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Measurement of the Panofsky Ratio Spuller, Joseph Edward 1977

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A MEASUREMENT OF THE PANOFSKY RATIO by JOSEPH EDWARD SPULLER B.Sc, SUNY at Stony Brook, New York, 1971 M.Sc., Un i v e r s i t y of B r i t i s h Columbia, 1974 A thesis submitted i n p a r t i a l f u l f i l m e n t of the requirements for the degree of Doctor of Philosophy i n the Department of Physics We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA March 15, 1977 (c) Joseph Edward Spuller, .1977 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of Brit ish 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 representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department The University of Brit ish Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date 99? J9?9 i ABSTRACT The TT p -> Tr°n and ir:p -> yn have been investigated for stopped pions i n a l i q u i d hydrogen target. In order to separate the i n f l i g h t ir° -* yY decay (55 to 83 MeV photons) from the r a d i a t i v e capture photons (129)MeV), a large Nal(Tl) c r y s t a l of dimensions 46 cm cj) x 51 cm was used as a t o t a l energy detector. The energy r e s o l u t i o n was 6% fwhm at 129 MeV, and t h i s resulted i n good separation between the ir° photons,and the r a d i a t i v e capture photons. We show that the value of the Panofsky r a t i o i s P = 1.546 ± 0.009, where the uncertainty i s 50% s t a t i s t i c a l and 50% systematic i n o r i g i n . (2.1 x 10^ stopped pions were observed.) Data on the timing response and data for the t a i l region of the energy response of the Nal detector are also presented. The l a t t e r was acquired through an n-y coincidence. The timing response i s shown to be a s u r p r i s i n g l y good 2 ns and t h i s was achieved by using constant f r a c t i o n discriminators. A discussion of low-energy pion r e l a t i o n s i s included, and by using these r e l a t i o n s h i p s , i t i s shown that there i s an inconsistency i n the determination of the s-wave component of pion production i n the TT d pp r e a c t i o n near threshold. The discrepancy i s resolved, and i n order to do t h i s , i t was found necessary to allow an energy dependence to the s-wave pion production amplitude. The low^energy pion r e l a t i o n s are also used to r e l a t e the Panofsky r a t i o to the s-wave l ^ - 3 ^ s c a t t e r i n g length. I t i s found that ^ - 3 3 ! = 0.261 ± 0.005 — . mTt+c i i TABLE OF CONTENTS Page ABSTRACT i TABLE OF CONTENTS i i LIST OF TABLES i i i LIST OF FIGURES iv ACKNOWLEDGEMENT ' v CHAPTER 1 INTRODUCTION 1.1 The Panofsky Ratio 1 1.2.1 Other Low-Energy Ratios 5 1.2.2 The Ratio T 5 1.2.3 The Ratio R 8 1.2.4 The Ratio S 9 1.3 S-Wave Production in pp -> i T +d 11 CHAPTER 2 EXPERIMENTAL EQUIPMENT 2.1 Pion Beam Line 23 2.2 Hydrogen Target 26 2.3 Electronics and Event Definition 27 2.4 Stopping the Pions 30 2.5 Nal Crystal 33 CHAPTER 3 EXPERIMENTAL RESULTS 3.1 Backgrounds 43 3.1.1 Procedure 43 3.1.2 Low-Energy Photons 45 3.1.3 Neutron Contaminants 45 3.1.4 Electrons 47 3.2 Photon Spectra 49 CHAPTER 4 DATA ANALYSIS 53 CHAPTER 5 CONCLUSIONS 65 BIBLIOGRAPHY- 67 APPENDIX A 70 LIST OF TABLES Page Table (1.1.1) Measurements of the Panofsky ra t i o 2 Table (1.2.1) Experimental values for the low-energy pion ra t i o s . . . 7 Table (1.3.1) Comparison^ of low.energy polynomial f i t s to a^.(pp TT d) ~x 15 Table (1.3.2) Representative f i t s to a^.(pp T r d ) using (poly-nomial + Breit-Wigner) .. .' . 20 Table (4.1) Corrections applied in determining the Panofsky ra t i o 55 Table (4.2) Normalization factor for determining the high-energy background. 59 Fi gure Fi gure Fi gure Figure Fi gure Figure Fi gure Fi gure Figure Figure Figure Fi gure Fi gure Fi gure Fi gure Fi gure Fi gure Fi gure Figure Fi gure Figure Fi gure i v LIST OF FIGURES Page 1 . 2 . 1 ) Relations between low-energy pion reactions 6 1 . 3 . 1 ) Comparison of polynomial f i t s to a^,(pp -> Tt +d) ...... 16 1 . 3 - 2 ) Comparison of (polynomial + Breit-Wigner) f i t to Gj (pp TT d) 19 2 . 1 . 1 ) M9 beam 1 ine 24 2 . 1 . 2 ) Experimental layout 25 2 . 3 . 1 ) Schematic diagram of electronics 29 2.4 .1) Stopped pion rate.measurement 32 2 . 5 . 1 ) Response of the Nal detector to 11 .7 MeV photons ... 36 2 . 5 . 2 ) 129 MeV photon from n-y- coincidence (Run 1) 37 2 . 5 . 3 ) 129 MeV photon from n-y coincidence (Run 2) 38 2.5.4) Log plot of Figure ( 2 . 5 . 2 ) 39 2 . 5 . 5 ) Log plot of Figure ( 2 . 5 . 3 ) 4 0 2 . 5 - 6 ) Timing response of the Nal detector 42 3 . 1 . 1 ) Target-Empty data 44 3 . 1 . 2 ) Neutron spectra as a function of timing cuts 46 3 . 1 . 3 ) Low energy electron spectrum 48 3 . 2 . 1 ) Photon spectrum for Run (1) 50 3 . 2 . 2 ) Photon spectrum for Run (2) 51 4 . 1 ) Low energy subtraction for the photon spectra 57 4 . 2 ) Inflight correction for the radiative capture photons 59 6.1) S-wave cross sections for (TT p -*• Tr°n) and (TT p -> yn) 71 6 . 2 ) Spectral functions used for i n f l i g h t corrections'... 73 V ACKNOWLEDGEMENTS I take great pleasure in acknowledging the guidance of Professor David F. Measday in not only the writing of this thesis, but more importantly, in helping me to understand and gain appreciation for physical phenomona. I would also lik e to thank Professors D. Axen, D. Beder, and G. Jones for their time and effort in reading this manuscript and as acting as members of the supervising committee. As is customary in this f i e l d of physics, the experimental work was a collaborative effort, and my colleagues a l l contributed to the f i n a l success of the experiment. These included D. Berghofer, Dr. M.D. Hasinoff, Dr. J.K.P. Lee, R. MacDonald, Dr. J-M. Poutissou, Dr. R. Poutissou, Dr. M. Salomon, and T. Suzuki. For the work concerning the s-wave production of pions, I had a considerable number of chats with Dr. A.W. Thomas and I.R. Afnan, and I am especially grateful for the suggestion of the use of a Breit-Wigner form for p-wave production. I am also thankful to the NRC of Canada for financial support during my work. It would not be f a i r to not express my appreciation to the many friends who have also supported me during my stay in Vancouver. In their own ways they have also helped me in this effort. INTRODUCTION §1.1 The Panofsky Ratio The Panofsky ratio (P) was f i r s t measured in 1951 by Panofsky et a l . (1951) by using a pair spectrometer to detect the photons from the stopped TT p reactions (ir p -> Tr°n) and (TT p -»• yn) . At this early date in the develop-ment of pion-nucleon physics, i t was not yet established that the (TT ,Tr°,TT+) TT — trio were the sam? members of an isospin T=l, J =0 tr i p l e t of strongly inter-acting particles. Indeed, the f i r s t measurement of the Panofsky ratio found P=0.94 ± 0.30, and this was a puzzle because P i s essentially the ratio of the strong interaction (TT p ->• ir 0n) to the electromagnetic interaction (TT p -> yn) [See Equ. (1.1.1).] The implication was that there was a large suppression of the strong interaction. Nowadays, i t is well known that there is a relatively weak TTN interaction near threshold, but this was not so in the early 1950's. The original measurement of Panofsky et a l . has been repeated by many groups, and there has been a substantial improvement in precision over the years. [See Table (1.1.1) for a summary of the measurements.] However, some of the experiments have been in error, and this includes the measurement of Panofsky et a l . (P = 0.94 ± O.3BO) and a later measurement by Fisgher et a l . (P = 1.87 ± 0.10; J.pischer et; al..,. 1958). The latter value of P was deter-mined by using a lead glass Cerenkov counter as a total energy counter. The value of P = 1.87 was untenable with certain low-energy pion relationships (see below), and this caused a whole new series of experiments. Among the new values for P, there was the result of CocQoni et a l . (1961) who found P = 1.533 ± 0.021, and this value has been the most precise determination of the Panofsky ratio prior to this experiment. Shortly after the measurement of Panofsky et a l . , Anderson and Fermi (H.L. Anderson and E. Fermi, 1952) discussed a possible relation among low energy pion processes. By using the principles of detailed balance and charge 2 Values of Panofsky ratio 1 r 1 r k . w J I 1 1 I I I t I 0.6 0,8 1.0 U 1.4 1/5 1.8 2.0 Values of Panofsky ratio Table (1 .1 .1) Measurements of the Panofsky ratio Group Techn ique Value (a) Panofsky et a l . (1951) pai r spectrometer 0.94 ± O.3'0* . (b) Cassels (1957) Cerenkov-total energy 1.50 + 0 .15 (°) 'f&r.her et ;£b; .(1958) .Cerenkov-total energy 1.87 ± 0 . 1o " (d) Koller and Sachs (1959) Cerenkov-total energy 1 .46 ±0.10 (e) Kuehner et a l . (i960) pair spectrometer 1.60 ± 0 .17 (f) Derrick et a l . (1960) bubble chamber 1.51 ± 0 .10** (g) Samios (i960) bubble chamber 1.62 ±0.05 ** (h) Jones et a l . (1961) Cerenkov-total energy 1.56 ± 0.05 (i) Cocconi et al . (1961) Nal detector 1.533 ± 0.021 (j) Ryan (1963) pai r spectrometer 1 .51 ± 0.04 (k) This work Na1 detector 1.546 ± 0 .009 Weighted average 1.544 + 0.008 not included in weighted average detect the internal conversion electrons 3 independence of the strong interaction, they developed a simple relation-+ ship between the Panofsky ratio, the TT p elastic scattering amplitude, and the pion photoproduction amplitude. This relation may be used to test either the principle of detailed balance or, more l i k e l y ^ h e principle of isospin symmetry in the TTN system. The seemingly unrelated electromagnetic and strong interactions can be shown to be related by the following argument. Suppose that the Panofsky ratio which is understood to be p = P T°n) (1.1.1) CJ(TT p Y n) has the well defined limiting value = q(TT~p -*• Tr°n) (1.1.2) q _ ' - * TT 0 a(ir p -> yn) where q - is the cm. momentum of the incident ir , and cu is the transition TT rate. Furthermore, suppose that the ratio R which is defined as R _ a(yn -»• TT~P) (1.1.3) ^y ->• 0 a(yp -> Tf +n) also has a well defined value at threshold. It follows from equations (1.1.2) and (1.1.3) that a(TT~p -> Tr°n) = P«R-aT(yp -> Tr+n) (1.1.4) where indicates the time reversed reaction. By using the principle of the invariance of the strong interaction under the discrete operation of time re-versal, cfT is related to a via the equation a"^ (yp -> Tr+n) = g(iT +n -> yp) = 2 2 a(yp -> TT+ri) . (1.1.5) Equation (1.1.5) also follows from the weaker principle of detailed balance. It i s well known that in the isotopic spin decomposition of the TTN scattering 4 amplitude that the s-wave partial amplitude for fr-p -> TT°n near threshold can be expressed /~2/3(a - a ) to give 1 3 o(ir~p ->• Tr°n) = 4TT*(V /v ) • I ^-f- (a - a )| , (1.1.6) o - 3 i 3 where a and a are the isospin 1/2 and 3/2 s-wave TTN scattering lengths, 1 3 respectively; and V q and v_jare the cm. velocities of the i r 0 and I T - respect-ively. Now we suppose that the cross section for the s-wave photoproduction of T T + has the low energy behaviour f a(yp ir+n) = 4ir* q + | E + | 2 (1.1.7) where E + is the electric dipole transition amplitude. (For a complete dis-o cussion of the photoproduction amplitude, there is the review ar t i c l e by S.L. Adler, 1968). Equations (1.1.4) - (1.1.7) can be combined to give the relation 2 1 2 la - a | = 9-P-R |E +I (1.1.8) 1 3 1 J V V 0 o 1 Hence, one may use the experimental values of P, R, and E* to determine |a - a |. Now one can compare this result with the independent determination of |a - a | which comes from knowing |a | and |a + 2a | from the analysis of low energy ir +p and T7~p elastic scattering data. If an inconsistency i s found which can not be explained by either the uncertainties in the data or by the extrapolations to threshold, then one could question the vali d i t y of isospin symmetry in the TTN system. Since the experimental situation in the early part of the 1950's was unclear, Anderson and Fermi could not form quantitatively accurate conclusions with their analysis. However, today the experimental situation i s in a better position. For example, Rasche and Woolcock, 1976 have recently calculated a value for P in which they found that they need to treat the problem in a coupl-5 ed channel model in order to get agreement with P = 1.533 ± 0.021 (Coccbni et a l . , 1961). Their model coupled the (TT p -> TT p), (TT p Tr°n), and (ir p -»- yn) channels and used Coulomb modified s-wave TTN scattering lengths. The objective of this thesis i s to present a new experimental value for the Panofsky ratio, and i t w i l l be shown that i t is the most precise de-termination of the ratio P = CJ(TT~P -» Tr°n) (1.1.9) W(TT p -> yn) + U ( T 7 p -»• e +e n) for s-wave capture of the TT from atomic orbitals. With the advent of the new meson f a c i l i t i e s (LAMPF, SIN, and TRIUMF), this new value of the Panofsky ratio w i l l provide an important data point for checks on inf l i g h t measurements of the absolute cross section for IT p -> Tr°n and TT p -> yn. We conclude the thesis by returning to equation (1.1.8) to check the charge independence hy-pothesis. §1.2 Other Low Energy Pion Relations Besides the low energy pion relationship which connects P and R [Equation (1.1.8)], there are other relations which have been found among various low energy pion processes. The relations are given in Fig. (1.2.1), and the present experimental situation for the measurable ratios is given in Table (1.2.1). The most recent discussion of these relations has been given by Rose (CM. Rose, 1967), and further discussion may be found in an ar t i c l e by Cassels (J.M. Cassels, 1959). Some of these relationships are based on procedures such as detailed balance, (D.B.); charge independence, (C.I.); or extrapolation to zero energy, (E.Z.E.); a l l of which are nowadays quite well understood. §1.2.1 The Ratio T Since the value of T is not accessible by experimental means because 6 Figure (1.2.1) <r{y p - ~ 7r + n) }R cr(yr)-~if~ p) |D.B. c r ( 7 r - p ^ n 7 ) cr 7 cr o j { 7 r " p — n y ) |T a)(7r""d — nn/) | s co(7r""d — nn ) | E . Z £ . cr(7r~d — nn) |c.i. c r ^ d — p p ) C r ( 7 T ~ p — n 7 T ° ) I E . Z . E . CD (7r " " p —n77°) D.B. c r ( p p — T r ^ d ) Figure (1.2.1) Relations between low-energy pion reactions; C.I. = charge independence, D.B. = detailed balance, E.Z.E. = extrapolation to zero energy, The parameters R, P, T and S are the ratios for the processes as noted. 7 Table (1.2.1) The Low Energy Pion Ratios Symbol Definition Experimental Value K to(Tr~d •> nnir 0) co(Tr~d -> nnY) (5.8 ± 1.0 x IO- 4) 1 P to(Tr~p -> Tr°n) to(Tr~p -> ya.-) + o)(Tr~p -> e^e~ij> (1.533 ± 0.021) 2 (1.546 ± 0.009) 3 R o(y.nTT-p) a(yp TrTh) (1.34 ± 0.02) 4 S co(iT-d nil) u)(Tr~d -> nhy) (3.16 ± 0.12) 5 (2.89 ± 0.09) 6 T co(Tr~d nny) IO(TT~P n'y) (0.83 ± 0.08) 7 1. R. MacDonald et a l . (1977). 2. V.T. CocPoni et a l . (1961). 3. This experiment. 4. M.I. Adamovich et a l . (1975). 5. J.W. Ryan (1963) . 6. P.K. Kloeppel (1969). 7. R.H. Traxler (1962). This is a theoretical value. 8 of the very fast absorption rate of the ir , this number has only been deter-mined theoretically. Cassels has shown that a simple theoretical expression for T i s 3 T = io(TT~d ->• nny) = (2/3) (o(ir~p ->- ny) BP-S'1 o = .81 (1.2.1) where B^ and B^ are the Is Bohr r a d i i of the atomic IT p and i r _ d systems, re-o o spectively. The basic assumptions which led to equation (1.2.1) were (1) the impulse approximation is valid, (i.e. during the time interval in which the pion i s absorbed by the proton in the deuterium nucleus, the neutron behaves as a spectator) ,,and (2) the basic process of yp T + n i s described by the matrix element k(a*e) S(p + p - p _ -p ) where k is a constant, a is the spin of the nucleon, and e i s the polarization vector of the photon. The value of T has also been calculated by Traxler (R. Traxler, 1962), and he found that T=0.83 ± 0.08. Traxler's calculation differed from Cassels' theory in that he corrected for nuclear effects (i.e. N-N interactions in both i n i t i a l and f i n a l states, and Fermi motion of the proton), and for this reason we quote his value in Table 1.2.1. §1.2.3 The Ratio R Since R is defined as the ratio R q + threshold = g ( Y n * ^"P} ' Y ~ 4-a(yp •+ it n) i t - .can not^beadirectlysdete arexnof f reeco.neutron targets .to measure-the Y,n •Ss.TTdptireactdona 2.,. However, this problem has been circumvented by Adamovich et a l . (M.I. Adamovich et a l . , 1975) by measuring the relative yields of i r - / i v + in the yd •+ TTNN reaction for a number of different photon energies in the threshold region. Extrapolation to threshold was done by using 9 the theory of Baldin (A.M. Baldin, 1958) which corrected for Coulomb effects in the f i n a l state and residual NN interactions in both the i n i t i a l and f i n a l states. Adamovich et a l . have found that R = 1.34 ± 0.02, but the quoted error does not contain any estimate of the uncertainty which is introduced by the zero energy extrapolation technique. §1.2.4 The Ratio S In order to directly determine the value for S, where g = c o ( T r~d -> nn) , co(Tr~d ->- nny) i t i s necessary to measure the absolute rate of the neutrons from the (Tr~d -> nn) channel, and this has well known experimental d i f f i c u l t i e s . In order to circum-vent the neutron detection problem, a technique was exploited by Ryan (J.W. Ryan, 1963) which required that an equal number (N ) of TT~ stop in hydrogen and deute-rium targets. By measuring the number of photons (N and N ) in the reactions: 2 4 N = N (TT~P -* Tr°n) + N (iT~p -> yn) 1 2 N = N (TT -d -> nn) + N (Tr~d -* nny) , 3 4 Ryan showed that S = (1 + P) (N /N ) - 1, 2 1+ where N , N , N , and N are the number of interactions in the indicated channels. 1 2 3 4 Hence, the neutron detection problem is avoided, but there is the problem in de-termining the absolute value of the relative rate for stopping I T - in hydrogen and deuterium. Ryan found that S = 3.16 ± 0.12; but i f this value of S i s compared to Kloeppel's measurement of S = 2.89 ± 0.09 (P.K. Kloeppel, 1964), there i s a slight disagreement. Kloeppel's experimental technique consisted of stopping a TT - beam in a bubble chamber and measuring the internal conversion electrons. He was then able to deduce the value of S from the theoretical branching ratio of 10 [(u-d e-e+ + n+n)/(Tr _d -* y+n+n)] = 0.69% (D.W. Joseph, 1961). Thus Kloeppel avoided the "stopped TT rate" problem which is inherent in Ryan's technique. Nevertheless, Ryan's procedure is sound, and i f care is taken, accurate measure-ments can be made with this technique. As Ryan has pointed out in his paper, there i s the further complicat-ions of determining the purity of the deuterium target and the concurrent problem of the TT~ being transferred from the ir -p to the i r - d molecule as the Tr~p molecule diffuses through the liquid deuterium. This effect was thought to be small by Ryan since the transfer rate for u~p to u~d was calculated to be fylO10 sec - 1 (S. Cohen et a l . , 1959) and the Tr~p absorption rate was %10 1 3 sec - 1 (M. Leon and Hi' Bethe, 1962) . Ryan does not mention the more serious d i f f i c u l t y of the hydrogen contaminant being in the molecular form HD and the possibility of the TT~ condensing onto the deuterium nucleus with a higher probability than condens-ing onto the proton 1. This process would occur during the time interval in which the pion is i n i t i a l l y captured by the HD molecule to form a molecular orbit. At present there are no measurements or theoretical estimates of this phenomena, and i t is an open problem. Since Ryan's hydrogen contamination in the deuterium target was only 2.25% and his f i n a l uncertainty for the value of S was ±4.0%, his value is not seriously affected. Nowadays, the purity of D^  is 99.97%, and this need not be a problem in future determinations of S. 1. We have looked at this problem for the case of various concentrations of [H^], [HD], and [D-], and we have found indications that such a process can occur, although the evidence is not conclusive. There w i l l be future studies of this process by the TINA group at TRIUMF (spokesperson, Prof. D.F. Measday). 11 §1.3 On The S-Wave Pion Production In The (pp -> T^d) Reaction} In Rose's (1967) paper i t was shown that the values of R, S, and T r e l a t e cr(yp i r + n ) to a(pp ->- Tf +d) v i a the equation / + s H U M „ , .TT. a ( Y P - T r n ) = n n " ' (1 + ^ ) X < X + m i T / M ) - a ( P P + Tr+d) (1.3.1) (1 + mTr/2M) where m^  and M are the masses of the pion and nucleon, r e s p e c t i v e l y . This equation was then used to show the inconsistency between the yp T r + n data which gave a(yp •> TT+n) = (0.201 ± 0.005)n mb experimentally, and the a(pp Tr +d) data of Crawford and Stevenson (F.S. Crawford and M.L. Stevenson, 1955) which gave a(yp -> Tr +n) = (0.111 ± 0.02)n mb, where n i s the cm. momentum of the pion i n units of m^c. In an e f f o r t to remove the discrepancy Rose made a new measurement of a ( T r +d pp) for 0.15 < n < 0.5 ( i . e . 2 -> 16 MeV incident T T + k i n e t i c energies), and h i s data was analyzed with the equation a(/rnd pp) = \ P 2 (an + gn 3) (1.3.2) 3 I n where P p i s the cm. momentum of the proton i n units of m^c. He found that ct= 0.240 ± 0.02 mb and 6= 0.52 ± 0.2 mb. This phenomenological equation had f i r s t been suggested by Watson and Brueckner (K.M. Watson and K.A. Brueckner, 1951) and l a t e r reviewed by Rosenfeld (A.H. Rosenfeld, 1954) and by Gell-Mann and Watson (M. Gell-Mann and K.M. Watson, 1954). The theory assumed that the TTN force was of f i n i t e range and also that i s o s p i n , angular momentum, and 1. The r e s u l t s of t h i s section of the thesis have been published ( J . Spuller and D.F. Measday,' 19 7<|) . 12 parity were conserved. These assumptions lead to an s-wave production of pion which goes as o^n and p-wave production which goes as a'vn 3. Hence, a(pp -> TT+d) = an + Bn3 , (1.3.3) and equation (1.3.2) follows by detailed balance. The value of a found by Rose satisfied equation (1.3.1), but a more recent determination of a and 8 was done by Richard-Serre et a l . (Richard-Serre et a l . , 1970) who found a = 0.18 ± 0.02 mb and 8= 0.95 ± 0.15 mb. This new value of a renewed the inconsistencies in equation (1.3.1). Since Reitan (A. Reitan, 1965) had shown that a could only depend slightly on energy, the analysis of Richard-Serre et a l . used the same phenomonological form as equation (1.3.2) to determine their values of a and 8. Now a recent calculation by Afnan and Thomas (I.R. Afnan and A.W. Thomas, 1974) has shown that a can vary quite significantly with energy, even near threshold. Although their results differ according to the nucleon-nucleon potential used to derive the deuteron wave function, nevertheless a common feature i s that a f a l l s and almost linearly with increasing n for 0<u<0.6. If one inspects the f i t s by Richard-Serre et a l . , i t i s quite appar-ent that the very-low-energy data l i e above the f i t t e d curve, while the medium-energy data (nfy0.4) f a l l below. Afnan and Thomas themselves noted that their energy dependence of a could explain the discrepancy between the high value of a obtained from the low-energy data of Rose (0.15<n<0.5) and the low value of a obtained by Crawford and Stevenson (1955) at a higher energy (0.38<n<0.58). The Crawford and Stevenson analysis covered a narrow energy band, and their value of a should not be taken as conclusive because the data at this energy are not sensitive to s-wave production. The analysis of Rose was f i r s t done with his own data over .15<n<.5 in order to determine that a=0.24 ± 0.02 mb, but his value of 3=0.52 ± 0.2 mb is not acceptable when higher-energy data 13 are included. He then proceeded to f i t higher-energy data (n < 1.6) by fixing a, but this is logically unsatisfactory within the framework of the model of Gell-Mann and Watson which has a and g as constants. The best value of a i s that obtained by Richard-Serre et a l . who applied the model consistently over an extended data set (0.15 < n < 2.1) to find ct=0.188 mb and 8=0.90 ± 0.16. I have thus returned to this problem to see i f i t was possible to reconcile the low-energy pion cross sections. I started off using the data compilation of Richard-Serre et a l . as a basis. This compilation consisted of most of the data existing at the time except for gentle pruning of very old data which either was incomplete in the angular distribution of completely i n -consistent with the majority of the data. I have also included the the datum of Dolnick (1971) at n=1.0 which was only available in a preliminary form at the time of Richard-Serre et al.'s (1970) analysis. Now the most v i t a l set of data i s the very-low-energy experiment of Rose; although I have c r i t i c i s e d his analysis, nevertheless his data remain the keystone of my discussion. At the time my analysis was done the recent data of Preedom et a l . for T7+d -> pp at = 40, 50, and 60 MeV (B.M. Preedom et al.,1976) and Aebischer et a l . (1976) for pp -> Tr +d at E p = 398, 455, 497, 530, and 572 MeV were only available in preliminary forms; they are not included in my f i t s to the data. We have also omitted the datum of Axen et a l . at E =47.5 MeV who measured a ( T r + d pp) = 5.9 ±0.2 mb. A l l these new data points are consistent with this analysis to within the experimental uncertainties. 1.3.1 Data Fitting F i r s t I tried to reproduce the CERN f i t s of Richard-Serre et a l . to check the data set and the least squares f i t t i n g routine. I thus f i t the data 14 to the form (1.3.1) Table 1.3.1 gives the CERN f i t 2 together with the same data set; the agreement is adequate and the samll discrepancy i s probably due to minor differences in the f i t t i n g techniques. One slight difference i s that I have applied the Coulomb corrections of Reitan to the data directly; this procedure is possible because the difference between the s-wave and p-wave Coulomb corrections is negligible i n the energy region where the data exist. fore does my f i t A; when i t i s included I obtain f i t B, Table 1.3.1., and note that i t makes l i t t l e difference to the numerical value of the parameters, but i t does add considerably to the x 2« In order to test for the energy dependence of a, I allowed a to have an energy dependence of the form A (1232)] i t was f i r s t decided to follow Richard-Serre et a l . , using a poly-nomial expansion However, i t was found that this allowed too. much freedom in the f i t t i n g function when a l l the coefficients were let free. It was then decided to investigate the prediction of Afnan and Thomas that a had a negative gradient. The data were fi t t e d over n<_1.55 and included the datum of Dolnick. Holding ct^  fixed at the value of -0.2 gives the following result, f i t C: The CERN f i t 2, of course, omitted the datum of Dolnick and so there-a = aQ + a^n. (1-3.4) To approximate the shape of 'the peak in the cross section [related to the 3 = B0 + B^n + B 2n 2. (1.3.5) r. P. 2 a ( T r d -> pp) = (2/3) P [ (0.247 ± 0.017)n -0.2n2 + n (0.6 ±0.3) n3 + (1.0 + 0.5)n4 - (0.6 ± 0.2) n 5 ] mb, 15 Table (1.3.1) CERN 2 Fi t A Fi t B a 0.188 ±0.024 B 0 0.90 ± 0.16 62 -0.050 ±0.045 X2/v £.05 0.186 ±0.010 0.93 ± 0.06 •0.07 ± 0.03 1.1 0.180 ± 0.011 0.94 ± 0.07 •0.09 ± 0.03 1.4 Table 1.3.1 Comparison of low-energy polynomial f i t s to Eq. (1.3.1) for n< 1.55. The f i r s t column i s f i t 2 of Richard-Serre et a l . The second column is my f i t A using the same parameterization, while the third colomn i s the same as f i t A except the data set includes the cross section of Dolnick. 16 Figure (1.3.1) LABORATORY PION ENERGY (MeV) 0 2 10 20 50 100 E .10 Q. a + z: o j~ o LU 00 CO o rr o O 8 CERN fit 2 fit C, present work Rose • Sachs A Fields • Durbin & Dolnick Stadler © Meshcheryakov A Crawford & Stevenson H3 f ,1 0 Figure (1. our f i t C. J I I I 0.5 1.0 1.5 V 3.1) Comparison of the CERN f i t 2.of Richard-Serre et a l . with The experimental data points have been corrected for Coulomb e f f e c t s . 17 where Pp i s the proton cm. momentum in units of m^c. The x /v was 1.1 for 26 degrees of freedom. Although this does not represent an improved (nor worse) f i t than that of Richard-Serre et a l . , as judged by a x 2 c r i t e r i a , i t does indicate that the discrepancy among the processes (yp iT +n) , ( i r + d -> pp) and (TT ±P -> Tr±p) can be removed by allowing a significant energy dependence to the a parameter. Figure (1.3.1) shows a comparison between f i t 2 of Richard-Serre et a l . , and our f i t C. The difference i s very small, and we are forced to conclude that the value of ct Q i s sensitive to the model used in the data analysis. It i s also clear that i f one wishes to distinguish between these solutions using total cross section data only, then a factor of ten improvement in the experiments w i l l be needed. (This is the reason that the exclusion of the new data of Preedom et a l . and Aebischer et a l . does not affect my results.) Since the use of nonorthogonal polynomials does not lead to easily interpretable results, I was not able to obtain consistent values of the coef-ficients by systematically increasing the domain of the f i t t i n g region, and correspondingly, the order.; (This does not help as i t would s t i l l be neces-sary to extract the physical quantity of interest in a model dependent fashion.) I was then led to try a different parameterization in an attempt to decouple the s-wave and p-wave parts of the cross section. I chose to add on a Breit-Wigner function to a low-order polynomial. The kinematics were handled in a r e l a t i v i s t i c manner, and the appropriate penetrability factors were included to ensure that the Breit-Wigner term would have the proper threshold behavior for p-wave pions (e.g., a^ri 3for I T K ) ) . This was done by introducing a pen-etrabi l i t y factor (nR) 2^ /D^  for the angular momentum barrier i, where R i s the radius of interaction. The D,£ were approximated by (R.D. Tripp, 1966) : 18 DQ = 1 = 1 + ( n R ) 2 D2 = 9 + 3(nR)2 + (nR)4. The phase space introduces another factor proportional to n so that r = Y [ ( n R ) 2 £ + 1 / % ] , where y i s assumed to be a constant. The f i t t i n g function for the reaction (Tr +d -»- pp) was therefore P 2 o = j — (a 0n + o^n2 + a 2n 3 + o^n4) n 2 + G(s)ir,x^+ r e I r r ( 1 > 3 > 6 ) (E-E R)2 + r 2 /4 r e l = Y e l [ (nR) 3/(i + (nR)2) ], r r = ky T (RPp), r T = Y t [ (nR ) 3 / d + (nR)2) + k ( R P P ) ] , R = channel radius in units of : 1h/i%c Pp = cm. momentum of proton in units of m^ c, E R = resonant energy, G(s)= = | (23^ + 1) (2Sd +1) k = fraction of total width from ir+d p+p channel where 19 Figure (1.3.2) LABORATORY PION ENERGY (MeV) 10 50 100 150 200 300 400 Figure (1.3.2) A comparison of our f i t F with CERN f i t 3 and with existing data for the total cross section for (ir+d -> pp) . The dotted portion of CERN f i t 3 specifies a region of extrapolation. The experimental data points have been corrected for Coulomb effects. Table 1.3.2 FIT ao (mb) a l (mb) °2 (mb) a 3 (mb) Y e l K c 2 ) E (HeV) R k X2/V D 0 .29 ± 0.05 -0.8 ± 0.3 0.02 ± 0.1 0.5 + 0.1 0.57 + 0.04 2181 ± 7 1.4 0.05 1.2 E 0 .30 ± 0.05 -0.8 + 0.3 0.3 + 0.1 -0.01 + 0.04 0.5 + 0.1 0.56 + 0.05 2181 + 7 1.4 0.05 1.2 F 0 .27 ± 0.04 "0,5 ± 0.2 0.05 ± .0.1 -0.03 + 0.1 0.06± 0.15 0.71 ± 0.06 2183 + 8 1.2 0.05 1.2 G 0 .26 ± 0.04 -0.4 + 0.2 0.08 + 0.06 0.6 + 0.15 0.92 + 0.08 2179 ± 7 1.0 0.05 1.35 Table 1.3.2 Representative f i t s to the t o t a l cross section for the reaction ( i r a p+p) using a low-order polynomial with a Breit-Wigner function, i . e . , Eq. (1.3.6). Quoted errors correspond to one standard deviation. 21 I am assuming that the widths for most of the reactions (e.g.., IT d •> Tr+pn) have the same energy dependence as the elastic channel, but the reaction Tr +d -> pp is treated differently because the phase space and penetrability of the (p+p) channel w i l l have l i t t l e energy dependence. It should be emphasized that the choice of the Breit-Wigner function i s predicated on the notion that the p-wave ird interaction i s strongly influenced by the resonant p-wave TTN interaction. The situation i s undoubtedly more complicated than this, and i t w i l l not be possible to ascribe physical significance to the reduced widths. This i s not a serious fault as I merely wish to investigate the behavior of the a 0 coefficient for low values of n . The penetrability factor for T r was set equal to 1 because Pp = 2.63 ( %370 MeV/c), even for the case of n=0. i t was not possible to allow the par-ameter k to be free in the f i t t i n g routine, and I set i t to various values between 0.03<k<0.07, these limits being educated guesses. It was found that the coefficients were not sensitive to i t s value. Furthermore, the parameter R was treated in a similar fashion. With this parameterization i t was possible to f i t over the region n<2.46 (41 data points) and obtain quite reasonable re-sults (x 2/v %1.2). This is a much larger f i t t i n g domain than that used in any other analysis. Some representative f i t s are given in Table (1.3.2), and a plot of f i t F i s given in Figure (1.3.2). A common feature of a l l the f i t s was the larger value of a 0 as compared to Richard-Serre et a l . In conclusion i t can be affirmed that the experimental data for the total cross section prefer a negative value of c|, as predicted by Afnan and Thomas. According to the other assumptions made in the analysis one can obtain a value of a 0 between 0.2 mb and 0.3 mb, and so i t is probable, although not certain, that the low energy relations can now be satisfied. To f i x the value of a 0 more firmly i t i s possible that careful measurements of the absolute d i f -ferential cross section for the reaction (pp -* ir +d) w i l l help i f accurate 22 measurements can be made from threshold at 287.5 MeV up to about 300 MeV ( i . e . , n_<0.3). However, i t i s more l i k e l y that a d e f i n i t i v e analysis w i l l have to await experiments using polarized beams with polarized targets, so that the s-wave and p-wave IT production can be disentangled. 23 §2 Experimental Equipment In this experiment I have measured the relative rates for the charge exchange reaction Tr~p ->• Tr°n and the radiative capture reaction Tr~p -> yn for a T T - which came to rest in a liquid hydrogen target. This was done by measur-ing the energy spectra of the photons emanating from the target with a large 45.7 cm (j) x 50.8 cm Nal(Tl) crystal, and thereby, deducing the number of TT° from the TT° -> yy decay. Since the TT0 from the charge exchange reaction at rest has a velocity of (v/c) fy 0.2 and an isotropic angular distribution, the u ° photons are Doppler shifted to give photons which have a uniform distribut-ion between 55 and 83 MeV. Of course, the radiative capture process results in a mono-energetic photon of 129 MeV, and the chief experimental task is to sep-arate these two different reactions on the basis of energy resolution. For this reason the Nal crystal was crucial to the measurement, and I devote a separate section on i t s performance (i.e., energy and timing responses). §2.1 The Pion Beam Line This experiment was performed at the TRIUMF Project, and in particular, the experiment was set up in the meson ha l l where I used the stopped TT/U channel (M9) which takes pions off the T2 target. The protons were accelerated to 500 MeV, and the beam had a flux of about 5 x 10 1 2 protons/sec (0.8 micro-ampere) . The protons were focussed onto a 10 cm beryllium target which was chosen as i t gives a relatively large number of TT - per incident proton. The secondary beam was focussed for negatively charged particles with a series of magnets. See Figure (2.1.1) The beam focussing was achromatic, and i t s profile was defined by a beam collimator which was set to a 10 cm width in the bending plane. This resulted in a 13 cm fwhm spread in the beam at the hydrogen target. The magnets were set to give a beam of momentum 24 Figure (2.1.1) prolons 1  M9 B EAM LINE Figure (2.1.1) Layout of M9 channel. The B3 dipole is a 30 cm gap cyclotron magnet, and i t was not used in this experiment. See Figure (2.1.2) for a de-tailed layout of the target area. 25 Figure (2.1.2) Neutron counter C3 1.4mm A l vacuum vesse C1 CO Hydrogen A l target Lead collimator^ ^ 25cm Iron shielding Figure (2.1.2) Layout of the target area. C0(15 cm x 15 cm x 0.16 cm), Cl(10 cm x 10 cm x 0.16 cm), C2(6.4 cm<j> x 0.16 cm), C3(20 cm x 20 cm x 0.32 cm) plastic counters were used to define the incident TT- beam. The C4 counter was used as a charge i d e n t i f i e r . 26 130 MeV/c (51 MeV Tr -), but there was a large momentum spread (%15% fwhm) which was a consequence of the large beam collimator; but this was tolerable as i t was necessary to keep the TT" flux high. (It was about 80 x 103/sec.) The beam's mean energy was determined for the magnet settings by range-energy and time of f l i g h t (tof) techniques and has an experimental uncertainty of 4% (i.e., 51 ± 2.0 MeV for the case of TT~) . The electrons in the beam constituted 22% of the total flux, while the u~ contaminant was only about 6%. These estimates come from the time of fl i g h t measurements and do not include the number of u~ which have as their source the inf l i g h t decay TT- u~ +v. The distance from the last bending magnet to the hydrogen target was 4.5.meters, and i t i s estimated that 50% of the incid-ent TT~ decayed before hitting the hydrogen target. In order to define the entrance of a pion into the hydrogen target, three plastic counters (NE 110) were placed on the upstream side of the target. (See Figure (2.1.2).) The C2 (6.35 cm <j> x 0.16 cm) counter placed the tightest constraint on the incident pion's trajectory, and i t s small size was chosen in order to be sure that the pion hit the 11 cm <f> hydrogen flask. Between the Cl and C2 counters there was a 2.86 cm slab of aluminum which was used to degrade the incoming pions. The C3 counter was used as a veto counter and was located 45 cm from the back end of the target. A l l the plastic counters were viewed by RCA 8575 phototubes. §2.2 Hydrogen Target The hydrogen target was cooled by a CTi (Cryogenic Technology, Inc., of Waltham, Mass.) Cryodyne Helium Refrigerator (model 1020). This unit has a two stage cooling cycle with the f i r s t stage cooling the thermal radiation shield of the condenser, while the second stage cooled the H 2 gas to liquid temperatures. The f i r s t stage could handle a 25 watt heat load at 77 K, while the condenser was able to handle a 10 watt load at 20 K. By thermally 27 isolating the target flask (1.2 l i t r e s ) in a vacuum and surrounding the flask with 10 layers of 6.4 microns of aluminized Mylar sheets, the heat load on the condenser at equilibrium was kept to about 1-2 watts. The target was run slightly above 1 atmosphere, in order to completely avoid the possibil-i t y of leakage of air into the hydrogen reservoir. The target flask was constructed out of Mylar sheets and had cylind-r i c a l symmetry. (There was a slight dome shape to the front end of the tar-get.) The target was 15 cm long and had a diameter of 11 cm. The side walls were 360 microns while both the front and back were 250 microns. Surrounding the flask there was a 1.40 mm aluminum vacuum vessel except at the rear of the target where there were two layers of 250 microns of Mylar. [See Figure (2.1.2).] §2.3 Electronics and Event Definition A schematic of the electronics used in this experiment i s shown in Figure (2.3.1). For the sake of simplicity, the electronics may be divided :p into 5 categories: 1. identify a possible TT - stop in the liquid hydrogen target, 2. determine energy of the photon, 3. determine the time of fl i g h t of the photon, 4. process data in PDP11-40 computer to produce on line information, 5. record data on DEC tape for future analysis. An event was defined as a Cl*C2»C3«(NaI or neutron counter) coincidence. The occurrence of an event resulted in the clipped anode pulse (250 ns) of the summed Nal signal ibeiing recorded in an LRS 2248 ADC CAMAC module. A fast gate (^0 ns) was provided by the event definition logic in order that the di g i t a l encoding of the Nal could proceed. There was also a CAMAC scaler which was set i f the C4 counter fired, thus indicating i f the particle which had entered 28 the Nal crystal was charged or uncharged. This scaler was reset to zero after each event. The event rate was recorded as well as the C1-C2-C3 coin-cidence rate in CAMAC scalers, and these scalers were operative as long as the computer was free to accept events, otherwise they were inhibited. The event signal also served to start the CAMAC TDC. module. Three stop signals were recorded: 1. C2 2. Nal 3. Neutron counter. Each of these signals came from a constant fraction discriminator?. (Ortec 463). The actual time spectra for the photons or neutrons were determined by using the C2 counter as a start signal, since the event signal had poor timing quality. The computer read the CAMAC modules and then reset the appropriate electronic unl't?s to their i n i t i a l states so that they could be ready to accept a new event. The maximum data taking rate was about 15 events/sec, and dead time in the reading cycle was negligible. After completion of the read cycle, the computer processed the data to form histograms. During this time interval the CAMAC crate was inhibited, but the data taking rate was sufficiently slow that this did not seriously impede the experiment. In fact the on-line his-tograms were quite useful and necessary for understanding the progress of the experiment. After the accumulation of 145 events, the computer wrote a buffer containing these events onto a DEC tape. The mechanical operation of moving the tape drive added considerably to the dead time and gave a 40% lower data taking rate than otherwise could have been realized. The DEC tapes were later transferred to another magnetic tape which was compatible with the IBM 370 system at the University of British Columbia Computing Centre. 29 Figure (2.3.1) C 3 N a l Anolqgue S igna l I Beam Counters C.2 C l C O Disc. LRS-621 CAMAC C o m q ADC LRS 2 2 4 8 Disc, (con. frac) ORTEC 463 Gate if i< v Logic OR LRS-465 Unit LRS429 LRS 3 65 OR LRS429 Con. F rac Disc. ORTECl 4 63 EVENT -4v-Con. F r a c Disc. ORTEC 4 6 3 Logic Unit L R S 3 6 5 C A M A C Sca l e r El iot -SR 1608 Comp. k t a r i C A M A C I ^ o p stop stop Fa s t Amp. LRS 133 B Fa s t Amp LRS 133B Fan Out LRS 128 •'r cl ip 2 5 0 ns TDC LRS 2 2 2 8 C o m p . Disc. C r on . 151 C A M A C Fan Out LRS128 r S c a l e r E l io t -SR 1608 Comp . COMPUTER CYCLE (PDP 11/40) Fan In Fan In Disc. LRS LRS LRS 621 127 B 4 2 8 Na l Anode Signals Neutron C o u n t e r C 4 Cha r ge I den t i f i e r 1) L A M issued by s ta r t s ignal of TDC 2) INHIBIT a l l C A M A C modules 3) READ and RESET 4) P roces s data 5) ENABLE sy s tem 30 §2.4 Stopping the Pions Since the incident pions had a mean energy of 51 ± 2.0 MeV, i t was necessary to put an aluminum slab in the beam line in order to degrade the energy of incident pions to 20 MeV. (The 15 cm long liquid hydrogen target would stop a 20 MeV TT".) A measurement of the C0-C1-C2-C3/C0-C1 ratio was done as a function of the degrader thickness. The results are shown in Figure (2.4.1). On the basis of these results a 2.86 cm aluminum slab was chosen as giving the maximum stopped TT - rate. However, the determination of the absolute stopped rate of the pions from the scaler readings was more dif -f i c u l t because of the following systematic uncertainties: 1. no i n i t i a l particle identification, and more seriously 2. less than 4TT solid angle covered by the C3 counter. The fact that the C0*C1«C2«CT did not select the type of i n i t i a l particle (i.e., e~, u~, TT~) led to the problem of understanding the different dE/dx behavior for the incident particles. However, the more serious problem was the fact that the CT veto counter was located about 45 cm from the back end of the target, and there was the possibility of divergence and multiple scat-tering in the beam such that the incident particle would pass through the tar-get but miss the C3~ counter. Furthermore, there were vent lines etc. in which the pions could have easily stopped. In order to estimate the number of pions stopped in the hydrogen target per incident pion, a Monte Carlo simulation of the experiment was done. The following factors were included: 1. beam spread in momentum space (15% fwhm and assumed Gaussian with mean momentum of 130 MeV/c) 2. beam divergence of the line (1-2 degrees) 3. dE/dx i . Al degrader 31 i i plastic counters i i i flask and vacuum vessel of target iv hydrogen .-. -(liquid at 22. K and 1.1 atm) v air 4. multiple scattering (Coulomb only) in the i Al degrader i i plastic counters i i i A l vacuum vessel The dE/dx functions were taken from the TRIUMF Kinematics Handbook, and I used the following formula for the multiple scattering angle 9: e l / e = \t\UMeV/c ^ [ ± + 1 / g ^ ^ / ^ j x [1 + M2/EMg] where M = mass of pion Z = charge of pion 6 = velocity of pion (c = 1) P = momentum of pion (lab) (MeV/c) Mg= mass of scatterer E = total energy of incident pion ;(MeV) L r= radiation length of material (e.g., Al = 9.0 cm) L = length of material traversed. This formula may be found in the recent "Review of Particle Properties" (Particle Data Group, 1976), and should be accurate to about 10-20%. There was no attempt to include range and dE/dx straggling in the calculation as the beam's width was sufficiently broad that in comparison these effects were of second order in importance. 32 Figure (2.4.1) CC0°C1° C2-C3)/(C0-C1) R R T 1 0 0.30 d 0.20 0.10 0.00 0.00 1.00 2.00 3.00 4.00 DEGRADER THICKNESS (cm.) Figure (2.4.1) Data used to determine the optimal thickness of the aluminum degrader. The broad peak i s a consequence of the large energy spread i n the incident TT beam. 33 It was found that the principle cause of the beam's divergence was the 2.86 cm Al degrader which gave a cr (assume Gaussian distribution of the multiple scattering angle) of 5° for a pion which would stop in the middle of the target. It was also found that about 10% of the incident pions would cause a CO*C1*C2-TTJ coincidence but not enter the target because of stopping in the C2 counter or stopping in the Al vacuum vessel. Since the spread in energy (20% fwhm) was so large, the pion stopped rate was not sensitive to the mean energy of the incoming pions, and this explains the f l a t top in Figure (2.4.1). It was found that (26 ± 5)% of the incident pions stopped in the liquid hydrogen target, where the error i s based on reasonable changes of the input parameters (e.g., beam energy and width) and a 10% uncertainty in the method of using the Monte Carlo technique. These two errors were added linearly. §2.5 Nal Crystal The most important piece of equipment which was used in this experi-./ mentwas a large Nal(Tl) crystal with a diameter of 45.7 cm and a length of 50.8 cm. It was hermetically sealed in an aluminum can. The crystal was purchased from the Harshaw Chemical Company of Ohio, U.S.A. and is to function as an efficient detector for photons of a few hundred MeV in a series of experiments at TRIUMF. (e.g., u -*• e y, TT -*• evy) . The crystal was viewed by seven RCA 4522 phototubes of diameter 12.7 cm.; .these1"'ph:o'totube%s2'have a rise time of 2.0 ns. The anode pulses of the individual tubes were added together to give a summed pulse which could be clipped to about 150 ns duration with no deterioration of the energy res-olution. The rise time of the summed pulse was about 50 ns, and this was primarily due to the variation in path length for light before i t reaches the phototube. Experience with a 25.4 q> cm x 25.4 cm Nal crystal had shown 34 that improved energy resolution could.be realized by using several phototubes to gather the light instead of one large phototube. (See M.D. Hasinoff et a l . , 1974 .) This is due to the non-uniformity of the photo sensitive area of large tubes. Of course, the introduction of several tubes to view the light adds to the d i f f i c u l t i e s of operation, and one of these is the balancing of the tubes. For r low-energy"'tests' t n e crystai, %the tubes were balanced by using a well collimated y-ray source which entered the crystal at i t s front on the central axis. For higher energy i t was found that i t would be feasible to use minimum ionizing cosmic rays (primarily muons) which traverse the Nal crystal. By suitably placing two plastic s c i n t i l l a t o r s , i t was possible to select those muons which traverse perpendicular to the axis of the crystal and through i t s centre. These muons leave about 220 MeV and had a fwhm of about 14% in the energy spectra. Without the two defining plastic counters, i t was about 20% fwhm. This technique suffers from a counting rate problem, but i t does have the attractive feature of providing uniform illumination of the crystal. Another technique to match the gains of the phototubes was to use the light emitting diode which was installed near the front of the Nal crystal. The pulser which drove the LED was adjusted to mimic the pulse of a photon inter-action in the crystal, but the LED proved to be more convenient as a monitor of gain shifts than as a device for matching gains of the tubes. This was mostly due to the fact that the LED was not centrally located, and that i t was not a diffuse source of light. Several tests were done with a 1 3 7 C s source (662 keV photon) to test the uniformity of the crystal as a function of the position of the entering photon. Since the range of a 662 keV photon (as defin-ed by flux f a l l i n g to 1/e) in Nal is only about 3.5 cm, the results test only the outer edges of the crystal. It was found that the crystal had a uniformity 35 of ±2% which was within the design specifications of ±4%. It is d i f f i c u l t to predict the behavior of the crystal for high energy photons on the basis of the low-energy uniformity tests because the manufacturer, i s free to polish surfaces (etc.) to meet the design specifications, and this might not optimize the uniformity of response for the deeper portions. In order to test the response of the Nal to higher energy photons, the crystal was tested with the 11.7 MeV photons from the 1 1B(p»y) 1 2C(4.44 MeV) reaction, and at this energy the photon develops a legitimate shower. Figure (2.5,.1) shows the energy spectra which was found. There was a small 5 cm lead collimator which defined the entrance of the photon, and this resulted in a 7.7% fwhm. For the experimental set-up for measuring the Panofsky ratio there was a neutron counter (NE 213 liquid s c i n t i l l a t o r of dimensions 13 cm <j> x 5 cm) with which y-n coincidences could be made between the Nal crystal and the neutron counter. By using this technique i t was possible to get a clean empirical determination of the Nal response function of a mono-energetic 129 MeV photon. In the energy region below 80 MeV, there was some contamination due to the 7T~p -> Tr°n reaction. Data was taken for two different lead collimators: 1) 25 cm c|> for Run (1) and 2) 15 cm <)> collimator for Run (2). Figures (2.5.2) and (2.5.3) show the results for Run (1) and (2), respectively. It was found that the larger collimator resulted in a 6.0% fwhm resolution, while the smaller collimator had 4.9% fwhm. The f u l l width tenth maximum (fwtm) was also determined for the measured line shapes. For a Gaussian function one would expect the ratio (fwtm/fwhm)=l.7. It was found that this ratio was 1.88 ± 0.04 for the 11.7 MeV photon, while i t was 2.15 ± 0.10 and 2.05 ± 0.10 for Run (1) and Run. (2), respectively. Outside this region there was evidence of a low energy t a i l . 36 3220 Figure (2.5.1) ' 8 'B(p,Y) , aC 2760 E r = 11.7 MeV CO ZD o 2300 1840 1380 920 460 7,7 % — * 0 90 • - 8 115 • 140 165 CHRNNEL NUMBER Figure (2.5.1) Response of the Nal c r y s t a l to a mono-energetic photon of 11.7 MeV. 37 Figure (2.5.2) PHOTON-NEUTRON COINCIDENCES FOR RUN 1 525 450 375 300 in 225 150 H 75 0 T 1 I I 50 75 100 125 150 PHOTON ENERGY (MEV) Figure (2.5.2) Response of the Nal crystal to 129 MeV mono-energetic photons, The data were recorded by requiring an n-y coincidence from the stopped TT~P yn reactions of Run (1). 3« PHOTOrf-NEUTRON COINCIDENCES FOR RUN2 300 c n 240 1 8 0 12( 6 0 0 50 if i i -1 75 100 125 150 PHOTON ENERGY (MEV) Figure (2.5.3) Response function of the Nal crystal .to 129 MeV mono-energetic photons. The data were recorded by requiring an n-y coincidence from the stopped Tf -p •> yn reactions of Run (2) . 60 OQ C i-l K> VD -P-8 0 100 120 140 Energy (MeV) Line shape for 129 MeV photons Figure (2.5.4) Log plot of the response function of Figure (2.5.2). The dashed l i n e i s an estimate of the t a i l shape. The error bars i n the x direction indicates regions where i t was necessary to average over the observed counts. 40 Figure (2.5.5) LOG PLOT OF 129 MEV PHOTON FOR RUN 2 1000 100 •z. ZD O O 10 0.1 80 105 130 155 PHOTON ENERGY (MEV) Figure (2.5.5) Log plot of the response function of Figure (2.5.3). The error bars in the. x direction indicate regions in which i t was necessary to average over the observed counts. 41 In this experiment i t was necessary to understand the t a i l of the 129 MeV photon in order that the TT° photons and the radiative photons be cleanly separated. It was observed that i f one plotted the y-xi coincidence data on a log scale there was clear indications that the t a i l was predominantly exponential in character. Figures (2.5.4) and (2.5.5) show these results for Run (1) and Run (2), respectively. Both Run (1) and (2) indicated that the t a i l ^ is'. of the form exp, (n E) where n =0.14 ±0.01 MeV. It did not depend on the collimator size to within the experimental uncertainties. The 129 MeV response function for the Nal crystal was found in an empirical fashion, However, i t was necessary to extrapolate the t a i l region to zero energy, and the exponential character was assumed to dominate in this region. It is estimated that less than 0.16% of the response function was below 90 MeV. The line shape for the 129 MeV photon was used to determine the line shape for the lower energy photons of the n 0 decay, and this was done by folding a Gaussian distribution into the 129 MeV response function in order to get the appropriate fwhm of these lower energy photons. These line shapes were then convoluted with the theoretical uniform distribution of the i\° decay photons to get an estimate of the t a i l region. Another important feature of the Nal crystal which turned out to be of great use in removing neutron associated events was the timing response. (See Chapter (2.3) for details of the electronics and logic.) Figure (2.4.6) shows the timing response for the 129 MeV photons and the 55-83 MeV TT° photons, and the fwhm for these two cases was 1.8 and 2.2 ns, respectively. Furthermore there was a slight shift of 0:8ns of the two spectra. The shift i s such that the lower energy photons took effectively longer to be recorded, and the larger fwhm of the TT 0 photons i s consistent with this observed walk. 30000 42 tfa¥reHWirik RESPONSE I \ fj* photon / i . ~ 129 MeV " 80 90 100 l i O 120 130 CHANNEL NUMBER (2.5.6) Timing resolution of the Nal c r y s t a l . 43 § 3. Experimental Results §3.1 Backgrounds §3.1.1 Proceedure Experimental data were collected for two different configurations of the Nal-H^ target system. Run (1) was taken with the Nal crystal at 91 cm from the target center, and the aperture of the lead collimator was 25 cm$. Run (2) differed from Run (1) in that the Nal-H^ target separation was a considerably larger 193 cm while the collimator size was smaller (15 cmcf>). The raison d'etre for these choices was that they would aid in understanding systematic uncer-tainties associated with the geometry of the collimators, etc. In both exper-imental runs, the Nal crystal was placed at 90° with respect to the incoming pion beam in order to minimize any electron contaminants. In addition to the 25 cm of steel which shielded the front of the Nal crystal, there was always at least 20 cm of lead which shielded any photons which might come from the aluminum degrader (from TT° decays, etc). A background run was also taken with the target empty except for some residual gas which was l e f t i n the flask. Figure (3.1.1) shows the results of this run. The principle mechanism which caused this background was the scattering out of the pions into the material of the cryostat and the sub-sequent radiative capture process in the heavy elements. Although these rad-iative capture reactions occur only about 1-2% of the time for nuclei with A ^ 25, they tend to have a high probability of carrying off most of the pion's rest mass energy, and this means that care i s needed not to confuse them with the 129 MeV photon of the Tr~p yn reaction. The timing cuts for the back-ground data were chosen to be identical to that of Runs (1) and (2), so that the same neutron contaminants would be in a l l the photon spectra. Neutrons are a worry because they can not be vetoed by the C4 charge identifying counter, and timing requirements must be relied upon in order to remove this contaminant. 44 Figure (3.1.3) in c o o -C a. o — « 1 I — H i 1 I * I a CD m CO LO 00 o U3 IO ro a to to syjnoo U J CD a: LU ZZ. U J X Q_ Figure (3.1.3) Target-empty Data The solid line i s an estimate of the shape of the spectrum with the 7T -p -> TT°n and Tr -p -> yn reactions removed. These reactions occurred because of the residual H 2 gas l e f t in the target flask. 45 Hence, the timing cuts of the background data were chosen to be identical to those of the photon spectra of Runs (1) and (2). (See Chapter (3.1.3) where the choice of the timing cuts i s explained.) §3.1.2 Low Energy Photons The chief concern during the collection of data was to understand the sources of the low energy photons. This problem was dealt with by taking a series of short runs with various placements of lead shielding blocks in order to help identify their origin. It was found that the number of low energy phot-ons (Ey ^  40 MeV) varied by about 10% with the different placements of the lead blocks. However, by removing a l l the lead, except for the lead which shielded the aluminum degrader, there was a 30% reduction in the low energy photons. The principle cause of the low energy photons was found to be the lead collimat-or of the Nal crystal, and i t was not possible to shield against this source. This was later confirmed in an off-line analysis of the data where i t was found that Run (2) [15 cm<|> collimator] had a larger low energy background than Run (1) [25 cm<j> collimator]. This was due to the fact that the ratio of the collimator edge to the solid angle of the collimator goes as 1/r, where r i s the radius of the collimator, The effect was found to give an ambiguous definition of the Nal's solid angle at the 1% level (for "the case of the 25 cf>cm collimator). In the measurement of the relative yields of the TT° photons and the 129 MeV photon, this ambiguity was of no concern. However, the low energy back-ground was a problem in the sense that i t extended underneath the region where the TT° photons occurred, and this required special care in the data analysis. Indeed, i t was found that this background was the principle uncertainty in estim-ating the number of TT° per stopped T T - . §3.1.3 Neutron Contaminants As previously discussed in Chapter (2.5), the Nal crystal had a timing resolution of 2.0 ns fwhm for the high energy photons. In the case of Run (1), 46 Figure (3.1.2) PHOTON SPECTRA FOR VARIOUS TIME CUTS , — 1 1 — — r 1 — ^ I 0 . 2 . 8 ns • o o « a a t i •••'••v».i i " < ••'.'•'* 1 — - — L 7T' photons 4- 4 "s 129 MeV photons i . l«*»**«.r»«.»^  l ».......J '""'i—it—JL— 6.0 ns ••••• . . «r . . . . - l ' I .....j....... 1 7.6 ns f * * I I I""- ---1-/ 9.2 ns H neutrons * • - . a 0 25 50 75 100 125 150 175 PHOTON ENERGY (MEV) 47 the distance from the center of the target to the front face of the Nal crystal (not the back end of the collimator) was 1.2 metres, and this gives a time of f l i g h t of 4.0 ns for a photon. For a 70 MeV neutron which traveled over the same fl i g h t path, the tof would be 11.0 ns. (The high energy neutrons are the chief worry; they are produced i n the Tr~A •> (A-2) + n+n reactions which occur in the cryostat walls.) Hence, the At between the neutrons and photons was 7.0 ns, and the timing resolution of 2.0 ns fwhm was ample to reject the vast majority of the neutrons. Figure (3.1.2) expli c i t l y shows the energy spectra of the neutrons as a function of time cuts, where t=o i s defined as the arrival of the photons. It can be seen that the neutrons are faintly v i s i b l e at At=6.0 ns, and become quite obvious at At=7.6 ns. This i s in good agreement with the predicted 7.0 ns time difference. Hence, the neutrons were removed by placing a time cut at 5.5 ns. Another feature of Figure (3.1.2) i s that the TT 0 photons arid the radiat-ive capture photon are both present up to At=4..4 ns, but at At=6.0 ns the radiat-ive capture photon has disappeared, while the TT 0 photon remains. Hence, there was a possibility of introducing a bias in determining the number of TT° photons as compared to the number of radiative capture photons. However, by estimating the number of TT° photons which were missed by the choice of the At=5.5 ns tim-ing cut, i t was found that the bias would be about 0.075%, and this i s neglig-ible for the purposes of this experiment. §3.1.4 Electrons Since the electrons have v/c ^ 1, they can not be removed by timing re-quirements, and the charge identifying C4 counter was used to remove this con-taminant from the photon spectra. (See Figure (2.1.2) for the layout.) On the other hand, the electrons themselves can be selected and we show the results in Figure (3.1.3). LOW ENERGY ELECTRON SPECTRUM 40 60 80 100 320 140. ' CHANNEL NUMBER % • Figure (3.1.3) The bump i n the spectrum's due to the I T 0 e+e'+^decay and also the external conversions of TT° photons. The so l i d l i n e estimates the shape of the spectrum without these electrons. 49 The three principle sources of electrons are 1) pair production, 2) TT° -> e+e""+ y, and 3) TT~p -> e +e~n. The dominant source of pair production was the high energy photon interactions in the lead collimator of the Nal crystal. The evidence for this conclusion is based on the observation that Run (2) had a larger electron contaminant than Run (1). As has been discussed in Chapter (3.1.2), this can be explained by the smaller collimator of Run (2). It should be emphasized that both the low-energy photons and e* were caused by the same compound process of the photon producing pairs (or e- producing brem-sstrahlung) i n material other than the Nal crystal. Thus, they both have a similar energy dependence, and so I used the energy spectra to estimate the energy behavior of the low energy photons which extended under the TT 0 photon peak. The solid line i n Figure (3.1.3) represents an estimate of the energy dependence of the low-energy background. It was obtained from the electron spectra by removing the e± pair from the TT° -»- e+e~y^decay. (An iterative procedure was used to remove the e- pairs from the 55 -* 83 MeV region.) It was not possible to measure the branching ratio for the TT° e +e -y because there was significant conversion of the photon in the 1.4 mm aluminum vacuum wall #1%) • §3.2 The Photon Spectra The photon spectra for Run (1) and Run (2) were found by removing the electron contaminant through the charge identifying C4 counter and requiring the time of f l i g h t to correspond to a particle with v/c=l. (The details have been discussed in Chapter (3.1).) The results of Run (1) and Run (2) are shown in Figures (3.2.1) and (3.2.2), respectively. Prior to this experiment, the most precise value for P was the experi-ments of Cocconi et a l . (1961) who found P=1.533 ± 0.021. Their technique was very similar to the one used i n this experiment, but the crucial difference was \ Counts 6 0 0 0 4 0 0 0 , 2 000 0 0 -This experiment — C o c c o n i et al. \ \ 6 % v 2 d I ' \ 20 4 0 6 0 8 0 100 120 140 Photon energy (MeV) Figure (3.2.1) Photon Spectrum of Run (1). The dashed l i n e i s the re s u l t of Cocconi et a l . for same measurement. CO o C J 3000 2625 2250 1875 1500 1125 750 375 0 0 PHOTON SPECTRUM OF RUN (2) 25 50 75 100 125 PHOTON ENERGY (MEV) F i g u r e (3.2.2) Photon Spectrum of Run (2). •4J9 % 150 52 that they used a Nal c r y s t a l with a dimension of 20 cm<j> x 20 cm which i s a factor of 10 smaller, i n volume than the 46 cm<t> x 51 cm Nal c r y s t a l used i n thi s experiment. Furthermore, we observe about 2.1 x 10^ stopped T T _P reactions which i s a factor of 20 greater than Cocc°ni et a l . In order that a v i s u a l comparison of the precisions can be made, a sketch of the results of Gocconi et al.'s measurement has been superimposed on the results of our Run (1) i n Figure (3.2.1). I t can be seen that the results of this experiment are con-siderably more precise. 53 §4. Data Analysis The objective of thi,s experiment was to measure the reactions: (1) TT~p -> Tr^n (2) TT~P -*• yn (3) TT~P e~e* + n for stopped pions i n a l i q u i d hydrogen target. Reaction (1) produced a neutron of 0.4 MeV and a 2.9 MeV'ir0 which decayed i n f l i g h t to the f i n a l channels: Cla) y + Y [98.83%] Clb) e +e~ + Y I 1.17%] Clc) e +e~ + e+e- I 3.5 x 10 _ 3%] CParticle Data Group, 1976). Channel (la) gave r i s e to a photon energy spectrum which was uniformly distributed between 55 and 83 MeV. The photons i n channel (lb) have a similar energy d i s t r i b u t i o n because the t o t a l energy of the e* pair has a strong tendency to be equal to m^0c2/2 i n the rest frame of the TT°, and also, the e* pair tend to come off i n the same dir e c t i o n . [See the theory of D.W. Joseph (1960) or the experimental results of M. Derrick et a l . (1960) and N.P. Samios (I960).] Channel ( l c ) i s neglected i n t h i s analysis because of i t s very small branching r a t i o . The e^ pair i n reaction (3) are due to the internal conversion of the 129 MeV photon of reaction (2), and they have strong tendency to be emitted with 129 MeV. The branching r a t i o of t h i s process has been calculated by D.W. Joseph (1960), and i t was found to be 0.71%. The Panofsky r a t i o i s defined as _ (la) -f (lb) + (lc) (4.1) V (2) + (3) which i s consistent with the d e f i n i t i o n of Cocconi et a l . (1961). In order to determine the value for P, the photon spectra of Runs (1) and (2) were used to separate the different channels on the basis of energy resolution. Since the 54 Nal detector had a f i n i t e energy resolution, the T T 0 photons (N^) are defined as being between 40 and 90 MeV, while the 129 MeV photons (U^) are defined as being between 90 and 160 MeV. Hence, an equation for P is P" - (%)CNa/N2) (4.2) where the factor of (%) i s to correct for the two photons in channel (la), but this equation needs to be corrected for the electron events which were vetoed by the charge identifier. F i r s t , i t i s noted that N 2 = (la) + Os) (lb) N 2 = (2) Now, P i s defined as P' = t _ (la) + (lb) (2) + (3) and this may be rewritten as (lb) _,. ,(la) + igCib), . r l +(ja)+ % ( l b ) 1  P " L (2) J 1 , (3) J 1 + T2)" It follows from simple algebra that P . = P" (1+ P') (4-4) ' (1 + p ) ( l + hp') where p and p' are the ratios [(3)/(2)] and f (Lb)/'(lk) J , respectively. Since the equation (4.4) includes background events, the f i n a l equation for P is p . p . . r 1 + £BKG(N}) (4.5) L l + £BKG(N 2) J The following background sources were considered. iTable (4.1) summarizes the results]: (a) Low-energy Background: The connection of this background to the e± spectra has been discussed in Chapter (3.1.2), where i t was shown that the li k e l y cause RUN 1 RUN 2 Counts a low energy BKG b target-out BKG ci ir° inflight NI 172186 -1.540 (.20 ) -0.105 (.020) -0.594 (.120) N2 55186 0.0 -1.03 (.20 ) -0.543 (.120) NI 83484 -3.10 (.40 ) -0.240 (.05) -0.590 (.120) N2 26758 0.0 -2.46 (0.50 ) -0.538 ( .120) in C o o u SH O U C2 Rad. cap. inflight 0.0 -0.157 (.030) 0.0 dj IT 0 tail +0.204 (.06 ) 0.0 +0.09 (.03 ) d 2 130 Mev tail -0.087 (.03 ) +0.27 (.09 ) -0.048 (.016) -0.138 ( .030) 0.0 0.149 ( .09 ) -2.12 (.24 ) -1.46 (.25 ) -3.80 (.42 ) -2.99 ( .53 ) See the text for explanation of the sources in the corrections. of the corrections. The numbers in brackets are the uncertainties 56 of the low energy photons was the high energy photon interactions in the lead.collimator of the Nal crystal (i.e., pair production and bremsstrahlung of the e- pair) . Since the principal source of energetic y - r a y s was the hydro-gen in the target, the target-empty data does not give a good estimate of this background. Hence, there were two unknowns for this correction: 1) the energy dependence and 2) the normalization factor. The energy dependence was assumed to be similar to that of the e- spectra, and the normalization was found by minimizing x2> where 90 MeV 25 N d a t a ( E ) - [aN b k g(E) + 2>N*°(E)] N d a t a ( E ) N (E) = number of counts in spectra N b k g(E) = background function NTr°(E) => counts due to IT0 photons a = free parameter h - free parameter (The procedure for getting N11 (E) has been explained in Chapter (2.5).) It was found that this background was of the order of 1.5% and 3.0% for Runs (1) and (2), respectively, which is a small but significant correction. Figure (4.1) shows the low energy background which was found by using this procedure. The f i t can be seen to be quite good. The computer program gave an 8% uncertainty in determining this back-ground (i.e., a + a ->(x2/v) +1) where v i s the number of degrees of freedom); but i t is f e l t that a more reasonable estimate of the uncertainty would be ±15% which takes into account a possible systematic uncertainty i n the choice of the N^k^ function. LOW ENERGY BACKGROUND 500 400 \— 300 200 h -100 20 30 40 50 60 70 80 90 100 PHOTON ENERGY (MEV) Figure (4.1) The low-energy photon background subtraction. The solid line i s an estimate of the background due to low energy photons. Note that ir 0 photons extend up to 5000 counts. See Figure (3.2,1). 58 (b) High-energy Background: Most of this background came from the radiative capture of the TT" in the cryostat walls, and the target-empty data provided the best estimate of the energy dependence of this background. [chapter C3.1.1).] The normalization was arrived at by forming the ratio a(E) for a number of different photon energies in the energy range 90 < Ey. _< 115 MeV. The ratio a(E) is defined as f i 1 '^empty N d a t a ( E ) - [N t a ± 1(E) + N l n f(E)] inf N (E) = number of counts due to inflight TT p ->• Tf°n reactions (discussed below) N^k^(E) = high-energy background function emptv N (E) = number of counts i n the target-empty data del £ 3 , * N (E) = number of counts in spectrum t a i l N (E) = number of counts due to t a i l of 129 MeV photon 2 f data*? , , ta i l , ? , , i n f s ? a * <=> (a ) + (o ) + (c ) . The results of this calculation are shown in Table (4.2). Since the ratio a(E) was found to be independent of energy to'within the experimental uncertainties, i t was affirmed that a consistent interpretation for the photon spectrum in the 90 < Ey < 115 MeV region was that these counts were due to 1) the high-energy background, 2) the t a i l of the 129 MeV photon, and 3) the TT° -> YY decay of infl i g h t charge exchange. This technique also provided for an estimate of the uncertainty in normalizing the high-energy background subtraction which i n -corporates the uncertainties in the t a i l correction and the inflight correction. 59 TABLE (4 .2 ) N o r m a l i z a t i o n f o r t h e h i g h - e n e r g y backg round RUM (1) E (MeV) N d a t a N t a M N i n f N e m p t y a ( E ) i 9 0 2 3 2 81 8 9 6 5 1.0 ± 0.3 1 9 5 2 5 0 118 72 71 0.9 ± 0.3 9 9 2 8 4 173 5 8 8 0 o'.6 ±o:k 10^t 3 9 5 26k k6 89 1.0 ±Q.k 108 621 4 2 6 30 9 6 1.8 ± 0.8 1 1 3 1 1 3 5 9 7 7 10 100 1.5 ±0.9 7(17" = 0 .95 ± 0 ;20 RUN (2) 89 124 10 40 65 1.2 ±0.2 Sk 1^5 2 5 35 71 1.2 ± 0.2 9 8 149 35 28 80 1.1 ±0.2 103 172 56 2 2 8 9 1.1 ±0.2 107 198 93 15 96 1 L 0 ±0.2 112 288 167 5 100 1.2+0.2 i W = 1.1 ± 0.1 60 (c) I n f l i g h t Corrections: In general, the r a t i o W ( T T -P -> Tr°n)/W(TT -P yn) i s dependent on energy, and a c o r r e c t i o n . i s needed to remove the background which a r i s e s from the I n f l i g h t T r ~ p reactions. In order to c a l c u l a t e the TT° photon spectrum and the r a d i a t i v e photon spectrum, the s-wave cross section formula of Rasche and Woolcock (1976) were used. Since the maximum energy of the TT - beam was about 30 MeV at the front end of the target, the p p a r t i a l wave amplitude was also included i n the c a l c u l a t i o n f o r the t o t a l cross section of Tr -p -»- Tr°n. (M. Salomon 1974.) The number of photons produced from the i n f l i g h t charge exchange react-ion was calculated by i n t e g r a t i n g over the TT 0 angular d i s t r i b u t i o n and f o l d i n g the energy d i s t r i b u t i o n of the incident I T - beam. (See Appendix A for the r e s u l t s . ) The c a l c u l a t i o n included the dE/dx losses of the incident ir - beam and also a stopped ir - d i s t r i b u t i o n i n the hydrogen target, but t h i s l a t t e r con-s i d e r a t i o n was not too c r u c i a l since only 25% of the beam stopped i n the hydro-gen target. (See Chapter 2.4.) The i n f l i g h t TT° gave r i s e to photons whose energy extended up to 115 MeV, and hence, there was a c o r r e c t i o n (c.^) to both and N 2. The i n f l i g h t r a d i a t i v e process only contributed a c o r r e c t i o n ( c 2 ) to N 2. A 20% uncertainty was ascribed to t h i s c o r r e c t i o n , and t h i s i s based on the quoted 10% uncertainty i n the s-wave formula of Rasche and Woolcock (1976) and a 15% uncertainty i n the i r " stop rate. A check on t h i s c o r r e c t i o n was done by by comparing the number of predicted i n f l i g h t photons i n the energy i n t e r v a l 140 _< Ey < 160 MeV with the observed events. There was agreement to within 10%. [ See Figure (4.2).] (d) T a i l Corrections: The t a i l region of the 129 MeV photon was determined experimentally by the TT-p -> yn coincidence data, and t h i s provided a means of determining the low-energy t a i l of the TT° photons. (See Chapter (2.4) for co 40 c O O 20 0 r ~ 130 61 Figure (4.2) I n f l i g h t Cor rec t ion \ \ 129 MeV ^ — photons inflight 150 170 Channel Number Figure (4.2) The inf l i g h t radiative capture correction. The solid line is the estimate of the in f l i g h t correction and served as a check on the normalization of the subtraction. 62 the details). There was good separation between the ir°photons and the radia-tive capture photons so that this was a small 0.2% correction. A 30% uncertain-ty i s ascribed to this correction which is based on the statistics of the line shape measurement and the extrapolation of the t a i l to zero energy. Since the estimate of the low-energy t a i l of the ir° photons is quite similar to the t a i l of the 129 MeV photon, the uncertainty in correcting the ratio N 1 / N 2 would be smaller. (e) External Conversion: There was a small probability (1%) of conversion + of the photons into e pairs in the 1.4 mm aluminum vacuum flask, and these events are rejected by the C4 charged particle identifier. This effect can be easily becorrected byrcalculating the flux which i s absorbed by the aluminum via the equation: N = N(incident) exp(-ax), where x is the amount of matter 'which the photon traverses and a i s the cross section for pair production. Using a(129 MeV y) = 0.59 ^ and a(55-83 MeV y) = 0.66 ^ , we obtain a small multiplicative correction of Ce=0.9986. (Particle Data Group, 1976; they give the values of a and •: rvalue of the rad-iation length of aluminum = 9.0 cm). (f) Random Backgrounds: The random evehitsunwe'res found by placing time cuts in the time of f l i g h t spectra of the photons such that their velocity would correspond to values greater than the speed of light. It was found that the number of randomsvents was only 0.12% of the true events. Their effect is included in the empirical background functions used in corrections (a) and (b). By inserting the corrections into equation (4.5), i t is found that P(run 1) = 1.547 ± 0.010 P(run 2) = 1.543 ± 0.015 Average = 1.546 ± 0.009 The uncertainty in P was found by adding the s t a t i s t i c a l uncertainties . in quadrature with 63 the uncertainties in.the background subtractions. The uncertainty i n the treated as indicated. Now t h i s procedure of adding the uncertainties i n quadrature i s a c t u a l l y conservative, since many of the background subtractions tended to cancel each other. However, i n view of the f a c t that some of the errors are of a systematic nature, i t i s f e l t that t h i s r e s u l t s i n a r e l i a b l e estimate of the uncertainty. (The dominant uncertainty i n the f i n a l r e s u l t f o r the Panofsky r a t i o was the s t a t i s t i c s and the background subtractions (a) and (b). Since Run (1) and Run (2) were consistent with each other, i t was decided to take a weighted average to obtain P = 1.546 ± 0.009. t h i s i s whether there was any s i g n i f i c a n t capture from a p-state atomic o r b i t a l . Now, t h i s problem was f i r s t examined by Brueckner, Serber, and Watson (K. Brueckner, R. Serber, and K. Watson, 1951) for the case of the TT d mesonic atom, and they showed that c o r r e c t i o n (d 2) had a p o s i t i v e c o r r e l a t i o n ( i . e There i s one question l e f t i n understanding our measurement of P, and T(n,p) V<f>n,p(0) |2 a where absorbtion rate hydrogen wave function f o r p r i n c i p l e quantum number n and o r b i t a l angular momentum e. t o t a l cross section for theoe wave v _ cm. v e l o c i t y of the TT cm. momentum of the I T - . 63a In order to evaluate these expressions f o r the s and p absorption rates of the i r , the following low-energy formula f o r are used. v 2 a = 4TT — I f I S V 1 s 1 cr = 8TT P p o Tr m oc V 17 . , 2 ( V i i m - c l MUTT ' f = ^ - (0.265) U/m c f p = ^ (0.220) V m f c The sc a t t e r i n g lengths were taken from the recent compilation of Nagels et a l . (1976), and only the P 3 3 amplitude i s included i n the equation f o r a . P Putting these expressions into the equations f o r T ( n , Z ) , we obtain T(n,s) = (1 + 1/P) ' r n = 1.0 x 10 15 v n 1 - 1 —jr sec 3J and T(n,p; = 5.6 x 10E f n T - i ; - j J~5—| sec 1 n-The T(n,s) absorption rate f o r the ir°has been m u l t i p l i e d by the (1 + 1/P) factor i n order to correct for the s-wave r a d i a t i v e capture channel. We do not include a s i m i l a r c o r r e c t i o n f o r the p-wave absorption rate because the p-wave TTN i n t e r a c t i o n i s much stronger than the electromagnetic i n t e r a c t i o n . Now, i f we assume that the n 2 atomic states are equally populated, then an estimate of the f r a c t i o n f of p-wave absorption i s P f = 3r(n,p) a - 6 n 2 - 1 5 p r(n,s) ) , B X L U 1 , J • 64 However, i t has been shown by M. Leon and H.A. Bethe that there i s some depletion of the s states for n < 7 (M. Leon and H.A. Bethe, 1962), and this means that the assumption of equal population of the n 2 is not valid. The reason for the depletion of the s states i s that the absorption of the TT becomes competive with the Stark-mixing rate for low values of n. Indeed, Leon and Bethe have shown that the TT is most likely to be captured from an n = 3 (or 4) level and that only 4% of the TT—manage to reach the n = 2 state. Now, the electromagnetic 2p -> Is transition rate i s about 1.5 x IO 1 1 sec , so that even i f there was 100% depletion of the 2s state, the fraction of p state capture would s t i l l be negligible (i.e., f = 4 x 10 5 ) . Hence, we may be certain that the measured value of the Panofsky ratio may be interpreted as occurring only from atomic s states. 1. The effect of the depletion of the s states was pointed out to the author by M. Leon (private communication) who himself estimated f should be corrected by a factor of three or so. This gives f = 2.0 x 10 - 5 which is s t i l l negligible. 65 5. Conclusion In this chapter we return to equation (1.1.8) to check the charge independence hypothesis of the strong interaction, having finished the presen-tation of the experimental data. This equation relates the s-wave scattering length of the ir~p •+ Tr°n reaction to the e l e c t r i c dipole photoproduction ( E + ) o amplitude for the yp iT +n reaction v i a the r e l a t i o n i n - a o l 2 = 9 P R — ^ — |E 1 3 1 m -uv .+ 1 2 ro +v„ TP 0 where a.^ and a 3 arethe T = 1/2 and T = 3/2 TTN scattering lengths, respectively. Putting i n the values of P = 1.546 ± 0 . 0 0 9 , R = 1.34 ± 0.02 (M.I. Adamovich et a l . , 1975) and E J = (2.83 ± 0.05) x I O - 2 m"| (M.I. Adamovich et a l . , 1970), we obtain | a i-a 3|= 0.263 ± 0.005 (5.1) (We now use the units of length = 1>/m^ .fC.) Before comparing this r e s u l t to other values of |a 1-a 3| from i n f l i g h t Tr^p scattering data, a few comments are i n order. F i r s t , i n deriving equation ( 1 . 1 . 8 ) , i t was assumed that v a (s-wave) = 4 IT ( ^ A ) 3 2 ( a r a 3 ) | 2 (5.2) instead of P o J / 2 1 o(s-wave) = 4rr (^)- '| " | ( a i - a 3 ) | 2 (5.3) Now, equation (5.2) and (5.3) d i f f e r i n the way i n which the phase space i s handled, so that equation (5.2) already has an ad hoc correction for the isospin breaking effect of the mass differance between the i r + and TT° . I f equation (5.3) i s used to determine the s-wave scattering lengths, then i t i s found that la - a . l = 0.267 ± 0.005 (m + ) - 1 which i s . a s i g n i f i c a n t difference. Besides 1 1 0 ' TP the mass ef f e c t , i t might be supposed that there are other Coulomb effects which need to be considered. Now, both the values of R and E + have had Coloumb o 66 corrections applied, so that they do not need further modification. In our measurement of P, the i n i t i a l states were identical for both the Tr~p Tr°n and 7r~p -> yn reactions, so that any Coulomb distortion of the i n i t i a l wave function would apply equally to both channels. Furthermore, in both the (•rr0n) and (yn) channels, there are no f i n a l state electromagnetic interactions. Now a more detailed calculation of the Coulomb and mass difference effects has been carried out by Rasche and Woolcock (1976), and they suggest a 2.5% change in the s-wave scattering length in order to correct for the isospin breaking effects. (This correction applies to equation (5.3).) If we use their suggest-ed correction factor, we then get |a 1~a 3| = 0.261 ± 0.005. This i s in good agreement with the prediction of equation (5.2) [i.e., |a^—a3| = 0.263 ± 0.005]. We can now compare our results for ja^-agl with those which have been determined from the inflight n^p elastic scattering data. The two most recent analyses^?- of the low-energy pion scattering data give laj-agi = 0.262 ± 0.004 (D.V. Bugg et a l . , 1973) and |a 1-a 3| = 0.259 ± 0.004 (W.S. Woolcock,^1974). Now, these results have assumed charge independence in order to derive the l a ^ a g l scattering length, whereas our value of [aj-agl determines i t from the Tr~p -»- Tr°n reaction. However, i t should be mentioned that both Bugg et a l . and Woolcock needed to apply Coulomb corrections, andcthey differed in the manner of making these corrections. This is the reason for their two different results. (Woolcock claims Bugg's technique was unsatisfactory.) Our result of ^ - a g l = 0.263 ± 0.005 (or |a - a j = 0.261 ± 0.005 IS/m^+c) is not sufficiently precise to settle the discrepancy, and this i s mainly due to the uncertainty in the E* amplitude. Nevertheless, we can see that our re-sults do support the charge independence hypothesis of the strong interaction amplitude to the 2.5% level. 67 BIBLIOGRAPHY M.I. Adamovich, V.G. Larionova, A.I. Lebadev, S.P. Kharlamov, and F.R. Yagodina, Sov. Jour. Nucl. Phys. 1_1, 369 (1970). M.I. Adamovich, V.G. Larinova, and S.P. Kharlamov, Sov. Jour. Nucl. Phys. 20, 28 (1975). S.L. Adler, Ann. Phys. 50. 189 (1968). D. Aebischer, B. Favier, G. Greeniaus, R. Hess, A Junod, C. Lechanoine, J.-C. Nikles, D. Rapin, and D.W. Werren, Nucl. Phys. B106, 214 (1976). I.R. Afnan and A.W. Thomas, Phys. Rev. CIO, 109 (1974). H.L. Anderson and E. Fermi, Phys. Rev. 86, 794 (1952). D. Axen, G. Duesdieker, L. Felawka, Q. Ingram, R. Johnson, G. Jones, D. LePatourel, M. Salomon, W. Westlund, and L. Robertson,wNucl. Phys. A256, 327 (1976). A.M. Baldin, Nuovo ' Cimento 13, 569 (1958). D.V. Bugg, A.A. Carter, and J.R. Carter, Phys. Lett. 44B, 278 (1973). K.A. Bruekner, R. Serber, and K.M. Watson, Phys. Rev. 81_, 575 (1951). W.F. Cartwright, C. Richman, M.N. whitehead, and H.A. Wilcox, Phys. Rev. 91, 677 (1953). J.M. Cassels, Nuovo Cimento Supplemento 259 (1959). D.L. Clark, A. Roberts, and R. Wilson, Phys. Rev. 85,523 (1952). V.T. Cocconi, T. Tazzini, G. Fidecaro, M. Legros, N.H. Lipman, and A.W. Merrison, Nuovo Cimento 22^ 494 (1961). S. Cohen, D.L. Judd, and R.L. Riddel, Jr., Lawrence Radiation Laboratory Report UCRL-8391, (1959) unpublished. F.S. Crawford, Jr., and M.L. Stevenson, Phys. Rev. 97_, 1305 (1955). M. Derrick, J. Fetkovich, T. Fields, and J. Deahl, Phys. Rev. JL20, 1022 (1960). C.L. Dolnick, Nucl. Phys. B22, 461 (1971). R. Durbin, H. Loar, and J. Steinberger, Phys. Rev. 83, 646 (1951); 84, 581 (1951). J. Fischer, R. March, and L. Marshall, Phys. Rev. 109, 533 (1959). 68 M. Gell-Mann and K.M. Watson, Ann. Rev. Nucl . S c i . 4_, 219 (195*0 . M.D. H a s i n o f f , S.T. Lim, D.F. Measday, and T.J. M u l l i g a n , Nucl. I n s t r . and Meth. U]_, 375 (197*0 . D.P. Jones, P.G. Murphy, P.L. O ' N e i l l , and J.R. Wormald, Proc. Phys. Soc. (London) A77, 77 (1961). D. W. Joseph, Nuovo Ci men to 22_, 494 (1961). E. L. K o l l e r and A.M. Sachs, Phys. Rev. 116, 760 (1959). E.K. K l o e p p e l , Nuovo Cimento 34_, 11 (1965T J . A. Kuehner, A.W. Merrison and S. Tornabene, Proc. Phys. Soc. (London) A73.-545 (1959). M. Leon and H.A. Bethe, Phys. Rev. 127, 636 (1962). R. MacDonald, Ph.D. t h e s i s , U n i v e r s i t y of B r i t i s h Columbia, ( r e s u l t s to be published in Phys. Rev. L e t t . ) . M.M. Nagels, J . J . de Swart, H. N i e l s e n , G.C. Oades, J.L. Petersen, B. Tromborg, G. Gustafson, C. J a r l s k o g , W. P f e i l , H. P i l k u h n , F. S t e i n e r , and L. Tauscher, Nucl. Phys. Bl09, 1 (1976). W.K.H. Panofsky., R.L. Aamodt, and H. Hadley, Phys. Rev. 8l_, 565 (1951). P a r t i c l e Data Group, Rev. Mod. Phys. 48, No. 2, P a r t i I, 1 (1976). B.M. Preedom, C.W. Darden, R.D. Edge, T. Marks, M.J.M. Saltmarsh, E.E. Gross, CA. Ludeman, K. Gabethuler, M. Blecher, K. Gotow, P.Y. B e r t i n , J . A l s t e r , R.L. Burman, J.P. Perroud, and R.P. Redwine, Phys. L e t t . 6j>B, 31 (1976). G. Rasche and W.S. Woolcock, Helv. Phys. Acta 49, 565 (1976); 4_9, 455 (1976); 4_9, 435 (1976). A. R e i t a n , Nucl. Phys. B11, 170 (1969). C. Richard-Serre, W. H i r t , D.F. Measday, E.G. Michael i s , M.J.M. Saltmarsh, and P. Skarek, Nucl. Phys. B20, 413 (1970). CM. Rose, Phys. Rev. 154, 1305 (1967). A.H. Rosenfeld, Phys. Rev. 9_6, 139 (1954). J.W. Ryan, Phys. Rev. 130, 1554 (1963). A.M. Sachs ,H';;Wi n i c k , and B.A. Wooten, Phys. Rev. 109, 1733 (1958). M. Salomon, TRIUMF Report TRI 74-2, (1974) unpublished. N.P. Samios, Phys. Rev. L e t t . 4, 470 (1960). 69 J. Spuller and D.F. Measday, Phys. Rev. D12, 3550 (1975). H.L. Stadler, Phys. Rev. 9_6_, 496 (195*0. R. Traxler, Lawrence Radiation Laboratory Report UCRL-10^17, (1962), unpubli shed. R.D. Tripp, in Strong Interactions, Proceedings of the International  School of Physics,"Enrico Fermi ," Course XXXI I I, edited by L.W. Alvarez (Academic Press, New York, 1966). K.M. Watson and K.A. Brueckner, Phys. Rev. 8_3_, 1 (1951). W.S. Wool cock, Nucl. Phys. B75, ^55.(1974). CM. York CM. York, W.J. Kernan, and E.L. Garwin, Phys. Rev. 119, 1096 (1960). 70 APPENDIX In order to calculate the Tr~p •> yn and iT~p -> Tf°n inflight corrections, we used the low-energy cross section formula of Rasche and Woolcock (1976). These equations are: On = 4IT q q - 1 Co 2 A ~ 2 A ' -ik(A ;' A '. -A A ) uo_ H 0 H ^ U " 0-.... v - . o-v YY Y~ Y° .2 av = 4TT k q - 1 CO 2|A| - 2|A - i q (A A -A A ). 12 Y- 1 1 1 y- oo Y _ ° - Y° where q 0, q, and k are the cm. momentums of the TT°, TT-, and photon, respectiv-ely. The CO 2|A| - 2 factor corrects for the distortion of the incoming plane wave in the Coulomb f i e l d of the proton. The variable A was expanded in a power series of q 2; that is A = a + aq 2, where a and a are 3 x 3 constant matrices which describe the Tr-p -»• T f p , ir -p -> Tr°n, Tr~p -> yn, Tr°n ->• yn, Tr°n Tr°n, and Y n Y n channels (i.e., A , A Q , A , A , A and A , respectively). It is worthwhile to point out that the y- Y° 0 0 YY A matrix which Rasche and Woolcock suggest in their paper has had Coulomb cor-rections applied, and the reader i s refered to their paper for the f u l l dis-cussion. We have evaluated these expressions and show the results in Figure (6.1) The rise in the cross section as q o is due to the exothermic character of a the reactions, but as q -> o, the ratio — a -> P quite rapidly, so that this is aY-no real worry. In order to evaluate the spectral function for the photons from the charge exchange reaction, the following integral was done. d V= _ „ f _ % w(n,k)dn', (6 .D dkdfi where G i s a normalization factor, k i s the photon's lab energy, ft i s the angle of the Nal crystal, fl' i s the cm. angle of the TT° , and W(ft,k) i s the S - W a v e T o t a l C r o s s S e c t i o n s 0 10 20 K i n e t i c E n e r g y o f yr" ( lab) 30 MeV ut"e ( 6 . 1 ) . The calculated values for the (TT p -*• ir°n) and (TT p -> yn) reactions using the formula of Rasche and Woolcock. 72 probability of a photon entering ft+d# with energy k+dk. For the case of s-wave the formula reduces to da yy _ G a(ir p •> Tr°n) (6.2) dkdfi eOY 0k"Y(l-8cosa) where BQ i s the cm. velocity of the TT0, YO = *> ^  -*-s t n e velocity of the cm. in the lab., y = ( l ^ B 2 ) - ^ , a i s the angle of the Nal detector, and k" is the energy of the photon in the TT° rest frame. (For higher values of I, there i s an analytic formula. For example, see C.M. York et a l . , I960.'.) In order to determine the yield of photons from the inf l i g h t charge exchange reaction, equation (6.2) was used as input to the following integral which was done numerically. dN Y = G dkdfi E . F(q;q,0> ( f ) d W g dEdq, E max where F i s the gaussian distribution of the i n i t i a l pion beam, and dE/dx i s the energy loss function of the TT- in the liquid hydrogen. A similar calculat-ion was done for the inf l i g h t ir""p -»- yn reactions, and the results are shown in Figure (6.2) for the case of Run (1). Now, i t can be seen that the TT0 photons extended up to ^115 MeV, and hence, a subtraction i s required for both the TT° and radiative capture photons. (This reduces the sensitivity of our value of P to the in f l i g h t corrections.) Before concluding, we return to equation (6.2) and point out that the TT0 photon spectrum i s uniformly distributed, but depends on the angle of the Nal detector. In order to get to the situation of an isotropic d i s t r i b -ution of incident TT" (or for that matter, a random angular distribution of T r -p mesonic atoms), equation (6.2) should be averaged over the Nal's laboratory angle. 40-4 30H | 20-O o 0 nr 20 Inflight Reactions T > ° photons r 60 T l 129 MeV photons inflight infl ight 80 100 120 Photon Energy (MeV) 1 4 0 1 6 0 ^4 Figure (6.2) Inf 1ight s p e c t r a l f u n c t i o n s . 

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