 38 SrO 30 13 A1 20 3 30 42 Mo02 30 14 SiO 30 42 Mo03 30 14 sio 2 30 48 CdO 30 15 P2°5 30 82 P b 3 ° 4 30 22 TiO 30 Table I I I : L i s t of Targets (Run #2) z Target Mass(g) Z Target Mass(g) 13 A l ( l ) 15.0 26 F e3<\ 49.2 13 Al(2) 29.0 27 CoO 49.5 26 F e ( l ) 21.3 28 NiO 25.3 26 Fe(2) 42.2 29 CuO 52.3 26 Fe(3) 63.6 29 Cu 20 57.4 29 Cu 25.1 30 ZnO 35.7 42 Mo 35.6 34 Se0 2 42.7 48 Cd 67.3 38 SrO 31.1 50 Sn(l) 25.0 39 Y2°3 17.3 50 Sn(2) 50.2 40 Z r 0 2 61.0 79 Au 32.7 41 NbO 52.0 82 Pb 55.2 41 Nb 20 5 27.0 42 Mo02 47.4 11 Na 20 2 34.3 42 Mo03 38.0 20 CaO 22.6 47 Ag 20 36.8 21 S c 2 0 3 26.1 48 CdO 56.9 22 T i 0 2 35.0 50 SnO 73.8 23 V2°5 25.4 50 Sn0 2 35.3 24 Cr0 3 31.2 51 sb 2o 3 27.6 25 MnO 48.7 52 Te0 2 56.0 25 Mn02 60.3 56 BaO 71.6 25 Mn 20 3 38.9 56 Ba0 2 37.9 26 F e 2 0 3 28.1 57 L a 2 0 3 42.7 Table III (continued) : L i s t of Targets (Run #2) z Target Mass(g) Z Target Mass(g) 58 Ce0 2 53.2 82 PbO 59.7 59 P r 6 ° l l 27.5 82 Pb0 2 43.7 62 Sn^Og 32.1 82 P b 3 0 - 56.4 74 WO 2 50.6 83 B i 2 0 3 81.0 80 HgO 48.4 Table IV : Meaning of the Symbols In Logic Diagram Symbol Name Model -\fl0fl0r- Delay TRIUMF B 007 D Lower Edge Discriminator LRS-821 Z DL Lower Discriminator Level LRS-821 Z DH Higher Discriminator Level LRS-821 Z C Constant F r a c t i o n Discriminator ORTEC-EGG 934 COI Coincidence Logic LRS-465 LRS-365 AL F Logic Fan-in/Fan-out LRS-429 A LF Linear Fan-in/Fan-out LRS-428 F PUG P i l e up Gate Generator EGG-GP100/NL GG Gate Generator LRS-222 s r s9 Counter names -»-o A n t i coincidence o-»- Inverted output PHR Pulse Height Rejection Inc u Incident muon (1.2.3) Stop u Stopped muon (1.2.3.4.5) Good \i Good muon 5.6-9. p. el e c t r o n event Good e Good electron event 2nd e Second electron event 31 Figure 6 : E l e c t r o n i c l o g i c . 32 s a t i s f i e d , where 4(5) means an an t i coincidence of S^Sg) s i g n a l . The stopped muon si g n a l was fed into a p i l e up gate generator G l , which i n turn generated a gate for 30 us. If two stopped muon signals enter Gl within 30 us of each other, the second s i g n a l extended the gate generated by the f i r s t s i g n a l for another 30 us, and at the same time another gate would be generated for 30 us at the P output. The stopped muon was rejected i f 1. There was a muon within 30 us p r i o r to or a f t e r the " f i r s t " muon. The r e j e c t i o n was ca r r i e d out using the p i l e up s i g n a l . 2. There was a muon s i g n a l with a pulse height higher than 350 mV, which indicated a double muon or a slow muon. 3. The MBD (Microprogrammed Branch Driver) was busy. If the stopped muon was not rejected, i t was considered to be a good muon and the si g n a l was sent to a gate generator to produce a muon gate (G2). This gate remained open for about 30 us i f there was no second muon. The good muon signal was also sent to the clock as a start s i g n a l . The detection of an electron was s i g n a l l e d by a (5,6), (5,7), (5,8) or (5,9) coincidence. I f an electron a r r i v e d inside the muon gate G2, i t was considered as an electron event. The electron s i g n a l was sent to a gate generator to produce an electron gate (G3). The electron s i g n a l was also sent to the clock as a stop s i g n a l . The electron gate remained open u n t i l the end of the muon gate. I f there was another electron event (second electron) i n s i d e the electron gate, then the event was rejected. If there was no second electron, then the electron event was considered as a good electron event. Whenever there was a good electron event, the time difference 33 between the s t a r t and stop signals from each telescope was stored i n a histogram by a PDP-11/40 computer. There were 4000, 8 ns channels i n every histogram. For the second run a second histogram was recorded for each telescope with 1000 channels of 1 ns each. Thus an event i n the f i r s t microsecond was entered into two histograms. At the end of the muon gate, i f there was a good electron event, a LAM s i g n a l was generated by a C212 unit and activated the MBD. If there was no good electron event, a "no electron" s i g n a l was generated to reset the MBD at the end of the gate. Since i t took about 4 us for the MBD to produce a busy s i g n a l , a protection gate (G4), 10 us wide, was created at the end of the muon gate i f there was good electron event, to i n h i b i t the clock and the muon gate. 2.7 Run Procedure At the beginning of the experiment, the detection system was c a l i b r a t e d by measuring the p o s i t i v e muon l i f e t i m e which was measured to be 2197.7 ± 2.6 ns ( f o r the f i r s t run and 2196.2 ± 2.7 ns for the second run) i n good agreement with the present world average 7 of 2197.138 ± 0.065 ns. Then the beam was switched to negative muons by changing the p o l a r i t y of the magnets of the M20 beam l i n e . The measurements were c a r r i e d out for 8 metal targets and 21 oxides during the f i r s t run (16th to 20th July 1982) and for 12 metal targets and 42 oxides during the second run (22nd to 29th A p r i l 1983). In the course of the experiment, the beam was switched twice to nT with the veto 4.5 removed, to obtain the channel corresponding to t=0. At the end of the run the time histograms were transferred to tape for l a t e r analysis, 34 which we s h a l l discuss i n the next chapter. 2.8 Run Record Total number of incident muons, stopped muons, electrons, good electrons, rejected muons (slow and double muons) and second electrons are l i s t e d i n tables V and VI. From tables V and VI the following observations can be made. 1. During the f i r s t run the stopping rate was around 50-60% of the incident f l u x except for thin metals where i t was around 30%. During the second run the stopping r a t i o was around 70-80%. 2. Of a l l the muons stopped i n the target, double or slow muons were less than 2%. 3. The second electrons did not cause a serious problem; le s s than 4% of the t o t a l electrons detected had to be rejected due to the presence of a second e l e c t r o n . 4. The event rate for oxides with low Z was between 150 to 200 per second but for oxides with high Z i t was less than 100 per second (because f o r heavy elements most p,- i n t e r a c t with the nucleus and produce neutrons, and therefore do not produce decay electrons) at 25 uA proton beam current. 35 Table V : Run Records (Run #1) z Target Incident u Stopped p Total e Good e PHR 2nd e 13 A l ( l ) 2322413 727065 226217 203201 17382 4184 13 Al(2) 1434938 829792 229343 205483 11052 3812 26 F e ( l ) 4399634 1736581 270093 241976 33597 4372 26 Fe(2) 3983488 2764509 276292 245210 30045 3775 29 Cu(l) 2029281 940701 119489 107156 17525 1716 29 Cu(2) 2176643 1643645 125603 111753 18236 1443 42 Mo 4504420 2449743 245698 219952 31188 3048 79 Au 2157128 665507 108898 92325 15325 1570 11 Na 20 2 3500121 2183396 970420 863841 27214 18816 12 MgO 4710647 2146182 949674 850149 34514 18349 13 A1 20 3 2680401 1720493 731868 658007 21766 11703 14 S10 2687076 1653560 648355 581076 14 S i 0 2 3442575 2228023 962241 863908 27437 15706 15 P2°5 3303829 2051336 906132 806363 24778 17599 22 TiO 5623968 2887537 723581 645620 40873 12068 22 T10 2 4873787 2937622 914326 814658 36907 15963 24 Cr0 3 4669617 2666789 837383 744921 35211 14957 25 MnO 5273043 2957815 662905 592194 37770 10229 25 Mn0 2 3267158 1750555 457274 409996 27066 6802 26 F e 2 0 3 5087172 2867021 709587 632514 36319 11753 26 F e 3 ° 4 5144732 2690756 672678 599650 39184 11250 36 Table V (continued) : Run Records (Run #1) z Target Incident u Stopped u Total e Good e PHR 2nd e 29 CuO 4884807 3520017 445440 398281 42500 6085 29 Cu20 7172670 3280075 565992 504648 51824 9025 30 ZnO 4148748 2321480 478609 429026 34244 6859 38 SrO 4102898 2226473 572614 511178 31633 9726 42 Mo02 4731431 2579856 653799 584043 32760 10687 42 Mo03 4411558 2415074 688009 613489 30862 11262 48 CdO 6574672 3174555 584091 521733 50629 8246 82 PbgO, 5091072 2170299 427188 379308 35697 7082 37 Table VI : Run Records (Run #2) z Target Incident u Stopped u Total e Good e PHR 2nd e 13 A l ( l ) 2582801 1605371 509104 442403 56920 10728 13 Al(2) 2516671 2006435 560707 485852 54930 10901 26 F e ( l ) 4483051 3125821 495454 428852 96658 8720 26 Fe(2) 4370586 3548518 410729 352400 65511 6638 26 Fe(3) 6292692 5305668 540025 463661 130289 7706 29 Cu 4014995 2970755 412563 353437 61700 7688 42 Mo 4841702 3738080 393200 336674 71506 6134 48 Cd 5834133 4822682 392641 334569 84388 5692 50 Sn(l) 4183747 2844529 392115 339122 61682 6379 50 Sn(2) 7877643 6193734 541174 461424 293010 8169 79 Au 2421566 1680135 234073 190713 30120 7133 82 Pb 5746412 3859528 405230 346682 7661 11 Na 20 2 3269535 2180267 1010899 818479 41252 46176 20 CaO 3470157 2650015 760053 647846 18083 21 S c 2 0 3 3713176 2785515 831809 684232 44514 28904 22 T i 0 2 3968745 3219478 940064 761287 46471 35513 23 V2°5 2322239 1777977 548643 449711 28191 19389 24 C r 0 3 2896629 1900541 657014 547360 35717 21924 25 MnO 2749171 2848874 742178 556726 47776 38186 25 Mn0 2 4051917 2745415 820728 664104 54050 32059 25 Mn 20 3 4045842 3150678 810121 671490 42854 24168 38 Table VI (continued) : Run Records (Run #2) z Target Incident u Stopped u Tot a l e Good e PHR 2nd e 26 F e 2 0 3 4410886 3420312 819558 680976 44374 24196 26 F e3°«. 3499356 2574661 689263 562764 44076 23172 27 CoO 4030394 3208058 746025 553686 27484 30764 28: NiO 6475430 4488475 1083060 883429 87458 38316 29 CuO 7108506 5699308 1045043 891963 131294 19448 29 Cu 20 4482892 3301746 667083 549233 54021 20098 30 ZnO 6135094 4652072 1028085 876789 119932 20701 34 Se0 2 5432586 3664782 1178969 961974 68830 43381 38 SrO 3631168 2850167 783067 674250 53891 15928 39 Y2°3 3943542 2856872 800805 659739 47378 27136 40 Z r 0 2 4322184 3048902 907062 727493 56307 36034 41 NbO 5183045 3566937 787749 664055 65997 19496 41 Nb 20 5 2467584 1871369 544117 444399 29408 19124 42 Mo02 5025801 3307615 974266 791381 59677 37127 42 MoO 3 3724249 2901956 806922 661899 43965 26703 47 Ag 20 7777706 6111691 1064579 866847 99196 32047 48 CdO 5519581 3724587 783280 668542 96619 15753 50 SnO 7316411 5420318 1046042 893949 125037 19877 50 Sn0 2 4171636 2749721 806416 660712 57278 28180 51 sb2o3 3560378 2737295 665139 546576 43598 20882 52 Te0 2 6022478 4019331 1145986 928872 74820 42440 56 BaO 6760067 5019106 1064270 902086 116667 21792 39 Table VI (continued) : Run Records (Run #2) z Target Incident u Stopped u Total e Good e PHR 2nd e 56 Ba0 2 5624475 3611407 1063925 871351 72119 37559 57 L a 2 0 3 4930418 3758116 790571 654004 54281 22335 58 Ce0 2 6656780 4380972 1102352 889121 81800 40935 59 P r 6 ° l l 5966681 4496522 1067732 872933 73795 34572 62 Sm 20 3 5816559 4253574 847781 702065 61556 24030 74 WO 2 7189905 4563554 1024250 825478 87986 37526 80 HgO 6587192 4047513 822623 670962 87872 27423 82 PbO 6472223 4715775 773673 657100 101121 14160 82 Pb0 2 6061473 3751910 985738 796011 73959 37232 82 Pb 30, 1624693 1182089 243349 197147 19393 8012 83 B i 2 0 3 8578644 6461110 1177949 956883 101341 35821 40 CHAPTER 3  Data Analysis 3.1 Data Analysis Procedure A l l data analysis was performed on the TRIUMF PDP-11/60 computor. A computor program c a l l e d MINUIT1*0 which was developed at CERN, was used f o r a chi-squared minimisation. The p o s i t i v e muon histograms were f i t t e d to a decay curve, N (t) = B + a.exp(t/T ) e g r m here Bg i s the time independent background T m i s the p o s i t i v e muon l i f e t i m e and a i s the amplitude at t=0. The negative muon histograms f o r metals and oxides were f i t t e d to a decay curve N ( t) = B + a .exp(t/T ) + a .exp(t/T ) + a .exp(t/T ) e g z e z o r o c v c where a z i s the amplitude for the metal at t=0 T 2 i s the l i f e t i m e of negative muons i n the metal a Q Is the amplitude for oxygen at t=0 T Q i s the l i f e t i m e of negative muons i n oxygen a c i s the amplitude for background carbon at t=0 and T c i s the l i f e t i m e of negative muons i n carbon. In the case of metals a Q was set to zero. In the above i t i s assumed that f o r the elements considered, the u~ has a simple exponential l i f e t i m e . This i s true for oxides, but 41 for f l u o r i n e and chlorine there i s a hyperfine e f f e c t 4 1 which complicates the disappearance r a t e . Such elements were not used i n this work. In a l l the above three cases the whole spectrum was f i r s t f i t t e d to obtain the constant background. The r a t i o of constant background to the amplitude at t=0 was around 8 x 10~ 3 f o r p o s i t i v e muons and 1.8 x 10" 3 for negative muons. These background events came mainly from random coincidences with charged p a r t i c l e s scattered around i n the M20 area. Figures 7 and 8 show t y p i c a l decay electron spectra for negative muons i n an oxide with low Z and an oxide with high Z. For negative muons, a f t e r 16 microseconds, the spectrum consists only of background events. Hence the rest of the analysis was done only for the f i r s t 16 microseconds of the spectrum. A l l the decay electron spectra for metals were f i r s t analysed. In t h i s analysis there were 5 parameters: Bg, oc2, T z, <xc and T c. It was found that i f a l l the parameters were fl o a t e d i n the chi-squared minimisation, d i f f e r e n t f i t s with very good chi-squares could be obtained for d i f f e r e n t sets of parameters. This caused a serious problem i n the a n a l y s i s . Since we were only interested i n the amplitudes at t=0, we decided to f i x the parameters T 2 and T c at the values already a v a i l a b l e i n the l i t e r a t u r e 2 2 and Bg at the e a r l i e r estimated value. Since T z and T c were not very close, there was no s i g n i f i c a n t c o r r e l a t i o n between the amplitudes az and occ. With the f i x e d parameters i t was possible to obtain a good c h i -squared minimisation f i t with r e l i a b l e values for az and a c . In the analysis of the decay electron spectra for oxides there 0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 Time in /LLS (120 ns/bin) Figure 7 : The decay electron spectrum for negative muons i n Na,0 0.0 2.5 5.0 7.5 10.0 12.5 15.0 Time in /zs (120 ns/bin) F i ? u r e 8 : T h e d e c a y electron spectrum for negative muons In PbO. 44 were seven parameters : Bg, ocz, T z, a Q , T Q, a c and T c. As was done i n the case of metals, T z, T Q and T c were f i x e d at the values obtained i n an e a r l i e r measurement i n TRIUMF 2 2, and Bg at the estimated value. This l e f t us with three parameters az, aQ and a c . Since the l i f e t i m e s of negative muons i n oxygen and carbon are close to each other (1795 ns & 2026 ns), there was a f a i r l y s i g n i f i c a n t c o r r e l a t i o n between <x0 and a c , which i n turn made f i t t i n g d i f f i c u l t . Hence the amplitude for the carbon background had to be predetermined. This was done using the information a v a i l a b l e from the carbon backgrounds i n the metals. C a l c u l a t i o n of the carbon background w i l l be discussed i n the next section. With Bg, T 2, T Q, a c and T c f i x e d , the decay electron spectra for negative muons i n oxides were analysed to obtain ocz and a Q . Figures 7 and 8 show the "best" f i t to data ( f o r Na 20 2 and PbO) obtained using the above method. I t can be seen that f o r t>15us, the f i t t e d l i n e i s s l i g h t l y above the mean of the data points. This i s believed to be due to 1. The heavier weighting from the points above the l i n e with smaller errors, and 2. The constraints brought about by f i x i n g some of the parameters. 45 3.2 The Carbon Background The background events i n the decay electron histograms are due to negative muons which stop i n : 1. The beam defining counter S 3 2. The wrappings a f t e r the S 3 counter 3. The polyethelene bag containing the target material 4. The wrappings i n front of the counter. Polyethelene and wrapping materials consist only of carbon and hydrogen. I f the u~ stops i n the hydrogen, i t i s immediately transferred to carbon. Therefore the background i s e f f e c t i v e l y pure carbon. The sources of carbon events before the target are always the same for a l l the targets. On the other hand, the sources of carbon events a f t e r the target might be d i f f e r e n t for d i f f e r e n t targets, because the number of muons which go through the target and reach the rear wall of the polyethelene bag and/or the wrappings i n front of the veto counter depends on the thickness of the target. In f a c t , as shown in f i g u r e 9(a), the t o t a l number of carbon events (normalised to the number of stopped muons) i s a smooth function of target thickness (given i n terms of equivalent thickness of carbon where we have corrected f o r the s l i g h t l y d i f f e r e n t dE/dx). In figu r e 9(a), the data points are those obtained from pure metal targets i n run #1, and the smooth l i n e i s the best f i t to the data produced by the following r e l a t i o n (obtained using a chi-squared minimisation method) 10.92 N(c) = - 1.85 x + 0.32 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 3.0 Target Thickness (C-g/cm2) Figure 9 : The dependence of carbon normalisation on the target thickness, (a) for run #1 (b) for run #2. 47 where N(c) = normalised carbon events x = equivalent target thickness. During the second run two d i f f e r e n t beam tunes were used. The second tune was used halfway through the run to accommodate another experiment on the beamline M20 B. Figure 9(b) shows the carbon background during beam tune #1, as a function of target thickness. The corresponding funtion i s 3.65 N(c) = — = + 2.80 x + 0.08 For tune #2 only two data points from metals were a v a i l a b l e to obtain the carbon background. As a chi-squared minimisation was not possible, a function N(c) = A/x 2 + B was assumed and the parameters A and B were found to be 1.72 and 4.52 res p e c t i v e l y , using the two data points. The v a l i d i t y of the above assumption i s j u s t i f i e d as following: 1. The atomic capture r a t i o s of CuO and SrO were measured twice, with the two d i f f e r e n t beam tunes. When the two d i f f e r e n t carbon backgrounds were used i n the an a l y s i s , i d e n t i c a l r e s u l t s were obtained within e r r o r s . 2. Out of the 21 oxides measured i n the f i r s t run, 15 were repeated i n the second run. Twelve of these 15 were done while running with the tune #2. Through the above assumption f i v e of the r e s u l t s obtained i n the f i r s t run could be reproduced. Two disagreed, but were i n agreement with the l i t e r a t u r e values of the capture r a t i o s measured using the mesic x-ray method. For the remaining f i v e there were no l i t e r a t u r e values. This w i l l be discussed i n d e t a i l i n chapter 4. 48 The equivalent target thickness (target thickness i n terms of equivalent thickness of carbon) for each target was calculated using the method of the reference 42. The range R x ( i n g/cm2) of a negative muon with energy x ( i n MeV) i n a target X (X = Z ( 1 ) Z ( 2 ) eg. Cu„0 : Z ( 1 ) = Cu, Z ( 2 ) = 0, n,=2, n 0=l) nj^ n 2 2 ' ' l » 2 ' i s given by M R = — ~ • X x Q2 x A 3 3 and log X - log < - > + J I a m n ( l o g I a d j ) m ( l o g x ) n n=0 m=0 J where (a) M i s the mass of the muon ( i n units of proton mass) Q i s the charge of the muon. (b) f o r an element: I a c j j = 12 Z + 7 Z < 13 =9.76 Z + 58.8 Z" 0' 1 9 i f Z > 13 fo r a compound: 1 0 8 " - ^ n < " , i (Throughout the c a l c u l a t i o n s , i n most cases, the experimental values of Iadj (taken from l i t e r a t u r e ) were used. This r e l a t i o n was only used i n case of those elements that experimental values were not a v a i l a b l e . ) Z I n i Z (c) <-> = where A. i s the atomic weight. S n A 1 i 49 ( d) «mn a r e l i s t e d i n table VII. Using the above formula and R x of the target, the corresponding % value was c a l c u l a t e d . Having T , the formula was reused to c a l c u l a t e the range R c of the muon i n carbon, i . e . the equivalent thickness of the target i n terms of grams of carbon per cm2. The target thicknesses of metals and oxides are l i s t e d i n tables VIII and IX. 50 Table VII : The Values of C o e f f i c i e n t s a m n n m 0 1 2 3 0 -8.0155 1.8371 4.5233 X 10-2 -5.9898 X 10" 3 1 3.6916 x 1 0 _ 1 -1.4520 x 10-2 -9.5873 X lO" 1* -5.2315 X 10_1» 2 -1.4307 x l O - 2 -3.0142 x 10-2 7.1303 X 10~ 3 -3.3802 X 10 _ l t 3 3.4718 x 1 0 - 3 2.3603 x 10" 3 -6.8538 X lO" 1* 3.9405 X 10-5 51 Table VIII : Target Thicknesses (Run #1) z Target actual thickness, R 2 X (g/cm ) equivalent thickness, R 2 C (carbon g/cm ) 13 A l ( l ) 0.88 0.76 13 Al(2) 1.71 1.48 26 F e ( l ) 1.25 0.95 26 Fe(2) 2.49 1.92 29 Cu(l) 1.48 1.09 29 Cu(2) 3.22 2.39 42 Mo 2.03 1.34 79 Au 1.36 0.69 11 Na 20 2 1.76 1.64 12 MgO 1.18 1.09 13 A1 20 3 1.76 1.64 14 SiO 1.71 1.59 14 S i 0 2 1.88 1.76 15 P2°5 1.76 1.64 22 TiO 2.00 1.64 22 T i 0 2 1.71 1.48 23 Cr0 3 1.88 1.64 25 MnO 1.80 1.43 25 Mn02 1.94 1.62 26 F e2°3 1.71 1.43 26 Fe 30 1 + 2.22 1.85 52 Table VIII (continued) : Target Thicknesses (Run #1) z Target actual thickness, R 2 X (g/cm ) equivalent thickness, R 2 C (carbon g/cm ) 29 CuO 2.50 1.98 29 Cu 20 2.50 1.92 30 ZnO 1.71 1.34 38 SrO 1.76 1.27 42 Mo02 2.22 1.64 42 Mo03 2.07 1.59 48 CdO 2.31 1.56 82 P b 3 ° 4 2.40 1.36 53 Table IX : Target Thicknesses (Run #2) actual equivalent z Target thickness, R thickness, R (g/cm 2) x 2 C (carbon g/cm ) 13 A l ( l ) 0.89 0.77 13 Al(2) 1.74 1.51 26 F e ( l ) 1.20 0.92 26 Fe(2) 2.37 1.82 26 Fe(3) 3.57 2.78 29 Cu 1.51 1.11 42 Mo 2.00 1.32 48 Cd 3.78 2.43 50 S n d ) 1.40 0.85 50 Sn(2) 2.82 1.76 79 Au 1.84 0.96 82 Pb 3.10 1.62 11 Na 20 2 2.08 1.92 20 CaO 1.33 1.19 21 S c 2 0 3 1.63 1.41 22 T i 0 2 2.06 1.76 23 V2°5 1.54 1.32 24 Cr0 3 1.95 1.70 25 MnO 2.86 2.31 25 Mn02 3.65 3.07 25 Mn 20 3 2.29 1.88 54 Table IX (continued) : Target Thicknesses (Run #2) Z Target actual thickness, R 2 X (g/cm ) equivalent thickness, R 2 C (carbon g/cm ) 26 F e 2 0 3 1.70 1.41 26 Fe 30 i t 2.89 2.39 27 CoO 3.00 2.43 28 NiO 1.53 1.25 29 CuO 3.08 2.43 29 Cu 20 3.59 2.78 30 ZnO 2.10 1.64 34 Se0 2 2.67 2.05 38 SrO 1.83 1.32 39 Y 2 0 3 1.05 0.77 40 Z r 0 2 3.81 2.92 41 NbO 3.25 2.35 41 Nb 20 5 1.69 1.29 42 Mo02 2.87 2.16 42 Mo03 2.38 1.85 47 Ag 20 2.23 1.44 48 CdO 3.56 2.43 50 SnO 4.61 3.12 50 Sn0 2 2.21 1.54 51 sb2o3 1.67 1.13 52 Te0 2 3.50 2.39 56 BaO 4.21 2.68 55 Table IX (continued) : Target Thicknesses (Run #2) Z Target actual thickness, R 2 X (g/cm ) equivalent thickness, R 2 C (carbon g/cm ) 56 Ba0 2 2.30 1.54 57 L a 2 0 3 2.67 1.73 58 Ce0 2 3.32 2.23 ° 59 P r6°ll 1.67 1.09 62 Sm 20 3 1.94 1.23 74 wo2 3.37 2.08 80 HgO 3.03 1.70 82 PbO 3.73 2.08 82 Pb0 2 2.91 1.70 82 P b 3 < \ 3.42 1.95 83 B i 2 0 3 4.91 2.87 56 3.3 Bound Muon Decay The decay of a bound muon has three d i f f e r e n t c h a r a c t e r i s t i c s as compared to that of a free muon'13. They are: 1. A negative muon i n the K-orbit of the muonic atom has a lower decay p r o b a b i l i t y than a free muon because the coulomb binding energy reduces the amount of energy a v a i l a b l e i n the decay. The decay rate, Rd(+) « (m^) 5 for a free muon Rd( -) " ( mu - B.E.) 5 f o r a bound muon where m^ i s the mass of the muon and B.E. i s the binding energy of the K-orbit. It has been shown1*3 that the r a t i o Q of the decay rate of bound muon to that of the free muon i s a function of Z. Rd(-) • Q(Z).R d(+) (3) A simple Z dependence of Q has been obtained by Uberall as Q(Z) = 1 - 0 . 5 (Z/137)2 (4) 2 . The motion of the bound negative muon i n the K-orbit, causes a Doppler e f f e c t i n the decay electrons. This broadens the energy spectrum of the of the decay electrons and thus causes the maximum energy of the decay electrons to st r e t c h beyond the cut o f f energy of those electrons f o r free muons. Figure 10 shows the decay electron spectra from free muons and from bound muons i n Pb, Sb, Zn, Fe and Mg. The above mentioned e f f e c t s can be seen very c l e a r l y i n the f i g u r e . 3. The nuclear coulomb f i e l d a f f e c t s the decay p r o b a b i l i t y of the negative muon. This causes the peak of the decay electron spectrum to s h i f t towards the low energy region as the atomic number of the Figure 10 : Theoretical spectra of decay electrons from negative muons In lead antimony, zinc, Iron and magnesium. 58 target nucleus Increases. This too Is shown In figu r e 10. 3.4 The Atomic Capture Ratio The r e l a t i o n between the t o t a l number of muons N^-(Z), captured by an atom with atomic number Z, and the t o t a l number of detected decay electrons N e-(Z), i s R.(-) N _(Z) = - . N (Z). E . E„(Z) R c(+) + R d(-) * R d ( ~ ) l = Q(Z).Rd( +)} i s t n e decay rate of the bound muon R c i s the nuclear capture rate Q(Z) i s the Huff factor given by equation (4) E^ i s the counter e f f i c i e n c y including the e f f e c t of the lim i t e d s o l i d angle EjCZ) i s the co r r e c t i o n for the loss of low energy decay electrons. R c + Rd(-) i s defined as t o t a l disappearance rate, R f The l i f e t i m e of the negative muon i n the atom t ( - ) , i s given by T ( - ) = 1/Rt Hence the number of muons captured by the atom with atomic number Z i s given by N (Z) N _(Z) ? (5) * T ( - ) . Q(Z). E 1 . E 2 ( Z ) . R d(+) The negative muons stopped i n an oxide Z m0n are captured either by the metal or by the oxygen, and the r a t i o of atomic capture p r o b a b i l i t i e s per atom i s defined as 59 n. N _(Z) A ( i ) - * (6) m. N _(0) u where N^_(i) i s the number of muons captured i n atom i (given by equation 5). On the other hand N e_(Z) = az. T z (7) N e-(0) - o 0 . T 0 (8) where az and a Q are the amplitudes of the metal and the oxygen component at t=0 i n the decay spectrum of the oxide. From equations (5) - (8) we get n. Q(0). a . E(0) A ( £ ) ~ (9) m. Q(Z). a Q . E(Z) 3.5 Correction E 2 for the l o s t Electrons The need for c o r r e c t i o n E 2 a r i s e s from the f a c t that some low energy decay electrons are l o s t In the target and the counters, and are not accounted f o r . I t i s shown i n figure 10 that the peak of the decay electron spectrum i s s h i f t e d towards the low energy end, i n d i c a t i n g that more low energy electrons are produced by heavier n u c l e i , hence bigger corrections are needed for higher Z's. The corrections are made by estimating the threshold energy necessary for a decay electron to reach the outermost counter i n the electron telescopes by determining the energy losses i n the target, p l a s t i c bag, mu metal and the cup and c y l i n d r i c a l counters. Some t y p i c a l values of the threshold energies are about 17 MeV for Pb.,0. and 8.5 MeV for CrO, . The threshold energy of 17 MeV for 60 PbgO^ corresponds to a 20% loss of the decay electrons from Pb and 6.3% from 0. This gives an o v e r a l l c orrection of 13.7% for the atomic capture r a t i o . For Cr0 3 1.7% of electrons from Cr and 1.1% of electrons from 0 are l o s t , giving an o v e r a l l c orrection of 0.6%. 3.6 Counter Asymmetries Considerable care was taken to ensure that the counter array was symmetrical and that the target was centered within the array. The mu metal s h i e l d reduced the magnetic f i e l d at the target l o c a t i o n but the remnant f i e l d was s u f f i c i e n t to rotate the muon spin a l i t t l e . In s p i t e of these precautions, some r e s i d u a l asymmetries were apparent. For example one histogram (bottom) normally had 20% more counts than the others. Also the l i f e t i m e s on the various telescopes were co n s i s t e n t l y high or low because of the precession of the muon spin. To ensure that the f i n a l answer was reasonable a l l four telescopes were always analysed independently and any large differences checked. To v e r i f y that an average of the telescopes was legitimate, a check was always made with a u + beam for which spin e f f e c t s are s i x times more severe. The l i f e t i m e of the u + obtained by averaging the telescopes was always i n agreement with the world average, within the er r o r s . 3.7 Lifetime Analysis The l i f e t i m e s of negative muons i n heavy metals were obtained using the 1 ns per channel histograms. The decay curve (2) of section 3.1 was f i t t e d to the histograms following the same procedure mentioned 61 i n section 3.1 except for the following: 1. The l i f e t i m e T 2 was not f i x e d but was obtained from the c h i -squared minimisation f i t . 2. The time independent background Bg and carbon background a c were obtained from the corresponding 8 ns histograms. 62 CHAPTER 4  Experimental Results 4.1 Lifetime Measurements Our r e s u l t s for l i f e t i m e s of negative muons i n n u c l e i are l i s t e d i n table X. Past measurements are also shown along with our r e s u l t s . The errors quoted are those due to s t a t i s t i c s . Most of the present r e s u l t s for heavy elements are i n very good agreement with the previous measurements c a r r i e d out at TRIUMF 2 2. For l i g h t e r elements (Z<30) the present r e s u l t s are s l i g h t l y higher. Three notable disagreements between the two sets of data are the l i f e t i m e s i n Mo, Nb and B i . But, the present r e s u l t for Mo (104.2 ± 0.9 ns), though s l i g h t l y higher, i s i n good agreement with the re s u l t s of Sens et a l . 2 3 and Eckhause et al.** 4* It should be pointed out that: 1. The "1 ns per bin" histograms of t h i s experiment, as compared to the "8 ns per bin" histograms of the previous measurement, have a better r e s o l u t i o n and i n the case of heavy elements, the l i f e t i m e s could be extracted with a much higher p r e c i s i o n . However, for l i g h t e r elements, the number of channels i n "1 ns per bin" histogram (1000) do not extend far enough i n time (lusec) to produce a very accurate r e s u l t . 2. For the "atomic capture r a t i o " measurements, which were the main objective of the present work, i t was not e s s e n t i a l to have very high s t a t i s t i c s , so our l i f e t i m e measurements s u f f e r from r e l a t i v e l y poor s t a t i s t i c s . 63 Table X : Negative Muon Lifetimes z Element Present measurement T r i u m f ( 8 0 ) 2 2 Eck(66) l t' f F i l ( 6 3 ) ^ 5 Sens^)1*6 26 Fe 206.1 + 1.9 206.0 + 1.0 206.7 + 2.4 201 + 4 27 Co 197.7 + 3.4 185.8 + 1.0 184.0 + 1.7 28 Ni 163.9 + 2.8 156.9 + 1.0 159.4 + 3.1 154 + 3 29 Cu 167.9 + 1.7 163.5 + 1.0 163.5 + 2.4 164.0 + 2.3 160 + 4 30 Zn 168.1 + 2.0 159.4 + 1.0 161.2 + 1.1 161 + 4 34 Se 164.1 + 3.8 163.5 + 1.0 163.0 + 1.2 38 Sr 140.6 + 3.1 134.1 + 2.5 130.1 + 2.3 39 Y 125.1 + 3.3 120.2 + 1.4 40 Zr 110.1 + 2.8 110.0 + 1.0 110.8 + 0.8 41 Nb 101.1 + 1.6 92.7 + 1.5 92.3 + 1.1 42 Mo 104.2 + 0.9 99.6 + 1.6 103.5 + 0.7 105 + 2 47 Ag 86.7 + 0.9 87.0 + 1.5 88.6 + 1.1 88.7 + 0.9 85 + 3 48 Cd 90.5 + 0.8 90.7 + 1.5 90.5 + 0.8 95 + 5 50 Sn 91.7 + 0.7 92.1 + 1.5 89.9 + 1.0 51 Sb 91.4 + 2.2 94.1 + 1.7 91.7 + 1.1 56 Ba 96.2 + 1.5 96.6 + 1.5 94.5 + 0.7 57 La 85.9 + 1.6 89.9 + 0.7 58 Ce 86.5 + 1.7 83.3 + 1.0 84.4 + 0.7 59 Pr 70.1 + 1.3 72.1 + 0.6 62 Sm 76.2 + 1.3 79.2 + 1.0 74 W 78.3 + 1.6 78.4 + 1.5 74.3 + 1.2 81 + 2 64 Table X (continued) : Negative Muon Lifetimes z Element Present measurement Triumf(80) 2 2 Eck(66)t»1» Fil(63) 1* 5 Sens(59) 4 6 80 Hg 74.0 ± 1.4 76.2 ± 1.5 76.2 ± 1.5 82 Pb 75.9 ± 1.2 75.4 ± 1.0 73.2 ± 1.2 74.9 ± 0.4 82 ± 5 83 Bi 65.3 ± 1.2 74.2 ± 1.0 73.3 ± 0.4 79 ± 5 65 4.2 X-ray Analysis At the end of the experiment, a series of x-ray analyses were c a r r i e d out i n order to determine the l e v e l of p u r i t y of our targets. Samples from the b o t t l e s were tested on two powder diffractometers ( P h i l i p s XRD PW 1050/65), one i n the laboratory of R. Haering at the Physics Department, the other at the Department of Geological Sciences. The samples were i r r a d i a t e d by x-rays from a Cu target (40kV/20mA) with a Ni f i l t e r . The r e s u l t i n g x-ray d i f f r a c t i o n spectra were compared with standard spectra to decide on the contents of the samples. In a l l cases the spectra could be explained using only those heavy elements claimed to be present by the manufacturers. Therefore, i n those cases where the r e s u l t s of the x-ray analysis agreed with the l a b e l l i n g on the b o t t l e , we r e l i e d on the manufacturers for the degree of p u r i t y of the target samples (nearly all>99%). For other cases we found d i f f e r e n t type of oxide or sometimes hydroxides i n the sample but always of the same metal. The r e s u l t s of these investigations indicated that 7 out of the 48 oxides were contaminated. The following are some of the findings of the a n a l y s i s . 1. Three oxides ( L a 2 0 3 , Se0 2 and BaO) had absorbed water. The b o t t l e l a b e l l e d L a 2 0 3 contained LaOOH. Se0 2 was i n f a c t Se0 2 + H 3Se0 3 and the BaO b o t t l e contained Ba(0H) 2.H 20. 2. Three oxides (SrO, CuO and W02) were i n fact a mixture of d i f f e r e n t oxides of the same element. The b o t t l e l a b e l l e d SrO contained SrO, S r 0 2 and Sr(OH) 2.2H 20. CuO was mixed with Cu 20 and the W02 b o t t l e contained W o n0 c o as w e l l . 66 3. The bo t t l e l a b e l l e d Mn 20 3 did not contain any Mn 20 3; i t was MnO instead. 4. The bo t t l e l a b e l l e d PfgO^ contained P r 0 2 . It was decided to discard a l l the atomic capture r a t i o s measured for the above mentioned oxides with the exception of Pr0 2- Of course these contaminations did not a f f e c t our l i f e t i m e measurements, as the only foreign element present i n these oxides was hydrogen; a muon captured i n hydrogen i s immediately transferred to ei t h e r a metal, or an oxygen atom. 4.3 Atomic Capture Ratio Measurements The r e s u l t s of the atomic capture r a t i o measurements f or negative muons i n oxides for the two runs are l i s t e d separately i n table XI. The errors quoted are systematic errors a r i s i n g from the s e n s i t i v i t y of the atomic capture r a t i o s to the carbon background. S t a t i s t i c a l errors are not included as they are only about 2%. For each oxide , the atomic capture r a t i o was determined independently from the spectrum of each electron telescope, and a scatte r i n the value of the atomic capture r a t i o was observed. This scatter, which i s believed to be due to the precession of the muon i n the target caused by the presence of a low i n t e n s i t y magnetic f i e l d , was very small for the oxides of l i g h t elements (within the systematic errors quoted), but r e l a t i v e l y large for the oxides of the heavier elements; B i 2 0 3 showed the highest scat t e r (~30%). The atomic capture r a t i o s given i n table XI, i n each case, are the average values of those obtained from the four telescopes. From table XI, i t i s clear that the re s u l t s of the second run Table XI : Atomic Capture Ratios from the 2 Runs z Oxide Results of run//l Results of run#2 11 Na 20 2 0.99 ± 0.04 1.00 ± 0.05 12 MgO 0.93 ± 0.05 13 A 1 2 0 3 0.94 ± 0.03 14 SiO 0.98 + 0.03 14 S i 0 2 0.90 ± 0.03 15 P2°5 0.97 ± 0.03 20 CaO 1.69 ± 0.09 21 S c 2 0 3 2.23 ± 0.25 22 TiO 2.65 ± 0.15 22 T i 0 2 2.81 ± 0.08 2.71 ± 0.14 23 V2°5 3.04 ± 0.16 24 Cr0 3 3.57 + 0.15 3.15 ± 0.13 25 MnO 3.61 ± 0.20 2.89 ± 0.14 25 Mn02 3.75 ± 0.15 3.28 ± 0.15 26 F e 2 0 3 3.64 ± 0.13 3.48 ± 0.24 26 F e 3 ° - 2.69 ± 0.11 2.36 ± 0.13 27 CoO 3.09 ± 0.15 28 NiO 2.35 ± 0.19 29 Cu 20 2.63 ± 0.19 2.20 ± 0.25 30 ZnO 3.93 ± 0.25 3.17 ± 0.16 39 Y 2 0 3 3.17 ± 0.42 40 Z r 0 2 2.89 ± 0.13 68 Table XI (continued) : Atomic Capture Ratios from the 2 Runs z Oxide Results of run#l Results of run#2 41 NbO 3.44 + 0.20 41 Nb 20 5 3.57 + 0.19 42 Mo02 3.70 ± 0.15 3.08 + 0.14 42 Mo03 4.07 ± 0.15 3.86 + 0.18 47 Ag 20 3.95 + 0.52 48 CdO 3.83 ± 0.25 2.59 + 0.21 50 SnO 2.74 + 0.13 50 Sn0 2 3.40 + 0.16 51 sb 2o 3 3.64 + 0.24 52 Te0 2 3.30 + 0.15 56 Ba0 2 3.16 + 0.15 58 Ce0 2 4.89 + 0.27 59 P r 0 2 5.00 + 0.44 62 Sm 20 3 6.24 + 0.50 80 HgO 4.88 + 0.42 82 PbO 4.30 + 0.24 82 Pb0 2 4.65 + 0.25 82 P b 3 ° 4 5.87 ± 0.33 4.19 + 0.59 83 B i 2 0 3 6.15 + 0.53 69 are s l i g h t l y lower than those of the f i r s t run. The difference between the two sets increases with the Increase i n the Z value. During the second run 13 measurements were repeated, 5 of which are i n good agreement with those of the f i r s t run. Those which disagreed, except fo r CdO and MnO are a l l s t i l l within three standard deviations. The main differ e n c e between the two runs i s In the size of the target sample. During the f i r s t run, as i t can be seen from table I I , almost a l l the target samples had the same weight. With t h i s f i x e d weight, the l i g h t e r oxides would f i l l the polyethelene bags completely whereas the heavier oxides, once i n target p o s i t i o n , had a tendency to sag inside the bag causing a considerable amount of uncertainty In the target thickness, i . e . , the measured target thickness was le s s than the actual thickness. Hence the estimated carbon background was s l i g h t l y higher, which i n turn reduced the amplitude for oxygen, r e s u l t i n g i n a higher value for the atomic capture r a t i o . During the second run t h i s deficiency was improved by f i l l i n g the bag instead of f i x i n g the mass of the oxide. This gave a much more precise value for the target thickness and hence the carbon normalisation. Due to the above considerations, the r e s u l t s of the second run are believed to be more r e l i a b l e , e s p e c i a l l y for the heavier elements. 4.4 Comparison with the Past Measurements Our f i n a l r e s u l t s for the atomic capture r a t i o s of negative muons i n oxides are l i s t e d i n table XII. Previous measurements are also given i n t h i s table. A l l the previous r e s u l t s have been obtained using the mesic x-ray method, the most comprehensive set being due to von Egidy et a l . 1 * 7 The main features of table XII are as follows. 70 Table XII : Atomic Capture Ratios of Negative Muons i n Oxides z Oxide Present measurement EgidyCSl) 1* 7 DanielC??) 1* 8 Knight(76)' t 9 Zinov(66) 2 , + 11 Na 20 2 1.00 + 0.05 0.99 ± 0.05 12 MgO 0.93 + 0.05 0.89 ± 0.05 0.83 ± 0.07 13 A1 20 3 0.94 + 0.03 0.74 ± 0.04 0.85 ± 0.06 14 SiO 0.98 + 0.03 0.96 ± 0.05 14 S i 0 2 0.90 + 0.03 0.84 ± 0.04 0.79 ± 0.07 15 P2°5 0.97 + 0.03 1.00 ± 0.05 20 CaO 1.69 + 0.09 1.71 ± 0.09 1.45 ± 0.09 1.36 ± 0.10 21 S c 2 0 3 2.23 + 0.25 2.78 ± 0.20 22 TiO 2.65 + 0.15 2.64 + 0.19 22 T i 0 2 2.71 + 0.14 2.70 ± 0.13 1.87 ± 0.19 1.90 ± 0.10 2.70 ± 0.20 23 V2°5 3.04 + 0.16 2.86 ± 0.20 2.68 ± 0.14 3.10 ± 0.20 24 Cr0 3 3.15 + 0.13 3.52 ± 0.18 25 MnO 2.89 + 0.14 25 Mn02 3.28 + 0.15 2.60 ± 0.19 26 F e 2 0 3 3.48 + 0.24 3.21 ± 0.20 2.43 ± 0.24 26 F e3 ° « t 2.36 + 0.13 27 CoO 3.09 + 0.15 28 NiO 2.35 + 0.19 29 Cu 20 2.20 + 0.25 3.8 ± 0.9 30 ZnO 3.17 + 0.16 3.06 ± 0.24 2.66 ± 0.32 39 Y2°3 3.17 + 0.42 2.19 ± 0.16 2.07 ± 0.13 1.83 ± 0.12 40 Z r 0 2 2.89 + 0.13 2.62 ± 0.19 2.38 ± 0.16 71 Table Xll(continued) : Atomic Capture Ratios of Negative Muons i n Oxides z Oxide Present measurement Egidy(81)' t 7 D a n i e l ( 7 7 ) 4 8 Knight (76) 4 + 9 Zinov(66) 2 1 + 41 NbO 3.44 + 0.20 41 Nb 20 5 3.57 + 0.19 2.95 ± 0.23 42 Mo02 3.08 + 0.14 42 Mo03 3.86 + 0.18 3.60 ± 0.29 3.48 ± 0.23 47 Ag 20 3.95 + 0.52 3.83 ± 0.32 48 CdO 2.59 + 0.21 3.14 ± 0.25 2.50 ± 0.28 2.47 ± 0.22 6.7 ± 1.5 50 SnO 2.74 + 0.13 50 Sn0 2 3.40 + 0.16 3.02 ± 0.23 3.17 ± 0.24 51 sb2o3 3.64 + 0.24 3.52 ± 0.28 3.48 ± 0.35 52 Te0 2 3.30 + 0.15 3.22 ± 0.25 56 Ba0 2 3.16 + 0.15 2.84 ± 0.36 2.23 ± 0.24 3.28 ± 0.30 58 Ce0 2 4.89 + 0.27 4.50 ± 0.55 3.7 ± 0.4 59 PrO 2 5.00 + 0.44 62 Sm 20 3 6.24 + 0.50 4.40 ± 0.74 3.09 + 0.34 80 HgO 4.88 + 0.42 82 PbO 4.30 + 0.24 4.88 ± 0.55 4.1 ± 0.4 5.8 ± 0.7 82 Pb0 2 4.65 + 0.25 5.03 ± 0.58 4.1 ± 0.4 4.17 ± 0.30 82 P b 3 0 4 4.19 + 0.59 83 B i 2 0 3 6.15 + 0.53 3.77 ± 0.43 3.1 ± 0.4 4.3 ± 0.5 72 1. Our set of data contains 10 new measurements. They are MnO, FegO^, CoO, NiO, NbO, Mo02, SnO, Pr0 2, HgO and PbgO^. 2. A comparison of our re s u l t s with those of von Egidy et a l . shows that, i n 21 out of 29 cases of common measurements, the two sets of data are i n very good agreement. 3. Our r e s u l t f o r A1 20 3, disagrees with von Egidy et a l . ' s r e s u l t , but i s i n good agreement with that of Zinov et a l . 2 i + , which i s the only other mesic x-ray measurement a v a i l a b l e . In the case of CdO, again our r e s u l t disagrees with that of von Egidy et a l . , but i t i s i n very good agreement with the r e s u l t s of both Daniel et a l . 1 * 8 and Knight et a l . 1 * 9 4. In a l l the cases of disagreement (except for B i 2 0 3 ) , the difference between ours and the previous measurements i s less than three times the combined e r r o r s . 5. Although our re s u l t s obtained from the l i f e t i m e method are i n very good agreement with mesic x-ray r e s u l t s , i t can be seen that i n most cases our r e s u l t s tend to be s l i g h t l y higher (but i t should be noted that the r e s u l t s of von Egidy et a l . are cons i s t e n t l y higher than those of Daniel et a l . ) . 4.5 Comparison with T h e o r e t i c a l Models The r e s u l t s of our measurements along with the predictions of the t h e o r e t i c a l models of F e r m i - T e l l e r 2 5 , Vasilyev et a l . 3 2 , D a n i e l 2 9 and Schneuwly et a l . 3 5 are shown i n figure 11. For Daniel's and Schneuwly's models, the t h e o r e t i c a l values were taken from references 38 and 35 r e s p e c t i v e l y . The periodic o s c i l l a t i o n s of atomic capture r a t i o s with the atomic number Z, f i r s t observed by Zinov et a l . 2 * * In 20 40 60 80 100 Atomic Number (Z) Figure 11 : Experimental atomic capture ratios In oxides (from this work) and the theoretical predictions. 74 1966, can be c l e a r l y seen i n our data. From figure 11 i t i s r e a d i l y seen that the "Z-law" by Fermi and T e l l e r i s quite inadequate i n explaining our experimental r e s u l t s . Deviation of the predictions from experimental data increases with Z. It i s only for the oxides with 21 < Z < 27 that a good agreement between t h e o r e t i c a l values and the experimental data can be seen. The modified Z-law, obtained e m p i r i c a l l y by Vasilyev et a l . 3 2 , describes the data somewhat better than the Fermi-Teller model but i t s predictions are much smaller than the measured values. Moreover, neither of these models can produce the o s c i l l a t o r y pattern of the experimental data. The models proposed by D a n i e l 2 9 and Schneuwly et a l . 3 5 turn out to be the most favoured ones. For oxides with Z<18, Schneuwly's model produces the "best" f i t while for oxides with 18 < Z < 56 Daniel's model has better p r e d i c t i o n s . For oxides with Z>57 none of the models are capable of describing the experimental capture r a t i o s . On the other hand, both of these models seem to predict the periodic o s c i l l a t i o n s f a i r l y w e l l . In reference 30, Horvath and Entezami have been able to improve on some of the e x i s t i n g models. Their "modified" Schneuwly model produced a chi-square of 13.36 per degree of freedom for 127 experimental data a v a i l a b l e for oxides. Using the t h e o r e t i c a l capture r a t i o s given i n t h i s model, our experimental r e s u l t s produce a reduced chi-square of 14.3. I t should be noted that the parameters of the "modified" model were obtained without including the r e s u l t s of the present work. Hence, the fact that, using these parameters, our data produce a c l o s e l y comparable chi-square value shows that our r e s u l t s are very much i n agreement with the previous measurements. 75 4.6 Chemical or S o l i d State E f f e c t s i n Atomic Capture In our i n v e s t i g a t i o n of atomic capture r a t i o s i n oxides we have searched for a possible chemical or s o l i d state e f f e c t i n the capture mechanism. In eight cases atomic capture r a t i o s were measured for two d i f f e r e n t oxides of the same element. In three cases (TiO and T i 0 2 , NbO and Nb 20 5, PbO, Pb0 2 and P b ^ ) per atomic capture r a t i o s were found to be the same within e r r o r s . In the remaining 5 cases (SiO and S i 0 2 > MnO and Mn02, F e 2 0 3 and FegO^, Mo02 and Mo03, SnO and Sn0 2) the capture r a t i o s seemed to depend on the type of the oxide. This e f f e c t was f i r s t observed by Zinov et a l . 2 1 * , and may be a t t r i b u t e d to the s o l i d state structure of the oxide and the type of the chemical bonds involved. A c a r e f u l look at these f i v e sets shows that except for the case of SiO and S i 0 2 , the oxide i n which the metal has the higher valency (or the oxide Zm0n with larger n/m r a t i o ) exhibits a higher atomic capture r a t i o . This i s true even In the case of the f i r s t three sets (TiO and T i 0 2 , NbO and Nb 20 5, PbO and Pb0 2) though they are the same within e r r o r s . Hence, i t can be saf e l y stated that there i s evidence to the e f f e c t that the atomic capture r a t i o i n Z m 0 n can vary a l i t t l e with the valency of the metal. 76 CHAPTER 5 Dependence of Atomic Capture Ratio on Density 5.1 Evidence f o r a Dependence on Density Almost a l l the t h e o r e t i c a l models proposed up to now define the atomic capture r a t i o as a function of the atomic number Z. The Z-dependence , as seen from experimental data, has an o s c i l l a t o r y pattern which can be reproduced, to some extent, by models proposed by D a n i e l 2 9 and Schneuwly et a l . 3 5 From figure 11 i t can be seen that, for the region of 18 < Z < 56, Daniel's model follows the experimental data very c l o s e l y . In his model, i n addition to a Z-dependence, Daniel has assumed a dependence of the atomic capture r a t i o on the atomic radius R(Z). From our experimental data we have observed that, i n addition to the Z-dependence (which i s well established), the capture r a t i o has a near l i n e a r dependence on the density of the oxide. In figures 12(a) to 12(c) the capture r a t i o i s plotted as a function of the density f o r monoxides,dioxides and sesquioxides, respectively; the d e n s i t i e s of the oxides were taken from the Handbook of Chemistry and Physics 5 0. The dotted l i n e i s only to guide the eye. From these three p l o t s , a l i n e a r dependence i s c l e a r l y evident. The small o s c i l l a t i o n s , that can mainly be seen i n the case of monoxides, can be at t r i b u t e d to the Z-dependence. In figure 12(b) (dioxides), the two points that stand out are Ce0 2 and P r 0 2 which belong to the lanthanide group. Very s p e c i a l properties of the lanthanides compared to the other elements, are * believed to be responsible for the behaviour of these two points. 77 o a: u i _ a. o O 9 10 11 12 Density ( g / c m 3 ) Figure 12 : The dependence of atomic capture ratio on density. (a) Monoxides (b) Dioxides (c) Sesquioxides. The dotted straight lines are only to guide the eye, and the deviations from them are due to the elements with Z<18 and lanthanides.(See text for details). 78 Another s l i g h t deviation from a st r a i g h t l i n e i s observed i n the oxides with Z<18. This can be explained as follows. In an oxide, both the oxygen and the metal contribute towards i t s density. For oxides with a higher atomic number, the contribution from the metal dominates over that of the oxygen, but for l i g h t e r oxides, the contribution from oxygen i s comparable to that of the metal. Further d e t a i l s on th i s e f f e c t w i l l be discussed l a t e r i n t h i s section. Following the observations from figure 12, a search was i n i t i a t e d i n order to formulate an empirical expression for atomic capture r a t i o s In oxides. For t h i s end, we used our own data and the "world" oxide data given i n reference (30), except for gases and the oxides of Be and B. BeO and B 20 3 were not included because of the very high contribution from oxygen towards t h e i r density and the gases were discarded because they do not have a c r y s t a l l i n e structure and also because t h e i r density depends on temperature and pressure. Altogether 163 experimental values were used i n the f i t . For oxides with Z>18, except for lanthanides, a l i n e a r dependence on density was assumed. A Z-dependence was also included, and a function of the form „b A = a pZ where a and b are parameters and p i s the density of the oxide, was f i t t e d to the data, using a chi-squared minimisation. For the "best" f i t the parameters were found to be: a = 0.772 ± 0.002 b = -0.123 ± 0.001 7 9 For oxides with Z<18 and for lanthanides a non-linear dependence was assumed in the form of -0 123 A = 0.772 p ( l + ap)Z * This form of the function turned out to be the "best" after many t r i a l s with different functions containing Z and p. This function was fitt e d to the two sets of data ( Z<18 and lanthanides) independently, producing the best f i t value for a as: a = -0.160 ± 0.001 for Z<18 = 0.026 + 0.003 for lanthanides. Hence, an empirical description of atomic capture ratio for negative muons in oxides was formulated as: -0.123 A = 0.772 p(l + ap)Z where a = -0.16 for Z<18 = 0.026 for lanthanides = 0 for the rest. (16) This function produces a reduced chi-square (chi-square per degree of freedom) of 13.11 for the 163 data points used. This can be compared with the work of Horvath and Entezami 3 0 where they improved on some of the existing theoretical models by f i t t i n g them to the global data. Their modified Schneuwly model produced a reduced chi-square of 13.36 for 127 data points. The capture ratios calculated using the above formula, experimental capture ratios and the corresponding chi-square values for 163 oxides are listed in table XIII. A careful study of the table shows that the highest chi-squares were produced by trioxides and pentoxides. This was expected because in trioxides and pentoxides the 80 Table XIII : Chi-squares for Oxides z Oxide Experimental Calculated Chi-Square 11 * Na 20 2 0.99 + 0.04 0.89 6.5 12 MgO 0.83 + 0.07 0.87 0.3 MgO 0.83 + 0.04 0.87 1.0 MgO 0.80 + 0.02 0.87 12.2 MgO 0.89 + 0.05 0.87 0.2 * MgO 0.93 + 0.05 0.87 1.4 13 A1 20 3 0.65 + 0.06 0.86 12.3 A 1 2 ° 3 0.85 + 0.06 0.86 0.0 A 1 2 ° 3 0.84 + 0.03 0.86 0.4 A1 20 3 0.74 + 0.04 0.86 9.0 * A1 20 3 0.94 + 0.03 0.86 7.1 14 SiO 0.96 + 0.05 0.78 12.4 * SiO 0.98 + 0.03 0.78 42.7 Si0 2 0.57 + 0.05 0.85 31.8 Si 0 2 0.79 + 0.07 0.85 0.8 Si0 2 0.86 + 0.07 0.85 0.0 Si 0 2 0.96 + 0.04 0.85 7.3 Si0 2 0.84 + 0.04 0.85 0.1 * S i 0 2 0.90 + 0.03 0.85 2.6 15 P2°5 0.93 + 0.10 0.82 1.3 P2°5 0.87 + 0.03 0.82 3.1 P2°5 1.00 + 0.05 0.82 13.4 * P2°5 0.97 + 0.03 0.82 26.0 20 £. -J CaO 1.36 + 0.10 1.76 16.2 CaO 1.45 + 0.09 1.76 12.0 CaO 1.71 + 0.09 1.76 0.3 * CaO 1.69 + 0.09 1.76 0.6 21 Sc 20 3 2.78 + 0.20 2.02 14.5 * Sc 0 3 TiO 2.23 + 0.25 2.02 0.7 22 2.64 + 0.19 2.60 0.0 * TiO 2.65 + 0.15 2.60 0.1 TiO 2 2.70 + 0.20 2.25 5.1 TiO 2 1.90 + 0.10 2.25 12.1 TiO 2 1.87 + 0.19 2.25 4.0 T i 0 2 2.17 + 0.11 2.25 0.5 TiO 2 2.70 + 0.13 2.25 12.1 * T i 0 2 2.71 + 0.14 2.25 10.9 23 V2°3 2.19 + 0.18 2.56 4.1 V2°«» 2.28 + 0.23 2.28 0.0 V2°4 2.70 + 0.19 2.28 4.9 V2°5 3.10 + 0.20 1.76 44.6 V2°5 2.68 + 0.14 1.76 42.8 V2°5 2.86 + 0.20 1.76 30.0 * V2°5 3.04 + 0.16 1.76 63.6 24 Cr0 3 2.96 + 0.20 1.41 60.1 Cr0 3 3.52 + 0.18 1.41 137.4 Cr0 3 3.23 + 0.22 1.41 68.4 * Cr0 3 3.15 + 0.13 1.41 179.1 Table XIII(continued) : Chi-squares for Oxides z Oxide Experimental Calculated Chi-Square 24 C r 2 ° 3 3.00 + 0.17 2.72 2.7 C r 2 ° 3 2.04 + 0.11 2.72 38.3 C r 2 ° 3 2.63 + 0.13 2.72 0.5 C r 2 ° 3 3.45 + 0.25 2.72 8.5 C r 2 ° 3 2.65 + 0.20 2.72 0.1 25 * MnO 2.89 + 0.14 2.83 0.2 Mn02 3.00 + 0.17 2.61 5.2 MnO j 2.60 + 0.19 2.61 0.0 * MnO 2 3.28 + 0.15 2.61 19.7 26 F e 2 ° 3 2.43 + 0.24 2.71 1.4 F e 2 ° 3 3.21 + 0.20 2.71 6.2 * F e 2 ° 3 3.48 + 0.24 2.71 10.3 ¥e3°k 2.36 + 0.13 2.68 6.0 27 * O * T CoO 3.09 + 0.15 3.32 2.4 C o 2 ° 3 3.04 + 0.29 2.67 1.7 Co .,0, 3.70 + 0.38 2.67 7.4 28 * NiO 2.35 + 0.19 3.42 31.6 29 CuO 6.14 + 0.85 3.27 11.4 CuO 3.60 + 0.40 3.27 0.7 CuO 4.06 + 0.23 3.27 11.9 CuO 3.26 + 0.23 3.27 0.0 Cu20 3.80 + 0.90 3.06 0.7 * Cu20 2.20 + 0.25 3.06 11.9 30 ZnO 2.22 + 0.06 2.85 110.2 ZnO 2.66 + 0.32 2.85 0.4 ZnO 2.39 + 0.10 2.85 21.2 ZnO 3.06 + 0.24 2.85 0.8 * ZnO 3.17 + 0.16 2.85 4.0 31 Ga 20 3 2.77 + 0.20 2.97 1.0 32 GeO 2 2.90 + 0.21 3.14 1.4 33 As 2 0 3 3.39 + 0.25 1.88 36.6 34 Se0 2 2.72 + 0.20 1.98 13.8 38 SrO 2.12 + 0.11 2.32 3.3 39 Y2°3 1.83 + 0.12 2.46 28.0 Y2°3 2.07 + 0.13 2.46 9.2 Y2°3 2.19 + 0.16 2.46 3.0 * Y2°3 3.17 + 0.42 2.46 2.8 40 Zr0 2 2.38 + 0.16, 2.75 5.2 Zr0 2 2.62 + 0.19 2.75 0.4 * Zr0 2 2.89 + 0.13 2.75 1.2 41 NbO 3.44 + 0.20 3.57 0.4 N b 2 ° 5 2.95 + 0.23 2.19 11.0 * N b 2 ° 5 3.57 + 0.19 2.19 53.1 42 * Mo02 3.08 + 0.14 3.15 0.3 Mo03 3.48 + 0.23 2.29 27.0 Mo03 3.60 + 0.29 2.29 20.5 * Mo03 3.86 + 0.18 2.29 76.5 46 PdO 3.57 + 0.44 4.19 2.0 Table XHI(continued) : Chi-squares for Oxides z Oxide Experimental Calculated Chi-Square 47 Ag20 3.83 + 0.32 3.43 1.5 * Ag20 3.95 + 0.42 3.43 1.5 48 CdO 6.70 + 1.50 3.33 5.0 CdO 2.47 + 0.22 3.33 15.4 CdO 2.50 + 0.28 3.33 8.8 CdO 3.14 + 0.25 3.33 0.6 * CdO 2.59 + 0.21 3.33 12.5 49 I n 2 ° 3 2.94 + 0.28 3.43 3.1 I n 2 ° 3 2.92 + 0.31 3.43 2.7 50 * SnO 2.74 + 0.13 3.08 6.8 Sn02 3.17 + 0.24 3.32 0.4 SnO 2 3.02 + 0.23 3.32 1.7 * SnO, 3.40 + 0.16 3.32 0.3 51 Sb2<53 2.79 + 0.14 2.70 0.4 S b 2 ° 3 3.48 + 0.35 2.70 5.0 S b 2 ° 3 3.52 + 0.28 2.70 8.6 * S b 2 ° 3 3.64 + 0.24 2.70 15.4 Sb 0Oc 1.73 + 0.09 1.81 0.8 52 Te0 2 3.22 + 0.25 2.70 4.4 * Te0 2 3.30 + 0.15 2.70 16.2 56 BaO 2.27 + 0.22 2.69 3.7 BaO 1.45 + 0.18 2.69 47.5 * BaO 2 3.16 + 0.15 2.33 30.3 57 L a 2 ° 3 2.21 + 0.25 3.57 29.8 L a 2 ° 3 2.73 + 0.33 3.57 6.5 58 Ce0 2 3.70 + 0.40 3.96 0.4 Ce0 2 4.50 + 0.55 3.96 1.0 * Ce0 2 4.89 + 0.27 3.96 11.9 59 * Pr0 2 5.00 + 0.44 3.75 8.1 60 N d 2 ° 3 5.13 + 0.35 4.01 10.2 62 Smo0, 3.09 + 0.34 4.72 23.0 Sm203 4.40 + 0.74 4.72 0.2 * S m 2 ° 3 6.24 + 0.50 4.72 9.2 63 E u 2 ° 3 3.60 + 0.40 4.11 1.6 E u 2 ° 3 4.34 + 0.46 4.11 0.3 64 Gd 20 3 5.52 + 0.33 4.09 18.8 66 Dy 20 3 4.70 + 0.50 4.33 0.5 Dy 20 3 5.79 + 0.61 4.33 5.7 70 Y b 2 ° 3 3.18 + 0.34 5.20 35.3 Y b 2 ° 3 6.85 + 0.42 5.20 15.4 71 L u 2 ° 3 4.40 + 0.50 5.36 3.7 L u 2 ° 3 5.31 + 0.58 5.36 0.0 73 T a 2 ° 5 6.00 + 0.60 3.73 14.3 T a 2 ° 5 7.20 + 0.73 3.73 22.5 74 W 0 3 4.70 + 0.50 3.26 8.4 wo3 5.75 + 0.67 3.26 13.9 80 * HgO 4.88 + 0.42 5.00 0.1 81 T1 20 3 4.00 + 0.50 4.34 0.5 83 Table XIII(continued) : Chi-squares for Oxides z Oxide Experimental Calculated Chi-Square 81 T1 20 3 4.81 + 0.57 4.34 0.7 82 PbO 4.56 + 0.53 4.28 0.3 PbO 5.80 + 0.70 4.28 4.7 PbO 4.10 + 0.40 4.28 0.2 PbO 4.88 + 0.55 4.28 1.2 * PbO 4.30 + 0.24 4.28 0.0 * PbO j 4.65 + 0.25 4.21 3.1 * *h3°k 4.19 + 0.59 4.09 0.0 83 B i2°3 4.30 + 0.50 3.83 0.9 B 12°3 3.10 + 0.40 3.83 3.4 B i2°3 3.77 + 0.43 3.83 0.0 * B 12°3 6.15 + 0.53 3.83 19.1 90 Th0 2 2.90 + 0.40 4.38 13.6 Th0 2 3.57 + 0.50 4.38 2.6 92 u o 2 3.60 + 0.40 4.85 9.8 u o 2 4.65 + 0.55 4.85 0.1 uo 3 6.00 + 0.50 3.23 30.8 * denotes results of the present measurement. 84 contribution of oxygen towards their density is considerably higher compared to the other oxides and the linear relationship may not be a valid assumption. Removing the very few trioxides and pentoxides from the data set decreases the reduced chi-square value from 13.11 to a very low 8.52. In figure 13, the calculated capture ratios produced by our empirical formula (shown as dots) are plotted alongside our experimental data. It is interesting to see that the calculated values follow the experimental data very closely and the characteristic oscillations are very well reproduced. Note that the calculated values of capture ratios for trioxides and pentoxides are not included in this plot. 5.2 Atomic Capture Ratios for Chlorides The global atomic capture ratio data for chlorides were also tested for a possible dependence on density. These data were taken from reference (30) with the exception of the gases and the chlorides of L i and Be. The same procedure for oxides was adopted for a chi-squared minimisation f i t and the corresponding parameters were given as: a = 0.859 ± 0.008 b = -0.124 ± 0.003 and for Z<18: a = -0.221 + 0.003 There were no lanthanides among the chloride data. The empirical formula for atomic capture ratio of negative muons in chlorides, then, took the form of: A = 0.859 p(l + ap)Z~°* 1 2 4 a = -0.22 for Z<18 = 0 for the rest. 8 Figure 13 : Experimental and theoretical atomic capture ratios for negative muons in oxides. The experimental values are those of the present measurement and the theoretical values (dots) are calculated from formula 16. co 86 This function produced a reduced chi-square of 6.58 for the 61 data points used. For the same data points, the modified version of the Schneuwly et a l . model produced a reduced chi-square of 8.58 (see ref 30). The capture ratios calculated using the above formula, experimental capture ratios and the corresponding chi-squares for 61 chlorides are listed in table XIV. In this table the AgCl data point (0.80 ± 0.20) stands out as a major contributor to the reduced chi-square; i t has a chi-square of 116.9. Comparing i t with the other experimental data in that region, this point can be safely discarded as a mistake. With this data point removed, the reduced chi-square drops to the remarkable low value of A.61. It should also be noted that the next highest contributor to the total chi-square is WClg. 5.3 Discussion on the Empirical Formulae The empirical formulae for both oxides and chlorides show explicit and identical dependence on Z, i.e., A = a p(l + ap)Z in both cases, with different values of a and a for oxides and chlorides. In the case of oxides, we noticed that the parameter a of the formula shows a direct relationship to the Z-value of oxygen, i.e., for oxygen we have: z-0.124 b ( g ) - l / 8 B 0 7 n but on the other hand, the parameter a for oxides is given by a = 0.772 ± 0.002 With this in mind, the empirical formula for oxides could be simplified as: Table XIV : Chi-squares for Chlorides z Chloride Experimental Calculated Chi-Square 11 NaCl 1.05 ± 0.08 0.72 17.0 NaCl 0.68 ± 0.04 0.72 1.0 NaCl 0.79 ± 0.03 0.72 5.4 NaCl 0.68 ± 0.06 0.72 0.4 NaCl 0.71 ± 0.07 0.72 0.0 NaCl 0.69 ± 0.02 0.72 2.2 NaCl 0.77 ± 0.13 0.72 0.1 NaCl 0.79 ± 0.03 0.72 5.4 12 MgCl 2 0.76 ± 0.04 0.71 1.4 13 A1C13 0.63 ± 0.21 0.70 0.1 A1C1 3 0.66 ± 0.02 0.70 4.4 AICI3 0.47 ± 0.25 0.70 0.9 19 KC1 1.16 ± 0.03 1.18 0.6 KC1 1.16 ± 0.11 1.18 0.0 KC1 1.15 ± 0.05 1.18 0.4 KC1 1.19 ± 0.12 1.18 0.0 KC1 1.14 ± 0.02 1.18 4.6 KC1 1.14 ± 0.15 1.18 0.1 KC1 1.14 ± 0.06 1.18 0.5 20 CaCl 2 1.56 ± 0.17 1.27 2.8 CaCl 2 1.27 ± 0.03 1.27 0.0 CaCl 2 1.37 ± 0.24 1.27 0.2 CaCl 2 1.41 ± 0.07 1.27 3.8 23 v c i 3 1.96 ± 0.36 1.75 0.4 V c l 3 2.03 ± 0 . 1 1 1.75 6.6 24 CrCl 3 1.91 ± 0.35 1.60 0.8 C r C l 3 2.15 ± 0.10 1.60 30.4 25 MnCl 2 2.29 ± 0 . 3 2 1.72 3.2 MnCl 2 2.48 ± 0.14 1.72 29.8 26 FeCl 3 2.01 ± 0.35 1.66 1.0 27 CoCl 2 2.09 ± 0.29 1.92 0.4 28 N i C l 2 2.34 ± 0.31 2.02 1.1 30 ZnCl 2 2.17 ± 0.30 1.64 3.1 37 RbCl 1.78 ± 0.11 1.54 4.9 RbCl 1.75 ± 0.18 1.54 1.4 RbCl 1.53 ± 0.19 1.54 0.0 40 ZrCl^ 2.06 ± 0.43 1.52 1.6 41 NbCl 5 3.25 ± 0.58 1.49 9.2 42 MoCl 5 3.22 ± 0.58 1.58 8.0 46 PdCl 2 2.75 ± 0.36 2.14 2.9 47 AgCl 0.80 ± 0.20 2.96 117.0 AgCl 2.50 ± 0.25 2.96 3.4 48 CdCl 2 2.26 ± 0.26 2.34 0.1 CdCl 2 2.22 ± 0.30 2.34 0.2 50 SnCl 2 1.98 ± 0.22 2.09 0.2 SnCl 2 2.04 ± 0.31 2.09 0.0 SnCl^ 2.36 ± 0.40 1.18 8.7 51 SbCl 3 2.55 ± 0.39 1.66 5.2 Table XIV(continued) : Chi-squares for Chlorides z Chloride Experimental Calculated Chi-Square 51 SbCl 5 2.30 + 0.45 1.23 5.6 52 TeCl^ 2.07 + 0.50 1.72 0.5 55 CsCl 1.75 + 0.09 2.08 13.8 CsCl 2.07 + 0.21 2.08 0.0 CsCl 2.21 + 0.23 2.08 0.3 56 BaCl 2 2.67 + 0.33 2.04 3.6 73 TaCl 5 3.68 + 0.62 1.86 8.6 74 wci 6 7.20 + 0.93 1.77 34.1 80 HgCl 2 3.91 + 0.44 2.71 7.4 81 T1C1 3.69 + 0.37 3.49 0.3 82 PbCl 2 3.16 + 0.24 2.91 1.1 PbCl 2 4.23 + 0.45 2.91 8.6 83 B i C l 3 2.55 + 0.41 2.36 0.2 89 A(|) = 0.772 p(l + « p)Z~ 1 / 8 ~178 p ( 1 + a p )^Z^ = 0.6 P ( l + a p ) ( | ) 1 / 8 where 0 = 8 is the Z-value for oxygen. It was very interesting to find that in the case of chlorides, a similar approach produces an identical result for the empirical formula, i.e., Aff^) = 0.6 p(l + a p ) f f 1 ) 1 / 8 where CI = 17 is the Z-value for chlorine. Hence, the empirical formula for the atomic capture ratio can be generalised as: A(|l) = 0.6 p ( f-2) 1 / 8 for Z>18 = 0.6 p(l + a p ) ^ ) 1 / 8 for Z<18 and lanthanides z l where for Z<18 a = -0.16 for oxides = -0.22 for chlorides and for lanthanides a = 0.026 (oxides) This general formula was also tested against 30 data points for fluorides with Z>18, taken from reference (30). It produced a chi-square (per data point) of 11.45 as compared to 13.9 obtained for the same data set in the above reference. Removing trifluorides, tetrafluorides and pentafluorides reduces the chi-square to 7.08. Unfortunately, at the present time, due to the lack of sufficient experimental data in the region of Z<18 and for the 90 lanthanides, a general expression for parameter a cannot be found. The empirical formula may slightly deviate for the chemical compounds with higher valencies ( i . e . for oxides, chlorides,... in which oxygen, chlorine ... make a significant contribution towards the density). But again, due to the lack of experimental data for such compounds this cannot be established at this time. 91 CHAPTER 6  Summary Negative muons from the muon channel (M20) at TRIUMF were stopped in 41 oxides in order to measure the atomic capture ratios using the lifetime technique. The experimental method was to detect the decay electrons and to use the unique lifetime signature to identify the element which captured the muon. This was the f i r s t time that such a large number of atomic capture ratios were measured using this method. The data set is second only to the measurements of von Egidy et a l . 1 * 7 who used the mesic x-ray method, by which they measured atomic capture ratios for 57 oxides. This large number of measurements by one group has the advantage of eliminating the uncertainties due to systematic errors associated with the measurements of different experimental groups. Though the two methods have their own systematic errors, a comparison of our results with those of von Egidy et a l . shows that, out of 29 common measurements, in 21 cases the two sets are in good agreement. For the x-ray measurements a serious problem is the energy dependence of the detector's efficiency as the energy of the Lyman x-rays increases rapidly with Z. For the lifetime measurement the major correction is that for the muon absorption which reduces the number of decay electrons; in the experiment i t s e l f there is a serious d i f f i c u l t y with the carbon background, and with discrepancies between different counters. Considering the very different problems associated with each method, the reasonable agreement that we observe i s quite satisfying. 92 We have been able to considerably improve on the results of the previous measurements at TRIUMF51. In order to attain this, the most important improvements made in the experiment method have been: 1. The definition of t=0 more precisely. 2. The estimation of the carbon background in a more reliable way. 3. The positioning of each target in exactly the same place relative to the experimental set up. In our measurements of atomic capture ratios we have included 10 new target materials, v i z : - MnO, F e ^ , CoO, NiO, NbO, Mo02, SnO, Pr0 2, HgO and Pb 30 l t. We have observed evidence for the existence of some chemical or solid state effects in the atomic capture process. In 8 cases, atomic capture ratios were measured for two different oxides of the same element (TiO and Ti0 2 , SiO and Si0 2, MnO and Mn02, Fe 20 3 and F e ^ NbO and Nb 20 5, Mo02 and Mo03, SnO and Sn02, PbO and Pb0 2). In almost every case, the atomic capture ratio was higher for the oxide with a larger number of oxygen atoms per metal atom. We have also observed that the atomic capture ratio has a very simple linear relationship with the density. For oxides, chlorides and fluorides this relationship takes the form A(fl) = 0.6 p ( § 2 ) 1 / 8 for Z >18 z 2 z 1 It is a worthwhile project to study theoretically how the density can be related to the chemical bonds and the solid state structure of the chemical compounds and to investigate how this can affect the atomic capture process. 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