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 in the Department of Physics We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August, 1977 C?) Randy N. MacDonald, 1977 In presenting th i s thes is in pa r t i a l fu l f i lment of the requirements for an advanced degree at the Univers i ty of B r i t i s h Columbia, I agree that the L ibrary sha l l make it f ree ly ava i l ab le for reference and study. I further agree that permission for extensive copying of th is thesis for scho lar ly purposes may be granted by the Head of my Department or by his representat ives. It is understood that copying or pub l i cat ion of th is thes is for f inanc ia l gain sha l l not be allowed without my writ ten permission. Department of Phvslcs The Univers i ty of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date August 3,1977 i ABSTRACT By using a pair of large Nal spectrometers in a coincidence con-figuration we have observed the charge exchange of stopped ir in deuterium ir + d - * - 2 n + T T ° . We have measured the branching ratio of this reaction R _ C J ( I T d -> 7T°nn) to Crr~d -> a l l ) and find R = (1.45 ± 0.19) x 10 - I t. This measurement is the f i r s t 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 - R - 1.59 x 10 H ) . i i TABLE OF CONTENTS Page ABSTRACT i TABLE OF CONTENTS i i LIST OF TABLES iv LIST OF FIGURES v ACKNOWLEDGEMENT v i i CHAPTER I INTRODUCTION 1 CHAPTER II THE EXPERIMENT II. 1 Kinematic Considerations 7 II-.2 The Spectrometers 15 11.3 The Experimental Arrangement.: 18 11.4 Electronics and Data Collection 21 11.5 Summary 24 CHAPTER III THE DATA ANALYSIS 111.1 The Off-line Analysis 25 111.2 Fitting the Histograms 37 III.2.A The Coincidence Spectrum 39 III.2.B The Singles Spectrum 42 111.3 The Coincidence Efficiency 47 111.4 Corrections for In-Flight Interactions 56 III.4.A Charge Exchange 58 III.4.B Radiative Capture 62 111.5 The Final Analysis 63 CHAPTER IV DISCUSSION IV. 1 The Charge Exchange Branching Ratio 67 i i i Page LIST OF REFERENCES 74 APPENDIX I NEUTRAL PION KINEMATICS AI.l Gamma Ray Doppler Shift . 77 AI.2 The y-spectrum Resulting from Isotropic IT 0 Decay 79 AI.3 The Second-Gamma-Ray Cone Angle 84 APPENDIX II THE ELEMENTARY COINCIDENCE EFFICIENCY 86 APPENDIX III THE DEUTERIUM CHARGE EXCHANGE PHASE SPACE 90 APPENDIX IV THE REDUCTION OF THE EFFECTIVE HYDROGEN CONTAMINATION. 95 i v LIST OF TABLES Table Page 1.1 Previous work with stopped pions in deuterium 3 III.5.1 Summary of Data (used in calculating branching ratios)... 64 V LIST OF FIGURES Figure Page o II. 1.1 " decay in the laboratory 8 II. 1.2 The y-ray spectrum of stopped pions i n hydrogen 10 II.1.3 Theoretical y-ray spectrum of stopped pions in deuterium.... 10 II. 1.4 The second y-ray cone 11 II. 1.5 Coincidence lineshapes for 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 constitution 18 II. 4.1 Schematic of Electronics 22 III. 1.1 Typical 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 in TINA and MINA 29 111.1.4 Centroid and zero point corrections In TINA and MINA 34 III. 1.5 Effect of Gain and Zero Stabilization 35 III. 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 41 111.2.3 Fitting the y-ray singles spectrum 43 111.2.4 Estimated Number of ir° singles y-rays as a function of the range of f i t 44 111.3.1 I T 0 Decay on the collimator axis 48 111.3.2 TT° Decay off the Collimator Axis 50 v i Figure Page III.3.3 Coincidence Efficiency as a function of displacement along the collimator axis 52 III.3.A Coincidence efficiency as a function of angular displace-ment from the collimator axis 53 111.4.1 Liquid Deuterium Target 56 111.4.2 ir~ momentum byte and In f l i g h t charge exchange 60 111.4.3 Calculated Lineshape for 18.9 MeV ir~ 60 IV.1.1 Relations between low energy pion reactions 68 AI.1.1 The I T 0 decay in the lab frame 7 7 AI.2.1 Center of mass - lab transformations 79 AI.3.1 ir° decay in the lab frame 8 ^ AII.l The second y-ray cone • 8 7 All, 2 Intersection of two cones 8 7 AIII.l Deuterium Charge Exchange Final State 90 v i i ACKNOWLEDGEMENTS Like most experiments in this f i e l d this one was performed by a group of experimenters and I am grateful to each of them for the part he played. Included in this 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 efforts made by my supervisor, Dr. M.D. Hasinoff, and Dr. D.F. Measday who have read this work in i t s draft form and made -many useful suggestions. I am grateful for many useful discussions with Dr. D.S. Beder. In particular the work presented in Chapter IV and .Appendix II has been derived from his notes. I am also grateful to Dr. A.W. Thomas for his assistance with the calculations presented in Appendix III. I thank D. Garner for his invaluable help with the ySR groups computing system. Finally I thank my wife Hilary for her part in typing the manuscript. I dedicate this work to my parents and Mr. J. Stanton. 1 CHAPTER I INTRODUCTION When negative pions interact at rest in deuterium three processes are energetically allowed. 1. IT + d -»• 2n absorption 2. ir +d-»-2n + Y radiative capture 3. i r ~ + d 2n • + ir° 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 for the charge exchange reaction in deuterium (PA51, CH55) i t had not been observed. In fact, the charge exchange reaction for pions at rest had only been observed in hydrogen (PA51, C061) and helium-3 (ZA65, TR74). It i s common to define the ratios h)(ir d -» 2n) w(ir""d -> 2ny) and K = a) (IT d -> 2mr°) U)0r~d ->• 2ny) or R = e>Qr~d ->• 2nTr°) u (Tr~d ->• a l l ) 2 where co i s the rate f o r the s p e c i f i e d r e a c t i o n . The previous experimen-t a l r e s u l t s are summarized i n table 1.1. I t i s i n t e r e s t i n g to note that those authors who have used l i q u i d or gas targets have a l l measured a negative (although consistent with zero) branching r a t i o . The f a i l u r e to observe the charge exchange r e a c t i o n i n deuterium i s a r e s u l t of the i d e n t i c a l p a r i t y of the TT~*and ir° (PA51, CH55, TA51, BR51). In hydrogen the absorption process (ir p -*• n) i s forbidden by momentum-energy conservation and the allowed reactions are: 4. ir + p n + y r a d i a t i v e capture, 5. IT + p - * - n + i r ° charge exchange. In helium-3 the corresponding processes are: r a d i a t i v e capture, charge exchange, and the breakup r e a c t i o n s : 8. + 3He n + d > 9. TT~ + 3He ->- P + n n > 10. v~ + 3He -*• Y + n + d > 11. 1T~ + 3He ->- Y + P + n + n . In a l l cases the u capture proceeds p r i m a r i l y from an s state (LE62). In the case of hydrogen and helium-3 the nuclear spin and p a r i t y Is V*"* In the case of deuterium i t i s 1 +. For pseudoscalar pions we then have a t o t a l i n i t i a l spin and p a r i t y h~ i n the case of hydrogen and helium-3. 6. ir + 3He •> 3H + y 7. ir " + 3He -*• 3H + ir° EXPERIMENTER VALUE OF S VALUE OF R VALUE OF K Panofsky, Aaaodt 2.36 i 0.5 -0.007 * 0.020 - 0.023 t 0.072 and Hadley (PA51) Chlnovsky onij 1.5 t 0.8 -8.3 x lO"* -0.0034 t 0.0043 1 10.5 x 10"* Steinberger (CH34, (CP.55) Kuehner, M e r r i s o n 2.39 i 0.36 and Tornabene (KU39) Ryan (RV63) 3.16 1 0.10 KLoeppel (KL64) 2.89 : 0.09 -5.7 x 10"" -2.3 x 10" 3 ± 6.7 x 10"* ± 2.7 x 10" 3 P e t r u k l n and <10~ 3 « 4 x lO" 3* Prokoshkin (PE64) METHOD OF OBSERVATION Compare Y-ray y i e l d s from * d and IT p r e a c t i o n s ; gas t a r g e t ; p a i r spectrometer. S: Compare Y " r a y s i n g l e and neutron 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 t a r g e t ; l i q u i d s c i n t i l l a t o r f o r neutrons; l e a d 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 t e l e s c o p e f o r Y - r a y s . Compare Y"* ay 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 t a r g e t ; p a i r spectrometer. 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 t a r g e t ; p a i r spectrometer. Y-ray ( i n t e r n a l conversion) y i e l d s from it d r e a c t i o n s i n a bubble chaaber. Coincidence Y-ray y i e l d s of L1D1 lead g l a s s d e t e c t o r s . A W i t h no spin-flip the neutral pion w i l l be emitted in an s state. In the case of deuterium however the i n i t i a l state is 1 . In the f i n a l state the two neutrons, being identical 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 singlet s wave (*s) and a t r i p l e t p wave ( 3p). In the case of a 1 s di-neutron wave function the neutral pion must be emitted in a p state to conserve angular momentum. This results in a f i n a l state with positive parity and hence i t is forbidden. In the case of a 3p di-neutron wave function the neutral pion must also be emitted in a p state relative to the rdi-neutron in order to conserve parity. Since the reaction i s only slightly exothermic (Q = 1.10 MeV) the reaction rate is greatly retarded. Of course, from an hi s t o r i c a l perspective the above argument has been reversed and in fact i t was the non-observation of a significant charge -exchange rate i n deuterium which led co the conclusion that the TT and ir° must both be pseudoscalar particles (CH54, CH55). For scalar mesons the absorption reaction (1) is 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 ar-gument for the reaction p + p •+ d + i r + to determine the ratio 'S' for scalar pions absorbed from an atomic p-state and found a value which is a factor of 30 too small to account for the experimental value deter-mined by Panofsky et a l (PA51). On the other hand the calculated 'S' ratio for pseudoscalar pions, which can be captured from an s-wave atomic state, 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, parities and coupling theories. In the calculation mesons of both 5 positive and negative parity were considered. Negative mesons of spin 0 and 1 were considered; however the neutral meson was known to decay into 2 Y ~ r a y s and hence only spin zero neutral mesons had to be con-sidered. In the case of scalar ir the absorption reaction i s prohibited from the s-state and Tamor puts an upper limit of 4% on the number of mesons absorbed from atomic orbitals with higher angular momentum in essential agreement with Brueckner et a l . If the TT~ was a vector meson Tamor predicts a ratio S=55, again in distinct conflict with Panofsky's data, S = 2.36 ± 0.5. The calculated ratios for both pseudoscalar and pseudo-vector mesons were compatible with the experimental value of S. For a pseudovector IT he finds a negligible value of 'K' for either parity of i r 0 and so nothing can be inferred from Panofsky's data. For a pseudoscaiar TT the calculated value of 'K' is negligible in the case of pseudoscalar ir° but K s 0.1 for scalar ir°. Thus the experimental data of Panofsky, and later data of Chinowsky et a l (CH55) indicate that i f the u is a pseudoscalar particle then the ir° must also be a pseudo-scalar particle. We have now observed the charge exchange reaction in deuterium and have measured the ratios K and R to be K = (5.71 ± 0.82) x 10 _ l t R = (1.45 + 0.19) x 10~h This is in agreement with a recent theoretical result 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 is described in Chapter II and the details of the data analysis are reported in Chapter III. Chapter IV includes a brief discussion of the theore-tical values as well as the relationships of these values with other low energy pion data. The details of some calculations have been placed in appendices to avoid interrupting the flow of ideas. Appendix IV is 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 in 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 principle advantages over a singles technique. F i r s t , the dominant y-ray singles background from the radiative 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 n(d) 1.1 MeV) over the hydrogen charge TT° exchange TT°(Tii0(p) = 2.9 MeV). To understand these advantages i t i s useful to f i r s t consider the y -ray spectra which one would observe from the interaction of stopped ir i n hydrogen. As i n the case of deuterium the y ~ r a d i a t i o n arises from the radiative capture of the pions and also from the decay of the neutral pions formed in the charge exchange process. The radiative capture y-vay is mono-energetic at 129.4 MeV. The neutral pion from the charge exchange reaction i s likewise mono-energetic with T n ( P ) = 2.90 MeV. Since the reactions take place at rest there is no TT preferred direction and the distributions of these Y— rays and pions w i l l be isotropic. If one now observes the y-decay of the TT° in the lab frame, one sees a Y~ r ay with an energy which has been Doppler shifted by an amount which is determined by the ir° momentum In the lab frame. In Appendix 1.1 i t i s shown that p * 1 m^^ Y ° 2 E„ -P^osO (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 distribution of y-rays between the energy limits and Detector —CD Figure II.1.1 Tr°decay in the. laboratory rest frame determined by 0 = 0 and 0 = rr; i . e . * - 1 % 2 % H 2 E ± P or E . ^ ± ^TT H,L 2 (II.1.2) In the case of hydrogen = 54.9 MeV and E R = 83.0 MeV. Further, as i s shown in Appendix 1.2, for an isotropically distributed mono-chro-matic TT° the observed y-ray spectrum i s the box spectrum dlly dE Y 0 for E y < E L < < K for E L - E - Eg 0 for E > Eg . These features can be seen in the measured y-ray spectrum from stopped ir in hydrogen (figure II. 1.2). The deviations from a perfect rectangle and delta function are consistent with the f i n i t e resolution function of the spectrometer. In the case of stopped ir in deuterium the situation i s compli-cated by the 3-body f i n a l state (ir°nn and ynn). The Y-ray from the radiative capture channel is no longer mono-energetic but is 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 attractive low energy s-wave neutron-neutron interaction) than a simple phase-space calculation would predict there is s t i l l sufficient contribution in the medium energy range ('WO MeV) to obliterate the details 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 in the hydrogen contamination (nominally 0.3%) of the deuterium target. Thus the separation of the small deuterium TT° Y~"ray spectrum from the radiative capture background 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 in the 55 MeV to 83 MeV region only 3% is from charge exchange in deuterium. Fortunately both of these backgrounds can be eliminated with a coincidence arrangement. That the radiative capture Y-rays are elimi-nated with a coincidence technique is obvious. That the deuterium 10 Figure II.1.2 The Gamma Ray Spectrum of Stopped Pions in Hydrogen 3000U > (0 cd OS 2000H 1000-TT 4p->n+rr 20 Uy+Y • • •• — r 40 — r ~ 60 ' Tr~+p-+n+Y T I— 80 100 Gamma Ray Energy (MeV) .0 — I — 120 140 Figure II.1.3 Theoretical Gamma Ray Spectrum of Stopped Pions in Deuterium .a cd ,o o i-i Cu u Cd rH CO Pi rr +d->-2n+Y Gamma Ray Energy (MeV) 11 ir° decays may be separated from the hydrogen TT° decays i s a fortunate, consequence of the difference in Q-values for the two reactions and the good energy resolution of the Nal spectrometers. If one knows the energy of the ir° before i t decays and defines both the direction and energy of one of the decay Y -ray s then the direction of the TT° momentum and the direction of the second Y~ r ay a r e determined up to an angle about the direction of the f i r s t Y~ r ay- (figure II.1.4) The angle 0 between the TT° and the f i r s t y ~ r a y i s determined by (II.1.1) and using the conservation laws i t can be shown (Appendix 1.3) that the angle ij) between the two y ~ r a y directions i s given by the equation: Y TT Y The second Y -ray must l i e on the surface of a cone of half-angle (j) generated by a rotation about the axis of the f i r s t y-vay. It Is clear Figure II.1.4 The second y~ray cone 12 that for detectors which subtend small solid angles there w i l l only be a coincidence i f <t) = 0 or cos(J> = 1. From (II. 1.3) we find that this requirement means g ± 7>v N 2 We r e c a l l (from (II.1.2)) that these are the maximum and minimum limits of the rectangular Y-ray spectrum. That i s they correspond to the cases o where the 11 is travelling in the forward or backward directions 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 and 83.0 MeV from the hydrogen decays ( ^ ( p ) = 2.90 MeV). Since the deuterium T T ° ' S 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 limit 59.4 MeV and 76.7 MeV (T o(d) = 1.1 MeV). IT max Thus the Y-rays from the hydrogen I T 0 decays w i l l be separated in energy from the Y-rays which result from the deuterium ir° decays. Of course in any experiment i t is necessary to hav.e sources and detectors of f i n i t e size and the actual spectra obtained in each case must be determined by an integration over the detector surfaces and the target volume. This has the effect of allowing the hydrogen coincidence peaks to encroach somewhat on the deuterium coincidence spectrum. Detailed calculations of the coincident lineshapes and efficiencies have been performed for hydrogen and deuterium and are presented in chapter III of this work. The lineshapes shown in figure II.1.5 are the results for the geometry used i n the present experiment and demonstrate the energy separation of the deuterium and hydrogen events. It i s significant to note that the Relative Probability of Detection ET 14 net coincidence detection efficiency for deuterium TT° decays i s 6.7 times larger than the corresponding efficiency for hydrogen TT° decays. 15 2. The Spectrometers The experiment was made feasible by the av a i l a b i l i t y of two large sodium iodide spectrometers, TINA and MINA. TINA is a cylindrical block of Nal 45.7 cm. in diameter and 50.8 cm. in length. It i s viewed by seven matched 12 cm. photomultiplier tubes (RCA 4522). MINA is a c y l -inder 35 cm. in diameter and 35 cm. long and i s also viewed by four matched 12 cm. photomultiplier tubes (RCA 4522). The sodium iodide crystals are essentially 100% efficient and have sufficient resolving power to separate the coincident gamma rays originating from deuterium from those originating from hydrogen. Furthermore, the timing resolution of 2.0 nsec (FWHM) obtained with constant fraction discriminators makes i t possible to separate the y-rays from the large neutron background produced by the absorbtion and radiative 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 relation-ship with energy one would anticipate an energy resolution of about 6.4% in TINA and 7.9% in MINA in the 55 MeV - 83 MeV range. In fact the resolution obtained in this range in the present experiment was about 10% in TINA and considerably larger than 10% in MINA. The increase in width in 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 encoun-tered over a comparatively long period of time. The large width in MINA was found to be a result of insufficient voltages on the primary dynode stages of the photomultiplier tubes. The resolution in TINA was s t i l l TINA Time Spectrum (.n~d -»• anything) 8k •• • • • • • • 4k • • 1.8 nsec.i * _ FWHM — [ * " " . • • • o CO 05 • • • • c lb 20 3b CO 4-1 c MINA Time Spectrum (^ "d + anything) CJ • • 2k • • • • 2.2 nsec. . *i # . FWHM )| |< . • • • • io 26 3b Time (nsec.) Figure II.2.1 Time Response of the Nal Spectrometers* 400 TINA Energy Spectrum 144 MeV Electrons 300 • s 200 »|/ |< 4.4% FWHM • 0 100 > a • • «..••— CO • U § 200 ' 110 120 130 140 150 160 _ MINA Energy Spectrum . 125 MeV Electrons • 100 • ^ ji 5.3% FWHM < • • •A" I I i l l ! 110 120 130 140 150 160 Energy (MeV) . Figure II.2.2 High Energy Response of the Nal Spectrometers. 17 sufficient to separate the y-rays from the deuterium and hydrogen charge exchange. The resolution in MINA was not sufficient in this respect. However MINA was used to provide a test of the total energy, ^TINA + M^INA = ^ + a n c* t o ^ o r m a t w o dimensional histogram of the coincidence events (N, (EJJ^,^) ). 18 3. The Experimental Arrangement Figure II.3.1 shows the configuration of our apparatus i n the TRIUMF M-9 area. The pion beam was produced by a 1 uA beam of 500 MeV protons which strikes a 10 cm. beryllium target. The M-9 beam li n e was tuned for 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 consist of 76% pions, 18% electrons and 6% muons (Figure II.3.2). The incident beam i s defined by the plastic s c i n t i l l a t o r s labelled Cl and C2 and degraded i n a 2.9 cm. sheet of aluminum. 90,000 - •• • • • 10 cm. Beryllium 130 MeV/C - TT"(76%) Target 60,000- • CO 4 J *H 0 • e~ (18%) B P QJ & 30,000-• • • • • • • • p" (6%) . 4 1 • • • • • 1 10 i 20 Time of Flight (nsec.) 1 ' ' 30 Figure II.3.2 The beam constitution The target flask was a cylinder 15.5 ± 0.5 cm. long and 11.1 cm. in diameter giving 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 flask was wrapped in 10 layers of 6.4 x 10 Vm. aluminized mylar to reduce the heat load. The deuterium was made by electrolyzing deuterium oxide which had a •C3 shielding (lead or steel) Liquid Deuterium H-C2 Aluminum Degrader "C l TT"( 130MeV/c) — Co Plastic S c i n t i l l a t o r s CO 15 cm. x 15 cm. x 0.16 cm. C l 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 vo e Experimental Layout 20 0.2% (by atom) H 0 comtamination. The ionic content was enhanced by making a solution of solium peroxide with the deuterium oxide. The resulting gas was analysed by means of a mass spectrometer. The hydrogen contamination was found to be 0.3% to 0.4% although, due to calibration d i f f i c u l t i e s in the low mass region, the r e l i a b i l i t y of this measurement i s questionable. Thus the gas was taken to have a 0.3% hydrogen conta-minat ion. Beam particles which did not stop i n the target were vetoed by the plastic s c i n t i l l a t o r C3. With this arrangement we stopped 25% to 50% of the incident pion beam in the target li q u i d . The s c i n t i l l a t o r s in the pion beam were hidden from the Nal detectors with lead shielding to reduce the contribution of charge exchange in hydrogen to a minimum. The plastic s c i n t i l l a t o r s C4, C5, and C5' served as charged particle tags to distinquish 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). That i s an event was defined as C1'C2'C3'(T + M). The pion stop rate was typically 2 x 10k sec" 1 and the event rate was about 60 sec 1 . 21 4. Electronics and Data Collection Figure II.4.1 i s the schematic of the electronics. The outputs of the TINA and MINA phototube bases were brought separately into the counting room on low loss cables (FM-8) to allow the phototube gains to be balanced. The signals were then summed in active fan-in modules (LRS 127B, LRS 428) inside the counting room. The summed spectrometer signals were s p l i t in isolated output fanout modules (LRS 128) and pro-cessed for pulse height and timing information. A l l of the data was digitized in CAMAC modules interfaced through a standard CAMAC crate to a PDP-11-40 computer. The pulse height information was processed in two channels of the LRS octal ADC (model 2248). The timing information was processed in two channels of an LRS octal TDC (model 2228). Since we are interested in examining only "events" i t was necessary to start the TDC with the "event" signal. Because the timing quality of this signal was poor a third TDC channel was used to record the time of passage of the pion through C2 as a zero time reference. Hence the relevant times are t„,T... - t„„ and t„,.„, - t„„. The timing of a l l three stop signals TINA C2 MINA C2 was done with constant fraction discriminators (Ortec 463). The charged particle tags C4 and C5 + C5' were fed into CAMAC scalers and a charged particle was identified by a non-zero count in the appropriate scaler. 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 this varied considerably depending upon the type of event this time did not —3 —1 exceed 10 sec. Thus the dead time for an event rate of 60 sec was less than 6%. The software processing involved two stages. Fir s t the 2 2 C3 Beam Counters c , 2 c i C O N a l Analogue Signal I CAMAC Disc. LRS-621 , Disc, (coa frac) ORTEC 463 Comp ADC LRS 2248 Gate LRS-465 clip Logic Unit LRS 365 OR LRS429 Con. Con. Frac. Frac. Disc Disc. ORTEC ORTEC 463 463 I Fast Fast Amp. Amp LRS LRS 133 B 133B I Fan Out LRS 128 2 5 0 ns Fan Out LRS128 Fanln LRS 127 B clip 250 ns 1 " Fan In LRS 428 TINA Disc LRS-621 ± OR LRS429 Disc. LRS-621 Logic Unit LRS-365 EVENT Logic Unit LRS365 Visual iScaler C A M A C Scaler Eliot-SR 1608 Comp. C A M A C Stop stop stop TDC LRS 2228 Comp Disc. Cron. 151 CAMAC Disc. LRS 621 Scaler Eliot-SR 1608 Comp. Disc. LRS 621 MINA Fan In LRS 127 B C4 Anode Signals' Charge Identifiers COMPUTER CYCLE (PDP11/4Q) 1) LAM issued by start signal of TDC 2) INHIBIT all CAMAC modules 3) READ and RESET 4) Process data 5) ENABLE system 23 event was analysed and binned into time and energy histograms and second, the event was recorded on magnetic tape for a more complete off-l i n e analysis. In the on-line analysis seven one-dimensional 256 channel histo-grams were formed. These were the TINA/MINA time spectra, the TINA/MINA y-ray energy spectra, the time difference spectrum ( T ^ j ^ ~ TMINA^ a n (* the TINA/MINA coincident y-ray energy spectra. 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 later the data were transferred to IBM compatible 9 track magnetic tape. The Dectape has the disadvantage of being a rather inefficient medium for 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 selectively. Two modes of data recording were used. In the "singles" recording mode events were selected on the basis of their time of f l i g h t ; data for 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 in TINA were recorded. In the "coincidence" recording mode only coincident events were recorded but, with the excep-tion 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. A total of 2.2 x 10^ TT stops and 6,153,962 events were observed. Of these half were observed during singles recording runs and half during coincidence recording runs. 25 CHAPTER III THE DATA ANALYSIS 1 . The Off-line 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 particular i t was possible to apply gain and zero channel stabilization to the data during the re-analysis and thereby significantly improve the energy resolution. It was also possible to generate various histograms which were not available in the on-line analysis and to adjust the definition of y ~ r a y s a n Q neutrons on the basis of the time spectra. The time spectra of TINA and MINA for a single run are shown in figure III.1.1 with the time definition of the y-rays marked. As can be seen there was good separation between the neutrons and y-rays in TINA and the overlap may be considered to be negligible. In MINA the separation was not quite as good. This was part i a l l y due to the smaller distance from the target and part 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 7.2 nsec. between the y-rays and the high energy neutrons. 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 in terms of the low gain on the primary dynode stages of the photomultiplier tubes. This same problem also resulted in poor energy resolution in the MINA spectrometer. Even with the reduced separation in MINA the overlap of neutrons and y-rays was negligible. 26 .2K| o o> in c \ in | m Ul TINA Time Spec t r um ( run 12 ) e | ^ — 7.2 n s e c . — • neutrons > • k-1.8 nsec. a'-rays , ».....»«»» 0.0 5.0 10.0 15.0 time of f l ight t ( nsec. ) 4K o in c £ 2K c QJ > U J MINA Time Spec t rum ( run 1 ) N 6 1 n s e c—^| neutrons—> —$| k—2.3 nsec. • 1 • y-rays * • • • • * , » r 0.0 5.0 10.0 15.0 time of f l i gh t . t ( nsec. ) F i g u r e III.1.1 Typical Na l Time Spect ra 27 The energy spectra of the singles Y-rays in TINA and MINA are shown in figure III.1.2. In the TINA spectrum there is an obvious shoulder at 55 MeV due to the charge exchange box spectrum from the small hydrogen content of the target. As mentioned earlier deuterium charge exchange Y-rays comprise only 3% of the events in this region. The high energy edge of the box spectrum is hidden i n the rapidly rising radiative capture spectrum. In the MINA spectrum the energy resolution was not sufficient to see any indication 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 radiative capture centroids for individual runs. These values are plotted as a function of run number in figure III.1.3. The curve drawn on these points serves only to guide the eye. One would expect that the width of a peak would be increased by at least an amount comparable to the zero sh i f t . In TINA this was about 4 MeV and in MINA 7 MeV. This was a serious problem in view of the small energy separation between the hydrogen and deuterium charge exchange lineshapes (about 5 MeV). One of the beauties of an experiment which records raw data from events on magnetic tape is that i t i s possible to compensate for this type of problem in a dynamic way. In order to do this the energy para-meters were re-defined as: = ^T^^T ~ ^ T^ * Event s / MeV Ev e n t s / M eV o NJ C- CT> O O O Q O ^_ c m <D g - i <Q :^ 73 fD PJ C L DJ < O PJ o " a o c to i in —1» —» O m 3 •3 C H fD O - i a I D <D <^ C »-* fD — i C 3 £ 3 Events /MeV ro t o o Events / MeV 83 o , 0 - 0 _ © ' 0 ^ J I L. ^ o R u n N o " 12 ' o - 0 16 3-4" TINA Zero Point Instability ^ 2 <5 3d C/3 O P-, o u CU M 5' -1 -2 ft f \ • • , Run No. 12 / O 1 No<f6 MINA Zero Point Instability A A o 4 12 , 16 ^0-0-- - o Run No. TINA Centroid Instability > 3d w 3 -2 o M 4-1 C o _4 Figure III.1.3 Centroid and zero point instabilities in TINA and MINA J — 1 — • — t V .1, i t 1 i o o 12 Run No. MINA Centroid Instability 16 30 where F^ , and are the energy data for TINA and MINA recorded for a particular event. and Z^ are zero point adjustments associated with TINA and MINA respectively and and G^ are adjustable gain factors. Dynamic gain stabilization was achieved by continuously modifying the gain and zero parameters in such a way as to correct for the gain and zero shifts in the original data. It was necessary to define the cen-troid energies (C°) and zero point energies (Z°) which were to be main-tained. The choice of these values i s entirely arbitrary. However once they have been determined 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 fixed 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 = 7.62 (channel number) , The fixed centroid energies were taken to be the average centroid position for run / / l corrected for the i n i t i a l zero parameters (Z^ and Z^); 31 a 132.27 (channel number) , ** 173.54 (channel number) . With these values for 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 CT = CT * °M = CM* The gain parameters were defined to be 5 1 GT = C T ; . °M = C M Hence the i n i t i a l value of the gain parameters was 1.0. A y~ r ay 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 at half-maximum of the fixed centroid energy. For such events the centroid parameter was adjusted by an amount AC where AC = k c(E'- 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 available for one spectrometer when the event was the result of a response in the opposite spectrometer. Because there was no "stop" signal for such events they could be recognized by an overflow in the corresponding time data. For zero energy events the zero parameter was adjusted by an amount AZ, AZ = k z(E - Z) , where i s a small ( <<1) constant. Of course the adjustments were applied after the event i t s e l f had been analysed to avoid any auto-biasing effects. Suitable values for k c and k z were found to be 10 and 10 4 respectively. The variation of the parameters was monitored as the stabilized analysis was done. The results are plotted in figure III.1.4. The indication from these graphs is that for both TINA and MINA most of the observed fluctuation was due to the zero s h i f t . The effectiveness of the stabilization may be measured by comparing the centroid positions for individual runs with and without the stabilization. These centroid positions have been plotted in figure III.1.5. In TINA the improvement i s seen to be a factor of 3. In MINA the improvement is more than a factor of 2. The residual differences in centroid position among the runs were corrected by stretching a l l of the spectra to give them a common centroid. The same stabilization and stretching procedure was applied in the case of the coincidence spectra. The usefulness of this procedure is clearly seen in the improvement of the TINA coincidence spectrum (figure III.1.6). The MINA coincidence spectrum is also shown 33 here and i t is seen that the separation between the hydrogen and deu-terium charge exchange events i s s t i l l very poor. Centroid Correction (Channels) Zero Point Correction (Channels) l—• o • ro CO to to o >— to to to to •*1 H -00 C n n o CD 3 r t H O P 3 CU N ro i-i o •xl o H» 3 O O H H fl> O r t H -O 3 ca H-3 > CO 3 CU 2 n n> 3 r t >i O O O H H ft) O O 3 I H M c o N fD l-l O T) O H» 3 r t O o H H ID O It H» O 3 A M 15 Centroid Correction (Channels) i I—' O f O I • 1 1 • • • • • • n • n> • 3 * r t **** • ro • • • • CU * • • • C? OO - • • i-l • ro * o • r t • * H - — o * 3 I-* to - • • • MIN Run • > 53 • • O • • • t *-* • • • • ts CD i i o ••d o n o l i H m o rt H» O 3 > lo COh c ? 3 S3 O Zero Point Correction (Channels) I—• I—• f tr-• V O O i — to 1 > — i 1 r-to • • * * — • — 11 - • *—• —• • • o • • • • • • pa • c 3 • S3 - • • O • vC 35 TINA centroid shift O - Before Stabilization > \ ' \ s- After Stabilization > 2 h / \ & 1 ,/ \ \ s \ Run No. + V / \ V / . \ 10 14 16 I I \ / • V ' o u u c o-4 > MINA centroid shift o + — . ' 1 16 S nl ' ,J J 4 - i - J U It 1 I 1 l i I L ~ 0 | 2 T i p 1 13 8 10 12 14 Run No. /j O Before Stabilization w + \ / ' + After Stabilization -2 - v+1 8 <> *J / g / u o -4h / / V o Figure III.1.5 Effect of Gain and Zero Stabilization 36 Y - Y Coincidences TINA (Not Stabilized) 83 MeV Gamma-Ray Energy (MeV) 20 c CD 10 X L 55 MeV Y - Y Coincidences TINA (Stabilized) 83 MeV ID, 50 75 Gamma-Ray Energy (MeV) 100 20 j CO •u g 101 55 MeV Y - Y Coincidences MINA (Stabilized) 83 MeV n „ 1 np-in Gamma-Ray Energy (MeV) Figure III.1.6 Gamma-Ray Coincidence Spectra 37 2. Fitting the Histograms Two procedures were used to determine the branching ratio for charge exchange in 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 relative efficiencies for observing these processes. ? T T D NY( T r~ d Ynn) S + 1 (III.2.1) where: is the number of coincidence y-rays due to charge exchange in deuterium is the efficiency for observing deuterium charge exchange coincidence y-rays £ i s the efficiency for observing singles y-rays s Ny(Tr~d -> ynn) is the number of radiative capture y-rays observed. The second approach was the comparison of the relative number of charge exchange events due to deuterium and due to hydrogen. The charge exchange branching ratio in deuterium may then be calculated from the known charge exchange branching ratio in hydrogen i f the effective concentration of hydrogen in the target i s known. ^ - V H ^ F 1 < M - 2 - 2 > TTD TT 38 where is the charge exchange branching ratio in hydrogen C. H i s the effective concentration of hydrogen in the target H i s the number of coincidence y-rays due to charge exchange TI i n hydrogen 'ITH i s the efficiency for observing hydrogen charge exchange coincidence y-rays. The effective concentration of hydrogen in 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 ex-pected to be reduced in the liquid state due to d i s t i l l a t i o n . (The boiling point of hydrogen i s about 3.2K. lower than that of deuterium at normal pressures (MC64). A second possible effect was the preference of stopped IT captured in molecular H - D orbits to form TT""-D atomic states rather than TT - H atomic states. The effect of preferential TT capture on high Z nuclei in hydrogenous compounds is well known (KR68, PE69, P073). In the case of deuterium hydride ( H - D ) the TT bound states are sli g h t l y deeper (-6%) on the deuteron than on the proton. These effects which reduce both the effective concentration of hydrogen in the liquid and the actual concentration in the liquid from the concentration in the gas are discussed more f u l l y in Appendix I V . It was possible to directly measure the effective hydrogen concentration in the target liquid 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 for C„ from the y _ ray singles n spectrum. 39 A. The Coincidence Spectrum Even with stabilization to correct the gain and zero shifts 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 ratio in hydrogen (SP77). The geometry of the two experiments was identical except for a 3% difference in the distance of the TINA collimator 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 in figure III.2.1. It 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 75 100 Gamma Ray Energy (MeV) Figure III.2.1 - F i t of the Hydrogen Peaks. 40 was Important to not consider the events due to deuterium charge ex-change. Hence the central region was omitted from the f i t t i n g procedure and only the regions shown as (Li,L 2) and (1,3,1^) on figure III.2.1 were considered. To test the effect of any deuterium events in these regions the limits L 2 and L 3 were varied over a wide range of energies and the effect 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 negligible variation of the amplitude with varia-tions of the exterior l i m i t s . The slight variations in the amplitude with the variation of the interior limits, L 2 and.1,3, are seen to be well within the error limits set for the amplitude. We conclude that the dependence of the f i t upon these parameters and upon any deuterium events in the regions defined by these parameters i s negligible. In the region between the limits and there were 182.0 ± 13.5 events -of which we determine 22.2 ± 2.3 were due to charge exchange in hydrogen. This leaves 159.8 ± 13.7 events in this region due to charge exchange in deuterium. 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 result was that there are 8.2 ± 14.5 deuterium events under the hydrogen peaks. Thus the total number of deuterium charge exchange events in the coincidence spectrum was 168.0 ± 20. The total number of events from hydrogen charge exchange was determined directly 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. 41 220 200 o o w o o 8 180] •H l-l •H Range of F i t (20.L2) (59,75) O O S t a t i s t i c a l Error oo o u 32 36 40 44 Limit of F i t - L2 (channel number) cu w 220K a CU o M CU 2001 o o o o Range of F i t (20,39) (L3.75) 180 St a t i s t i c a l Error 56 60 64 68 72 Limit of F i t - L3 (channel number) Figure III.2.2 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 in the region between 20 MeV and 95 MeV with a smooth estimate of the background and Ynn t a i l (f(E)) and an empirical charge exchange y-ray spectrum from hydrogen (g(E)). The hydrogen charge exchange spectrum used i s shown in figure II.1.2. The experimental configuration of the hydrogen target and the TINA spectrometer was the same for the present experiment except for a 3% difference in 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) = a e b E + I c , E _ i + dg(E). i=0 The f i r s t term was included to approximate the radiative capture y-ray t a i l . The second term was included to f i t the rising low energy back-ground. The last term represents the contribution of charge exchange events to the spectrum in this region. A grid search technique for the parameter b was combined with a linear least 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 parameter 'd' was tested for i t s dependence upon the limits of the f i t t i n g range and upon different choices of f(E). The value of 'd' was found to be relatively 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 limit (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 43 800 400 co B CD ' o * o a.. Functions Used to F i t the Data P 6 O 'O o *o © o o 6 a o o i". Svb'o 0 0 0 ° o o o o Data Background F i t to Data CEX Spectrum f i t to data 50 100 Gamma-Ray Energy (MeV) 40 b. Difference of Data and f i t 20 > CD CO u C cu -20l O o o o o o o o o 0 ° ° o o CP o o o o © o o o o o O o ° o ° 0 o o % o 0 o Figure III.2.3 Fitting the y-ray Singles Spectrum 44 4k 3k co «s 2k o 0) 60 c .a CD 60 H J8 O M a) .o I *o CJ 4-1 cd M 3k 4-> (0 w 2k I 1 III HI I « 1 Lower Limit = 20 90 95 i4- TD3" Upper Limit of F i t (channel No.) T 1 iii 111 Upper Limit = 95 value chosen 15 20 25 30 35 Lower Limit of F i t (Channel No.) Figure III.2.4 Estimated Number of TT Singles y-rays as a Function of the range of f i t 45 upon the parameter 'd'. Inclusion of positive powers of energy in the polynomial was found to give generally unsatisfactory f i t s to the data as evidenced by significant structure in 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 largely on the basis of the variation of the parameter with the low energy limit of the f i t t i n g range. The effective concentration of hydrogen (i.e. the relative 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° V T T P -»• Tr°n) 1 H ~ 2 N(TT — d -v ynn) R R(S + 1) where is 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) is the number of y-rays from ir - radiative capture on deuterium. The charge exchange y-rays in the singles spectrum are due to both charge exchange in hydrogen and charge exchange in deuterium. The number of singles y-rays due to charge exchange in hydrogen may be extracted by referring to the coincident y-ray spectrum (figure III.2.1) from which the ratio 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 , - , o . . NV(TT p -»• Tr°n) + Ny(TT d Tr°nn) N (TT P TT n) = -3f c F + t T ' -Using the values for the efficiencies presented in chapter III.3 and the f i n a l values of D and H presented in chapter III.5 we find TT TT F - 0.132 ± 0.020, N (ir~p Tr°n) = 2911 ± 585, C„ = 0.00145 ± 0.00029 . n Thus the effective 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 Efficiency Because the reactions take place at rest the y-rays are isotro-pi c a l l y distributed i n the lab frame. The singles efficiency i s then simply the solid angle factor for the collimator of the spectrometer. For TINA £ t(T) = (3.29 ± 0.03) x 10~3 s and for MINA K (M) = (2.11 ± 0.02) x 10~3 The efficiency calculation for coincidence y-rays i s somewhat more involved due to the correlation between the direction of the two y-rays. It i s shown in appendix 1.3 that for a TT° of total energy E^ and momen-tum 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~ ^ )" ? 1 (IH.3.1) Y ir Y whose axis i s the direction of the f i r s t Y _ r ay- For a fixed direction and energy of the f i r s t y-ray the efficiency for 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 probability for detecting a coincidence can be shown (appendix II) to be d P c = / c o s i - c c s ^ c o s ^ W d N , ^ d E d T ( I Ii.3.2) c 2TT^ V sinl})^ 5 1 1 1^! / dE^ dT^ cj y TT where fa is determined by (III.3.1), fa a n d ^12 a r e determined by the dN geometry (figure III.3.1), -^f- is the charge exchange singles Y - r a y (box) y spectrum (AI.2.1) and -rrr11- i-s t n e v° energy spectrum. In the case of D TTT charge exchange in hydrogen the TT° i s monochromatic. Hence (P) = cT(To - T W ) 49 where T Q = 2.90 MeV is the kinetic energy of the TT° . In the case of deuterium the TT° has a continuous energy spectrum from 0 MeV to TMAX " 1 , 1 M e V 8 i v e n by (Appendix III) d I* \^ MAX J \^ \MXJ The net probability of detecting a coincidence y-ray of energy Ey, i.e. the coincidence lineshape is just the 3-fold integration /-(£ ) . 4£c . f f -Los" 1/cos(j) 2 - cosfacosfoAd^dN^ d dE Y ' T t JK c 2 ? C O S I sin^zsinf))! 1 dE^ d T c f l w (III.3.3) The net probability for detecting a coincidence y-ray of any energy is the further integration, 5 c(x = 0,y = 0,z = 0) = 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 is 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 elliptic 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 will 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 this factor as a correction. In the present case the maximum deviation that we need consider i s determined by the size of o the target flask and was -6 . Hence the error that we make in consider-ing off axis points i s less than 1%. The total coincidence efficiency for the target then i s Kc - / V 5 c ( x , y . « ) ^ - - 4 V (III.3.4) where i s the density of stopped TT at the target position (x,y,z). In the case of the deuterium coincidence efficiency equation (III.3.4) i s seen to be effectively a 7-fold integration. In the case of the hydrogen coincidence efficiency 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 calculation economical the functions 5„(a.d = 0) 51 and 5C(<* = 0,d) were evaluated at several values of a and d (where a and d are the angular deviation and decay collimator distance pictured in figure III.3.2; note that because of the symmetry around the collimator axis we may express £c(x,y,z) as ? c(a,d)). These functions are shown plotted in figure III.3.3 and III.3.4. The functions S c(a = 0,d) are well approximated by straight lines in this region of d. In fact 5 c(a = 0,d) w i l l have a maximum value when the solid angles of the two collimators are equal. This corresponds to a target position 10.4 cm. closer to MINA than was used in the present case, and we see that the function is indeed rising for positions closer to MINA. We assumed that this relationship would also be valid for small values of ot $ 0 and hence . d ) = gfa,0) g(0 , d ) e t a ,a; £(o,o) The function £. (a = 0,d) was evaluated from the straight line f i t to the points plotted in figure III.3.3. The function £(a,d = 0) was obtained by means of linear interpolation between the data points plotted in figure III.3.4. The TT stopping distribution, w a s presumed to be a function of 'y' only (the beam direction). It was taken to be a Gaussian d i s -tribution whose centroid was determined by the range of 51 MeV ir in the s c i n t i l l a t o r s , degrader, vacuum jacket, target windows and target liquid (ME74). The width of the Gaussian was determined by assuming that 49% of the "stopped pions" actually stop in the target l i q u i d . The resulting distribution i s shown in figure III.3.5. The fact that this estimate for the stopping distribution is very crude need not be of great concern as i t w i l l be shown that the shape of the stopping 52 1.72 1.68L-Hydrogen Coincidence Efficiency (x I0~k) 1.64 -6 ~3r 0 2 4* 6 Displacement (cm.) <— TINA 1.34 1.33/f / / / / / 1.32' -6 Deuterium Coinci- / MINA—> dence Efficiency/ ( x l O 3) - 2 — Q ^ r Displacement (cm.) Figure III.3.3 Coincidence Efficiency as a function of displacement along collimator axis 53 Hydrogen Coincidence Efficiency I (Relative to Singles) 4.2% ^ 4.0% h ^ir -O _L_ 6^ -6 0° Angular Displacement 32% o „-o-d—o -^ ^ 28% / o 26% 24% Deuterium Coincidence Efficiency (Relative to Singles) ' 0 _L -6° -4o -20 0° 20 Angular Displacement 40 60 Figure III.3.4 Coincidence Efficiency as a function of angular displace-ment from collimator axis 54 distribution (at least for wide distributions) has l i t t l e effect on the total efficiency. The chosen distribution serves the purpose of this demonstration. The only remaining consideration in evaluating equation (III.3.3) to determine the total efficiency i s the volume of integration to be used. Again, we found that we could tolerate a rather crude descrip-tion of the active volume since the total efficiency was rather insen-s i t i v e to this 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 primarily due to multiple scattering in the degrader and i s well represented by a Gaussian distribution with a 6° standard deviation (SP77). For the purpose of this calculation the beam was con-sidered to uniformly f i l l a cone of 6° half-angle. The radius of this cone exceeded the target radius at approximately the collimator axis. The error in the calculated efficiency 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 size of C2 and a f u l l y illuminated target volume. The errors in the efficiency calculation are summarized in the following table. The error due to f i n i t e binning in the numerical integration i s large in the case of deuterium because of the integration over the pion energy distribution. In general the larger errors In the deuterium calculation are due to the more extreme variation of the function £ c(a,d) with a. The error due to the stopping distribution was deter-mined by considering the difference in efficiency using the Gaussian distribution for and using a constant stopping distribution. The values determined for the coincidence efficiencies are 55 'TTH (1.74 ± 0.026) x IO - 4 'TTD (1.166 ± 0.042) x IO"3. The coincidence lineshapes (equation (III.3.3)) have also been evaluated for deuterium and hydrogen. In the case of hydrogen the integration over the target volume has been performed. In the case of deuterium o the lineshape i s for a TT decay at the centre of the target. These lineshapes have been plotted in figure II.1.5. ERROR IN EFFICIENCY CALCULATION DUE TO: 1. Numerical Integration 2. Non-circular intersection of cones 3. Interpolation of Jj(cc,d) 4. Beam volume 5. Target length 6. Stopping distribution DEUTERIUM EFFICIENCY 2.2% 0.5% 1.5% 1.7% 1.6% 0.7% HYDROGEN EFFICIENCY 0.1% 0.5% 1.0% 0.9% 0.3% 0.1% TOTAL ERRORS 3.6% 1.5% 56 4. Corrections for In-Flight Interactions The probability that a ir which enters the target with kinetic energy T Q w i l l interact in f l i g h t in a volume element Adx at position 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 for the interaction. I—x—H Figure III.4.1 Liquid Deuterium Target The probability for observing the interaction i s dP o(T o,x) = po(T)5(T,x)dx and the total probability for observing an interaction while in f l i g h t i s 57 x P tr S m / rpo(T)5(T,x)dx (III.4.1) o' x=0 where x r i s the range of the pion in the target. To determine the function T(x) we note that for pions of kinetic energy less than -50 MeV the stopping power in deuterium (ME74, TR76) is approximated to about 10% by With T measured in MeV dT ,,,-b (III.4.2) - 3 — = aT dx a = 10.87 b = 0.805. Integrating equation (III.4.2) gives T(x) = (T? - k x ) 1 / c where c = 1.805 and k = 19.63. Then 58 A. Charge Exchange In appendix III i t i s shown that for small energies a ( T ) = d (III.4.3) The constant 'd' was determined from the data of Rogers and Lederman (R057) at 85 MeV. For T and Q measured i n MeV d = 2.05 x 10~6mb . The function S C(T) has been evaluated at energies T = 0.0 MeV, 8.9 MeV and 18.9 MeV. It i s found that this data f i t s the function «c < T ) - ! ¥ t ( i i i - 4 - 4 ) to better than 10%. Substituting (III.4.2), (III.4.3) and (III.4.4) into (III.4.1) and performing the integration gives P O(T Q) = 1.56 x I O " " 1 ^ - 3 0 5 jl.O + 3.92^-y 5.60 ( j ~ J + 3-31(TJ (III.4.5) To make use of this relationship we must know the distribution of the pion kinetic energies at the face of the target. We took the original momentum distribution to be Gaussian with a centroid at 130 MeV/c and a FWHM of 15%. (The M9 momentum byte has been measured at T - = 30 MeV for a 10 cm. Be target and 10 cm. horizontal s l i t s as 15% FWHM (BR76)). 59 This momentum distribution was divided into equal bins and for each bin the energy at the target face and the number of in-flight charge exchange Y-rays expected from the bin were determined. The momentum distribution and the expected number of in - f l i g h t charge exchange y-rays for each bin are plotted in figure III.4.2. Also shown on the abscissa are the target entry energies for the given momentum bin. There are a total of 17.2 coincidence Y-rays expected from charge exchange in 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 in 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 in the previous section and is plotted in figure III.4.3. Lineshapes for higher energy interactions w i l l have an even larger energy spread and a greater dip in the central region. Thus the use of the 18.9 MeV lineshape to estimate the contribution from in f l i g h t events under the hydrogen and deuterium peaks i s expected to give a result that i s somewhat too large. The contribution under the hydrogen peaks is expected to be less 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 realized that many of the approximations used to arrive at these figures are rather crude. For example, the stopping power, as given by equation (III.4.2) is expected to be good to only about 10% in the 30-50 MeV region and i t i s this energy region which largely determines the in- f l i g h t contribution. Furthermore the extrapolation of the low energy cross section relationship (equation III.4.3) to energies as large as 85 MeV is questionable. Despite these shortcomings the calculation does point out that the in f l i g h t charge exchange contribution 60 6 0 c o o rt u cu cu 6 0 C C H c cu cu n o 5 CJ o «4-l > o cu cu m J u cu 4.0 3.0 2.0 1.0 Target Entry Energy (MeV) 7.5 18.6 30.2 40.1 48.5 / Momentum D i s t r i -bution 2.67 x 10 9 TT" In Flight Charge Exchange Interactions / \ \ / i / o A \ s \ no T 130 150 Pion Momentum (MeV/c) Figure III.4.2 ir Momentum Byte and In f l i g h t Charge Exchange G cu u cu a 20 \-10 Measured Coincidence Spectrum (TINA) Calculated In Flight \ Lineshape (18.9 MeV) 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 in f l i g h t charge exchange coincidence y-rays to have energies above the high energy hydrogen coincidence peak. The previous calculation suggests that we should find 5 y-ray events in this region. In fact we see 11 events. This discrepancy may be interpreted either as an indication that our calculation i s low by a factor of two or that the 18.9 MeV lineshape i s not a good description of the "average" lineshape for the energy region above 18.9 MeV. The former interpretation w i l l increase our corrections by a factor of two; the latter could decrease them consider-ably. The correct interpretation of course is some combination of these two. (It is clear that the 18.9 MeV lineshape has too l i t t l e weight in 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 this region, so this alone can not account for the discrepancy). In any case the size of the correction i s less than the error in the number of events in either the hydrogen or the deuterium coincidence spectrum and so i t i s expedient to avoid these uncertainties simply by applying large error limits to the corrections. These are taken as 2.0 ± 2.0 in the case of the correction to the deuterium coincidence counts and 7.8 ± 7.8 in the case of the correction to the hydrogen coincidence counts. 62 B. Radiative Capture The determination of the correction due to in f l i g h t radiative capture i s more straight forward. We again make use of equation (III.4.1) to determine the probability of observation. In this case however the efficiency i s just the singles efficiency and i s not a function of energy. Furthermore, the cross-section now is given by 0 < I ) . a ( ^ i 2 and in this case Q = 136 MeV. The effect of this i s a rather slow variation of the cross section with energy in the range of interest. 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 rising for energies below 40 MeV care was taken to avoid integrating beyond the end of the target. The net result of the calculation was an expected contribution of 8100 ± 4400 radiative capture events from ir In f l i g h t . This re-presents a 1% correction to the total number of radiative capture events. 63 5. The Final Analysis The following table (Table III.5.1) summarizes the measured and calculated quantities which were used to determine the charge exchange branching ratio from our data. Using the ratio of the charge exchange rate to the radiative capture rate (equation III.2.1)) we find Using the ratio of the charge exchange rate i n hydrogen to the charge exchange rate in deuterium (equation (III.2.2)) we find In determining the errors assigned to this second number i t was necessary to consider the negative correlation between the errors in D and H TT TT and between the errors in £ „ and £ ~. The correlation in the D and ^TTH ^TTD TT H terms arises because the sum of the two terms i s constrained. The correlation in the £ T T and £ terms i s a result of the opposite cur-TTH ^TTD vatures of the functions £ „(a) and £ „(<x) (figure III.3.4). Furthermore TT h 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 in the weighting. In detail = (1.46 ± 0.19) x 10" R2 = R„C = (1.16 ± 0.33) x 10" k 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 Number of Deuterium Radiative Capture Events. Corrected for in - f l i g h t radiative capture, contributions from deuterium and hydrogen charge exchange and hydrogen radiative capture. C„ - (1.45 ± 0.29) x 10~3 Effective concentration of hydrogen. H K - (3.29 ± 0.03) x 10~3 TINA singles y-ray efficiency, s £ _ = (1.166 ± 0.042) x 10~3 Efficiency for observing deuterium charge TTD exchange Y~ r ays. 5 „ = (1.740 ± 0.026) x 10-1* Efficiency for observing hydrogen charge exchange y-rays. S = 2.97 ± 0.17 Ratio of ir" absorption and radiative capture rates i n deuterium (world average (excluding CH54)). R^ = 0.607 ± 0.002 Charge Exchange branching ratio in hydrogen (Most recent result (SP77)). 65 R TTD 1 W! + W2 Wl _ Es ] 1 N^ O d •> ynn)ys + 1 + W 2 RH CH H TfH o r where Fx = *° Nydr'd -> ynn) S + 1 F* - VH I T 1 TT and the weighting factors Wj_ and W2 are assigned in the usual fashion. That i s Wl = dpjr and W2 = ^ . When written i n this format i t i s clear that the relatively large errors in C H and H w i l l result in the second method being weighted very l i g h t l y compared with the f i r s t . In fact the average value, which I quote as the f i n a l result i s R = 1.45 x 10~^± 0.19 x 10"** • From this we determine 66 K *> 5.76 x IO"1* ± 0.71 x 10~4 where the relative error is slightly decreased.because the value of ' S' i s eliminated from the calculation. 67 CHAPTER IV DISCUSSION 1. The Charge Exchange Branching Ratio The branching ratio for pion charge exchange in deuterium ties in with other low energy pion results. In particular, in a recent calculation Beder (MA77) has used the impulse approximation to relate the charge exchange rate in deuterium to the hydrogen charge exchange scattering length. In a recent analysis of T r +d -*• pp cross section data Spuller (SP75) presents a figure which demonstrates the relation-ships between the low energy pion data. This is shown in figure IV.1.1 with the relationship of the deuterium charge exchange rate included. Following Beder's calculation (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> is given in terms of the |i> -»• |f> S-matrix as w(f) = / dP f [ U r O W p f - P ± ) ] ~ 1 |<f !S| ±>|2 (IV.1.1) where dp^ is 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 in terms of a complete set of tf~d plane wave states. In the IT d rest frame <f|s|i> «\, J <f |s|Tr(q)d(-q)><Tr(q)d(-q)|i(atomic)>d 3q 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 in r space. This transform 68 trCyp-Tr+n) f R c r i y n — -rr p ) |D.B. c r ( 7 r ~ p — n y ) c j ( 7 r ~ p — n y ) IT ts \X) \ ti v j - — : n : / }j EL . 2. EL*. cr(7r~d — nn) fc . i . criir^d — pp) Jim cr(pp— T T d) K crt/T^p — 7T~p) fC.L cr(7r~p— n7T°) ^ EL*. 2. E. 0j(7T*"p — n7T°) IA. -» C U ( T T d — n n 7 T ° ) 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 ratios (experimental or calculated) for the processes as no ted., Figure- IV.1.1 Relations Between Low Energy Pion Reactions 69 i s n e g l i g i b l e f o r |q| >> — where i n the present case, a i s the TT Bohr radius. The f i r s t term i s the conjugate f o u r i e r transform of a function i n r space which has a range of approximately the nuclear p o t e n t i a l and hence w i l l be nearly constant over the short range (q - 0) of the second term. Thus we may s i m p l i f y expression IV.1.1 by removing the f i r s t term from the i n t e g r a l . <f|s|i> * <f |s|ir(0)d(0)> J<Tr~(q)d (-q) | i (atomic )>d 3q Using t h i s approximation i n "equation IV.1,1 gives u «u / dp £ [ ( 2 i r ) V<P f - pJJ^Hf |s|Tr(0)d(0)>|2 Q<7r~(q)d(-q)|i(atomic)>d ; = q^ O ^a^"d f) ^/ <Tr~(q)d(-q) | i(atomic)>d 3qj where a i s the free 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 i n i t i a l plane wave TT d states to i n t e r a c t . A l l of the d e t a i l s of the i n i t i a l atomic state are wrapped up i n the i n t e g r a l . Since we are con-s i d e r i n g the r a t i o of t r a n s i t i o n rates to d i f f e r e n t f i n a l states the e f f e c t s of the coulomb d i s t o r t i o n of the i n i t i a l atomic state do not enter. In p a r t i c u l a r , o % ii™ qa(Tr~d -> Tr°nn) W(TT d -» TT nn) _ q->0 . (IV.1.2) w(ir~d -> nn) i i m ( i r~ d + j q+0 M The charge exchange c r o s s - s e c t i o n may be expressed i n terms of the S-matrix. 70 ~ qo(ir d ir nn) » Jd^dj^^ J 7 2 i r ) ^ ( P £ " JgJ ^ n n l s l i r - d H * - * (IV.1.3) where q', k j , k 2 are the f i n a l TT° and two neutron momenta respectively. The S-matrix i s anti-symmetrized in kj and k 2. The capture from the i n i t i a l state i s s wave so parity conservation and angular momentum conservation results in 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 ^.|S| i s equivalent to |S|2 spins ignoring spins. The f i n a l factor of h avoids double counting the identical nucleons. We now define a reduced scattering matrix S by ex p l i c i t l y including the energy conservation delta function; v i z . ' <f |s~|i> = £2TTcf(Ef - E ± ) j" 1<f js|i>. We expand the reduced matrix element in terms of a complete set of n and p plane wave states. <Tr°nn|s"|Tf~d> » (2TT)~ 6Jd 3td 3's <TT°k2k1 | s"|ir~n(t)p(s)>3tt(t)p (s) |d> - <Tr°k2ki |?|ir"n(t)p(s)><n(t)p(s) |d> In the impulse approximation the i n i t i a l neutron takes only a spectator role. Hence ^ t i ^ n & J l S ^ - n i t ) ? ^ = (2TT) 3cf 3(^i " t)<TT 0n(k 2)|s; N|TT"p(1)> C ( 2 I T ) 3 J 3 ( £ I - t ) c f 3 ( q " + ^ - q -"s) 8* f (u~p -> ix°n) " J 2 E (TTU ) • 2 E N (E2) ' 2 E C O • 2 E P (s ~~ w h e r e t a n d s a r e t h e m o m e n t a o f t h e n e u t r o n a n d p r o t o n p l a n e w a v e s t a t e s , V J ^ i s t h e t o t a l e n e r g y i n t h e p i o n - n u c l e o n c e n t e r o f m a s s a n d f (IT p -*• Tr ° n ) i s t h e h y d r o g e n c h a r g e - e x c h a n g e s c a t t e r i n g a m p l i t u d e . I n t h e l i m i t q -> 0 W ^ = M + u ( t h e sum o f t h e n u c l e o n a n d p i o n m a s s e s ) a n d f (TT p ->• Tr°n ) i s r e p l a c e d b y t h e h y d r o g e n c h a r g e e x c h a n g e s c a t t e r i n g l e n g t h , a (IT p -»• i r ° n ) W i t h t h e s e c o n s i d e r a t i o n s we d e t e r m i n e f r o m e q u a t i o n (IV.1.3) t h a t " l in i j qa(Tr~d -»• T r ° n n ) a 2 ( u - p - T T % ) J d q ' k ( q ' ) q'2| F(q')|2 . (IV.1.4) VM where F(q) = J j j ( k r ) j 1 (^|) <t^(r)r2dr, k-fMCQ-i^-)} 5* , Q = r e a c t i o n Q - v a l u e (1.1 M e V ) , u = n e u t r a l p i o n r e d u c e d m a s s , (> d = u n i t n o r m a l i z e d d e u t e r o n w a v e f u n c t i o n . (*e<a + B ) Vs e - " r - e'* >2ir(o - B ) z r a - 1 = 4.3 fm B = 7a 72 A correction may be made for double scattering. This i s of order |a(Tr~n TT~n) + a(Tr°n T r°n)| a(u~p •> Tr°n)<<|) r ( 1 = 1) | —^—| ^> and contributes a -1% correction to the impulse approximation amplitude. Using the most recent result for the hydrogen charge exchange scattering length (a(Tr~p •+ Tr°n) = 0.175 fm (NA76)) in equation (IV. 1.4) we find l'5jJ-qa0r"d -»• ir 0nn) = 0.0358 MeV mb (IV. 1.5) The denominator of equation (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 latter cross-section has been analysed in detail by Spuller and Measday (SP75) who f i t Coulomb corrected data to a function of the form a( T r +d + pp) - | ^ (an + 0(n 2) J where P i s the proton center of mass momentum in units of M c,a is a f i t t i n g parameter and n = q/y. They find the preferred range for a to be 0.25 mb - a - 0.29 mb. In the limit that q -> 0 this corresponds to 162 MeV-mb - lim qa(Tr~d nn) - 187 MeV mb. (IV.1.6) 73 The charge exchange branching ratio i s given by ^ = to (TT d -> Tr°nn) # (O(TT d ->• nn) . io(Tr~d -+ nn) co(Tf~d ->• a l l ) The f i r s t factor i s determined by equation (TV.1.2) and the subsequent results (IV.1.5) and (IV.1.6). The second factor i s determined from empirical value for S (2.97 ± 0.17). Thus the theoretical calculation predicts a branching ratio 1.39 x 10~k - R - 1.59 x I O " 4 which i s in good agreement with our experimental value R = (1.45 ± 0.19) x 10*"1* . It should be pointed out that using the earlier 'recommended' value of Pilkuhn et a l (PI73) for the hydrogen charge exchange scattering length (a(Ti~p -»• Tr°n) = 0.193 ± 0.013 fm) results in a branching ratio 1.67 x 10~ 9- R - 1.91 x 10"1* in distinct conflict with the present value. 74 LIST OF REFERENCES BE76 D. Beder, Private Communication BE77 D. Beder, Private Communication BI76 J.A. Bisterlich, 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. Lett. 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 (1954) CH55 W. Chinowsky, J. Steinberger, Phys. Rev. 100, 1476 (1955) C061 V.T. Cocccni, T. Tazzini, G. Fidecaro, M. Legros, N.H. Litman, A.W. Merrison, Nuovo Cimento 2_2, 494 (1961) HA65 R.P. Haddock, R.M. Salter Jr., M. Zeller, J.B. Czirr, D.R. Nygren, Phys. Rev. Lett. 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. Merrison, S. Tornabene, Proc. Phys. Soc. (London) 73, 551 (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. Lett, (accepted for publication 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. Schiff, 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. Bisterlich. K.M. Crowe, N. de Botton, J.A. Helland, Phys. Rev. Lett. 32_, 1268 (1974) TR76 T.G. Trippe et a l . , "Review of Particle Properties", Reviews of Mod. Phys. 48, no.2, part II (April 1976) 7 6 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. Filippov, 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° in the lab frame i s pictured in figure AI.l y Figure AI.1.1 - The TT° decay in the lab frame We choose the x axis to l i e along the direction of one of the gamma rays with the origin at the position of the n° decay. The statements of conservation of energy and momentum are: P„ + P, = P = P cosO 2X I X TTX TT P,v - P = P sino 2y ^y TT P. + (P • 2 + P„ 2 f i - (M 2 + P 2)** lx v 2x 2y Tf TT We square and add (1) and (2) to obtain 78 P, 2 + P, 2 + P„ 2 + 2P, P = P 2 2x lx 2y lx 2x TT and subtract this from the square of (3) to find 2P. ((P„ 2 + P, 2) 1 5 - P ) = M 2 l x v v 2x 2y ' 2xy TT We now substitute from (1) and (3) for the components of P 2 < 2 P l x ( ( M T r 2 + " P l x " V O S 0 + Plx> * V P M 2 lx 2 (M 2 + P - P cos0 -TT Tf TT Now P = E and (M 2 + P 2 ) ? s = E . Hence ix yi 1 1 ^ ^ 79 2. The y-spectrum Resulting from Isotropic u° Decay Let tiny be the number of y-rays per unit pion solid angle in the lab frame d n = & Y d « TT We f i r s t consider those pions whose momentum is contained in the elemental solid angle dtt^. In the pion rest frame the y~ r ays are isotropically distributed. Hence d n " k d Q Y Y or dnX-< - 2nk (1) d(cos0 ) i where the prime denotes the pion rest frame. In the lab frame this d i s -tribution i s pushed forward i n the direction of the pion momentum. Figure AI .2.1 - Center of Mass - Lab transformations 80 The situation i s depicted in figure AI.2.1. The pion rest frame i s moving with velocity 6 with respect to the lab frame and 6' = -8 11 LAB TT For the y-ray in the pion rest frame (with c = 1) v x = c o s 0 ' In the lab frame v x = cosG The r e l a t i v i s t i c velocity addition equations connect these two velocities. v * 8' x - LAB v = x i - a ; v , LAB x Then V „ ' - PT'A-D cose' - B;,„ c o s 0 = — X - ' L A B LAB x LAB so 0 C O S © + BTT COS0 = ^ J -1 + p C O S © and conversely c o s 0 ^ . cose- - B, (2) 1 -.6 cos© TT 1 81 In the lab frame dn y dny . d (cos©") dcos© dcosQ' d(cosO) From (2) d(cos0") _ 1 - B^2 d(cos0) (1 - B^cos©^ and with (1) we arrive at d(cos0) c K ( l - B^cos©)* This i s the distribution of y-rays as a function of y-ray angle with respect to the TT° direction. Note that the angle cf the pion with respect to the gamma ray direction ©^ i s just -0. Since cos©^ = cos© we have for a fixed gamma ray direction d"y = 2 T T k 1 " ^ d(cos© ) (1 - 8 cos© ) z dft TT TT TT TT where the f i n a l term i s the distribution of the pion in the lab frame. For an isotropic distribution K dfi d % r T - 2TrkK * - BIT* (3) d(cos© ) Z 1 T K K (1 - g COS0 )* TT T f TT 82 Now dry dEy" dn d (cost" ) TT dcosO 1 EY x. = From appendix 1.1 1 MTT2 v _ - * i-rr 2 E - P cosG (4) TT TT TT from which we derive dEy = 1 P A 2 d(cos0 ) 2 - P cosO ) z * (5) IT IT TT Combining (3) and (5) with the identity (E - P cos0 ) = E (1 - BcosO ) IT TT TT TT IT we obtain dn _ 4 T f k K(l — g 7 T) 2E 7 T 2 = 4 i r k K d~E~7 fJT2 p w Y TT TT TT dNv The total number of y rays observed per unit energy interval i s -r=r- where dNy _ r dny , dEy " J dEy ^TT dNy = 16iT 2kK dEy P„ which i s independent of the y-ray energy. This formulation i s valid as long as (4) i s va l i d , i.e. for the energy range between E^ and E^ where 83 fi H,L 2 dN Clearly -r=¥- = 0 outside of these li m i t s . Hence for an isotropically distributed mono-energetic TT° we observe a y-ray spectrum: d^ dE. 16^ 2kK Err + PTT ^ < Ey < = H " 2 E > E„ y H 84 3. The Second-Gainma-Ray Cone Angle The angle in the lab frame between the two y-rays from the decay of a ir° with total 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 find i s (j), shown in figure AI.3.1. y -y P 0 TT y / \ a -> Pl 0 TT X Figure AI.3.1 i r ° Decay in the I.ab Frame The statements of conservation of energy and momentum are Fl = P cosa'+ P2COsCp P sina = P2Sin(l) (1) (2) Pl + P 2 = (M 2 + P 2)'* = E * TT TT TT (3) From (2) cosa = ± (1 - *L \ 2 s i n 2 ^ Using this (1) becomes 85 \ sin2!)))15 + P^ cos<() . P 2 + P 2cos 26 - 2P P cos(t) = P 2 ( 1 - f l 2 - ) sin2<l)) . 1 2 r 1 2 r * Vv j p 2 + p 2 _ 2 p p cos(l) = P 2 . 1 2 1 2 T ' Substituting from (3) for P^ we get P 2 + (E - P ) 2 - 2P (E - P )cos(j) = P 2 1 * 1 1 TT i TT i P i 2 + (E,, - P i ) 2 - P^2 . C O S <P = - i 2P(E - P ) T 1 TT ] / Since E = P 1 1 c o s (K = E l 2 + (ET^ - E ; ) 2 - P f f 2 = C ° S ( P 2E v.E - E ) 2E (E - E ) 1 1 IT 1 1 TT 1 or C O S 2 = 2VEH1E - E ) = S i4 1 1 1 1 where i s the angle between the two y-rays, 8 6 APPENDIX II The Elementary Coincidence Efficiency For a TT° of energy (T^,T^ + dT^) the probability of observing a decay y-ray of energy (E ,E + dE ) i n collimator 1 solid angle dfi y y y C j i s dP (T ) = 2 ^ d E ^ (AII.l) s v TT dEy y 4TT dN where - j ^ - i s the normalized singles y-ray box spectrum (appendix 1.2) and the factor 2 i s included to account for 2 y-rays per i r ° decay. The probability for a i r 0 to have energy (T^.T^ + dT^) i s where is the normalized TT° energy spectrum. For such a TT° decay i t TT i s shown in appendix 1.3 that the second y-ray l i e s on the surface of a cone of half-angle ((), given by c o4-2VKX - v • ( A I I , 3 ) Furthermore, because the angular distribution of the i r 0 momentum is isotropic, the second Y-ray probability i s uniformly distributed over the surface of the cone. Thus the probability for detecting the second y-ray i n spectrometer 2 i s determined by the amount of this cone which intersects collimator 2. For the special case that the IT 0 decay occurs on the axis of collimator 2 this i s just the intersection of two confocal circular cones (figure AII.l) 8 7 Figure AII.l The Second y - r a y cone §12 * s fche angle between the f i r s t y ~ r a y and the collimator 2 axis, i.e. this i s the angle between the axes of the collimator cone and the second y-vay cone. <|>1 is the half angle of the second y-ray cone and i s given by (AII.l) (j>2 is the half angle of the collimator cone. Consider the intersection of the two cones and the surface of a unit sphere with center 'o' at the focus of the cones (figure All.2) Figure A l l . 2 Intersection of two cones • 88 The probability of detecting the second y-ray is just P 2 = 20/2n. = O/TT where 0 is the angle between the OAX and OAB planes. If e^, efi, &x are the unit vectors from the origin, 'o' to the specified points then OAB = sin(J)i i s the unit normal to the OAB plane and 6 A X ft. OAX sin(|>i Is the unit normal to the OAX plane. Then n - • * ( 6A x V • ( §A x V cos© = n Q A 7 ) fiQAX Sin(j)i2sin(l>i The numerator may be expanded using standard vector identities to give <*A * V <*B ' V " <*A * V < SB ' V cos© = sin(t>i 2 sin(()] The dot products are easily evaluated and cos _ cos(>2 - cosOicosO 12 ® sin(>i 2 s i n ( ) i The probability for detecting the second y-ray becomes P . - £ . l ^ e - l / c o s ^ - c 0 8 ? 1 ; 0 8 ^ ) (AH.3) l 2 s i n < p i J j*. B JO. = —cos / 1—-—rr •2 TT TT I sin(() and the probability of detecting a coincidence from a TT" of energy o 89 (T ,T + dT ) with one y-ray energy (E ,E + dE ) in the element of IT ir TT' 1 J Y Y Y solid angle dft in collimator 1 i s just the product of (AII.l), 1^ (All.2) and (All.3); d P - * cos-1 (COS^ : c°H>l">s4>12 )*§f. J d O dEdT . c 2TTZ I sin(pi2Sin<|)i J dEy dT^ C j y TT 90 APPENDIX III The Deuterium Charge Exchange Phase Space The rate for a transition 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|HraT|i>|2dp where dp i s the density of f i n a l states. For charge exchange in deu-terium we have a three body f i n a l state (figure AIII.l) and in the center of mass frame dp = d 3P d 3P d 3P cf(P +P +P )X(E. - Ec) TT ni n 2 TT nj n 2 ^ i f n 2 * Figure AIII.l - Deuterium Charge Exchange Final State It i s more convenient to use the set of vectors (p,q,k) defined by 91 P = P IT 9 = P_ - P n k = p + p n 1 n 2 Then dp = d 3pd 3qd 3k S(p + k) £(E± - E f) The two neutrons are constrained to be in a relative p-state and the ir° - (nn) system must also be in a relative p-state. For small values o f p and q the matrix element may be approximated as < * I H ^ I ^ = c|pq| where c is just a constant of proportionality. The spectrum of the pions is just = c/|pq|2d3kd3qp2dO S\? + k) < f(E ± - E f) dp For small values of momentum the f i n a l energy i s "T 2m 4m 4m f f n n v v ir . n n 92 4m ' * TT n n Now = so the f i n a l integration gives " n 2 /o_ _ \ \ \ 3 / 2 £ - 1 6 , c(2 Vp'.(4» n(E 1 - | - * £ ± = . r v n TT With the identification 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 M A X ~2V ~ fii we see dw „ n / 2 „ 2 " \ 3 / 2 dp"^ p M p M A X " P 2 ) Now dq) _ djo_ dp _ dtj_ 2 1 1 ^ dE dp dE dp p n TT Hence the energy spectrum of the pions i s d a ) « 3 / 2 2\ d E * P ( P M A X " P > 93 o r £ L - = C E 3 / 2 ( E , - E ) 3 / 2 dE TT i Tf where C is a normalization constant. The total rate i s to where J J i C E 3 / V - E ) 3 / 2 d E E =0 Tf If we let n = E 1 I/E i the integral becomes co = 4c/n 3 / 2 (l - r O ^ d n Hence w ^ E^. The i n i t i a l energy E i is the sum of the i n i t i a l kinetic energy 'T' in the center of mass system and the Q value for the reaction. Thus co (T + Q> . Now the cross section and rate are related by 0 co where v i s the i n i t i a l pion velocity. Then for low energies v = (2T/m) and 94 where d i s just a constant of proportionality. 95 Appendix IV The Reduction of the Effective Hydrogen Contamination In Section III.2 i t was pointed out that there was a discrepancy in the effective concentration of hydrogen gas as determined from the number of hydrogen charge exchange events in the TINA singles spectrum and the hydrogen concentration in the gas phase determined from the quality of the deuterium oxide used to form the gas and a subsequent mass spectrometer measurement. These concentrations were C R (effective) = 0.00145 ± 0.00029 and C R (gas phase) = 0.003 ± 0.001 respectively. Thus the reduction in effective hydrogen concentration l i e s between 10% and 70%. It i s interesting to note that we are not alone in our observation of this phenomenon. The negative values of the charge exchange branching ratio, 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 results was presented in Table 1.1. Furthermore, in a recent report by Bisterlich et a l (BI76) on the ^It should be mentioned here that Petrukin does not give enough detail of this measurement to definitely decide that they also have not measured a negative value for R. 96 reaction ir + 3H -*• nnny in liquid tritium there is a clear indication of 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 in the gas phase. There are two mechanisms which might contribute to this observed decrease in the effective hydrogen concentration. The f i r s t hypothesis i s that simple d i s t i l l a t i o n has taken place. The boiling point of liquid deuterium i s about 3.2K higher than that of liquid hydrogen at atmospheric pressure and d i s t i l l a t i o n may be used to separate these two isotopes (MC64). The action of this mechanism could be supported by the non-negative value of R reported by Petrukin and Prokoshkin(PE64) who have used a solid LiD target to make their measurement. This hypothesis i s not well 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 is 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 this measurement. The second hypothesis is that we have observed the preferential absorption of pions into deuterium atomic states rather than hydrogen atomic states. The effect of preferential TT capture on high Z nuclei in hydrogenous compounds i s well known (KR68, P073, PE69). For a hy-drogen-like atom the energy levels are given by (SH68 ) E = - (A.IV.l) n 2-tVnz where the reduced mass, u, which is slightly greater for the TT d system 97 than for the TT p system, gives r i s e to bound states that are slightly deeper (=6%) on the deuteron than on the proton. Ponomarev (P073) has suggested the model of "Large Mesic Molecules" to account for the strong preference of the pion to be captured by the high Z fraction of hydrogenous compounds of the form Z^H^. The model presumes that the free pion i s captured by the Z mH n system by displacing one of the systems electrons. Since the only electron of the hydrogen atom i s tied up In a molecular orbit the only pos s i b i l i t i e s are for the pion to be captured into either molecular orbits or isolated orbits of the Z atom. The probability for the former process is given by "1 n + mZ Those pions captured into isolated atomic orbits of the Z atom ultimately undergo nuclear capture on the Z nucleus. Those pions captured into the common molecular orbits undergo transition to either the isolated Z atomic orbits or isolated hydrogen atomic orbits. The probability for the latter process is W2 where W2 = 1/Z2 . The TT'"" systems which are thus formed move through the surrounding matter and undergo collisions with other nuclei. During these collisions the probability of transfer of the pion to a Z nucleus is proportional to the concentration of these nuclei. 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 collisions and the rate for these processes i s thought to be proportional to the con-centration of hydrogen nuclei. W c " B n H where n i s the number of hydrogen nuclei per unit volume and 8 i s a n constant of proportionality. The probability for the meson to be cap-tured by the hydrogen nucleus then i s W3 = -r J = (1 + XC) 1 3nH + an z where C = n„/n„ and X = a/8. In the case of most hydrides the constant L n 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 - WlW2 = anZ _ 2/(n + mZ) (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 real deuterium nucleus. Unfortunately one can not apply equation A.IV.2 directly to such a nucleus. F i r s t of a l l for this H-D system the argu-ments used to determine Wj_ would not apply. In this case in fact only molecular orbits would be expected to occur since both available elec-trons are consumed in molecular orbits. Thus Wi * 1. Further, the argument that W3 plays no role i s based on the smallness of XC. In the present case the concentration i s very large, C = n„/n = 332 £ r l and comments by Ponomarev (P073) indicate that X = kZ where k could be of the order of 0.3. Thus W3 « (1 + kCZ)" 1 and even the small value of Z w i l l be significant. In this case W = WiW2W3 = a/Z 2 (1 + kCZ). The constant of proportionality may be determined by noting that as 100 Z -*• 1 the probability, W, must be determined by the concentration of hydrogen. That i s 1 w(z - i) - - — = TVF • V nz 1 + C In the present case a = 0.3 and with Z = 1.034 we find that the probab-i l i t y for nuclear capture on hydrogen (i.e. the "effective hydrogen concentration") i s W = 0.00269 , or about a 10% reduction from the actual concentration. Whereas this result 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 in our measurement of the effect are very large and the pos 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 careful measurements of the effect using a gas target so d i s t i l -lation effects 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. |
IsShownAt | 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 |
AggregatedSourceRepository | DSpace |
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