CHARGE EXCHANGE OF STOPPED ir IN DEUTERIUM by RANDY NEIL MACDONALD B.Sc., University of Alberta, Edmonton, Alberta, Canada M.Sc, McMaster University, Hamilton, Ontario, Canada A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN THE FACULTY OF GRADUATE STUDIES i n the Department of Physics We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August, 1977 C?) Randy N. MacDonald, 1977 In p r e s e n t i n g t h i s thesis an advanced degree at the L i b r a r y s h a l l I f u r t h e r agree for fulfilment of the requirements f o r the U n i v e r s i t y of B r i t i s h Columbia, make i t freely available that permission for I agree reference and f o r e x t e n s i v e copying o f this that study. thesis s c h o l a r l y purposes may be granted by the Head of my Department or by h i s of in p a r t i a l this representatives. thesis It is understood that copying or p u b l i c a t i o n f o r f i n a n c i a l gain shall written permission. Department of Phvslcs The U n i v e r s i t y o f B r i t i s h 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date August 3,1977 Columbia not be allowed without my i ABSTRACT By using a pair of large Nal spectrometers f i g u r a t i o n we have observed i n a coincidence con- the charge exchange of stopped ir i n deuterium ir +d-*-2n+TT°. We have measured the branching r a t i o of t h i s r e a c t i o n R _ CJ(IT d -> 7T°nn) to Crr~d -> a l l ) and find R = (1.45 ± 0.19) x 10 . -It This measurement i s the f i r s t observation of pion charge exchange at r e s t i n deuterium and represents an increase i n s e n s i t i v i t y of a factor of 40 over previous measurements. The measured value of R agrees well with the recent t h e o r e t i c a l r e s u l t of Beder(1.39 x 10 - R - 1.59 x 10 ) . H ii TABLE OF CONTENTS ABSTRACT Page i TABLE OF CONTENTS i i LIST OF TABLES iv LIST OF FIGURES v ACKNOWLEDGEMENT v i i CHAPTER I INTRODUCTION 1 CHAPTER I I THE EXPERIMENT II. 1 Kinematic Considerations II-.2 The Spectrometers 15 11.3 The Experimental Arrangement.: 18 11.4 E l e c t r o n i c s and Data C o l l e c t i o n 21 11.5 Summary 24 CHAPTER I I I 7 THE DATA ANALYSIS 111.1 The O f f - l i n e Analysis 25 111.2 F i t t i n g the Histograms 37 III.2.A The Coincidence Spectrum 39 III.2.B The Singles Spectrum 42 111.3 The Coincidence E f f i c i e n c y 47 111.4 Corrections for In-Flight Interactions 56 III.4.A Charge Exchange 58 III.4.B Radiative Capture 62 111.5 The F i n a l Analysis 63 CHAPTER IV DISCUSSION IV. 1 The Charge Exchange Branching Ratio 67 iii Page LIST OF REFERENCES APPENDIX I 74 NEUTRAL PION KINEMATICS AI.l Gamma Ray Doppler S h i f t . AI.2 The y-spectrum Resulting AI.3 The Second-Gamma-Ray Cone Angle from Isotropic I T Decay 0 77 79 84 APPENDIX I I THE ELEMENTARY COINCIDENCE EFFICIENCY 86 APPENDIX I I I THE DEUTERIUM CHARGE EXCHANGE PHASE SPACE 90 APPENDIX IV 95 THE REDUCTION OF THE EFFECTIVE HYDROGEN CONTAMINATION. iv LIST OF TABLES Table 1.1 Page Previous work with stopped pions i n deuterium III.5.1 Summary of Data (used i n c a l c u l a t i n g branching r a t i o s ) . . . 3 64 V LIST OF FIGURES Figure Page o I I . 1.1 " decay i n the laboratory 8 I I . 1.2 The y-ray spectrum of stopped pions i n hydrogen 10 II.1.3 Theoretical y-ray 10 I I . 1.4 The second y-ray cone 11 I I . 1.5 Coincidence lineshapes f o r pion charge exchange 13 11.2.1 Time Response of the Nal spectrometer 16 11.2.2 High energy response of the Nal spectrometer 16 11.3.1 The experimental layout 19 11.3.2 The beam c o n s t i t u t i o n 18 I I . 4.1 Schematic of E l e c t r o n i c s 22 spectrum of stopped pions i n deuterium.... I I I . 1.1 T y p i c a l Nal time spectra 26 111.1.2 Radiative capture y-xays from deuterium 2S 111.1.3 Centroid and zero point i n s t a b i l i t i e s i n TINA and MINA 29 111.1.4 Centroid and zero point corrections In TINA and MINA 34 I I I . 1.5 E f f e c t of Gain and Zero S t a b i l i z a t i o n 35 I I I . 1.6 Gamma ray coincidence spectra 36 111.2.1 F i t of the hydrogen peaks 39 111.2.2 Estimated number of hydrogen y-ray coincidences as a function of the range of f i t 111.2.3 111.2.4 F i t t i n g the y-ray singles spectrum 41 43 Estimated Number of ir° singles y-rays as a function of the range of f i t 44 111.3.1 I T Decay on the collimator a x i s 48 111.3.2 TT° Decay o f f the Collimator Axis 50 0 vi Figure Page III.3.3 Coincidence E f f i c i e n c y as a function of displacement 52 along the collimator axis III.3.A Coincidence e f f i c i e n c y as a function of angular d i s p l a c e ment from the collimator a x i s 53 111.4.1 Liquid Deuterium Target 56 111.4.2 ir~ momentum byte and In f l i g h t charge exchange 111.4.3 Calculated Lineshape for 18.9 MeV ir~ IV.1.1 Relations between low energy pion reactions AI.1.1 The I T decay i n the lab frame AI.2.1 Center of mass - lab transformations AI.3.1 ir° decay i n the lab frame 60 68 0 AII.l The second y-ray All,2 Intersection of two cones AIII.l 60 cone Deuterium Charge Exchange F i n a l State 7 • 7 7 9 8 ^ 8 7 8 7 90 vii ACKNOWLEDGEMENTS Like most experiments i n t h i s f i e l d t h i s one was performed by a group of experimenters and I am g r a t e f u l to each of them f o r the part he played. Included i n t h i s group were D.Berghofer, Dr. M.D. Hasinoff, Dr. D.F. Measday, Dr. M. Salomon, Dr. J . Spuller, T. Suzuki, Dr. J.M. Poutissou, Dr. R. Poutissou, Dr. P. Depommier, and Dr. J.K.P. Lee. I appreciate the e f f o r t s made by my supervisor, Dr. M.D. Hasinoff, and Dr. D.F. Measday who have read t h i s work i n i t s d r a f t form and made -many useful suggestions. Dr. D.S. Beder. I am g r a t e f u l for many useful discussions with In p a r t i c u l a r the work presented i n Chapter .Appendix I I has been derived from h i s notes. IV and I am also g r a t e f u l to Dr. A.W. Thomas f o r h i s assistance with the c a l c u l a t i o n s presented i n Appendix I I I . I thank D. Garner f o r h i s invaluable help with the ySR groups computing system. part i n typing the manuscript. Mr. J . Stanton. F i n a l l y I thank my wife H i l a r y f o r her I dedicate this work to my parents and 1 CHAPTER I INTRODUCTION When negative pions i n t e r a c t at r e s t i n deuterium three processes are e n e r g e t i c a l l y allowed. 1. IT + d -»• 2n 2. ir + d - » - 2 n + Y 3. ir~ + d absorption 2n • + ir° r a d i a t i v e capture charge exchange. Of these only the f i r s t two have been previously observed (PA51, CH54, KU59, RY63, KL64). Although searches had been made f o r the charge exchange reaction i n deuterium (PA51, CH55) i t had not been observed. In f a c t , the charge exchange reaction f o r pions at r e s t had only been observed i n hydrogen (PA51, C061) and helium-3 (ZA65, TR74). It i s common to define the r a t i o s h)(ir d -» 2n) w(ir""d -> 2ny) and K = a) (IT d -> 2mr°) U)0r~d ->• 2ny) R = e>Qr~d ->• 2nTr°) u (Tr~d ->• a l l ) or 2 where co i s the r a t e f o r the s p e c i f i e d r e a c t i o n . t a l r e s u l t s a r e summarized i n t a b l e 1.1. t h o s e a u t h o r s who have used negative The The p r e v i o u s experimen- It i s interesting that l i q u i d o r gas t a r g e t s have a l l measured a (although c o n s i s t e n t w i t h zero) branching failure to note to o b s e r v e the charge ratio. exchange r e a c t i o n i n d e u t e r i u m i s a r e s u l t o f t h e i d e n t i c a l p a r i t y o f the TT~*and ir° (PA51, CH55, TA51, BR51). I n hydrogen the a b s o r p t i o n p r o c e s s (ir p -*• n) i s f o r b i d d e n by momentum-energy c o n s e r v a t i o n and t h e a l l o w e d r e a c t i o n s a r e : 4. ir + p n + y radiative 5. IT + p - * - n + i r ° charge capture, exchange. In helium-3 t h e c o r r e s p o n d i n g p r o c e s s e s a r e : and 6. ir + H e •> H + y radiative 7. i r " + H e -*• H + ir° charge 3 3 3 3 capture, exchange, the breakup r e a c t i o n s : 8. + 3 He He 9. TT~ + 3 10. v~ + 3 11. 1T~ + 3 In a l l c a s e s the u n + d P + n H e -*• Y + n Y + P He ->- ->- capture proceeds > n > + d > + n + n . p r i m a r i l y from an s s t a t e (LE62). In the case o f hydrogen and helium-3 the n u c l e a r s p i n and p a r i t y In the case o f deuterium a total initial i tis 1 . + I s V*"* F o r p s e u d o s c a l a r p i o n s we then have s p i n and p a r i t y h~ i n t h e case o f hydrogen and helium-3. EXPERIMENTER VALUE OF S VALUE OF R VALUE OF P a n o f s k y , Aaaodt 2.36 -0.007 * 0.020 - 0.023 t 0.072 Compare Y-ray y i e l d s from * d and IT p r e a c t i o n s ; g a s t a r g e t ; p a i r spectrometer. -8.3 x l O " * 1 10.5 x 10"* -0.0034 t 0.0043 S: Compare Y " y s i n g l e and n e u t r o n c o i n c i d e n c e y i e l d s from * d r e a c t i o n s K: Compare Y-ray s i n g l e s and Y-ray c o i n c i d e n c e y i e l d s from it d r e a c t i o n s ; l i q u i d target; l i q u i d s c i n t i l l a t o r f o r neutrons; lead c o n v e r t e r - p l a s t i c s c i n t i l l a t o r telescope for Y y s . and H a d l e y i 0.5 K METHOD OF OBSERVATION (PA51) C h l n o v s k y onij Steinberger 1.5 t 0.8 (CH34, r a - r a (CP.55) Kuehner, M e r r i s o n 2.39 i 0.36 Compare Y " * y y i e l d s from n d and IT p r e a c t i o n s ; l i q u i d spectrometer. target; pair Ryan (RV63) 3.16 1 0.10 Compare y-ray y i e l d s from ir~d and Ti~p r e a c t i o n s ; l i q u i d spectrometer. target; pair K L o e p p e l (KL64) 2.89 : 0.09 and Tornabene a (KU39) -5.7 x 10"" -2.3 x ± 6.7 x 10"* P e t r u k l n and Prokoshkin <10~ (PE64) 3 10" ± 2.7 x «4 x lO" * 3 3 10" Y-ray ( i n t e r n a l c o n v e r s i o n ) y i e l d s from it d r e a c t i o n s i n a b u b b l e c h a a b e r . 3 C o i n c i d e n c e Y - r a y y i e l d s o f L1D1 lead glass detectors. A W i t h no s p i n - f l i p the neutral pion w i l l be emitted i n an s state. the case of deuterium however the i n i t i a l state i s 1 . In In the f i n a l state the two neutrons, being i d e n t i c a l fermions, must have an a n t i symmetric wave function. The only p o s s i b i l i t i e s allowed are a s i n g l e t s wave (*s) and a t r i p l e t p wave ( p ) . 3 In the case of a s di-neutron 1 wave function the n e u t r a l pion must be emitted i n a p state to conserve angular momentum. This r e s u l t s i n a f i n a l state with p o s i t i v e p a r i t y and hence i t i s forbidden. In the case of a p di-neutron wave function 3 the neutral pion must also be emitted i n a p state r e l a t i v e to the rdi-neutron i n order to conserve p a r i t y . s l i g h t l y exothermic (Q = 1.10 MeV) Since the reaction i s only the reaction rate i s greatly retarded. Of course, from an h i s t o r i c a l perspective the above argument has been reversed and i n f a c t i t was the non-observation of a s i g n i f i c a n t charge -exchange rate i n deuterium which led co the conclusion that the TT and ir° must both be pseudoscalar p a r t i c l e s (CH54, CH55). For scalar mesons the absorption reaction (1) i s forbidden for a ir i n an s-wave atomic state but could occur for capture from a p-wave -atomic state. Brueckner et a l (BR51) have used a detailed balance a r - gument for the reaction p + p •+ d + i r for scalar pions absorbed + to determine the r a t i o 'S' from an atomic p-state and found a value which i s a factor of 30 too small to account for the experimental value determined by Panofsky et a l (PA51). On the other hand the calculated 'S' r a t i o for pseudoscalar pions, which can be captured from an s-wave atomic s t a t e , was found to be compatible with the experimental value. Tamor (TA51) has calculated the rates for the three deuterium reactions (1,2,3) using a l l reasonable combinations of meson spins, p a r i t i e s and coupling theories. In the c a l c u l a t i o n mesons of both 5 p o s i t i v e and negative p a r i t y were considered. Negative mesons of spin 0 and 1 were considered; however the neutral meson was known to decay into 2 Y ~ y s and hence only spin zero neutral mesons had to be conr a sidered. In the case of scalar ir the absorption reaction i s prohibited from the s-state and Tamor puts an upper l i m i t of 4% on the number of mesons absorbed from atomic o r b i t a l s with higher angular momentum i n e s s e n t i a l agreement with Brueckner et a l . If the TT~ was a vector meson Tamor predicts a r a t i o S=55, again i n d i s t i n c t c o n f l i c t with Panofsky's S = 2.36 ± 0.5. The calculated r a t i o s for both pseudoscalar and pseudo- vector mesons were compatible with the experimental value of S. a pseudovector of i r 0 data, IT For he f i n d s a n e g l i g i b l e value of 'K' for either p a r i t y and so nothing can be inferred from Panofsky's data. For a pseudoscaiar TT the calculated value of 'K' i s n e g l i g i b l e i n the case of pseudoscalar ir° but K s 0.1 for scalar ir°. Thus the experimental data of Panofsky, and l a t e r data of Chinowsky et a l (CH55) indicate i f the u that i s a pseudoscalar p a r t i c l e then the ir° must also be a pseudo- scalar p a r t i c l e . We have now observed the charge exchange reaction i n deuterium and have measured the r a t i o s K and R to be K = (5.71 ± 0.82) x R = (1.45 + 0.19) x 10 _lt 10~ h This i s i n agreement with a recent t h e o r e t i c a l r e s u l t of Beder (MA77) 6 1.39 x 10~ k < R < 1.59 x 10" * . 1 The present work is a detailed description of the experimental work and the data analysis which derives these ratios. The experiment i s described i n Chapter II and the details of the data analysis are reported in Chapter III. Chapter IV includes a brief discussion of the theoret i c a l values as well as the relationships of these values with other low energy pion data. The details of some calculations have been placed i n appendices to avoid interrupting the flow of ideas. Appendix IV i s a discussion of possible explanations for the anomolously low hydrogen contamination which was observed in the liquid deuterium target. Future experimental studies of the effects described i n this appendix have been planned (me77). 7 CHAPTER II THE EXPERIMENT 1. Kinematic Considerations The experimental method employs a coincidence technique to observe the 2y decay of the neutral pion. The coincidence technique has two p r i n c i p l e advantages over a singles technique. y-ray singles background F i r s t , the dominant from the r a d i a t i v e capture process which over- whelms any small charge exchange component i s eliminated and second, the coincidence geometry greatly favors the detection of the deuteron charge exchange TT° (0.0 MeV < T ( d ) n 1.1 MeV) over the hydrogen charge TT° exchange TT°(T (p) = 2.9 MeV). To understand these advantages i t i s ii0 u s e f u l to f i r s t consider the y r a y spectra which one would observe from - the i n t e r a c t i o n of stopped ir i n hydrogen. As i n the case of deuterium the y ~ d i a t i o n a r i s e s from the r a d i a t i v e capture of the pions and also r a from the decay of the neutral pions formed i n the charge exchange process. The r a d i a t i v e capture y-vay i s mono-energetic a t 129.4 MeV. The neutral pion from the charge exchange reaction i s likewise mono-energetic T TT( P ) = 2.90 MeV. n with Since the reactions take place at r e s t there i s no preferred d i r e c t i o n and the d i s t r i b u t i o n s of these Y—rays and pions w i l l be i s o t r o p i c . If one now observes the y-decay of the TT° i n the lab frame, one sees a Y~ y with an energy which has been Doppler s h i f t e d by ra an amount which i s determined by the ir° momentum In the lab frame. In Appendix 1.1 i t i s shown that p * Y 1 m^^ ° 2 E„ - P ^ o s O (II.1.1) where 0 i s the angle between the observed y-ray and the pion momentum (figure II.1.1). Thus we see a continuous d i s t r i b u t i o n of y-rays between the energy l i m i t s and —CD Detector Figure II.1.1 Tr°decay i n the. laboratory r e s t frame determined by 0 = 0 and 0 = rr; i . e . * - 1 %H % 2 2 E ± P ^ ^TT or E In the case of hydrogen H,L . = 54.9 MeV ± 2 and E (II.1.2) R = 83.0 MeV. Further, as i s shown i n Appendix 1.2, f o r an i s o t r o p i c a l l y d i s t r i b u t e d mono-chromatic TT° the observed y-ray spectrum i s the box spectrum dlly dE Y 0 for E y < E K for E L < - E < - Eg 0 for E L > Eg . These features can be seen i n the measured y-ray spectrum from stopped ir i n hydrogen (figure II. 1.2). The deviations from a perfect rectangle and d e l t a function are consistent with the f i n i t e r e s o l u t i o n function of the spectrometer. In the case of stopped ir i n deuterium the s i t u a t i o n i s complicated by the 3-body f i n a l state (ir°nn and ynn). The Y-ray from the r a d i a t i v e capture channel i s no longer mono-energetic but i s spread over a range of energies from 0 MeV to 131.5 MeV. Although the Y-ray spectrum i s more sharply peaked at high energies (due to the a t t r a c t i v e low energy s-wave neutron-neutron interaction) than a simple phase-space c a l c u l a t i o n would predict there i s s t i l l s u f f i c i e n t contribution i n the medium energy range ('WO MeV) to o b l i t e r a t e the d e t a i l s of the Y-ray spectrum from the TT° decay (figure II.1.3). Furthermore, the singles y-ray decay spectrum of the deuterium TT° w i l l be superimposed on the box spectrum created by the charge exchange i n the hydrogen contamination (nominally 0.3%) of the deuterium target. Thus the separation of the small deuterium TT° Y~" y spectrum from the r a d i a t i v e capture background ra and hydrogen contamination would be a formidable problem. With hind- sight i t i s possible to say that of the y-ray spectrum measured i n the 55 MeV to 83 MeV region only 3% i s from charge exchange i n deuterium. Fortunately both of these backgrounds can be eliminated with a coincidence arrangement. That the r a d i a t i v e capture Y-rays are e l i m i - nated with a coincidence technique i s obvious. That the deuterium 10 Figure II.1.2 The Gamma Ray Spectrum of Stopped Pions i n Hydrogen 3000U • • •• TT 4p->n+rr ' Uy+Y .0 Tr~+p-+n+Y > 2000H (0 cd OS 1000- 20 —r —r~ T 80 60 Gamma Ray Energy 40 I— 100 (MeV) — I — 120 140 Figure II.1.3 Theoretical Gamma Ray Spectrum of Stopped Pions i n Deuterium rr +d->-2n+Y .a cd ,o o i-i Cu u Cd rH CO Pi Gamma Ray Energy (MeV) 11 ir° decays may be separated from the hydrogen TT° decays i s a fortunate, consequence of the d i f f e r e n c e i n Q-values f o r the two reactions and the good energy r e s o l u t i o n of the Nal spectrometers. If one knows the energy of the ir° before i t decays and d e f i n e s both the d i r e c t i o n and energy of one of the decay Y r a y - of the TT° momentum and the d i r e c t i o n of the second then the d i r e c t i o n s Y~ y ra up to an angle about the d i r e c t i o n of the f i r s t Y~ yra a r e determined (figure II.1.4) The angle 0 between the TT° and the f i r s t y ~ y i s determined by (II.1.1) r a and using the conservation laws i t can be shown (Appendix 1.3) that the angle ij) between the two y ~ y d i r e c t i o n s i s given by the equation: r a Y TT Y The second Y r a y must l i e on the surface of a cone of half-angle (j) - generated by a r o t a t i o n about the a x i s of the f i r s t y-vay. Figure II.1.4 The second y~ray cone I t Is c l e a r 12 that f o r detectors which subtend small s o l i d angles there w i l l only be a coincidence i f <t) = 0 or cos(J> = 1. From (II. 1.3) we f i n d that t h i s requirement means ± 7> g N v 2 We r e c a l l (from (II.1.2)) that these are the maximum and minimum l i m i t s of the rectangular Y-ray spectrum. That i s they correspond to the cases o where the 11 i s t r a v e l l i n g i n the forward or backward d i r e c t i o n s along the axis of the detectors. Hence, with small detectors and a small source one would expect to see only y-rays at energies 54.9 MeV 83.0 MeV from the hydrogen decays ( ^ ( p ) Since the deuterium T T ° ' S = 2.90 and MeV). have a continuous spectrum due to the 3 body f i n a l state we would also expect the coincident Y r ^ y spectrum to be continuous with maximum l i m i t 59.4 MeV and 76.7 MeV (T o(d) = 1.1 IT max _ MeV). Thus the Y-rays from the hydrogen I T decays w i l l be separated i n energy 0 from the Y-rays which r e s u l t from the deuterium ir° decays. Of course i n any experiment i t i s necessary to hav.e sources and detectors of f i n i t e size and the actual spectra obtained i n each case must be determined by an integration over the detector surfaces and the target volume. This has the e f f e c t of allowing the hydrogen coincidence peaks to encroach somewhat on the deuterium coincidence spectrum. Detailed c a l c u l a t i o n s o f the coincident lineshapes and e f f i c i e n c i e s have been performed for hydrogen and deuterium and are presented i n chapter I I I of t h i s work. The lineshapes shown i n figure II.1.5 are the r e s u l t s for the geometry used i n the present experiment and demonstrate the energy separation of the deuterium and hydrogen events. It i s s i g n i f i c a n t to note that the Relative P r o b a b i l i t y of Detection ET 14 net coincidence detection e f f i c i e n c y f o r deuterium TT° decays i s 6.7 times larger than the corresponding e f f i c i e n c y for hydrogen TT° decays. 15 2. The Spectrometers The experiment was made f e a s i b l e by the a v a i l a b i l i t y of two large sodium iodide spectrometers, TINA and MINA. TINA i s a c y l i n d r i c a l block of Nal 45.7 cm. i n diameter and 50.8 cm. i n length. I t i s viewed by seven matched 12 cm. photomultiplier tubes (RCA 4522). MINA i s a c y l - inder 35 cm. i n diameter and 35 cm. long and i s also viewed by four matched 12 cm. photomultiplier tubes (RCA 4522). The sodium iodide c r y s t a l s are e s s e n t i a l l y 100% e f f i c i e n t and have s u f f i c i e n t resolving power to separate the coincident gamma rays o r i g i n a t i n g from deuterium from those o r i g i n a t i n g from hydrogen. Furthermore, the timing r e s o l u t i o n of 2.0 nsec (FWHM) obtained with constant f r a c t i o n discriminators makes i t possible to separate the y-rays from the large neutron background produced by the absorbtion and r a d i a t i v e capture channels. (The f l i g h t path of about 1 m. gives 6 nsec time separation between the Y-rays and the high energy neutrons). Figure II.2.1 shows the time spectrum of TINA and MINA and the separation of the Y-rays and neutrons. Figure II.2.2 shows the energy response of TINA and MINA to beams of high energy electrons. On the basis of an inverse square root r e l a t i o n - ship with energy one would anticipate an energy r e s o l u t i o n of about 6.4% i n TINA and 7.9% i n MINA i n the 55 MeV - 83 MeV range. In fact the r e s o l u t i o n obtained i n t h i s range i n the present experiment was about 10% i n TINA and considerably larger than 10% i n MINA. The increase i n width i n TINA may be attributed to the use of a large collimator (25 cm. diameter) and the problems of gain and zero i n s t a b i l i t i e s encountered over a comparatively long period of time. The large width i n MINA was found to be a r e s u l t of i n s u f f i c i e n t voltages on the primary dynode stages of the photomultiplier tubes. The resolution i n TINA was s t i l l TINA Time Spectrum (.n~d -»• anything) TINA Energy Spectrum 144 MeV Electrons 400 •• • • 8k s •• 300 • • • • 4k _ 1.8 nsec.i FWHM [ • * " " • . • 100 > a • • CO 05 c lb CO 20 MINA Time Spectrum 4-1 c CJ 3b (^"d + anything) | < • • • • • o »|/ 200 * — • 4.4% FWHM 0 «..••— ' 110 CO •U § 200 • 120 130 140 150 160 _ MINA Energy Spectrum . 125 MeV Electrons • • • • • • • 2k 2.2 nsec. . *i FWHM )| |< • # . ^ 100 . • <• • • • io ji 5.3% FWHM •A" 26 3b Time (nsec.) Figure II.2.1 Time Response of the Nal Spectrometers* I 110 I 120 130 i 140 l l 150 Energy (MeV) . Figure II.2.2 ! 160 High Energy Response of the Nal Spectrometers. 17 s u f f i c i e n t to separate the y-rays from the deuterium and hydrogen charge exchange. The r e s o l u t i o n i n MINA was not s u f f i c i e n t i n t h i s respect. However MINA was used to provide a test of the t o t a l energy, ^TINA + ^MINA = ^ + coincidence events anc * t o ^ o r ma t w o dimensional histogram of the (N, ( E J J ^ , ^ ) ). 18 3. The Experimental Arrangement Figure II.3.1 shows the c o n f i g u r a t i o n of our apparatus i n the TRIUMF M-9 area. protons The pion beam was produced by a 1 uA beam of 500 MeV which s t r i k e s a 10 cm. beryllium target. The M-9 beam l i n e was tuned f o r 51 MeV negative pions (momentum - 130 MeV/c). The beam contamination under these conditions was measured by time of f l i g h t and the beam was found to c o n s i s t of 76% pions, 18% electrons and 6% muons (Figure II.3.2). The incident beam i s defined by the p l a s t i c l a b e l l e d C l and C2 and degraded 90,000 - i n a 2.9 cm. sheet of aluminum. 10 cm. Beryllium Target 130 MeV/C •• • • 60,000- • - TT"(76%) • 0 CO e~ (18%) • 4J *H P B 1 • QJ • & 30,000• • • scintillators p " (6%) . 4 • • • 1 10 •• • • • i 20 Time of F l i g h t Figure II.3.2 1 (nsec.) ' ' 30 The beam c o n s t i t u t i o n The target f l a s k was a c y l i n d e r 15.5 ± 0.5 cm. long and 11.1 cm. i n diameter g i v i n g a volume of 1.7 l i t e r s . The walls were 0.036 cm. mylar and the entrance window was 0.024 cm. mylar. The f l a s k was wrapped i n 10 l a y e r s of 6.4 x 10 Vm. aluminized mylar to reduce the heat load. The deuterium was made by e l e c t r o l y z i n g deuterium oxide which had a •C3 shielding (lead or steel) Liquid Deuterium H-C2 vo Aluminum Degrader "Cl TT"( 130MeV/c) — Co Plastic Scintillators CO 15 cm. x 15 cm. x 0.16 cm. Cl 10 cm. x 10 cm. x 0.16 cm. C2 6.35 cm'. •<) x 0.16 cm. C3 20 cm. x 20 cm. x 0.32 cm. C4 30.5 cm. x 30.5 cm. x 0.32 on. C5 25.4 cm. x 50.8 cm. x 0.32 cm. C5* 25.4 cm. x 50.8 cm. x 0.32 cm. Beam pipe Window e Experimental Layout 20 0.2% (by atom) H 0 comtamination. The i o n i c content was enhanced by making a solution of solium peroxide with the deuterium oxide. r e s u l t i n g gas was analysed by means of a mass spectrometer. contamination was found to be 0.3% The The hydrogen to 0.4% although, due to c a l i b r a t i o n d i f f i c u l t i e s i n the low mass region, the r e l i a b i l i t y of t h i s measurement i s questionable. Thus the gas was taken to have a 0.3% hydrogen conta- minat ion. Beam p a r t i c l e s which d i d not stop i n the target were vetoed by the p l a s t i c s c i n t i l l a t o r C3. With t h i s arrangement we stopped 25% to 50% of the incident pion beam i n the target l i q u i d . The s c i n t i l l a t o r s i n the pion beam were hidden from the Nal detectors with lead shielding to reduce the contribution of charge exchange i n hydrogen to a minimum. The p l a s t i c s c i n t i l l a t o r s C4, C5, and C5' served as charged particle tags to d i s t i n q u i s h electrons from the y-rays entering the spectrometers. A stopped pion was defined by the coincidence condition C1'C2-C3. An event was defined by the coincidence of a stopped pion and a signal from either TINA (T) or MINA (M). C1'C2'C3'(T + M). That i s an event was defined as The pion stop r a t e was t y p i c a l l y 2 x 10 the event rate was about 60 sec 1 . k sec" 1 and 21 4. E l e c t r o n i c s and Data C o l l e c t i o n Figure II.4.1 i s the schematic of the e l e c t r o n i c s . The outputs of the TINA and MINA phototube bases were brought separately into the counting room on low l o s s cables (FM-8) to allow the phototube gains to be balanced. The signals were then summed i n a c t i v e fan-in modules (LRS 127B, LRS 428) inside the counting room. The summed spectrometer signals were s p l i t i n isolated output fanout modules (LRS 128) and processed for pulse height and timing information. A l l of the data was d i g i t i z e d i n CAMAC modules interfaced through a standard CAMAC crate to a PDP-11-40 computer. The pulse height information was processed i n two channels of the LRS o c t a l ADC (model 2248). The timing information was processed i n two channels of an LRS o c t a l TDC (model 2228). Since we are interested i n examining only "events" i t was necessary to s t a r t the TDC with the "event" s i g n a l . Because the timing q u a l i t y of this signal was poor a t h i r d TDC channel was used to record the time of passage of the pion through C2 as a zero time reference. are t„,... - t„„ and t„,.„, - t„„. TINA C2 MINA C2 T Hence the relevant times The timing of a l l three stop signals was done with constant f r a c t i o n discriminators (Ortec 463). The charged p a r t i c l e tags C4 and C5 + C5' were fed into CAMAC scalers and a charged p a r t i c l e was i d e n t i f i e d by a non-zero count i n the appropriate s c a l e r . In addition CAMAC scalers were used to record the number of ir stops and the number of events observed. Once an event had been accepted by the CAMAC - PDP-11-40 computer system no further events could be accepted u n t i l that event had been processed by the software. Although the time required to accomplish t h i s varied considerably depending upon the type of event t h i s time did not —3 exceed 10 —1 sec. Thus the dead time for an event rate of 60 sec was l e s s than 6%. The software processing involved two stages. F i r s t the 22 Beam Counters C3 c , Disc. , Disc, LRS-621 (coa frac) ORTEC 463 N a l Analogue Signal I CAMAC ADC LRS C o m p 2248 Gate LRS-465 Logic Unit LRS 3 6 5 ci CO Disc LRS-621 Disc. LRS-621 2 ± Logic Unit LRS-365 OR LRS429 Visual iScaler OR EVENT LRS429 CAMAC Scaler Logic Unit LRS365 Con. Con. Frac. Frac. Disc Disc. ORTEC ORTEC 463 463 Fast Amp. LRS 133 B clip 2 5 0 ns Fanln LRS 127 B TINA Stop stop I stop Fast Amp LRS 133B Disc. Cron. 151 Fan Out LRS128 clip 2 5 0 ns1 " Fan In Disc. LRS LRS 428 621 MINA A n o d e Signals' Comp. CAMAC I Fan Out LRS 128 EliotSR 1608 Disc. LRS 621 Fan In LRS 127 B Comp CAMAC Scaler EliotSR 1608 Comp. COMPUTER C Y C L E (PDP11/4Q) 1) L A M issued by start signal of TDC 2) INHIBIT all C A M A C modules 3) READ C4 TDC LRS 2228 and RESET 4) Process C h a r g e Identifiers 5) ENABLE data system 23 event was analysed and binned into time and energy histograms and second, the event was recorded on magnetic tape f o r a more complete o f f - l i n e analysis. In the on-line analysis seven one-dimensional grams were formed. 256 channel h i s t o - These were the TINA/MINA time spectra, the TINA/MINA y-ray energy spectra, the time d i f f e r e n c e spectrum the TINA/MINA coincident y-ray energy spectra. ( T ^ j ^ ~ MINA^ T a n ( * In addition one two dimensional 64 x 64 channel histogram was formed. The axes were the TINA coincidence y-ray energy and the MINA coincidence y-ray energy. The data recording was done on Dectape and l a t e r the data were transferred to IBM compatible 9 track magnetic tape. The Dectape has the disadvantage of being a rather i n e f f i c i e n t medium f o r storage of a large amount of data and because we had only a f i n i t e number of these small tapes the data recording had to be done s e l e c t i v e l y . of data recording were used. Two modes In the " s i n g l e s " recording mode events were selected on the basis of t h e i r time of f l i g h t ; data f o r neutrons not being recorded. In general the recording threshold was set well into the neutron time of f l i g h t peak to ensure that a l l y-rays were recorded. In one run a l l neutron events i n TINA were recorded. In the "coincidence" recording mode only coincident events were recorded but, with the except i o n of one run, neutron coincidences as well as gamma ray coincidences were recorded. i 24 5. Summary The data were taken over a period of about 68 hours beginning on May 28, 1976. The data were recorded in 11.runs of singles recording mode and 6 runs of coincidence recording mode. Except for one case in which two short coincidence recording runs were taken consecutively the coincidence recording runs were alternated with singles recording runs. This was done to minimize the effect of gain and zero shifts in the pulse height data. were observed. A total of 2.2 x 10^ TT stops and 6,153,962 events Of these half were observed during singles recording runs and half during coincidence recording runs. 25 CHAPTER I I I THE DATA ANALYSIS 1. The O f f - l i n e Analysis Because the data were recorded on magnetic tape i t was possible to do a more refined data analysis than that provided by the on-line computer. In p a r t i c u l a r i t was possible to apply gain and zero channel s t a b i l i z a t i o n to the data during the re-analysis and thereby s i g n i f i c a n t l y improve the energy r e s o l u t i o n . I t was also possible to generate various histograms which were not available i n the on-line analysis and to adjust the d e f i n i t i o n of y ~ y r a spectra. s a n Q neutrons on the basis of the time The time spectra of TINA and MINA for a single run are shown i n f i g u r e III.1.1 with the time d e f i n i t i o n of the y-rays marked. As can be seen there was good separation between the neutrons and y-rays i n TINA and the overlap may be considered to be n e g l i g i b l e . the separation was not quite as good. In MINA This was p a r t i a l l y due to the smaller distance from the target and p a r t i a l l y due to an increased peak width. TINA was 122.9 cm. from the center of the target and there was a time separation of neutrons. 7.2 nsec. between the y-rays and the high energy MINA was 103.2 cm. from the target center giving a time separation of 6.1 nsec. These times are indicated on figure III.1.1 and are seen to be consistent with the leading edge of the neutron peak. The increased width of the MINA y-ray peak can be understood i n terms of the low gain on the primary dynode stages of the photomultiplier tubes. This same problem also resulted i n poor energy r e s o l u t i o n i n the MINA spectrometer. Even with the reduced separation i n MINA the overlap of neutrons and y-rays was n e g l i g i b l e . 26 TINA .2K| o o> Time S p e c t r u m |^— 7.2 ( run 12 ) n s e c . — • in c neutrons \ in | m k-1.8 e > • nsec. Ul a'-rays , 0.0 ».....»«»» 5.0 10.0 time of f l i g h t MINA Time S p e c t r u m N 4K 6 1 o 15.0 t ( nsec. ) ( run 1 ) n s e c—^| neutrons—> • in c £c 2K —$| k—2.3 nsec. QJ > UJ • • 0.0 y-rays * •••*,» 5.0 r 10.0 time of f l i g h t Figure 1 III.1.1 15.0 . t ( nsec. ) Typical N a l Time S p e c t r a 27 The energy spectra of the singles Y-rays i n TINA and MINA are shown i n figure III.1.2. In the TINA spectrum there i s an obvious shoulder at 55 MeV due to the charge exchange box spectrum from the small hydrogen content of the target. As mentioned e a r l i e r deuterium charge exchange Y-rays comprise only 3% of the events i n t h i s region. The high energy edge of the box spectrum i s hidden i n the r a p i d l y r i s i n g r a d i a t i v e capture spectrum. In the MINA spectrum the energy r e s o l u t i o n was not s u f f i c i e n t to see any i n d i c a t i o n of the charge exchange box spectrum. It has been mentioned previously that there were problems with gain and zero s t a b i l i t y over the duration of the experiment. The extent of the problem i s seen by examining the zero point energies and r a d i a t i v e capture centroids for i n d i v i d u a l runs. These values are plotted as a function of run number i n f i g u r e III.1.3. points serves only to guide the eye. The curve drawn on these One would expect that the width of a peak would be increased by at l e a s t an amount comparable to the zero s h i f t . In TINA t h i s was about 4 MeV and i n MINA 7 MeV. This was a serious problem i n view of the small energy separation between the hydrogen and deuterium charge exchange lineshapes (about 5 MeV). One of the beauties o f an experiment which records raw data from events on magnetic tape i s that i t i s possible to compensate for this type of problem i n a dynamic way. In order to do this the energy para- meters were re-defined as: = ^T^^T ~ ^T^ * E v e n t s / MeV o c NJ O Q C- Ev e n t s / M eV CT> O ^_ O O m <D g 73 -i <Q ^: fD PJ CL DJ < O PJ o o "a c to E v e n t s /MeV E v e n t s / MeV ro i in —» —1» O 3 a m •3 CH O fD -i ID <D ^< C »-* fD —i C 3 £3 to o 83 ^ 2 <5 ,0-0_©' 0 3d ^ o ft f J I L. ^ 12 o R u n No \ , Run No. " ' o - 0 16 • • 12 C/3 o<f6 N / O 1 5' -1 O P-, o u CU M 34" -2 MINA Zero Point Instability TINA Zero Point Instability A o 4 > 3d A 12 , 16 ^0-0-- - o Run No. TINA Centroid Instability V J—1—•—t w 3 o M 4-1 C i t 12 1 i Run No. -2 o _4 Figure III.1.3 .1, o o MINA Centroid Instability Centroid and zero point instabilities in TINA and MINA 16 30 where F^, and are the energy data f o r TINA and MINA recorded f o r a p a r t i c u l a r event. and Z^ are zero point adjustments associated with TINA and MINA r e s p e c t i v e l y and and G^ are adjustable gain f a c t o r s . Dynamic gain s t a b i l i z a t i o n was achieved by continuously modifying the gain and zero parameters i n such a way as to correct f o r the gain and zero s h i f t s i n the o r i g i n a l data. I t was necessary to define the cen- t r o i d energies (C°) and zero point energies (Z°) which were to be maintained. The choice of these values i s e n t i r e l y a r b i t r a r y . once they have been determined However the gain (G) and zero (Z) parameters should be chosen so that the i n i t i a l centroid and zero point energy of the data w i l l correspond to the chosen fixed values (C° and Z°). The f i x e d zero energies were taken to be = 0.0 (channel number) , = 0.0 (channel number) , The i n i t i a l zero parameters were then chosen as the average zero point energy of run #1; Z,j, = 19.03 (channel number) , Z^j = (channel number) , 7.62 The fixed centroid energies were taken to be the average centroid p o s i t i o n for run / / l corrected f o r the i n i t i a l zero parameters (Z^ and Z^); 31 132.27 (channel number) , ** 173.54 (channel number) . a With these values f o r the fixed centroid energies the appropriate choice for the i n i t i a l values of the centroid parameters (C^ and C^) were C T = C T * °M = M* C The gain parameters were defined to be G T 5 C = . ; T °M = C 1 M Hence the i n i t i a l value of the gain parameters was 1.0. A y~ y ra e v e n t w a s recognized as a high energy y - r a y i f E" was within one half width a t half-maximum of the fixed centroid energy. For such events the centroid parameter was adjusted by an amount AC where AC = k ( E ' - C) , c k c being a small (<< 1) constant, and the gain was re-defined with the new value of C. 32 Except In the case of a coincidence event zero energy events were a v a i l a b l e for one spectrometer when the event was the r e s u l t of a response i n the opposite spectrometer. Because there was no "stop" s i g n a l for such events they could be recognized by an overflow i n the corresponding time data. For zero energy events the zero parameter was adjusted by an amount AZ, AZ = k ( E - Z ) , z where i s a small ( <<1) constant. Of course the adjustments were applied a f t e r the event i t s e l f had been analysed to avoid any autobiasing e f f e c t s . and 10 4 Suitable values f o r k respectively. and k z were found to be 10 The v a r i a t i o n of the parameters was monitored as the s t a b i l i z e d analysis was done. III.1.4. c The r e s u l t s are plotted i n figure The i n d i c a t i o n from these graphs i s that for both TINA and MINA most of the observed f l u c t u a t i o n was due to the zero s h i f t . effectiveness of the s t a b i l i z a t i o n may The be measured by comparing the centroid positions f o r i n d i v i d u a l runs with and without the s t a b i l i z a t i o n . These centroid positions have been plotted i n figure III.1.5. the improvement i s seen to be a factor of 3. more than a factor of 2. In TINA In MINA the improvement i s The r e s i d u a l differences i n centroid p o s i t i o n among the runs were corrected by stretching a l l of the spectra to give them a common centroid. The same s t a b i l i z a t i o n and stretching procedure was applied i n the case of the coincidence spectra. The usefulness of t h i s procedure i s c l e a r l y seen i n the improvement of the TINA coincidence spectrum (figure III.1.6). The MINA coincidence spectrum i s also shown 33 here and i t i s seen that the separation between the hydrogen and deuterium charge exchange events i s s t i l l very poor. Centroid Correction — l• •*1 (Channels) o • Zero P o i n t C o r r e c t i o n to o CO ro (Channels) to to to >— to to H- 00 C n n lo N fD l-l n n> O 3 T) rt >i O O H» 3 rt o CD 3 rt H O P 3 CU N O O O H H o H H co ft) COh ID O It H» O O O 3 3 I A H M M ro o •xl o 15 i-i c? H» 3 3 S3 O O O H H fl> O rt HO 3 ca Centroid Correction (Channels) Zero P o i n t C o r r e c t i o n I—• i H- O I—' 3 • • > n n > 3 • 1 o ro CU • ••• •• • • * - • • i-l * — o I-* 3 MIN Run > 53 to > * * • - • • • • • • • •• O *-* • r- 1 to * * — — -• *—• —• 11 • n o • li H m o rt H» O • o 3 • • > •• • pa c 3 t • tr-• — i ••d **** • H- i— O o * ro o rt •• ii • • • OO 1 CD rt C? V O ts • • • 3 2 1 I CO CU fO (Channels) f to I—• S3 • •• O • vC - • • •• • • 35 TINA centroid > > h 2 \ / I \ Before S t a b i l i z a t i o n s- After S t a b i l i z a t i o n \ ,/ \ V + I \ / 1 O\ ' & shift s \ \ V Run . \ / No. 14 10 16 •V ' / o u u c o-4 MINA centroid > S ~ shift + nl ' ,J 4-i-J J 0 | 2 Tip /j w \/' +1 + -2 - 1 U 1 It 13 8 O + I 1 —. l 10 g / o / -4h / V o Figure III.1.5 L 12 After S t a b i l i z a t i o n / <> u I 14 Run No. Before S t a b i l i z a t i o n v 8 *J i o Effect of Gain and Zero S t a b i l i z a t i o n ' 1 16 36 Y - Y Coincidences TINA (Not Stabilized) 83 MeV Gamma-Ray Energy 55 MeV (MeV) Y - Y Coincidences TINA (Stabilized) 83 20 MeV c CD 10 ID, XL 50 100 75 Gamma-Ray Energy 20 j 55 MeV (MeV) Y-Y Coincidences MINA (Stabilized) 83 CO •u g 101 n „ MeV 1 np-in Gamma-Ray Energy Figure III.1.6 Gamma-Ray Coincidence Spectra (MeV) 37 2. F i t t i n g the Histograms Two procedures were used to determine the branching r a t i o f o r charge exchange i n deuterium. The f i r s t was the comparison of the number of deuterium charge exchange coincidences to the number of radiative capture events tempered with the r e l a t i v e e f f i c i e n c i e s f o r observing these processes. ? TTD N Y( ~ T r d Ynn) S + 1 (III.2.1) where: i s the number of coincidence y-rays due to charge exchange i n deuterium i s the e f f i c i e n c y f o r observing deuterium charge exchange coincidence y-rays £ s i s the e f f i c i e n c y f o r observing singles y-rays Ny(Tr~d -> ynn) i s the number of r a d i a t i v e capture y-rays observed. The second approach was the comparison of the r e l a t i v e number of charge exchange events due to deuterium and due to hydrogen. The charge exchange branching r a t i o i n deuterium may then be calculated from the known charge exchange branching r a t i o i n hydrogen i f the e f f e c t i v e concentration of hydrogen i n the target i s known. ^ - V H ^ F TTD 1 TT < - - > M 2 2 38 where i s the charge exchange branching r a t i o i n hydrogen C.H i s the e f f e c t i v e concentration of hydrogen i n the target H i s the number of coincidence y-rays due to charge exchange TI i n hydrogen 'ITH i s the e f f i c i e n c y f o r observing hydrogen charge exchange coincidence y-rays. The e f f e c t i v e concentration of hydrogen i n the target was not t r i v i a l to determine. F i r s t , the concentration of hydrogen i n the gas was pected to be reduced i n the l i q u i d state due to d i s t i l l a t i o n . ex- (The b o i l i n g point of hydrogen i s about 3.2K. lower than that of deuterium at normal pressures (MC64). stopped IT A second possible e f f e c t was the preference of captured i n molecular rather than TT - H atomic states. H-D o r b i t s to form TT""-D atomic states The e f f e c t of p r e f e r e n t i a l TT capture on high Z n u c l e i i n hydrogenous compounds i s well known (KR68, PE69, P073). In the case of deuterium hydride ( H - D ) the TT bound states are s l i g h t l y deeper (-6%) on the deuteron than on the proton. These e f f e c t s which reduce both the e f f e c t i v e concentration of hydrogen i n the l i q u i d and the actual concentration i n the l i q u i d from the concentration i n the gas are discussed more f u l l y i n Appendix I V . It was possible to d i r e c t l y measure the e f f e c t i v e hydrogen concentration i n the target l i q u i d by examining the singles spectrum for charge exchange events. In summary i t was necessary to extract the numbers D ^ and H ^ from the y-ray coincidence spectrum and a value f o r C„ from the y a y n spectrum. _ r singles 39 A. The Coincidence Spectrum Even with s t a b i l i z a t i o n to c o r r e c t the gain and zero s h i f t s the coincidence y-rays from hydrogen charge exchange s t i l l encroach s i g n i f i c a n t l y into the region of the deuterium charge exchange events. In order to extract the number of deuterium events and the number of hydrogen events from the coincidence spectrum the hydrogen component was f i t with an empirical hydrogen coincidence lineshape. This lineshape was measured i n conjunction with a re-measurement of the Panofsky r a t i o i n hydrogen (SP77). The geometry of the two experiments was identical except f o r a 3% d i f f e r e n c e i n the distance of the TINA c o l l i m a t o r from the target (113.0 cm. instead of 110.2 cm.). The f i t of the empirical lineshape i s shown superimposed on the TINA coincidence data i n f i g u r e III.2.1. I t can be seen to give a very good f i t to the hydrogen charge exchange peaks. In f i t t i n g the amplitude of the empirical function i t 75 Gamma Ray Energy 100 (MeV) Figure III.2.1 - F i t of the Hydrogen Peaks. 40 was Important to not consider the events due to deuterium charge exchange. Hence the c e n t r a l region was omitted from the f i t t i n g and only the regions shown as ( L i , L ) and 2 considered. the l i m i t s L (1,3,1^) procedure on figure III.2.1 were To test the e f f e c t of any deuterium events i n these regions 2 and L 3 were varied over a wide range of energies and the e f f e c t of the f i t was observed. In figure III.2.2 the amplitude of the empirical lineshape i s plotted as a function of the values I^ and L . 3 As expected there was n e g l i g i b l e v a r i a t i o n of the amplitude with v a r i a tions of the exterior l i m i t s . The s l i g h t v a r i a t i o n s i n the amplitude with the v a r i a t i o n of the i n t e r i o r l i m i t s , L 2 and.1,3, are seen to be w e l l within the error l i m i t s set f o r the amplitude. We conclude that the dependence of the f i t upon these parameters and upon any deuterium events i n the regions defined by these parameters i s n e g l i g i b l e . the region between the l i m i t s and In there were 182.0 ± 13.5 events - of which we determine 22.2 ± 2.3 were due to charge exchange i n hydrogen. This leaves 159.8 i n deuterium. ± 13.7 events i n t h i s region due to charge exchange The number of deuterium events under the hydrogen peaks was determined by subtracting the f i t i n these regions from the data. The r e s u l t was that there are 8.2 ± 14.5 deuterium events under the hydrogen peaks. Thus the t o t a l number of deuterium charge exchange events i n the coincidence spectrum was 168.0 ± 20. The t o t a l number of events from hydrogen charge exchange was determined d i r e c t l y from the f i t to be 195.6 ± 15.6 where the error includes the s t a t i s t i c s of the empirical spectrum. 41 220 Range of F i t (20.L2) (59,75) 200 O o o w o •H l-l •H 32 u o Statistical Error 8 180] oo o O 36 40 Limit of F i t - L2 44 (channel number) cu w 220K a CU o M CU o 2001 o Range of F i t (20,39) (L3.75) Statistical Error 180 56 Figure III.2.2 o o 60 64 68 Limit of F i t - L3 72 (channel number) Estimated Number of Hydrogen y-ray coincidences as a function of the range of the f i t . 42 B. The Singles Spectrum To extract the number of charge exchange Y-rays from the TINA singles Y-ray spectrum the data were f i t t e d i n the region between 20 MeV and 95 MeV with a smooth estimate of the background and Ynn tail (f(E)) and an empirical charge exchange y-ray spectrum from hydrogen (g(E)). The hydrogen charge exchange spectrum used i s shown i n f i g u r e II.1.2. The experimental configuration of the hydrogen target and the TINA spectrometer was the same f o r the present experiment except f o r a 3% difference i n the distance separating them (113.0 cm. compared with 110.2 cm. i n the present case). The function used to f i t the background was f(E) The f i r s t tail. ground. = ae b E + I c,E i=0 _ i + dg(E). term was included to approximate the r a d i a t i v e capture y-ray The second term was included to f i t the r i s i n g low energy backThe l a s t term represents the contribution of charge exchange events to the spectrum i n t h i s region. A grid search technique f o r the parameter b was combined with a l i n e a r l e a s t squares technique for parameters a, c and d. The data and the f i t t i n g functions are shown- i n figure III.2.3a. Figure III.2.3b shows the difference between the data and the f i t . the The parameter 'd' was tested for i t s dependence upon l i m i t s of the f i t t i n g range and upon d i f f e r e n t choices of f ( E ) . The value of 'd' was found to be r e l a t i v e l y independent of the high energy l i m i t of the f i t t i n g range but dependent upon the low energy l i m i t (figure III.2.4). Good f i t s to the data were achieved for any inverse polynomial of second order or larger and the order had l i t t l e influence 43 ' o * a.. Functions Used to F i t the Data P o 6 800 o o o © o o O 'O *o 6 a 400 co B CD i". Data Svb'o oo o o Background F i t to Data 0 0 0 ° CEX Spectrum f i t to data 50 Gamma-Ray Energy 40 20 b. Difference of Data and f i t O > o CD CO o o © o o o o Oo° o o o CP o u C cu o o oo Figure III.2.3 o0°° o o -20l 100 (MeV) % o0 o F i t t i n g the y-ray Singles Spectrum o o o ° 0 o 44 4k I I III 1 HI 1 « 3k co «s o Lower Limit = 20 2k 0) 90 60 c 95 i4- Upper Limit of F i t .a TD3" (channel No.) CD 60 H J8 O T 1 M a) .o I *o CJ 4-1 cd M 3k 4-> iii 1 Upper Limit = 95 11 (0 w 2k value chosen 15 Figure III.2.4 20 25 30 35 Lower Limit of F i t (Channel No.) Estimated Number of TT the range of f i t Singles y-rays as a Function of 45 upon the parameter 'd'. Inclusion of p o s i t i v e powers of energy i n the polynomial was found to give generally unsatisfactory f i t s to the data as evidenced by s i g n i f i c a n t structure i n the difference between the f i t and the data. The f i n a l value of the parameter *d' was d » 0.0190 ± 0.0035. The error was assigned l a r g e l y on the basis of the v a r i a t i o n of the parameter with the low energy l i m i t of the f i t t i n g range. The e f f e c t i v e concentration of hydrogen ( i . e . the r e l a t i v e proporti of TT which are captured by hydrogen atoms), C^, may be determined from the number of singles y-rays from hydrogen and deuterium. r _ 5° H ~ 2 V T T P -»• Tr°n) 1 N ( T T d -v ynn) R R (S + 1) — where i s the concentration of deuterium, taken as 1.0 N (ir p Tr°n) i s the number of y-rays from ir charge exchange on hydrogen N^(TT d -*• ynn) i s the number of y-rays from ir r a d i a t i v e capture on - deuterium. The charge exchange y-rays i n the singles spectrum are due to both charge exchange i n hydrogen and charge exchange i n deuterium. The number of singles y-rays due to charge exchange i n hydrogen may be extracted by r e f e r r i n g to the coincident y-ray spectrum (figure III.2.1) from which the r a t i o 46 NyQir'd -v Tr°nn) SrH = may be determined. Then the number of singles Y _ r a y s due to hydrogen i s just , - N (TT P , o . . NV(TT p -»• Tr°n) + N (TT d TT n) = -3f c T y F + t Tr°nn) '- Using the values f o r the e f f i c i e n c i e s presented i n chapter III.3 and the f i n a l values of D and H TT presented i n chapter III.5 we f i n d TT F - 0.132 ± 0.020, N (ir~p Tr°n) = 2911 ± 585, C„ = 0.00145 ± 0.00029 . n Thus the e f f e c t i v e concentration of hydrogen i s found to be reduced by a factor of two from the nominal concentration of 0.3%. 47 3. The Coincidence E f f i c i e n c y Because the reactions take place at r e s t the y-rays are i s o t r o - p i c a l l y d i s t r i b u t e d i n the lab frame. The singles e f f i c i e n c y i s then simply the s o l i d angle factor f o r the collimator of the spectrometer. For TINA £ (T) s t and = (3.29 ± 0.03) x 10~ 3 for MINA K (M) = (2.11 ± 0.02) x 10~ 3 The e f f i c i e n c y c a l c u l a t i o n f o r coincidence y-rays i s somewhat more involved due to the c o r r e l a t i o n between the d i r e c t i o n of the two y-rays. It i s shown i n appendix 1.3 that f o r a TT° of t o t a l energy E^ and momentum and f i r s t y-ray of energy E^ the second y-ray must l i e on the surface of a cone of half-angle (j>i, cosh - ^ 2E (E~ ^ Y ir ( whose axis i s the d i r e c t i o n of the f i r s t Y )" ? Y _ r a y- (IH.3.1) 1 For a fixed d i r e c t i o n and energy of the f i r s t y-ray the e f f i c i e n c y f o r detecting the second y-ray i s determined by the amount of the y-ray cone which intersects the collimator. In figure III.3.1 we consider the case of a TT° decay on the axis of the spectrometers. 48 The p r o b a b i l i t y f o r detecting a coincidence can be shown (appendix II) to be dP c c = 2TT^ /cosi-ccs^cos^WdN,^ V sinl})^ 5 1 1 1 ^! / dE^ dT^ where fa i s determined by (III.3.1), fa a n d ^12 d E d T cj a r e y TT (II i.3.2) determined by the dN (figure III.3.1), -^f- i s the charge exchange singles Y y (box) y spectrum (AI.2.1) and -rrr - iv° energy spectrum. In the case of geometry - r a 11 D T s t n e TT charge exchange i n hydrogen the TT° i s monochromatic. ( ) = cT(T P o TW) Hence 49 where T Q = 2.90 MeV i s the kinetic energy of the TT° . In the case of deuterium the TT° has a continuous energy spectrum from 0 MeV to MAX " T 1 , 1 M e V 8 i v e n b y (Appendix III) * d I \^MAX J \^ \MXJ The net probability of detecting a coincidence y-ray of energy Ey, i.e. the coincidence lineshape i s just the 3-fold integration /-(£ ) . 4£c . f dE ' T Y J t f K -Los" /cos(j) - cosfacosfoAd^dN^ 2? I sin^zsinf))! 1 dE^ d T c 1 2 C O S c d f l w (III.3.3) The net probability for detecting a coincidence y-ray of any energy i s the further integration, 5 (x = 0,y = 0,z = 0) = c j c£(E )dE . Y where the co-ordinates of the decay have been included to remind us that we are considering a special case. Equation III.3.2 i s based upon the intersection of two confocal circular cones (the collimator cone and the second Y-ray cone). If the decay does not occur on the axis of the collimators then the collimator cones are no longer circular but rather e l l i p t i c cones. For small deviations from the collimator axis however the cones may be considered to be approximately circular and the equation applied. In this case we expect that the probability w i l l be reduced by a factor of - cosa x cosB (figure III.3.2) and 50 Figure III.3.2 ir° Decay off the Collimator Axis we can apply t h i s factor as a c o r r e c t i o n . In the present case the maximum deviation that we need consider i s determined by the s i z e of o the target f l a s k and was -6 . Hence the error that we make i n considering o f f axis points i s l e s s than 1%. The t o t a l coincidence efficiency for the target then i s K c where - / 5 (x,y.«)^--4V V (III.3.4) c i s the density of stopped TT a t the target p o s i t i o n (x,y,z). In the case of the deuterium coincidence e f f i c i e n c y equation i s seen to be e f f e c t i v e l y a 7-fold integration. hydrogen coincidence (III.3.4) In the case of the e f f i c i e n c y the integration over the TT° energy spectrum i s t r i v i a l however we s t i l l have a 6-fold numerical integration to perform. In order to make the c a l c u l a t i o n economical the functions 5„(a.d = 0) 51 and 5 (<* = 0,d) C were evaluated at several values of a and d (where a and d are the angular deviation and decay collimator distance pictured i n figure III.3.2; note that because of the symmetry around the collimator axis we may express £ (x,y,z) as ? ( a , d ) ) . c These functions are shown c plotted i n f i g u r e III.3.3 and III.3.4. The functions S (a c well approximated by straight l i n e s i n t h i s region of d. 5 (a c = 0,d) = 0,d) are In f a c t w i l l have a maximum value when the s o l i d angles of the two collimators are equal. This corresponds to a target p o s i t i o n 10.4 closer to MINA than was used i n the present case, and we see that the function i s indeed r i s i n g for positions closer to MINA. cm. We assumed that t h i s r e l a t i o n s h i p would also be v a l i d f o r small values of ot $ 0 and hence . d ) = gfa,0) g(0,d) £(o,o) eta,a; The function £. (a = 0,d) was evaluated the points plotted i n f i g u r e III.3.3. from the straight l i n e f i t to The function £(a,d = 0) was obtained by means of l i n e a r i n t e r p o l a t i o n between the data points plotted in figure The TT III.3.4. stopping d i s t r i b u t i o n , of 'y' only (the beam d i r e c t i o n ) . t r i b u t i o n whose centroid was w a s It was presumed to be a function taken to be a Gaussian d i s - determined by the range of 51 MeV the s c i n t i l l a t o r s , degrader, vacuum jacket, target windows and liquid (ME74). The width of the Gaussian was ir in target determined by assuming that 49% of the "stopped pions" a c t u a l l y stop i n the target l i q u i d . The r e s u l t i n g d i s t r i b u t i o n i s shown i n f i g u r e III.3.5. The f a c t that t h i s estimate for the stopping d i s t r i b u t i o n i s very crude need not of great concern as i t w i l l be shown that the shape of the stopping be 52 Hydrogen Coincidence E f f i c i e n c y (x I0~ ) k 1.72 1.68L- 1.64 -6 ~3r 0 2 Displacement <— TINA 4* 6 (cm.) Deuterium C o i n c i - / dence E f f i c i e n c y / (xlO ) MINA—> 3 1.34 1.33/f / / / / -6 / 1.32' -2— Q Displacement Figure III.3.3 ^ r (cm.) Coincidence E f f i c i e n c y as a function of displacement along collimator axis 53 Hydrogen Coincidence Efficiency (Relative to Singles) I 4.2% ^ -6h ^ir 4.0% -O _L_ 0° Angular 6^ Displacement Deuterium Coincidence Efficiency (Relative to Singles) 32% „ - o - d — o -^ ^ 28% o '0 26% / o 24% -6° -4o -20 Angular Figure III.3.4 0° _L 20 40 60 Displacement Coincidence E f f i c i e n c y as a function of angular d i s p l a c e ment from collimator axis 54 d i s t r i b u t i o n (at l e a s t for wide d i s t r i b u t i o n s ) has l i t t l e e f f e c t on the total efficiency. The chosen d i s t r i b u t i o n serves the purpose of t h i s demonstration. The only remaining consideration i n evaluating equation (III.3.3) to determine the t o t a l e f f i c i e n c y i s the volume of integration to be used. Again, we found that we could t o l e r a t e a rather crude d e s c r i p - t i o n of the active volume since the t o t a l e f f i c i e n c y was rather insen- s i t i v e to t h i s consideration. The volume of integration i s that volume of the target which i s also i n the volume of the TT beam as defined by the counter 'C2'. The beam divergence i s p r i m a r i l y due to multiple scattering i n the degrader and i s well represented by a Gaussian d i s t r i b u t i o n with a 6° standard deviation (SP77). For the purpose of t h i s c a l c u l a t i o n the beam was con- sidered to uniformly f i l l a cone of 6° half-angle. The radius of t h i s cone exceeded the target radius at approximately the collimator a x i s . The error i n the calculated e f f i c i e n c y due to the estimate of the beam volume was determined by considering the two extreme cases of a c y l i n - d r i c a l beam volume the s i z e of C2 and a f u l l y illuminated target volume. The errors i n the e f f i c i e n c y c a l c u l a t i o n are summarized i n the following table. The error due to f i n i t e binning i n the numerical integration i s large i n the case of deuterium because of the integration over the pion energy d i s t r i b u t i o n . In general the larger errors In the deuterium c a l c u l a t i o n are due to the more extreme v a r i a t i o n of the function £ (a,d) with a. c The error due to the stopping d i s t r i b u t i o n was deter- mined by considering the difference i n e f f i c i e n c y using d i s t r i b u t i o n for and using a constant stopping d i s t r i b u t i o n . The values determined f o r the coincidence e f f i c i e n c i e s are the Gaussian 55 'TTH (1.74 ± 0.026) x I O - 4 (1.166 ± 0.042) x IO" . 3 'TTD The coincidence lineshapes (equation (III.3.3)) have also been evaluated for deuterium and hydrogen. In the case of hydrogen the i n t e g r a t i o n over the target volume has been performed. In the case of deuterium o the lineshape i s f o r a TT decay at the centre of the target. These lineshapes have been plotted i n f i g u r e II.1.5. ERROR IN EFFICIENCY DEUTERIUM HYDROGEN CALCULATION DUE TO: EFFICIENCY EFFICIENCY 1. Numerical Integration 2. Non-circular i n t e r s e c t i o n 2.2% 0.1% 0.5% 0.5% of cones 3. Interpolation of Jj(cc,d) 1.5% 1.0% 4. Beam volume 1.7% 0.9% 5. Target length 1.6% 0.3% 6. Stopping d i s t r i b u t i o n 0.7% 0.1% 3.6% 1.5% TOTAL ERRORS 56 4. Corrections f o r In-Flight The energy T Interactions p r o b a b i l i t y that a ir which enters the target with k i n e t i c Q w i l l i n t e r a c t i n f l i g h t i n a volume element Adx a t p o s i t i o n x In the target (figure III.4.1) i s given by a P j C T ^ x ) = po(T)dx , where p i s the density of deuterium atoms i n the target and a(T) = o(T(x)) i s the cross section f o r the i n t e r a c t i o n . I—x—H Figure III.4.1 The Liquid Deuterium Target p r o b a b i l i t y f o r observing the i n t e r a c t i o n i s d P ( T , x ) = o(T)5(T,x)dx o and is o p the t o t a l p r o b a b i l i t y f o r observing an i n t e r a c t i o n while i n f l i g h t 57 P tr S m o' where x r x / po(T)5(T,x)dx x=0 r i s the range of the pion i n the target. (III.4.1) To determine the function T(x) we note that for pions of k i n e t i c energy l e s s than -50 MeV the stopping power i n deuterium (ME74, TR76) i s approximated to about 10% by dT ,,,-b = aT dx (III.4.2) - 3 — With T measured i n MeV a = 10.87 b = 0.805. Integrating equation (III.4.2) gives T(x) = (T? - k x ) where c = 1.805 and k = 19.63. Then 1 / c 58 A. Charge Exchange In appendix I I I i t i s shown that f o r small energies a(T) = d The constant 'd' was determined (R057) at 85 MeV. (III.4.3) from the data of Rogers and Lederman For T and Q measured i n MeV d = 2.05 x 10~ mb . 6 The function S ( T ) has been evaluated a t energies T = 0.0 MeV, 8.9 MeV C and 18.9 MeV. It i s found that t h i s data f i t s the function «c< -!¥t (iii T ) to better than 10%. -4 4) Substituting (III.4.2), (III.4.3) and (III.4.4) into (III.4.1) and performing the integration gives P (T ) O Q = 1.56 x IO"" ^1 3 0 5 jl.O + 3.92^-y 5.60 (j~J + - (TJ 3 31 (III.4.5) To make use of t h i s r e l a t i o n s h i p we must know the d i s t r i b u t i o n of the pion k i n e t i c energies at the face of the target. We took the o r i g i n a l momentum d i s t r i b u t i o n to be Gaussian with a centroid at 130 MeV/c and a FWHM of 15%. for (The M9 momentum byte has been measured a t T - = 30 MeV a 10 cm. Be target and 10 cm. horizontal s l i t s as 15% FWHM (BR76)). 59 This momentum d i s t r i b u t i o n was divided into equal bins and f o r each b i n the energy at the target face and the number of i n - f l i g h t charge exchange Y-rays expected from the bin were determined. The momentum d i s t r i b u t i o n and the expected number of i n - f l i g h t charge exchange y-rays for each bin are plotted i n figure III.4.2. Also shown on the abscissa are the target entry energies for the given momentum b i n . of There are a t o t a l 17.2 coincidence Y-rays expected from charge exchange i n f l i g h t . We have used equation (III.4.5) to estimate that of these only 1.7 coincidence y-rays occur from i n f l i g h t interactions below 18.9 MeV. A charge exchange coincidence Y-ray lineshape for 18.9 MeV incident ir~ has been calculated using the methods outlined i n the previous section and i s plotted i n f i g u r e III.4.3. Lineshapes for higher energy interactions w i l l have an even larger energy spread and a greater dip i n the c e n t r a l region. Thus the use of the 18.9 MeV lineshape to estimate the contribution from i n f l i g h t events under the hydrogen and deuterium peaks i s expected to give a r e s u l t that i s somewhat too large. The contribution under the hydrogen peaks i s expected to be l e s s than 7.8 Y-ray events. The contribution under the deuterium data i s expected to be less than 2.0 Y-ray events. It i s r e a l i z e d that many of the approximations used to a r r i v e at these figures are rather crude. For example, the stopping power, as given by equation (III.4.2) i s expected to be good to only about 10% i n the 30-50 MeV region and i t i s this energy region which l a r g e l y determines the i n - f l i g h t contribution. the Furthermore the extrapolation of low energy cross section r e l a t i o n s h i p (equation III.4.3) to energies as large as 85 MeV i s questionable. Despite these shortcomings the c a l c u l a t i o n does point out that the i n f l i g h t charge exchange contribution 60 Target Entry Energy 7.5 18.6 30.2 (MeV) 40.1 / 4.0 60 48.5 In F l i g h t Charge Exchange Interactions Momentum D i s t r i bution 2.67 x 10 TT" 9 \ co 3.0 / o rt \ / u cu i cu 60 C C H / 2.0 o c cu cu n o 5 CJ o «4-l A 1.0 > \ o cu cu m J u cu s no Figure III.4.2 20 \- \ T 130 150 Pion Momentum (MeV/c) ir Momentum Byte and In f l i g h t Charge Exchange Measured Coincidence Spectrum (TINA) G cu 10 \ Calculated In F l i g h t Lineshape (18.9 MeV) u cu a 100 Figure III.4.3 Calculated Lineshape for 18.9 MeV TT" 61 i s dominated by ir energies greater than 18.9 MeV. Thus, on the basis of figure III.4.3, we would expect about 30% of the i n f l i g h t charge exchange coincidence y-rays to have energies above the high energy hydrogen coincidence peak. The previous c a l c u l a t i o n suggests that we should find 5 y-ray events i n t h i s region. This discrepancy may In fact we see 11 events. be interpreted either as an i n d i c a t i o n that our c a l c u l a t i o n i s low by a factor of two or that the 18.9 MeV lineshape i s not a good d e s c r i p t i o n of the "average" lineshape for the energy region above 18.9 MeV. The former i n t e r p r e t a t i o n w i l l increase our corrections by a factor of two; ably. two. the l a t t e r could decrease them consider- The correct i n t e r p r e t a t i o n of course i s some combination of these (It i s clear that the 18.9 MeV lineshape has too l i t t l e weight i n the region above the high energy hydrogen coincidence peak. However the actual lineshape can not have more than 50% of i t s area i n t h i s region, so t h i s alone can not account f o r the discrepancy). In any case the size of the c o r r e c t i o n i s l e s s than the error i n the number of events i n either the hydrogen or the deuterium coincidence spectrum and so i t i s expedient to avoid these u n c e r t a i n t i e s simply by applying large error l i m i t s to the c o r r e c t i o n s . These are taken as 2.0 ± 2.0 i n the case of the c o r r e c t i o n to the deuterium coincidence counts and 7.8 ± 7.8 counts. i n the case of the c o r r e c t i o n to the hydrogen coincidence 62 B. Radiative Capture The determination of the c o r r e c t i o n due to i n f l i g h t r a d i a t i v e capture i s more straight forward. We again make use of equation (III.4.1) to determine the p r o b a b i l i t y of observation. In t h i s case however the e f f i c i e n c y i s j u s t the singles e f f i c i e n c y and i s not a function of energy. Furthermore, 0 < I ) and i n t h i s case Q = 136 MeV. the cross-section now i s given by . a ( ^ i 2 The e f f e c t of t h i s i s a rather slow v a r i a t i o n of the cross section with energy i n the range of i n t e r e s t . Again 'a' was taken from the data at 85 MeV where the cross section i s 1.1 ± 0.6 mb (R057). Since the cross section i s r i s i n g f o r energies below 40 MeV care was taken to avoid integrating beyond the end of the target. The net r e s u l t of the c a l c u l a t i o n was an expected of 8100 ± 4400 r a d i a t i v e capture events from ir In f l i g h t . contribution This r e - presents a 1% c o r r e c t i o n to the t o t a l number of r a d i a t i v e capture events. 63 5. The F i n a l Analysis The following table (Table III.5.1) summarizes the measured and calculated quantities which were used to determine the charge exchange branching r a t i o from our data. Using the r a t i o of the charge exchange rate to the r a d i a t i v e capture rate (equation III.2.1)) we f i n d = (1.46 ± 0.19) x 10" Using the r a t i o of the charge exchange rate i n hydrogen to the charge exchange rate i n deuterium (equation (III.2.2)) we find R 2 = R„C = (1.16 ± 0.33) x 10" k In determining the errors assigned to t h i s second number i t was necessary to consider the negative c o r r e l a t i o n between the errors i n D and H TT and between the errors i n £ „ and £ ~. ^TTH H TT The c o r r e l a t i o n i n the D and ^TTD TT terms a r i s e s because the sum of the two terms i s constrained. c o r r e l a t i o n i n the £ TT TTH and £ The terms i s a r e s u l t of the opposite cur- ^TTD vatures of the functions £ „(a) and £ „(<x) (figure III.3.4). TT h Furthermore ITU i t i s not legitimate to simply consider the weighted mean of the two numbers as an average value because the two numbers are not independent. To obtain a properly weighted average value we must separate the common terms and not consider them i n the weighting. In d e t a i l 64 TABLE III.5.1 - Summary of Data D = 166.0 ± 20.0 Number of Deuterium Coincidence Events. Corrected TT for i n - f l i g h t charge exchange. = 187.8 ± 17.4 Number of Hydrogen Coincidence Events. Corrected for i n - f l i g h t charge exchange. N^O-d ynn) = 806122 ± 8000 Events. Number of Deuterium Radiative Capture Corrected f o r i n - f l i g h t r a d i a t i v e capture, contributions from deuterium and hydrogen charge exchange and hydrogen r a d i a t i v e capture. C„ - (1.45 ± 0.29) x 10~ H K E f f e c t i v e concentration of hydrogen. 3 - (3.29 ± 0.03) x 10~ TINA singles y-ray e f f i c i e n c y , s £ _ = (1.166 ± 0.042) x 10~ E f f i c i e n c y f o r observing deuterium charge 3 3 TTD exchange Y ~ y s . ra 5 „ = (1.740 ± 0.026) x 10 * -1 E f f i c i e n c y f o r observing hydrogen charge exchange y-rays. S = 2.97 ± 0.17 Ratio of ir" absorption and r a d i a t i v e capture rates i n deuterium (world average (excluding CH54)). R^ = 0.607 ± 0.002 Charge Exchange branching r a t i o i n hydrogen (Most recent r e s u l t (SP77)). 65 1 R TTD W! + W Wl 2 ] _ Es 1 N^O d •> ynn)ys + 1 + W 2 H H H R TfH C or where Fx = *° Nydr'd -> ynn) S + 1 F * - VH I T 1 TT and the weighting factors Wj_ and W 2 are assigned i n the usual fashion. That i s Wl = dpjr and W 2 = ^ . When written i n t h i s format i t i s c l e a r that the r e l a t i v e l y large errors in C H and H w i l l r e s u l t i n the second method being weighted very l i g h t l y compared with the f i r s t . In f a c t the average value, which I quote as the f i n a l r e s u l t i s R = 1.45 x 10~^± From t h i s we determine 0.19 x 10"** • 66 K *> 5.76 x IO" * ± 0.71 x 10~ 1 4 where the r e l a t i v e error i s s l i g h t l y decreased.because the value of ' S' i s eliminated from the c a l c u l a t i o n . 67 CHAPTER IV DISCUSSION 1. The Charge Exchange Branching Ratio The branching r a t i o for pion charge exchange i n deuterium t i e s i n with other low energy pion r e s u l t s . c a l c u l a t i o n Beder In p a r t i c u l a r , i n a recent (MA77) has used the impulse approximation to r e l a t e the charge exchange rate i n deuterium to the hydrogen charge exchange scattering length. In a recent a n a l y s i s of T r d -*• pp cross section + data Spuller (SP75) presents a f i g u r e which demonstrates the r e l a t i o n ships between the low energy pion data. This i s shown i n f i g u r e IV.1.1 with the r e l a t i o n s h i p of the deuterium charge exchange rate Following included. Beder's c a l c u l a t i o n (BE76, MA77) we note that the capture rate to a f i n a l state |f> from an i n i t i a l state |i> i s given i n terms of the |i> -»• |f> S-matrix as w(f) = / dP f [UrOWpf - P ) ] ~ |<f !S| ±>|2 (IV.1.1) 1 ± where dp^ i s the density of f i n a l states and P^ and P^ are the f i n a l and i n i t i a l four-momenta.. We may expand the S-matrix element i n terms of a complete set oftf~dplane wave states. In the IT d r e s t frame <f|s|i> «\, J <f |s|Tr(q)d(-q)><Tr(q)d(-q)|i(atomic)>d q 3 where q i s the TT" momentum. W e recognize the second term as the Fourier transform of the i n i t i a l atomic wave function i n r space. This transform 68 trCyp-Tr+n) crt/T^p — 7T~p) fC.L f R c r i y n — -rr cr(7r~p— n7T°) p) |D.B. cr(7r~p— ^ EL*. 2. E. n y ) 0j(7T*"p — n7T°) IA. cj(7r~p — n y ) IT ts \X) \ ti vj - — -» C U ( T T : n : / 2. EL*. cr(7r~d — nn) }j EL . fc.i. criir^d — pp) Jim nn7T°) LEGEND C. I. «= charge independence D. B. = detailed balance E. Z.E. '= extropolation to zero energy I.A. = Impulse Approximation R,P,T,S,K, are the r a t i o s (experimental or calculated) for the processes as no ted., c r ( p p — T T d) Figure- IV.1.1 d— K Relations Between Low Energy Pion Reactions 69 i s n e g l i g i b l e f o r |q| >> Bohr r a d i u s . The f i r s t — where i n the p r e s e n t c a s e , a term i s the c o n j u g a t e f o u r i e r t r a n s f o r m of a f u n c t i o n i n r space which has a range o f a p p r o x i m a t e l y p o t e n t i a l and hence w i l l be n e a r l y c o n s t a n t over of the second the f i r s t term. term from * <f u «u / dp [ ( 2 i r ) V < P £ a d the s h o r t range (q - 0) the i n t e g r a l . Using t h i s approximation ^ ^" the n u c l e a r Thus we may s i m p l i f y e x p r e s s i o n IV.1.1 by removing <f|s|i> = q^O i s the TT f - f |s|ir(0)d(0)> J<Tr~(q)d (-q) | i (atomic ) > d q 3 i n "equation IV.1,1 g i v e s pJJ^Hf |s|Tr(0)d(0)>| 2 Q<7r~(q)d(-q)|i(atomic)>d ) ^/ <Tr~(q)d(-q) | i ( a t o m i c ) > d q j 3 where a i s the f r e e p a r t i c l e c r o s s - s e c t i o n (without coulomb e f f e c t s ) f o r initial p l a n e wave TT d s t a t e s to i n t e r a c t . initial atomic sidering A l l o f the d e t a i l s o f t h e s t a t e a r e wrapped up i n the i n t e g r a l . the r a t i o o f t r a n s i t i o n r a t e s t o d i f f e r e n t f i n a l s t a t e s the e f f e c t s of the coulomb d i s t o r t i o n o f the i n i t i a l enter. S i n c e we a r e c o n - atomic s t a t e do n o t In p a r t i c u l a r , o % ii™ W(TT d -» TT nn) _ q->0 w(ir~d -> nn) i The charge S-matrix. i qa(Tr~d -> Tr°nn) m q+0 ( i r M ~ + d . (IV.1.2) j exchange c r o s s - s e c t i o n may be expressed i n terms o f the ; 70 ~ qo(ir d Jd^dj^^ J72ir)^(P where q', k j , k The 2 £ ir nn) » JgJ ^ n n l s l i r - d H * - * " are the f i n a l TT° and two neutron momenta r e s p e c t i v e l y . S-matrix i s anti-symmetrized i n k j and k . The capture from the 2 i n i t i a l state i s s wave so p a r i t y conservation conservation (IV.1.3) and angular momentum r e s u l t s i n a t r i p l e t spin state for the f i n a l nucleons. 2 Since we do. not look at spin here 1/3 ignoring spins. i s equivalent to |S|2 The f i n a l factor of h avoids double counting the i d e n t i c a l nucleons. explicitly ^.|S| spins We now define a reduced scattering matrix S by including the energy conservation d e l t a function; v i z . <f |s~|i> = £2TTcf(E - E ) j " < f j s | i > . ' 1 ± f We expand the reduced matrix element i n terms of a complete set of n and p plane wave states. <Tr°nn|s"|Tf~d> » ( 2 T T ) ~ J d t d ' s <TT°k k | s"|ir~n(t)p(s)>3tt(t)p (s) |d> 6 3 3 2 - 1 <Tr°k ki |?|ir"n(t)p(s)><n(t)p(s) |d> 2 In the impulse approximation the i n i t i a l neutron takes only a spectator role. Hence = (2TT) cf (^i " t)<TT n(k )|s; |TT"p(1)> ^ t i ^ n & J l S ^ - n i t ) ? ^ (2IT)3J (£I 3 C " 3 0 2 - t)cf (q"+^ 3 J 2 E (TT ) • 2 E 3 U N - q -"s) 8 * (E ) ' 2 E C O • 2 E (s 2 P N f (u~p -> ix°n) ~~ where VJ^ is is the W ^ is t and the are total hydrogen = M + u replaced With s these t h e momenta energy the the sum o f the the hydrogen considerations neutron pion-nucleon charge-exchange (the by in of scattering nucleon charge and proton center of plane mass and amplitude. In pion masses) and exchange we d e t e r m i n e and scattering from equation the wave states, f (IT p -*• T r ° n ) limit q -> 0 f (TT p ->• T r ° n ) length, a (IT p -»• i r ° n ) (IV.1.3) t h a t " l i n i j qa(Tr~d -»• T r ° n n ) a (u-p-TT%)J d q ' k ( q ' ) q ' 2 | F(q')|2 2 VM where F(q) = Jjj ( k r ) j 1 (^|) . <t^(r)r dr, 2 k-fMCQ-i^-)} * , 5 Q = u reaction = neutral (> d = unit Q-value pion (1.1 reduced normalized MeV), mass, deuteron (*e<a + B ) V e - " - e'* s >2ir(o - a - 1 = 4.3 B = 7a fm B) z r r wave function. (IV.1.4) 72 A c o r r e c t i o n may be made for double s c a t t e r i n g . |a(Tr~n TT~n) + a(Tr°n T r ° n ) | a(u~p This i s of order •> Tr°n)<<|) ( 1 = 1) | —^—| r ^> and contributes a -1% c o r r e c t i o n to the impulse approximation amplitude. Using the most recent r e s u l t for the hydrogen charge exchange s c a t t e r i n g length (a(Tr~p •+ Tr°n) = 0.175 fm (NA76)) i n equation '5jJ-qa0r"d -»• ir nn) l 0 The denominator of equation (IV. 1.4) we find = 0.0358 MeV mb (IV. 1.5) (IV.1.2) i s evaluated by assuming that for Coulomb corrected cross-sections a(ir d -> nn) = a(u d -»• pp). + The threshold behavior of the l a t t e r cross-section has been analysed i n d e t a i l by Spuller and Measday (SP75) who f i t Coulomb corrected data to a function of the form a(Tr d + where P + pp) - | ^ (an + 0 ( n ) J 2 i s the proton center of mass momentum i n units of M c,a i s a f i t t i n g parameter and n = q/y. They f i n d the preferred range for a to be 0.25 mb - a - 0.29 mb. In the l i m i t that q -> 0 t h i s corresponds to 162 MeV-mb - lim qa(Tr~d nn) - 187 MeV mb. (IV.1.6) 73 The charge exchange branching r a t i o i s given by ^ = to (TT d -> Tr°nn) # (O(TT d ->• nn) io(Tr~d -+ nn) The f i r s t factor i s determined r e s u l t s (IV.1.5) and (IV.1.6). . co(Tf~d ->• a l l ) by equation (TV.1.2) and the subsequent The second factor i s determined empirical value f o r S (2.97 ± 0.17). from Thus the t h e o r e t i c a l c a l c u l a t i o n predicts a branching r a t i o 1.39 x 10~ IO" - R - 1.59 x k 4 which i s i n good agreement with our experimental value R = (1.45 ± 0.19) x 10*"* . 1 It should be pointed out that using the e a r l i e r 'recommended' value of Pilkuhn et a l (PI73) f o r the hydrogen charge exchange scattering length (a(Ti~p -»• Tr°n) = 0.193 ± 0.013 1.67 fm) r e s u l t s i n a branching r a t i o x 10~ - R - 1.91 x 10" * 9 i n d i s t i n c t c o n f l i c t with the present value. 1 74 LIST OF REFERENCES BE76 D. Beder, Private Communication BE77 D. Beder, Private Communication BI76 J.A. B i s t e r l i c h , S. Cooper, K.M. Crowe, F.T. Shively, E.R. G r i l l y , J.P. Perroud, R.H. Sherman, H.W. Baer, P. Truol, Phys. Rev. L e t t . 36, 942 (1976) BR51 K. Brueckner, R-. Serber, K. Watson, Phys. Rev. 81_, 575 (1951) BR76 D. Bryman, Triumf Internal Report, Feb 19, 1976 CH54 W. Chinowsky, J . Steinberger, Phys. Rev. 95, 1561 CH55 W. Chinowsky, J . Steinberger, Phys. Rev. 100, 1476 C061 V.T. Cocccni, T. T a z z i n i , G. Fidecaro, M. Legros, N.H. Litman, A.W. HA65 (1954) (1955) Merrison, Nuovo Cimento 2_2, 494 (1961) R.P. Haddock, R.M. Salter J r . , M. Z e l l e r , J.B. C z i r r , D.R. Nygren, Phys. Rev. L e t t . 14, 318 (1965) KL64 P.K. Kloeppel, Nuovo Cimento 34_, 11 (1964) KR68 Z.V. Krumshtein, V.I. Petrukhin, L . I . Ponomarev, Yu.D. Prokoshkin, Soviet Physics JETP 27_, 906 (1968) KU59 J.A. Kuehner, A.W. (London) 73, 551 Merrison, S. Tornabene, Proc. Phys. Soc. (1959) LE62 M. Leon, H.A. Bethe, Phys. Rev. 127, 636 (1962) MA77 R. MacDonald, D.S. Beder, D.C. Berghofer, M.D. Hasinoff, D.F. Measday, M. Salomon, J . Spuller, T. Suzuki, J.M. Poutissou, R. Poutissou, P. Depommier, J.K.P. Lee, Phys. Rev. L e t t , (accepted f o r p u b l i c a t i o n 1977) 75 MC64 M. McClintock, Cryogenics, Reinhold Pub. Corp., New York (1964) ME74 D.F. Measday, M.N. Menard, J.E. Spuller, TRIUMF Kinematic Handbook ME77 D.F. Measday, Private communication. NA76 M.M. Nagels et a l . , Nucl. Phys. B109, 1 (1976) PA51 W.E.H. Panofsky, R.L. Aamodt, J . Hadley, Phys. Rev. 81_, 565 (1951) PE64 V.I. Petrukin, Yu.D. Prokoshkin, Nuclear Physics 54_, 414 (1964) PE69 V.I. Petrukin, Yu.D. Prokoshkin and V.M. Suvorov, Sov. Phys. JETP 28, 1151 (1969) PI73 H. Pilkuhn, W. Schmidt, A.D. Martin, C. Michael, F. Steiner, B.R. Martin, M.M. Nagels, J . J . de Swart, Nucl. Phys. B65, 460 (1973) P073 L . I . Ponomarev, Annual Reviews of Nuclear Science, 1973 R057 K.C. Rogers, L.M. Lederman, Phys. Rev. 105, 247 (2957) RY63 J.W. Ryan, Phys. Rev. 130, 1554 (1963) SH68 L . I . S c h i f f , Quantum Mechanics, McGraw-Hill Book Co., New York (1968) SP75 J . Spuller, D.F. Measday, Phys. Rev. D12, 3550 (1975) SP77 J . Spuller l . , 3 . , Submitted to Physics Letters (1977) 2., Private communication TA51 S. Tamor, Phys. Rev. 82, 38 (1951) TR74 P. Truol, H.W. Baer, J.A. B i s t e r l i c h . K.M. Crowe, N. de Botton, J.A. Helland, Phys. Rev. L e t t . 32_, 1268 (1974) TR76 T.G. Trippe e t a l . , "Review of P a r t i c l e Properties", Reviews of Mod. Phys. 48, no.2, part I I ( A p r i l 1976) 76 WA51 K.M. Watson, R.N. Stuart, Phys. Rev. 82, 738 (1951) YA50 C.N. Yang, Phys. Rev. 77_, 242 (1950) ZA65 O.A. Zalmidoroga, M.M. Kulyukin, R.M. Subyaev, I.V. Falomkin, A.I. F i l i p p o v , V.M. Tsupko-Sitnikov, Yu.A. Shcherbakov, Soviet Physics JETP 21_, 848 (1965) 77 APPENDIX I NEUTRAL PION KINEMATICS 1. Gamma Ray Doppler Shif t The decay of the TT° i n the l a b frame i s pictured i n f i g u r e A I . l y Figure AI.1.1 - The TT° decay i n the l a b frame We choose the x a x i s to l i e along the d i r e c t i o n of one of the gamma rays with the o r i g i n a t the p o s i t i o n of the n° decay. The statements of conservation of energy and momentum are: P„ + P, 2X = P IX TTX = P cosO TT P, - P = P sino 2y ^y TT v P. + (P • lx 2x v 2 + P„ f i - (M 2 2y We square and add (1) and (2) to obtain 2 Tf + P 2)** TT 78 P, 2 2x + P, 2 lx + P„ 2 + 2P, P = P lx 2x 2y 2 TT and subtract t h i s from the square of (3) to f i n d 2P. ((P„ + P, 2 ) - P ) = M lx 2x 2y ' 2x TT 2 15 2 v v y We now substitute from (1) and (3) f o r the components of P 2 P lx Tr ( ( M 2 " lx " V P + M 2 (M 2 + P P lx -TT Now P ix = E and (M yi 11 2 + P ) ^ 2 ? s = E . ^ O S 0 + P lx> * 2 Tf Hence - P cos0 TT V 2 < 79 2. The y-spectrum Resulting from Isotropic u° Decay Let tiny be the number of y-rays per unit pion s o l i d angle i n the lab frame dn = Y & d« TT We f i r s t consider those pions whose momentum i s contained i n the elemental s o l i d angle dtt^. distributed. In the pion r e s t frame the y ~ y s are i s o t r o p i c a l l y r a Hence dn Y " k d Q Y or X-< - 2 n k d(cos0 ) (1) dn i where the prime denotes the pion r e s t frame. In the lab frame t h i s d i s - t r i b u t i o n i s pushed forward i n the d i r e c t i o n of the pion momentum. Figure A I . 2 . 1 - Center of Mass - Lab transformations 80 The s i t u a t i o n i s depicted i n figure AI.2.1. The pion rest frame i s moving with v e l o c i t y 6 with respect to the lab frame and 6' = -8 LAB TT For the y-ray i n the pion r e s t frame (with c = 1) 11 v x = c o s 0 ' In the lab frame v = cosG x The r e l a t i v i s t i c v e l o c i t y addition equations connect these two v e l o c i t i e s . v v* 8' x - LAB = x i - a; v , LAB x Then V cos0=— „ ' - PT'A-D X cose' - B;,„ - ' LAB L LAB x A B so 0 COS0 = + BTT COS© 1 + ^Jp COS© and conversely c o s 0 ^ . cose- - B, 1 -.6 cos© TT (2) 1 81 In the lab frame dny dcos© dny . d (cos©") dcosQ' d(cosO) From ( 2 ) d(cos0") _ 1 - B^ d(cos0) (1 - B^cos©^ 2 and with ( 1 ) we a r r i v e a t d(cos0) c K ( l - B^cos©)* This i s the d i s t r i b u t i o n of y-rays as a function of y-ray angle with respect to the TT° d i r e c t i o n . Note that the angle c f the pion with respect to the gamma ray d i r e c t i o n ©^ i s j u s t -0. Since cos©^ = cos© we have f o r a fixed gamma ray d i r e c t i o n d "y = d(cos© ) 2 TTk 1 " ^ (1 - 8 cos© ) TT TT z TT dft TT where the f i n a l term i s the d i s t r i b u t i o n of the pion i n the lab frame. For an i s o t r o p i c distribution K dfi d % r T d(cos© ) TT - Z 2TrkK 1 T K K * - IT* B (1 - g COS0 ) * Tf TT (3) 82 Now dry dEy" dn d (cost" ) TT dcosO x. = Y 1E From appendix 1.1 v 1 - _ 2 E MTT * i-rr 2 - P cosG TT TT (4) TT from which we derive dEy = 1 d(cos0 ) 2 Combining PA - P cosO ) 2 IT IT (3) and (5) with the i d e n t i t y * z TT (E (5) - P cos0 ) = E (1 - BcosO ) IT TT TT TT IT we obtain _ 4TfkK(l dn d~E~7 Y — g ) E 2 7 T 2 7 T fJT TT = 4irkK p 2 TT w TT dNv The t o t a l number of y rays observed per u n i t energy i n t e r v a l i s -r=r- where dNy _ r dny , dEy " J dEy ^TT dNy dEy 16iT kK 2 = P„ which i s independent of the y-ray energy. This formulation i s v a l i d as long as (4) i s v a l i d , i . e . for the energy range between E^ and E^ where 83 H,L fi 2 dN C l e a r l y -r=¥- = 0 outside of these l i m i t s . Hence f o r an i s o t r o p i c a l l y d i s t r i b u t e d mono-energetic TT° we observe a y-ray spectrum: d^ dE. 16^ kK 2 ^ E y < E y < > E„ H =H " Err + 2 PTT 84 3. The Second-Gainma-Ray Cone Angle The angle i n the lab frame between the two y-rays from the decay of a ir° with t o t a l energy i s completely determined by the energy E and the energy of one of the y-rays E i . The angle which we wish to f i n d i s (j), shown i n f i g u r e AI.3.1. y -y P / \ TT 0 y -> Pl a 0 X TT i r ° Decay in the I.ab Frame Figure AI.3.1 The statements of conservation of energy and momentum are Fl P cosa'+ P2COsCp = (1) (2) P sina = P2Sin(l) Pl + P * 2 = (M 2 (3) + P )'* = E 2 TT TT TT From (2) cosa = ± (1 - Using t h i s (1) becomes *L \ 2 s i n 2 ^ 85 \ sin !))) 2 P + P c o s 6 - 2P P cos(t) = P 2 2 1 2 2 1 r 2 p + p 2 2 1 r _ 2 2 * + P^cos<() . 15 ( 1 - f l - ) sin <l)) . p p cos(l) = P 2 1 2 2 V 2 v j . 2 ' T Substituting from (3) f o r P^ we get P 2 1 + (E - P ) * C O S < Since E 1 = P 1 i P 2 - 2P ( E - P )cos(j) = P 1 TT i TT 2 P i + (E,, - P i ) - P^ 2P(E - P ) 2 = 2 i . 1 TT ] / 1 c o s ( C ° K S ( P E l + (ET^ - E ; ) - P 2E v.E - E ) 2 = 2 1 2 ff = 2E ( E - E ) 1 IT 1 or C O S where 2 T 2 = 2VE 1E H 1 1 1 E ) = 1 i s the angle between the two y-rays, Si 4 TT 1 1 86 APPENDIX II The Elementary Coincidence E f f i c i e n c y For a TT° of energy (T^,T^ + dT^) the p r o b a b i l i t y of observing a decay y-ray of energy (E ,E + dE ) i n collimator y y y is dP (T ) = 2 ^ d E s TT dEy v 1 s o l i d angle dfi Cj ^ y (AII.l) 4TT dN where - j ^ - i s the normalized singles y-ray box spectrum (appendix and the factor 2 i s included 1.2) to account f o r 2 y-rays per i r ° decay. The p r o b a b i l i t y f o r a i r to have energy (T^.T^ + dT^) i s 0 where i s the normalized TT° energy spectrum. For such a TT° decay i t TT i s shown i n appendix 1.3 that the second y-ray l i e s on the surface of a cone of half-angle ((), given by co 4-2VKX - v • (AII,3) Furthermore, because the angular d i s t r i b u t i o n of the i r momentum i s 0 i s o t r o p i c , the second Y-ray p r o b a b i l i t y i s uniformly d i s t r i b u t e d over the surface of the cone. Thus the p r o b a b i l i t y f o r detecting the second y-ray i n spectrometer 2 i s determined by the amount of this cone which i n t e r s e c t s collimator 2. For the s p e c i a l case that the IT decay occurs 0 on the axis of collimator 2 t h i s i s j u s t the i n t e r s e c t i o n of two c i r c u l a r cones (figure A I I . l ) confocal 87 Figure A I I . l §12 * s The Second y - r a y cone he angle between the f i r s t y ~ y and the collimator 2 a x i s , fc r a i . e . t h i s i s the angle between the axes of the collimator cone and the second y-vay cone. <|>1 i s the h a l f angle of the second y-ray cone and i s given by (j>2 i s the half angle of the collimator cone. (AII.l) Consider the i n t e r s e c t i o n of the two cones and the surface of a u n i t sphere with center 'o' at the focus of the cones (figure A l l . 2 ) Figure A l l . 2 Intersection of two cones • 88 The p r o b a b i l i t y of detecting the second y-ray i s j u s t P where 0 i s the angle between the OAX and OAB = 20/2n. = O/TT 2 If e^, planes. e , fi are the unit vectors from the origin, 'o' to the specified points OAB i s the unit normal to the OAB = Is the unit normal to the OAX OAX plane. • * Q A 7 ) X Then (6 fi A sin(|>i A V x QAX S V • A in(j)i2sin(l>i (§ x be expanded using standard vector i d e n t i t i e s to give <*A * V cos© = <*B ' V " <*A sin(t>i 2 sin(()] The dot products are e a s i l y evaluated cos ® The p r o b a b i l i t y for detecting V * ' <B S V and _ cos(>2 - cosOicosO 12 sin(>i 2 i i s n ( ) the second y-ray becomes . B - JO. £ =. —cos l ^ e - l // c o s—-—rr ^ - c ? ; j*. •2 TT TT I sin(()l s i n < p i P 0 8 1 2 and then sin(J)i 6 The numerator may x plane and ft. n cos© = n & the p r o b a b i l i t y of detecting 1 0 8 ^ ) J o a coincidence from a TT" of energy (AH.3) 89 (T ,T IT ir + dT ) with one y-ray energy J 1 TT' (E ,E Y Y + dE ) i n the element of Y s o l i d angle dft i n collimator 1 i s just the product of ( A I I . l ) , ^1 (All.2) and ( A l l . 3 ) ; dP c - * cos-1 ( ^ sin(pi2Sin<|)i : c°H>l">s4>12 )*§f. J dEy COS 2TT Z I J d dT^ O dEdT Cj y TT . 90 APPENDIX I I I The Deuterium Charge Exchange Phase Space The rate f o r a t r a n s i t i o n from some i n i t i a l state |i> to a f i n a l state |f > i s given by d u = {^<f|H |i>|2dp raT where dp i s the density of f i n a l states. For charge exchange i n deu- terium we have a three body f i n a l state (figure A I I I . l ) and i n the center of mass frame dp = d P d P 3 3 TT d P cf(P + P +P )X(E. - E ) ni n TT nj n ^ i f 3 c 2 2 n * 2 Figure A I I I . l - Deuterium Charge Exchange F i n a l State It i s more convenient to use the set of vectors (p,q,k) defined by 91 P =P 9 = IT P_ - P k = p n +p n 1 n 2 Then dp = d p d q d k S(p 3 3 3 + k) £(E ± - E ) f The two neutrons are constrained to be i n a r e l a t i v e p-state and the ir° - (nn) system must also be i n a r e l a t i v e p-state. For small values o f p and q the matrix element may be approximated as < * I H ^ I ^ = c|pq| where c i s just a constant of p r o p o r t i o n a l i t y . The spectrum of the pions is j u s t = c/|pq| d kd qp dO 2 dp 3 3 2 S\? + k) < f ( E ± - E ) f For small values of momentum the f i n a l energy i s "T v 2m ff v 4m n 4m ir n .n n 92 Now = " TT n /o_ _ 4m ' * n so the f i n a l i n t e g r a t i o n gives n 2 £ - 1 6 , c(2 p'.(4» (E V n - |- 1 r \\\3/2 * £ ± = . n TT v With the i d e n t i f i c a t i o n of the reduced mass of the TT°-(nn) system as 2m m n TT V = 2m + m n TT and noting that the maximum pion momentum i s given by P 2 MAX ~ i ~2V fi we see dw n / 2 „ dp"^ p M p „2"\ / 3 M A X " P 2 2 ) Now dq) dE n _ djo_ dp dp dE TT _ dtj_ 2 1 1 ^ dp p Hence the energy spectrum of the pions i s « d E * d a 3/ ) P ( P 2 MAX " P 2\ > 93 or £L-=CE (E, - E ) dE TT i Tf 3 / 2 where C i s a normalization constant. J E Tf J i CE 3 / 2 The t o t a l rate i s to where 3 / V -E) 3 / 2 dE =0 If we l e t n = E / E the i n t e g r a l becomes 1I i co =4c/n 3 / 2 (l - r O ^ d n Hence w ^ E^. The i n i t i a l energy E i i s the sum of the i n i t i a l k i n e t i c energy 'T' i n the center of mass system and the Q value f o r the r e a c t i o n . Thus (T + Q> co . Now the cross section and rate are related by 0 where v i s the i n i t i a l pion v e l o c i t y . and co Then f o r low energies v = (2T/m) 94 where d i s j u s t a constant of proportionality. 95 Appendix IV The Reduction of the E f f e c t i v e Hydrogen Contamination In Section III.2 i t was pointed out that there was a discrepancy i n the e f f e c t i v e concentration of hydrogen gas as determined from the number of hydrogen charge exchange events i n the TINA singles spectrum and the hydrogen concentration i n the gas phase determined from the q u a l i t y of the deuterium oxide used to form the gas and a subsequent mass spectrometer measurement. C These concentrations were ( e f f e c t i v e ) = 0.00145 ± 0.00029 R and C respectively. R (gas phase) = 0.003 ± 0.001 Thus the reduction i n e f f e c t i v e hydrogen concentration l i e s between 10% and 70%. It i s interesting to note that we are not alone i n our observation of t h i s phenomenon. The negative values of the charge exchange branching r a t i o , R, presented by a l l the previous authors (PA51, KL64, CH55) except Petrukin (PE64)^ indicate an over estimation of the hydrogen background subtraction. A summary of these r e s u l t s was presented i n Table 1.1. Furthermore, i n a recent report by B i s t e r l i c h et a l (BI76) on the ^ I t should be mentioned here that Petrukin does not give enough d e t a i l of t h i s measurement to d e f i n i t e l y decide that they also have not measured a negative value for R. 96 reaction ir + H -*• nnny i n l i q u i d t r i t i u m there i s a c l e a r i n d i c a t i o n of 3 over-substraction of the hydrogen component of the y-ray spectrum. In a l l these cases the hydrogen background has been subtracted by considering only the absolute hydrogen contamination i n the gas phase. There are two mechanisms which might contribute to t h i s observed decrease i n the e f f e c t i v e hydrogen concentration. i s that simple d i s t i l l a t i o n has taken place. l i q u i d deuterium i s about 3.2K hypothesis The b o i l i n g point of higher than that of l i q u i d hydrogen at atmospheric pressure and d i s t i l l a t i o n may two isotopes (MC64). The f i r s t be used to separate The action of t h i s mechanism could be these supported by the non-negative value of R reported by Petrukin and Prokoshkin(PE64) who have used a s o l i d LiD target to make their measurement. This hypothesis i s not w e l l supported by the measurement of Panofsky et a l (PA51) who have measured a negative value for R even with the use of a gas target, however the error of the measurement i s so large that the d i s t i l l a t i o n hypothesis could hardly be rejected on the basis of t h i s measurement. The second hypothesis i s that we have observed the p r e f e r e n t i a l absorption of pions into deuterium atomic states rather than hydrogen atomic states. The e f f e c t of p r e f e r e n t i a l TT capture on high Z n u c l e i i n hydrogenous compounds i s well known (KR68, P073, PE69). For a hy- drogen-like atom the energy l e v e l s are given by (SH68 ) E = n (A.IV.l) 2-tVn z where the reduced mass, u, which i s s l i g h t l y greater for the TT d system 97 than for the TT p system, gives r i s e to bound states that are s l i g h t l y deeper (=6%) on the deuteron than on the proton. Ponomarev (P073) has suggested the model of "Large Mesic Molecules" to account f o r the strong preference of the pion to be captured by the high Z f r a c t i o n of hydrogenous compounds of the form Z^H^. presumes that the free pion i s captured by the Z H m one of the systems electrons. n The model system by d i s p l a c i n g Since the only electron of the hydrogen atom i s t i e d up In a molecular o r b i t the only p o s s i b i l i t i e s are f o r the pion to be captured into either molecular o r b i t s or isolated o r b i t s of the Z atom. The p r o b a b i l i t y for the former process i s given by "1 n + mZ Those pions captured into isolated atomic o r b i t s of the Z atom ultimately undergo nuclear capture on the Z nucleus. Those pions captured into the common molecular o r b i t s undergo t r a n s i t i o n to either the isolated Z atomic o r b i t s or isolated hydrogen atomic o r b i t s . the l a t t e r process i s W 2 The p r o b a b i l i t y for where W 2 = 1/Z . 2 The TT'"" systems which are thus formed move through the surrounding matter and undergo c o l l i s i o n s with other n u c l e i . During these c o l l i s i o n s the p r o b a b i l i t y of transfer of the pion to a Z nucleus i s proportional to the concentration of these n u c l e i . 98 where n i s the number of Z nuclei per unit volume and a i s a constant of proportionality. The de-excitation of the n-p system and subsequent nuclear capture by the proton are also enhanced by these c o l l i s i o n s and the rate for these processes i s thought to be proportional to the concentration of hydrogen nuclei. W where n n c " Bn H i s the number of hydrogen nuclei per unit volume and 8 i s a constant of proportionality. The probability for the meson to be cap- tured by the hydrogen nucleus then i s W3 = -r 3n where C = n„/n„ and X = a/8. L n H J = + an (1 + XC) 1 z In the case of most hydrides the constant X i s believed to be small and hence i t i s thought that atomic,transfer of the pion w i l l not contribute greatly to the overall probability of pion capture on the hydrogen nucleus. This probability then i s W - W = anZ /(n + mZ) _2 Wl 2 (A. IV. 2) which gives a good f i t to the data of Krumshtein et a l (KR68) with the constant a - 1.28. We see from equation A.TV.l that we could construct a gedanken 99 nucleus, D , to replace the deuterium nucleus where the mass would be the same as the hydrogen mass but the value of Z would be Z - 1.034. Such a nucleus would have the same TT- atomic structure as the r e a l deuterium nucleus. Unfortunately one can not apply equation A.IV.2 d i r e c t l y to such a nucleus. F i r s t of a l l for this H-D ments used to determine Wj_ would not apply. system the argu- In this case i n f a c t only molecular o r b i t s would be expected to occur since both a v a i l a b l e e l e c trons are consumed i n molecular o r b i t s . Thus Wi * 1. Further, the argument that W3 plays no r o l e i s based on the smallness of XC. In the present case the concentration i s very large, C = n„/n £ = 332 rl and comments by Ponomarev (P073) indicate that X = kZ where k could be of the order of 0.3. Thus W « (1 + kCZ)" 3 1 and even the small value of Z w i l l be s i g n i f i c a n t . W = WiW W = a/Z 2 3 The constant of p r o p o r t i o n a l i t y may 2 In this case (1 + kCZ). be determined by noting that as 100 Z -*• 1 the p r o b a b i l i t y , W, must be determined by the concentration of hydrogen. That i s w(z - i) TVF • 1 - - — V z n = 1 + C In the present case a = 0.3 and with Z = 1.034 i l i t y for nuclear capture on hydrogen we f i n d that the probab- ( i . e . the " e f f e c t i v e hydrogen concentration") i s W = 0.00269 , or about a 10% reduction from the actual concentration. Whereas t h i s r e s u l t i s i n keeping with the range of our observed reduction i t would be premature to state that this i s the mechanism which we have observed. The errors i n our measurement of the e f f e c t are very large and the p o s s i b i l i t y of d i s t i l l a t i o n i s by no means excluded. It w i l l be necessary to make c a r e f u l measurements of the e f f e c t using a gas target so d i s t i l l a t i o n e f f e c t s may be eliminated before any firm statement can be made. Such a study i s being planned for the near future (ME77).
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Charge exchange of stopped π⁻ in deuterium MacDonald, Randy Neil 1977
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Title | Charge exchange of stopped π⁻ in deuterium |
Creator |
MacDonald, Randy Neil |
Date Issued | 1977 |
Description | By using a pair of large Nal spectrometers in a coincidence configuration we have observed the charge exchange of stopped π⁻ in deuterium π⁻ + d → 2n + π°. We have measured the branching ratio of this reaction [equation omitted] and find R = (1.45 ± 0.19) x 10⁻⁴. This measurement is the first observation of pion charge exchange at rest in deuterium and represents an increase in sensitivity of a factor of 40 over previous measurements. The measured value of R agrees well with the recent theoretical result of Beder(1.39 x 10⁻⁴ ≤ R ≤ 1.59 x 10⁻⁴). |
Subject |
Deuterium ions Mesons |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-02-22 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0085184 |
URI | http://hdl.handle.net/2429/20737 |
Degree |
Doctor of Philosophy - PhD |
Program |
Physics |
Affiliation |
Science, Faculty of Physics and Astronomy, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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