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A helium high temperature drift chamber Huang, Jiachang 1998

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A Helium High Temperature Drift Chamber By Jiachang Huang B .Sc , Tsinghua University, 1988 M . S c , Institute of High Energy Physics, 1991 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in T H E F A C U L T Y OF G R A D U A T E STUDIES Department of Physics We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y OF BRITISH C O L U M B I A September 1998 © Jiachang Huang 1998 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department The University of British Columbia Vancouver, Canada Date D i e 3 , DE-6 (2/88) Abstract We have built and operated a high temperature T P C which was integral to a po-larized target originally designed for a triton detector at T R I U M F . The T P C is the key part for the improvement of the measurement accuracy in the study of the spin dependence of the reaction: + 3He —y 3H + v^. This spin dependence is very sensitive to the induced pseudoscalar form factor, Fp, which is very important for our understanding of strong interactions at low energies, but about which relatively little is known experimentally. The nuclear capture rate in 3He is proportional to (1 + AVPV cos 6), where Pv is the muon polarization and 6 is the angle between the muon polarization and the direction of the recoil triton. The last E683 data run at T R I U M F gives Av = 0.59 ± 0.09 ± 0.10 using a Princeton ionization chamber as the triton detector. The design, construction and development of the T P C is the main emphasis of this thesis. A n extensive study of the T P C performance as a function of the electric field at various temperatures and different helium-nitrogen mixtures has been made. The properties of electron transport and gas amplification in mixtures of helium with a variety of molecular additives have been studied and compared with a standard argon based gas mixture. The optimized operating condition using helium/nitrogen(97:3) mixture achieves an angular resolution of about ± 1 degree. The directional informa-tion can be obtained by fitting the anode pulse shapes based on a detailed model of the detector. Future improvements of the detector will make the T P C a candidate for a new generation of this experiment, which should then provide a more precise measurement of Av and hence, a more precise form factor, Fp. Thus, it will provide a better test of our understanding of QCD at low energy. in Contents Abstract " List of Tables viii List of Figures ix Acknowledgements xn 1 Introduction 1 2 Theory 5 2.1 History 5 2.2 Muon Capture 6 2.2.1 Muon Capture by a Proton 6 2.2.2 The Conserved Vector Current Hypothesis(CVC) . . 8 2.2.3 The Partially Conserved Axial Current Hypothesis (PCAC) 9 2.3 Models of Muon Capture in Nuclei 11 2.3.1 Muon Capture by 3 He in the Elementary Particle Model 11 2.3.2 Muon Capture by 3He in the Impulse Approximation 15 iv 2.4 Comparison to Other Experiments 17 2.4.1 Ordinary Muon Capture(OMC) 17 2.4.2 Radiative Muon Capture(RMC) 19 2.4.3 Summary 20 3 Detection of Charged Particles 22 3.1 Single Particle Detection 22 3.2 Energy Loss Due to Electromagnetic Interactions 25 3.3 He Stopping Cross Section 28 3.4 Total Ionization 30 3.5 The Motion of Ions and Electrons in Gases 31 3.5.1 Diffusion 31 3.5.2 Mobility 32 3.5.3 Drift Velocity 34 3.5.4 Electron Avalanche 40 3.6 Proportional Chamber 42 3.6.1 General Characteristics 42 3.6.2 Time Development of the Signal 44 3.7 Multiwire Proportional Chamber 47 3.8 Drift Chamber 51 3.9 Time Projection Chamber(TPC) . 52 3.10 Chamber Gas 53 4 Triton Detector 55 v 4.1 Design Considerations 55 4.2 T P C 56 4.2.1 T P C Development . 58 4.2.2 Mechanical Design and Construction 61 4.2.3 High Voltage 62 4.2.4 Gas System 66 4.3 Rubidium 67 4.4 Oven 71 4.5 Electronics and Data Acquisition 72 4.6 Gas Gain Calibration 74 5 Data Analysis 77 5.1 Ionization Measurement 78 5.1.1 Gas Gain vs Angle, Temperature and High Voltage . 78 5.1.2 Gas Gain for He - N2 Mixtures 80 5.1.3 Comparison of He Mixtures and Ar Mixtures . . . . 84 5.1.4 Energy Resolution 87 5.1.5 Breakdown 89 5.2 Direction Measurement 91 5.2.1 Full Width vs. Angle and Temperature 91 5.2.2 Pulse Fitting 94 5.2.3 Angular Resolution 101 5.3 Comparison Between the T P C and the Ion Chamber 105 vi 6 Conclusion 108 Appendices A Program for the Pulse Fitting I l l B i b l i o g r a p h y 1 2 1 V l l List of Tables 2.1 Summary of gp measurements 21 3.1 Properties of different gases used in proportional chambers 26 3.2 Parameters for the Andersen-Ziegler formula 29 3.3 Experimental mobilities of several ions in different gases 33 3.4 The fundamental processes in a wire chamber 51 5.1 VT, p and a for different helium/nitrogen gas mixtures 83 5.2 x2 f ° r s i x pulses 100 5.3 Angular resolution of the T P C 104 5.4 The operating conditions for the T P C 105 vm List of Figures 2.1 The Feynman diagram for muon capture by a proton 7 2.2 The pion exchange diagram for nucleon muon capture 10 2.3 Energy level diagram for muonic 3He 14 3.1 Definition of the scattering cross section 23 3.2 Stopping power of 3 He and 4 He 29 3.3 Drift velocity of electrons in argon, and in argon with small admixtures. 36 3.4 Drift velocity of electrons in pure gases 37 3.5 Characteristic energy of electrons in pure gases 37 3.6 Drift velocity of electrons in argon based mixtures 38 3.7 Characteristic energy of electrons in argon based mixtures 38 3.8 Drift velocity of electrons in helium based mixtures 39 3.9 Characteristic energy of electrons in helium based mixtures 39 3.10 Proportional chamber 42 3.11 Time development of a pulse in a proportional chamber 47 3.12 A multiwire proportional chamber 49 ix 3.13 Schematic diagram of a T P C 52 4.1 Mechanical design of the time projection chamber 57 4.2 Comparison of two generations of T P C detectors 60 4.3 Time projection chamber 61 4.4 The electric field equipotentials and the electron drift lines for the T P C . 64 4.5 Anode signal for different emission angles 65 4.6 Vacuum/gas handling system 67 4.7 Energy level diagram for optically pumped Rb 68 4.8 Transmission scan of the probe laser at 200°C 71 4.9 Electronic readout system 72 4.10 Preamplifier circuit diagram 73 4.11 Circuit used for absolute gain calibration 75 4.12 Decay time vs. effective resistance 76 5.1 Gas gain as a function of emission angle 78 5.2 Gas gain as a function of temperature 79 5.3 Gas gain as a function of anode high voltage 80 5.4 Gas gain as a function of anode high voltage for different He — N2 mixtures 81 5.5 In M as a function of V0 for different He — N2 mixtures 82 5.6 Threshold voltage Vj and a as a function of helium/nitrogen gas mixtures. 82 5.7 M as a function of VQ for He/N2 (97:3) mixture with different anode wire diameters 83 x 5.8 vs. Inn for different helium/nitrogen gas mixtures 84 5.9 Gas gain as a function of high voltage for different mixtures of helium. 85 5.10 Gas gain as a function of high voltage for different mixtures of helium and argon 85 5.11 Comparison of the standard drift chamber gas mixture with the gas mixture used in the T P C 86 5.12 Signal charge and amplitude histogram 88 5.13 Energy resolution as a function of emission angle 89 5.14 Charge at breakdown for different gas mixtures 91 5.15 Signal width histogram 92 5.16 Signal width as a function of emission angle 93 5.17 Signal width as a function of temperature 93 5.18 Skewness histogram 95 5.19 Transient response of a RC coupled amplifier used in the circuit simu-lation 97 5.20 T P C pulses for two ionization events 98 5.21 T P C pulses for four ionization events 99 5.22 The contour of one sigma error for emission angle and pulse starting time 103 5.23 Ion chamber pulses for two triton events 107 XI Acknowledgements I am very grateful to my supervisor, Prof. Michael D. Hasinoff for his guidance and support during my thesis work at T R I U M F . I have learned a great deal from what he taught me about doing research. This thesis would have been hard to read without his patience, especially for me who comes from China. His influence upon me will be beneficial to my future career. I would like to thank two men whose work was indispensible to this experiment. I am indebted to Wolfgang Lorenzen for his contribution to the design and construction of the drift chamber, which is the critical part for the progress of this work. We have spent many enjoyable evenings and weekends working together in the early stages of the detector development. The building of such a complicated detector would have been considerably more difficult without Bi l l Cummings, who was involved in many of the design decisions. Our project has benefited enormously from his knowledge about all details of the experiment. It is not possible for me to list all the people who have made contribution to this project. I apologize in advance for any omissions. I want to thank everyone in the E683 collaboration. My sincere thanks are due to the late Prof. Otto Hausser for his continuous interest and support as well as many valuable suggestions. I would especially like to thank Erik Saettler for his assistance with the electronics setup and data analysis. I would also like to extend my appreciation to Pierre Amaudruz for his assistance with the data acquisition system setup. I express appreciation to our Princeton collaborators. The useful suggestions from Paul Bogorad and Xuejun Wang xn is much appreciated. I would also like to thank Robert Henderson, Wayne Faszer, Robert Openshaw, Grant Sheffer, Peter Vincent and Marielle Goyette in the Detector Development Lab for their outstanding technical support. Marielle was always there to help when things got tough. I appreciate T R I U M F machine shop members for their contribution in machining the apparatus, especially Norman, who is so nice to work with. I'm thankful to George Takacs and Tom Inglis in the vacuum group for the loan of the vacuum pump. I also thank Phil Levy and John Behr for their assistance with the laser system setup. I would particularly like to thank my parents who have encouraged me from the very beginning. X l l l Chapter 1 Introduction Since the fundamental question about the structure of the nucleon remains unresolved, and the related QCD effects at low energies are still not very well understood, a new current may be written in terms of four coupling constants, gv, gM, 9A, 9P, which are a function of q2, the square of the four momentum transfer. Experimentally, the pseudoscalar form factor, gp, is less well determined than the other form factors. Using the chiral Ward identities, and applying QCD corrections, the value of gp has been predicted to three percent accuracy.^ Therefore, an accurate experimental determination of gp will provide a stringent test of QCD at low energies. In nuclear muon capture or beta decay, form factors, Fa, are often used as well as the coupling constants. The relation between ga and Fa is as follows: generation of experiments is very welcome In a semileptonic weak interaction, the 9v{q2) = Fv(q2) (1.1) (1.2) 9T(q2) 2MFT{q2) (1.3) miFs{q2) (1.4) 1 2 CHAPTER 1. INTRODUCTION gP{q2) = miFP(q2) (1.5) gA(q2) = FA(q2) (1.6) where mi, M are the mass of the lepton and the nucleon, respectively. The ga(q2) can be considered essentially as constants over the limited range of q2 in muon capture. The study of the spin dependence of the reaction is a promising candidate for a precise measurement of gp, because it is very sensitive to the pseudoscalar form factor, Fp. The theory will be discussed in chapter 2. The production of spin polarized muonic 3He is necessary for the study of reaction (1.7). A technique developed by the Princeton-Syracuse-LANL group yielded an average polarization of 26.8 ± 2.3 % in 3He® which was about ten times larger than with the previous techniques.t4,5! The polarization was transferred from optically pumped Rubidium vapor to the muonic helium by spin exchange collisions. The experiment demands building a detector which serves both as a stopping target to produce the polarized muonic 3He atoms and a high temperature wire chamber to detect the recoil tritons emitted in reaction (1.7). This thesis will focus on the design, construction, operation and development of such a target/detector system. Progress in experimental particle and nuclear physics has been closely linked to improvements in detector technology. The first multiwire proportional chamber was constructed and operated by Charpak and his collaborators in 1968.^ Wire chambers have very good time resolution and position accuracy, and they can be self-triggered.^ Their use has spread from high-energy and nuclear physics to many other fields such as astrophysics, medicine and biology J 8 ' Wire chambers are one dimensional devices; hence to determine both x and y coordinates, a second wire chamber must be used. This requirement makes the experimental arrangement more complicated and reduces the solid angle subtended by the detector. The time projection chamber (TPC), ^ 3He 3H + ut (1.7) 3 invented in 1974 by Nygren, is a nearly ideal detector. The T P C is essentially a large cylinder filled with a gas, usually a mixture of argon and ethane. Uniform electric and magnetic fields are applied parallel to the axis. There is no E x B type force so that it is possible to have the ionization electrons drift over large distances to the end caps. Futhermore, the strong magnetic field considerably reduces the diffusion by causing the electron trajectories to be tiny spirals along the magnetic field lines. The ends of the cylinder are then covered by sector arrays of proportional wire chambers followed by cathode pads. At a collider machine, the detector is positioned so that its center is at the interaction point. The T P C thus subtends a solid angle close to 47T. Ionization produced by charged particles passing through this volume drifts towards the endcaps where it is detected by the anode wires as in a multiwire proportional chamber. This yields the position of a spatial point projected onto the endcap plane in two dimensions. The third dimension, along the cylinder axis, is given by the drift time of the ionization electrons. The total charge collected at the endcaps is proportional to the energy loss of the particle. The momentum of the particle is obtained from the curvature of its trajectory in the magnetic field. This information can be used to identify the particle. Hence a T P C allows a three-dimensional reconstruction of the interaction patterns, provides particle identification and a full kinematical analysis of the event is possible. TPCs are now used in many nuclear and high energy laboratories such as C E R N , Fermilab, S L A C , L B L and CESR, etc. The T R I U M F T P C , used in the N —> e - Af experiment, was one of the first large T P C detectors used in a physics experiment. The principal physical process for the operation of such a device will be given in chapter 3. Over the last few decades a substantial amount of work has been done towards understanding and optimizing the properties of wire clmmbers. Generally speaking, there are five main factors which can be optimized: chamber electrode geometry, readout electronics, primary ionization, drift and diffusion of the electrons and the gas amplification process. The most challenging aspect of this thesis was the design and construction of a T P C capable of operating under extreme conditions such as high temperature, high pressure and high Rubidium density. 4 CHAPTER 1. INTRODUCTION Three generations of our detector were built. The first generation of the detector was based on a gas scintillation time projection chamber(GSPC). Extensive studies on the light amplification dependence on pressure, gas composition and the electric field applied to the gas were performed at room temperature.t10^ However, we were not able to observe the anode signals at temperatures above 180°C due to the large breakdown noise. The second generation of our detector was a conventional T P C . The main im-provement is discussed in section 4.2.1. Although a new oven was built to provide a more uniform temperature distribution, and all the internal grids were made of stainless steel wires, very serious problems remained to be solved. The third generation detector has been successfully run at high temperatures in the absence of Rubidium vapor. A l l the details is discussed in chapter 4. The anode signal as a function of emission angle, temperature, high voltage and gas composition has been studied extensively. The data are presented and analyzed in chapter 5. These data show good energy resolution and angular resolution. If the problem of T P C operation in the Rubidium environment can be overcome, this detector can be considered as a promising candidate for the next generation of experiments. Experiment E683 has been undertaken at T R I U M F to measure the vector ana-lyzing power Av. The apparatus was based on an ionization chamber,^ and muonic 3He was produced by stopping muons in high pressure 3He gas which was polarized by collisions with laser optically pumped Rubidium vapor. The data analysis gives:f12' Av = 0.59 ± 0.09(stat.) ± 0.l0(syst.) (1.8) This is consistent with the theoretical prediction(see section 2.3.1), which assumes Fp = 20. This present result doesn't give any improvement for the quantitative knowledge of Fp or gp, but the technique has been shown to have the potential to provide a very precise measurement method for the pseudoscalar form factor in the future. Chapter 2 Theory 2.1 History The muon was discovered by Anderson and Neddermeyer'1^ in 1938 in a study of cosmic-ray particles using a cloud chamber. What we now call the muon was first thought by most physicists to be the particle predicted by Yukawa's meson theory of the strong nuclear force. In 1947, experiments which stopped the muons in carbon^14! suggested that the decay probability was about equal to the capture probability. Fermi and Teller's theoretical analysis^15' " indicated that the interaction of mesotrons with nucleons according to conventional schemes is many orders of magnitude weaker than usually assumed." Therefore, the muons were not the Yukawa mesons. By the time accelerator sources of muons were available, the essential properties of the muon as a lepton had been found - (a mass of 207 electron-masses, a charge of ± e , a half-life of 2.20 x 1 0 - 6 sec, and a spin of | , thus obeying Fermi statistics). The first comprehensive summary of the theory of muon capture can be found in the Tiomno and Wheeler's paper J 1 6 ' The elementary process was expressed as (i~ + P -> fi0 + N where fio is the Pauli neutrino. Muon decay, muon capture and nuclear beta decay 5 6 CHAPTER 2. THEORY had nearly equal coupling constants, which were extremely small compared to those of the strong interaction. The three types of weak interaction processes constituted the well known universal Fermi interaction. 2.2 Muon Capture 2.2.1 Muon Capture by a Proton In the framework of the standard model,t 1 7 - 1 9] the leptonic electro-weak interaction is mediated by the exchange of charged(VK±) or neutral(Z°, 7) vector bosons. When the momentum transfer, q2, is much less than M^, where Mw is the mass of the W vector boson, the weak interaction can be very successfully described by the vector minus axial vector (V-A) current for the weak leptonic sector. The simplest nuclear muon capture process which can be studied experimentally is \T +p —• Vft + n The Feynman diagram for this reaction is shown in figure 2.1. This semileptonic weak interaction can be represented by a fermion current-current interaction, where the existence of the intermediate vector bosons is ignored. The current becomes somewhat more complicated due to the presence of the strongly interacting particles. The interaction between a quark and a lepton involves QCD effects. Assuming Lorentz invariance, the general forms of the weak hadronic vector and axial vector currents are:!20! J M = + K (2-1) T/ 1 • 9M V • 9s /0 0 \ ZM A» = 9A1^ + ^Trn^^lh + —?„75 (2.3) ZM 2.2. MUON CAPTURE 7 V u n P Figure 2.1: The Feynman diagram for muon capture by a proton. where q^ = k' — k^ is the four-momentum transfer, M is the nucleon mass and m M is the mass of muon. The six form factors are: the vector gv, the axial vector gA, the weak magnetism gM, the scalar gs, the induced tensor g?, and the pseudoscalar gp. These form factors are all functions of the transferred momentum q2. Since gut, 9s, gx and gp do not exist in purely weak interactions, they are induced by the strong interactions. The gp term is proportional to q^, which is substantially larger in muonic processes than in beta decay. Equations (2.2) and (2.3) can be simplified by various assumptions of the symme-tries possessed by the hadronic current. For example, time reversal invariance requires that all six form factors are real. G-parity is defined as charge conjugation plus a rotation of 180° in isospin space. Under the G-parity transformation for vector and axial vector current, gv, gA, 9M and gp are even, but gs and gr are odd. Since G-parity is conserved in the strong interaction, G-invariance of the weak hadronic current gives 9T{q2) = 0 (2.4) (2.5) 8 CHAPTER 2. THEORY 2.2.2 The Conserved Vector Current Hypothesis(CVC) The eonserved vector current hypothesis(CVC), proposed by Feynman and Gell-Mann,t 2 2l guided by the analogy between the electromagnetic current and the weak vector current, states that the divergence of the vector current is zero. = 0 (2.6) This means the weak current is conserved. Thus the C V C hypothesis also predicts 9s(q2) = 0. Since isospin is a good symmetry for the strong interaction, another statement of C V C is called the isotriplet vector current hypothesis(IVC). The IVC states that the vector parts of the charge raising(n —• p) and charge lowering(p —• n) currents and the isotriplet part of the electromagnetic current form an isotriplet of currents corresponding to the components 7 3 , 7 + and I~ respectively, where 7 * = 7 i ± 7 2 . This hypothesis follows from the 577(2) x U(l) nature of electroweak interactions. The isovector part of the proton electromagnetic current is J ^ F ^ + Z K ^ F ^ ^ - (2.7) and the neutron current is J ^ F ^ + i ^ F ^ ^ (2.8) The IVC predicts gv = Fl - F? (2.9) 9M = K'F* - KNF? (2.10) where Kp(nn) are the proton(neutron) anomalous magnetic moments,^ KP = 1.793, KN = —1.913, respectively. 2.2. MUON CAPTURE 9 From elastic scattering of electrons on protons and neutrons,t24' gv(q2) and <7m(<72) can be expressed as a function of q2 with ry and rM as parameters. gv(q2) = gv(0)(l + rW/V (2-11) 9M(q2)=gM(0)(l + rW/G) (2.12) where gv(0) = 1 and gM(0) = 3.706, r\ = 0.576 / m 2 , r2M = 0.771 fm2. Evaluating (2.11) and (2.12) at q2 = 0.954m2 for muon capture by 3He gives gv(q2 = 0.954m2) = 0.974 ±0 .001 (2.13) 9M(q2 = 0.954m2) = 3.576 ± 0.001 (2.14) 2.2.3 The Partially Conserved Axial Current Hypothesis (PCAC) The partially conserved axial current hypothesis(PCAC), proposed by Gell-Mann and Levy,t 2 5l relates the divergence of the axial vector current to the pion field with a proportionality factor measured in pion beta decay: d^Afj, = m 2 / ^ (2.15) where /„. is the pion decay constant and <f)v is the pion field. The axial current is conserved in the limit when the pion mass, m x , is equal to zero. If it is assumed that the one-pion exchange mechanism is dominant at low q2, (see the single pion exchange Feynman diagram in figure 2.2), then solving the Klein-Gordon equation for the pion field in the nucleon yie lds :^ 1 which gives, at q2 = 0, 2MN gA(q2) ~ q 2 ^ q2 + ml (2.16) 10 CHAPTER 2. THEORY Figure 2.2: The pion exchange diagram for nucleon muon capture. <M(0) = (2.17) This is the famous Goldberger-Treiman relation.t27' Combining equations (2.16) and (2.17) gives, 9 p t f ) * ? ^ ( - ? ^ + 9 A t f ) ) (2-18) ql qz + m* From the neutrino-nucleon elastic scattering experiments, [28] ^ 2 > = ( T 7 W ( 2 ' 1 9 ) where M\ = (1.08 ± 0.04)GeV 2, is determined by a fit to the experimental cross-sections with a parameter of M\. Substituting equation (2.19) into equation (2.18) g ives^ gptf) ^ 2m,MN 2m,MN _q^ _ 9A(0)-qi + m l + q2 U i + M 2 J i j [ Z - Z U ) At q2 = 0.954m2 for muon capture by 3He, equation (2.20) gives gP{q2 = 0.954m2) V , „ — ^ ^ 6 . 2 2 (2.21) 9A <22 = 0 V ; 2.3. MODELS OF MUON CAPTURE IN NUCLEI 11 From neutron beta decay experiment:^23] gA(0) = -1.2573 ± 0.0028 (2.22) Thus, for 3 He the P C A C predicts: gP(q2 = 0.954m2) ~ -7.82 (2.23) 2.3 Models of Muon Capture in Nuclei 2.3.1 Muon Capture by 3He in the Elementary Particle Model There are two alternative methods used to describe muon capture in nuclei. They are called the elementary particle model(EPM) and the impulse approximation(IA). The elementary particle model(EPM), proposed by K i m and Primakoff , ' 2 9 ' 3°] treats the nuclei as elementary particles. This approach avoids any assumptions about nu-clear wave functions. The nuclear transition matrix elements are expressed in terms of known kinematic quantities and the nuclear form factors. The form factors, as a function of q2, reflect the complexity of the internal nuclear structure. The observ-ables are then expressed in terms of these form factors, which are derived independent of nuclear models. The study of muon capture in complex nuclei involves many uncer-tainties arising from the approximate treatment of the nuclear many-body problem. This makes the interpretation of the weak hadron form factors in nuclei very difficult. In the E P M , the triton and 3He are assumed to be an isospin doublet. In the absence of second class currents, the charge changing weak current of the 3He —> 3H transition is given by h = "(fc') ( Fvl» + i-^-cr^qu + FAWs + — ^75 ) u(k) (2.24) where u(k) and u(k') are the Dirac spinors for the nuclei, M3 is the trinucleon bound state degenerate mass, and Fy, FM, FA and Fp are the nuclear form factors. 12 CHAPTER 2. THEORY Then CVC(or IVC) predicts FV = 2F?\q2) - F?(q2) (2.25) FM = *HeF2Hetf) - KTF?(<?) (2.26) where F\ and F2 are the electromagnetic form factors. KHS(KT) is the 3He(3H) anoma-lous magnetic moment/3 1! KHe = _8.3689, KT = 7.9173. From elastic electron scattering on 3He and 3 i f / 3 2 ! Fv(q2 = 0.954m2) = 0.834 ± 0.001 (2.27) FM(q2 = 0.954m2) = -13.969 ± 0.052 (2.28) The relationship between Fp(q2) and i 7 ! ^ 2 ) i s / 3 3 ! where FA{q2 = 0.954m2) = 1.05 ± 0.01 Thus P C A C predicts: FP(q2 = 0.954m2) = 20.7 ± 0.2 (2.30) Equation (2.24) can be reduced using Pauli spinors, and the related matrix is given (2.31) where ai(h) is the Pauli spin matrix which operates on the leptonic(hadronic) spinors and k is the direction of the neutrino momentum in the final state. The effective couplings Gv, GA and Gp are functions of the nuclear form factors and kinematic quantities. GV = FV (1 + - F M ^ - ( U + "~M3) = 0.85 ± 0.01 (2.32) 2.3. MODELS OF MUON CAPTURE IN NUCLEI 13 GA = FA u> + M3 [FV + FM) = 1.29 ± 0 . 0 1 (2.33) GP = (Fv + FM + FA- FP) = 0.603 ± 0.01 (2.34) where the neutrino energy v = 103.22 MeV, u = y/v2 + M | = 2811.3 MeV. The value of Gp only holds when Fp takes on the P C A C value. The triton differential rate is given by [34] R(e) = T, m = -rri1 + A " p v c o s 6 + oAtpt@ C O § 2 6 - ! ) + ^ A P A ) (2.35) a(cosp) z / where 9 is the angle between the triton direction and the polarization axis. Pv is the muonic 3 He vector polarization, Pt is the tensor polarization, P A is the deviation of the hyperfine singlet state population: P V = JV(1 ,1) -JV(1 , -1) (2.36) P T = 7 V ( l , l ) ' ± i V ( l , - l ) - 2 7 V ( l , 0 ) (2.37) P A = N(l, 1) + i V ( l , 0) + N(l,-1)- 3W(0,0) = 1 - 4A^(0,0) (2.38) where N(F, M) are the population densities of the muonic 3He hyperfine states. The analyzing powers Av, At and A A were found to be Av = (Gv + GA — Gp) /G0 (2.39) At = -4GP(Gv + GA)/(3Gl) (2.40) AA = - 2 -Gp(Gy + GA) — GA{GV + Gp — GA) IGl (2.41) and T q — C RG^ (2.42) 14 CHAPTER 2. THEORY +1 0 -1 0 Figure 2.3: Energy level diagram for muonic 3He. where R is not a function of the form factors. Gl-Gv + 2G2A + (GA - GP)2 (2.43) C is the muon overlap reduction factor which takes into account the non-pointlike nature of the nucleus. In experiment E683, [ 3 5 ] the muonic helium was polarized by collisions with opti-cally pumped Rb(see section 4.3 for the details). Some of the electronic polarization was transferred to the muonic helium through the hyperfine coupling (see figure 2.3). The transitions for the muonic 3He atoms from the | 1,-1 > hyperfine state to the | 1,0 > state were driven at the same rate as those from the | 1,0 > state to the | 1,1 > state. Thus, the vector polarization was much larger than the tensor polar-ization. For a Pv of 10 %, Pt is about 0.23 %. [ 3 5 ] The | 0,0 > state was unaffected, resulting in P A = 0. Ignoring the tensor and A terms, equation (2.35) gives m-R(Q + *) = A P c o s 9 ( 2 4 4 ) The E P M predicts:^ Av = 0.524 ± 0.006 (2.45) 2.3. MODELS OF MUON CAPTURE IN NUCLEI 15 r 0 = (1497 ±21) sec'1 (2.46) Av is more sensitive to Fp than r 0 . The predicted sensitivities to the value of Fp are = 11 % (2.47) Observation of the asymmetry Av is advantageous experimentally because large vector polarizations, P„, can be achieved and measured accurately via the muon decay electron asymmetry. 2.3.2 Muon Capture by ^He in the Impulse Approximation The impulse approximation(IP) is based on the assumption that the nucleus consists of physical nucleons interacting with leptons in the same fashion as free nucleons. The hadronic current is taken as a simple sum of single nucleon currents. Meson exchange and other many-body effects are neglected. The IP is appropriate for the study of nuclear models and structures. The calculated matrix elements are rather sensitive to the details of the wavefunction used. Calculations in medium and heavy nuclei are very difficult due to the accuracy of the nuclear wavefunctions. Since the wavefunctions for both 3He and the triton are relatively well-known, the IP has been used!34) to relate the pseudoscalar nuclear form factor, Fp, to the nucleon form factor 9P-The q2 dependence of the form factors based on the IP can be expressed as^34] Fv(q2) = gv(q2)[l}° (2.49) FM(q2) = 3 (gy(q2) + gM{q2)) [<r]~ ~ 9v(q2){l}° ~ ^ ^ [ x P ] 1 ' 1 (2-50) 16 CHAPTER 2. THEORY FA(q2) = gA(q2)[a}- (2.51) FP(q2) = 3gP(q*)[a}+ + $V21^gA(q2)[a}2'1 (2.52) 9 where the reduced matrix elements [ l ] ° , [<r]_, [cr] + , [cr]2'1 and [ iP] 1 ' 1 are q2 dependent. Highly sophisticated and reliable wavefunctions for 3He and 3H, which are based on accurately solving the three body Schrodinger equation, were used for calculating the matrix elements.^34! The [ iP] 1 ' 1 term contributes to Fy + FM at about the 0.06 % level and can be safely ignored. The other matrix elements have been calculated at q2 = 0.954m2(34] J. ft [a]0'1 = -0.809 (2.53) [a]2'1 = -0.014 (2.54) [a}+ = [a] 0' 1 + V 2 M 2 ' 1 = -0.828 (2.55) [a]' = [a]0'1 - yfiTSi*]2'1 = -0.799 (2.56) [1]° = 0.853 (2.57) Hence the impulse approximation predicts T 0 = 1304 sec'1 (2.58) Av = 0.568 (2.59) These values disagree quite strongly with the E P M one. This is because of the ne-glect of the meson exchange correction(MEC). The recent calculations performed by Congleton and Truhlikf3 6' including the M E C correction to the impulse approximation give T 0 = (1514 ± 32) sec'1 (2.60) Av = 0.515 ±0 .004 These corrected values are in good agreement with the E P M values. (2.61) 2.4. COMPARISON TO OTHER EXPERIMENTS 17 2.4 Comparison to Other Experiments 2.4.1 Ordinary Muon Capture(OMC) Muon capture in hydrogen is certainly the most appropriate choice for the quanti-tative measurement of gp. However, there are substantial experimental problems to overcome.f37! The capture probability is less than 10~3 times the muon decay prob-ability in hydrogen, so it is necessary to work at high hydrogen concentrations. The transfer rate from ftp atoms to deuterons and heavier atoms is also very high. This requires an ultrapure target. The muons stopped in a liquid hydrogen target quickly form a [ip atom in either the singlet or triplet state. The conversion rate from the triplet state to the singlet state depends on the hydrogen density. The O M C process in the experiments with liquid hydrogen^ 3 7 - 4 0! involves a very rapid formation of pfj,p molecules(r(/ip —• pfip) — 500 ns) in the ortho(triplet) state. The transition rate between ortho and para molecules is difficult to determine accurately, but gp is very sensitive to this factor. For the experiments using hydrogen gas/ 4 1! the atom pfi~ is formed. At the high gas densities required for the experiments, the lifetime of the triplet state is about 10 nanoseconds, so all p\i~ captures occur from the singlet state. The singlet capture rate is not very sensitive to gp. Most of the early experiments were performed by detecting the outgoing neutrons, which were very hard to detect. The accuracy of these measurements was limited by the elaborate Monte Carlo calcu-lations for the detector neutron efficiency. The most accurate measurement of O M C , performed by a very precise comparison of the fj,^ lifetimes, gives 42 % uncertainty for gp. The world average of five different O M C experiments on hydrogen yields gp/9A = 6.9 ± 1.5 (2.62) in excellent agreement with the P C A C prediction, albeit, within large errors. In the reaction [i~ + 3He —> 3H + z/M, the recoil 3H can be easily identified by its unique energy, 1.9 MeV. The negative muons were stopped in the high pressure helium gas targets, and the capture rate was measured by detecting the recoil tritons by a 3 He diffusion cloud chamber^42! or by a gas scintillation method J 4 3 , 4 4 ! The best value 18 CHAPTER 2. THEORY for the r 0 measurement is from the PSI^ 4 5 ' experiment. A high pressure ionization chamber was used to measure the ratio between the number of muons stopped and the number of tritons created in their chamber. Analysis of their data gives V0 = im±3(stat.)±3(syst.) (2.63) The value of To corresponds to: Fp(q2 = 0.954m2) = 20.8 ± 2.8 (2.64) in good agreement with the P C A C prediction, however, the uncertainty of Fp is much larger than that of To- From equations (2.47) and (2.48), To is less sensitive to Fp than Av. For example, 1% uncertainty in a To measurement gives 10% uncertainty for Fp, but 1% uncertainty in a Av measurement gives about 3% uncertainty for Fp. A measurement of Av combined with the existing To value will increase the sensitivity to FP, - 49 % (2.65) 1QA v rp Therefore, it is time for a new experiment to be designed to determine Av accurately in order to provide a more precise value for Fp. The experimental data accumulated on muon capture in nuclei is very abundant. The accuracy of extracting gp from muon capture on heavy nuclei is not very good due to the uncertainty in the various models employed to compute the different quantities measured in the experiments. The determination of gp/ gA is only applicable for a few specific nuclei and it is somewhat dependent on the model chosen. Measurements of the 12B recoil polarization following muon capture on 12C (p- + 1 2 C -> 12B + givet46! FP(q2)/FA(q2) = 1.03 ± 0.14 (2.66) in agreement with the E P M P C A C prediction of FP = 0.99. The IA yields gP(q2)/gA(q2) = 9-0 ±1.7 (2.67) 2.4. COMPARISON TO OTHER EXPERIMENTS 19 in reasonable agreement with P C A C . However, the other measurements'47! give dif-ferent values of gp/gA depending on the choice of the nuclear response function. The ratio of the 160 muon capture rate and the 1 6 A^ beta decay can also be used to extract a 9P/SA value (n~ + 1 6 0 —> 16N + v^,). The recent experiment based on a comparison of O M C on 1 6 0 to the 0" -»• 0+ beta decay of 1 67V gives'48! gp/gA ~ 13 ± l (2.68) This result is not in agreement with P C A C (see table 2.1). 2.4.2 Radiative Muon Capture(RMC) Radiative Muon Capture(RMC) is basically an O M C process accompanied by a bremsstrahlung photon. Ai(Z) + /i- -• + ^ + 7 The ratio of the radiative capture rate to the ordinary capture rate is of the order 1 0 - 4 — 1 0 - 5 . This means it is much easier to perform such measurements in heavy nuclei than in hydrogen because the muon capture rate is approximately proportional to Z4S491 There are a number of data obtained from muon capture on C, 0 , A l , Si, Ca, etcJ 5 0" 5 3] In heavy nuclei, the experimental results are somewhat inconclusive. The final states are unresolved and have all been summed together. Several assumptions must be made to complete such calculations. There are two nuclear models used to deter-mine values of gp for R M C on oxygen.I 5 1 , 5 2! Using the semi-phenomenological nuclear excitation model,' 5 4! gp/gA — 7.3 ± 0.9, which is in agreement with the P C A C esti-mate. However, when the modified impulse approximation'5 5! ] s u s e d to extract gp from the same experimental data, a much larger value is obtained, gp/gA = 13.6±2.0. R M C on hydrogen is the ideal experiment for the extraction of gp. The sensitivity to gp in R M C is more than 3 times that in O M C . However, the experimental problems are formidable because of the extremely low partial branching ratio(2.1 x 1 0 - 8 for 20 CHAPTER 2. THEORY E~y > 58 MeV) and large background sources. The extraction of gp from the data depends on a knowledge of the relative fraction of muons in the ortho and para p\ip state. A recent experiment at T R I U M F obtains/56^ gP(q2 = 0.88m2) — ^ = 9.8 ± 0 . 7 ± 0 . 3 2.69 9A(0) where the first error is the statistical plus systematic uncertainties and the second error is due to the uncertainty in the decay rate of the ortho to para pfip molecule. This value of gp disagrees with the prediction of P C A C and pion-pole dominance. 2.4.3 Summary Only for muon capture on the proton is it possible to extract a value for gp that is completely free of the complications of nuclear structure. However, the problems from the molecular structure of the hydrogen and the small R M C branching ratio result in the situation that the experimental uncertainty for gp is considerably larger than the theoretical one.^ Moreover, the R M C measurement yields a gp value which is a factor of about 1.5 times the expected value. The extraction of gp from R M C on heavy nuclei requires a correct treatment of the nuclear response function. Unfortunately, the different nuclear models do not all provide the same gp values when compared to the available experimental data.!47! The main results of gp measurements for light nuclei are summarized in table 2.1. Muon capture on 3 i f e has many advantages for Fp studies. The reaction product(a triton) is a monoenergetic charged particle and is thus easy to detect. The wavefunc-tions can be calculated accurately so that it is possible to determine gp from an Fp measurement. Although very precise measurements of To have been performed,t45! they are limited in their ability to extract a precise value for Fp, because To is not very sensitive to Fp. From equations (2.47) and (2.48), Av is more sensitive to Fp by a factor of 3.5 than To- From the measurements, Congleton and Truhlikt 5 7! found that the 19 % uncertainty in gp results almost entirely from the 2 % theoretical un-certainty in the calculation of the rate. If Av were precisely measured, then a precise 2.4. COMPARISON TO OTHER EXPERIMENTS 21 value for gp could be obtained. The first successful measurement of the Av was made by E683 collaboration.'12^ Although there is still a large error for the extraction of Fp, the same experimental technique can be used in the future to obtain a more precise result. Table 2.1: Summary of gp measurements. The R M C in heavy nuclei measurements are not included. " C G " refers to Christillin and Gmitro' 5 4! and "GOT" refers to Gmitro et al. ' 5 5 ! The 1 6<3(RMC) values were obtained by Armstrong et al . ' 5 1 ! f r 0 m an experiment using the T R I U M F T P C detector. The semi-phenomenological nuclear excitation model(CG) and the modified impulse approximation model(GOT) were both used to extract the gp/gA values. These do not agree each other although the data fits the two models equally well. Thus these gp/gA values obtained are strongly model dependent. Experiment Measurement Theoretical Value Capture on xH(OMC) gP(q2)/gA(0) = 6.9±l.5 6.77 Capture on 1H(RMC) <7P(<72)/<7A(0) = 9 .8±1 .0 6.77 Capture on 3He(OMC) FP(q2)/FA(q2) = 20.8 ± 2 . 8 20.7 Capture on 12C(OMC) gp(q2)/gA(q2) = 9.0 ±1.7 7 Capture on 12C(OMC) FP(q2)/FA(q2) = lM±0.U 0.99 Capture on 160(OMC) 9p(q2)/9A(q2) = U±l 7 Capture on 160(RMC)(CG) (GOT) 9p(q2)/9A(q2) = 7.3 ± 0 . 9 7 9p(q2)/9A(q2) = 13.6 ± 2 . 0 7 Chapter 3 Detection of Charged Particles 3.1 Single Particle Detection The detection of a single charged particle requires a transfer of some of the particle's energy to the detector, but usually the total amount of energy deposited in a detector by one particle is too small for it to be detected. In order to bring the signal amplitude above threshold, a suitable detection material has to be chosen. The charged particle may excite atoms to emit photons, or ionize atoms to release electrons. Photomulti-pliers can convert light into a measurable electrical signal with a gain of up to 109 J 5 8 ! Both pulse amplitude and timing information can be recorded. The detection range covers the electromagnetic spectrum including the near infrared, visible and ultravi-olet. Scintillation counters are often joined to a photomultiplier tube through a light pipe to serve as a trigger for the event. Semiconductor detectors or gas-filled detec-tors are usually used for ionization detection. Semiconductor detectors give excellent energy resolution. Gas-filled detectors such as wire chambers have excellent spatial resolution and good time resolution and they can also be self-triggering. The primary signals can be fed to suitable electronic circuits for amplification and shaping. After 22 3.1. SINGLE PARTICLE DETECTION 23 digitization of these analog signals, they are recorded by a data acquisition system, and then processed by computers, normally both on-line and offline. Neutrons can be detected through nuclear reactions which result in charged particles J 5 9 ! Gamma-rays can be detected using the secondary electrons from the photoelectric and Compton effects or electron-positron pair production as well as by detecting protons from the nuclear photoelectric effect. In order to detect a particle efficiently, a large interaction probability is required. This probability can be expressed in terms of a cross section. It is assumed that a monoenergetic particle beam with uniform flux, F, impinges on a target as shown in figure 3.1. The flux, F, of the incident beam is defined as the number of particles per Incident monoenergetic beam Scattered beam Figure 3.1: Definition of the scattering cross section. unit area per unit time. The particles are scattered by the number NT of independent scattering centers in the target by an angle 6 into the solid angle <ifi. The number dN recorded per unit time is proportional to the solid angle dfl: dN = FNTo(6)dtt (3.1) 24 CHAPTER 3. DETECTION OF CHARGED PARTICLES The differential cross section for the process is defined as a { 6 ) ~ ~aW ( 3 - 2 ) This is the average fraction of the particles scattered into dO, per unit time per unit flux F. In many cases a(9) can be calculated theoretically and measured experimentally so that they can be compared with each other. a(9) is normally a function of both energy and angle. Sometimes the angle information is not so important. The total cross section can then be related to the interaction probability: , = (3.3) where a has units of area(ie, cm2). If the incident beam is scattered by a target with area A and thickness dx, then integration of (3.1) over all solid angles gives dn = F(noAdx)a = nriQodx (3-4) where no is the number of scattering centers per unit volume. The number of incident particles per unit time will then be n = FA. In muon capture by 3 He, the final product is a triton, which is a charged particle in a monoenergetic final state. Thus it is relatively easy to detect. In order to use equation (2.44) to measure the experimental Av value, the triton emission angle 9 has to be determined. This can be done by ^ measurement as discussed in the next section. The ionization distribution for a triton event is proportional to (^f)/sin# as shown in section 5.2.2. Once all the difficult technical challenges are overcome with the Rb vapour inside the hot target vessel, the study of the \T 3He —> 3H reaction will be a very promising candidate for a precise measurement of Fp. 3.2. ENERGY LOSS DUE TO ELECTROMAGNETIC INTERACTIONS 25 3.2 Energy Loss Due to Electromagnetic Interac-tions Measurement using the electromagnetic interaction is very important in nuclear and particle physics experiments. A knowledge of the interactions for charged particles and photons as they pass through matter is a crucial part in the design and the evaluation of most experiments. The main energy loss for a charged particle traversing the material is due to the interaction with the atomic electrons through Coulomb forces. The electromagnetic field of the particles can excite the atomic electrons from their ground state energy level to various higher energy levels or even ionize the electrons. The mean rate of energy loss (or stopping power) is given by the Bethe-Bloch formula, dE 2irNAe* Z z2 ( 2mc2ji2EM 2 S\ dx mc2 rA/32\ I2(l-{32) r Zt where — dE is the energy lost in a distance dx, NA is Avogadro's number, m and e are the electron mass and charge, z and /3 are the charge and the velocity (/? = v/c) of the incoming particles, Z, A and p are the atomic number, atomic weight and density of the absorbing material, 8 is the density correction, and S is the shell correction. The maximum energy transfer EM is given by EM = 2mc2 ' (3.6) M2 + 2Em + m2 v ; where E, M are the energy and mass of the incident particle. For heavy particles, where M2 ^> 2Em, we have 2mc2/32 EM C t - £ - (3.7) The mean excitation potential of the atoms of the absorbing material is 7, which is theoretically defined as n In 7 = J ] / , - I n ^ (3.8) 26 CHAPTER 3. DETECTION OF CHARGED PARTICLES where Ei and /,• are all possible energy transitions and corresponding oscillator strengths for the atoms of the absorbing material. This equation is very difficult to use except for the simplest atoms. Values of / from experimental data can be fitted to an empirical formula:'60! /< = (11.2 + 11.7 Z) eV, Z < 1 3 (3.9) 7> = (52.8 + 8.7 Z) eV, Z > 13 (3.10) Table 3.1: Properties of different gases used in proportional chambers' Z A P /o / Wi (xl0-3g/cms) (eV) (eV) (eV) H2 2 2.016 0.0837 15.43 20.4 36.3 He 2 4.003 0.165 24.59 38.5 42.3 N2 14 28.013 1.166 15.58 97.8 34.65 o2 16 31.999 1.331 12.08 115.7 30.83 Ne 10 20.18 0.838 21.56 133.8 36.4 Ar 18 39.95 1.662 15.76 181.6 26.3 Kr 36 83.8 3.478 14.00 340.8 24.05 Xe 54 131.3 5.485 12.13 508.8 21.9 CH4 10 16.043 0.668 12.71 44.13 27.1 C2H§ 18 30.070 1.250 11.50 48.07 24.38 C02 22 44.011 1.829 13.79 102.35 32.8 H20 10 18.015 0.986 x 103 12.61 81.77 29.9 The properties of different gases used in proportional chambers are given in ta-ble 3.1. Io is the first ionization potential of the atoms of the gaseous material. For the same incident particles, ^ depends primarily on p^ , therefore, high atomic number, high density materials provide the greatest stopping power. For the same material, ^ primarily depends on z2, which means that particles with higher charge 3.2. ENERGY LOSS DUE TO ELECTROMAGNETIC INTERACTIONS 27 result in the greater energy loss. Usually, it is more convenient to express ^ in units of mass thickness. Equation (3.5) can be written as dE where dX = pdx, and /(/?) is a function of the incident particle velocity. — is dE ' dX a function of the charge and j3 instead of the mass of the incoming particles. d Edecreases inversely proportional to (3 , and reaches a minimum when /9 ~ 0.96. At high energy, there is a logarithmic increase o f - | f with increasing energy until saturation (Fermi plateau) appears due to the polarization effect. 8 and S are corrections to the Bethe-Bloch formula which are important at high and low energy, respectively J 6 1 ! In addition to the density and shell effects, the Bethe-Bloch formula can be extended by including other corrections such as higher-order terms in the scattering cross-sections, higher-order QED processes and electron capture at low velocities. With the exception of electron-capture effects for heavy ions, these effects are all less than one percent J 6 2 , 6 3 ! The — |f formula can be applied to gas mixtures. dE v—\ „ . dE. .„ _ „ x s * = £ * < 5 * f c ( 3 - 1 2 ) 1=1 where 8i is the percentage of the ith gas in the mixture. X^iLi «^ = 1 The range of a particle, defined as the distance beyond which no particles can penetrate, is obtained by integrating equation (3.5) f° dT R= / A (3-13) where Tt- is the initial kinetic energy. In deriving equation (3.5), the assumption is made that the velocity of the incident particle is large compared to that of the atomic electrons. When the particle is moving slow enough to capture electrons, the Bethe-Bloch formula breaks down. 28 CHAPTER 3. DETECTION OF CHARGED PARTICLES 3.3 He Stopping Cross Section The stopping of ions in matter is still not fully understood.'64! At high energies (/? > 0.1), the energy loss can be accurately described by the Bethe-Bloch formula to a few percent. At low energies, any theoretical calculation is limited due to the inability to calculate the effective charge of an ion as it moves through the medium. A formula has been derived from the Thomas-Fermi statistical model of the atoms. — ^  is proportional to the ion velocity, thus, proportional to \[~E. However, it is not as accurate as the Bethe-Bloch formula. The intermediate region is not covered by any theories at all. It is necessary to obtain an accurate knowledge of the stopping cross section for helium or triton ions in helium in order to simulate the distribution of the energy loss for the charged particles in the present experiment. There is a large collection of He experimental stopping and range data for He. After a large variety of different fitting functions were tried, a semi-empirical formula was obtained by Andersen and Ziegler,'641 = Sf0VJ + S£gh (3.14) where Siow = AiEA2 (3.15) Shigh = ^-Hl + ^ + AsE) (3.16) where the electronic stopping power is S = - g in units of 10- 1 5 eV cm2, E is the energy of the incoming particles in units of keV. The energy ranges are 10 keV to 1 MeV for protons and 1 keV to 10 MeV for helium ions. A\ to A5 are the best fit parameters of the data for each element or molecule. The experimental data can be represented within an accuracy of about 2.5 percent using the Andersen-Ziegler formula. The Ziegler parameters are given in table 3.2. 3.3. HE STOPPING CROSS SECTION 29 Table 3.2: Parameters for the Andersen-Ziegler formula.^ A, . A2 A3/IOOO A4/IOOO A5 x 1000 H2 0.242 0.716 4.07 85.55 19.55 He 0.408 0.646 6.19 130.0 44.07 N2 2.096 0.536 31.34 18.11 4.37 o2 2.422 0.502 34.41 25.88 4.34 Ne 1.838 0.528 35.24 37.96 5.092 Ar 2.907 0.579 107.8 4.478 -0.544 Kr 2.527 0.662 301.3 1.466 -0.211 Xe 2.218 0.734 137.9 1.878 2.72 CH4 3.247 0.573 234.7 1.00 0.222 C02 6.927 0.495 272.6 2.00 0.496 Figure 3.2: Stopping power of 3 He and 4 He. The stopping power of 3 He at any velocity is identical to that of 4 He. Following the transformation of E(4He) = ^E(3He), the stopping power of 3 He at any energy can be obtained. If E < 0.6MeV, the 3 He range will be larger than the 4 He range. If E > 0.6MeV, the 3 He range will be smaller than the 4 He range. The transformation for protons and tritons is E(3H) = SE^H). 30 CHAPTER 3. DETECTION OF CHARGED PARTICLES Figure 3.2 shows the stopping powers of 3 He and 4 He. The triton is lighter than 4 He so that the stopping power is smaller, therefore the range of the triton is greater than that of 4 He for the same initial kinetic energy. When a charged particle is stopped in a gas, the total energy will be lost in the gas through collisions with neutral gas molecules. Along the path of the particle, there are photons emitted and electron-ion pairs are liberated. If the energies of the primary electrons are above the ionization threshold, then secondary ionization can occur. The sum of the two ionization processes is called the total ionization. The total number of ion-electron pairs created is given by where AE is the total energy loss in the gas, and Wi is the effective average energy to produce one pair, which depends on the gas instead of the energy of the incident particle. Table 3.1 gives the measured values of Wi for different gases in the order of 20-40 eV. W{ is greater than Io due to the excitation process. The calculation can be applied to a gas mixture according to a simple composition law. where Si is the percentage of the ith gas in the mixture. Since Wi is a constant for many gases, the deposited energy is proportional to the number of ion pairs formed. Certain transitions between energy levels in atoms and molecules are forbidden according to selection rules in atomic physics. However, energy levels can be excited by electron collision, from which a transition to the ground state by emission of a photon is unable to take place immediately. These levels are called metastable. The lifetime of metastable levels can exceed that of normal excited levels by several orders of magnitude. Helium has a metastable state (2sHe) at 19.8 eV, which is higher than 3.4 Total Ionization (3.18) 3.5. THE MOTION OF IONS AND ELECTRONS IN GASES 31 the ionization levels of many other gas molecules used in the mixtures in table 3.1 (15.58 eV for N2, 12.71 eV for CH4, 11.50 eV for C2H6, etc.). Helium atoms that are excited to such metastable states are able to transfer energy by collision to other molecules. For helium gas with an admixture A, the reaction 2sHe + A —>• lsHe + A+ + e~ takes place and this strongly modifies the secondary ionization yield. This is known as the Penning effect. lsHe denotes the ground level of helium atom. If the concentration of A exceeds a few times 1 0 - 4 , W{ changes from 42(pure He) to 28 eV, independent of the chemical nature and the concentration of A J 6 6 ! 3.5 The Motion of Ions and Electrons in Gases 3.5.1 Diffusion In the absence of an electric field, the charges released by ionization in a gas diffuse radially outward from their point of creation. They lose their energy by multiple collisions with the gas molecules, and eventually come into thermal equilibrium with the gas. The displacement due to the random collisions is the process of diffusion. The velocities of these ions and ionization electrons both obey a Maxwell distribution. The energy distribution of these charges is given by F(e) = Cji exp{~) (3.19) where T is the temperature, k is the Boltzmann constant, and e is the energy of the charges. At room temperature, er = |/sT ~ 0.04eV. In the kinetic theory of gases, if it is assumed that no additional ions are produced and no recombination takes place, the distribution of charges as a function of space and time is expressed as dN Nn *2 32 CHAPTER 3. DETECTION OF CHARGED PARTICLES where iVo is the total number of charges at time t = 0, x is the distance from the origin after a time t, and D is the diffusion coefficient. The rms spread in one or three dimensions is a(x) = V2Dt or a{r) = V6Dt (3.21) respectively, where r is the radial distance. D can be calculated from kinetic theory. D = (3,2) iyJisFoo V m where m is the mass of the particle, P is the pressure, and <70 is the total cross section for a collision with a gas molecule. The order of magnitude of the atomic cross section is 10- 1 6 cm 2 .! 6 1! A positive ion can recombine with an electron, a negative ion or extract an elec-tron from the metal walls. An electron can recombine with a positive ion, attach to a electronegative molecule, or come into the walls. The electronegative gases are O 2 , H2O, SO2, CCI4, etc. The probability of attachment is very low for all noble gases, hydrogen, nitrogen and hydrocarbon gases. 3.5.2 Mobility When an electric field is applied across the gas volume, the ionization electrons and positive ions are accelerated by the electric field between collisions with the gas molecules. The equilibrium between acceleration and collision gives a steady global velocity along the field lines towards the anode or the cathode superimposed on the random thermal motion. The average velocity of the motion is called the drift velocity, w. From the kinetic theory, the mobility of a charge is defined as M = I (3-23) where E is the electric field strength. For positive ions, the drift velocity is pro-portional to E. This formula holds up to high values of the electric field (several 3.5. THE MOTION OF IONS AND ELECTRONS IN GASES 33 K V / ( c m atm)). For a given E, the mobility is inversely proportional to the pressure. Due to their large mass and large scattering cross section, the average energy of ions is equal to their thermal energy. The relationship between mobility and the diffusion coefficient is known as the Einstein relation, ° = *I (3.24) \i e Table 3.3: Experimental mobilities of several ions in different gases^67'6^ Gas Ions Mobility (cm2V~1sec~1) He He+ 10.7 He N+ 22.4 He cot 20.2 Ar Ar+ 1.62 Ar 2.33 Ar cot 2.30 Table 3.3 gives the mobility of several ions drifting in the gases normally used in proportional and drift chambers. The mobility calculation can be applied to gas mixtures according to a simple mixture rule:'6 9! 1 n (3.25) From equations (3.23), (3.24) and (3.21), aion(x) = <2kTx eE (3.26) Therefore, the rms linear diffusion is independent of the nature of the ions and the gas. 34 CHAPTER 3. DETECTION OF CHARGED PARTICLES 3.5.3 Drift Velocity Unlike ions, the mobility of electrons is not constant except at very low electric field values. The elastic cross section of electrons is much smaller than that of ions. Elec-trons have a long mean free path and obtain substantial energy between collisions in an external field. The energy loss of an electron in a collision with a gas molecule is also smaller than that of an ion. As a result, the mean energy of the electrons can exceed the thermal energy by several orders of magnitude. In a uniform electric field, the electrons acquire a constant drift velocity, which is given by:' 7 0 ' w = r^-Er (3.27) 2m v ; where r is the mean time between collisions of electrons and gas molecules, r is a function of E. The drift velocity and diffusion depend strongly on the detailed structure of the elastic and inelastic electron cross section of the molecules in the gas. The behaviour of electrons in gases can be described by a more rigorous theory.'711 The energy distribution function of the electrons can be written as: F(e) = Cjiezp(- f — ^ % — ) (3.28) where A(e) is the mean fraction of energy lost by an electron at the time of a collision, A(e) is the mean free path between collisions, = (3-29) and a(e) is the collision cross section, and N, the Loschmidt number is given by P 273 N = 2.69 x — — molecules/'m3 (3.30) If the elastic and inelastic cross-sections are known, F(e) can be calculated and the drift velocity, w and diffusion coefficient, D are then given by 3.5. THE MOTION OF IONS AND ELECTRONS IN GASES 35 D(E) = J ^\{e)vF(e)de (3.32) where v = (2e/m)^ is the instantaneous velocity of electrons of energy e. The calcu-lation can be extended to gas mixtures n (7(e) = <We) (3-33) i=l n a(e)A(e) = 5,-a,-(e)A,-(e) (3.34) i=l The characteristic energy, can be defined as ED(E w(E) « - (3.35) From equations (3.35) and (3.21), Thus gases with a low characteristic energy at high electric fields are promising can-didates for high resolution detectors. Drift velocity, diffusion and the characteristic energy of electrons are very impor-tant transport parameters for the operation of a high precision drift chamber. A computer program D R I F T - V E l / 7 2 ] allows us to evaluate different characteristic pa-rameters of electrons in a gas or gas mixture. The program includes elastic collisions, the influence of the thermal energy of the molecules of the gas and inelastic collisions of the excitation type. The program does not consider collisions with ionization, thus it can not be applied to energies above approximately 10 eV. The behaviour of the drift velocity and diffusion depends very strongly on the gas nature due to the very different values of the elastic and inelastic electron cross sections of the gas molecules as a function of energy. The addition of very small qucintities 36 CHAPTER 3. DETECTION OF CHARGED PARTICLES O.O 0 . 2 0 . 4 0 . 6 O . S 1.0 1.2 1.4 1.6 1.8 2 . 0 E l e c t r i c f i e l d ( K V / c m ) Figure 3.3: Drift velocity of electrons in pure argon, and in argon with small admix-tures of N2, CH4 and C2H6. The very large effect on the velocity for small additions is apparent. of one gas to another may substantially modify the drift properties, as shown in figure 3.3. The computed drift velocity and the characteristic energy for several pure gases and gas mixtures are shown in figure (3.4-3.9), Argon based mixtures allow sufficiently fast drift velocity (4-10 cm//js) and low electron diffusion (100-250 for one cm of drift). The drift velocity of helium based mixtures is smaller. The drift velocity increases rapidly as a function of the electric field strength up to fields between 300 and 400 V / c m . Beyond 400 V / c m the velocity slowly rises to a value of the order of 2 cm/^s for high values of E. In helium mixtures, the cross section is dominated by the almost energy independent elastic scattering cross section of helium. A l l mixtures of helium with hydrocarbons show similar velocities of the order 1-2 cm//is, approximately increasing with the square root of E/N}73^ The drift velocity increases with the helium fraction in the mixture. The addition of very small fractions of another gas to pure helium or argon can dramatically change the drift properties as shown in figure (3.9). 3.5. THE MOTION OF IONS AND ELECTRONS IN GASES 37 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1.0 1.2 1.4 1.6 1.8 2 . 0 E l e c t r i c f i e l d ( K V / c m ) Figure 3.4: Drift velocity of electrons in pure gases 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1.0 1.2 1.4 1.6 1.8 2 . 0 E l e c t r i c f i e l d ( K V / c m ) Figure 3.5: Characteristic energy of electrons in pure gases. 38 CHAPTER 3. DETECTION. OF CHARGED PARTICLES 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1.0 1.2 1.4 1.6 1.8 2 . 0 E l e c t r i c f i e l d ( K V / c m ) Figure 3.6: Drift velocity of electrons in argon based mixtures CD C P CD e CD _o ~co q3 -+—> CJ o o o A r / C H 4 ( 9 0 : 1 0 ) ' A r / C 2 H 6 ( 9 0 : 1 0 ) A r / l s o b u t a n e ( 9 0 : 1 0 ) A r / N 2 ( 9 0 : 1 0 ) | - A r / C 0 2 ( 9 0 : 1 0 ) A r / M e t h y l a l ( 9 0 : 1 0 ) 0 . 0 0 . 2 0 . 4 0-.6 0 . 8 1.0 1.2 1.4 1.6 1.8 2 . 0 E l e c t r i c f i e l d ( K V / c m ) Figure 3.7: Characteristic energy of electrons in argon based mixtures. 3.5. THE MOTION OF IONS AND ELECTRONS IN GASES 39 ' ~r i i 1 1 1 1 1 1 1 r 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1.0 1.2 1.4 1.6 1.8 2 . 0 E l e c t r i c f i e l d ( K V / c m ) Figure 3.8: Drift velocity of electrons in helium based mixtures H e / C H 4 ( 9 0 : 1 0 ) H e / C 2 H 6 ( 9 0 : 1 0 ) H e / l s o b u t a n e ( 9 0 : 1 0 ) |~ H e / C 0 2 ( 9 0 : 1 0 ) H e / M e t h y l a l ( 9 0 : 1 0 ) ~i i i i 1 1 1 r 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1.0 1.2 1.4 1.6 1.8 2 . 0 E l e c t r i c f i e l d ( K V / c m ) Figure 3.9: Characteristic energy of electrons in helium based mixtures. 40 CHAPTER 3. DETECTION OF CHARGED PARTICLES 3.5.4 Electron Avalanche If the electric field is sufficiently strong, the electrons can receive enough energy to excite or ionize the gas. When the energy of an electron exceeds the first ionization potential of the gas, the result of a collision can be an ion pair. e~ + A -> ( A + + e~) + e~ where A is the gas molecule. Thus the number of electrons increases after the collision. When the secondary electrons ionize other molecules, an electron avalanche can occur. From equation (3.4), if n is the number of electrons per unit volume, the increase in the number after a distance dx is: dn = nnoadx (3.37) where n0 is the number of atoms per unit volume, a is the cross section for ionization of the atoms, n^a can be considered as the number of ion pairs produced per unit length of drift. This is known as the first Townsend coefficient a, which itself is a function of x, because it varies with electric field which is a function of x in many cases. Equation (3.37) can be rewritten as dn = na(x)dx (3.38) By integration n = n0e!*? a ( x ) d x or M = — = a ( x ) d x (3.39) n 0 where n0 is the number of primary electrons, M is the multiplication factor or gas gain. The charge grows exponentially with distance resulting in a very large gain. The Townsend coefficient depends on the gas, electric field and pressure.'74^ If the Townsend coefficient is known as a function of electric field, the multiplication factor can be computed. However, this function is usually unknown. Many theoretical models have been developed for calculating a for different gases. A model by Korff gives 3.5. THE MOTION OF IONS AND ELECTRONS IN GASES 41 where A and B are constants depending on the gas. It is assumed that the value of a is low. In the same region, the cross section for an electron to ionize an atom is roughly proportional to the mean energy of the electron. The Townsend coefficient can be written as a = aN = ksN (3.41) where & is a constant depending on the gas. Many efforts have been made for the determination of a and a lot of data exist for a wide variety of gases and mixtures. In the avalanche process, light quanta are emitted by molecules excited by electron impact or as a result of recombination. These quanta can release photoelectrons at the cathode. They might contribute to the secondary ionization and spread the discharge over the distance between the cathode and the anode to cause breakdown of the chamber. A limit for multiplication before breakdown is given by the Raether condition M < 108 or ax < 20 The breakdown limit of the measurements was found to have the same charge limit under completely different experimental conditions such as pressure, gas composition, electrode configuration, primary ionization, etc. Fonte et alJ7 6^ have made a quantitative study of the breakdown phenomena. An-alytical expressions were given for a and for the ultraviolet light production efficiency as a function of the electric field and gas mixture. These expressions established the constant fast breakdown limit on the basis of a space charge induced electric field distortion. 42 CHAPTER 3. DETECTION OF CHARGED PARTICLES 3.6 Proportional Chamber 3.6.1 General Characteristics Figure 3.10: Proportional chamber, (a) A parallel plate proportional chamber; (b) A cylindrical proportional chamber. The two basic electric field configurations of proportional chambers are shown in figure 3.10. Figure 3.10 (a) shows parallel anode and cathode plates with a gas in the region in between. The electric field is uniform and perpendicular to the plates. The electrons generated by an ionizing event will be accelerated towards the anode plate. The maximum voltage induced on the anode depends not only on the total number of ion pairs produced in the chamber, but also on the length of the path and thus on the initial position of the primary charges. There is no proportional relation between the size of the voltage pulse and the energy deposited, however, the energy deposited is proportional to the time integral of the voltage pulse. This disadvantage can be overcome by using a cylindrical geometry as shown in figure 3.10 (b). The anode is a thin metal wire at the center of a metal cylinder which serves as the cathode. The voltage on the wire is positive with respect to the cylinder. The electric field and potential can be expressed as E(r) = £ £ i (3.42) (3.43) 3.6. PROPORTIONAL CHAMBER 43 where a is the radius of anode wire, b is the radius of outer cylinder, SQ is the dielectric constant, C = (2ireo)/ln(fe/a), which is the capacitance per unit length, Vo is the high voltage applied, and r is the radial distance from the wire. The electric field is proportional to r _ 1 . In the region with large r, where the field is relatively low, the electrons drift towards the anode without charge multiplication. The electric field close to the anode wire surface (within a few wire diameters) is so strong that electron avalanche takes place when the electrons enter this region. Since the drift velocity of the electrons is much greater than that of the ions, there is a "drop-like" distribution of charges with the electrons at the head and the ions in the rear. The electrons are collected quickly and the ions drift towards the cathode at decreasing speed due to the decrease in the field strength as you move away from the anode wire. In a proportional chamber, the gas multiplication is a constant. The gain does not depend on the amount of charge present and the size of the charge signal is proportional to the initial ionization. This requires that any modification of the electric field near the wire caused by the space charge can be neglected. This condition is fulfilled for low gains. The average energy e gained from the electric field E between collisions with the mean distance a - 1 is E/a. From equation (3.41), a = VkNE (3.44) From equation (3.42), (3.44) can be rewritten as kNCVp 1 a ( r ) = V ^ o " ^ ( 3 - 4 5 ) The combination of (3.45) and (3.39) gives f r c /kNCVp \j M = eJa V 2« 0 rdr (3.46) The range of the integration is from a to rc, the radius at which multiplicative collisions begin. rc can be related to the threshold voltage VT, at which the electric field close to the surface of the anode is large enough to cause the process of multiplication. rc = ^-a (3.47) 44 CHAPTER 3. DETECTION OF CHARGED PARTICLES The integral in (3.46) gives an analytic expression, M = e V »«o ^ J (3.48) Hence, the gas gain increases exponentially with an increase in the voltage. The agreement is very good for moderate gains (< 104). A good assumption for helium based gases is a — constant. The similar method as above gives M = e a a { V o l V T ~ a ) (3.49) The gas gain depends on the chamber geometry, operating voltage, gas mixtures and pressure. At low and moderate electric fields the gain is dependent on a function of EIP, which is known as the reduced field. The mean energy gained by the electrons between collisions is determined by the reduced field. It is often assumed that near the anode wire, the first Townsend coefficient is proportional to the pressure and a function of E/P: & *,E. p = Hp) (3-50) In the literature there are many different empirical formulae given for the gas gain equation. A l l of these formulae were obtained using different parameterizations of a/P on E/P, and reduce to the general form:f77l l n M = / ( | ) (3.51) 3.6.2 Time Development of the Signal Since the avalanche begins at a distance of a few wire radii, the whole multiplication process takes place in less than 1 ns. When the electrons and ions are separated due to the electric field, two cylindrical shells of charge appear. The electron shell moves quickly towards the anode wire, leaving the positive ion shell drifting slowly 3.6. PROPORTIONAL CHAMBER 45 towards the cathode. The signal on the electrodes comes from the induction due to this process as the explaination given below J 7 4 ' Suppose that there is a charge Q at a distance r from the anode wire. The potential energy of the charge is U = QV(r) (3.52) When the charge moves a distance dr, the change in potential energy is dU = Q^P-dr (3.53) dr The electrostatic energy for a cylindrical capacitor is ^LCV02, where L is the length of the cylinder. According to the law of energy conservation: dU = LCVodV = Q^^-dr (3.54) dr The induced signal is Q dV(r) d V = LCV0^r-dr ( 3 " 5 5 ) Assuming that the multiplication takes place at a distance A from the wire, the induced voltages from the electrons and positive ions are, respectively, and y + Q f < m d r = _ _ « _ l n * (3.57) LCV0Ja+x dr 2ire0L a + X y ' The total signal induced on the anode is V = V+ + V- = --Q-lnb- = --^- (3.58) 2ire0L a LC and their ratio is 46 CHAPTER 3. DETECTION OF CHARGED PARTICLES For our triton detector, a — 12.7fim,b = 6.35 mm, and A = l ^ m , hence the final contribution from positive ion movement is 81 times that from electron movement. The major part of the signal comes from the contribution of ions. Integration of equation (3.55) gives: From equation (3.23) the ion velocity is * + E II+CVQ 1 where p+ is the mobility of the positive ions. Intergration gives Combining this with equation (3.60) gives V(t) = - T - V l n ( l + ^-2t) = - - ^ y l n ( l + f) (3.63) Awe0L TT60Pa2 4TTS0L t0 The total drift time of the ions, T, is obtained from r(T) = 6, = irepPjb2 - a2) V+CV0 (3.64) Equation (3.63) gives the characteristic shape of the voltage signal as shown in fig-ure 3.11. The voltage at first increases linearly with time and later logarithmically until the positive ions completely disappear at time T. The typical value for the rise time is about 1 /is compared to the 300 ^s for T. Therefore, for high count rate op-eration, it is necessary to differentiate the pulses by a short time constant (r = RC) circuit. To determine the voltage pulse at the output of an amplifier, the transient response of the amplifier must be considered.^ The transient response r(t) is defined as the output voltage caused by applying a unit step voltage at t — 0 to the input. The 3.7. MULTIWIRE PROPORTIONAL CHAMBER 47 0 50 100 150 200 250 300 t (jis) Figure 3.11: Time development of a pulse in a proportional chamber. The pulse shape obtained with several differentiation time constants is also shown. shape of the output voltage V^(^) for an arbitrary input voltage Vi(t) starting at t = 0 is W ) = j T ^ p r ( * - * > ' (3-65) For the resistance-capacitance coupled amplifier, the response function is r(t) = e~r (3.66) If M is very large, the contribution to the pulse due to the motion of the primary electrons and ions is negligible. 3.7 Multiwire Proportional Chamber Although proportional chambers are widely used in the measurement of the energy loss of charged particles, their capabilities are limited if the determination of the 48 CHAPTER 3. DETECTION OF CHARGED PARTICLES particle trajectory is required. The invention of the multiwire proportional chamber provides a good solution. A multiwire proportional chamber consists of an anode plane with parallel and equally spaced wires centered between the two cathode planes. The chamber geometry is shown in figure 3.12 (a). Typical values for the anode wire spacing range from 1 to 5 mm. The distance between anode and cathode ranges from 5 to 10 mm. The wires are maintained at a constant potential. The electric field equipotential lines are shown in figure 3.12 (b). An expression for the electric field was given by Erskine'79^ with the definition of V(a) = V0, V(L) = 0, E(x, y) = —-^(1 + tan 2 — tanh 2 — )* (tan 2 — + tanh 2 — ) " (3.67) 2eoS S S S S y ( l > y) = ^ o ( 2 ^ _ i n ( 4 ( s i n 2 — + sinh 2 ^ ) ) ) (3.68) 47T£o S S S where s is the wire spacing, L is the distance between the anode and the cathode. The capacitance per unit length is given by: C = (2 t t£o) / (— - I n — ) (3.69) s s where a is the radius of the anode wire. The capacitance increases with increasing wire spacing, s. Since a <C s, C is smaller than 2SQSJL. For the symmetry lines x = 0 and y = 0, equations (3.67) and (3.68) can be written as E y = E(0,y) = ^ c o t h ^ (3.70) CVn ITT Ex = E(x,0) = ^ c o t — (3.71) ZSQS S If y > s, then coth(7ry/s) ~ 1, E> = < 3 - 7 2 > Far enough away from the anode plane, the field lines are parallel and almost constant. In this region, electrons drift towards the anode wires. 3.7. MULTIWIRE PROPORTIONAL CHAMBER 49 — r X 1 . e i o 1 . -H i . 1 6 5 4 3 2 1 -1 -9 8 7h 6 5h 4 3 2 l h WIRE D R I F T L I N E PLOT P a r t i c l e ID= E l e c t r o n Gas: Argon 50% Ethane 50% o Co x - a x i s [cm] Figure 3.12: (a) Definition of parameters in a multiwire proportional chamber, (b) The electric field lines in a multiwire proportional chamber. 50 CHAPTER 3. DETECTION OF CHARGED PARTICLES If y <C s, then E(x,y) = ^ - (3.73) where r = - y / x 2 + y2. Near the anode wires, the field lines are identical to that of a cylindrical proportional chamber in the multiplication region. The solution of the drift equation (3.23) for positive ions with the expression (3.67) for the electric field is:' 8 0! x(t,a) = (s/2ir) arccos ((1 + cos2a)e _ , 4 t - cos 2a) (3.74) y(t, a) = (s/2ic) arccos h ((1 — cos 2a)eK t + cos 2a) (3.75) where a is the angle for ions leaving the origin, 0 < a < 2TT, and K = ( 3 - 7 6 ) If it is assumed that the initial angular distribution is uniform, the induced pulse shape can be obtained from equations (3.68) and (3.75), integrating from a = 0 to a = 2ir, Vi(t) = Q ATVEQL 2TTL/S — In ( (4 + 4cosh -Kt) sinh —nt (3.77) The typical time development of the process in a wire chamber is summarized in table 3.4. If the electrons created by a primary charged particle are in a constant field region, they will drift without multiplication along the field lines toward the nearest anode wire until they reach the high field region, where they will multiply. The final signal induced on the anode wire is negative, and proportional to the amount of ionization deposited by the particle through the chamber. The position of the wire which records the signal is a measure of the position of the particle. The spatial resolution of a multiwire proportional chamber is one-half of the anode wire spacing. The signal from one anode plane gives one coordinate of the ionizing event. If two or more chambers are aligned to measure the trajectory of a particle as a telescope, the track of the particle can be reconstructed. 3.8. DRIFT CHAMBER Table 3.4: The fundamental processes in a wire chamber^81' 51 t fundamental process 0 ns primary interactions 0.01 ns primary and secondary interactions 10 ns electron drift 20 ns first electron avalanche 100 ns full avalanche, photon emission 110 ns secondary electron drift 200 ns secondary avalanche, ion drift 2 [is ion drift 100 US ion drift and diffusion 200 /is ion neutralization, secondary electron and photon emission 3.8 Drift Chamber The spatial resolution for a multiwire proportional chamber is limited by the wire spacing. However, it is possible to get an improved resolution using a drift chamber. A drift chamber consists of a region of moderate electric field followed by a propor-tional region. Field shaping wires, stripes or electrodes provide the desired electrical configuration. The spatial information can be obtained by measuring the electrons' drift time. Electrons created by an ionizing event at time t0 drift with velocity w, and reach the anode wire at time t\. The track length is given by x = 1'wdt (3.78) For a constant drift velocity, x = w(h - t0) (3.79) This requires external trigger device such as a scintillator to determine to- The signal at the anode gives t\. 52 CHAPTER 3. DETECTION OF CHARGED PARTICLES A linear space-time relationship can be obtained for a uniform electric field. Ve-locity saturation is useful in a drift chamber. For some gas mixtures, the electrons' drift velocity is nearly constant even if the electric field strength is not completely uniform. This reduces the dependence of drift velocity on variations in field, pressure, temperature and gas composition. 3.9 Time Projection Chamber(TPC) sector wire chambers ionizing particle Electric Field Magnetic Field anode sense wire cathode "pads" Figure 3.13: Schematic diagram of a T P C . A Time Projection Chamber(TPC) is a three-dimensional tracking detector. This device makes full use of the ideas of both the multiwire proportional chamber and the drift chamber. Its structure is sketched in figure 3.13. When a high voltage is applied, a uniform electric field is built up along the axis. There is an axial magnetic field to give curved tracks for momentum measurement. Ionization electrons created by the primary charged particle passing through the chamber drift towards the end plates, where they are detected by the sector arrays of anode wires and cathode pads 3.10. CHAMBER GAS 53 with two dimensional read-out. The third dimension is given by the drift time of the ionization electrons. The particle trajectory is sampled by the anode wire. This allows a reconstruction of the particle path. The energy deposited by the charged particles can be obtained from the signal amplitude. There is a grid at ground potential just before the anode wires. This prevents the ions from the avalanche from drifting back into the chamber by capturing them. Such positive ions can accumulate and distort the field due to the space charge effect. The grid separates the drift region from the avalanche region and allows independent control of each region. For the triton detector designed for E683 at T R I U M F , a simple small TPC-like detector was constructed. The details will be given in the next chapter. 3.10 Chamber Gas A suitable gas mixture is critical for the operation of a gas-filled detector with a high charge gain. Noble gases are the principal component of chamber gas mixtures. In the gas multiplication process, the field strengths for the noble gases are much lower than those for the complex molecules. Argon gives more primary ionization than helium and neon. Although krypton and xenon can give more ionization than argon, they are too expensive. Pure argon cannot be operated with gains of more than about 103 —104 without causing a continuous discharge. Excited argon atoms formed in the avalanche can return to the ground state only through a radiative process. The minimum energy of the emitted photon(11.6 eV) is greater than the ionization potential of any metal cathode. The electrons extracted from the cathode can cause further avalanches. Unlike argon, polyatomic molecules such as methane and ethane have many non-radiative rotational and vibrational excited states. Argon atoms in excited states may be de-excited through collisions with such molecules. The photons emitted from the argon atoms can also be absorbed by the same molecules. Only a small addition of this quenching gas allows gains in excess of 106 before breakdown in a argon gas chamber. 54 CHAPTER 3. DETECTION OF CHARGED PARTICLES The presence of a quenching gas also provides a more stable detector operation. The addition of a small amount of an electronegative gas to the argon mixture allows a gain up to 107. Charpak and collaborators'74^ found such a high gain mix-ture: argon:isobutane:freon 70:29.6:0.4. A large variety of gases and gas mixtures are currently used in wire chambers. One example of a standard drift chamber gas is argomethane 50:50 at normal pressure. Argon based mixtures are the best choice for drift chamber gases.'66' They allow high gas amplification, fast drift velocity(4-10 cm/fis) and low single electron longitudinal diffusion( 100-250 /xm for one cm of drift). Considerable improvement of the spatial resolution can be reached by using gases with low diffusion, like C02 or DME((CH 3 ) 2 0) . With such gases resolutions as good as 30 nm have been reached.'82' There has been a recent interest in helium based gas mixtures.I 7 3 ' 8 2 - 8 4 ! The ad-vantages of a helium gas mixture are long radiation length(5299 m) compared to argon(110 m), good momentum resolution and low synchrotron radiation absorption. These are important selection parameters for tracking detectors built for several pro-posed high intensity facilities^, T-charm and B-meson factories). Helium mixtures have been used as the drift gas for a triton T P C detector which is being designed here at T R I U M F for a polarized fi~ capture experiment. The various reasons for this choice of the detector will be explained in the next chapter. Chapter 4 Triton Detector 4.1 Design Considerations In order to study the asymmetry in muon capture by polarized muonic 3 He, the muon stopping target and the triton detector had to be combined together. The following requirements must be fulfilled: 1. Stop the muons in a low density 3 He gas target 2. Produce polarized muonic helium 3. Measure the muonic helium polarization 4. Identify the tritons and measure their direction of emission Since the stopping power depends on p^ and /? as discussed in the last chapter, and helium has a very low atomic number, high p and small f3 are required to stop a sufficient number of muons. Therefore, five atmospheres of helium gas had to be placed in the detector. In order to maximize the number of muons stopping in the gas volume, the energy of the muon beam was decreased as low as possible. A 50 pm thick, 3.8 cm diameter stainless steel window allowed the low momentum(23 MeV/c) muon 55 56 CHAPTER 4. TRITON DETECTOR beam to enter the target/detector. The experimentally determined burst pressure for this window is in excess of thirty atmospheres. The muonic helium was polarized by collisions with laser optically pumped Rubid-ium vapor. The detector had to be heated to a temperature above 200° C to obtain the Rubidium number density needed for optical pumping with high power GaAlAs Diode laser arrays^85] tuned to the 795 nm Rubidium D l line. The laser power is approximately 28 W. A probe laser was used to measure the Rubidium density. Glass windows were installed in the detector for the entering laser beam. The muonic helium polarization was measured by monitoring the electrons from muon decay using an electron telescope. The telescope consisted of five parts. There was a scintillator, two delay line wire chambers, an electron absorber to eliminate low energy electrons, followed by a second scintillator. The wire chambers were the standard T R I U M F delay line chambers.t86' These were used to study the positional dependence of the muon polarization within the target volume. The tritons were detected by a T P C inside the target vessel. The T P C consisted of four wire planes connected to high voltage feedthroughs. The integration of the anode pulse as a function of time is proportional to the total number of electron-ion pairs created, which helped to identify the triton events. The detailed shape of these pulses was able to provide directional information by the location of the Bragg peak(see fig 4.5). The T P C was operated at high temperature, high pressure and high Rubidium density. To our knowledge, this was the first successful operation of a high temperature drift chamber. 4.2 T P C Figure 4.1 is a cross section of the experimental setup for the T P C test. The chamber consisted of a stainless steel cross of 10.2 x 10.2 cms capable of holding pressures up to ten atmospheres. Two laser entrance windows mounted on the top and the bottom flanges allowed a probe laser beam to pass through the detector for Figure 4.1: Mechanical design of the time projection chamber. 58 CHAPTER 4. TRITON DETECTOR measuring the Rubidium number density. There was another laser entrance window on the left side flange for admitting the optical pumping laser light into the detector. The muon window was attached to the right side flange. This flange also held the high voltage feedthroughs. Eight feedthroughs were welded into the flange on a 7.6 cm diameter circle. The Rubidium reservoir was connected to the bottom flange. There were two valves welded directly to the bottom flange for filling the detector with gas. The T P C was mounted on the right side flange. It was designed to be taken apart easily for repairing if there was any failure of the chamber operation. The entire assembly was placed inside an oven which was capable of heating the detector to temperatures up to 250° C. 4.2.1 T P C Development Two generations of the T P C were built prior to the one described in the next sec-tion. The initial design was a gas scintillation time projection chamber (GSTPC)' 1 0 ^ instead of a conventional T P C , because it offered the advantage of obtaining position information without introducing a large number of electrical feedthroughs into the hot reaction vessel. Gas scintillation in helium-dominated gas mixtures has been used very successfully at PSI^ 8 7 ' to obtain very clean triton spectra from muon capture by 3He. Since helium emits light at A = 50 — 100 nra, a small amount of nitrogen had to be added to shift the wavelength to the 300-380 nm regionJ88^ The scintillation light could then be measured using U V sensitive phototubes. The T R I U M F G S T P C was tested with a collimated alpha source of 241Am. The primary scintillation light was observed by two phototubes situated above and below the ionization region indicat-ing the creation of an ionization event. A coincidence between the two phototubes served as a trigger for the electronics. The third phototube was used to detect the secondary scintillation light produced by electron-atom collision in the "light" gap to obtain the signal containing the information about the emission angle and energy of the alpha particle. The GSTPC performed well during the test runs at room tempera-ture. The light yield dependence on pressure and concentration of the gas constituents 4.2. TPC 59 was studied extensively. For the high temperature tests, it was found that the light amplification was reduced significantly.'89J The second generation design was a conventional T P C to avoid the difficulties associated with the reduced light output at high temperatures as well as with the large thermal gradients.'9^ In general, the resolution of the light signal is two times better than that of the charge signal, because the light is produced over the entire region of the light gap, whereas charge amplification occurs only very close to the wire surface. Hence the noise gets amplified linearly in light amplification, but exponentially in charge amplification. However, for the conventional T P C , the design of the entire detector could be significantly simplified since no light has to be detected. This could result in much better temperature uniformity inside the oven. At the same time, it was not expected that there would be as large a reduction in the signal amplitude at high temperatures as we measured with the light amplification detector. The main problem for the T P C operation was the current leakage of the high volt-age power supply at high temperature. As a consequence, the signals were distorted and covered by breakdown pulses. In order to solve this problem, a third generation T P C was designed as will be discussed in the next section. By applying high voltage up to 8 K V to the second generation T P C exposed to air in a dark room, the ma-jor sparking points were found. The high voltage breakdown happened between the stainless steel wire frames as indicated in figure 4.2(generation 2) due to the path of the breakdown being considerably shorter than the distance between the anode and the ground wire plane. Inner and outer insulation steps were added to the old macor holder as shown in figure 4.2(generation 3) at a price of shrinking the size of the wire planes in order to increase the path between the two stainless steel frames. With this design, the leakage current at high temperature dropped from 1 uA to 0.05 fiA, sim-ilar to what was observed at room temperature. The most important improvement was to double the anode wire spacing which produced only a slight increase in the capacitance. As a consequence, the operating high voltage for the same amplification gain decreased by a factor of 1.5. Although there was still some noise present, it could be excluded from the electronic readout system by using a coincidence between the 60 CHAPTER 4. TRITON DETECTOR £X j z > o i n t z s 3 - 8 c m .3 m m 4 . ^ 7 <S i n m • ^ ^ ^ g r o u n d a n o d e ^ e n « o n T" X . c m 3 . 8 c m —I Figure 4.2: Comparison of two generations of T P C detectors for the amplification gap. signals from the anode and the ground. 4.2. TPC 61 4.2.2 Mechanical Design and Construction direction of the drift field 3.8 cm 7.6 cm cathode mirror ground field shaping wires anode Figure 4.3: Time projection chamber (1:1). Figure 4.3 illustrates schematically the T P C configuration. Because the T P C had to be within the vessel operated with the hot Rubidium vapor environment, the materials used to construct it needed to be chosen very carefully. The T P C was configured as four circular wire planes: cathode, ground, anode and mirror plane, which were interconnected by four macor rods. Macor, a machinable ceramic material, was chosen for the support rods, spacers and wire frame holders because it did not interact with Rubidium. The wire planes and frames were made of stainless steel. They had the same thermal expansion coefficient so that there was no warping of the wires on the grids at the required high temperature. The grids were made in a winding machine. A suitable tension ensured that the wire 62 CHAPTER 4. TRITON DETECTOR plane was kept as flat as possible. High temperature conducting epoxy was used to secure the wires in place on the wire planes after the curing process, which had to be done twice at 110°C and 250°C, was completed. The chamber was divided into two regions-drift and amplification. The drift region, extending 9.9 cm from the cathode plane to the ground plane, had ten circular wires acting as field-shaping rings which were made out of 0.5 mm thick stainless steel wire. The 2.3 cm triton tracks easily stopped in this region, which allowed the production and collection of enough primary ionization. The amplification region had a symmetric structure with an distance of 6.4 mm between the ground plane and the anode plane or between the anode plane and the mirror plane. The anode wire diameter was 25 um with an equal wire spacing of 1.6 mm. The other wire planes had a wire diameter of 125 um with a spacing of 0.8 mm between the wires. The wires were provided with special surface treating,t91' and they all looked very smooth under an electron microscope. The wires on all grids were aligned to obtain uniform transmission. An alpha source of 2 4 1 A m ' 9 2 ' in a stainless steel collimator was mounted on one of the macor rods. The source was placed inside the chamber to avoid any absorption or scattering through the window. The collimator was at the center of the drift region with a 1.6 mm diameter hole at its center. The emission angle could be adjusted freely to cover 180 degrees relative to the direction of the drift field. The accuracy of the angular scale was ±2 degrees. The source emitted alpha particles of 5.49 MeV energy at a rate of about 2 Hz. It was electrodeposited and could be heated to 250°C —300°C. 4.2.3 High Voltage The wire grids were electrically connected to the ceramic insulated feedthroughs by welding them onto the 0.5 mm thick stainless steel wires. The electric field was produced by the four wire grids across which a voltage from a high voltage power supply followed by a low pass filter with a time constant of about 0.1 s could be applied. Typical values for Vc, Vg, Va and Vm were -4.2 kV, 0 kV, 2.9 kV and -0.5 k V respectively. The four wire planes were maintained parallel by the well machined 4.2. TPC 63 macor spacers of the same length to insure an uniform electric field. Field shaping rings were necessary to further ensure an uniform electric drift field of 438 V / c m . Unfortunately, there were only a limited numbers of feedthroughs available. The Garfield simulation program'9^ was used to optimize the number and position of these field shaping rings. A good drift field could only be created if a minimum of ten field shaping rings were used. However, these rings needed to be at four different voltages which just matched the number of available feedthroughs. A resistance network was made to provide the high voltage required for the cathode and the field shaping rings and to keep the current at a low value, typically about 30 \iA. The ground wire plane was capable of reducing the influence of the anode wire on the uniform electric field in the drift region. The mirror plane was very useful to get a more uniform electric field distribution on the anode wires. Without this, the electric field around the anode wires would be distorted by the space charge effect due to the charge accumulation from the avalanche at the anode wire. The electric field equipotential lines for the T P C are shown in figure 4.4 (a). A good drift field configuration was achieved with this configuration. When a 1.9 MeV triton was ejected in the drift region, about 62000 ionization electrons would appear along the track. The triton slowed down and then stopped in the gas. The ionization density is proportional to the ^ f ° r l ° w e n e r gy> thus most electrons were produced near the end of the track. The electrons drifted with a constant velocity along the electric field lines toward the amplification gap, where an avalanche occured in the high field region(4.57 kV/cm). The electron drift lines for the T P C are shown in figure 4.4 (b). From the detailed pulse shape of the anode signal, one can in principle determine in which direction the triton was emitted. When looking at the charge signal on the anode as a function of time, each event exhibited a characteristic Bragg distribution as shown in figure 4.5. If the largest intensity of electrons arrived at the beginning of the pulse as shown in figure 4.5 (a), then the triton was travelling towards the amplification gap. If the largest part of the signal appeared at the end of the pulse as shown in figure 4.5 (c), this indicated that the triton was travelling away from the gap. If the ionization electrons all arrived at the 64 CHAPTER 4. TRITON DETECTOR TRACK-DRIFT LINE PLOT P a r t i c l e !•= E l e c t r o n Gas: Argon 50% Ethane 50* x-axis [cm] Figure 4.4: (a) The electric field equipotential lines for the T P C . (b) The electron drift lines for the T P C . 4.2. TPC 65 Figure 4.5: T P C anode signals for different emission angles, a) shows an event in which the particle is moving towards the amplification gap. b) shows an event in which the particle is moving perpendicular to the direction of the drift field, c) shows an event in which the particle is moving away from the amplification gap. 66 CHAPTER 4. TRITON DETECTOR same time, then the triton track was perpendicular to the direction of the drift field. Such pulses were sharp in time and showed no Bragg peak (see figure 4.5 (b)). Hence the pulse amplitude as a function of time contained information on the direction of the triton track. The width of the pulse is proportional to the length of the projection of the track onto the drift field direction. 4.2.4 Gas System Because the chamber had to be operated at 200° C , it was very important to maintain a very clean system to get the T P C working. The presence of any electronegative pol-lutants such as 0 2 , H20, N20, NO, S02 and HCl etc. was fatal to the operation of the detector. The drift electrons would have been captured by these impurities before their arrival at the amplification region. The chamber was filled using the vacuum/gas handling system shown in figure 4.6. The system consisted of stainless steel tubings, valves and standard V C R fittings. A He — N2 mixture was chosen as the drift gas. The gas bottles were connected via pressure regulators to the gas handling system through flexible gas lines. By mixing the gas online it was convenient for us to change the nitrogen concentration. A molecular pump was used to pump out the vessel by attaching it as close as possible to the vessel. The vessel was constructed using standard ConFlat and V C R sealing technologies. The gas inlet and outlet valves were welded directly to the bottom flange together with a Rubidium reservoir and a thermocouple holder. The whole system was leak checked to better than 1 x 10~9 Torr, which was the most sensitive scale for the leak detector. A l l components of the chamber were cleaned with acetone and ethanol. A l l the small pieces were cleaned in an ultrasonic container. Gloves were used when handling the components. After the chamber was assembled in the clean room, it was attached to the vacuum system. The entire detector was baked out at 230°C for twenty-four hours under vacuum. Then it was flushed with He gas, and after a second pump-out cycle it was filled with nitrogen to a certain level and then filled with helium to the 4.3. RUBIDIUM 67 DETECTOR I I HELIUM PURIFIER - M -NITROGEN PURIFIER - N -PRESSURE GAUGE - M -HELIUM NITROGEN TO VACUUM PUMP Figure 4.6: Vacuum/gas handling system. desired pressure. The two partial pressures were measured to an accuracy of 1 Torr using two B A R A T R O N gauges. 4.3 Rubidium The muonic helium was polarized by collisions with optically pumped RbJ3J Rb atoms with a Si ground state and a Pi excited state were illuminated by a circularly polarized laser beam which propagated along the direction of a small magnetic field (see figure 4.7). Rb atoms in the | | , \ > Si ground state cannot absorb light to jump to the | | , | > Pi excited state due to the spin of the photon. However, ground state Rb atoms in the | | , — | > Si state can absorb a photon and jump to the | | , | > 68 CHAPTER 4. TRITON DETECTOR Sj 1 \ X=l 95 nm < + • + 2 J 2 Figure 4.7: Energy level diagram for optically pumped Rb. Excited-state polariza-tion can be transferred to the ground-state by spontaneous emission resulting in a repopulation. P i excited state. Rb atoms in the | | , | > Pi excited state decay very quickly (lifetime = 28 ns) and fall back to either the | | , — | > Si state or the | | , | > Si ground state. If there were no relaxation mechanisms, almost all the Rb atoms would eventually be pumped into the | | , | > Si state. Hence the Rb atoms are polarized by the application of the laser light. The relaxation is caused by collisions with 3He and Rb atoms. The Rb spin destruction rate'3 5' is given by, TSD = (2 x 1(T 1 8 cm3/s) [3He] + (8 x 1(T 1 3 cm3/s) [Rb] (4.1) For the present experimental conditions, TSD — 10 3/s. In order to achieve high Rb polarization with the laser, the photon absorption rate should be equal to the Rb spin destruction rate: ^ L = YSD[Rb]Vs (4.2) where Piaser is the laser power, v is the laser frequency and V/ is the fiducial volume. Hence Piaser should be at least 10 W. Therefore, the present experiment should achieve ~ 100 % Rb polarization. When a muon is captured by helium, a (u~He)+ ion is formed after the atomic cascade. This process also causes the depolarization of the muons. Due to the large mass of the muon as compared to that of the electron, its Bohr orbit around the nucleus is very small. As a consequence, the (u~He)+ ion can 4.3. RUBIDIUM 69 be considered as a pseudonucleus like a proton. A molecular ion He(fi~ He)+ will be formed after a few nanoseconds.t11' This ion will be dissociated and neutralized in about a microsecond by collisions with the polarized Rb vapor. The polarization is transferred directly from the Rb atom through the reaction Rb T +(p-He)+He -> Rb+ + He + (p-He)+e~ | (4.3) where the arrows indicate the spin polarization of the transferred electron. Immedi-ately after the charge exchange collision, the polarized electron will share its polar-ization with the pseudonucleus through the hyperfine interaction. The neutral atom He)+e~ is formed, and further polarized by spin exchange collisions with Rb: Rb T +(^~He)+e- | Rb | +{u~He)+e- | (4.4) The polarization time scale is comparable to the muon lifetime (2.2 ps). Thus the polarized muonic 3He is usually produced before the nuclear capture process occurs. Using this technique, an average muonic 3He polarization of 26.8 ± 2.3%[3' has been achieved by the Princeton-Syracuse-LANL group. The muonic helium polarization can be expressed as:^ P(t) = PaV(l - aQe-^ - bae-t0") (4.5) where the subscript a = 3 (a = 4) for 3He (4He). The constant n represents the fraction of the muonic 3'4He atoms that are formed when the Rb is 100 % polarized. The parameter P Q , the maximum theoretical polarization, is equal to 3/4 (1) for the case of 3He (4He). Note that for 3He, the singlet state |0,0 >, which is formed 25 % of the time (see section 2.3.1), cannot be polarized. ^a (f3a) is the rate for process (4.3) ((4.4)). aa and ba are the coefficients related to ^ a and fla. With r) = 0.70, the maxmium polarization for muonic 3He, -P(oo), is 53 %. Because the branching ratio for the reaction (1.7) is only 0.3 %, most of the muons stopped in helium will decay via: pT —> e~ + ve + z/M 70 CHAPTER 4. TRITON DETECTOR The electron is more likely to move opposite to the direction of the muon polarization than parallel to it due to the helicities of the decay leptons. The decay electrons were detected by the electron telescope(see section 4.1), and the muon polarization was then extracted from the electron asymmetry Nu + Nd v ; where Nu(Nd) is the number of decay electrons emitted parallel (antiparallel) to the Rb polarization direction. It was more difficult to obtain the required Rb density in a metal chamber than in a glass cell. A large quantity of Rb reacted with the impurities inside the chamber. Before the Rb vapor density became stable, considerable Rb coated the stainless steel walls of the chamber. Thus a very large amount of Rb had to be loaded into the detector. Four 99.99 % pure 1 gram Rb ampoules'94' were broken open and dropped into a glass tube with a narrow neck. The tube was then sealed off by heating its neck with an oxygen/gas torch.' 9 5 ' The glass tube was wrapped in a fine stainless steel mesh and then it was put into the Rb reservoir of the chamber, which was made from a 1.3 cm diameter stainless steel tube with a 0.25 mm wall thickness. The end of the tube was welded together like a tube of tooth paste. The chamber was pumped and baked at 230°C for 24 hours. After the chamber was cooled down, 1 atmosphere N2 was added. The glass tube was squeezed from the outside releasing the Rb. The mesh acted as a wick for the Rb allowing it to diffuse throughout the chamber.'11' Then the gas was pumped out, He/N2 (97:3) gas mixture was filled in and the chamber was heated up. The Rb number density was measured'96' by scanning through the Rb D l line with a Ti:Sapphire laser and measuring the transmitted laser light vs wavelength. This laser has a F W H M line width of about 0.001 nm (500 MHz) , which is much smaller than the total absorption width of the Rb D l line (0.3 nm). Figure 4.8 shows one of the transmission scans, which indicates a Rb density — 8 x 10 1 3 c m - 3 at 200°C. 4.4. OVEN 71 792 793 794 wavelength (nm) 798 Figure 4.8: Transmission scan of the probe laser at 200°C 4.4 Oven The detector and oven were mounted on a stainless steel cart which could be rolled around. The oven had an octagonal cross section to allow the electrons to exit through perpendicular surfaces. The walls of the oven had a sandwich design with an inner conducting surface (Cu) backed by an Omega Heater'9 7' followed by thermal insulation and an outer (Al) wall. The wall panels could be made very thin with a strong frame between the panels. The insulation behind the wall was used to decouple the oven inside from room temperature. The current for heating the top, side and bottom wall was provided by variac transformers to keep the temperature gradients across the chamber as small as possible. Copper-constantan thermocouples were used at ten different places to monitor the chamber temperature. One of the thermocouples was installed inside the vessel to measure the temperature of the detector. The maximum temperature difference was 20° C across the chamber when the temperature inside the vessel was 200°C. The temperature were held stable to within 1°C. The Rb reservoir was heated separately by wrapping heater tape around it. Teflon insulated, teflon jacketed coax cables and high temperature M H V feedthroughs'98' were chosen for the T P C signal and high voltage cables. There were three windows at the top, side and 72 CHAPTER 4. TRITON DETECTOR bottom of the oven each matched to the three laser entrance windows of the vessel. The laser entrance windows had to withstand a pressure of 8 atmospheres, sustain repeated cycling to 200° C and be of high optical quality. This design is similar to that of an ion chamber constructed by the Princeton group.'1 1' The windows were made by Larson Electronic Glass' 9 9 ' with a 7056 glass-to-Kovar metal seal mounted on a ConFlat flange. The expansion coefficients of the metal and the glass were well matched. The thickness of the glass was 14 mm for the 4.8 cm diameter window and 7 mm for the 2.4 cm diameter window. The experimentally determined burst pressure for these windows is in excess of 20 atmospheres. The two windows at the top and bottom of the vessel provided a Ti:Sapphire laser beam path through the chamber. The Rubidium number density could be estimated by measuring the laser beam transmission as a function of frequency. 4.5 Electronics and Data Acquisition DETECTOR CAMAC CRATE M. Q +H.V. PREAMP FAN ~" TFA \-j DISCR R. BOX TFA i PREAMP FAN DISCR STOP G.G. PC DELAY -H.V. O Figure 4.9: Electronic readout system. Figure 4.9 shows the electronic readout in the self-triggering mode. The signals from the anode or the ground plane were input into a preamplifier based on the DF 1001 hybrid circuit ' 1 0 0 ' and sent to the T F A (ORTEC 474 Timing Filter Amplifier). The preamp circuit diagram is shown in figure 4.10 and consists of a diode protected 4.5. ELECTRONICS AND DATA ACQUISITION 73 +6V O-GND O-SIG ID-TEST O -12V O-470 Q. .01 nF BAV99 S .01 \xF 470 O. 24K 27K 51Q BFT25 BFT25 51fi BFR93A BAV99 7A /ts.OIHF <27K < 18K -A/V 220 n 2.7K 39f> .1 \iF -> I O OUTPUT 330 O. -6V Figure 4.10: DF1001 Preamplifier. input and two cascaded emitter followers in the output. The output saturation am-plitude is 180 mV. The hybrid power dissipation is typically 24 mW. The preamplifier had a 20 ns rise time, therefore, the avalanche signal shape was relatively unaffected by the readout electronics. The signals from the T F A with a 200 ns integration time constant passed through a linear fanout to distribute three identical signals to a waveform digitizer (LeCroy 2261 ICA) , a discriminator and a digital oscilloscope (Tektronix-520 500 MHz memory oscilloscope). The signals from both anode and ground were discriminated and directed to a logic unit to fulfill a coincidence require-ment. The threshold levels of the discriminators were adjusted to keep the noise rates as low as possible. The amplitude of the signal as a function of time could be observed directly from the oscilloscope. The readout was delayed by about 22 /is by a dual gate generator. The final logic signal was also used for the stop or trigger signal of the four channel waveform digitizer.' 1 0 1 ' Each channel provided 11-bit analog-to-digital conversion and a clock rate up to 10 MHz. The output had 1 mV amplitude reso-lution with a full scale range of 2 V , and a variable baseline continuously adjustable from -2 V to 0 V . The digitizer sampled the signal at 100 ns intervals for a full scale interval of 32 /is. The digital data was then loaded into memory for C A M A C readout. 74 CHAPTER 4. TRITON DETECTOR Programmable logic allowed any part of the record in memory to be accessed from the 2261 Memory. The waveform digitizer was mounted in a C A M A C crate with a C A M A C controller (3922 Parallel Bus Crate Controller). SUSIQ' 1 0 2 ' controlled the C A M A C modules, wrote raw data to disk, processed data events and compiled the histograms. SUSIQ is a data acquisition system and event analyzer for P C based systems. It uses a high level language for acquisition def-inition and event analysis. The data acquisition was performed on an event-by-event basis. The package has facilities for conditional branching, logic tests, mathematical calculations and C A M A C access, etc. A User Customized SUSIQ can be generated by using the F O R T R A N code written by the user and the compilation and linking tools provided by SUSIQ. The user subroutines written by the author contained the exact details of data acquisition including online single pulse and histograms monitoring, data manipulating and the second level trigger cuts. The spectra and variables could be easily defined and modified. The data(inverted and with the offset subtracted) were written out to ASCII files. The files were transferred via T R I U M F ethernet to a U N I X workstation to be analyzed off-line with P A W . [ 1 0 3 ] 4.6 Gas Gain Calibration In order to calibrate the gas multiplication factor of the gas mixtures for the T P C , a collimated 2 4 1 Am alpha source was installed in the chamber. The alpha particles with different emission angles to the drift field were stopped in a gas mixture, and the total energy E of the particle was deposited. Since the total number of ion pairs produced by an alpha particle in the drift region is proportional to the energy loss of the particle within the chamber, and the gas gain is a constant, the total charge induced on the anode is a measure of this energy loss. From equation (3.17), the total ionization for an alpha particle is Mr. 4.6. GAS GAIN CALIBRATION 75 The gas gain can be expressed as [104] M = CeffAWi (4.7) gT eE where A is the time integral of the pulse shape from the T P C anode, g is the total gain for the preamplifier and T F A , and T is differential time of the preamplifier. CC T Calib. Pulse cable scope Rd Cca T Rs < T C s Figure 4.11: Circuit used for absolute gain calibration. The gain calibration is achieved by injecting square pulses on a known capacitor whose value is close to the capacitance of the T P C . The output pulse can be examined directly on a Tektronix 520 scope. The equivalent circuit is as shown in figure 4.11. The effective resistance and capacitance are Reff = RdRsRv ^ ^ RdRs + RdRv + RSRV Ceff = Cc + Cs + Cca (4.9) where Rd is the discharge resistor, Rs is the input impedance of the scope, Rv is the variable resistor, Cc is the capacitance of the amplification gap, Cs is the capacitance 76 CHAPTER 4. TRITON DETECTOR Figure 4.12: Decay time vs. effective resistance. of the scope, Cca is the distributed capacitance of the cables from the chamber to scope. If Vp is the peak-height of the observed voltage pulses, then the charge injected to the capacitor is Q ~ CeffVp (4.10) The value of C e / / can be determined from the decay time of the signal. T = ReffCeff (4.11) where Reff could be varied by changing the resistor Rv. The graph of r as a function of Reff is shown in figure 4.12. By fitting the experimental data T to (4.11), C e//=242 Pf-As an example, an alpha particle from a 2 4 1 A m source was stopped in a He/N? (97:3) mixture. E = 5.486 MeV, and W, = 30.7 eV. This results in 178700 drift elec-trons for each alpha particle. For 2.9 kV anode voltage at 200°C, A = 13.915 V us. For our setup, g = 6138, and T = 0.1 ps. Hence M = 192. Chapter 5 Data Analysis The data analysis is based on the raw data collected per event, which contain the in-duced charge signal within a 30 fts time window, sampled in 100 ns steps. A pedestal value determined from the average signal value before the actual pulse begins is sub-tracted from each event. The analysis of the data was performed using software packages supported by the T R I U M F computing services group. By linking user writ-ten application routines with these standard software packages, it was possible to read the data and construct histograms and various scatterplots. The analysis of the data reduces all the experimental data to essentially four histograms, the charge, width, amplitude and skewness spectra. The first three histograms have been used as a win-dow for data acquisition to provide a powerful background rejection. The function of the fourth spectrum will be discussed in section 5.2.2. 77 78 CHAPTERS. DATA ANALYSIS 5.1 Ionization Measurement 5.1.1 Gas Gain vs Angle, Temperature and High Voltage The data taken for this section are from five atmospheres He/N2 (97:3) mixture. The gas gain was determined using the calibration method discussed in the last chapter. Each data point represents a data set with about six percent statistical error. 3 0 0 2 5 0 ; 2 0 0 n i i i i i i r - 6 0 - 5 0 - 4 0 - 3 0 - 2 0 - 1 0 0 10 2 0 3 0 4 0 5 0 6 0 angle ( d e g r e e ) 3 0 0 2 5 0 : 2 0 0 150 100 i i i i i i i r 6 0 - 5 0 - 4 0 - 3 0 - 2 0 - 1 0 0 10 2 0 3 0 4 0 5 0 6 0 angle ( d e g r e e ) Figure 5.1: Gas gain vs angle at 23°C and 200°C. The data for gas gain as a function of emission angle of the alpha particle is shown in figure 5.1. If the track of an alpha particle is perpendicular to the direc-tion of the drift field, the emission angle is defined as zero degree. The angles are positive(negative) for the tracks heading towards(away from) the amplification gap starting from the zero degree line. The data were taken at an anode voltage of 2.9 K V in angular steps of 10 degrees at room temperature and 20 degrees at high tempera-ture. The total charge is independent of the emission angles within the uncertainties. At 200° C , the average charge gain for all angles agrees with the room temperature data within their uncertainty. With the large error bars, the loss of the drift electron over the whole drift field at 5 atm appears to be negligible. The data for the gas gain as a function of temperature is shown in figure 5.2. The 5.1. IONIZATION MEASUREMENT 79 : 200 H 0 5 0 degree 1 1 50 100 150 200 temperature (° C) 300 250 200 150 i 100 250 50 100 150 200 250 temperature (° C) 50 300 250 : 200 150 H 100 100 150 200 temperature (° C) 250 0 50 100 150 200 temperature (° C) 250 Figure 5.2: Gas gain vs. temperature for +50, +10, -10, -50 degree. data were taken at an anode voltage of 2.9 K V in steps of 50°C at emission angles of ±10 and ±50 degrees. The errors at 100°C or 150°C are the same or smaller than at 200°C, but all are larger than at room temperature. The total charge is independent of the temperature within the uncertainty. The data for the gas gain as a function of high voltage at room temperature and 200°C, respectively are shown in figure 5.3. The gain increases exponentially with increasing voltage. These data were taken in steps of 100 V . At relatively low voltage, the gas gain is almost independent of the emission angle. The difference in gain for the different emission angles becomes larger at higher voltages until breakdown finally occurs. The separation is much bigger at 200°C. 80 CHAPTER 5. DATA ANALYSIS " - 5 0 degree at 23°C * -10 degree at 23°C « +10 degree at 23°C * +50 degree at 23°C 8 9 i ? IP » I 8 1 5000 4000 3000 1000 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4 j V n (KV) _L _L _L o - 5 0 degree at 200°C « -10 degree at 200°C « +10 degree at 200°C * +50 degree at 200°C 8 S D . | t ' I I 1 8 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 V 0 (KV) Figure 5.3: Gas gain vs anode high voltage for 50, -50, 10, -10 degree. 5.1.2 Gas Gain for He - N2 Mixtures Figure 5.4 shows the gains measured as a function of high voltage for several different He — N2 mixtures at 200° C. The nitrogen concentration varies from one percent to six percent. Two data sets were obtained from two separate measurements to check the reproducibility of the results. They are in good agreement within the several percent uncertainty. These curves show the gain increasing in an exponential manner with anode voltage. In order to obtain the same gain, the high voltage required is higher for the larger nitrogen concentration. This can be explained by the nitrogen having a cooling effect on the drifting electron, absorbing energy in non-ionizing collision mode and thus decreasing the number of ionizing collisions.'7 5' For a fixed anode voltage, the gas gain increases when the nitrogen concentration is decreased. For the triton detector, a low nitrogen concentration is preferred since 5.1. IONIZATION MEASUREMENT 81 4000 3500 3000 2 2000H 1500 1000 500 charge 1st (6% N ) charge 2nd (6% r l ) charge 1st (5% N j charge 2nd (5% f l ) charge 1st (4% N j charge 2nd (4% i) charge 1st (3% N j charge 2nd (3% f f ) charge 1st (2% N j charge 2nd (2% f j ) charge 1st (1% N]f charge 2nd (1% f } ) t i t i * • • ~ i r~ 4.4 4.7 2J 2.3 .6 2.1 3.2 3.5 V 0 (KV) 4.1 Figure 5.4: Two charge tests for He — N2 mixture at 200°C. the measurement is focused on the muon capture on helium rather than nitrogen, and a lower voltage is also more compatible with the Rubidium vapour environment. However, as the nitrogen concentration is decreased, the system becomes increasingly more unstable, and the maximum electric field, where no breakdown occurs, becomes smaller. To ensure a stable operation of the T P C at high temperatures, three percent nitrogen concentration was chosen. For this gas mixture, the high voltage should be great enough to overcome the noise level, but relatively low for the Rubidium vapour environment. The data can be fitted to equation (3.49), which can then be rewritten as: l n M = p ( ^ - l ) (5.1) where p = aa. 82 CHAPTER 5. DATA ANALYSIS Figure 5.5 shows In M versus Vo for each of the different He — N2 mixtures. The six curves are separated by one percent steps in the nitrogen concentration and vary with VQ. The slopes of the curves decrease as the nitrogen concentration increases. The values of Vr and a are plotted in figure 5.6 respectively as a function of nitrogen concentration n. The highest nitrogen concentration gives the highest Vr- The resul-° charge (6% N;) • charge (5% N2) » charge (4% N2) ° charge (3% N2) x charge (2% N2) • charge (1% N2) 2 % „ 3 % 4 % 5 % 6% 1% / / y • i 1 1 i 1 1 1 1 1 i I 2.0 2.3 2.6 2.9 3.2 3.5 3.8 4.1 4.4 4.7 5.0 V 0 ( K V ) Figure 5.5: InM as a function of V0 for different He — N2 mixtures at 200°C; the solid lines are fits to equation (5.1). 05 .06 .07 Figure 5.6: Values of the threshold voltage Vr and a used in equation (5.1) as a function of helium/nitrogen gas mixtures. 5.1. IONIZATION MEASUREMENT 83 tant two parameter fit (p,Vr) is summarized in table 5.1 including a, which can be calculated from p. Using p, it is possible to calculate M(VQ) curves for various anode wire diameters using equation (3.49). The results for Yie/N2 (97:3) mixture for 10, 20, 30 /mi anode wires are shown in figure 5.7. It can be seen that M increases as the wire diameter decreases. Table 5.1: Vy, p and a for different helium/nitrogen gas mixtures. n(N2) VT(KV) P a(cm *) 0.02 1.135 4.283 337 0.03 1.177 3.571 281 0.04 1.210 3.185 251 0.05 1.322 3.249 256 0.06 1.434 3.319 261 10= 10' 10" 10' 10 charge (2a=10 /j.m) charge ( 2 a = 2 0 yum) charge ( 2 a = 3 0 fim) 2.6 2.8 3.0 3.2 3.4 V 0 ( K V ) 3.6 3.8 4.0 Figure 5.7: M as a function of VQ for different anode wire diameters. An expression can be found by fitting the curve for Vj versus nitrogen concentra-tion n. VT 0.19426 Inn + 1.0995 a (5.2) 84 CHAPTER 5. DATA ANALYSIS Figure 5.8 shows a very good linear relationship. This means Vj increases much more slowly than n. If the Townsend coefficient for the He — N2 mixture is known, then the threshold voltage Vj can be calculated using (5.2). u-|—: 1 1 1 1 1 1 r -4.0 -3.8 -3.6 -3.4 -3.2 -3.0 -2.8 -2.6 In n Figure 5.8: ^ vs. Inn for different helium/nitrogen gas mixtures. 5.1.3 Comparison of He Mixtures and Ar Mixtures The data taken for mixtures of helium with three percent of nitrogen, methane and ethane at both room temperature and 200° C as a function of high voltage are shown in figure 5.9. Methane and ethane mixtures are very similar except in the higher voltage region, where the maximum voltage for ethane is greater than that for methane. Ethane is an excellent quenching gas. Nitrogen has a higher voltage threshold, therefore it requires a higher voltage to obtain a measurable signal. The gain is smaller than that of the other two gas mixtures at the same high voltage. The gas gain increases by a factor of 1.1 as the temperature is increased from 23° C to 200°C. Figure 5.10 is a comparison of the gas gain for helium and argon based mixtures 5.1. IONIZATION MEASUREMENT 85 10' 10J 102H 10 "i 1 1 1 1 r 90 100 110 120 130 140 E/N (X 10-1 7Vcm2) He+3%C H (5atm, 23°C) He+3%CH (5atm, 23°C) He+3%N (5atm, 23°C) He+3%C H (5atm, 200°C) He+3%CR (5atm, 200°C) He+3%N,(5atm, 200°C) 150 160 170 Figure 5.9: Charge gain for He mixtures at 23°C and 200°C. i f A E/P (V c m : ' t o r r Ar+3%C H (1atm, 23°C) Ar+3%CH (1atm, 23°C) Ar+3%N (latm, 23°C) He+3%C H (5atm, 23°C) He+3%CH (5atm, 23°C) He+3%N (5atm, 23°C) 50 100 150 200 250 Figure 5.10: Charge gain for Ar and He mixtures at 23°C 86 CHAPTER 5. DATA ANALYSIS with three percent of nitrogen, methane and ethane at room temperature. The data for the argon based mixture were taken at one atmosphere compared to that for the helium based mixtures at five atmospheres. The measurements were limited by the range difference for the alpha source between the helium and argon based mixtures. The high voltage thresholds are relatively low for argon based mixtures. The argon-ethane mixture has the lowest high voltage threshold and the highest gain. A l l the curves show a linear behaviour of a. This demonstrates a linear response of the T P C up to the breakdown limit. 10 10 J 10' 10 He+3%N A r + 5 0 % C H 2 6 0 ~KX) 150 20o" E/P (V cm" ' t o r r - ' ) 50 250 300 Figure 5.11: Comparison of the standard drift chamber gas mixture with the gas mixture used in the T P C . The data taken for the argon-ethane mixture are shown in figure 5.11. The ar-gon/ethane (50:50) mixture is the standard gas mixture used in most drift chambers. It is known that, in an argon plus quencher gas mixture, increasing the amount of quencher decreases the gain so that one requires a higher voltage to restore the gain. It is difficult to compare the results from different publications. The quenching gas concentration, electronics response, noise level, anode wire size and quality and other parameters rarely have equal values in different experiments. After using the Garfield 5.1. IONIZATION MEASUREMENT 87 program to simulate the field configuration in the literature,' 1 0 5 ' and finding the equiv-alent surface field of the anode wire, the gain for this mixture gives the same order of magnitude as the literature value. For example, the gain is 2.73 x 103 with a surface field of 209 K V / c m for our T P C compared to a gain 1.1 x 103 with a surface field of 209 K V / c m for the chamber described in the literature.' 1 0 5 ' 5.1.4 Energy Resolution Since the 2 4 1 Am source emits a monoenergetic alpha beam, the width of the charge histogram provides a quantitative measure of the energy resolution of the detector. Figure 5.12 shows the charge and amplitude histograms obtained at five atmospheres pressure with a helium/nitrogen (97:3) mixture at a gas gain M = 185. The total counts can be obtained by integrating the area under the entire spectrum. Since the number is large, the fluctuations can be treated as Gaussian. The energy resolution is defined as R = J _ = 2 ^ Qo Qo where Q0 is the peak centroid, T is the full width at half maxmium of the peak, and a is the standard deviation of the peak. The energy resolution is governed mainly by the statistical fluctuations in the number of ion pairs produced and the fluctuations in the multiplication process. The resolution for this data set is 15 percent(FWHM). Figure 5.13 shows the energy resolution as a function of emission angle at temper-atures of 23°C and 200° C. There is a deterioration of the resolution by about seventy percent with increasing temperature. 88 CHAPTER 5. DATA ANALYSIS C O (J 3 5 3 0 2 5 2 0 15 1 0 e n t r i e s = 3 0 0 mean = 6 0 . 4 6 4 resolut ion = 1 5 % j i I I i I i I i U i I i L_i I i i L 9 0 8 0 7 0 6 0 5 0 4 0 3 0 2 0 1 0 11 e n t r i e s = 3 0 0 mean = 2 2 . 5 0 3 resolut ion = 1 5 % j i LJJ LU i I I_I i I i i i i L 0 2 0 4 0 6 0 8 0 1 0 0 c h a r g e 0 2 0 4 0 6 0 8 0 1 0 0 a m p l i t u d e Figure 5.12: Signal charge and amplitude histogram for a 50 degree emission angle at 200°C for a He/N2 (97:3) mixture and M=185. 5. J. IONIZATION MEASUREMENT 89 -10 0 10 angle (degree) 2 0 3 0 4 0 5 0 Figure 5.13: Energy resolution vs angle at 23°C and 200°C. 5.1.5 Breakdown As discussed in chapter 3.5.4, the breakdown phenomena are related to the photon feedback process. The excited atomic levels of carbon C*(6.43, 7.46 and 7.94 ev), nitrogen iV*(8.3, 10.0 and 10.3 ev) or hydrogen if*(10.2 ev), etc., are usually respon-sible for photon emission.t106' These elements are typical constituents of gases used in detectors. For the triton detector with He — N2 mixture, the excited N2(N2 and iV~2+) emits light at 316, 337, 358, 381 and 406 nm. The ultraviolet light yield for the N;(C3UU) - N;(B3ug) + hv(uv) transition depends on the rates of the following reactions which involve N2 He* + N2 - > Ar 2 * (C 3 I i u ) + He He* + He + N2 -> He*2 + N2 He\ + N2 -* 2He + JV2 He* + N2 -• N;(B3UG) + He 90 CHAPTER 5. DATA ANALYSIS N;(CSUU) + N2 -> N;(B3U9) + w 2 where N%(C3IIU) and 7V£ (B3HG) are two different energy levels for nitrogen molecule. The classification of the electronic states for the molecules is analogous to the classi-fication for the atomic energy states. In a diatomic molecule, there is axial symmetry about the internuclear axis. As a consequence, the component of the orbital angular momentum of the electrons about the internuclear axis is a constant of the motion. Then the corresponding molecular state is designated a S, II, A , $, ... state. The electronic states of nitrogen molecule are distinguished by a letter X(ground state), a, b, c, ...(excited singlet states), A, B, C , ...(excited triplet states), whose order is usually the order of increasing energy. The subscripts, g and u are even and odd parity states respectively. The first of these reactions is responsible for the increase in U V yield, whereas all the others quench the U V yield. The U V light emission by the avalanches can produce photoelectrons from the metallic electrodes or from the walls. These photoelectrons trigger secondary avalanches. The positive ions created in the avalanche can extract electrons from the cathode when they neutralize there, generating secondary avalanches. The contribution of secondaries to the signal may cause the following undesirable effects:!76' • the good localization of the initial avalanche is worsened as a result of broadening of the avalanche lateral size • the signal extends in time more than a single avalanche signal • the energy resolution deteriorates due to the large contribution to the signal from the secondary processes • the positive feedback process makes the amplification unstable and limits the maximum attainable gain Figure 5.14 shows the total charge as a function of nitrogen concentration for He — N2 mixtures near breakdown. The breakdown mechanism is dominated by the 5.2. DIRECTION MEASUREMENT 91 Quencher C o n c e n t r a t i o n Figure 5.14: Charge at breakdown vs quencher concentration for different gas mix-tures. total amount of charge, up to 10 1 0 electrons in the avalanches.'107' This limit depends slightly on the concentration of the quench gas and on the gas composition. Adding more nitrogen helps to make the chamber operation more stable. When the anode voltage is increased to the breakdown value, an increasing current appears. The current quickly becomes unstable and a spark occurs. The Bragg shape of the pulses is distorted due to the space charge effect. This is not suitable for the T P C operation. 5.2 Direction Measurement 5.2.1 Full Width vs. Angle and Temperature Figure 5.15 shows the width histogram at 200°C. It was fitted to a Gaussian function to find the centroid and width of the peak. The resolution is 13% (FWHM). The data for the pulse width as a function of emission angle of the alpha particles at room temperature and 200°C are shown in figure 5.16. The width is defined as the 92 CHAPTER 5. DATA ANALYSIS o 1OO 150 w i d t h w i d t h Figure 5.15: Signal width histogram with 9 = 50 degree at 200°C for a He/N2 (97:3) mixture and M=185. full width of the anode pulse in units of fis. For an ideal case: Wi = - | s i n 0 | (5.4) w where L is the track length of the alpha particle, w is the drift velocity of the electrons, 9 is the emission angle. Clearly, the width, Wt, increases with increasing j sin#|. The data for the width as a function of temperature are shown in figure 5.17. The width is almost independent of the temperature within the big error bar. In the triton detector, the width depends on the gas nature, T P C geometry and drift field distribution, etc. It is obvious that the width is not sensitive enough to use for the determination of the emission angle. Therefore, it is necessary to extract directional information by fitting the pulse shape for each event. 5.2. DIRECTION MEASUREMENT 93 150-135 -120-105 -— 90 -d 75 -X 60 -tim 45 -30 -15 -0 --1 I I I I I I I I 1 I ° width (23°C) 1 1 - 6 0 - 5 0 - 4 0 - 3 0 - 2 0 - 1 0 0 10 20 30 40 50 60 angle (degree) - 6 0 - 5 0 - 4 0 - 3 0 - 2 0 - 1 0 0 10 20 30 40 50 60 angle (degree) Figure 5.16: Signal width vs angle at 23°C and 200°C. 50 100 150 200 temperature (°C) 50 100 150 200 temperature (°C) 250 50 100 150 200 temperature (°C) 50 100 15^ 200 temperature (°C) 250 Figure 5.17: Signal width vs. temperature for +50, +10, -10, -50 degree. 94 CHAPTER 5. DATA ANALYSIS 5.2.2 Pulse Fitting The detailed time structure of the anode pulse contains the directional information of the ionization event. In this section, the fitting procedure is focused on the He/A^ (97:3) mixture, although the method can be extended to other mixtures. When an ion passes through the gas in the detector, it is stopped in the drift region, leaving ion-electron pairs along the track. The ionization density distribution can be obtained from equations (3.14-3.17). The related parameters are given in table 3.1 and table 3.2. The avalanche takes place in the region closest to the anode wire, where the electric field is inversely proportional to the distance from the wire. The contribution to the final signal from the primary electrons can be neglected due to the gain. (M = 185 for the normal operation) The anode pulse shape depends on the positive ions drifting away from the anode wire as shown in chapter 3, section 3.6.2. The ion cloud moves along the local field lines, starting close to the avalanche wire and ending up on the cathode wires. The voltage input to the preamplifier at time t is given by equation (3.77), where k is proportional to n+. k should be constant for all events unless the operating parameters change. Any perturbation of the electrostatic field of the chamber caused by the ions is neglected. The theoretical pulse shape is convoluted with the amplifier response function. In order to apply equation (3.65), we must first differentiate equation (3.77) with respect to t, dVi _ QK 2 cosh \nt - 1 ^ At STTEQI sinh | / d When the ions are very close to the wire, equation (5.5) has to be modified by the introduction of a time constant to with an order of magnitude of ns. Since the signals were only sampled each 100 ns, the actual moment of the avalanche could not be determined more precisely. The final output voltage pulse shape can be found using equation (3.65). 5.2. DIRECTION MEASUREMENT 95 entries^ 300 i eon — 0.5758 igle = 50 degree , n n -O. 5 s k: e w m < ;ntries = 3QO -mean O. 1 1 07 Dngle = — 50 dogri -O. 5 Figure 5.18: Skewness histograms for 9 = 50 degree and 9 = —50 degree. This procedure was programmed for a computer fit to determine the emission angle for a given pulse using the least-squares method.t10^ The pulses can be separated into two groups with positive and negative angles by calculating the skewness, skew, which is defined in the usual way: skew(V1...VN) = ^ J2(^~) (5-6) »=i ^ ' where the distribution's standard deviation, TV and the mean of the distribution is given by (5.7) V = ^ f^- (5.8) If skew > 0, then 9 > 0. If skew < 0, then 9 < 0. Figure 5.18 shows the skewness histograms for 9 — 50 degrees and 9 = —50 degrees. They can be distinguished from each other very well. 96 CHAPTER 5. DATA ANALYSIS The pulse shape was fitted based on the mathematical model of the ionization along the ionization track and the behaviour of the detector. The advantage of the fitting technique is that the pulse is characterized by parameters which have phys-ical meaning. Also the parameters included in the function can be chosen a priori based upon experimental conditions. These parameters can become a powerful tool for systematic studies. A subroutine was written for the initial ionization density distribution calculation in the drift region. This provides a numerical table of energy loss vs travel distance. This distribution has to be projected onto the anode plane as a function of time. The projection depends on the drift velocity and emission angle. Hence, the final fitting function is an implicit function of the emission angle. The theoretical anode pulse shape is determined by the induced signal from the ions, and further modified by the readout electronics. The RC-coupled amplifier has a limited rise time. The effect can change the simple RC transient response function r(t), and give a leading edge to the input pulse. Instead of characterizing the amplifier by its complex dependence of gain on frequency, it is more convenient to find r(t) by a mixture of a differential and an integral circuit. The shape of r(t) is shown in fig-ure 5.19. The frequency band width is determined by two time constants ts and tj, which related to the cut-off at high and low frequencies respectively. The response function is given by: r(t) = - e _ 7 7 (5.9) where ts = 300 ns and tj = 20 ns for the circuit simulation. Combining equations (3.65), (5.9) and (5.5) gives the final pulse amplitude: = c f< / (e-ff _ e - ^ m , i ) d t (5.10) J0 y sinh^/d J where C is an adjustable constant related to the pulse amplitude. The effect due to the motion of ions in the electric field provided by the wire planes contributes to the hyperbolic term for the integral function. The electronic response to this effect produces the exponential term. Q is proportional to S/ sin 6, where S is the electronic stopping power(see section 3.3). The ionization distribution can be considered as a 5.2. DIRECTION MEASUREMENT 97 0 100 200 300 400 500 600 700 800 900 1000 t ( n s ) Figure 5.19: Transient response of a RC coupled amplifier used in the circuit simula-tion. sum of discrete charges, and the superposition of the discrete charges determines the pulse shape. The calculation has to be done by my computer program(see Appendix A) . This fitting function (5.10) can be evaluated numerically. Figure 5.20 shows fits to two pulses with 9 = ±50 degrees. The sharp leading edge for 0 = —50 degrees is due to the size of the collimator. The fitting of the tail of the pulses is not very good. This can be explained due to the fact that this contribution to the signal comes from the drift electrons close to the edge of the drift region where the electric field is deformed. The overshooting part of the pulse is due to the T F A , and is not well fitted. Since the threshold for the noise was about 15 divisions, the pulse fitting starts and stops at this level. The floating parameters for the fits are the starting time of the 98 CHAPTER 5. DATA ANALYSIS Figure 5.20: T P C pulses for two ionization events. The solid lines are the raw data. The dashed lines are the fits to the pulses. a)shows the anode signal for 9 = 50 degrees. b)shows the anode signal for 9 = —50 degrees. 5.2. DIRECTION MEASUREMENT 99 Figure 5.21: T P C pulses for four ionization events. The solid lines are the raw data. The dashed lines are the fits to the pulses. a)shows the anode signal for 0 = 30 degrees. b)shows the anode signal for 9 = 10 degrees. c)shows the anode signal for 9 = —30 degrees. d)shows the anode signal for 0 = —10 degrees. 100 CHAPTER 5. DATA ANALYSIS pulse and the emission angle 0. The pulse amplitude can be fixed or varied. The drift velocity w and the time constant ts and tj of the detector and electronics are fixed at values determined by making several fitting passing over a portion of the full data set. The value of w is in agreement with the calculation value using the program, DRIFT_VEL. w = 2.7 mm/ps is for the normal operating condition for the T P C . The fits for 6 = ±30, ±10 degrees show a fair agreement as shown in figure 5.21. For high electric fields the ion drift velocity is lower than the value obtained by assuming a constant mobilityM09^ The deviation from a constant mobility occurs only very close to the wire. Within one sampling interval the ions will be out of this region. The quality of the fit is expressed by X2 = 2 " } (5-11) where Aexpt — A\lt is the difference between the measured and the fitted value of the signal and a is the sigma of the measured signal samples due to noise. The values of X2 for the six pulses shown in figure 5.20 and figure 5.21 are summarized in table 5.2. Table 5.2: \ 2 f ° r s ^ x pulses. emission angle (degree) x 2 X2/N 50 83.3 0.824 30 58.7 0.776 10 53.2 0.858 -10 45.1 0.959 -30 58.3 0.870 -50 63.7 0.838 Generally, time projection chambers record signals from the sector arrays of anode wires and cathode pads together with the drift time of the ionization electrons, and they can reconstruct many simultaneous three dimentional tracks. Our drift chamber 5.2. DIRECTION MEASUREMENT 101 has only one anode. The track fitting scheme discussed in this thesis is a much more cost effective way of reconstructing a single track. 5.2.3 Angular Resolution Suppose that each data point A*xpt has a measurement error crt- that is independently random and distributed as a Gaussian distribution around the model A[xt. Then the probability of the data set is the product of the probabilities of each point. ) 2 (5-12) i = l where pj are the parameters to be fitted in the model. The optimum values of pj are obtained by minimizing x2 with respect to each parameter using equation (5.11), This equation is a set of coupled equations with k unknown parameters pj. x2 n a s to be treated as a continuous function of the k parameters, describing a hypersurface in a k-dimensional space, where the minimized value of x2 c a n be- searched. For a sufficiently large data set, small random deviations converge to a Gaussian distribution for each parameter centered on the values, p'j, that minimize %2, P{pi) = Aexp (-iP3~Jj)2^J (5-14) where A is a function of the other parameters except pj. Combining equations (5.11), (5.12) and (5.14) gives, X2 = % ^ + C (5.15) where C is a function of <r,- and the parameters pi for / ^ j. Thus x2 va,ries as the square of distance from a minimum, and an increase of one cr in the parameter 102 CHAPTER 5. DATA ANALYSIS from p'j at the minimum increases x2 by 1. The measurements fall within ±cr of the true value 68 percent of the time. The contour plot of x2 a s a function of two parameters is generally approximately elliptical at a constant value of x2 = Xmin + 1 near the minimum. To find the overall limits on a single parameter, it is necessary to reminimize the function at each step with respect to all the other parameters, corresponding to the new value of the parameter. The C E R N program MINUITt 1 1 1 ! was used for minimizing x2 a n d estimating the uncertainty in the best parameter values. MINUIT was conceived as a tool to find the minimum value of a multi-parameter function and analyze the shape of the func-tion around the minimum. A subroutine called F C N can be defined by the user to calculate x2 a s defined in equation (5.11). Using MINUIT commands, the user can request MINUIT to minimize F C N with respect to the parameters, that is, to find those values of parameters which give the lowest value of x2• One-standard-deviation value is defined by setting E R R to 1. The MINUIT processor M I G R A D is the best minimizer for nearly all functions. The minimizing technique in M I G R A D is a stable variation of the Davidon-Fletcher-Powell variable-metric algorithm, which converges to the correct error matrix as it converges to the function minimum. M I G R A D is a method with inexact line search, a stable metric updating scheme, and it also checks for positive-definiteness. Its main weakness is that it depends heavily on a knowl-edge of the first derivatives, and fails miserably if they are very inaccurate. The first derivatives are calculated using a five-point-approximation formula^108] by modifying FCN(see Appendix). MINUIT needs to be guided to the right minimum with the F I X and R E L E A S E commands. The MINUIT processor MINOS can calculate parameter errors taking into account both parameter correlations and non-linearities. It can only operate once a good minimum has already been found, and the error matrix has been calculated, so the MINOS command normally follows a M I G R A D command. The predictions of MINOS give reliable information about parameter errors. When the problem is very non-linear (i.e. F C N is far from a quadratic function of its param-eters), MINOS is needed in order to indicate the correct parameter errors. MINOS calculates the errors by tracing the contours of the function. 5.2. DIRECTION MEASUREMENT 103 0.05 0.04 0.03 H 0.02 (/) 3.-1 -0.01 -0.02 --0.03--0.04 -0.05 a, =0.027 us tp r ff(=-0.82 deg (7, =-0.034 us tn r (7. =0.88 deg 48.0 48.5 49.0 49.5 50.0 50.5 510 515 52.0 0 (degree' Figure 5.22: The contour of one sigma error for emission angle 9 and pulse starting time t. 6 = (49.8±g;l) degree, (t = 0.00±g;g|) (is. 104 CHAPTER 5. DATA ANALYSIS Table 5.3: Angular resolution of the T P C for six pulses. The resolution is defined as one standard deviation errors. The Minuit follows the \ 2 function out from the minimum of x2 to get a contour of x2 + 1 f ° r two free parameters 9 and t. The error intervals are assymmetric due to nonlinearities. emission angle (degree) resolution (degree) negative error positive error 49.8 -0.8 +0.9 29.4 -0.6 +1.0 10.5 -0.5 +0.6 -9.7 -1.1 +1.2 -29.8 -0.9 +0.5 -49.7 -0.9 +1.4 The pulse fitting parameters are the emission angle 0, the starting time t, and the pulse amplitude A. Figure 5.22 shows the contour for the variation of x2 a s a function of 9 and t. MINOS holds A at its optimum value and varies 9 and t to obtain increases in x2 by 1 from the minimum value. Because the pulse fitting problem is highly nonlinear, this contour is not exactly on ellipse. The extreme contour points, whose x-coordinates, relative to the x-coordinate at the minimum(#), are the negative and positive errors of 6 respectively. These errors are defined as the angular resolution. The angular resolutions for different 9 values are summarized in table 5.3. They are all close to ± 1 degree, which shows the very good resolution of the drift chamber detector. This high quality of the fits confirms our ability to reconstruct the event track parameters from the pulse shapes. 5.3. COMPARISON BETWEEN THE TPC AND THE ION CHAMBER 105 5.3 Comparison Between the T P C and the Ion Chamber The operating conditions for the T P C are summarized in table 5.4. Table 5.4: The operating conditions for the T P C . Mechanical parameter anode wire diameter 25.4 um cathode wire diameter 127 um anode — ground plane distance 6.35 mm ±0.02 mm drift field distance 9.9 cm ±0.1 cm Electrical parameter anode voltage 2.90 KV ±0.01 kV cathode voltage -4.20 KV ±0.01 kV ground voltage 0.00 KV ±0.01 kV Electronics parameter preamplifier rise time 20 ns TFA integration time 200 ns waveform digitizer range 30.0 us ±0.1 fis gas parameter gas mixture He, iV2(3%) < 1% gas pressure (23 °C ) 5 atm < 1% temperature parameter temperature 200 °C ± 1 °C The first measurement of the spin dependence of reaction (1.7) was performed at the T R I U M F cyclotron based on a combined polarized target/ionization chamber, which was designed and built by our Princeton collaborators.'11^ The chamber was inside a stainless steel vessel, about 5 liters in volume, filled with helium, nitrogen and Rb metal. The entire device was enclosed in an oven and operated at about 200°C, resulting in a Rb density of about 10 1 5 c m - 3 . Muons that stopped in the target formed muonic helium atoms, which were then polarized by collisions with optically pumped Rb vapor. Two high powered GaAlAs Diode laser arrays were used to polarize the Rb in the 100 cm 3 fiducial volume located inside the ion chamber. The energy and recoil direction of the tritons were measured by the ion chamber. The ion chamber consisted of three wire planes: cathode, grid and anode. The direction of 106 CHAPTER 5. DATA ANALYSIS the tritons was determined by analyzing the shapes of the anode pulses, which then were fitted to a mathematical model using a least squares method. Two typical pulses are shown in figure 5.23. The upper pulse has a faster rise time at the beginning and a slower rise time near the peak. This indicates that the triton is travelling toward the anode. The lower one is reversed, and indicates that the triton is travelling away from the anode. The dotted lines are the fits to the muon and the triton pulses. The fit determines the value of cos 9 and whether the triton travels towards or away from the anode. The maximum voltage peak from the anode depends on the energy of the triton. Compared to the ion chamber, the T P C has many advantages: Firstly, The operation voltage of the ion chamber is about 1200 volts at the cath-ode. The low voltage means that very careful attention has to be paid to the detection low-noise signal. The fluctuation of the primary ionization dominates the energy res-olution and sets the upper limit at about 30%.' 3 5 ' For the T P C , the higher gas gain gives much better energy resolution which is improved by a factor of two. Secondly, it is easy for T P C pulse to distinguish between the positive and negative direction due to the obvious Bragg shape, which can even be seen by eye without any difficulty. However, for the ion chamber, a slightly increase in the noise make it very difficult to distinguish positive or negative emission angles. Careful shielding and a very clean power supply through an isolation transformer has to be used to minimize the noise. Each pulse was fitted assuming that it could be either positive or negative due to the ambiguous curvature of the leading edge as shown in figure 5.23. Pulses with small values of A ^ 2 had to be rejected, because the noise could distort the curvature. Thirdly, the angular resolution for the T P C is about ± 1 degree which is a consider-able improvement over the ± 4 ' 1 2 ' degrees resolution achieved for ion chamber. Again, this demonstrates that the T P C is a more promising candidate for improvement in the measurement of Ay, and hence Fp. 5.3. COMPARISON BETWEEN THE TPC AND THE ION CHAMBER 107 3 0 50 100 150 200 250 300 Q Time (100 us Full Scale) r • i i ' I i ' i L_J I L _ l I I 1 I I I I I I I I I I I I U - J 0 50 100 150 200 250 300 Time (100 us Full Scale) Figure 5.23: Ion chamber pulses for two triton events, a) shows the inner anode signal for a triton moving toward the cathode, b) shows the inner anode signal for a triton moving away from the cathode. The solid line is actual data, whereas the cross-hatched and shaded regions show the fits to the muon and the triton signals respectively. The sum of the two fits, which closely follows the data, is shown with a dashed line. Chapter 6 Conclusion Generally, helium based gas mixtures are used less often in wire chambers, because of their low drift velocity and low ionization, which can limit the precision of the angular and energy resolution of the detector as compared to argon based gas mixtures. The high temperature operation with the Rb vapor in the present T P C detector caused the high voltage breakdown problems. The high gas pressure and high temperature operating conditions made the design considerations very difficult. Although the ex-perimental requirements are very complicated, it was a significant achievement to get the first helium high temperature T P C running at T R I U M F . Determining the emis-sion angles from the anode signals is one of the critical and more difficult tasks. The pulse fitting program developed in this present work has included almost all aspects of the physics taking place inside the wire chamber such as electron-ion pair production and distribution, electron drift and multiplication in the gas mixture, electric field configuration, ion motion induced signal and finally the electronic response of the cir-cuit. The integration of all these factors has given a very nice fit to the anode pulses, although this work is time consuming. As has been discussed in the last section, a high temperature T P C has many advantages over an ion chamber. A very good en-ergy resolution and angular resolution can be achieved while the T P C is operated at 200°C. However, the Rb problem remains to be solved. A number of improvements 108 109 have been suggested for the development of the T P C . Firstly, the distance between the ground and the anode plane can be decreased. This helps to lower the anode voltage. However, a thinner stainless steel frame can not keep its shape and will allow the wire plane to lose tension. In addition, spot welding doesn't work. A good idea would be to replace the wires with mesh. With a simple mechanical design using very small screws to clamp the mesh between two piece of stainless steel frames, the two wire planes can be very close. At the same time, the drift field length can also be reduced by a factor of 0.3. In addition, the mesh has a larger capacitance than the wire plane. This should also decrease the operating voltage. Secondly, the size of the detector should be larger in order to minimize the edge effect and increase the wire plane capacitance resulting in further improvements to the resolution and a decreased operating voltage, respectively. Thirdly, the preamplifier can be modified to give a balance for rise and fall time, thus improving the angular resolution of the T P C . Fourthly, a refinement for the pulse fitting is to introduce a reponse function to simulate the preamplifier and express this function for a step function input in terms of three time parameters and one amplitude parameter.'1 1 0' A new experiment using the technique described in this thesis is planned for the near future. Other improvements should allow us to make a much more accurate measurement of Fp. • Powerful new diode laser arrays, up to 150 Watts, are now available. Thus, a much higher muon polarization can be achieved. • Design a chamber incorporating more segmentation of the electrodes. This will help to determine the event position resulting in a significant decrease of the systematic uncertainties. • More running time will decrease the statistical uncertainties. The data sample 110 CHAPTER 6. CONCLUSION from the last E683 running at T R I U M F only represented four days running due to the complexity of the whole experiment. Thus, it is practical to obtain a large increase in the number of events. From section 2.3.1, the relative error for the vector analyzing power Av is the sum of the relative errors for the muonic 3He vector polarization P„, the triton asymmetry as defined in equation (2.44) and cos#. A higher laser power will certainly help to improve the Pv measurement accuracy due to the larger polarization. The E683 data' 3 5 ' gave only 11 % for Pv, which is less than the upper theoretical limit, 26.8 %. This is due to the attenuation of the light intensity inside the fiducial volume. A larger polarization will, of course, decrease the relative error for Pv. This error also depends on the Monte Carlo simulation for the electron decay asymmetry data which takes account of the detailed experimental geometry of the detection system. The triton asymmetry measurement contributed an additional 9 % error due to the ambiguous curvature problem in distinguishing positive and negative emission angles for the Princeton ionization chamber. Our detector can totally eliminate such problems with the obvious Bragg shape. The uncertainty in cos 9 is proportional to the angular resolution. With our drift chamber, the angular resolution has been improved by a factor of 4. If the final error source comes from the angular resolution only, the relative error for Av can be obtained within about 1 % accuracy. This will produce an Fp value with about 7 % accuracy compared to the best Fp value of about 13 % accuracy.'4 5' For the Princeton ionization chamber, the best possible accuracy limit for Fp value is about 15 %. The possibility of extracting an Fp value with a very good accuracy has been considered a great advantage for our detector. The technique presented here will play an important role in future studies of Fp and gp and should eventually result in measurements which will quantitatively test our understanding of QCD at low energy. Appendix A Program for the Pulse Fitting e x t e r n a l f e n o p e n ( u n i t = 1 , f i l e = ' m a i n . d a t ' , s t a t u s = ' o l d ' ) o p e n ( u n i t = 2 , f i l e = ' m a i n . o u t ' , s t a t u s = ' o l d ' ) o p e n ( u n i t = 3 , f i l e = ' r p 5 0 . d a t ' , s t a t u s = ' o l d ' ) o p e n ( u n i t = 4 , f i l e = ' e l o s t . d a t ' , s t a t u s = ' o l d ' ) o p e n ( u n i t = 5 , f i l e = ' f i t . d a t ' , s t a t u s = ' o l d ' ) c a l l m i n t i o ( 1 , 2 , 6 ) c a l l m i n u i t ( f c n , 0 ) s t o p end s u b r o u t i n e f c n ( n p a r , g r , f v , x v , i f l a g ) i m p l i c i t d o u b l e p r e c i s i o n ( a - h , o - z ) p a r a m e t e r ( i a = 3 0 0 , i b = 4 , i g = 2 0 ) d i m e n s i o n g r ( * ) , x v ( * ) d i m e n s i o n a ( i a ) , b ( i a ) , p x ( i b ) , p y ( i b ) d i m e n s i o n d a ( i a ) , d b ( i a ) , d c ( i a ) , d d ( i a ) , d e ( i a ) , d f ( i a ) i n c l u d e ' p . d a t ' i f ( i f l a g . e q . 1 ) t h e n do j = l , i g r e a d ( 3 , ' ( l O f 6 . 0 ) ' ) ( d a ( i ) , i = l , i a ) end do r e a d ( 4 , ' ( i 7 , 3 x , f 1 5 . 8 ) ' ) ( a ( i ) , i = 2 , 2 1 0 ) do i = l , i a 111 112 APPENDIX A. PROGRAM FOR THE PULSE FITTING i f ( d a ( i ) . g e . t h ) t h e n k l = i go t o 10 end i f end do 10 t l = 1 . 1 5 d 2 do i = k l , i a i f ( d a ( i ) . l t . t h ) t h e n k r = i go t o 50 end i f end do 50 h = d a ( k l ) do i = k l , k r i f ( d a ( i ) . g t . h ) t h e n h = d a ( i ) k m = i end i f end do i = k m do w h i l e ( d a ( i ) . g e . t l ) i = i - l end do k l = i + l i = k m do w h i l e ( d a ( i ) . g e . t l ) i = i + l end do k r = i - l w = k r - k l q=a0 t = a 0 do i = k l , k r q = q + d a ( i ) - y t = t + ( d a ( i ) - y ) * f l o a t ( i ) end do t = t / q x=a0 do i = k l , k r x = x + ( d a ( i ) - y ) * ( f l o a t ( i ) - t ) * ( f l o a t ( i ) - t ) end do x = d s q r t ( d a b s ( x / q ) ) 113 sk=aO do i = k l , k r s k = s k + ( d a ( i ) - y ) * ( f l o a t ( i ) - t ) / x * ( f l o a t ( i ) - t ) / x * ( f l o a t ( i ) - t ) / x end do s k = s k / q t n = u / b n a k = v / t n a ( l ) = e i n l = k r n r = 2 * k r - k l n m = 2 * k r - k m end i f a g = x v ( l ) * p i / l . 8 d 2 d x = a l * d / d s i n ( a g ) l c = i n t ( p + r / d c o s ( a g ) ) do i = l , i i x = d x / c * f l o a t ( i ) + f l o a t ( l c ) i f ( i d i n t ( x ) . g e . i i ) go t o 100 do j = l , i b p x ( j ) = f l o a t ( j + i d i n t ( x ) - 1 ) p y ( j ) = a ( j + i d i n t ( x ) - l ) end do c a l l p o ( p x , p y , i b , x , y a , a e ) b ( i ) = y a end do 100 i = i - l do i l = i , 2 , - l d e ( i l ) = ( b ( i l - l ) - b ( i l ) ) * o / g end do do i j = l , i a d d ( i j ) = a 0 b ( i j ) = a 0 end do d t = a l t = t n do j = l , i m m2= l aa=a2 x l = e x p ( a k * t ) x 2 = l . d 0 / x l y l = ( x l - x 2 ) * a 2 y 2 = ( x l + x 2 ) * a 2 114 APPENDIX A. PROGRAM FOR THE PULSE FITTING a b = a 2 * d t * ( e x p ( a 3 * d t / t s ) - e x p ( a 3 * d t / t f ) ) * ( 2 . d 0 * y 2 + a 3 ) / y l do m = l , i i d=aO do m l = l , m 2 t a = d t * a a * f l o a t ( 2 * m l - l ) t l = d t - t a x l = e x p ( a k * ( t a + t ) ) x 2 = l . d 0 / x l y l = ( x l - x 2 ) * a 2 y 2 = ( x l + x 2 ) * a 2 t 2 = d t * a a * ( e x p ( a 3 * ( t l + t ) / t s ) - e x p ( a 3 * ( t l + t ) / t f ) ) + * ( 2 . d 0 * y 2 + a 3 ) / y l d = d + t 2 end do d f ( j ) = a 2 * x + d i f ( d a b s ( ( d f ( j ) - x ) / x ) . g t . e e ) t h e n x = d f ( j ) a a = a 2 * a a m2=m2*2 e l s e go t o 200 end i f end do 200 d t = d t + a l end do i f ( s k . g t . a O ) t h e n i c = l d d ( l ) = d e ( i ) * e e l s e d t = a l do m = i c , l , - l d d ( m ) = d e ( i ) * d e x p ( - l . * d t / s * d t / s ) d t = d t + a l end do end i f do m = i , 2 , - l j = i - m + i c + l d d ( j ) = d e ( m ) end do d t = a 0 m 3 = l x=a0 i f ( s k . g t . a O ) t h e n i l = i d i n t ( b 2 * x v ( l ) * x v ( l ) + b l * x v ( l ) + b 0 ) do m = i l , i e d d ( m ) = d d ( i l ) * d e x p ( a 3 * d t / t p * d t / t p ) d t = d t + a l end do do j = k l , k r d b ( j ) = a O do m 4 = l , m 3 d b ( j ) = d b ( j ) + d f ( m 4 ) * d d ( m 3 - m 4 + l ) end do m3=m3+l i f ( d b ( j ) . g t . x ) t h e n x = d b ( j ) j j = j end i f end do e l s e i l = i d i n t ( c 2 * x v ( l ) * x v ( l ) + c l * x v ( l ) + c 0 ) do m = i l , i e d d ( m ) = d d ( i l ) * d e x p ( a 3 * d t / f * d t / f ) d t = d t + a l end do i l = i d i n t ( d 2 * x v ( l ) * x v ( l ) + d l * x v ( l ) + d 0 ) do m = i l , i e dd(m)=aO end do i l = i d i n t ( e l * x v ( l ) + e O ) do j = n l - i l , n r b ( j ) = a O do m 4 = l , m 3 b ( j ) = b ( j ) + d f ( m 4 ) * d d ( m 3 - m 4 + l ) end do m3=m3+l end do do j = n l - i l , n r i 2 = 2 * n l - j d b ( i 2 ) = b ( j ) i f ( d b ( i 2 ) . g t . x ) t h e n x = d b ( i 2 ) J j = i 2 end i f APPENDIX A. PROGRAM FOR THE PULSE FITTING end do end i f do j = k l , k r i 2 = j " ( j j - k m ) d c ( i 2 ) = d b ( j ) end do do j = l , k l - l d c ( j ) = y end do do j = k r + l , i a d c ( j ) = y end do do j = k l , k r d c ( j ) = d c ( j ) / x * x v ( 3 ) + y i f ( d c ( j ) . l t . y ) d c ( j ) = y end do i d = i d i n t ( x v ( 2 ) ) do j = k l , k r x = f l o a t ( j ) + x v ( 2 ) - i d do j j = j - l , j + 2 p x ( j j - j + 2 ) = f l o a t ( j j + i d ) P y ( j j - j + 2 ) = d c ( j j + i d ) end do c a l l p o ( p x , p y , i b , x , y a , a e ) d b ( j ) = y a end do do j = l , k l - l d b ( j ) = y end do do j = k r + l , i a d b ( j ) = y end do i f ( i f l a g . e q . 2 ) t h e n g r ( k l ) = ( - 2 . 5 d l * d b ( k l ) + 4 . 8 d l * d b ( k l + l ) - 3 . 6 d l * d b ( k l + 2 ) + + 1 . 6 d l * d b ( k l + 3 ) - 3 . d O * d b ( k l + 4 ) ) / l . 2 d l g r ( k l + l ) = ( - 3 . d 0 * d b ( k l ) - l . d l * d b ( k l + l ) + 1 . 8 d l * d b ( k l + 2 ) + - 6 . d O * d b ( k l + 3 ) + 1 . d O * d b ( k l + 4 ) ) / l . 2 d l g r ( k r - 1 ) = ( - 1 . d O * d b ( k r - 4 ) + 6 . d O * d b ( k r - 3 ) - 1 . 8 d l * d b ( k r - 2 ) + + I . d l * d b ( k r - l ) + 3 . d 0 * d b ( k r ) ) / 1 . 2 d l g r ( k r ) = ( 3 . d 0 * d b ( k r - 4 ) - 1 . 6 d l * d b ( k r - 3 ) + 3 . 6 d l * d b ( k r - 2 ) + - 4 . 8 d l * d b ( k r - l ) + 2 . 5 d l * d b ( k r ) ) / 1 . 2 d l do j = k l + 2 , k r - 2 117 g r ( j ) = ( d b ( j - 2 ) - 8 . d 0 * d b ( j - l ) + 8 . d 0 * d b ( j + l ) - d b ( j + 2 ) ) / 1 . 2 d l end do end i f f v=aO do j = k l , k r f v = f v + ( d a ( j ) - d b ( j ) ) * * 2 end do f v = f v / ( z * z ) i f ( i f l a g . e q . 3 ) t h e n w r i t e ( 5 , ' ( 1 0 ( d 8 . 3 , l x ) ) ' ) ( d b ( j ) , j = l , i a ) w r i t e ( 5 , * ) k l , k m , k r , h , s k , q , w end i f r e t u r n end c d a t a d r i v e n mode s e t t i t l e f i t t o a p u l s e p a r a m e t e r s 1 ' a n g l e ' 5 . d l l . d - 1 2 ' t i m e ' O.dO l . d - 1 3 ' a m p l i t u d e ' 2 . 2 8 d 2 l . d O s e t e r r o r d e f 1 s e t eps l . d - 4 s e t g r a d i e n t 1 s e t s t r a t e g y 2 f i x 2 f i x 3 m i g r a d r e l e a s e 2 m i g r a d m i n o s s t o p APPENDIX A. PROGRAM FOR THE PULSE FITTING p a r a m e t e r ( p . d a t ) t h = 1 . 2 d 2 u = 2 . 1 7 7 5 d - 2 v = 3 . 1 0 9 d - 4 d = 2 . 6 6 d - 3 t f = 2 . d - 2 t s = 3 . d - l p i = 3 . 1 4 1 6 d 0 e i = 5 . 4 8 6 d 0 . b n = 9 . 1 d 0 i i = 2 0 9 im=15 o = l . d 6 g = 2 . 8 d l p = 3 . 9 6 9 d 0 r = 5 . d 0 c = 2 . d - 4 i c = 1 5 i e = 1 2 0 e = 8 . 9 4 8 4 d - l s = 9 . d - l f = 7 . d O y = l . d 2 t p = 2 . d 0 t n = 7 . d 0 aO=0.dO a l = l . d - l a 2 = 5 . d - l a 3 = - l . d 0 b 2 = 5 . d - 3 b l = 5 . d - l b 0 = 1 . 1 5 d l c 2 = - 1 . 1 2 5 d - 2 c l = 1 . 2 d 0 c 0 = 1 . 6 1 2 5 d l d 2 = - 1 . 3 7 5 d - 0 2 d l = 1 . 9 d 0 d 0 = 2 . 2 3 7 5 d l e l = 3 . 5 d - l e 0 = - 8 . 5 d 0 z = 5 . 8 0 9 d 0 119 c i n t e r p o l a t i o n s u b r o u t i n e p o ( x a , y a , n , x , y , d y ) i m p l i c i t d o u b l e p r e c i s i o n ( a - h , o - z ) d i m e n s i o n x a ( 4 ) , y a ( 4 ) , c ( 4 ) , d ( 4 ) n s = l d i f = a b s ( x - x a ( l ) ) do i = l , n d i f t = a b s ( x - x a ( i ) ) i f ( d i f t . I t . d i f ) t h e n n s = i d i f = d i f t end i f c ( i ) = y a ( i ) d ( i ) = y a ( i ) end do y = y a ( n s ) n s = n s - l do m = l , n - l do i = l , n - m h o = x a ( i ) - x h p = x a ( i + m ) - x w = c ( i + l ) - d ( i ) d e n = h o - h p i f ( d e n . e q . O . d O ) p a u s e ' f a i l u r e i n p o l i n t ' d e n = w / d e n d ( i ) = h p * d e n c ( i ) = h o * d e n end do i f ( 2 * n s . I t . n - m ) t h e n d y = c ( n s + l ) e l s e d y = d ( n s ) n s = n s - l end i f y = y + d y end do r e t u r n end 120 APPENDIX A. PROGRAM FOR THE PULSE FITTING c e l o s t . f ( i o n i z a t i o n e n e r g y ) d i m e n s i o n d e ( 5 0 0 ) , s i ( 5 0 0 ) o p e n ( u n i t = 4 , f i l e = ' e l o s t . d a t ' , s t a t u s = ' n e w ' ) a l = 0 . 4 0 8 a 2 = 0 . 6 4 6 a 3 = 6 . 1 9 a 4 = 1 3 0 . a 5 = 4 4 . 0 7 b l = 2 . 0 9 6 b 2 = 0 . 5 3 6 b 3 = 3 1 . 3 4 b 4 = 1 8 . 1 1 b 5 = 4 . 3 7 w h e = 4 2 . 3 w n 2 = 3 4 . 6 5 e i = 5 . 4 8 6 d x = 0 . 0 0 0 2 u t = 1 0 0 0 . o = 1 0 0 0 0 0 0 . p l = 1 2 . 9 8 8 8 p 2 = 0 . 4 0 3 0 7 4 do i = l , o s l h e = a l * ( e i * u t ) * * a 2 s h h e = a 3 / e i * l o g ( l . + a 4 / e i + a 5 * e i ) s h e = l . / ( 1 . / s l h e + 1 . / s h h e ) s l n = b l * ( e i * u t ) * * b 2 s h n = b 3 / e i * l o g ( l . + b 4 / e i + b 5 * e i ) s n = l . / ( l . / s l n + l . / s h n ) s i ( i ) = ( s h e * p l + s n * p 2 ) * d x d e ( i ) = ( s h e * p l / w h e + s n * p 2 / w n 2 ) * d x * o e a = e i - s i ( i ) w r i t e ( 4 , ' ( l x , e l 5 . 8 ) ' ) e a e i = e a i f ( e a . l t . 0 . 0 0 1 ) go t o 100 end do 100 c o n t i n u e s t o p end Bibliography [1] H. 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