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Pion transfer in gaseous hydrogen Noble, Anthony James 1986

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PION TRANSFER IN GASEOUS HYDROGEN by ANTHONY JAMES NOBLE •Sc. (Math/Phys.), The University of New Brunswick, 1983 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES Department of Physics . We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA January 1986 © Anthony James Noble , 1 986 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of P H Y S I C S  The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date 13/01/86 DE-6(3/81) ABSTRACT. The experiment consisted of stopping negative pions i n a high pressure gas target to measure the transfer rate ir-p •*• ir-d i n mixtures of H2, D2 and HD gas. The gamma rays from the decay of the I T 0 i n ir-p •*• n + I T 0 were detected i n coincidence using two large sodium iodide c r y s t a l s . The p r o b a b i l i t y that a pion be transferred to a deuteron from a pionic hydrogen complex was described i n terms of a phenomenological model parameterized by B and A . F i t s to the data yielded B = 0.77 ±0.14 and A = 0.21 ±0.04. These values implied that the hydrogen capture r a t i o i n an equal mix of H2 and D2 was F(H 2D 2) = 0.45 ±0.01. The capture r a t i o f o r HD was measured to be F(HD) = 0.355 ±0.021. The r a t i o F(H 2D 2)/F(HD) indicated that there was l i k e l y to be i n t e r n a l transfer i n the breakup of ir-HD favouring the ir-d complex at about a 60% l e v e l . - i i i -TABLE OF CONTENTS. Abstract i i Table of Contents i i i List of Tables v List of Figures v i Acknowledgements v i i i 1. Introduction and Motivation 1 2. Theory 4 2.1 I n i t i a l Slowing Down 5 2.2 Coulomb Capture 6 2.3 De-excitation 16 2.4 Transfer 19 2.5 Nuclear Capture 22 2.6 The H2D2 and HD Systems 25 2.7 Summary of Previous Hydrogen Measurements 29 3. The Experiment 31 3.1 Overview of Experimental Setup 32 3.2 Target Design 33 3.3 Selection of Beam Parameters 35 3.4 Gas Preparation, Handling and Mixing 38 3.5 Electronics and Data Acquisition 42 - i v -4. Analysis 47 4.1 Determination of Gas Concentration 48 4.1.1 V i r i a l Corrections 49 4.1.2 Concentration Error Analysis 53 4.2 Analysis of the Data 54 4.2.1 Time Cuts 55 4.2.2 RF Cuts 57 4.2.3 Energy Cuts ..59 4.2.4 The Stop Definition 65 4.2.5 Fitting the Stop Region 71 4.2.6 Comment on Error Analysis and Combination of Data. 76 4.3 Calculation of the Hydrogen Capture Ratio 78 5. Results and Discussion 83 5.1 A Comparison of the Results to the Literature 87 5.2 Limits on the Molecular Breakup Rates in HD 88 5.3 The Future of These Measurements 90 5.4 Summary 91 References 93 - v -LIST OF TABLES. Table 4.1 V i r i a l y corrected deuterium gas concentration 52 Table 4.2 Stopping profile of target 65 Table 4.3 Hydrogen capture ratios 80 Table 5.1 Comparison of results with previous measurements 87 - v i -LIST OF FIGURES. Figure 2.1 Periodicity of the muon capture ratio in oxides 11 Figure 2.2 H Z T T - Potential 13 Figure 2.3 Rate diagram for HZir- system 14 Figure 2.4 Rate diagram for H2D2 mixtures 26 Figure 2.5 Rate diagram for HD 26 Figure 3.1 Experimental setup, viewed from above 33 Figure 3.2 Target design 34 Figure 3.3 Scintillator detail 35 Figure 3.4 TRIUMF M13 beam line 37 Figure 3.5 Apparatus for HD manufacture 39 Figure 3.6 Gas Manifold 40 Figure 3.7 Electronics diagram 43 Figure 4.1 TINA time spectrum 56 Figure 4.2 RF time spectrum 58 Figure 4.3 TINA coincidence spectrum 60 Figure 4.4 TINA singles spectrum 61 Figure 5.4a F i t to TINA radiative capture peak 63 Figure 5.4b F i t to MINA radiative capture peak ....64 Figure 4.6 Scatterplot of ES3 and ES4 for pulser strobe 67 Figure 4.7 Scatterplot of ES3 and ES4 for non-pulser strobe 68 Figure 4.8 Schematic of ES3 vs. ES4 69 Figure 4.9 Energy distribution of S3 before cuts 73 Figure 4.10 Energy distribution of S4 before cuts 73 Figure 4.11 Energy distribution of S3 after cuts 74 - v i i -Figure 4.12 Energy distribution of S4 after cuts 74 Figure 5.1 Hydrogen capture ratio in mixtures of H2 and D2 84 Figure 5.2 Transfer probability in mixtures of H2 and D2 85 Figure 5.3 Limit of molecular breakup rate 89 - v i i i -ACKNOWLEDGEMENTS. I would like to thank my supervisor, Professor D.F. Measday for his help and encouragement throughout the analysis and during the preparation of this thesis. I would also like to thank Dr D. Horvath, Dr M. Salomon and Dr K. Aniol for discussions about the analysis and their tireless help during the shift work. I am also indebted to Shirvel Stanislaus, Dr F. Entezami, Dr J. Smith and Dr D. Livesey for their assistance during the experiment. I would like to thank NSERC for their financial assistance during my term as a masters student. Finally, I would like to thank my friends, especially Shirley, who suffered with patience and understanding during the dark age of the writing of my thesis. Chapter 1 Introduction and Motivation. When a pion stops in matter i t is involved in interactions at molecular, atomic, and nuclear levels. The capture mechanism can be subdivided into many stages, and i t is the job of the physicist who sees only the f i n a l products to unravel these mutually interrelated processes. Hence experiments have focused their attention on pions stopping in groups of similar compounds so as to eliminate as many parameters as possible. From such experiments and extensive theoretical studies a phenomenological model describing the l i f e history of pions in matter can be developed. By studying the pionic capture mechanism one can determine the relative strengths of competing processes like radiative and Auger capture or co l l i s i o n a l l y induced transfer and Stark enhancement of nuclear capture. The monitors of a pion's progress in a condensed medium which are available to experimentalists are the X-Rays emitted as the exotic pionic atom de-excites as i t cascades down towards the nucleus and the diverse reaction products from nuclear capture. The knowledge of the pion stopping schedule has proved to be of interest to more than just the academic. At f i r s t glance one would suppose that a pion would stop in a mechanical mixture of gases like N 2 + 2 H 2 in much the same way as for a chemical compound like hydrazine ^H^. Experiment shows us however that this is not the case. There is a substantial suppression of capture on the hydrogen in hydrazine due to the proximity of the nitrogen atoms. In this way pionic chemistry has proven to be a useful tool in probing the electronic structure of molecules. - 2 -It is also clear that the atomic shell structure i s strongly correlated to the capture probability. This was f i r s t observed with muons in oxides 1 and has since been seen in many other compounds. In these compounds i t is seen2 that the oscillatory character of the capture probability is similar to observed oscillations of stopping power which can be explained by examining the electronic shell structure. In mixtures of hydrogen isotopes i t is found that a stopped y- can form muonic molecules from which the nuclei can fuse, an exothermic reaction which has been proposed as a macroscopic source of energy. By examining pionic capture one can deduce information about analogous processes in the muon catalyzed fusion reactions. The longlife time and absence of a strong interaction allow a muon to catalyze as many as 1 0 0 fusions in mixtures of deuterium and tritium. In the theory of muon induced fusion ddy and d t u complexes are formed via an excited resonant molecule of the form [ddu -d2e]. The resonant mechanism greatly enhances the rate of formation. The ddu and d t p can then fuse yielding neutrons, helium, and often ejecting the muon. This muon may then recycle and catalyze more fusions. Knowledge of the transfer process between dy and ty is c r i t i c a l to the understanding of the overall mechanism, and i t is hoped that studies of pions w i l l help clear up some of the mysteries. At a more practical level, in the medical treatment of deep seated brain and pelvic tumours pions have proven to be very useful. Since upon being captured by nucleus the entire pion rest mass Is given up to heavy fragments one can maximize the damage to the tumour while minimizing the effective dosage to the intervening healthy tissue. This - 3 -is in contrast to the traditional method of treatment by y -ray irradiation for which the relative body dosage f a l l s off with depth, so that the overlying healthy tissue is more heavily damaged than the tumour. A better understanding of the pion stopping mechanism w i l l allow optimization of the pion therapy technique. In this experiment we studied the stopping mechanism of pions in mixtures of H2, D2 and HD gases. We investigated very simple molecules in an attempt to determine fundamental parameters which would allow us to extract real physics from the data. The general procedure to date in the literature appears to be the systematic measurement of a whole host of complex molecules from which general trends can be determined but no simple physics principle. Experimentally we measured the fraction F X of stopped pions which were captured on the proton. We determined a functional relationship of FH D f° r t n e relative H,/D, concentration. By 2 2 z z measuring Fy^ we re-examined the previously observed3 anomalously high F H D / F H D ratio. - 4 -" Chapter 2 Theory. During the i n i t i a l capture process a negative pion can replace an electron about a nucleus to form an exotic atom. Since the Bohr radius i s inversely proportional to the mass, and a pion is much heavier than an electron, the pionic orbits eventually become very small. The lifetime of pionic hydrogen is in the order of a picosecond, (depending on the environment). In order to discuss the processes of pion capture we subdivide the stopping mechanism into several stages. In most cases these stages w i l l be competing with other processes. Here we try to present the possible mechanisms and discuss the theoretical and experimental predictions which support or contradict them. We w i l l also examine some of the models which have been developed to f i t the wealth of data available. To get an understanding of the entire process and because most of the data are not of this form, we w i l l not limit discussion to mixtures of hydrogen isotopes. The primary source of energy loss for the pion is in the f i r s t stage where i t is slowed from a r e l a t i v i s t i c velocity to that of the thermal electrons by Coulomb interactions, principally by ejecting electrons from atoms. It may then be captured into excited mesomolecular or mesoatomic orbits via an Auger transition. The f i n a l distribution of the pion w i l l be determined by c o l l i s i o n a l l y induced de_excitations or transfers and the density of electronic states about each atom. Once the pion reaches the lower energy states i t s wave function begins to overlap with the nucleus and i t can then be captured via the strong interaction. - 5 -§ 2 . 1 I n i t i a l Slowing Down. In the f i r s t stage, where the pion has lots of energy, small amounts of energy are given up to the electron cloud of the medium in the form of e x c i t a t i o n or d i s s o c i a t i o n . The process can be parameterized by the velocity and charge of the pion, v and z respectively, and by the number of electrons i t sees which is given by N, the number of atoms per cm3 and Z the atomic number of the medium. The only other parameter is how tightly bound the electrons are which is given by the average ionization potential I. Then the energy loss i s given by the Bethe-Bloch equation1*; _ dE 4Tie 4z 2ZN „ ,„ . . = 7— • B , (2.1) dX m v z e where B is the stopping number and is a logarithmic term which varies slowly with energy. ln(2m v 2) B = ln(l-B 2) - B2 • (2.2) This i s the simplest form of B, more elaborate treatments have added correction terms for shell effects, density effects, and other higher order corrections 5. We see from (2.1) that there is a linear dependence on Z so that in mixtures the pion w i l l be more likely to stop near the heavier atoms. Equation (2.1) is only valid for particle velocities greater than the thermal electron velocities; v > v t h « oc , (2.3) - 6 -where a is the fine structure constant, a = 1/137. Of course i t is also possible that the pion loses large amounts of energy by interacting strongly with a nucleus, but at these velocities the mean free path for hadronic interactions is very long and hence this process is relatively rare. §2.2 Coulomb Capture. By Coulomb capture one means the transition from the continuum to the discrete spectrum. This is by far the least well understood process in the overall picture of a stopping pion. This is due to the fact that we do not understand the transition mechanism and we do not have a good feeling for energy losses at low energies. It is s t i l l an open question whether one should treat the transfer as an adiabatic process or consider the transfer of small quanta of energy to vibrational and rotational molecular states. There has been much discussion about the competition between the slowing down and atomic capture processes, and in particular the energy of the pion when the transition occurs. Korenman and Rogovaya6 believe the energy is in the order of hundreds of eV whereas Leon 7 predicts that the transition occurs closer to 20 eV. This has interesting implications, especially for isotopic mixtures, as with energies of 10's of eV, vibrational and rotational differences could be significant in the stopping powers and cause an isotopic asymmetry in the i n i t i a l capture rate. We do know that when the pions have low enough energy, (10eV<E<10keV), most of them are captured into excited mesomolecular or mesoatomic states. The most likely process for this capture is the - 7 -emission of an Auger electron as this cross-section is much larger than that of radiative capture. We also know that throughout most of the capture process the pionic wavelength is short relative to the dimensions of the f i e l d through which i t travels and that the entire process time is short compared to the natural decay time. This process has been of interest for quite some time and so many models have been developed with somewhat limited success. The Fermi-Teller model8 was the f i r s t such model developed. Here the energy loss of a meson is considered to be the transfer of energy to conduction electrons in a degenerate electron gas. Following this line of thought, a Z dependence for the capture process is obtained. This Z dependence is derived by assuming to a f i r s t approximation that the probability of capture on an atom is proportional to the energy loss of the particle to that atom. If one defines by W(Z) the probability that the pion w i l l end up on an atom of atomic number Z, then one can define the atomic capture ratio by; A ( z 1 5 z 2 ) = WCZ^/WCZ^ . (2.4) Fermi and T e l l e r 8 derive; A C Z ^ Z ^ N-l w3/2 + 4,rZ 1e 3b 3 / 2(l-.8/X n i) = N-lw^ 2 + 4 1rZ 2e 3b^ 2(l-.8/X° 2) (2.5) where b is a constant from the s t a t i s t i c a l model of the atom proportional to the Bohr radius and X- is determined by boundary - 8 -conditions. In the approximation that the pion energy w is small, one gets, neglecting any Z dependence of XQ ; A(Z 1,Z 2) * Z 1/Z 2. (2.6) Numerous experiments followed which showed that the "Z law", was not a good description of the measurements. Zinov et a l . 1 studied the K-shell mesic X-rays in metal alloys and metallic halides and determined a capture ratio; A(Z 1 5Z 2) = 0.66 Z 1/Z 2 . (2.7) It was noted by Baijal et a l . 9 that a l l the experimental ratios could be described by; A(Z X,Z 2) = ( Z 1 / Z 2 ) n , 0.5 < n < 1.5 (2.8) Vogel et a l . 1 0 developed a new theory. They calculated the number of stops on each atom (N^) by integrating; N = /" n P(E) a i(E) dE , (2.9) 1 0 1 where P(E) was the flux of mesons of energy E, n^ was the number of atoms of species i and a* was the capture cross section. They extracted the values of a* by assuming a linear dependence on energy of the form; a 1 = Cx - C 2E (2.10) - 9 -where and C 2 were determined from the potential used in the Thomas-Fermi model of the atom. Their calculations predicted a capture ratio of the form; consistent with other models, but s t i l l not very useful. A different approach was taken by Dan i e l 1 1 . He determined the energy loss AW of a particle moving inside a screened spherical well and found; where VQ is the average electron velocity and is proportional to Z 2 / 3, and T is the pion kinetic energy. The integral is over the trajectory of the pion. This formulation yields; which f i t s rather nicely with the metal halides, but otherwise does a poor job, mainly because i t is so sensitive to the atomic radii which are somewhat ill-defined for atoms bound in molecules by ionic and/or covalent bonds. The importance of atomic structure has been clearly illustrated by Evseev et a l 2 . They point out that the stopping powers of low energy protons and alpha p a r t i c l e s are characterized by Z dependent oscillations- This has been known for quite some time and the A(Z X,Z 2) = ( Z 1 / Z 2 ) 1 . 1 5 (2.11) (2.12) A(Z X,Z 2) Z 1 / 3 l n ( . 5 7 Z ) Z 2 U i l n ( . 5 7 Z 2 ) (2.13) - 10 -oscillations have been explained by examining the idiosyncrasies of the electronic shell structure. These same periodic oscillations have been observed in negative muon capture in oxides and other compounds and i t was shown by Evseev et a l . that there was a strong correlation between the two. Figure 2.1 displays the periodicity of the muon capture ratio in oxides as a function of Z. The large points are predictions of an empirical formula developed by Stanislaus et a l 1 2 . The most successful model unt i l recently, as determined by a x 2 f i t to a l l data 1 3 was developed by Schneuwly, Pokrovsky and Ponomarev14 (S.P.P.). They considered molecules of the form Z^Z£. Let n and n be the effective numbers of core electrons, v and v the respective valencies, Then the total number of electrons is just; The probability of capture into mesoatomic and mesomolecular orbitals can then be given by; N T = k(n + v) + £(n*+v*) (2.14) cr = kn/Nx , (2.15a) a m = (kv + £v*)/NT (2.15b) From these one can write the atomic capture ratio as; A(Z,Z ) = I (a +o_w ) *—™~rr k (a +a w ) m n + 2vw * * * n + 2v w (2.16) as kv = tv . Here w and w are the probabilities that the pion w i l l FIGURE 2.1 Periodicity of the muon capture ratio in oxides. - 12 -de-excite into orbitals of Z or Z respectively. The effectiveness of electrons in a sub-shell j can be given by a function p(Ej) so that; n = Z p(E ) n (2.17) with Ej the binding energy of the jth sub-shell. In the S.P.P. model, assuming that only the less tightly bound electrons are important in the capture process, two p ( E j ) functions were tried. The "sharp boundary", P ( E j ) = i llS E? (2.18a) 0 Otherwise. and the "smooth boundary", 1 E. < E P ( E . ) = °M,v2s ^ (2.18b) exp( J(E j-E 0) 2/E 2 ;) otherwise where E Q and E c are adjustable parameters of the model. Fits to a l l the data have shown the smooth boundary approximation to be more successful. The S.P.P. model, and the large mesic molecule model are based on the belief that the pion is f i r s t captured into large mesomolecular orbitals rather than direct capture into atomic orbitals. Experimental evidence for this is strong. If one defines by F(x) the probability a pion w i l l end up on a hydrogen atom in some mixture x, then we have, for example 1 5* 3; - 13 -F(N2+2N2) F(N 2H k) 30 F( 2HD ) 1.23. (2.19) Here we see that the chemical bond has played an Important role In determining the ultimate fate of the pion. In recent work by von Egidy et a l . 1 6 the S.P.P model has been further developed. The muon capture ratios were calculated using a quantum mechanical approach. The interaction cross sections were calculated in terms of Z, the atomic binding energies, and the quantum numbers n and I. Their results reproduced the data quite well, except for low Z values where the S.P.P model, treating the valence electrons separately, did a better job. This was believed to be due to the increased role of the chemical bond in distorting the electron distribution. = 7 r» H FIG 2.2 HZir" Potential Thus the idea is that the pion is f i r s t captured in highly excited molecular states. In particular, i f we examine binary molecules of the form H Z , then an exotic H Z T T - molecule would be formed. The potential for the H Z T T - system is shown in figure 2.2. In order for a pion to occupy one of the orbitals i t must give up i t s energy to one of the electrons. Adiabatic capture has been proposed as the most likely process. The transfer of small amounts of energy implies that the - 14 -resulting orbit of the pion must be very similar in geometry to that of the electron knocked out. In the H Z system, since the only hydrogen electron i s locked up in the chemical bond, the pion must be captured into a mesomolecular state f i r s t unless one allows for radiative capture. The entire rate diagram for the H Z T T - is shown in figure 2.3. FIGURE 2.3. Rate Diagram for HZir" System. The radiative capture probability is given by D, and the molecular capture probability by P. The transitions from the mesomolecular state are given by Q and E as shown in the diagram. R is the probability of retention on the hydrogen. Let the cross-sections for atomic capture be given by o r and o z, where a r is the purely radiative capture on hydrogen. Let the molecular cross-section be given by o m. Then, i f the number densities of Z and H in a compound Z^ H^  are N z and N n one can write for P and D; ON, ON. P = o~N"+o"N~+o N * a"N.+o~N,+a~N " K ' m h r h z z m h r h z z However, we expect that ar « o m as Auger capture dominates - 15 -radiative capture in the energy region of interest, so D is expected to be negligible. Only in the very low lying states of de-excitation are radiative processes thought to play a role as w i l l be seen in section 2.3. Now we must consider possible isotopic effects. Calculations by Cohen et a l . 1 7 predict that there is an asymmetry in the number of mesons stopping on the various isotopes. Their calculations were done only for atoms, and the asymmetry is a manifestation of the different reduced masses and ionization potentials. For muons on H and D atoms they predict the probability of capture on the proton to be about 3% greater than that for the deuteron. To make this a more re a l i s t i c treatment molecular effects w i l l have to be included as there w i l l be dipole moments which w i l l tend to localize the pions about one atom. Now whilst predicting an i n i t i a l isotopic asymmetry, Cohen et a l . did not calculate any non-linearity in the concentration dependence. This is interesting to note as in the work by Petrukhin and Suvorov 1 8 i t was found that the capture probability could be described by; F(Z kH£) = (1+SC)-1 • (l+SC^ 3)"" 1 (2.21a) S = 7.1(Z 1 / 3 - 1) (2.21b) Here C is the relative concentration. Petrukhin and Suvorov believed that the second, non-linear term belonged to another process, the transfer process. Cohen et a l . believe that their treatment could add credence to the statement of Petrukhin and Suvorov, which is in contradiction to other opinions like that of Korenman and Rogovaya,19 who predict a large non-linearity in - 16 -the i n i t i a l capture rate. §2.3 De-excitation. When the pion is captured in excited mesoatomic or mesomolecular orbitals i t w i l l de-excite to the lower energy levels and eventually be captured into the nucleus. The radiative de-excitation and the transfer of energy to Auger electrons are the two competing processes of this mechanism. The probability for a transition from an excited state n' with angular momentum SL" and projection m' to the state n,£,m is a function of the transition energy AE. For radiative transitions, the probability WR i s , = 4 <d>2 (AE)3 WR 3 h* c? with the dipole matrix element being given by, <d>2 = < n',£',m*|r|n ,1 ,m >2 = Z - 2 (2.23) Atomic selection rules require that A£ = ±1 and Am = ±1,0 which limit the cascade and hence slow the radiative rate. The radiative rate for pionic hydrogen can be shown20 to be related to the radiative rate of hydrogen by i t s reduced mass p. We can write; r ' P = y r H . (2.24) rad irp rad The probability of an Auger process i s given by; - 17 -° < * f * f K e I * i ^ > 2 (2.25) where and <J> are the pion and electron wave functions, and the pion electron separation is r^g. A calculation of the matrix element (2.25) yields a ( A E ) ~ 1 / 2 dependence for W^ . Leon and Bethe 2 0 calculate the Auger rate to be; In the mesic atom the orbits are much smaller due to the increased mass of the meson. In hydrogen, the n = 16 pionic orbital is already smaller than that of the vacated K-shell electron orbital. In the higher orbitals, AE is very small between adjacent states, and since AE a Z 2, for the lighter elements the Auger process w i l l dominate. Only in the low valued n orbitals with large AE w i l l radiative transitions be of significance. Experimental evidence seems to support this, as Goldanskii et a l . 2 1 could not see the X-rays in carbon or oxygen for n values greater than Z. This implies that the probability of radiative capture, given by D in figure 2.3 is very small. In systems like HZ, i f the pion ends up on the hydrogen the resulting pit - atom w i l l be small and neutral. It can then easily penetrate other atoms and approach the nucleus. If the pw- is in an excited state, i t can give up large amounts of energy to Auger electrons of the host atom and de-excite. In mesic molecules the cascade time is shorter as we are no longer Aug a u ~ 2 A E " I / 2 . (2.26) - 18 -working in a central f i e l d , and hence the selection rules are somewhat weakened. In our standard compound Z^ Ify one can predict the ratio of the probabilities of transitions to the atoms from the molecular states N. W % n * " Z " 2 ( 2- 2 7> This implies that the Q probability of figure 2.3 is proportional to Z~2. Fits to data 2 2 in compounds of the above type have shown that F a Z - 3 which agrees well with the naive picture of P from the Fermi-Teller Z law and Q o Z - 2 from above. We note that we are s t i l l ignoring possible transfers between the atoms. However, controversy i s present again as Jackson et a l . 2 3 could find no evidence of a Q a Z~2 dependence when they measured the X-ray spectra of pion cascades in compounds of hydrogen, carbon and oxygen. An interesting feature of the mesomolecular system is that i t gives us a probe into the electronic composition of the molecule. There w i l l in general be some asymmetric electron distribution about the atoms. The form of this distribution w i l l be determined by the electronegativities of the individual atoms. Electronegativity is the capability of an atom to attract electrons to i t s e l f . There w i l l be a weighting of the electronic distribution about the more electronegative constituent. In purely covalent bonds there w i l l be no asymmetry relative to the single atoms but a dramatic effect in polar covalent and ionic bonds. Hence the effect of the bond and the resulting polarity is to reduce the capture probability on the less electronegative atom and to increase i t for the more electronegative - 19 -atom. In the model of the large mesic molecule, the ionicity i s included in the form of a constant a^ where L stands for a particular period in the periodic table. Then one can define the i n i t i a l capture probability P to be the probability that the pion stops in an orbital localized near the hydrogen atom. In equation (2.16) of the S.P.P. model there were probabilities w and w of de-excitation to atoms Z and Z . These probabilities can be determined from a knowledge of the valence electron density about each atom which i s given by the ionicity. The ionicity is related to a^ and can be calculated from the electronegativities. One can define localization parameters p and p which are equal to the electron density of the respective atom. p = (l-a)/2 , p* = (l+a)/2 , (2.28) where a is the ionicity. One can also define redistribution terms q and q* which correspond to the Q and 1-Q-E in figure 2.3 In the S.P.P. model the redistribution factors were taken to have a Z 2 dependence and w is given by; w = z 9-*-* • (2.29) pq + p*q* §2.4 Transfer. The possibility of transfer can play an important role in the - 20 -stopping schedule of mesons in matter. The transfer process is of the form; Z1 7 T~ + Z2 > Z l + Z2 1 T~ (2.30) In the example of H2 + D2 vs HD we saw that there was a molecular effect which was manifested in the fact that F(H2+D2 )/F(2HD) was not unity. One also finds that the absolute value of F(H2+D2) is 0.417, which is different from 0.5 showing a net transfer of the form pir- + d > p + dir -. In experiments21* with H2 + Z, where Z is an inert gas, this transfer has been seen very clearly. The probability R that the pion w i l l not be transferred to the heavier Z atom has been the subject of much debate21*. The transfer process is generally considered to be co l l i s i o n a l l y induced by the fast moving neutral piT - atom. In general the process is irreversible as the new de-excited state is more tightly bound and nuclear capture can proceed quickly. Since the transfer is a Coulombic attraction to the higher Z valued atom, i t tends to suppress the pion capture on hydrogen. In the H2 + Z experiments Petrukhin et a l . 2 1 * established this suppression and also measured the concentration dependence. They found that the capture probability was only a function of relative concentration. The transfer process is evident in gases and mixtures, but not so noticeable in bound systems like CH^  + Z [Ref 26] so for most complex molecules R is taken to be 1 to a f i r s t approximation. 2 7 In most cases transfer is not a large effect, and can only take place from excited states where the probability of transfer is - 21 -comparable with those for de-excitation and nuclear capture. The transfer in the pir- systems probably takes place from the n = 5-10 levels. To develop a phenomenological model for the transfer process one must include a l l po s s i b i l i t i e s . In terms of the cross sections 6 X for competing reactions we have; pion capture on proton due to radiative transitions in the pir" atom. pion capture on proton due to a co l l i s i o n with atom Z (or H). transfer from pir to a Z atom via a tunneling through the H-Z bond. collisionaly induced transfer to Z atom. 6 = r BZ(H) = X t x z -Thus the R probability can be written as; B + 6 »N + 8,»N,_ D = E h__2 (2 11 •> 8 + 8 -N + B «NU +X + X «N v J U r z z h h t z z which can be written in the following form; R = 6 /N + B C + 3, r h z h 3 /N + 8 C + B, + X /N. + X C r h z h t h z (2.32) Since experimental evidence shows only a relative concentration - 22 -dependence and hence no direct dependence on the terms £ r and X must be negligible. We can then write R in the standard form; 1 + B C R = z (2.33) 1 + (B + A )C z z where B z = B z/B h and A z = X Z/6 Z-It is very d i f f i c u l t to extrapolate this formulation to more complex molecules because there are so many interrelated competing effects like the structure of the Z-Z' and Z-H bonds. In the case of hydrogen and deuterium the transfer can go in either direction, although the net transfer w i l l s t i l l be to the deuteron. §2.5 Nuclear Capture. When the pion has de-excited to the lower lying levels of an atom, i t s wavefunction w i l l overlap with the nucleus and there w i l l be a high probability for nuclear capture. It is during this stage that we can probe the interactions between nuclei and pions. If we describe the pion wavefunction in an orbit n,£,m by t|»n£m(r) then we can calculate the overlap by integrating over the volume of the nucleus. This w i l l then give us the probability w of finding the pion in the nucleus. This can be solved, and in particular for pions which get captured primarily from the spherical £=0 states for low Z values, one obtains; w = J v I *<r) P dx (2.34) - 23 -w = 4(yr 0) 3zV3n 3 (2.35) In nuclei of low Z value, -for a given n, the p-state capture rate has been shown28 to be several orders of magnitude smaller than the s-state capture rate. Also we see from equation (2.35) that the very strong n dependence w i l l suppress any capture from the higher excited orbitals. There has been some confusion in the calculations of nuclear capture widths. In pionic X-ray experiments these calculations are v i t a l for the pion cascade codes. Most calculations are consistent for I > 2 but vary depending on how many terms of the pionic wavefunction are considered and what form of potential is used for 1=1 and 1=0. Recent calculations by Turner and Jackson 2 9 have improved the formulas for the p and s states by using a potential with experimentally determined parameters and by expanding the wavefunctions to higher order. In the case of hydrogen, the nuclear capture is easy to detect as the two main channels are; and the charge exchange reaction is strongly suppressed on a l l other nuclei except 3He. Since the p7T~ atom is so small and neutral i t can easily penetrate other atoms electronic shells and approach the nucleus and in the presence of this external coulomb f i e l d i t s energy levels may get mixed. The transition times between oscillating 1 states are extremely fast. For example,30 for n=6, T(6£-»-6(£-l)) is about p + ir —> n + y p + I T " —> n + I T 0 (2.36) > 2 Y - 24 -l x l 0 - 1 6 s e c . This allows the populated 1*0 states to spend some time in £=0 states and enhance nuclear capture. This i s known as Stark mixing. There is also the possibility for direct nuclear capture from mesomolecular states. This i s represented by probability E in figure 2.2. From figure 2.2 we see that the probability of capture on the proton is given by; F(H2+Z) = DR + PQR +PE (2.37) and using equations (2.20,2.31), and neglecting a r, one gets; 1 F = 1 + SC where S = a z/a m. Since for H2 + Z there is no chemical bond, one must have Q = 1-E. The only way that (2.38) w i l l f i t the data, which i s fit t e d by (2.21a), is for B z and E to be small. A small value for B z just implies that in a co l l i s i o n with a Z atom the pion is likely to be transferred. The smallness of E implies that direct capture from mesomolecular states is not a large contribution to the overall rate. It is interesting to note that none of the available gas data have shown a pressure dependence of the atomic capture or transfer rates over a range of about 10 to 100 atmospheres. It appears that the Stark enhancement of nuclear capture just offsets the increased probability of c o l l i s i o n a l l y induced transfer. The nuclear capture reactions that we need to understand for this experiment are the ir-p and n-d reactions. These w i l l be discussed in 1 + B C Q r+(B"+X")C + E z z (2.38) - 25 -some detail in chapter 3. The primary source of background comes from pion stops in the target which are essentially just stops on carbon or aluminum. In these reactions the pion couples to a nucleon-nucleon pair and the reaction becomes (ir-,2N). Specifically we have; ir" + " d " •»• n + n, (2.39a) and the much less likely; ir~ + p + p * p + n. (2.39b) From this we could anticipate that most of our background would be neutrons. §2.6 The H2D2 and HD Systems. The f u l l rate diagram for a pion stopping in a mixture of H2 and D2 gases is shown in figure 2.4 where the X's and B's refer to the following; ^mol _ direct nuclear capture from mesomolecular p,d states of hydrogen,deuterium. ^dir _ direct atomic capture of pion on atom of irp,ird hydrogen, deuterium. ^at _ transition from mesomolecular state to irp,ird atomic orbital. ^ _ transfer from p to d,(d to p) in c o l l i s i o n pd,dp with d,p. B C = irp + p -»• (n+ir° or n+y) PP P B ,C, = irp + d + (n+ir° or n+y) pd a 3. C = ird + p •»• (n+n or n+n+y) dp p - 26 FIGURE 2.5 Rate diagram for HD. - 27 -B J J C , = ird + d > (n+n or n+n+y) da a k .C , = I n i t i a l capture rate on H„,D.. P,d p,d * 2» 2 In the above, Cp ^ are the atomic molar fractions of protons and deuterons. If we assume that direct capture from the mesomolecular state i s small so that; xmol < K x at a n d xmol ^ xat ^ A Q ) p up d ird and from equation (2.19) we ignore the direct atomic capture rates then FJI j) can be written as; 2 2 k C B C +3 ,C , k C X, C F = _E_E EE_E__El_d + _2_p_ d 2 _ E ( , . H,D, T B C +B .C.+X ,C , T 6, C +B J JC,+X, C ^ - H L ' 1 pp p pd d pd d dp p dd d dp p where T = k p C p + kdC d . Then i f we define; 6 A X _ 8 B, C, z = A B = - 2- A = - E - K = ——E r = _4d c = --p pp pp dp dp p we can write; H2D2 1 + EC 1+BC+AC 1 + EC 1+K+rC The parameter E is expected to be very nearly 1 as i t is a measure of the asymmetry In the i n i t i a l capture rates of H2 and D2. It may have some concentration dependence i f the pion does get captured near 20 ev so that the difference in relative stopping powers between H2 and D 2 has a significant effect. We note that in the above formulation we have assumed that only one transfer can take place. From the experimental r e s u l t s 3 ' 3 1 i t appears that the reverse transfer is small so we can approximate FJJ n by; 2 2 - 28 -H2D2 " 1 + C 1 + BC + AC Now we can examine the HD rate diagram, shown in figure 2.5 where the parameters here have the same meaning as for H2D2 except that by definition C p = C d and; X m 0 ^ = molecular capture of pion on HD. ^mol _ direct nuclear capture on p,d from p,d HD mesomolecular states. Again, i f we assume that direct capture from the mesomolecular state i s small so that; X m o 1 « X and X ™ 5 1 « X . (2.44) p up d ird and from equation (2.19) we ignore the direct atomic capture rates then F^n can be written as; X 8 +6 , X , X F = *P . PP pd + ird . dp (2.45) HD X +X . B +8 .+X , X +X , Bj +B..+X, irp ird pp pd pd irp ird dp dd dp which can be reduced in the same fashion as equation (2.39) to; X 1 + B X , 1 F n n = n p • + * d • (2.46) X +X J 1+B+A X +X , 1+K+r irp ird irp ird By combining the experimental data from Ref (1) and Ref (31), the molecular break up ratio X^j /X^p for HD can be shown to l i e in the interval (1.46,1.54). From equation (2.25) we see that the ratio of radiative transition rates is just; - 29 -rnd f rTrp = u / u = 1.07. (2.47) rad rad nd Trp The Auger contribution is given from (2.26) as; Aug Aug irp T r d where we have used the fact that AE ~ y. Neither of the above processes explains the high ratio seen. Another p o s s i b i l i t y that may explain the effect i s the preferential capture to f i n a l states about deuterium in direct transitions from the mesomolecular state. This would require that the direct nuclear capture for deuterium be greater than that for hydrogen, and of the same order of magnitude as the molecular breakup rates. However as we have already pointed out there is no experimental evidence to support this hypothesis. §2.7 Summary of Previous Hydrogen Measurements. There have only been two measurements which concentrated their efforts on just hydrogen isotopes. Aniol et a l . 3 examined H 2 and D2 in a 50%-50% mixture and compared i t to HD. They determined; F H D = 0.417±0.004 2 2 F H D = 0.338±0.008 independent of pressure over the range 6 to 90 atm. Petrukhin and Prokoshkin 3 1 measured a range concentrations of H2 - 30 -and D „ , and f i t them to F J J Q as given in equation (2.43). A x 2 2 2 f i t to their data yielded; B = 1.3 ± 0.4 A = 0.4 ± 0.1 Similar data has been measured in the form of + Z mixtures. Results from these by Petrukhin et a l . 2 3 in an early experiment observed the transfer effect. They found; A = ( 0.7 ± 0.2 )Z B « A Here the low value of B just implies that in a co l l i s i o n of pionic hydrogen with a Z atom the pion w i l l have a very high probability of transfer. Later experiments by Petrukhin and Suvorov 1 8 determined that; S = 7.1( Z 1 / 3 - l ) A = SC 1' 3 where S is the i n i t i a l asymmetry term of equation (2.21) and is comparable to E in the H 2 D 2 system. The bulk of the rest of the experiments done in this f i e l d involve much more complicated molecules which can not easily be compared to the above data. - 31 -Chapter 3 The Experiment. The probability of transfer from hydrogen to deuterium can be measured experimentaly in mixtures of the two gases by counting the number of negative pions which stop on either a proton or a deuteron. The signal for nuclear capture on hydrogen is quite clean. From §2.5 we saw there were two main reactions, the charge exchange reaction, and the radiative capture reaction. The branching ratios for these are given by; 3 2 >33 I T - + p -»• n + T T ° ( 60.7 % ), (3.1) Tr- + p n + y ( 39.0 % ). (3.2) In deuterium there are again two main reactions 3 1*, namely; T T - + d •*• n + n T f - + d->-n + n + Y ( 73.7 % ) ( 26.1 % ) (3.3) (3.4) Although the pionic atoms ir-p and ir-d are small and neutral and as such are free to wander through the gas, for the purpose of kinematical calculations i t is sufficient to consider them to be at rest. In Tr-p radiative capture the single photon is monoenergetic at 129.44 MeV whilst the neutron has an energy of 8.9 MeV. The neutral pion decays almost instantly ( x = .83E-16 sec ) to two photons. Each of these photons has a center of mass energy of 67.5 MeV and are emitted back to back. Since the n° is semi-relativistic with a velocity of .2c, the lab photons range from 54.8 to 83.1 MeV. This energy range is referred to - 32 -as the pi-zero box. The neutron energy in this reaction is small at 0.42 MeV. The two neutrons of reaction (3.3) carry away 68.1 MeV back to back. Because of the strong f i n a l state interaction between the neutrons in reaction (3.4) the photon yield is peaked at about 130 MeV. [See Ref 35] . §3.1 Overview of Experimental Setup. The experiment was conducted at the Tri-University Meson Factory ( TRIUMF ) in May 1985. The high pressure gas target was positioned at the focus of the Ml3 pion beam line. The experimental setup i s shown in figure 3.1. The beam definition was formed by requiring a coincidence in s c i n t i l l a t o r s S1,S2 and S3 with s c i n t i l l a t o r S4 in anti-coincidence. A degrader was positioned between SI and S2 to slow the pions to the appropriate energy. SI was positioned near the beam snout and was a 41cm x 8cm x .65cm plastic s c i n t i l l a t o r . The degrader was a 2.75cm thick aluminum block and was positioned as close as possible to S2 and the target. S2 was a 3.8cm x 3.8cm x .16cm plastic s c i n t i l l a t o r and was positioned as near as the geometry would allow to the target. The photon detectors used were two large cylindrical sodium iodide crystals known as the TRIUMF Iodide of Natrium ( TINA ) and the Montreal Iodide of Natrium ( MINA ). TINA is the larger at 51cm x 46cm<(> and was located at right angles to the beam at the focus 98 cm from the target center. MINA measures 36cm x 36cm<J> and was at 180° to TINA through the target at a distance of 72 cm. These distances were selected to maximize the solid angle acceptance while s t i l l having a good time separation between photons and neutrons from the target. Both - 33 -TINA and MINA were shrouded by large Iron walls with opening orifices of 30ci»J>. This was done in an attempt to reduce background problems. In addition because of MINA's smaller diameter i t was necessary to i n s t a l l a 15cmJ> lead collimator. In order to veto charged events sc i n t i l l a t o r s were placed before the front faces of the crystals inside the iron boxes. FIGURE 3.1 Experimental setup, viewed from above. §3.2 Target Design. In order to have an appreciable number of pions stop in the target i t is desirable to have the gas as thick as possible which requires high gas pressures. However one would also like to have the gamma rays traverse through a minimal amount of material, so a compromise must be - 34 -made between wall thickness and density. The target vessel i t s e l f was turned from 7075-T6 aluminum which i s the alloy commonly used in aircraft applications due to i t s excellent strength-to-weight ratio. The brass flanges, see figure 3.2, formed a seal against the vessel and the light guides. External clamps were fastened to the light guides in an attempt to prevent inward movement when the target was under vacuum. B R A S S F L A N G E S A L U M I N U M T A R G E T V E S S E L S3 / , \ \ > ' \ \ V \ \ \ \ \ \ S3 L I G H T G U I D E S4 L I G H T G U I D E \ S I / A S E A L I N G O - R I N G S FIGURE 3.2 Target design. The light guides collected the light from the defining counter S3 and the veto counter S4 and transported i t to their respective photo multipliers. In an attempt to to reduce the hydrogenous background and minimize the number of stops in the sc i n t i l l a t o r i t s e l f , S3 was made - 35 -of a very thin ( 0.5mm ) deuterated plastic (CD) n. S4 was a 0.32cm ordinary plastic s c i n t i l l a t o r of closed cylinder shape with one face open. Scin t i l l a t o r detail is shown in figure 3.3. FIGURE 3.3 Scintillator detail. To prevent cross talk between the two scintillators an aluminum reflector was affixed to the end of S3. Tests were conducted with no beam to measure the cross talk. The Compton edges of 6 0Co and 1 3 7Cs were used to calibrate the two counters. Then using S3 as a trigger we examined the signal in S4. It was believed that the cross talk, i f present, occurred less than 5 percent of the time. §3.3 Selection of Beam Parameters. The TRIUMF cyclotron accelerates H~ ions which can be extracted by stripping off the electrons causing the protons to reverse their direction of curvature and leave the machine. The 500 MeV primary proton beam is then steered down a beam line and impinges upon production target Tl creating a copious supply of pions, neutrons and - 36 -secondary protons. The charged pions decay quickly to muons and the TT°+2Y decay generates electron pairs in any material. The Ml3 pion and muon beam line is shown in figure 3.4. It is tunable over a momentum range of 20 to 130 MeV/c. The pion f l u x 3 6 , Y(E) goes approximatly as, Y(E) « E 2 - 5 (3.5) at least up to energies of about 40 MeV. However above this energy i t flattens out reaching a maximum somewhere around 50 MeV. From a Monte Carlo analysis of the target as a sequence of degrading materials the fraction of pions stopping in the gas S(E) was seen to approximate to a function of the form, S(E) « E _ 1(AE/E) _ 1 (3.6) where the AE/E is the f u l l width half maximum ( FWHM ) of the distribution entering the gas. Hence the combined distribution took the form, S(E)Y(E) °= E _ 1 • 5(AE/E) - 1 (3.7) which indicated that a high energy i n i t i a l beam was favourable. Energy straggling in the degrader was a significant factor. Predictions for straggling due to large energy losses are d i f f i c u l t to make. Tschalar 3 7» 3 8 determined that AE/E could range from 5 to 15 percent. Taking 10 percent as a guide, and adding the variances, i t - 37 -FIGURE 3.4 TRIUMF M13 beam line. - 38 -appeared that the energy variance due to the beam momentum bite could go quite high without appreciably affecting the total energy spread. Another worry was the divergence of the beam due to multiple scattering in the degrader. Ideally at the focus the beam has 2.1cm FWHM horizontal and 1.3cm FWHM vertical profiles. Calculations of the standard deviation in the scattering angle indicated that the beam would not converge, but that this would not be a problem i f the degrader was close to the target. Based on the above analysis we planned to run with a pion energy of 40 MeV with s l i t s wide open and a 2.75cm thick aluminum degrader. It turned out that the s l i t s had been removed from the channel for a previous experiment so they were not replaced. The beam flux was about 60kHz through the defining counter S3. We then varied the momentum to maximize the I T 0 yield in hydrogen at 100 atm and the entire experiment was conducted using this single tune. §3.4 Gas Preparation, Handling and Mixing. The gases we used were isotopes of hydrogen, specifically H2,D2 and HD. The gases were bought in the case of H 2 and D2, with the manufacturers claiming purity levels of 99.9% and 99.5% respectively. The balance of the D2 was HD. The HD gas we manufactured ourselves. The apparatus for this is shown in figure 3.5. The HD is formed through the reaction 3 9, 4D20 + LiAlH^ •»• LiOD + A1(0D)3 + 4HD (3.8) with tetrahydrafuran ( THF ) being used as a catalyst. - 39 -Hg MANOMETER FIGURE 3.5 Apparatus for HD manufacture. The lithium aluminum hydride was mixed with the THF and then frozen by immersion in a liquid nitrogen ( LN2 ) bath. The entire manifold was then brought to vacuum by pumping and using an auxiliary pump in the form of a desiccant at LN2 temperature. This removed a l l undesirable gases. The mixture was then allowed to warm slowly and the D20 added in drops. The resulting gas was passed through the f i r s t cold trap and stored in the balloon reservoir. As the balloon f i l l e d the pressure could be read on the mercury manometer. As the pressure approached atmospheric the reaction mixture was again immersed in a LN 2 bath and the gas transferred via a glass cold trap and a s i l i c a gel cold trap into the reservoir. Mixing of gases and transfer to and from the target was accomplished using the gas manifold system illustrated in figure 3.6. After evacuating the lines a gas was transferred to the mixing vessel using either the natural bottle pressure or the compressor. The compressor was a single stage diaphragm compressor capable of pressures Q V A C U U M ^ T / C G A U G E T T / C T1 : <: <: *\'<: fu, ION G A U G E G A S I N L E T S u 6 COMPRESSOR V E N T - 41 -to 1300 atm. It pumped at lcm3/sec and so to make efficient use of time a reasonable back pressure was required. At any rate at least 3 atm backing pressure was required to operate the valves. After one gas was in the mixing vessel the lines were again pumped out and the second gas added to the lines with a positive pressure relative to that in the mixing vessel. This was done to reduce the error in concentration determination due to uncertain mixing in the narrow lines. The second gas was then added and the two allowed to mix. The pressure in the mixing vessel was measured using a Varian strain gauge transducer, accurate to better than .05 percent. It was noted that the pressure dropped gradually as a result of gas cooling, so presumably the gas was heated by the compressor. The ina b i l i t y to make consistent readings represented the largest error in the determination of relative gas concentration. After evacuating a l l the lines again the mixed gas was transferred to the target via the compressor and the s i l i c a gel cold trap. The f i n a l pressure in the target was read on a standard diaphragm gauge. We f i l l e d the target to 100 atm for a l l mixtures. In some cases, where the avai l a b i l i t y of gas dictated i t , upon completion of a run the mixture was returned to the vessel and further gas added. Samples of each gas were taken and their contents analysed on a mass spectrometer as an additional check of relative gas concentration. We measured a range of gas concentrations from pure D2 to pure H2. If we denote by C the molar fraction of deuterium then values of C measured were from 0.0 to 1.0 in steps of 0.1 with additional checks at 0.25, 0.45, 0.55 and 0.75. We also measured the HD gas. For pure HD the - 42 -molar fraction is just 0.5. A l l measurements were made at 100 atm gas pressure at room temperature. §3.5 Electronics and data Acquisition. A schematic of the electronics is shown in figure 3.7. In this diagram the squares labeled D or CFD represent discriminators. They are used to convert linear signals above a user set threshold into logic signals. The CFD's are constant fraction discriminators for which the timing of the output pulse is relatively independent of the size of the pulse, so these are used where the timing information is important. The triangles represent linear or logic fan-in/fan-out units. They are essentially OR gates with many outputs. The small triangles with variable arrows drawn through them represent amplifiers or attenuators. The circles indicate locations where items are to be read by the computer, into scalers, bit registers, ADC's or TDC's. The logical coincidence units are represented by the AND gates. The primary coincidence was the beam counters S1,S2 and S3. The veto S4 was not hard wired. In order to determine the total number of stops we sampled with a pulser the S1«S2«S3 coincidence. This became a strobe, known as the 'pulser' strobe, where by strobe we mean that an event has been tagged and a l l information about that event is read into TDCs and ADCs. Two other strobes were the 'singles' where the only requirements were that either TINA or MINA fired and there was a stop S1»S2'S3. The fourth type of strobe was a 'coincidence' which required that both TINA and MINA fired with the stop. The vetoes for TINA and MINA were not hard wired, but rather just registered as bits in the C212 for each event. - 43 -FIGURE 3.7 Electronics diagram. - 44 -There were two types of data read onto tape. The type 1 corresponded to strobe events. Whenever a strobe fired the ADC and TDC gates were set and the time and energy information for each counter read in. What follows is a summary of the information collected for each strobe. energies of the 7 TINA tubes, energies of the 7 MINA tubes, energies of the beam counters SI.... summed TINA energy, summed MINA energy, event bit pattern, time of flight to TINA, time of flight to MINA. time of RF strobe, two time delays, time of capacitive probe signal. 1) ETINA(l-7) 2) EMINA(l-7) 3) ES(4) 4) ETINA 5) EMINA 6) C212 7) TOFT 8) TOFM 9) RF(2) 10) CP Associated with each beam burst from the cyclotron is a signal. This signal is the 'RF' and has a period of 43ns. Also, in the beam line there is a capacitive probe which detects the presence of beam and issues a signal, called 'CP'. These signals are used to determine the beam content. The timing is determined by the S2 signal, so a l l times are relative to i t . The RF or CP spectra clearly show the difference in time taken for particles leaving the production target. In this way pions, muons and electrons can be separated by their time of f l i g h t . For a number of coincidences we set a bit i n a bit register, the C212, whenever a strobe fired. By examining the bit pattern we could then reconstruct the event. The bits that were collected into C212 are - 45 -listed below. 1) PULSER«S1»S2»S3 2) TINA*MINA*SI«S2»S3 3) TINA*S1*S2*S3 4) MINA*S1*S2*S3 5) S4 6) TINCH 7) MINCH Pulser sample of beam. Coincidence in TINA and MINA. TINA singles event. MINA singles event. Signal in S4. TINA charged signal. MINA charged signal. The type 2 events were just the scalers. They counted continuously only being written to tape when a register overflowed or at the end of a run. The scalers counted the number of signals for each item of interest. The following scalers were kept; 1) S1*S2*S3 5) TINA*Sl*S2*S3 2) S1*S2 6) MINA*S1*S2*S3 3) TINA 7) TINA*MINA*S1*S2*S3 4) MINA 8) PULSER*S1*S2*S3 The data acquisition was controled by a PDP 11/34 computer connected to a Camac logic controller, 2 disk drives and 2 tape drives. The data acquisition had priority over a l l other computer processes. Whenever a buffer was f u l l the buffer was transferred to tape. Only one event was handled at a time. If the computer was busy processing an event an inhibit was sent to the coincidence units to stop more events from piling up. When the computer was not busy, on-line analysis monitored the progress of the experiment. During the experiment we collected data in two different modes. In - 46 -one case we used a l l four strobes and gathered a l l information. This was known as the singles mode. In singles mode the data taking rate was very fast and contained few real events. Since we were primarily interested in coincidences from the ir° decay and wanted to get good statis t i c s on their number we limited the data taking by physically removing and terminating the two singles strobes from the electronics. This was known as coincidence mode. For each mixture we collected data in singles and coincidence mode. - 47 -Chapter 4 Analysis. The data was analysed using the TRIUMF VAX 8600 and VAX 780 computers. We used standard data manipulation programs with user supplied subroutines. The goal was to determine the hydrogen capture ratio F, the number of captures on a hydrogen in the target per stop in the gas. The number of stops was to have been calculated by counting the number of SI*S2«S3'N0T(S4) events. This turned out to be impossible as upon dismantling the target assembly at the end of the experiment a number of things were observed. F i r s t , the light guides must have slipped in when the target was evacuated as S3 was badly warped from pressure against S4. The veto counter S4 was not fixed to its light guide. However, due to the constrictive nature of the target geometry i t could not have moved far and so must have always been resting on the l i g h t guide with some o p t i c a l coupling. The movement of the sc i n t i l l a t o r s had two major repercussions. For one, with each new mix the degree of optical coupling between S4 and i t s light guide varied and thus so did the signal level. For another the two light guides were now in contact and depending on the level of contact there was a certain amount of cross talk between the two. This implied that S4 was no longer a perfect veto. The second strange effect noticed was that while S3 was warped, i t was s t i l l of good optical quality, whereas S4 was severely discolored on the surface. Presumably there was some reaction on the surface of S4 specific to this plastic rather than that of S3. It was not clear what caused the discolouration or when i t occurred as the target was not - 48 -disassembled u n t i l after a second experiment had been completed. The only gases that should have come into contact with the scintillators were Hj , D2 , HD, Ar, He and Xe. Also we used methanol to search for vacuum leaks and a brand name, "SNOOP", soap mixture to look for high pressure leaks. Both of these products were tested on fresh s c i n t i l l a t o r with null results. Another possibility considered was o i l from the compressor even though with such a low vapour pressure i t could only pass into the target in parts per million. Tests showed that o i l was not the culprit. Finally we wondered i f the hydrogen gases could be converting on the surface and causing the effect, but further tests showed that there was no conversion or discolouration in H2/D2 mixtures. The possibility of a contaminant in the HD gas was ruled out as HD from the same batch had been used in an earlier muon catalyzed fusion experiment. In that experiment any Z > 1 contaminant would have led to rapid muon transfer to the Z. This was not observed. §4.1 Determination of Gas Concentration. Originally we planned to determine the relative gas concentrations using both the measured pressures and a mass spectrometer analysis of the gas samples. Unfortunately the mass spectrometer was working poorly and the results were later found to be useless. In addition to ionizing the gases the molecules were being dissociated so that a l l the mass channels were being mixed. The only real problem this presented was that we could not determine the condition of the HD gas. It was not possible with our setup to manufacture pure HD. In the past our samples have had 93±2% HD with the remainder being two parts H 2 to one - 49 -part T>2 • Since the experiment which continued on after this consumed the remainder of the HD gas the exact concentration could not be determined. We surmised that the same conditions applied here and used these figures in our calculations. When measuring the gas concentration from the relative pressures there were two sources of uncertainty. First i t was important to realize that the readings had to made after the gases came to thermal equilibrium. When the gases were introduced to the mixing vessel i t took some time for the transducer reading to stabilize. This was attributed to the gas being heated as i t passed through the compressor. Since in the early going we did not recognize this effect and, in an attempt to optimize beam time, the mixing procedure was done as quickly as possible, there is an unavoidable uncertainty in the results. The other source of uncertainty comes from the transducer reading i t s e l f . Throughout the run we noticed that the calibration was fluctuating in an unpredictable manner. This was seen because, after every mix, the target was evacuated and the zero pressure readings did not agree. It was later discovered that this was due to problems with the power supply. We discovered ( too late ) that there must have been some strange ground loops in the local AC lines as when we used an extension cord and an outlet far from the experimental apparatus the problem was cured. §4.1.1 V i r i a l Corrections. It i s important to recognize that the determination of gas concentration from the pressures; - 50 -c d • «--h- ' <4-!> d p D2 2 i s only an approximation based on the ideal gas law PV = nRT. At pressures of 100 atm this is not such a good approximation. The correct treatment is to use the f u l l expansion observed for non-ideal gases; Pv = RT( 1 + § + ^- + ... ) (4.2) v v 2 where B and C are the second and third v i r i a l coefficients and v is the molar volume. It is sufficient to neglect the third coefficient and so we can write; Pv = RT( 1 + B'P ) (4.3) where now B' is the reduced coefficient B/RT. In theory we can calculate B' for a given mixture at a temperature T. Then i f we start with a mixture given by n d and n p we can write; P i(n d,n p)V = ( n d + n p )RT Q+B' ( n ^ n ^ P ^ n ^ ) ) (4.4) and upon adding a small amount of hydrogen, An p we get; P (n n ) n +n (1+B'(n ,n )P (n n )) _±__d__E_ = 5__E 2__E__±__a__E (A 5\ Pjr(n,,n +An ) n,+n +An (1+B*(n,,n +An )P^(n.,n +An )) v ' ' f v d' p p d p p d' p p' f v d' p p - 51 -In order to determine the concentration i t i s necessary to calculate the two second order v i r i a l coefficients. It i s sufficient to estimate the parameters of B 1(n d,np+An Q) by using the ideal gas law approximation. The calculation of B 1 is somewhat d i f f i c u l t . Various models are used to represent the molecular interactions. Among the best is the Leonard-Jones potential. A special form of this potential is often used to make the mathematics manageable. It is given by; r a 1 2 a 6 1 *(r) = 4e • 2 r j _ 2g. . ( 4 . 6 ) This i s a potential with a depth e and an attractive term for r > a and a repulsive term for r < a. Attacking this problem quantum mechanically gives the following expansion for B ' . B ' = B ' C £ ± A 3 B ' S + A 2 B * Q 1 + A 1 * B ' Q 2 + . . . (4.7) where B ' C £ is the classical limit and B ' Q J ^ and B'Q2 a r e t n e f i r s t and second order quantum corrections. The s t a t i s t i c a l term B ' S i s an added correction which takes into account the spin of the particles. It is positive for fermions. A is related to the molecular parameters e and a. For mixtures we calculate; B ' = B ' ^ 2 + B ' 2 X 2 2 + B ' ^ X ^ (4.8) where the X's are the molar fractions and B' 1 2 is estimated using the empirical mixing rules; a 1 2 = (a 1 a 2)/2 , e 1 2 = ( c ^ ) * ' 2 and A 1 2 is calculated using an average mass term. (4.9) Table 4.1 V i r i a l l y corrected deuterium gas concentration. Run #'s Pressure (psi) Concentration. Error 166,167 1500 0.995 ±0.005 62,63 1500 0.900 ±0.026 27,28,29 1500 0.811 ±0.032 168,169 1500 0.755 ±0.022 30,31 1500 0.713 ±0.034 32,33 1500 0.606 ±0.034 170,171 1500 0.559 ±0.016 35,36,37 1500 0.520 ±0.033 172,173 1500 0.460 ±0.014 38,39 1500 0.400 ±0.013 40,41 1500 0.316 ±0.011 174,175 1500 0.258 ±0.009 42,43 1500 0.216 ±0.009 44,45,46 1500 0.116 ±0.008 47,111,112,113 1500 0.000 ±0.005 The B' terms have been shown to be a converging series of gamma - 53 -functions multiplied by a power series in temperature with coefficients which have been tabulated 1* 0 • We calculated the the B' coefficients using the f i r s t 15 to 20 terms of the series depending on the rate of convergence. The v i r i a l coefficient for each mixture was calculated and the concentrations adjusted accordingly. The v i r i a l l y corrected concentrations and their associated errors are listed in table 4.1. §4.1.2 Concentration Error Analysis. The pressure was determined as a function of the transducer output voltage V and was given by; P = mV + b . (4.10) Then, since we are interested in the ratio C of i n i t i a l and f i n a l pressures, we write: C = ( mV± + b )/( mVf + b ). (4.11) Then the absolute error in C, AC is just given by; (AC) 2 = (Am) 2(6c) 2 + (AV ) 2(6C) 2 + (AV )(6C) 2 + (Ab) 2(«C) 2 . (4.12) 6m 6V± 6Vf <Sb where we have neglected the covariance terms. This is justified as the major error in the intercept was a result of the DC output from the power supply wandering whereas the error in the slope just represented the uncertainty in i t s measurement. - 54 -When the mixture was not a fresh mixture, but had been recirculated and additions made, the errors compounded. In these cases there was f i r s t an addition of D2, followed by an addition of H 2. If we label the intermediate pressure of this process P m then the fin a l concentration is just; P + ( C - 1 ) P. mV +b + (C.-l)(mVJ+b) r - m v i ' i _ m v i / v i ' , , i r»\ c _ ( 4 . 1 3 ) and the error becomes the obvious extension of (4.11); (AC) 2 = (Am)2r6C^,2 + (Ab) 2 r6C^ 2 + £(AV) 2 r6C^ 2 + (AC )2,6C >,2 (4.14) l-6mJ l6b J U v J 1 ^ c J where AC^ is the error determined for the previous mixture. §4.2 Analysis of the Data. We had to determine the number of pions which were captured on a proton in the gas. To do this we looked in TINA and MINA for the T T ° decay photons. To qualify as an event we required several things. First both particles reaching the Nal crystals were identified as photons. We also required that the event be neutral. This was done in software by requiring that neither of the veto counters of TINA or MINA fired. Then neutrons could be eliminated on the basis of the time of flight from the target to the detector. Secondly we demanded that the i n i t i a l particle entering the target was a pion. The beam content was determined by examining the RF time spectra and the non-pionic particles were discarded from the data set. A third condition imposed - 55 -was that the photons carry the correct amount of energy based on the kinematics outlined in chapter 3. In order to compare the number of captures on hydrogen between gas mixtures i t was necessary to normalize them to a common value. The normalization that we used was the total number of pions which stopped in the target. We then defined the probability of capture on pure hydrogen to be 1.0. In this way when we compared capture probabilities a l l solid angle and efficiency effects canceled. §4.2.1 Time Cuts. The start of the time gate was based on the arrival time of a strobe signal at the gate generator. The leading edge of the signal from the constant fraction discriminator of S2 determined the timing so a l l times were measured relative to the time a particle passed through S2. The timing stop was based on the arrival time of signals from the constant fraction discriminators of TINA and MINA. By examining the time of flight from S2 to the detector, neutrons and photons could be isolated from one another. A typical time spectrum for TINA is shown in figure 4.1. The photon peak i s easily distinquished from the neutrons. A clearer separation could be made at the expense of solid angle and thus rate, by moving the crystals further from the target. The cuts Imposed are shown on the figure as vertical arrows. The problem with the timing was that the strobes were not timed together so i t was necessary to have a different time window for each strobe. It was also a curious fact that the timing for the singles and coincidence runs were shifted by 1.5 ns. It appears that the number of - 56 -o o o FIGURE 4.1 TINA time spectrum. The first peak is the photon peak, whilst the second is the neutron peak. - 57 -strobes into the event fan-in unit affected the stagetime of the unit. The time of 1.5 ns is much longer than the manufacturers claimed time. In TINA the possible strobes were; TINA'MINA*SI*S2•S3 in both singles and coincidence modes, and TINA«S1*S2*S3 in singles mode only. In singles mode whenever there was a coincidence there had to be TINA*S1*S2«S3 strobe as well. Since the coincidence strobe arrived early i t dictated the timing in these cases. The same effect was seen with the MINA strobes, only here the TINA*MINA*SI*S2*S3 strobe was well timed to the MINA*S1*S2*S3 strobe, so i t was only necessary to have two time windows. One for singles mode, and one for coincidence mode. To try and make these cuts have the same efficiency the windows a l l had the same width and were centered about the photon peak centroid. The number of photons lost with the window size as shown in figure 4.1 is almost zero. §4.2.2 RF Cuts. As described e a r l i e r the RF and CP spectra allow us to discriminate between pions, muons and electrons in the beam. This is primarily of concern for the analysis of the randomly selected events which were sampled with the pulser. We shall refer to these events as pulser events. Cuts were made on the RF spectra for the other strobes as well although i t was very rare that a non pion could trigger an event which satisfied the other conditions in TINA and MINA. A typical RF spectrum is shown in figure 4.2. Most of the electrons have already been eliminated from the spectrum. This was done by raising the threshold of S2 so that only large non-electron pulses triggered the discriminator. The main peak in figure 4.2 contains the - 58 -FIGURE 4.2 RF time spectrum. Pions, muons and electrons are present. - 59 -pions, followed by a small muon peak. At the far edge some electrons appear. The pulser strobe was very early relative to the event strobes. This meant that we had to keep separate RF spectra for pulser strobes and event strobes. We recorded on tape the RF and CP times for pulser strobes and, with a different delay, the RF time for event strobes. The problem with the RF signal was that i t tended to wander quite a bit so each run had to be examined individually to determine the RF cuts. On several runs the RF peak shifted mid run and so portions of the data corresponding to the transition time had to be ignored. In some cases entire runs were useless. As we shall see the removal of muons and electrons is crucial to determining the number of stops in the target. The f i r s t pass at the data was to determine the RF cuts to be applied in the subsequent treatments, and to eliminate useless data. §4.2.3 Energy Cuts. The energy cuts were based on a knowledge of the kinematics and the spectra themselves. The kinematics were used to calibrate TINA and MINA. Using pure hydrogen we could clearly see the pi-zero box and the monoenergetic peak at 129.44 MeV. Because of the restricted geometry, only i f i t were approaching either TINA or MINA would a it" be detected in the coincidence mode. This is because neutral pions emitted at other angles would not have the correct photon opening angles to allow detection in both TINA and MINA. This means that only the lowest and highest energy photons are detected giving the characteristic coincidence spectrum shown in figure 4.3. In singles mode this is not the case and the entire spectrum is seen. A typical singles energy - 61 -FIGURE 4.4 TINA singles spectrum. - 62 -spectrum is shown in figure 4.4. The singles events are however contaminated with radiative capture from the deuterium or the containment vessel, so the cleanest identification of ir-p capture i s a coincidence event. The TINA and MINA energy resolution functions can be described by a gaussian distribution about the mean energy multiplied by an exponential drop-off at low energies. The gaussian term represents the natural line broadening and the exponential term represents the apparent lowering of energy due to bremsstrahlung from the shower escaping from the crystal without being detected. The un-normalized TINA resolution function is given by; F(E) = A exp((E-E(J)/B)( 1 - ERF((E-E0)/C) (4.15) where A is an amplitude term and B and C are representative of the widths of the peak on the low and high energy sides of the mean Ey. The energy spectra of TINA and MINA for pure hydrogen runs were fitted with this function to calibrate, the energy. The calibration was repeated a number of times throughout the experiment and found to be consistent. The f i t s to the energy spectra were done by f i t t i n g the end points of the TT° box in coincidence mode to 54.83 and 83.06 MeV and the single photon peak to 129.44 MeV. Fits to the single radiative capture peaks are shown in figure 4.5. From these we can see that the energy resolutions of TINA and MINA were 9 MeV and 8 MeV FWHM at 129 MeV respectively. The cuts which were applied to the pi-zero box are indicated in figure 4.3. As an additional energy cut we required that in each - 6 3 -FIGURE 4.5a F i t to TINA radiative capture peak. - 65 -coincidence one of the photons had to have low energy and the other had to have high energy. §4.2.4 The Stop D e f i n i t i o n . Because the damage i n the target to S3 and S4 brought the two s c i n t i l l a t o r s into contact, they were o p t i c a l l y coupled. This meant that S4 could no longer act as a simple veto as any s i g n a l that occurred i n S3 also appeared i n S4. Likewise there was s i g n i f i c a n t Tab! .e 4.2 Stopping profi. .e of target. Description of event. Raw energy ES3 ES4 With cross talk ES3 ES4 Approx. prob. TT- stops in S3 on carbon. med ium zero medium small 9% TT- stops in S3 on hydrogen. small zero small v. small <1% TT- stops in gas small zero small v. small 9% TT- stops in S4 on carbon. small large large large 80% Tf- stops in S4 on hydrogen. small medium large medium <2% Tf- stops in rear wall small small small small <1% non pions that escape cuts. ? ? ? ? -0% stops in front Al scat, into S3 ? 1 ? ? -0% stops in S2 scat, or random in S3,S4 ? ? 1 ? ~0% stops in rear Al random or scat into S3, S4. ? ? ? ? -0% - 66 -crosstalk from S4 into S3. In table 4.2 we l i s t the possible conditions in the target and estimate the relative contribution each w i l l make to the observed energy spectra of S3 and S4. In the following we shall refer to the energies in S3 and S4 as ES3 and ES4. The f i r s t thing to point out was that i t was impossible to directly compare the pulser strobes with any of the event strobes. This as we shall see was due to the fact that the pulser strobe created a timing situation so different that the ADC gate sampled a different part of the pulses for S3 and S4 and hence the apparent energies were different. In S4 particularly the gain shift was on the order of a factor of 2. The best way to examine the data was by making two-dimensional scatterplots of the energies in S3 and S4. These we shall refer to as ES3 vs ES4 scatterplots. A raw spectrum for the pulser type strobes is shown In figure 4.6. The same scatterplot is shown in figure 4.7 where we have required that at least one of the non-pulser strobes fired in anti-coincidence with the pulser strobe. For the purpose of discussion these scatterplots w i l l be referred to as the pulser and event scatterplots. The second pass through the data allowed us to remove the muons and electrons from the spectra. This removed the troublesome peaks outlined and labeled 3 and 4 in the pulser scatterplot. There was no noticeable effect in the event scatterplot. With the peaks 3 and 4 removed the two scatterplots have much the same appearance, except for the energy scaling. It was not at a l l apparent that the different energy scaling was due to a timing effect so we conducted a systematic - 67 -FIGURE 4.6 Scatterplot of ES3 and ES4 for pulser strobe. Peaks 1,2,3 and 4 are discussed in the text. If) . 10 s 3 FIGURE 4 . 7 Scatterplot of E S 3 and E S 4 for non-pulser strobe. Peaks 5 and 6 are discussed in the text. - 69 -search of the data to determine the nature of the peaks. The best source of information came from the data collected while doing the range curves. Since these runs varied in pion momentum we could approximate the location i n the target that each peak corresponded to. Referring to the schematic depicted in figure 4.8 there were approximately 8 peaks observed in a l l . These peaks had the following properties. ES3" — ES4 FIGURE 4.8 Schematic of ES3 vs. ES4. Fi r s t peaks B and C were eliminated when the RF cut was imposed. In addition they were only observed with the pulser and were favoured at low momenta. This led to the obvious interpretation that these peaks represented the beam contamination. These peaks were analogous to peaks - 70 -3 and 4 of figure 4.6. The exact origin of peaks A and E was not clear. They both appeared only in the very low momenta runs and also were recorded as rejected events. The most likely explanation was that they were neutrons in TINA and MINA from pions stopping in S2 and the front aluminum with some scattering or randoms in S3. The randoms rate was much higher since few of the pions were making i t to the gas. Peak D was only seen with an event and was maximized at the optimum momentum. This peak was thus identified as representing pions stopping in the gas or in S3. It turned out that since S3 was so thin there was not enough energy resolution to discriminate between stops in S3 and stops in the gas. This peak was analogous to peak 6 of figure 4.7. Peak F was very similar to D, having similar energy ranges and maximized at medium momenta. However i t was only seen with the pulser. It was given the same identification as peak D and corresponded to peak 2 of figure 4.6 Peaks G and H were also very similar to each other except for the large energy shift and the fact that G only occurred with a pulser strobe whereas H was associated with event strobes. Both of these peaks dominated at high momenta indicating stops in the veto counter. They are represented by peaks 1 and 5 in the pulser and event scatterplots respectively. Referring back to table 4.2, we reemphasize that the resolution of S3 prevented us from distinguishing between stops in S3 and stops in the gas. Also statistics did not allow us to observe different peaks for stops on carbon and hydrogen as the capture on carbon dominates that on hydrogen, and the energy resolution was probably not sufficient - 71 -anyway. Further tests supported the above conclusions. When looking at the background vacuum runs peaks D and F were greatly reduced relative to G and H indicating a loss of stops due to missing gas. In addition when searching for neutral pion events in pure deuterium the only events occurred in an energy band similar to H. These must have corresponded to stops on the hydrogen in the veto counter. When we examined the data from the second experiment where pure Xe was used, i t was seen that the peaks G and H virtually disappeared, indicating that the increased stopping power of the xenon was preventing pions from reaching S4. We f e l t that the above analysis was correct and that the only difference between figures 4.6 and 4.7 was due to the shift in energy caused by different arrival times of the ADC gate from the different strobes. We also believe that with the RF cuts the spectra are reasonably clean and only a very small fraction of the data could be attributable to the random events indicated in the last few rows of table 4.2. §4.2.5 Fitting the Stop Region. In the second pass through the data using the RF cuts determined from the f i r s t pass we collected into histograms and scatterplots a l l the available information about the energies deposited in the two sci n t i l l a t o r s S3 and S4. We also output the bin values of the histogram and scatterplot arrays so that they could be input into various f i t t i n g routines. We defined the stop region as the e l l i p t i c a l area in an ES3 vs ES4 - 72 -scatterplot corresponding to stops in S3 or the gas. The biggest problem with setting cuts on the stop region was that the cuts had to be different for each new gas mixture. This was because when gas was moved in and out of the target, the position of the veto counter S4 was altered, and so the degree of crosstalk between the two scintillators varied. In some cases the two peaks, representing the stop region and stops in S4, were reasonably well separated, but this was not always the case. Often the two peaks were very close together, making i t very d i f f i c u l t to f i t them. The f i r s t cuts made to the scatterplot and histogram arrays of the pulser and event strobes were rather broad two dimensional cuts to approximately isolate the region of interest. These cuts are referred to as the box cuts. Figures 4.9 and 4.10 are typical spectra of the energies ES3 and ES4 before the box cuts were made. The cuts applied are indicated by the box in figure 4.7. The results of the cuts to the data in figures 4.9 and 4.10 are shown in figures 4.11 and 4.12. The stop peak is now quite well defined. The data from the region to be fitted was input to a Minuit 1* 1 f i t t i n g routine. The data was f i t to a function composed of two gaussians plus a linear background. The f i r s t gaussian approximated the stop peak whilst the second was constrained to represent the t a i l of the S4 stopping peak. This was used as i t f i t the data well and was quite straightforward. The second gaussian told us what fraction of the stop peak was due to stops in S4. Other methods of f i t t i n g were tried. A three gaussian f i t where the additional gaussian allowed a separation to be made between stops in the gas and stops in S3 was unsuccessful as there were too many parameters, so that almost any f i t - 73 -E N E R G Y D I S T R I B U T I O N I 6000 20 0 0 _L S 3 R u n 3D T T " 200 400 600 800 E N E R G Y ( C h a n n e l s ) 1 ooo FIGURE 4.9 Energy d i s t r i b u t i o n of S3 before cuts, R u n 3D 4000 300 0 H 20 00 H t ooo H 1000 E N E R G Y C h a n n e l s FIGURE 4.10 Energy d i s t r i b u t i o n of S4 before cuts. - 74 -ES3. 6000 A F T E R S L A S H C U T S . J 1 i _ Ru n 30 5000 H Aooo H 3000 —{ 20 0 0 —\ 1000 —\ 200 400 600 800 E N E R G Y ( C h a n n e l s ) 1000 FIGURE 4.11 Energy d i s t r i b u t i o n of S3 af t e r cuts, ES4. 2500 2000 —\ 1 50 0 1000 —\ 500 A F T E R S L A S H C U T S . J _ J L R u n 30 200 E N E R G Y 60 0 ( C h a n n e l s ) 1000 FIGURE 4.12 Energy d i s t r i b u t i o n of S4 af t e r cuts. - 75 -could be had. Another method t r i e d was to remove the crosstalk, e f f e c t by f i t t i n g a parabola to the contours of an ES3 vs ES4 s c a t t e r p l o t and p r o j e c t i n g the data down to that l i n e . Then the converted data could be f i t using two gaussians as before. This method was a e s t h e t i c a l l y more pleasing but since the same r e s u l t s were obtained i n e i t h e r case i t was not employed. The important parameters derived from these f i t s were the mean and standard deviation, E Q and a r e s p e c t i v e l y . Both histograms, ES3 and ES4, were f i t and a stop defined by requiring that for a given event with energies ES3 and ES4 the following condition be met; With t h i s condition the data was constrained to l i e within the area of an e l l i p s e located at (E Q4,E 03) with major and minor axis of three standard deviations. For each mixture the data was f i t f o r both pulser and event strobes. Then these cuts were applied i n a t h i r d pass to the data. To determine the t o t a l number of stops i n the gas we had to scale the number obtained by f i t t i n g the pulser data. This was because we only sampled the beam with the pulser, and did not c o l l e c t information f o r each beam p a r t i c l e . The t o t a l number of stops NS was then given by the sum of the scaled pulser stops plus the number of stops which generated event strobes. The scale factor was given by the r a t i o of s c a l e r s , S c ( l ) / Sc(8) (see §3.5). Then the number of stops i s given + < 9.0 (4.16) by; - 76 -N S " f-fH NS . + NS (4.17) Sc(8) pul ev where NS p u^ and NS e y are the number of stops each for pulser or event strobes. §4.2.6 Comment on Error Analysis and Combination of Data. For a given concentration the number of pions stopping in the gas plus S3 region should have been a constant independent of whether we were running in singles or coincidence mode. This gave us a method to average the data. If NTT- was the number of incident pions coincident with the pulser then the ratio R; NS , should have been a constant. Since the ES3 and ES4 spectra were quite dissimilar for the two different data acquisition modes the f i t s varied slightly for what should have been equivalent data. In an attempt to average these runs we took a weighted average of the ratio (4.18) for runs of the same concentration. Then the weighted average <R> is given by; <R> = l-LUL**lll ( 4 1 9 ) < R > I (1/(AR)*) C 4 - i y j where AR i s the error in R. With this, the error in <R> is just; - 77 -( A < R > )2 = r ? f 7 i - - J I T (4.20) Then for a given run j , the t o t a l number of pulser stops i s just given by; <NS^ n> = <R> NTT- (4.21) pul and i n analogy with (4.17); N SJ = §£lU^ <NS J > + N SJ (4.22) Sc(8) J P U L From equation (4.18) the error AR can be approximated as; A D A N S 1 -- = 2 « i ( 4 .23 ) pul as the error i n NIT- was n e g l i g i b l e compared to the error of NSp U^. Then the problem was to determine the error associated with the f i t values N S p u l and N S e v « The Minuit routine calculated the best f i t to the data using a x 2 minimization technique. To c a l c u l a t e the associated error the parameter i n question was f i x e d to some value and the function minimized by varying the free parameters. The routine sought the l o c a t i o n at which the x 2 value had increased by 1.0. It found the l o c a t i o n on e i t h e r side of the mean where t h i s was true. These locations were the p o s i t i v e and negative e r r o r s . During the t h i r d pass at the data i t was necessary to determine - 78 -what e f f e c t these errors would have on the number of stops counted. To do t h i s the analysis program used the brute force method by c a l c u l a t i n g the true number of stops and the number for each of the other possible windows of d i f f e r e n t E Q or a. Then the errors i n the NS values were simply set to the maximum possible error (MPE) as determined by the greatest deviation between the true value and any of the other windows. The t o t a l error involved for a single measurement of the number of stops could now be calculated. A l l the errors for each variable i n equation (4.21) were added i n quadrature. We assumed Poisson s t a t i s t i c s when determining the errors of the scaler values, and as before ignored the error i n NTV- so that; A<NS ,> = <NS -> . (4.24) pul pul <R> §4.3 C a l c u l a t i o n of the Hydrogen Capture Ratio. In order to c a l c u l a t e the hydrogen capture r a t i o f o r a given mixture we counted the number of events stemming from a ir° decay and normalized them to the number of stops. To determine the number of events the f u l l energy, time and RF cuts were applied to the data. We also required that to be q u a l i f i e d as an event, i t had to be a stop. The background contribution was determined from empty target runs. The y i e l d of events per stop ( Y(x) ) for some mix x could then be given as; Y(x) = NTT°(X) / NS(x) (4.25) - 79 -where Nir° was the number of ir° events counted. If we define the capture rate on hydrogen to be identically 1.0, then the normalized capture ratio is given by; F(x) = ILZI-Z-ILP-I- ( 4.26) n 3 U Y(H2 ) - Y(0) where Y(0) i s the background contribution. The yield for the background runs and for pure deuterium was zero within the experimental errors, and so presented no problem to the data analysis. In addition for runs of the same concentration the yields Y(x) were calculated by taking the error weighted average. <v ( x )> = 0 _ X^)/iAY(x))£) <YQX)> £ ( i / ( A . Y ( x ) ) Z ) ^-tn and; (A<Y(x)»2 = l/( I( 1/(A Y(X))2 ) (4.28) The error AY(x) has been determined by adding the errors of NTT° and NS in quadrature with the error in Nir° just given by Poisson s t a t i s t i c s . Table 4.3 shows the fi n a l results for the capture ratios for each mixture and their associated errors. The value of F for the HD mix in table 4.3 has had an additional correction made to i t . The original uncorrected value Fyn was; - 80 -F° HD = .371 ± .011 (4.29) Table 4.3 Hydrogen capture ratios. Run #'s Concentration. F(H2D2),F(HD) Error 166,167 0.995 0.000 ±0.002 62,63 0.900 0.064 ±0.004 27,28,29 0.811 0.192 ±0.008 168,169 0.755 0.207 ±0.011 30,31 . 0.713 0.242 ±0.010 32,33 0.606 0.336 ±0.015 170,171 0.559 0.362 ±0.018 35,36,37 0.520 0.411 ±0.016 172,173 0.460 0.474 ±0.021 38,39 0.400 0.623 ±0.029 40,41 0.316 0.693 ±0.026 174,175 0.258 0.726 ±0.040 42,43 0.216 0.707 ±0.024 44,45,46 0.116 0.888 ±0.032 47,111,112,113 0.000 1.000 ±0.017 58,59 HD 0.355 ±0.021 This was corrected for the hydrogen and deuterium impurities. Since the impurity concentrations were small i t was safe to assume that the contributions to FHD from HD, H2 and D2 could be weighted by their relative concentrations. Then referring to figure 2.4 and - 81 -equation (2.42) we see that one can write; FHD ' CpdFHD + C p A + C d B ( 4 - 3 0 ) where A and B are given by; x m o 1 x a t X A . 2 + _2 E _ „ ( 4 . 3 1 ) xmol + x a t xmol + xat + x P P »P ^ P B = 1 » — (4.32) where now X p and X d are u-p and ir-d nuclear capture rates and X T and X T are the forward and reverse transfer rates. 1 2 Using the p r e d i c t e d values C p = 2C d and making the approximation that F H n (C=l) = ( A+B )/2 where C = Cd/Cp 2 2 we can write; FHD = CpdFHD +  2 C i \ l > ™ + C d A ( 4 ' 3 3 ) Averaging our result from chapter 5 with those from §2.7 yields; F u _ (1) = 0.430 ± .015 (4.34) H2D2 Term A is given by equation (2.43) as; - 82 -A = _ _ i j: B5_ 1 + " (§ "+ A)C which from our data from chapter 5 I f we use the value C p d = 0.93 ± .02 F„_ = 0.355 ± 0.021 nu where the errors have been added i n (4.35) must l i e i n the range 0.78 to 1.0. th i s gives the f i n a l r e su l t ; (4.36) quadrature. - 83 -Chapter 5. Results and Discussion. The hydrogen capture ratios that we measured as a function of gas concentration have been listed in table 4.3 and are presented graphically in figure 5.1. We see that the general trend is as expected, a monotonia increase in capture probability as the hydrogen content increases. The solid line represents the expected distribution of data for the case of no preferential i n i t i a l capture plus no transfer. We can clearly see that there is a significant deviation from this line. Since we believe that the i n i t i a l capture asymmetries are small we conclude that there is significant transfer occurring. We can calculate the probability of transfer given that the pion has been caught in an orbit about the proton. From the rate diagram of figure 2.4 we see that the above probability is given by; P = E2_2 C5 i) T 6 C + 6 ,C, + X .C, v J pp p pd d pd d which using the same notation as before is just; p _ AC /c o} T 1 + BC + AC v ' ' This can be expressed in terms of our measured quantity Ffl JJ 2 2 and using equation 2.43 as; PT " 1 " ( 1 + C ) FH 2D 2 (5.3) - 84 -FIGURE 5.1 Hydrogen capture ratio in mixtures of H 2 and D 2 . - 86 -In figure 5.2 we have plotted the transfer probability as a function of the relative concentration. The asymptotic behavior of the data is parameterized by the ratio A/(B+A) and is the probability that the pion w i l l be transferred to a deuteron in a coll i s i o n of pionic hydrogen with deuterium. In both figures 5.1 and 5.2 the dotted curve represents the best f i t to the data as determined by a x 2 minimization routine. The function was a f i t of n against concentration based on the 2 2 model developed in §2.6. The data was f i t with free parameters B' and A' to the function of equation (2.43). _ 1 + BJC . H2D2 1 + C r+"BTC +~A7C ^ } In order to relate B' and A' to the model parameters we must assume that the reverse transfer is small, that there is no i n i t i a l capture asymmetry, and that terms like the direct nuclear capture from the mesomolecular state and direct atomic capture contribute l i t t l e . Attempts were made to have more free parameters but these f i t s were not successful due to the scatter of the data. If we assume our model is correct then the values of B and A determined from the f i t are; B = 0.77 ± 0.14 A = 0.21 ± 0.04 The errors here were determined by the f i t t i n g routine Minuit in the same manner as outlined earlier. - 87 -§5.1 A Comparison of the Results to the Literature. We compare our results with those obtained by Petrukhin and Prokoshkin 3 1 and Aniol et a l 3 . Table 5.1 summarizes a l l the results. Table 5.1 Comparison of Results with Previous Measurements. Author B A F(H 2D 2) (a) F(HD) Petrukhin Aniol Current work 1.3 ±0.4 0.77 ± 0.14 0.4 ± 0.1 0.21 ± 0.04 .43 ± .02 .417 ± .004 .45 ± .01 .338 ± .008 .355 ± .021 (a) Taken at (C=l); measured by Aniol et a l , otherwise calculated. From the above table we see that our values for the parameters B and A are not in very good agreement with Petrukhin and Prokoshkin, but our errors are smaller. More importantly we point out that we calculate the asymptotic limit of the transfer probability to be PT(C->-<») = .21 ± .04 which is in very good agreement with the value from Petrukhin and Prokoshkin who calculate PT(C-»"») = .24 ± .07. Hence we are confident that this represents the true probability of transfer of a pion in the c o l l i s i o n w-p •*• d. Our capture ratio for H2D2 in equal parts is in agreement with that of Petrukhin et a l but the error bars are very large. Our result is not in very good agreement with the directly measured value of Aniol et a l , but a l l the results are clearly less than 0.5. The agreement is better in the case of the HD capture ratio, although again the - 88 -u n c e r t a i n t y i s l a r g e . We also note that our r a t i o of F J J n / F H J J i s i n good agreement with that of Aniol et a l . 2 2 We obtain 1.26 ± 0.13 compared to their value of 1.23 ± 0.03. §5.2 Limits on the Molecular Breakup Rates in HD. If we make the usual assumptions that direct atomic capture rates and direct nuclear capture from the mesomolecular states are small, but do include the possibility of reverse transfer we can write; FH 2D 2 " 2 1"+ B + A + 2 1 +~l +~T ' ( C = 1 ) ( 5 , 5 ) F HD - x r H - h + ( 1- x ) i-+-r+-f ( 5- 6 ) where X i s the HD molecular breakup rate; Trp i r d Then i f we c a l l the common terms in the above equations A 1 and A 2 then we can write; F = ( A + A 2 )/2 FHD X A l ^ 1 _ X> A2 we can solve for X and eliminate the term A1 to get; - 89 -FIGURE 5.3 Limit of molecular breakup rate. - 90 -x - i . (5.9) A plot of this curve is shown in figure 5.3. We see that for any reasonable value of the probability of breakup to a ir-p system is about 40%. Since we expect the reverse transfer to be small, in the range; A 2 e [0.0,0.2] , we find X e [0.314,0.397] . This implies that given a ir-HD system the pion w i l l transfer to the deuteron between 60 and 70% of the time. It is this intermolecular transfer that is attributed to the suppression of pion capture on hydrogen in HD relative to H 2 D 2 . It is possible of course to develop a theory where our assumed small terms actually play a significant role, but as pointed out earlier, there are experimental data on other molecules that indicate that these terms play a minimal role. §5.3 The Future of these Measurements. Since i t appears likely that experiments w i l l continue in this f i e l d , we would like to conclude with some comments on possible improvements which ought to be made to ensure better results. F i r s t in the experimental setup there must be a good stop - 91 -definition, so the operation of the internal scintillators i s crucial. Preferably they should be aluminized to prevent any cross talk. Since we have seen that our background is effectively zero, the addition of a very thin layer of aluminum to the scintillators should not be a problem. There should also be a convenient way of drawing a sample without having to break the vacuum in the rest of the system. This would be to allow for a quick analysis of the gas concentration. Also since i t appears that aluminum dissociates hydrogen the least i t would be a good idea to ensure that a l l mixing vessels and storage tanks are aluminum. It would also be useful to have a thermo-couple In the mixing chamber to ensure thermal equilibrium has been established before measuring the pressure. In the electronics the statistics could be improved by writing less to tape, so charged events should be hard wired and the 14 individual signals from TINA and MINA eliminated. This is mostly true for the singles rates as i f there had been enough real events on tape from the these runs the data could have been analyzed in another fashion. This method would have been to f i t the H2, D2 and background spectra individually and then do f i t s to the data to determine the relative amplitude of each of these components. Then knowing the branching ratios for the radiative capture rates the capture ratios could be calculated. Also with respect to the electronics i t would be nice to include timing information on both S3 and S4. §5.4 Summary• We measured the pion capture rate in mixtures of H2 and D2 and determined how the rate varied with concentration. We f i t the data to a - 92 -phenomenological model and determined the parameters. These gave us information about the transfer mechanism whereby the pion was observed to be preferentially captured on the deuterium. We also measured the capture rate in HD and probed the difference between i t s value and that of a mechanical mixture of equal parts H2 and D2. We surmised that the difference must have been due to the molecular breakup rate asymmetry. In conclusion we observed that the experiment was successful although the errors were necessarily large due to the damage in the target. The experiment should probably be repeated to reduce the errors and establish the true capture ratios. Means of improving the results in a subsequent experiment have been outlined. - 9 3 -References. 1. Zinov,V.G., Konin,A.D. and Mukhin,A.I. Sov. J. Nucl. Phys. 2, 613 (1966). 2. Evseev,V.S.,Mamedov,T.N.,Roganov,V.S. and Kholodov.N.I., Sov. J. Nucl. Phys. 35, 295 (1982). 3. Aniol,K.A., Measday,D.F., Hasinoff,M.D., Roser,H.W., Bagheri,A., Entezami,F., Virtue,C.J., Stadlbauer,J.M., Horvath,D., Salomon,M. and Robertson,B.C., Phys. Rev. A28, 2684 (1983). 4. Dearnaley.G. and Northrop.D.C., Semiconductor Counters for  Nuclear Radiations, Spon, (London), (1966). 5. Formulae and Methods in Experimental Data Evaluation. 1_, (Euro. Phys. Soc, CERN), (1984). _ 6. Korenman.G.Ya. and Rogovaya,S.I., Sov. J. Nucl. Phys. 22, 389 (1975). 7. 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Minuit, A System for Function  Minimization and Analysis of the Parameter Errors and Correlations. 

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