π⁰ Photoproduction from the Deuteron at Forward Angles in the Energy Range from 900 to 1400 MeV Citation Clinesmith, Carl Rudolph (1967) π⁰ Photoproduction from the Deuteron at Forward Angles in the Energy Range from 900 to 1400 MeV. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/ZGA0-4Y62. https://resolver.caltech.edu/CaltechETD:etd-04212005-111623 Abstract NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. The cross section for the reaction [...] was measured at the Caltech synchrotron. The [...] was detected by measuring its decay gamma rays with two lead glass, total absorption Cherenkov counters. The results are three angular distributions at k = 911, 1180, and 1390 MeV, at forward angles from 3 degrees to 90 degrees. The deuteron/proton ratio differs significantly from 2.0, but final state effects from the use of a deuteron target make impossible quantitative conclusions about the neutron cross section. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Physics) Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Physics Thesis Availability: Public (worldwide access) Research Advisor(s): Tollestrup, Alvin V. Thesis Committee: Unknown, Unknown Defense Date: 25 January 1967 Record Number: CaltechETD:etd-04212005-111623 Persistent URL: https://resolver.caltech.edu/CaltechETD:etd-04212005-111623 DOI: 10.7907/ZGA0-4Y62 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 1438 Collection: CaltechTHESIS Deposited By: Imported from ETD-db Deposited On: 22 Apr 2005 Last Modified: 15 Mar 2024 18:58 Thesis Files Preview PDF (Clinesmith_cr_1967.pdf) - Final Version See Usage Policy. 3MB Repository Staff Only: item control page

n° PHOTOPRODUCTION FROM THE DEUTERON AT FORWARD ANGLES IN THE ENERGY RANGE FROM 990 MEV TO 1400 MEV Thesis by Carl R}Clinesmith In Partial Fulfillment of the Requirements For the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1967 (Submitted January 25 , 1967) li ACKNOWLEDGMENTS This experiment was supervised by Professor Alvin V. Tollestrup. By his teaching I have made the transition from working book problems to doing experimental research, It is difficult to properly acknowledge also those things learned not from pedagogy but by example. The guidance of Professor Robert Walker in all phases of my graduate career has been invaluable. The continuing interest of Professor Robert Bacher is appreciated. The total experiment using hydrogen and deuterium targets was performed in collaboration with G. Laurie Hatch. The division of labor and material did not come about until the actual writing of the two theses. As such, the methods and results reported are his as much as mine, My first interaction with synchrotron physics was as an apprentice to Professor Richard Talman, Many of my attitudes were formed from working with him, From his days as a student, consultation with Professor Charles Peck has been useful. To him, and also to Drs. Joe Mullins and Owen Maloy, goes the credit for making the Phase I Synchrotron a working, 1500 MeV machine, Conversation with Drs. Donald Coyne, Stanley Ecklund, and Professor Donald Groom, while we were fellow students, made the solving of many problems a more pleasant task. Most of my knowledge of the theoretical aspects of photoproduction is the result of conversation with Dr, Jean Hebb. The hydrogen - deuterium target was designed by Mr, Earle Emery, and was constructed and maintained by him and by Mr, Richard Wileman, The Synchrotron crew, headed by Larry Loucks, and the lil machine operators Al Neubeiser, Frank Scarlino and Joe — Laurinovics, are of course essential to all experiments. Most of the mechanical apparatus was constructed by Walter A, Nilsson. Iam also especially grateful to Phyllis Nilsson, who made a significant contribution during the final phase of my career as a student. I would like to thank my wife Judy, my mother, the Woodrow Wilson Foundation, the IBM Corporation, the National Science Foundation, and the California Institute of Technology for financial support. iv ABSTRACT The cross section for the reaction y +d - n° +pt+n was measured at the Caltech synchrotron, The n° was detected by measuring its decay gamma rays with two lead glass, total absorption Cherenkov counters. The results are three angular distributions at k = 911, 1180, and 1390 MeV, at forward angles from 3 degrees to 90 degrees. The deuteron/proton ratio differs significantly from 2.0, but final state effects from the use of a deuteron target make impossible quantitative conclusions about the neutron cross section. . To Judy vi TABLE OF CONTENTS PART I... Introduction I, Description of Experimental Method Ii, Procedure IV. Data Reduction V. Cross Sections VI. Discussion of Results APPENDICES I, Experimental Apparatus IL. Data Reduction TIT, ‘Complete Counting Data Resolution Function Calculation V. Pion Pair Corrections REFERENCES PAGE 14 19 26 43 47 58 80 87 104 108 3a-c. 4a-c, 10, lla-c. 12a-c. 13. vii LIST OF TABLES Angular Distribution Parameters Counting Rates Deuteron Cross Section and Deuteron/Proton Ratio Deuteron - Proton Cross Section Difference Comparison of Cross Sections in Different Configurations Ratio of Background to Foreground Pion Rates Accidental Rates as a Function of Single Counter Rates Parameters at Experimental Points Cross Section and Resolution Function Parameters Run Configurations Complete Counting Data Proton and Deuteron Cross Sections and Efficiencies Pion Pair Corrections PAGE 15 17 28 31 61 62 64 710 75 77 81 84 107 10. lla-c. 12a-c. 13a-c, 14, 15. 16, Vili LIST OF FIGURES Beam Area Layout Experimental Arrangement Electronic Logic Calculation of Experimental Resolution Function Counting Rate as a Function of Cut-off Energy Angular Resolution Function for Point Detector Experimental Angular Resolution Function Energy Distribution of Coincident Gamma Rays Sum Spectrum From Recalibrated Matrix _ Observed and Computed Energy Spectrum for Detection of Single Pions Photoproduction Cross Sections, Y+Pon +p Deuteron- Proton Cross Section Ratio Photoproduction Cross Sections y +n - n +n Deuteron Overlap Integral Cherenkov Counter Assembly Cherenkov Counter Response PAGE 11 12 13 23 24 25 34 37 40 47 49 50 17... 18, 19, ix Cherenkov Counter Resolution Function Hydrogen Target Pion Pair Production PAGE 51 55 105 I, INTRODUCTION The single pion photoproduction reactions which can be measured in the laboratory are 1) ytpon den 2) y+p-n° +p 3) y+tnonm +p 4) ytnonen where the reactions from the neutron must use a deuteron target. A bremsstrahlung beam contains all incoming energies up to its maximum, so measurement of cross sections for the two particle final state requires identifying the particle and measuring its direction and energy. The ease and accuracy of magnetic spectrometer analysis make this method preferable whenever a charged particle is produced with sufficient energy. The magnetic spectrometer method has been used to measure n and 7 photoproduction cross sections detecting the pion, and for nr photoproduction from protons detecting the proton. The difficulty of detecting the low energy proton accompanying a mn produced in the forward direction limits the magnetic spectrometer method to angles greater than about 50 degrees in our energy range. The desire to measure small angle n photo- production from protons, as well as to measure the corresponding reaction from the neutron, leads to attempts to measure the Tm itself, Measurement of the n° was made practical by the development of total absorption Cherenkov counters as gamma-ray spectrometers. These counters allow measurement of the 7° alone by detecting the two decay gamma rays in coincidence and recording their pulse heights. The counters were first used to measure 1 production from complex nuclei in an attempt to measure the T° lifetime. (1) They : fe) . were then used to measure cross sections for 7 photoproduction (2) The increase in intensity of the CIT synchrotron, plus the from hydrogen in the forward direction. reaching of its designed endpoint energy of 1500 MeV, led to redoing the previous measurements from hydrogen, extending the data to higher energy, and doing an identical set of measurements from deuterium. The work was carried out at three mean photon energies, 911, 1182, and 1390 MeV, at 10 degree intervals from 0 to 90 degrees. This thesis reports the investigation of deuterium. The simultaneous (3) work on hydrogen is given in the thesis of G. Laurie Hatch. Tl, DESCRIPTION OF EXPERIMENTAL ME THOD The aim of the experiment is to measure the T° angular distribution in the reaction y +d — nm + (p +n) by detecting the two decay gamma rays from the n° in lead glass Cherenkov counters, (4) Briefly, these counters detect the Cherenkov light produced by the electromagnetic shower in the lead glass initiated by the incident photon or electron, The size of the counters is sufficient to contain the entire shower. The Cherenkov light is collected by nine 5'' RCA 7046 phototubes, the outputs of which are summed, The amount of Cherenkov light produced is directly proportional to the energy of the initiating particle, so the average pulse height produced is a linear function of the incident energy. The distribution in pulse height in the output of the counters from a beam of mono-energetic particles is approximately Gaussian. By testing with the mono-energetic electron beam the width of the distribution was found to correspond to production of about 200 photoelectrons per BeV. A non-shower producing particle such as a proton or cosmic ray muon passing through the counter produces the same amount of light as a 220 MeV electron. Since the average pulse height from the Cherenkov counter is a linear function of energy, the pulse height can be displayed in energy with just a scale change. We will use this scale throughout the thesis, usually depending upon the context to distinguish between reference to a particle energy or to a pulse height in the Cherenkov counter. The experimental layout is shown in Figure 1. In the usual manner, the bremsstrahlung beam from the synchrotron is incident on the target which contains liquid hydrogen or liquid deuterium, LAOAV VaAeuV WVAS YALAWVINVNO NOS TIM YaddOls WV3d SHOVEL AA TIONL SNI7 WV38 NOLOHd (LNOYG NI SYALNNOD OLSA) YS LNNOD AOWNSYAS AATIONL en es es ee ee ee ee 00% ,001 | aunol4 1a9uvL °H1 LANOVWN ONIdSSMS G NOY LOYHONAS ssavuos 2-22 7M SN ONIGTSIHS HOLVAITI0I— ~Yolviavu-—* The beam is then monitored directly by a Wilson quantameter, The produced T's decay essentially at their point of production, and each decay gamma ray is seen by the apertures, shown in Figure 2. Given that we have seen the decay gamma rays of a single n°, the kinematic variables which the system geometry defines are the pion direction and energy. In order to be able to detect both decay gamma rays, the pion direction must lie in the rectangle containing the apertures, which defines the 9 and » limits. To determine the energy limits of pions the system will detect, we consider the opening angle between the two decay gamma rays in the laboratory system. This angle has a minimum which increases with decreasing pion energy. .The maximum angle subtended by the apertures then defines the minimum energy pion the system can detect, which we call E ren the geometric cutoff energy. Having defined the angles by the setting, the maximum pion energy is given by the kinematics relation for the maximum photon energy. Hence the lead glass counter system will define the energy and angle acceptance without any reference to pulse-height information, although such reference will in fact be necessary to separate the reaction of interest from the detection of double T° production. With the system as used, the counter rectangle subtends a solid angle of about 20 millisteradians. The resolution function is chosen to sample the same photon energy limits by choosing the geometric cutoff energy to correspond to the same photon energy at all angles. The resolution function defined in this way has the same shape in photon energy at all angles. The Cherenkov counters were mounted one above the other on a trolley which could be moved to any angle at a constant radius about the target. The counters could be moved independently , V l CERENKOV == 4 COUNTERS LEAD APERTURES / / } FIGURE 2 EXPERIMENTAL ARRANGEMENT perpendicular to the beam plane to obtain the desired geometric cutoff energy. The lead houses which surrounded the Cherenkov counters eliminated beam associated particles not passing through the apertures. The apertures themselves were shaped on three sides and backed by a two scintillation counter telescope to veto charged particles. A gamma ray striking the side of the aperture has a high conversion probability so the veto counters accurately define aperture size as well. The sweep magnet was used at small angles to reduce charged particle rates in the veto counters. The electronics system used is given in Figure 3. A gamma ray is defined by a pulse in the Cherenkov counter with no signal from the aperture charged particle veto counter or from the cosmic ray veto counter. The discriminator also requires the Cherenkov counter pulse to be above a minimum height given by the decay kinematics of detectable pions. A T° event is defined by a coincidence between gamma rays in each Cherenkov counter, The pulse heights in each counter then define the X and Y addresses in the 32 x 32 matrix pulse-height analyser. The energy resolution function is calculated in Appendix IV, and shows the features previously discussed qualitatively. The efficiency for detecting pions produced into the counter rectangle € ) and the number of photons N(k) are given in Figure 4. In order to ignore the scale complication introduced by kinematics, the functions are given at 0 degrees where E =k. The total resolution function has the low energy side determined by «(E) and the high energy side determined by the bremsstrahlung spectrum N(k). The combination of the two gives the spectrum of detected pions S(k) = N(k) x e(k). The actual spectrum in pulse height is obtained using the MIXER v = é¢ Cc -—— DISC. COSMIC RAY 100-260 50k | MEV) coInc, Ee c | | tthe _ CHARGED | VVC} 28ke PARTICLE. CF : | COINC. 7 kc Y-RAY ] coInc, K— OTHER SIDE 7° | 70 SIDE A | ANALYZER a en cena wane) @ GATE SIDE B | FIGURE 3. ELECTRONIC LOGIC NOILONAA NOILNIOSSY WINAWIdAadxXs JO NOLLV INOS v AYNSIS NOILONNS NOILAIOSSY NOILONN4S NOILLNTOSSY JO SLNSNOdWOD | AAW ‘AOYMSN3 NOld ASW ‘ASYSINA NOld OOS! OO! OO! O0z! 00S! OOvl OOS! oozioSil I l 0 I I rain he -10'2 ~O'?¢ ov 4O0'p +09 —+0'9 os os WNYLOadS ool | NOLOHd —0'0! S1IOS AYVYELIEYV 10 response function of the counters R(E, E’), SE )= =[ N(E') x e(E') Xx RE, E') dE’. The angular resolution in the center of mass for the system is the result of two effects, the physical width of the counter apertures in the laboratory, and the variation in CM angle with energy at constant laboratory angle. These functions are given in Figures 6 and 7, The actual calculation of the functions takes into account the motion of the nucleons in the deuteron. The results differ from those for a stationary target as follows: 1) for a constant CM cross section the rate of production of pions in the laboratory is increased by less than 2 per cent, 2) the shape of the resolution function in energy as given by its first few moments is not changed by more than 2 per cent, 3) the angular resolution in the center of mass is increased by about 50 per cent. The choice of running parameters was dictated by the requirement of a reasonable counting rate. Figure 5 shows the system counting rate as a function of cutoff energy. The interval chosen for Eo = 1513 MeV was Eq= 1150 MeV. With this choice, the width of the resolution function defined as / ((K - K)? >» was about 80 MeV. For these limits, ¢(E) =6 x 10 “3 so the effective solid angle of the system is about 0,1 millisteradian for an energy acceptance of 160 MeV. The Caltech magnetic spectrometer measuring T's at the same energy has an angular acceptance of 0.1 millisteradians for an energy acceptance of 25 MeV. (5) The rather wide energy acceptance necessary is the principal defect of the system for kinematics definition used here. ASYANA O0OSI 4IO-1LND NOLOHd JO NOILONNS V SV 3LVY ONILNNOD GC auNdl4 AAW - AOSYANA NOLOHd oovI o0o¢I 00d cele)y J34d0-LND IWWYON-——= AAW sist = °3 oO = onp Ou = o fo) Nn fo) dig -wos6 -q7 / SINNOD 12 5.0 |- 4.0 PROTON ld —! < © ”Y ~ . <I oc E ac Lol -DEUTERON L | | | = 34 36 38 40 42 44 PION ANGLE, CM FIGURE 6 ANGULAR RESOLUTION FUNCTION FOR POINT DETECTOR 10.0 4.0;- ARBITRARY SCALE o (o) 2.0 [—- 13 0 | | | 34 36 38 40 42 44 PION ANGLE, CM FIGURE 7 EXPERIMENTAL ANGULAR RESOLUTION FUNCTION 14 Tt, PROCEDURE The experiment was run over an 18 month period at three endpoint energies, Eo = 1073, 1308, and 1513 MeV. The angular distributions were taken at 10 degree intervals in the center of mass from 0 degrees to 90 degrees. The distributions required several different configurations, with each point within a given configuration taken several times. The details of configuration and consistency checks between configurations are given in Appendix Il. The target condensed either hydrogen or deuterium with the same apparatus. The hydrogen and deuterium runs were taken alternately, providing a hydrogen-deuterium ratio free of most of the systematic errors. The background runs were taken throughout the running period, usually on the same day as the foreground. The 911 and 1182 MeV data show no statistically significant time variation in background rate for points taken in the same configuration. Some of the 1390 MeV data were taken with rather poor beam definition due to instabilities inherent in the synchrotron beam when running at an endpoint of 1513 MeV, For these runs it was necessary to compare the foreground runs with the background runs taken on the ‘same day, although in these cases the difference in the foreground rates was less than 20 per cent. The phototube gain was monitored by measurement of the pulse- height spectrum from the Cherenkov counters on cosmic ray muons. The measurement was done each day during the 12 hours while the experiment was not run, The mean pulse height was determined by comparison with the mono-energetic electron beam to 15 pareacg sorTsuy 98T 8T¢ 00¢ AW ‘XB Z/T UIPTM TINA GLET OSTT SLIT 0S6 066 004 ASIN ASL ‘AS190Uq ‘AS 10Uq uojoyg ueeyAl FJOND srojouBIeg uny T @Tavib OTST 3 80eT °S SOOT "T ASW UOT NGTAYSTT ‘AS.1OUT yurodpuy 16 correspond to 220 MeV electrons. The electronic gain was not a linear function of pulse height, but was measured daily by means of a simulated Cherenkov pulse produced by a pulser with a linear scale. The known linearity of the counters and these two measure- ments give daily energy calibration of the system. The rates encountered in the beam area are given in Table 2 for the various points in the system for the orn = 1308 MeV data from hydrogen, The counts were about 2/3 from deuterium, 1/3 from background. The hydrogen rates were about 0.8 times these. There are four electronic problems to consider: 1) accidentally vetoed gamma rays, 2) charged particles not vetoed due to dead time in the veto discriminator, 3) coincident gamma-ray events not recorded due to dead time in the analyser, and 4) accidental coincidences between the two Cherenkov counters. 1) and 2) arise from high rates in the aperture veto telescope. The ‘counting rate in a single scintillation counter is principally due to Compton electrons produced by the halo of low energy photons which fill the beam area. The use of a two counter telescope eliminates these electrons setting a low energy bias of about 15 MeV. The single counter rate is about 8 times the coincidence rate. The resulting rate keeps 1) and 2) less than 0.5% by measurement. The analyser had a 5 kilocycle acceptance rate so 3) is negligible. Angle, Lab 20 Mag On 20 Mag Off 42 Ei 0 f b Veto Scint. x 10 1060 540 240 120 1200 600 420 300 17 TABLE 2 Counting Rates = 1308 MeV = deuterium runs Coinc. 10° 110 40 34 19 346 170 65 50 = empty target runs Rate, Counts/Second Gamma >100 MeV x 103 10,0 5.3 4.8 1.6 3,0 1.7 2.6 0.2 Electrons >100 MeV x 103 3.0 0.5 12.0 10,0 2.8 0,45 0.8 0.1 Pions 7.3 4,0 2.9 0.3 1.7 0.5 0.8 0.1 18 The rate of accidental coincidences between the two Cherenkov counters is not negligible at some points and required constant monitoring at all times. The rate (y At 100 ns) - (yp) was monitored by a separate circuit. Each foreground run at which the rate made a correction necessary was followed by an accidental mt run, delaying the y A pulse through the normal rm coincidence circuit. The synchrotron intensity was kept at values such that the accidental coincidence rate did not exceed 10% of the foreground rate, and was usually less than 5% of the foreground rate. The 32 x 32 pulse-height matrix contained in the analyser was written onto paper tape for preliminary analysis by the Burroughs 220, and ultimately converted to IBM cards for the final analysis using the pulse height to energy calibration. 19 IV. DATA REDUCTION The system has produced a 32 x 32 matrix in pulse height of coincident gamma-ray pairs for each foreground, background and accidental run, The Cherenkov counter and electronic gain calibrations are used to generate a matrix in energy from the matrix in pulse height. The scale of these matrices is then converted to one linear in energy. The process of conversion and its associated errors is given in Appendix I. The resulting background and accidental matrices can now be subtracted from the corresponding foreground matrices to obtain the result from hydrogen or deuterium. The result next obtained from this matrix is the spectrum in total energy of the detected two gamma-ray events. This spectrum is given by the sum of the matrix diagonals, E S(E) =| ME - EY 0 E dE y2) Y 2? 2 ° The reactions which produce a significant number of events are. 1) the reaction of interest y +d - n° + (p +n) 2) single 7°'s from double r production, ytd- no + G+ N4N’) 3) events resulting from the detection of one gamma ray from each 1° iny+d- T+ + (p + n). 20 The desired reaction, 1), will contribute to the spectrum above the geometric cutoff energy. The detection of reaction 2) is a by- product of the large pion energy acceptance, which places the geometric cutoff energy well below the maximum pion energy in two pion production. Since a single n° is detected, reaction 2) only contributes events with total energies greater than the cutoff, For a given total cross section, the counting rate and the energy spectrum from these pions is calculated in Appendix V from two models, 1) pure phase space, and 2) resonant production by the reaction y + N > 7° + N*(1238 MeV), An upper limit for this reaction has also been obtained at one point by means of the synchrotron subtraction technique, The results for two pion contamination are given below: 1) phase space with a total cross section of 200 wbarns, which is 5 times the value measured for y + p - mn - _ (6) +7 +p” gives a correction of less than 5 per cent, 2) production of a n° and N*(1238 MeV) with a total cross section of 200 barns? 10 per cent at 0°, falling to 4 per cent at 50°, gives a correction of 3) the synchrotron subtraction at 40°, kay 1182 MeV, gives a correction of 3 + 3 per cent, 4) fitting of the computed spectrum to the observed - spectrum shows no statistically significant enhancement of the low energy cross section. This procedure puts an upper limit of about 0.05 ubarn on the cross section due to: multiple pions, or an upper limit of 20 per cent ai at 0°, falling to 4 per cent at 40°, Since it is not possible to make an angle dependent correction, no change has been made to the quoted cross sections. As such, they are the sum of all reactions y +N - 7" + N' producing . O,. . . . . single 7's into the given resolution function. The energy distribution of detected gamma-ray pairs in 3) can be understood qualitatively. Unlike single n° detection, there is no lower limit from the system geometry on the total energy of the two detected gamma rays, so the energy spectrum of these events extends from zero to its maximum, Furthermore, any photon of energy above the two pion threshold can produce events, so the spectrum will contain a large number of low energy counts. In the actual experiment a lower limit of about 100 MeV is given to the energy of each gamma ray by the bias set on the Cherenkov pulse into the coincidence circuit which defines a gamma-ray event, The maximum total energy can be understood as follows, The kinematics for producing a two pion system of mass 2m is very little different from that of a single pion at the same angle, or Eo + Eo = E A foreward going gamma ray carries almost all of the energy of the nm, e.g., at x = 500 MeV, EY (max) = 495 MeV. Hence reaction 3) can provide events with total energy almost up to that of single pion photo- production. The contribution of the third reaction above the single 2? detection cutoff requires a correction of about 10 per cent at 0° and 10° and is negligible at large angles. The details of the method of estimating the correction from the data are given in Appendix I. 22 The energy spectrum from each of these reactions is given in Figure 8. The pulse-height spectrum obtained by the data reduction is given in Figure 9. This spectrum shows a good separation between the single and double n° regions, making it a simple matter to determine the counting rate by counting no's above the cutoff. The more sophisticated procedure of using fitting functions for the two regions, together with the errors due to double T detection, is given in Appendix Il. Finally, a spectrum with an excellent separation of single no's is compared with the computed spectrum in Figure 10. 23 OOEI SAVY VAINVS LNAGISNIOD 40 NOILASINLSIG ADYANS 00zI ASW ‘ADYSNI Ooo! 006 008 8 S3YNdIs OOL 009 OOoll I UT 708 0? O'v og 00! ° a Ol-ASW/3LVY SNILNNOO O'vi 0°91 ° © Sdid 4 o'02. XIMLVN GSLVYSINVOSY WONS WAYLOTdS WNS 6 3YNOIS Aaw-43 . OOF) OOd! Ooo! 008 009 OOv C Ee en IL ] T | r T T j ; L. Pea aT La d3LovVYyLENs 3g on \ GNNOYNDNOVE 24 | (V1) 22 =“9 ; AAW 2811-4 4d0-LND DLaWOR39-———> _ bv aNnouoNovE l WLNAGIOOV 2¢. LEONVL AL ¢ly GNNOYDIYNOS ASW 008 3AO8V SYS99INL oe Sdid OOO!~-192 NNY ANNOYDaYOS Ol Oz o¢ Ov OG JIVOS AXVYLIGUV SNOIld ATONIS AO NOILOALAG HOS WOAYMLOAdS ADYANA GALNdWOD GNV GSAYaSdO Ol aYNDIS ASW - 73 OOS! © OOvI OO¢ | oozI Ooll erere) 006 008 eleva T T T T T S t (e) | I I roe y o | 1 =| 01 [e) —+ 02 ro) 0 ! ec [po] lop fo) | | —| OS (dV1) 022 =“9 | . . AAW 28ll = ; + 09 fe) | re} | OL, O1NVd JLNOW o ? log QaAy3Sgo + . =| | . 06 — ool SJIVOS AYVYELIEGYV 26 V. CROSS SECTIONS The results of the experiment are given in the tables which follow as the three directly measured quantities, the proton and deuteron cross section, and the deuteron/proton ratio. The proton cross sections have been taken from Reference 3. We have also given the result for a 'neutron" cross section assuming the simplest spectator model, for which the deuteron cross section is simply the sum of the free proton and neutron cross sections, o qi? p +O, The cross sections quoted are obtained from the counting rate at each point by the equation C =n x o(k,,9,) where 7 is the integral of the geometric resolution function at that point, as described in Appendix Ill. Summarizing the results obtained there, where T(k,9) is the resolution function, Cc =o (k, 6) T(k, 9) dkde 2 ~ 2 = nx {o(k,,8,) +25 (k-k)° >} . ak 'K8, The size of 5 7 /9 Ke? is estimated from the energy distribution data at 60° and 90°. The correction is apparently less than 5%. No attempt has been made to correct the data. The statistical errors - are: 1) counting statistics on foreground, background and accidental runs, 2) 3) 4) 27 calibration error, leading to incorrect combination of runs, subtraction of the (°°) spectrum, setting of the geometrical parameters. Systematic errors may be defined by o(k,6) = o(k,, a) x {1+ 8 (k, @)} . The sources of error and typical values are: 1) 2) 3) 4) 5) detection of single m°'s in double 7 production, -.05 < 6(k,@) < 0, deviation from the true value due to energy variation of cross section, | 6(k,@)| < 0.05, endpoint energy of synchrotron, |6| < .03 Mev™! (This correction is independent of angle to about 5%), absolute normalization of beam monitoring, |6| < 0.02 (This correction is time dependent, since the quanta- meter calibration may change. ), calculation of detection efficiency, including the small corrections, |6| < 0.02. 28 TABLE 3a Cross Section Data k= 911 MeV Deuteron Angie, Cross Section, Deuteron/Proton CM ubarns/steradian Ratio 3.5 1, 406 + 0.132 3.161 + 0, 394 11.2 1.831 + 0.124 3. 237 + 0, 407 20.3 2.688 + 0.127 3.841 + 0. 462 28.0 3.056 + 0.105 2.093 + 0.165 38.1 3.080 + 0.125 1, 921 + 0,126 47,9 3.001 + 0. 108 1, 842 + 0.096 57, 4 2.833 + 0.119 1.688 + 0,098 67.3 3.218 + 0,129 1,642 + 0.096 77,5 3.591 + 0.197 1.660 + 0,122 29 TABLE 3b Cross Section Data k = 1182 MeV Deuteron Angle, Cross Section, Deuteron/ Proton CM ubarns/steradian Ratio 3.7 0. 728 + 0,049 1.584 + 0. 183 11,4 1, 295 + 0. 161 2.162 + 0. 350 21.4 2.591 4+ 0. 122 2.334 + 0, 164 31.9 2.881 + 0. 206 2.036 + 0. 164 41,8 2,812 + 0, 214 1, 806 + 0. 153 51.6 2.045 + 0, 078 1,501 + 0,094 61.6 1,602 + 6.119 1,291 + 0.120 72.0 1,651 + 0. 106 1,602 + 0.145 81.6 1. 338 + 0.099 1,541 + 0, 158 91.0 1, 445 + 0.133 1,635 + 0, 190 30 TABLE 3c Cross Section Data k = 1390 MeV Deuteron Angle, Cross Section, Deuteron/ Proton CM .barns/steradian Ratio 3.1 1,155 + 0.159 1,858 + 0,401 10. 5 2,408 + 0, 246 ——-1, 987 + 0. 387 19.6 2.661 + 0,.213 2,192 + 0. 254 29.6 3.017 + 0.191 1,820 + 0.175 39, 2 2.166 + 0.118 1,597 + 0, 140 49.6 | 1.608 + 0.097 2.000 + 0. 229 60. 3 1,000 + 0.121 1.688 + 0, 333 70, 2 0.898 + 0.132 1. 681 + 0.356 80. 8 1,180 + 0. 164 1, 202 + 0, 224 90.7 1,244 + 0, 221 1, 264 + 0, 310 31 TABLE 4a Difference of Deuteron and Proton Cross Sections k = 911 MeV Angle, Cross Section CM ubarns/steradian Error 3.5 0. 961 + 0.175 11,2 1, 265 + 0, 230 20, 3 1, 988 + 0. 323 28, 0 1, 596 + 0, 240 38, 1 1.477 + 0. 202 47,9 1,372 + 0, 156 577. 4 1,155 + 0. 164 67.3 1, 258 + 0, 188 77,5 1, 427 + 0. 264 32 TABLE 4b Difference of Deuteron and Proton Cross Sections k = 1182 MeV Angle, Cross Section CM ubarns/steradian Error 3. 7 0, 268 + 0. 084 11,4 . 0. 096 + 0. 210 21,4 1, 480 + 0. 182 31.9 1, 466 + 0. 233 41.8 1, 255 + 0. 238 51.6 0. 683 + 0. 127 61, 6 0. 361 + 0. 149 72,0 0.620 + 0, 150 81.6 0. 470 + 0.137 91,0 0.561 + 0, 168 TABLE 4c 33 Difference of Deuteron and Proton Cross Sections Angle, CM 10.5 19.6 29, 6 39. 2 49, 6 60. 3 70, 2 80.8 90. 7 0. 1. 1. 1. 0. 0. k = 1390 MeV Cross Section ubarns/steradian 533 164 447 359 . 810 . 804 . 408 . 364 199 260 i+ + + I+ H+ I+ Error 0, 249 0. 480 0. 309 0, 291 0, 190 0. 184 0.197 0. 190 0, 220 0, 306 AAW IG =4 YVAN d+ 54-4 d+4 NOILOAS-SSOND NOLLONGOUOLOHd o4 PI] JYNIs SWO - STIONV NOILOINGOYd ot 08! 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[ I Ol ; T fe) ? ¥) > $ fe) ~ L i 40°20 1 a ae ; ~ oo -10°¢ 40 06 08 O02 AFNING =H 'U+ ot 2 U+4 NOILOSS SSOND NOILLONGOYdOLOHd o O€1 SUNDA WO ‘STONY off 09 OS Ov O€ O02 Ol | | i tT I I T I yS/q7-'(g)0 41 AANZ8Il= HSU ott u+A4 NOILOSS SSOYD NOILONGOYNdOLOHd »># Ge! AYNSIS WO ‘STONY oA 06 08 Ol OS OS | I Ov Of O02 | | Ol O | rT | us/q7t-"(9).0 42 ASWOGEI= SU+ ot a U+ A NOILOIS SSOND NOILONGONdOLOHd on WO “STONY o# OL 09 OS OV O€ Og 9g] AYNSIS Ol O'? ws /q7l—*(g).0 43 VI DISCUSSION OF RESULTS The analysis of results from deuteron targets utilizes the impulse approximation ® 9) which takes the T matrix elements for a bound nucleon target to be equal to those for a free nucleon, The condition for validity of this approximation is that the energy of the incident particle be large compared to the binding energy of the nucleons in the target particle. The deuteron binding energy is 2.2 MeV, while the incident photon or outgoing pion energy is about 1000 MeV. Clearly the impulse approximation can be used. The analysis must then take into account: 1) the interaction of the two nucleons in the initial and final states, 2) the interaction of the produced pion with the second nucleon, or multiple scattering. If 2) is neglected, the T - matrix may be written (8) Ty = ¢ |T,, + T li? We define T=2#+0-K D = average recoil of the two nucleons F(D) = f w4(e) exp - r) 44 U gf) = radial wave function of the deuteron. The cross section obtained from this definition for the T matrix is do 1. 2 2 #2 wy 12 1 _* ** bh ~ pt |K| +£i+|K lo + 2F(D)Re(3 K K,+ £4)! which is the sum of the free nucleon cross sections plus an interference term whose size depends upon the deuteron overlap integral F(D). F(D) is given in Figure 14 as a function of recoil D, and of CM pion angle for K = 1182 MeV. Since cross section data alone are insufficient to determine the four independent production amplitudes (2, K), the equation cannot be used to determine a neutron cross section. However, the size of the interference term may be estimated since 2”, |K|2< do /da. The rapid fall off of F(D) implies that the deuteron cross section is equal to the sum of the free proton and neutron cross sections to better than 5 per cent for CM angles greater than 20 degrees, so the simple "spectator model" is a valid approxi- mation here, (9, 10) The treatment of multiple scattering obtains the equation do _do® do? 1 do® doa Q Q pe a Qu Qu. fon where R is the radius of the scattering system defined by R= f RU“(R)dR and do? /do and do®/da are the production and scattering cross sections respectively. The principal contribution da? s from the scattering term dg” i do dg a2 dq” will arise from charged meson 45 production followed by exchange scattering, since the ratio of charged to neutral pion production cross sections is large. The size of the correction at 0° can be estimated as follows. The charged pions which can scatter with sufficient energy to be detected by the experimental resolution function define a production solid angle AQ). The solid angle for scattering is AN,, the solid angle D’ of the detector. (Note do®/ dq is in the laboratory frame.) Then the ratio of direct pions to scattered pions is doe ng x be I yx ag do do~* do” a= an Pi Rp? a0, Pan op, doe do° x AQ do° R? an™ 3K D ~~ da d The kinematics limits ¢_ +< 20°, so a0, « 0.5 steradians. Thus a= 25x 0.5x 1/100x 5=90.5. Multiple scattering effects can, then, give the observed y +d -—- n° +d'/y+p- mn + p ratio at small angles with o p* on To obtain a neutron cross section from the deuteron photoproduction data will require 1) all the scattering cross sections such as n +n- nm +p, which can be obtained from phase shift analyses giving both isospin 1/2 and isospin 3/2 states, 2) computer calculation of the contribution at each data point, and 3) estimation of ~ the higher order terms in the series. Previous experiments which have measured the ratio R, n° (d)/n°(p) are those of Chang“) and Bingham 7) , both in the energy region less than 1000 MeV. The results of Bingham show 46 the ratio at 60 degrees near 900 MeV to be about 1.95 +- 0.05, in disagreement with our result. The results of Chang show ratios more in accord with ours, but his results are at energy intervals which are too wide to permit detailed comparison. For example, at 870 MeV he finds R= 1, 71 +- .14 at 60 degrees, and R = 1.62 +- 0.2 at 90 degrees. The general character of the results obtained by this experiment shows a ratio which decreases with increasing angle. At small angles, the ratio decreases with increasing photon energy. In view of the final state interactions discussed above, we believe that even semi-quantitative conclusions about the neutron cross section will require more theoretical work on the effects introduced by the use of a deuteron target. 47 TWHOSLNI dVINSAO NOYSLNAG vl enbly A®W O8il=-y ‘FIONV WO SE Of Se og si of wv 2 Ol v 8) qT I i 1 T 1 A®W ‘d 11093y OSE 00] OS oleh, { OO€ OS2 002 Os] | | 48 APPENDIX I. EXPERIMENTAL APPARATUS A, Cherenkov Counters Each Cherenkov counter “) is constructed from two lead glass parallelepipeds 14 x 12 x 6 inches joined by q-bromo- naphthelene matching liquid to make a single optical block. The mounting house is made of soft iron, serving to shield the counter both optically and magnetically as well as support the lead glass and the 9 RCA 7046 phototubes which viewed it. The construction is shown in Figure 15. The Cherenkov counter is sufficiently large that, with the aperture used, the entire shower is contained in the glass. Statistical fluctuations on the shower production are therefore a small effect. The width of the pulse-height distribution is assumed to be due to the statistical production of photoelectrons by a constant number of Cherenkov photons. For n photoelectrons the pulse height is given by uy = an, and the width by o = a/n. The relation (u Jo)" =n allows. the determination of the number of photoelectrons produced by each tube, The results of the tests in the mono-energetic electron beam are shown in Figures 16 and 17. They show: 1) linear average response in energy, 2) a response function which is nearly Gaussian with a width corresponding to 220 photoelectrons/GeV. For each tube we have for the distribution of photoelectrons 1 - (n - n)?/ an, 2 1 -(u =U )"/2an, Jatn . 6] D(n) = 49: “ONILNONOW SENLOLOHd GNV NOILONYLSNOD 40 Sriviad ONIMOHS AIGWASSV YSLNNOD AOMNSYSD Gl AYNdIS aS AS SASS BEN \: mal i= =| z LL LLL LS, ZEIZITLG TE LELIGZ EN { ofr LI | Se NS { 5 a 4 7 4 iy - woorg SSv19 | ea Nor ANI ssuoe’ = MOCHIM, 3 asviawdS . iv oft PONLOLONE YZ vor > WA “D01g SssvI19 Faal WOLIANNOD AD¥VINOA HD . ; oongs —— t= LH MXOD = 2NZEAOIN ase 2ani—"7 a = ei St & PULSE HEIGHT-ARBITRARY SCALE LOO 90 80 70 60 50 40 30 20 Fr 10 50 | COUNTER A COUNTER B l ! | i | j | i | j Ol O02 03 O04 O05 O6 Of O08 O9 10 ELECTRON ENERGY - GEV FIGURE 16 C COUNTER RESPONSE GAUSSIAN FIT 20 30 40-7 SOF 60 F TOF 10 a 95 38 - 99 - 998 \~— i | 1 | J I | 1l0O-— iO 120 130 =: 140 150 I60 . PULSE HEIGHT | , Vv FIGURE 17 C COUNTER RESOLUTION FUNCTION 52 If the assumption of width source is true, minimum width of the distribution obtained by summing the outputs of the nine tubes is achieved by setting the average pulse height of each tube proportional to the number of photoelectrons, it produces, Thus Us = = B a JV i. = we have 1, an, My amn., (>) =(—) or N =n. In i actual test in the mono-energetic electron beam the array gave 220 photoelectrons, compared with 250 photoelectrons from summing the result of each tube. The decrease is most probably due to incorrect setting of the gain of the tubes, rather than statistics on the shower. The response and linearity of the counters have not been measured for mono-energetic photons. The mechanism of shower production by electrons is the same as that for photons so we have taken the results to be identical. The agreement of observed and computed spectra for photon energies near 650 and 1350 MeV (80°, 911 MeV and 10°, 1390 MeV) indicates that the conclusion is valid. A convenient scale on which to record pulse height will be in energy. Since the mean pulse height produced by the counter is a linear function of energy, Uo = KE os only a scale change is necessary. The distribution of pulse height can be written , ee BE)? ke Janke © E' is the output on the energy scale. We use this scale throughout the thesis, During the tests in the electron beam, the output of the R(E, E') = , where E is the photon energy, and counters on cosmic ray muons traversing the glass block was also measured, The energy equivalent of the produced Cherenkov light 53 was found to be 220 MeV for each counter. Since this number is a function only of the lead glass, its daily measurement provides a useful check on the gain of the phototubes, i.e., on the constant which will be used to connect the energy and pulse-height scales. In use, the counters were mounted one above the other on a trolley movable at constant radius about the target to any angle measured, The system, designed by Mr. Edward Taylor, was so delicately balanced that it could be moved at the rate of 10° per minute by a single graduate student using a four foot wrecking bar. Each counter could be moved independently in the vertical direction to obtain any desired displacement about the beam plane, Tape measure and fiducial marks allowed setting angles to better than 0. 1° and distances to 1/32 inch or better. The counters were surrounded on five sides by lead walls. The front walls were eight inches thick, the remaining walls four inches thick. With the apertures blocked, counting rates in the veto scintillators and the Cherenkov counter were negligible. The aperture size varied from 5" x 7 to 8" x 8", The variation in average pulse height as a function (3) and found to be less than + 4 per cent over the maximum aperture of position on the counter face was measured in the electron beam used, B,. The Aperture Veto Telescope Each aperture was backed by a two counter telescope to veto charged particles. The counters were separated by two grams of CH, absorber. The rate in a single scintillation counter is principally due to Compton electrons produced in the scintillator by the halo of low energy photons produced by the collimation of the bremsstrahlung beam. 54 These electrons, up to 15 MeV in energy, will be stopped by the absorber and hence do not contribute to the veto rate. The low energy photon halo is negligible at angles greater than 20° (lab). The efficiency of the veto system was measured by means of a small counter in coincidence with the Cherenkov counter and found to be better than 97 per cent at the highest rates encountered. The phototubes used were RCA - 7850. The capacitor bank supplying dynode power was boosted by external power supplies on the last two dynodes. At two megacycle counting rate, the drop in pulse height at the end of the 150 millisecond dump was measured to be less than 5 per cent. C. The Condensing Target The construction of the condensing target is shown in Figure 18. The target consisted of the vacuum chamber, the jacket, and the appendix itself. The housing windows and appendix end caps were of clear mylar, 5 and 1.5 mil thickness respectively. The clear mylar allowed visual inspection of the state of the appendix. In use, the jacket was filled with liquid hydrogen at atmospheric pressure. Supply was a commercial hydrogen dewar, pressure fed _to the jacket. The pressure feed valve was either on or off, and was controlled by the jacket temperature sense resistor. The sensitivity was variable. While operating, the valve was set such that the system did not oscillate, simply filling the jacket to replace hydrogen lost by boil off. Gaseous hydrogen or deuterium was put into the jacket under four pounds pressure. The rate of condensation with these parameters required about one hour to fill the appendix, and consumed about one liter of liquid hydrogen. The appendix was 15S JNIT WVW38 NOLOHd LA9YVL NSSONGAH 8] 3unNdls O-< YSAENVHD WANDA fon POWe- oH (SQN3_H.LOg) | (SGNa HLOg) YV TAN ,,SIO0'O UVTAW ,,SO0'O 56 quiescent after filling. The appendix sense resistor was used only to check that the target was full. Steady state hydrogen consumption was about 0.9 liters per hour. Hydrogen for the appendix was obtained directly from commercial cylinders and discarded after each day's run, The cost required the deuterium to be contained between runs. The method used was an inflatable bag of about 500 liters capacity under four pounds positive pressure. This system was adequate for the fill and boil off rate of the target. Purity of deuterium from all gases but hydrogen was maintained by discarding the last few liters of deuterium during boil off. The density of hydrogen and deuterium was calculated assuming the appendix temperature to be that of boiling hydrogen at Pasadena STP (750 mm, 23° C.) , or 20, 1°K, At this temperature, the densities“) are py = 70. 3 gm/liter, Pp = 169.7 gm/liter. 2 2 D. Calibration of the System The co-ordinates of the pulse-height matrix have been defined by the amplified and stretched Cherenkov pulse. From this matrix we wish 2 matrix with an energy scale which is the same for all runs which are to be combined. Let E = Output of phototubes on the energy scale V = Pulse height analysed a = Gain of phototubes in Volts/MeV G= Electronic gain of the system 57 The obvious equation relating these quantities is E = (V/a)/ G. The problem of obtaining the matrix in energy is non-trivial because 1) the electronic gain G is neither a linear function of E nor constant in time and may be different for the various configurations, and 2) the gain of the phototubes « is not constant in time. The calibration input at the phototube summer allows us to use a simulated Cherenkov pulse of known height to measure the electronic gain. The pulse was generated by an SKL pulser with an R-C network and duplicated the shape of the Cherenkov pulse sufficiently well to determine the gain over the range of energies measured. In particular, the extrapolation from the 220 MeV pulse height of cosmic ray muons to the maximum n° gamma-ray energy was assumed valid. The electronic calibration was measured daily, with short term drifts of 2 per cent observed. The stability of the SKL pulser and the ND analyser was checked by using a passive stretching network to produce 20 usec pulses which could be put directly into the analyser. The pulser analyser combination was found to be stable to better than 0.2 per cent. The average pulse height produced by cosmic ray muons was measured each day during the 12 hour period while the experiment was not run, Using the analyser in the 32 x 32 mode again allowed measurement of both counters at once. The muon pulse-height ‘spectrum is nearly Gaussian in shape with a width of 12 per cent, so the mean value determined by the 1200 counts collected in 12 hours is of sufficient accuracy. Given the gain as measured by the previous paragraph, and the energy equivalent for muons obtained by comparison with the electron beam, both factors G and q have been determined and the matrix can be converted to one in energy. 58 APPENDIX TI. DATA REDUCTION We consider here the process of converting the counting data into a cross section, with errors introduced. The procedure is: A) B) C) D) E) F) G) H) reduction of the output matrix in pulse height to a matrix in energy with a linear scale, analysis of the consistency of the foreground, background, and accidental runs with discussion of electronic problems, evaluation of the 1° spectrum from the foreground, back- ground, and accidental spectra as taken, to obtain the counting rate, computation of the detection efficiency, including small . ) corrections to the mn rate, computation of the quoted cross section from the counting rate and the resolution function, summary of the statistical errors, estimation of the size of the systematic errors, attempts to improve the energy resolution of the experiment, A. Reduction of the Output Matrix The raw data produced a 32 x 32 matrix in pulse height from the two Cherenkov counters. To be useful, this matrix requires considerable analysis which we describe in detail. The determination 59 of the matrix edges in energy is described in Appendix I, together with the uncertainty in the values. Given the edges in energy, the second step is generating a linear matrix with a given bin width from the nonlinear, nonequal matrix. To find the contribution to the new bin s from the given bin 2,2, a six parameter surface is determined by the number of counts in the nine bins shown, Then, an integral of this function between (a,, ay) and (b,, bo) gives the contri- bution. Proceeding likewise for bins (2,1), (3,1) and (3, 2), the number of counts in bin s is found. The initial choice of constant w connecting energy with pulse height is given by the day's cosmic ray cali- bration. However, the data itself, assuming a reasonably slowly varying cross section, can be used to find this ol number by comparing the observed spectra with the computed spectra of Appendix IV. We have used the following procedure to do this. 1) Using the calibration given by the day's cosmic ray run, the linear matrix is generated with the desired energy scale for each foreground (F), background (B) and accidental (A) run, The result for nm from deuterium (7) is then obtained. 2) The mean energy for each counter on (mr) is obtained for each (F) individually. This energy is given to about + 2%, which is better than the value given by the cosmic ray calibration. The photo- 60 tube gain is then changed to bring the mean energy in each counter to the value computed. From all (F) runs, we have the daily gain. The number is distributed about its mean with a width of about 2%. The (A) and (B) runs are then recomputed using the new gain figures. The procedure converges quickly to a consistent set of daily gain figures. B.. The Data 1. Foreground runs The foreground rates are given in Table 11 in Appendix II together with ye values for multiple runs. The overall 2 for the experiment is 122 for 134 degrees of freedom, but the values do not follow a x2 distribution very well. The cross comparison of runs done in the several configurations is given in Table 5. 2. Background data We present the background cross-section ratio in Table 6, The data showed no statistically significant time variation in back- ground rate for the 1073 and 1308 MeV data. Somewhat improved 2 values for individual (r) runs are obtained if foreground and back- ground runs are combined from the same configuration. The runs were done in such sequence that this is always possible. The 1500 MeV data show a pronounced difference in background for the small angle, wide radius configuration when compared with overlapping points at the same angle for the narrow radius. These points were taken during a period of poor beam collimation due to instabilities in the synchrotron circulating beam at an endpoint of 1513 MeV, 61 TABLE 5 Comparison of Experimental Configurations Counting Rates Pions/BIP Efficiency Cross Section Position 1 2 Ratio Ratio D,, 1078, 20° 1.86(0.08) 2.66(0.14) 1.33 1, 07(0, 08) 1073, 28° 1.41(0,05) 2.24(0,06) 1.52 1, 04(0. 05) 1512, 17° 0.54(0.04) 1.32(0.10) 2.38 1. 02(0, 12) H,, 1073, 28° 0,65(0.03) 0,99(0,04) 1,52 0. 99(0. 06) 1308, 11° 0,21(0.15) 0.34(0.03) 1.65 0. 98(0. 12) 1512, 11° 0,23(0,02) 0.48(0.03) 2.31 0, 92(0. 08) 1512, 17° 0,62(0.05) 0.26(0.03) 2.35 1. 02(0. 10) Average 1, 01(. 03) Ey = 1070 MeV Angle 0.0 5.0 10,0 15, 0 20.0 28, 0 35.0 42.0 49,0 Ratio 0,08 0.1 0.12 0.06 0.04 0, 04 0,053 | 0,053 0.067 62 TABLE 6 56, 0 Background/Foreground Ratio 1308 MeV Angle Ratio 0.0 0. 20 5.0 0. 20 10,0 0.085 17,0 0.042 22, 0 0,05 28.0 0.05 35.0 0.05 42.0 0. 06 49, 0 0.065 0, 064 1510 MeV Angle Ratio 0.0 0.15 5,0 0,125 9.5 0,145 15,0 0.150 20.0 0.074 26, 0 0, 032 32, 0 0, 09 39,0 0,10 46,0 0, 10 53.0 0.11 63 Comparison of the (mt) rates, subtracting backgrounds from the same configuration, gives reasonable x? values so no error has been added to the final rate, The ratio of background to foreground cross sections at wide angles show that the counters viewed about 0. 2 grams of material, The four mylar windows of the target contribute 0, 04 grams. The remaining 0.16 grams corresponds to 130 cm of air. The counter geometry shows this to be a reasonable figure. The rise in the forward direction is due to two effects. The secondary scrapers become visible from the apertures, and coherent production from the complex nuclei in the background material becomes large. 3. Accidental coincidences between the Cherenkov counters The rate of accidental 7° coincidences (y At 100 ns) - (yp) was monitored constantly. At points needing correction, the accidental n spectrum run was taken by delaying y A through the Tr coincidence circuit, The rates y ‘A and Yp were checked for consistency between the (F) and (A) runs. The gamma rate is governed principally by two variables. 1) The energy bias set on the Cherenkov counters by the discriminator gives a low energy cutoff to the analysed pulse height. 2) The beam alignment through the secondary scrapers affects the shape of the scattered photon spectrum, especially for low energies. It is not possible to express the accidental rate as a function of y ‘A and YB ina simple way. However, Table 7 shows that the simple product of the rates gives a correct measure to a factor of two which is sufficient to provide a check, and to prove that points for which a spectrum was not taken do not, in fact, need a correction. 10 15 20 28 10 15 20 28 R- = accidental rate YA BO rate in each Cherenkov Counter 3 a i R i 1.0 1.0 1.7 1.7 2.5 2.5 2.5 1.0 64 TABLE 7 Accidental Rates relative beam intensity proportionality constant Rate 0. 02 0, 07 0, 08 0,013 0. 01 0, 01 0. 12 0.13 0.09 0.014 | _ 2 =VYaXYpRxl Xo i Ya * YB Xx 25 70 120 19 10 20 150 180 120 20 0.8 1,0 0.7 0.7 1.0 0.5 0.8 0.7 0.8 0.7 65 ~ 4, Electronic problems _ We have considered accidental 7°'s in the third section. The problem of accidentally vetoed gamma rays is a straightforward correction, knowing the veto rate and the width of the veto discrimi- nator pulse, At the maximum veto rate of 200 ke and the 20 ns pulse width, both sides will veto 1.2 per cent of the gamma rays. The veto rate was monitored directly by means of coincidences between y and V. This monitor gave a correction of less than 2 per cent. The dead time in the veto discriminator was 30 ns per pulse. At the 200 kc veto rate, 1,2 per cent of the vetos are missed. With the veto removed on one side, the aa counting rate increase was less than 10 per cent at all points. The source of the increase is consistent with the detection of a single T°, one gamma ray of which converts in the aperture wall, Hence missed vetos can be ignored at all points. C. Evaluation of the n° Spectrum The three spectra which make up the data, foreground (F), background (B), and accidental (A) are shown in Figure 8, together with the resulting nm spectrum (r). The counting data have been tabulated in Table 11 for all the runs. The errors shown on the (F), (B), and (A) rates are their counting statistics. The number of counts is taken to be the number above the minimum in the spectrum, which usually occurs at an energy below which we calculate about 5 per cent of the counts lie. The numbers for (B) and (A) are taken at the same energy. This procedure gives a simple error analysis. The cali- bration for (B) and (A) was obtained as described in Section A) of this 66 Appendix, Since the rate depends on the energy scale, the errors introduced by the calibration procedure contribute to the counting rate error. The calibration error is statistically distributed with about 2 per cent width. The error introduced is given by the product (counts per MeV at cutoff) x (calibration error in MeV). These are combined and given in Table 11 in Appendix ITI, The final (1) spectrum does not show a complete separation between the 7m and mm regions. The correct procedure here is to fit to the complete spectrum a function in two parts, the single "> spectrum shape from Appendix IV, including single n° detection in double r production, and the semi-phenomenological shape for mone from Appendix V. However, the approximation of taking the number of counts above the minimum channel is good to better than 3 per cent except for the point (0°, 1073), the loss in aa being equal to the gain in won. The error in the exact procedure is due to counting statistics on (nr) below the cutoff. This number is quite well given by the number of counts per 25 MeV at the cutoff channel. The systematic error introduced by the choice of fitting function for (nr) is less than 1 per cent, by comparing several choices for the function. D. Calculation of the System Detection Efficiency 1, The geometric efficiency We define n A” the result of the analytic efficiency calculation "mM = the result of the Monte Carlo calculation ny = the true efficiency. 67 The detection efficiency at the various angles of the same configuration is given to about 10 per cent by = doom dp n(o_) =a x ( (9) )x = T disp dep where da mn! d %, b is the production solid angle transformation and dp/de is the decay probability for a n°, taking the decay angle » to be the mean angle of the aperture with the beam plane, This is the expression of the fact that the counters appear the same in the center of mass at all angles. (Note that this comparison is made with values of 7 A for the same photon cutoff energy, not the real values which vary slightly from one to the other.) The relation makes it reasonable to write n(9) = 6 x na (8) and find 8 by a minimum 2 method using the results of the Monte Carlo program which are distributed about the true value with known error, The errors on the Monte Carlo calcu- lation are typically on 2500 successes, or 2 per cent, so the value for 7 is certainly known to better than 1 per cent. 2. Corrections to the n° rate a) Effective aperture size The lead walls transmit some of the gamma rays, so the effective aperture size is greater than the geometrical aperture size, We have dn “2° dx, dn dn 1 ot a 1” dx} | x x 0) 3 lj o 68 where X14» Xo, Xg are the displacements at the outside, inside, and sides of the apertures. If the gamma ray makes an angle 6 with the lead wall, the traversed distance is 5 = x/sin@ and the absorption -7 § is o(8)=e where + = 1.4cm_ - To first order, then, _f dn _-xr/sme An =[ ax e dx or the effective aperture size is defined by Xy =X) + sine/r for each x, The apertures are shaped on sides Xo and X_ SO sing/t < 0.1 cm. Side Xy is not shaped and has a large derivative since the separation determines the cutoff energy. The efficiencies were calculated at each point with the corrected aperture size to give the correction. . fe) b) Conversion of t~’ gamma rays The materials which the gamma rays traverse are: the walls of the target and vacuum chamber, the air path, and the front scintillation counter, Referring to Figure 18, the target is of a complex shape requiring a ray trace at each angle to find the path length. (3) For example, at 28 degrees laboratory angle, traversing all of the aluminum layers gives a total path length of 0.62 cm, or an absorption of 5 per cent per gamma ray. The maximum absorption correction is 6.8 per cent per gamma ray. The results of the ray trace are integrated over the finite target to obtain the correction. 69 Taking + = 2.3 x 107° cm7! the absorption by the air path is 0. 23 per cent per 100 cm. , or 0.7 per cent per gamma ray. To be counted in the front scintillation counter the bias set on the counter pulse height required a path length of 0.3 cm for the produced electron pair. The length for conversion is then 0.85 cm, so taking ¢r = 0.185 om7! we have 1,6 per cent per gamma ray, c) Shadowing correction At the non-standard radii the apertures did not rotate about the target center, so at some angles not all of the apertures could be seen from the target. The correction is just geometric, Table 8 gives all the efficiencies with average angles, photon energy, and cutoff energy. Also given is the factor from the corrections just discussed which multiplies the geometric efficiency to obtain the system efficiency. Finally, the factors (12) of target material must be used. These are given by connecting BIP's with total beam energy and the actual amount 1,16 gm. hydrogen n(system) = No * { 2.795 gm. deuterium 13 X { 1.10 Machine BIP's per 10 Mev} . 70 TABLE 8a Parameters at Experimental Points Ey = 1170 MeV R = 110" Angle, Angle K E Efficiency Correction Lab CM v 0.0 3.5 925 925 0.511 0.955 5. 7 11,2 924 920 0. 503 0. 936 11.3 20, 3 925 907 0.477 0, 921 16. 0 28, 0 918 884 0, 457 0. 909 21,8 47.2 922 855 0. 413 0. 897 28,0 47,2 922 826 0, 352 0. 886 R = 82" 22.8 38,9 925 859 0. 656 0. 795 29.1 48.7 911 811 0. 566 0. 839 35,1 57.4 899 765 0. 469 0. 878 42,1 67,3 899 0. 349 0, 852 49,7 77.5 904 0, 249 0. 773 71 TABLE 8b Parameters at Experimental Points E, = 1308 MeV R = 110" Angle, Angle K E Efficiency Correction Lab CM T 0.0 3.7 1169 1169 0, 279 0, 959 5.7 11,4 1167 1160 0. 276 0, 949 11.3. 21.4 1171 1143 0, 254 0, 932 17.1 31.9 1175 1113 0, 223 0,917 22.8 41.8 1176 1071 0. 193 0, 904 28, 0 51.6 1180 1028 0,197 0. 898 35,0 61.6 . 1190 967 0,137 0, 894 42, 0 72, 0 1189 896 0.127 0,898 49, 0 81.6 1186 826 0,101 0, 900 0.071 0, 904 56. 0 91.0 1189 762 72 TABLE 8c Parameters at Experimental Points Ey = 1510 MeV R- = 140" Angle, Angle K E Efficiency Correction Lab CM 7 0.0 3.1 1382 1382 0.161 0, 968 5. 0 10.5 1385 1377 0.154 0.919 10.1 20. 2 1380 1349 0,157 0,851 15.3 30.1 1381 1312 0.143 0.772 R = 110" 9.5 19, 0 1387 1350 0, 334 0. 956 14.6 | 28. 8 1386 1323 0.311 0, 939 20.2 39.2 1384 1296 0, 280 0. 926 26.0 49,6 1388 1206 0, 214 0, 916 32, 4 60. 3 1385 1126 0.176 0, 908 638, 7 70.2 1385 1045 0,135 0. 909 46.1 80.8 1388 954 0.091 0,915 53.0 90.7 1385 872 0. 071 0. 919 73 E. The Quoted Cross Section We let C = the observed counting rate T(k,9)= the geometric resolution function o(k,@) = the differential cross section Then C = fos, a)T(k, oak de. The efficiency is defined by n (6) = T(k, 6)dk de the mean photon energy by k= <k) = 1/y [ kT (k, @)dk dé and the mean pion angle by 057 (8) =1/ fe T(k, 6)dk de. The cross section quoted is defined as o(k, 8) =C/n. We now examine the effect of the energy-angle dependence of the cross section. Expanding the cross section to second order, and dropping the *- notation, 2 o(k, 6) = o(k,,8) + 0, (k - ky) + 5, (8 - a) + Cg kg + & 2 + § 2 Ck’ kk ™ [99°99 * Then the counting rate is 1 2 1 2 Cen x {lk 00) +5 og Mk - ky ) +5 Tyg (6 - 8) > + oy 9 ((e- K)@ - 8) 74 where the linear terms are zero by definition. The moments of the function T are given in Table 9, together with the values for the cross section derivatives in the following three cases: 1) the cross section due to the third pion-nucleon resonance alone, taking an fp /2 Breit-Wigner resonance, 2) the data of Diebod‘t®) at 60 degrees, and 3) the data of this experiment at rather wide energy intervals. The corrections are Resonance Diebold This Expt 2 1) 548 o 1.0% : 0. 3% 1 2 2) 5 %q9 86 1.3% 3) 46. 52 14% 15% 6%, 5 Oe Otek 6 6 b Correction 1) may be ignored due to the small value of 8e 9? which reflects the fact that k and 6 are nearly independent variables for the two photon detection scheme, 2) is not quite ignorable, but the data comprise a distribution in 9 so the value of On, may be estimated and a correction made. Correction 3) due to energy variation alone ‘is not ignorable. If the cross section has the large but physically possible derivative indicated by the resonance, our data must be used with care. A systematic error has been added to the data for this effect. 75 TABLE 9 Cross Section and Resolution Function Parameters Parameters for k = 1180 6207 (8 - x.) Y- ¥,)) Moments 2 2 2 Sok V8 96 Pra ®om MeV-Deg Deg MeV 0 47,0 2.10 72.7 30 52, 4 1.57 71.0 60 67.1 1.58 67.8 90 90. 1 1.60 69.6 Derivatives Resonance Angle o o o CM. ke Te kk 0 0 3.8x 10-3 5x 1079 30 2.5x 1074 7.8 x 1073 5x 10-5 60 -2.3x 10-4 1.4x 10-3 5x 10-5 90 0 3.3x 10-3 5 x 1079 Diebold Data 60 - - 6x 107° This Experiment 0 25x 1074 2x 1073 2.14 107° 30 .8x 10-4 1x 10-3 1x 10-5 60 12x 10-4 1x 10-3 .5x 1075 90 | .2x 10-4 1x 1073 3.0 x 1075 76 F, Statistical Errors The errors discussed previously are given here together with typical values. 1) Counting statistics: 5 per cent on foreground, 3 per cent on background, and 2 per cent on accidentals, 2) Calibration error leading to error on counting rates of background and accidentals, 1 per cent 3) Uncertainty in making cutoff between n° and qn? regions, 3 per cent. G, Systematic Errors Two sources of internal systematic error have been discussed. In Appendix V the contribution from single n°'s in pion-pair production is estimated to be less than 5 per cent of the measured cross section, and to decrease with increasing angle. In Section F. of this appendix it was estimated that the correction from energy variation of the cross section is less than 5 per cent, although possibly much larger. Remaining sources of systematic error are related to the experimental work itself. 1) For the endpoint energy of the synchrotron, we have taken the value E (true) = E (set) x (1.023 + 0. 003) 44), The detection efficiency changes by about 2 per cent per MeV, or a total error of about 6 per cent. This correction is independent of angle for each distribution to about 10 per cent. 2) The absolute normalization of the quantameter is estimated to be 2 per cent, with long term drifts of the same order. 3) The integrator calibration is measured to 2 per 77 eTst €TST eTST €TST 80ET €TST S 8X9 OTT (OT/TT-S/0T STST &tct etct etSt 9 8X9 ShT G/0T-0¢/6 €LOT SLOT SIST SLOT SIST SLOT STST 80sT 80ST SLOT €IST 80sT GIST 808T G 7 8 x9Q LXQ OTT © 8 0¢/6-1/8 1/9-1/s 80ET 80ET 80ET ELOT 80ET €LOT €LOT €LOT €LOT €LOT SLOT & LX8 80ET 80ET 80ET 80&T 80ET G LX9 OLT OTT 1/S-@t/T JULOdpUm UBATD IO} SuOT}yeINSTJUOD uNY OT HIGVL 80ET 80ET 80ET 80€T 80ET J LXG OTT IT /T-T/01 9¢ GP GE 8c 0¢ ST oT G 0 o[suy # UOTTISOg oin.redy snipey a7eq 78 cent. These errors are considered in detail in Reference 3. The calculation of the detection efficiency has been discussed in Section EK. An upper limit on the error is 1 per cent. H. The Energy Dependence of the Cross Section All of the information obtained by the experiment is contained in the spectrum S(E) itself. An attempt to unfold the energy dependence of the cross section from the data may be made with varying degrees of sophistication, An easily interpretable method is to fit the observed spectrum with a several parameter computed spectrum, S (6) =B0, i [ @-£)"° ty@)R@, E)aEaE" . As usual, we define 2.8 (S ,6,)- 8(6,)) i=l o*(8.) and use least-squares method to find the O° We investigate this procedure numerically by generating spectra S54) for various cross sections, to see what errors Aa, to expect. The resulting error matrix does not depend very strongly on the choice of cross section, allowing us to write bay = 0,08 x /1000/N Aoiy = 0,10 x /1000/N x 10°" ub/Mev 79 hag = 0. 20 x (1000/N x 107" b/Mev" where N is the total number of counts in the spectrum. Recalling the values estimated from the data for the cross-section parameters, Qo = 0.3.x 107, Og = 0.1x 10°* and that N < 1000 counts, the difficulty of obtaining cross-section information is made apparent. We conclude the following: 1) 42 will be reasonable even for rapidly changing cross sections. 2) The sign of the slope Oo» and an idea of its magnitude, is obtainable, This will not, of course, change the mean cross section o(k,,6 3) 3) The second derivative is unobtainable so no correction can be made to the data, This is true even for Oe = 0.2 x 1074 ubarn/ Mev’, which would change the cross section by 0.1yubarn, or 10 per cent. 4) he errors on the data (og - me) are given by N= 500, so not even a slope may be reliably quoted. There are two additional sources of error other than counting statistics on the coefficients a 1) error in energy calibration of the two sides, and 2) error in the width of the response function R(E, E'). Since we are not going to try to obtain information from the deuterium data, we will not investigate these points. 80 APPENDIX fil, COMPLETE COUNTING DATA We present here the complete counting data of the experi- ment. Some of the points are the result of the combination of runs from different configurations. For presentation purposes, the rates and errors were normalized by use of the calculated efficiency ratio. The process of analysis made no attempt to combine back- ground runs widely separated in time, which causes rates used for deuterium to differ slightly from those used for hydrogen. As discussed in Appendix II, the background rates for the small angle Ey = 1513 MeV data were time dependent, which accounts for the rather large difference here between the hydrogen and deuterium rates. 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Yield of Pions in the Laboratory We wish to compute the yield of pions in the laboratory from a deuteron target with a constant center-of-mass cross section. We consider here the modification of kinematics due to the internal motion of the nucleons and due to the binding energy of the deuteron. We define the 4-vectors by = photon energy in lab k* = photon energy in cm = target nucleon in lab p* = target nucleon in cm = spectator nucleon in lab etc, recoil pion in lab recoil nucleon in lab H k p n i r s recoil spectator in lab b binding energy of deuteron in lab W = total cm energy and the angles by We take the binding energy of the deuteron into account in the conservation of energy. Define 88 b = (binding energy)/2 Po =M,-> n = -b Oo M, = (M, - M,)/2 M =M_-b, a so My = 2M, - 2b, and also M, =M,-e¢=M+b-e=M+re, We shall always talk of the target particle as a proton. We use the spectator model, which assumes that the matrix element describing the process T,, is a function of W only, and is the same as the free nucleon matrix clement for equal CM energy, and that the spectator nucleon does not contribute to the interaction except in conservation of energy, so the spectator momentum is unchanged, i.e., S=n. Except for the kinematic factors to be computed, then, the deuteron cross section is the sum of the free proton and free neutron cross sections. For momentum transfers larger than about twice the average internal momentum of the target nucleons, about 150 MeV, this result is obtained from the impulse approximation without multiple scattering corrections. The invariant definition of the cross section for a two-body reaction, k+p-n +r,is ap P12 2 HS 2.4 4 do = MyM, | erp Ol “1° )8 (e"- M6 “(n+r-k-p)| Ts] “dnd. ~ The quantity k-p is the usual flux factor KE + 8) for p=0. We consider the change to do(W) as a function of the target momentum p. The total energy is 89 w = (k + p)? = 2kM + M2 - 2k+- p and k+ p=kM+% cose ) but w2- 2 2 2 kM = Xe) k- p= (W"-M)/2 2(1 + # cose) and doing the integrals over the delta functions, we have wee. tw" 8 (SM) f (7x da* L so the center-of-mass cross section is not a function of the target momentum for a given total energy. The number of pions produced into the laboratory solid angle dq by N(k)dk photons from D(s a's nucleons is do * +o) _do 3 dN = do N(k)dkD(s)d‘s . for Q We take the parameters to be: 90 w = pion energy § —_ pion angle in the laboratory kK, = photon energy for production of a pion of energy w at angle 6 from a stationary target nucleon. The energy spectrum of pions is 218 Elz Qu = [9S S$ ney & d@a's evaluated for fixed (w,¢@) and integrated over the deuteron momentum distribution. The cross section quoted is defined from the total number of counts by N= oe.) | F(k, 8,9) dw d°s where da* B(K) dK F(k, 8,8) = Go- “K" qu DUS) - The procedure we use is to write for any kinematic variable F(x) = F@Q@ +6), where F , is the value of the variable for pion production from a stationary target and 6 is the correction which depends on the deuteron variables. The deuteron is parameterized in terms of the momentum and direction of the target nucleon (s,6,¢) and the kinetic energy T=s,- M. Since |s| < 200 MeV, the nucleon can be treated nonrelativistically, so T = s2/ 2M, 91 1) For example, to solve for the lab photon energy we use k+p+n=T+r+s lt+e k=k x ite =k x (1+e) 9 fe) 4% ((s,-M) (2M- w) - we + gs cosp)/(wM., - 2/2) Eo = -((s,- M - sCos5) - e)/(M, -w + qCose) where wM, - 2/2 k* M, -w + qCosé@ is the value of photon energy, given (w,@) and a stationary proton target. 2) Center-of-mass energy, We = Wwe+a oO . \ =e(W2-M)*- (ak +2ck - 2M )e- 57+ 2(k +ek )sCoss. 0 p O re) p oO 92 3) Solid angle transformation (do*/da) x dk/dw, for the invariant I, dq. di dat’ yak = da? sak /(1 +n) Oo n= k ,(sCos -e- (s, - M,))/(oM, - 2/2 da* _ da* ar ~ar | “+4 £0) Define Then 93 A Fay + d» Coss +a,Cosé +a, Cos" +a,Cos“6 +a .Cos 8 Cos 6 hs aM- pe/a Re (AM- 4 Ke) ol, = al i -\|- av, (k-“)ye hy > asl Ake wA| dg = A,r ak, s\i- so ay = nels] de = Rysakee | i} +e [x mdel ere a Mis |” dy = veuse salty ore | So we finally have lt+e da* _ 2 1 da du *o% (rye, )* Ot Tee F =: ale & 2 Next we write 94 ey = AL + B. cosp e,=C +D _ cosé 2 € € n =A_+B_coséd n n We take s = 100 MeV, S57 M= 10 MeV w =k = 1000 Mev, W2=3x 10° Mev” to get -6 -3 A, = 1/6 x 10 B= 1/5x 10 and with the order of the correction terms A.~ T/M B.~ S/M C.~ T/M Di~ S/M A ~ T/M B~S/M . ey, ~ 8/ - We expand to find e=A -C +B cosp-D_ cos6 Ee € € € + p? cos”s - BD cosg cos € 6 € 95 and 1fl+n)=1-n+ n2 keeping terms which are 0(1074). Combining all the binding energy terms gives Gp = (3 - 2A (K, - M) )e or Gp~ e/M = 10°? so we are justified in keeping only the first order terms. Since D(s) is isotropic, D(s)d°s = D(|s| )dsdo, . We also observe that the integrals i) cosédn, = ) cos 8dQ, =0., To order 1074, F=F x (1 + f) f=G,-A+ +i (B? - Bo +O +a.) +4 cose @ - Ba.) B “1° 3 27 %4 "S53 6 %37 Thus we have the yield of pions in the laboratory dN = Yj (1 + f) x C So.) x dw x D(| s|)ds * where Yo = oe oa Bi) evaluated for stationary target kinematics. Since f depends linearly on s and 8”, we can do the integral over s, [£D@) ds by replacing s by (s) and 3” by (87) to obtain Y= Y,x [+ #) Ds)ds. 96 B. “Acceptance Function _ In this section we compute the number 1, which connects the counting rate with a constant center-of-mass cross section. We also define the mean angle and energy kK and BO» and discuss the magnitude of error expected in equating o(k., 8) with o(k,,6.). The observable spectrum shape is also calculated. We define Y(E, 9) Yield of pions in the lab from a constant CM cross section, including the shape of the photon energy spectrum Cc = Counting rate k = Photon energy E = Pion energy E v Eo = Energy of the decay gamma rays Ey) ED Output of counters using energy scale R(E!,E) = Resolution function of Cherenkov counter. A quantity in the CM frame will be denoted by an *, e.g., k* = photon energy in CM, The following functions are useful: 1) T(k*,6*), the energy-angle resolution function in the CM for pions detected by the counter in terms of the quantities in which _ the cross section is usually given; 2) (E,6,0, tr), the probability of detecting a pion given by (E,9,) produced at a point r in the target; 97 | 3) V(E,, Ep; 6), the distribution of gamma-ray energies of detected pions at given angle 6; 4) » (configuration), the number which connects counting rate with the constant center-of-mass cross section, defined by C =x do/dn*, To find these functions analytically would require solving for a function @(£,6,0, E,E,,k, vr), Such a function is impossible to compute in practice, so we turn to two approximate methods. By use of a Monte Carlo program, the experiment may be duplicated on the computer to some statistical accuracy. The events are tallied by the parameters of interest. Clearly, all the functions above may be generated in this manner to an accuracy limited only by the amount of computer time available. The program for the configuration used by this experiment obtained about 1500 successes per minute, An approximate analytic function for n has (2) is of sufficient accuracy for most calculations. The advantage of been derived* “ which, when compared with the Monte Carlo program, this function is that a complete calculation may be done in about five seconds, The functions T and ¢« define n immediately, n= | T(k*,0*) dict de* =| Y(E,6)e(E,6) dEde which also relate T and ¢, _T(k*,e*) = Y(E,6)e(E, a) I Gao ox). 98 Defined in this manner T(k*,@*) includes the photon spectrum and the deuteron variables (a small correction to the free nucleon case) while ¢(E,6) does not. The counting rate in terms of T is given by C= | o (ek, 0*) TU, 0*) dk*do* so the quoted cross section is o (kt, @*) = C/y if ke and o* are the expectation values defined by T, 1 ka (kk) at *T (1k * ke (k*) = f eT (e ,o)dk*de . The Monte Carlo program will calculate these numbers easily. Next we consider the two dimensional distribution of gamma ray energies V(E,, E,). The observed matrix will use the response functions of each counter giving O(E, E}) = ) VE,,E)R, (Ej, £,)Ry(E5, Ep)dE dE, where Ry and Ro are Gaussian with widths hy and Hoe We wish _ the distribution of E’ = Ey + oP F(E' = Ey + E}) = [oe - E}, Ej)dEy . 99 The convolution of the two Gaussians gives V(E,, E,) (E'- (@,+E,))” exp - HE, + Hoo) 2K4E, + MoE o) dE dE, . “F(E’) = The widths of the two response functions were measured to be = 4,55 MeV (220 p.e.) hy Xo = 5.26 MeV (180 p.e.). Let he tH _ - 1 2. E = E, + E, 4 = 5 4,90 Ae HK _ _ 1 2_ A= Ei - E, § = 5 0,35 Then E)? y. 1 ¢ - - Ee Taking typical values for the variables, E= 1000 MeV ~<600MeV, xE = (70 Mev)” so (E' - E) ~ 200 MeV. Then 100 _ =l 2 i SA ~1 76a ,1;,6A cE te) Foe tg Ge) so the exponential term is 2 2 2 1 6A , 1,4A (E' - BE) (E' - E) aE ~je * 2 GcR) ) exP ~ “pg xP - gg Oe The counters are symmetric with respect to the production plane, so VE,, Eo) = V(E,, E,)- Noting A - -A the exponential is then 2 2 2 1 (E' - E) 1,6A (E' - E) ai exP - “3 [1 +5 (=) ] cos h( “ome 7 6A) . oye = 0,001 and cosh( ES EY 5a) = 2% EF The typical values give i (ohy” yp Vv § 5 OE 1+.05, The Monte Carlo program shows that, in fact, 90 per cent of the successes have A < 200 MeV, so the correction term is less than 2 per cent. The actual spectrum obtained will be oo (E' - (E, + E,) )° FE’) =] V@,,E,) exP = u(E, +E ) dE,dE, . Using the definition of S(E) in terms of V, S(E) = [ve - E,, E,)dE,. We obtain for the observed energy spectrum 101 ' 2 -F(E') = fs) exp © ~ E)'/2xE gp We next derive the parameters of the observed spectrum in terms of the cross section and the parameters of the resolution function. Since 2 ols) =o(k,) +22 C( - kK) + 2S (x - Kk) 2ake oF and E- E. ak kK)» we have o(E)=0,+8(B-E)+y(E-E,)* For the resolution function Y(E) we have | ¥@)aE =1, E,= [EY()az, Y S(E') =[o (E,E)Y(E)R(E,E')dE we have = | - = - f Oy (E - E_)"Y(E)dE, S {@ E_)°Y(E)dE. Then from [ S(@")aE =qaty of = u and (E E. u Oy + u Sy \ = Pim 8 2 Y 3 (E") = Eo EQ tiey +7 8y 2 on 2, 2 Ye 3 1 to t = & to After some algebra, o ((E E\) y oY +kE_ +k( Sy) 102 7 3 Ba 8 .y, 4. (8B 2 Yq 3 f ahs + rl Sy + a dy C Cy +) Sy ) . Acorollary of this formula is obtained by setting q = 1, 8 = y = 0, showing that the convolution of a distribution Y(E) of mean Ey and width o,, with a Gaussian Y response function of mean E, and width xE gives a distribution of mean Ey width o 2 oy" + kE O° For an estimate of the size of the corrections, we take the values again at 20 degrees, k = 1500 M 2 3 2,3. 4 3 max eV, Oy = 5.0x 10° MeV’, Sy =-3.5x 10° MeV, _d0 ak_ 4,-2 _ an74 . . _ |B | =k aE" 10", |y| = 10 ~ for which we obtain E, - Ei, = 35 Mev and ot” = 8420 = 0° ~ 2380, Since o,* 104 MeV, then o' = 92 MeV. Taking the values estimated from data we obtain Eo - Eh = 10 MeV and o 2 101 MeV. We apparently will find good agreement between observed and computed values for slowly varying cross sections. For plotting purposes we also have the spectrum in each counter, I(E) = [ UE, E,)de and the spectrum in A = Ey - Eo D(a) = f U(A+E,,E,)dE, . Note that the moments _. 2 2 re) re) and 108 2 2 op =o," top - (E- EB, VE - E,%) top B define the correlation coefficient for the matrix. 104 APPENDIX V, PION PAIR CORRECTIONS A. Detection of single n°'s in the reaction y+N- m+ (7 N') We choose the parameters of this reaction to be (E 0? k, T 2 2 Tt Gs M,), where M, = (E_ + Ey)” - Pe - Py, is the invariant mass of the pion nucleon system. Figure 19 shows this mass as a function of E, for various photon energies. The cross section is 2 given by a x tm, *)| M| 2 where q, 1 k* W | 1/2 2) _ 1 2 2) (2 2 f(A p= gl (a - (MO - MY") - (Mo +M_))1 and | Mj 2 is the sum over all the contributing matrix elements. We compute with two models: 1) The statistical model, in which | M| 2 is taken to depend on only W, and found from the total cross section. 2) The resonant production model, in which the cutoff on pion energy requires M A be near the mass of the first resonance N¥ (1238), and the mechanism of production to be y+N-10+N*, The matrix element then has a Breit- or? Wigner shape |M| 2 a) [(™M, - My) +t] The reaction y + P-r ° + (7 +N) has not been measured, much less the corresponding reaction from deuterium. We have measurements 105 1100 K =1300 MEV PION ENERGY, MEV S oO oO 900 YIELD OF PIONS 800 | | | | | 1050 1100 1150 1200 1250 1300 DELTA MASS, MEV PRODUCTION KINEMATICS, 6, = O° i.4ce RESONANT PRODUCTION 1.2 1.O- SINGLE PRODUCTION 0.8 fF 0.6 0.4 0.2. PHASE SPACE | 1 i — | 950. 1000. 1050. tl00 1150 1200 PION ENERGY, MEV FIGURE IS PION PAIR PRODUCTION 106 only of y+p-— n +m +p with total cross sections, and of y+p a nm + N*(1238), production of a pion and the first pion- nucleon resonance, with differential cross sections. As an upper limit, we take the cross sections for the corresponding mT reactions from deuterium to be equal to 5 times these. It is impossible to justify this, but there are several plausibility remarks. 1) Pion production by pions!) give comparable cross sections for all final states, The ratio (np = non°p)/(n p> mn’ nn) = 1/2 favors the charged mode. 2) Calculation from isospin in the decay of a pure T = 1/2 state intoa T = 3/2 state plus a pion, e.g., N*(1688) + N*(1238) +1, gives a ratio of 2/3 favoring the charged mode, 3) The one pion exchange, or Drell, diagram for two pion production is small for a n° in the forward direction. Using the results for the charged reactions, Table 12 for n's was calculated, The experiment provided two checks on multiple pion production. 1) Synchrotron subtractions at (40°, 1170) give a correction of (3.0 + 3.0) per cent. Most of the statistical uncertainty arises from the difficulty in subtracting off the contribution from nm production, 2) Fitting the observed spectrum to the computed spectrum shows there is no enhancement of the low energy side. The procedure will be inexact because of calibration uncertainty and difficulty in subtracting off the non events. An upper limit can be found of about 0.05 wbarns, which is a correction of 20 per cent at 0° and 3 per cent at 40°, 107 000 "0 Z00 “0 000 °0 200 0 0°06 800 0 TIO °O 800 “0 TIO °O 0°09 TTO “0 TIO °O TEO0 “0 TE0 “0 0°0€ 0£0°0 820 "0 ZrO “0 070 °0 0°0 AW OTST = 4 “€ 006 “0 T00 *O 000 *0 TOO °O 0°06 400 °0 PTO "O TIO “O Zc0 ‘0 0°09 0T0°O GTO “0 0€0°0 Zv0 “0 0°0€ 850 °0 ¢GL0°O 870 °0 290 “0 0°O | APW BOT = az TO0 *O TOO °O 20 ‘0 F0°0 0°06 LTO °0 €€0°0 G0 °O OT “0 0°09 0F0 0 Lb0°0 cr ‘0 bT “O 0 °0€ TIO 9T °O 9T °O Go 0 0°0 0°0 uolponpoig soedg uorjonporg gordg WO qery yueuosey estud yueuosey estud o[suy ‘aTsuy Ue AA) Ue A ‘oryexy uorjoeg ssory DATIETOY pr Pt + Pt ol}ey UOT}Ieg SssOrTy sureqosroru ggg = Lo SUOT}IeIIOD UoTJONporg ileg uotg 6. HIGVL 10, 11, 12. 108 REFERENCES Henry Ruderman, Ph.D. Thesis, California Institute of Technology (1962). R. M. Talman, Ph.D. Thesis, California Institute of Technology (1963). G. L. Hatch, Ph.D. Thesis, California Institute of Technology (1967). H. 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