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My thesis

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My master’s thesis

My master’s thesis

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  • 1. ABSTRACT Title of Thesis: Magnet Displacement in the GEp-III Experiment at Jefferson Lab Degree Candidate: Philip Charles Carter Degree and Year: Master of Science,  Thesis Directed by: Edward Brash, Ph.D., Associate Professor, Department of Physics, Computer Science and Engineering The goal of the GEp-III experiment at Jefferson Lab was to measure the ratio of the elec- tric and magnetic form factors of the proton, G E p /G M p , over a range of four-momentum- transfer-squared, Q ², from . to . (GeV/c)². In this experiment, high-energy electrons struck a proton target, causing the electrons and protons to scatter. Elastically scattered protons were analyzed using a magnetic spectrometer, which consisted of three quadru- pole magnets, a dipole magnet and a series of detectors. For an accurate analysis, the absolute positions of the quadrupole magnets, which each were roughly one meter in diameter, were needed to within a few millimeters. In order to measure these displacements, a series of measurements was taken of elastically scattered electrons traveling through the spectrometer. Using knowledge of the exper- imental geometry, together with this data, the most likely absolute positions of these magnets were determined.
  • 2. Magnet Displacement in the GEp-III Experiment at Jefferson Lab by Philip Charles Carter Thesis submitted to the Graduate Faculty of Christopher Newport University in partial fulfillment of the requirements for the degree of Master of Science  Approved: Edward Brash, Chair David Heddle Yelena Prok Brian Bradie
  • 3. © 2010 Philip Charles Carter
  • 4. ii DEDICATION Dedicated to my parents Paul and Sandra, to my sister Angie, and to my brother-in-law Cale. Their support in my seeking a master’s degree and their loyalty through all of the changes in my life have been invaluable.
  • 5. iii ACKNOWLEDGMENTS First, I would like to thank Edward Brash, my advisor and the chair of my thesis commit- tee. His guidance from the very start, both in my course work at cnu and in my thesis research, was essential in bringing my degree and thesis to completion. I would also like to thank the other members of my thesis committee, David Heddle, Yelena Prok and Brian Bradie, for taking the time to review my thesis and to sit on the committee for my thesis defense. Lubomir Pentchev, the expert on beam optics for the G E p series of experiments at Jefferson Lab, many times provided guidance and answered my questions on how to per- form my research. I owe much of my understanding of the topics discussed in this thesis to Lubomir. He also provided the cosy script used to model particle motion through the magnets of the spectrometer, a core component of the set of programs I used in my thesis research. I am indebted to Andrew Puckett for his assistance, who on multiple occasions an- swered my questions and provided valuable suggestions on how to continue my research. He also provided some of the figures used in this thesis. My time at Jefferson Lab and cnu would have been much less enjoyable had it not been for my friends in Newport News, especially Micah Veilleux, Jonathan Miller, Selina Maley and Megan Friend. Micah and Jonathan helped with brainstorming when writ- ing the code for my thesis and when writing the thesis itself, and Jonathan contributed information on the history of nucleon form factor studies when I was writing my thesis. Worthy of particular mention are my longtime friends Dan Braunworth, Peter Braun- worth and Jonathan Hopfer, who have stood by me and supported me longer than most. Of each of them, the proverb holds true: “there is a friend who sticks closer than a brother.” Alberto Accardi, whom I know from Jefferson Lab and from Our Lady of Mount Carmel Church, has proved to be a valuable friend to me as well. Multiple members of the faculty and staff at cnu assisted me in the completion of
  • 6. iv my degree in one way or another, especially Mary Lou Anderson and Pam Gaddis, the former and current secretaries of the physics department, and Lyn Sawyer from the Of- fice of Graduate Studies. I would also like to thank all of the professors from whom I took classes. Of course, without the efforts of the entire GEp-III collaboration, this experiment could not have been conducted. I am grateful for the opportunity to be a part of this collabora- tion and a contributor to the research done.
  • 7. v TABLE OF CONTENTS Section Page Dedication ii Acknowledgments iii List of Tables vii List of Figures viii Chapter 1: Introduction 1 1.1 History of nucleon structure studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  1.2 The GEp-III experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  1.2.1 Experimental techniques for determining proton form factors . . . . . .  1.2.2 The experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  1.2.3 Magnet position offsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  1.3 The physics behind GEp-III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  1.3.1 Proton form factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  1.3.2 Rosenbluth separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  1.3.3 Recoil polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  Chapter 2: Methodology 17 2.1 The optics data taken . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  2.1.1 Geometry of the experimental setup . . . . . . . . . . . . . . . . . . . . . . . . .  2.1.2 Primary goal of this research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  2.1.3 Description of data runs taken . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  2.2 Equations using the optics data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  2.2.1 Setting up the equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  2.2.2 Solving the equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  Chapter 3: Analysis 29 3.1 Measuring y PAW and φPAW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  3.1.1 Method of isolating the central sieve slit hole in the y PAW direction . . .  3.1.2 Methods of performing a cut on x PAW . . . . . . . . . . . . . . . . . . . . . . . . .  3.1.3 Fitting y PAW and φPAW and estimating errors . . . . . . . . . . . . . . . . . . . .  3.1.4 Details of fitting data for each magnet setting . . . . . . . . . . . . . . . . . .  3.2 Measuring δ using paw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  3.3 Modeling magnetic fields using a cosy script . . . . . . . . . . . . . . . . . . . . . . . .  3.4 Determining the beam position . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  3.5 Using survey data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  3.6 Performing checks on the data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  3.7 Solving for the quadrupole offsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  3.7.1 Considerations in minimizing the equations . . . . . . . . . . . . . . . . . . .  3.7.2 Method used for minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  3.7.3 Estimating errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 
  • 8. vi Chapter 4: Results and Conclusion 51 4.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  Appendix A: Plots of the Data Runs 58 Appendix B: Plots of the Fits 72 Bibliography 79
  • 9. vii LIST OF TABLES Number Page 2.1 Beam optics settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  2.2 List of optics runs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  3.1 Variable x PAW cuts used for each magnet setting and beam position x MCC , and corresponding cuts on y PAW and φPAW . . . . . . . . . . . . . . . . . . . . . . . . . . .  3.2 Fixed x PAW cuts used for each magnet setting and beam position x MCC , and corresponding cuts on y PAW and φPAW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  4.1 Final results for |y 0 fp | ≤  mm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  4.2 Final error values for each variable at y fp =  . . . . . . . . . . . . . . . . . . . . . . . . .  4.3 Final results and error estimations for µp G E p /G M p . . . . . . . . . . . . . . . . . . . .  4.4 Correlation coefficients of the final results at y fp =  . . . . . . . . . . . . . . . . . . . 
  • 10. viii LIST OF FIGURES Number Page 1.1 The spectrometer arm of the experimental setup . . . . . . . . . . . . . . . . . . . . .  1.2 The hms detector array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  1.3 Diagram of hms drift chambers as viewed from the target . . . . . . . . . . . . . .  1.4 Feynman diagrams of an elastic collision via one-photon exchange, and two corresponding collisions via two-photon exchange . . . . . . . . . . . . . . . .  1.5 Schematic representation of bulk charge and current density in the proton .  1.6 Rosenbluth separation data for G E p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  1.7 Rosenbluth separation data for G M p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  1.8 µp G E p /G M p data from the first two recoil polarization experiments at Jeffer- son Lab, compared to existing data from Rosenbluth separation . . . . . . . . .  2.1 Top view of target, sieve slit collimator and focal plane . . . . . . . . . . . . . . . . .  2.2 Comparison of central sieve slit hole at two beam positions . . . . . . . . . . . . .  3.1 Measured y PAW data using a variable x PAW cut and a fixed x PAW cut, and com- bined data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  3.2 Same as Fig. 3.1, zoomed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  3.3 Measured φPAW data using a variable x PAW cut and a fixed x PAW cut, and com- bined data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  3.4 Histogram of δ from the nominal setting at the final beam position . . . . . . .  3.5 Beam position readings projected to the target for the q1 data . . . . . . . . . . .  3.6 Diagram of relevant survey data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  3.7 Fit of y tgt vs. x MCC compared to the expected slope . . . . . . . . . . . . . . . . . . . .  3.8 Two fits of φtgt vs. x MCC compared to the expected slope . . . . . . . . . . . . . . . .  3.9 Linear fit of y PAW vs. x MCC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  3.10 Linear fit of φPAW vs. x MCC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  3.11 χ 2 /N dof when holding y fp , φfp or y tgt fixed while minimizing, accounting for δ terms in the analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 
  • 11. ix 3.12 χ 2 /N dof when holding y fp , φfp or y tgt fixed while minimizing, not accounting for δ terms in the analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  3.13 Selected minimizer results of each of the five minimized variables at y 0 fp = , compared to all results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  4.1 φ0 fp for sets of equations that include the dipole setting . . . . . . . . . . . . . . . .  4.2 y 0 tgt for sets of equations that include the dipole setting . . . . . . . . . . . . . . . .  4.3 s¹ for sets of equations that include the dipole and q1 settings . . . . . . . . . . .  4.4 s ² for sets of equations that include the dipole and q2 settings . . . . . . . . . . .  4.5 s ³ for sets of equations that include the dipole and q3 settings . . . . . . . . . . .  a.1 Dipole only, beam x = . (runs  and ) . . . . . . . . . . . . . . . . . . . .  a.2 Dipole only, beam x = . (run ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  a.3 Dipole only, beam x = . (run ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  a.4 Dipole only, beam x = −. (run ) . . . . . . . . . . . . . . . . . . . . . . . . . . . .  a.5 Dipole only, beam x = . (runs  and ) . . . . . . . . . . . . . . . . . . . .  a.6 Nominal, beam x = . (run  and ) . . . . . . . . . . . . . . . . . . . . . . .  a.7 Nominal, beam x = . (run ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  a.8 Dipole plus q1, beam x = . (runs  and ) . . . . . . . . . . . . . . . . . .  a.9 Dipole plus q2, beam x = . (run ) . . . . . . . . . . . . . . . . . . . . . . . . . .  a.10 Dipole plus q3, beam x = . (run ) . . . . . . . . . . . . . . . . . . . . . . . . . .  a.11 q1 reduced, beam x = . (run ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  a.12 q2 reduced, beam x = . (run ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  a.13 q3 reduced, beam x = . (run ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  b.1 Dipole only, beam x = . (runs  and ) . . . . . . . . . . . . . . . . . . . .  b.2 Dipole only, beam x = . (run ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  b.3 Dipole only, beam x = . (run ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  b.4 Dipole only, beam x = −. (run ) . . . . . . . . . . . . . . . . . . . . . . . . . . . .  b.5 Dipole only, beam x = . (runs  and ) . . . . . . . . . . . . . . . . . . . .  b.6 Dipole plus q1, beam x = . (runs  and ) . . . . . . . . . . . . . . . . . . 
  • 12. x b.7 Dipole plus q2, beam x = . (run ) . . . . . . . . . . . . . . . . . . . . . . . . . .  b.8 Dipole plus q3, beam x = . (run ) . . . . . . . . . . . . . . . . . . . . . . . . . .  b.9 q1 reduced, beam x = . (run ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  b.10 q3 reduced, beam x = . (run ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 
  • 13.  CHAPTER 1 Introduction The GEp-III experiment was conducted at Jefferson Lab in Newport News, Virginia. The goal of the experiment was to measure the ratio of the electric and magnetic form fac- tors of the proton, G E p /G M p , over a range of four-momentum-transfer-squared, Q ², from . (GeV/c)² to . (GeV/c)². The magnetic form factor of the proton is known to a preci- sion of a few percent over this range, so determining the ratio of form factors allows the extraction of the electric form factor. In this experiment, high-energy electrons struck a proton target, in the form of liquid hydrogen, causing the electrons and protons to scatter. Scattered protons from elastic collisions were detected after passing through a magnetic spectrometer, which consisted of three quadrupole magnets and a single dipole magnet. By measuring the position and angle of the proton using a series of detectors located near the focal plane of the spectrometer, the momentum of the proton can be determined, as well as the degree to which its spin precessed in the magnetic elements. In particular, a detailed knowledge of this spin precession was a crucial component in the extraction of the form factor ratio from the data. Specifically, the absolute positions of the quadrupole magnets, each of which were approximately one meter in diameter, were needed to within a few millimeters. In or- der to measure these displacements and rotations, a series of dedicated measurements were taken of elastically scattered electrons traveling through the magnetic spectrometer, with various magnetic field strengths in the magnetic elements. This is known as beam optics data, because the scattered particles are deflected as they pass through the series of magnets in a way analogous to light passing through a series of lenses. A sieve slit col- limator placed between the proton target and entrance to the first quadrupole magnet
  • 14.  of the spectrometer allowed only electrons incident at specific angles to pass through. Using the knowledge of the experimental geometry, together with these data, the most likely absolute positions of the magnets were determined. The GEp-III experiment, experiment number e-, was the third in a series of ex- periments to determine G E p /G M p at Jefferson Lab. The first experiment, which was pub- lished in February  [], measured the form factor ratio for values of Q ² between be- tween . and . (GeV/c)². GEp-II extended the measurement to . (GeV/c)² and was published in February  []. Data collection for GEp-III took place in  and . The beam optics data, necessary to determine the absolute positions of the quadrupole magnets, were taken in October . The final results of the GEp-III experiment were published in June  []. In addition, a fourth experiment is in the planning stages [] which is expected to extend the measurement of G E p /G M p to  (GeV/c)². 1.1 History of nucleon structure studies The atomic nucleus was discovered in  by Ernest Rutherford []. The nucleus was shown to have internal structure in , when Rutherford discovered the proton []. The neutron was discovered by James Chadwick in  []. In , Otto Stern measured the magnetic moment of the proton []. In this experiment, Stern found that the proton’s magnetic moment was not that of a point particle of the proton’s charge and mass; this discrepancy indicated that the proton had an internal structure. The magnetic moment of the neutron was measured in  by Luis Alvarez and Felix Bloch []. The electric and magnetic form factors of the proton were first measured in the s by Robert Hofstadter and Robert McAllister [] using the technique of Rosen- bluth separation. In their experiment, they also found the size of the proton to be about one femtometer. For his findings, Hofstadter won the Nobel Prize in physics in . Starting in the s, experiments revealed further evidence of composite nucleon structure, with the first direct evidence of quarks inside the proton published in  []. Multiple experiments conducted from the s until the present have used Rosenbluth separation to measure the electric and magnetic form factors of the proton and neutron.
  • 15.  As described in Sec. 1.3.2, form factor data for the proton is more easily measurable than for the neutron, and the proton’s magnetic form factor is more easily measured than its electric form factor for high values of Q ². Experiments to date have therefore provided a relative abundance of data for the magnetic form factor of the proton over a wide range of values of Q ², in comparison to the other electric and magnetic form factors. Recent measurements of G E p and G E n at Jefferson Lab have made significant contri- butions to the existing data for these form factors. As already discussed, the first mea- surement of G E p was published in  [], and measurements have continued with sub- sequent experiments. The most recent measurement of G M p at Jefferson Lab, experiment e-, will publish in the coming months []. 1.2 The GEp-III experiment 1.2.1 Experimental techniques for determining proton form factors The G E p experiments at Jefferson Lab, together with one other experiment carried out at Bates Laboratory, are currently the only experiments that have used the recoil polar- ization technique to determine the form factor ratio G E p /G M p . Previous experiments to determine G E p and G M p used the method of Rosenbluth separation, which does not ap- pear to provide reliable data for the electric form factor for Q ² values above  (GeV/c)². Rosenbluth separation has, however, been used to determine the magnetic form factor of the proton with good accuracy above  (GeV/c)². Recoil polarization can be used to de- termine the ratio G E p /G M p , and experiments using Rosenbluth separation have provided G M p , so the electric form factor G E p can be readily extracted. At present, Jefferson Lab is the only particle accelerator in the world that can pro- duce a beam with sufficient intensity and duty factor such that the recoil polarization technique can be used to determine G E p /G M p at high Q ²; the technique requires a highly polarized electron beam and high current in the energy range under study. Jefferson Lab can provide such a beam up to a beam energy of  GeV at  µA and –% polariza- tion. This allowed the GEp-III experiment to measure the form factor ratio up to a Q ² of
  • 16.  . (GeV/c)². For the GEp-IV experiment to take place, the lab must be upgraded to pro- duce a beam energy of  GeV. The reaction of interest in this experiment was the elastic collision described as fol- lows: ¹H(e , e p ) The target was made of hydrogen (¹H), and a polarized electron beam (e ) was used. The scattered electron (e ) and proton (p ) were detected in a lead-glass calorimeter and mag- netic spectrometer, respectively. In addition, the polarization of the scattered proton was measured. A series of cuts was placed on the data in order to select only elastic collisions. 1.2.2 The experimental setup The experiment was conducted in Hall C of Jefferson Lab. The electron beam left the ac- celerator and entered the hall, where it struck the proton target. The target consisted of a  cm long aluminum cylinder filled with liquid hydrogen, cooled to  K. Scattered elec- trons were detected using a lead-glass calorimeter, and scattered protons were detected in the High Momentum Spectrometer, or hms. The hms consisted of three quadrupole magnets and a dipole magnet that led to a detector hut (see Figs. 1.1 and 1.2). Inside the detector hut was a detector array consisting of two drift chambers for tracking, three scin- tillator hodoscopes for trigger and timing, and two focal plane polarimeters. In order to travel from the target to the hms detector hut, particles must pass through the quadrupole and dipole magnets. The dipole field strength was set at roughly . T. Given this field strength and the physical layout of the dipole magnet, protons from the target were deflected upwards by °, allowing them to enter the detector hut. Electrons were detected in a calorimeter, and the time of detection was recorded for both protons and electrons, in order to select elastic collisions. Other types of charged particles either would not bend at the correct angle when reaching the dipole magnet, or would have a different time of flight through the spectrometer and would be excluded as inelastic events. There was shielding along the direct line of sight between the target and the detector hut, so that particles could not enter the detector hut by that route.
  • 17.  Fig. 1.1: The spectrometer arm of the experimental setup, showing the target (yellow), three quadrupole magnets (gray), the dipole magnet (blue) and the detector hut (white). The detector hut contains the hms detector array, shown in the figure below. FPP1+FPP2 S1X+S1Y DC1+DC2 S0 CH2 Fig. 1.2: The hms detector array. Labeled components are the scintillator ho- doscopes (s1x, s1y and s0), the drift chambers (dc1 and dc2), the focal plane polarimeters (fpp1 and fpp2), and two blocks of analyzer material for the po- larimeters, made of CH² plastic.
  • 18.  The three quadrupole magnets of the hms focused the beam of scattered protons. The beam was focused in order to allow a wider scattering cross section of protons to reach the detector. The field strengths of the quadrupole magnets were individually adjustable, with nominal field strengths between . and . T. The hms was designed to accept a maximum central momentum of . GeV/c. The momentum bite is about % and the solid angle acceptance is about  msr. The nominal resolution is approximately .% in momentum,  mrad for both the in-plane and out-of- plane angles, and  mm for the interaction vertex reconstruction. Each of the drift chambers in the hms individually detect the position and, to a lesser resolution, the angle of any charged particle entering the detector. Data from both drift chambers is combined to infer the most likely track of each incident particle. The po- sition and angle information from the drift chambers is projected to the focal plane, an imaginary plane between the two drift chambers, yielding the focal plane coordinates x fp , y fp , θfp and φfp . The x -axis points down and the y -axis points to the left when facing the focal plane from the target. The angle φ is measured from the z -axis in the y z -plane and θ is measured in the x z -plane, where the axes form a right-handed coordinate system. The two drift chambers were approximately  cm apart in the z direction and detected particles over an area roughly  cm tall and  cm wide. The polarimeters determine the normal and transverse components of the spin of incident particles which scatter in the blocks of analyzer material placed before each polarimeter. The normal component of the particle’s spin in the detector precesses in the dipole magnet from the longitudinal component of the spin at the target, while to first order, the transverse component does not precess. For the beam optics study, the polarization data was not needed. Each drift chamber has six planes of signal wires, with the planes spaced . cm apart. Within each plane, the signal wires are spaced  cm apart. In order of increasing z coor- dinates (traveling downstream), the planes are designated x, y, u, v, y and x . Between each signal plane there are two planes of field wires. The x and x wires are horizontal and measure position in the vertical (dispersive) direction. These two planes of x wires are offset from each other to avoid a left-right ambiguity. The y and y wires are verti-
  • 19.  Y, Y′ U V Amplifier-discriminator cards X, X′ Fig. 1.3: Diagram of hms drift chambers as viewed from the target, showing the directions of the six planes of signal wires. The amplifier-discriminator cards are also shown. cal, measuring position in the horizontal direction, and are also offset from each other to avoid left-right ambiguity. The u and v planes are at ±° angles to the x and x wires. This concentration of near-horizontal wires gives the drift chambers better resolution in the dispersive direction, which allows for better reconstruction of the particle momentum. The drift chambers are diagrammed in Fig. 1.3. Each drift chamber was filled with a %/% argon-ethane mixture by weight. High- energy charged particles traveling through the drift chamber leave a trail of ionized gas particles. These ionized particles drift to the nearest signal wire where they cause a pulse which is detected by an amplifier-discriminator card. The particle’s trajectory can be re- constructed from the series of wires which sent a signal as the particle traveled through the drift chamber. Because there are six planes of wires in each drift chamber and only four coordinates to determine (x , y , θ and φ ), the particle’s trajectory is overdetermined. A position and angle is calculated for each drift chamber, and these coordinates are com- pared to determine whether an event in one drift chamber corresponds to an event in the other drift chamber. This method of reconstructing each particle’s trajectory allows
  • 20.  for tracking multiple particle trajectories at once. 1.2.3 Magnet position offsets As described in the previous section, the spectrometer arm of the experimental setup consisted of three quadrupole magnets, a dipole magnet and a detector hut (see Fig. 1.1). Scattered protons travel through the three quadrupole magnets, which focused the pro- ton beam. The protons then enter the dipole magnet, which bends the proton beam up by °, allowing the protons to enter the detector hut and be detected. The goal of this research is to determine the absolute displacements of the three quadrupole magnets. Any horizontal displacement would defocus the proton beam, introducing a horizontal bend to the central trajectory through the magnets. This could strongly affect the ratio of the transverse and normal spin components of the detected proton. These spin compo- nents are used to determine the ratio of the transverse and longitudinal components of the spin of the scattered proton at the target, which is proportional to the ratio G E p /G M p . A vertical displacement in the quadrupole magnets or dipole magnet would affect only the vertical bend angle, and this bend angle was measured and accounted for separately. A horizontal displacement in the dipole magnet would have almost no effect. Therefore, it was not necessary to investigate these offsets. 1.3 The physics behind GEp-III The GEp-III experiment was designed to probe the interior of the proton by observing the results from elastic collisions with polarized electrons. In general, when a high-energy electron collides with a proton, any number of interactions can occur. The simplest of these interactions is where the incident electron interacts with a proton, yielding an elec- tron and proton via one-photon exchange as shown in Fig. .(a). However, at increas- ingly high energies, the incident electron becomes more and more likely to destroy the proton, yielding scattered particles other than electrons and protons. These interactions, called inelastic collisions, are not useful to the analysis of the data in this experiment. Another possible interaction is that of an incident electron interacting with a proton via
  • 21.  e e e e e e γ∗ γ∗ γ∗ γ∗ γ∗ p p p p p p (a) One-photon exchange (b) Two-photon exchange (c) Two-photon exchange Fig. 1.4: Feynman diagrams of an elastic collision via one-photon exchange, and two corresponding collisions via two-photon exchange two-photon exchange, yielding a scattered electron and proton. Two such interactions are shown in Figs. .(b) and .(c). This two-photon interaction occurs much less fre- quently than the elastic one-photon interaction, but is still worthy of study. This is the subject of the GEp-γ experiment [], experiment number e-, the sister experiment to GEp-III. The GEp-III experiment itself focuses on elastic collisions via one-photon ex- change. One kinematic quantity of interest in describing the elastic collisions in this experi- ment is their four-momentum-transfer-squared or Q ², which has units of (GeV/c)². Q ² is calculated using Eqs. (.) to (.): ω = Ee − Ee (.) q = pe − pe (.) 2 2 2 Q = |q| − ω (.) where E e (E e ) is the energy of the incident (scattered) electron, and pe (pe ) is its momen- tum. In general, higher values of Q ² correspond to higher beam energy and a shorter wavelength for the incident electrons, which allows the electron to probe deeper into the proton, revealing the proton’s internal structure.
  • 22.  1.3.1 Proton form factors The physical property of the proton under investigation in this experiment is its Sachs electric form factor, G E p . Another property of interest is the Sachs magnetic form factor, G M p . The neutron has corresponding electric and magnetic form factors, G E n and G M n . The form factors are also designated G E and G M when in reference to either nucleon. The electric and magnetic form factors are among the simplest physics observables of the nucleon’s internal structure. They correspond to the Fourier transforms of the nucleon’s charge and current distributions, respectively. The electric and magnetic form factors are related to the Dirac and Pauli form factors according to Eqs. (.) to (.): G E (Q 2 ) ≡ F¹(Q 2 ) − τκF ²(Q 2 ) (.) 2 2 2 G M (Q ) ≡ F¹(Q ) + κF ²(Q ) (.) Q2 τ≡ (.) 4M 2 where F¹ is the Dirac form factor, F ² is the Pauli form factor, κ is the anomalous mag- netic moment of the nucleon, and M is the mass of the proton. These form factors are functions of Q ²; as indicated above, higher Q ² corresponds to probing deeper into the proton. Low values of Q ² correspond to bulk charge and magnetization distributions. A Q ² of . (GeV/c)² corresponds roughly to . fm, the radius of the proton. At Q ² = , F¹ = F ² = , so G E p =  and G M p =  + κp = µp , the magnetic moment of the proton (approx- imately . nuclear magnetons). Previous experiments have found that both the electric and magnetic form factors of the proton can be described by the dipole form given in Eq. (.): −2 Q2 G D (Q 2 ) = 1 + (.) Λ2 This corresponds to the charge and current densities of the proton falling off exponen- tially for distances far from the proton’s center. The dipole form holds approximately for Q ² less than about one (GeV/c)². In the equation, Λ² is a constant experimentally deter- mined to be . (GeV/c)². The exponential distribution of charge corresponding to the dipole distribution of
  • 23.  Charge or current density 0 0 Distance from center Fig. 1.5: Schematic representation of bulk charge and current density in the pro- ton, which cannot be accurate at the center form factors is diagrammed schematically in Fig. 1.5. However, it is impossible that this trend continues all the way to a radius of zero, because the derivative of exp(−r ) =  at r = . The derivative of the charge or current density of the proton would therefore be discontinuous in the center of the proton, which is unphysical. As a result, the ratios G E p /G D and G M p /µp G D , which are approximately equal to unity at low Q ², must deviate from unity at higher values of Q ². Both of these ratios have been measured to sufficiently high Q ² in previous experiments to demonstrate that this is indeed the case. For asymptotically large Q ² (greater than  to  (GeV/c)²), perturbative quantum chromodynamics (pqcd) predicts that the ratio G E p /G M p should become constant. In- termediate Q ² is the most theoretically challenging region, for which there are multiple conflicting theoretical models. It is therefore necessary to measure the ratio G E p /G M p in this range experimentally, to provide insight into which models may be correct. A por- tion of this intermediate region of Q ², . (GeV/c)² to . (GeV/c)², is the reign of study in the GEp-III experiment. 1.3.2 Rosenbluth separation One method of determining G E and G M is by Rosenbluth separation. In this technique, the cross section of elastically scattered protons is measured and compared to Eqs. (.) to (.):
  • 24.  dσ dσ 2 τ 2 1 = GE + GM (.) dΩ dΩ Mott ε 1+τ dσ α2 E e cos2 θe 2 = (.) dΩ Mott 4E e sin4 θe 3 2 −1 θe ε = 1 + 2 (1 + τ) tan2 (.) 2 Eq. (.) gives the Mott cross section, which is the expected cross section of a pointlike, spin-½ particle. The portion of Eq. (.) in square brackets, along with the factor of 1/(1 + τ), is the adjustment to Mott scattering owing to the internal structure of the nucleon. Eq. (.) defines ε, the longitudinal polarization of the virtual photon. The factor α is the fine-structure constant, approximately ¹⁄₁₂₇. E e is the beam energy, E e is the energy of the scattered electron, θe is the electron scattering angle in the labora- tory frame, and τ was given in Eq. (.). Because τ is proportional to Q ², G E dominates Eq. (.) for low Q ² and G M dominates for high Q ². Rosenbluth separation has been used effectively to determine G E p up to Q ² ≈ ; beyond this range, measurements of G E p were inconsistent even after account- ing for large uncertainties []. G M p has been measured to good accuracy up to Q ² ≈ . Figs. 1.6 and 1.7 show a representative sample of measurements of G E p and G M p obtained by Rosenbluth separation. Rosenbluth separation can also be used to determine the neutron form factors G E n and G M n . The neutron, however, is unstable when not in a nucleus, with a half-life of about  minutes. Because free neutron targets would decay quickly, the neutron must be studied indirectly using Rosenbluth separation on deuterium (²H). Such studies are more difficult than studies of the proton form factors, since free protons are readily available in the form of hydrogen. As a result, the neutron form factors are known much less precisely than those of the proton. [][] The electric form factor of the neutron is difficult to separate from the magnetic form factor when using Rosenbluth separation, in part because G E n is many times smaller than G M n . Values of G E n measured using Rosenbluth separation have been indistinguishable
  • 25.  1.6 1.4 1.2 GEp / GD 1 0.8 0.6 Andivahis et al, 1994 Berger et al, 1971 Borkowski et al, 1975 0.4 Christy et al, 2004 Janssens et al, 1966 Price et al, 1971 0.2 Qattan et al, 2005 Simon et al, 1980 Walker et al, 1994 0 0.01 0.1 1 10 Q2 (GeV/c)2 Fig. 1.6: Rosenbluth separation data for G E p [] from zero and have had large error bars []. This is another limitation of the technique of Rosenbluth separation. 1.3.3 Recoil polarization To determine G E p accurately at values of Q ² greater than  (GeV/c)², the technique of recoil polarization was developed, which was first used in the first G E p experiment at Jefferson Lab. In this technique, a polarized electron beam strikes an unpolarized target. The incident electrons transfer some polarization to the scattered protons. The ratio G E p /G M p can be determined by measuring the transverse and longitudinal components of the spin of the scattered proton, as described in Eqs. (.) to (.):
  • 26.  1.1 1 GMp/µpGD 0.9 0.8 Andivahis et al, 1994 Bartel et al, 1973 Berger et al, 1971 Borkowski et al, 1975 Christy et al, 2004 Janssens et al, 1966 0.7 Price et al, 1971 Qattan et al, 2005 Sill et al, 1993 Walker et al, 1994 0.1 1 10 Q2 (GeV/c)2 Fig. 1.7: Rosenbluth separation data for G M p [] ˆ θe E e + E e 2 I 0 Pl = k · he τ (1 + τ) tan2 GM (.) 2 M ˆ θe I 0 Pt = −2 k · he τ (1 + τ) tan G E G M (.) 2 I 0 Pn = 0 (.) 2 τ 2 I0 ≡ GE + GM (.) ε GE Pt E e + E e θe =− tan (.) GM Pl 2M 2 This reveals the relative sign of G E and G M , which is not possible using Rosenbluth sep- aration. Because good data exists for G M p up to Q ² ≈  (GeV/c)² and recoil polarization can accurately reveal G E p /G M p , the electric form factor of the proton can be readily ex- tracted. The term ε in Eq. (.) was given in Eq. (.). Eq. (.) follows immediately from Eqs. (.) and (.). The other equations are derived in references [], [] and []. Successfully measuring G E p /G M p using recoil polarization requires an electron beam
  • 27.  1.8 1.6 1.4 1.2 µp GE/GM p 1 p 0.8 Andivahis et al, 1994 Bartel et al, 1973 0.6 Berger et al, 1971 Christy et al, 2004 Crawford et al, 2007 Gayou et al, 2002 0.4 Jones et al, 2000 Jones et al, 2006 Maclachlan et al, 2006 0.2 Milbrath et al, 1998 Qattan et al, 2005 Ron et al, 2007 0 0.2 0.3 0.4 1 2 3 4 5 6 7 89 Q2 (GeV/c)2 Fig. 1.8: µp G E p /G M p data from the first two recoil polarization experiments at Jefferson Lab [][], compared to existing data from Rosenbluth separation [] with high current and high polarization. High current was necessary because at the beam energies under investigation, only a small percentage of collisions in the target result in elastic collisions; data collection at low current would take years or decades. Jefferson Lab can provide a beam up to  µA and –% polarization. The results of the two G E p experiments at Jefferson Lab prior to GEp-III have been published and are shown in Fig. 1.8. The data from Rosenbluth separation experiments are included for comparison. From this figure, recoil polarization clearly results in higher-quality data for G E p /G M p at values of Q ² greater than one. The ratio of the neutron form factors G E n /G M n can be determined by recoil polar- ization on a deuterium target, or by using a polarized target of deuterium or helium-. When using both a polarized beam and a polarized target, it is not necessary to measure the spin of the scattered particles. Instead, the beam polarization is reversed periodi- cally and the asymmetry in the scattering cross sections is measured. Experiments using
  • 28.  a polarized beam where either a polarized target are used or the spins of the scattered particles from the target are measured are called double polarization experiments. Like recoil polarization experiments for the proton, these experiments require a highly polar- ized electron beam with a high beam current. It is also possible to measure the proton form factor ratio G E p /G M p using a polarized target with a polarized beam.
  • 29.  CHAPTER 2 Methodology 2.1 The optics data taken 2.1.1 Geometry of the experimental setup As outlined in Chapter 1, to determine the horizontal displacements of the quadrupole magnets in the hms spectrometer, a series of beam optics runs was taken. In each run, the electron beam struck a carbon target which was roughly  mm thick. The beam en- ergy was chosen (. GeV) so that a large fraction of the collisions would be elastic, and thus the momentum of the scattered electron would be fixed for a given scattering an- gle. The hms was positioned at a .° angle to the electron beam. By adjusting the field strengths of the magnets in the hms, it was configured to accept electrons with a momentum of . GeV/c, i.e., the momentum of electrons scattered elastically at .°. Between the target and the magnets of the spectrometer arm was a sieve slit collimator, which allowed electrons through only at specific angles. Scattered electrons were chosen rather than protons for the optics runs because protons can travel through the metal of the collimator. This experimental setup therefore ensured that electrons originating from elastic collisions at a known location (that of the carbon target) and passing through one of the holes in the sieve slit collimator were detected in the detector hut. A carbon target was used because such targets can be thin—electrons can scatter anywhere along the intersection of the beam and target, so a thinner target limits one source of error in determining the location of the collision. Carbon also remains solid at high temperatures, allowing a higher beam current for more frequent collisions and better statistics. Fig. 2.1 diagrams the distances and angles of interest in the optics runs. The distance
  • 30.  Fig. 2.1: Top view of target, sieve slit collimator and focal plane. The quadrupole and dipole magnets are not shown in this diagram. The center of the central hole of the sieve slit is shown as a red circle. None of the distances are to scale, and the angles are greatly exaggerated, except for the hms angle (.°). The central axis of the spectrometer, y = 0 in the diagram, bends in the +x direction through the dipole magnet by the dipole bend angle. The part of the spectrom- eter axis from the target to the sieve slit is parallel to the ground, while at the focal plane, the spectrometer axis is at roughly a ° angle upwards. The spec- trometer coordinate system was considered to bend inside the dipole magnet. Quantities with a minus sign are negative as drawn. from the target to the sieve slit and the displacement of the central hole of the sieve slit from the spectrometer axis were measured in a survey. The figure also labels most of the quantities which will be discussed in the following sections. The position of the beam was recorded from three beam position monitors or bpms along the beam line. Also see the physical arrangement of the magnets in Fig. 1.1. The quantities y tgt and φtgt are the y and φ positions of the scattered particle at the target, respectively, and y fp and φfp are the corresponding positions at the focal plane. These four quantities are specified in the spectrometer coordinate system, the primary coordinate system used in this research. The x -axis points down, the z -axis points from the target to the focal plane, and the y -axis points to the left when facing in the posi- tive z direction. The beam position is given in the beam coordinate system using x MCC and z MCC , so named because the electron beam was controlled by the Machine Control Center, or mcc. The mcc coordinate system is offset from the spectrometer coordinate system by a .° angle. The x MCC -axis points roughly in the negative y direction of the
  • 31.  spectrometer coordinate system. The absolute position of the beam in the spectrometer coordinate system is related to the beam coordinate system by the unknown but small offset y 0 tgt . The quantity y beam is the same as x MCC , scaled to align with the spectrometer coordinate system. At the focal plane side, y PAW and φPAW are in the coordinate system used by the detector array. This coordinate system is supposed to be the same as the spectrometer coordinate system, but may be misaligned by a small amount, quantified by the offsets y 0 fp and φ0 fp . The spectrometer coordinate system was considered to bend inside the dipole magnet as a particle following the central trajectory would. Eqs. (.) to (.) describe the relationships between the quantities shown in Fig. 2.1, derived using the survey data and simple geometry: ∆x MCC φMCC = − arctan = 0.033◦ (.) ∆z MCC y beam = x MCC cos 12.01◦ + φMCC = 0.978x MCC (.) y tgt = y 0 tgt − y beam (.) y tgt + 0.24 φtgt = − arctan (.) y tgt tan 12.01◦ + φMCC + 1659.48 = −0.603y tgt − 0.145 = −0.603y 0 tgt + 0.589x MCC − 0.145 y fp = y PAW + y 0 fp (.) φfp = φPAW + φ0 fp (.) Eq. (.) gives φMCC (not shown in the diagram), the angle of the beam relative to the beam axis where x MCC = . This quantity was calculated for each run using the bpm data, and found to be roughly .° to the left of the beam axis for each run. Distances are in units of millimeters and angles are in units of milliradians, except where explicitly designated to be in degrees. 2.1.2 Primary goal of this research There are seven unknown quantities to be determined in the beam optics studies. The primary three quantities solved for were the misalignments in the y direction (using the spectrometer coordinate system in Fig. 2.1) of the three quadrupole magnets: s¹ is the y
  • 32.  displacement of the first quadrupole magnet (q1), and similarly for s ² and s ³. Another quantity to be solved for is the expected angular deflection in the y z -plane of a particle entering the quadrupole magnets along the spectrometer axis (x = , y = ). If all three quadrupole magnets were aligned perfectly, such a proton would travel along the central axis of the spectrometer without deflecting. Because of misalignments in the quadrupole magnets, there can be a deflection, φbend (not shown in the diagram). The final three unknown quantities are shown in Fig. 2.1: y 0 tgt , y 0 fp and φ0 fp . The quantity y 0 tgt is a measure of any misalignment in the y direction between the beam co- ordinate system and spectrometer coordinate system. The quantities y 0 fp and φ0 fp are measures of any misalignment in the hms detector array relative to the spectrometer co- ordinate system. These three quantities could either be fixed at zero (or another value) while solving for the quadrupole offsets, or allowed to vary. In particular, y 0 fp and φ0 fp are expected to be very small, so it was possible either to allow these parameters to vary to see if the solutions found give small numbers for the parameters, or to hold these values fixed at zero to facilitate finding a solution for the quadrupole offsets and for y 0 tgt . As shown in Eq. (.), the form factor ratio G E p /G M p is proportional to the ratio of the scattered proton’s spin components, Pt /Pl . At Q ² = 8.5, the highest energy setting of the experiment, Pt /Pl ≈ ₀₀₀₅ . This makes G E p /G M p close to zero in this Q ² range (see ⁰⁷ Fig. 1.8), and very sensitive to Pt . If all of the magnets of the hms were aligned perfectly, the longitudinal component of the proton’s spin at the target, Pl , would precess to the normal component at the hms detector, while the transverse component Pt would not precess at all. With horizontal magnet misalignments, particles experience a horizontal deflection φbend and Pl at the target precesses slightly to the transverse direction at the detector. This small precession can combine with the small component of Pt , changing the measurement of Pt by a large percentage. As a result, an accurate knowledge of the horizontal positioning of the quadrupole magnets is essential to accurately determine Pt and therefore G E p /G M p .
  • 33.  Tab. 2.1: Beam optics settings. All quadrupole field strengths are relative to nom- inal. Dipole field strength was always nominal. q1 field strength q2 field strength q3 field strength Nominal 1 1 1 q1 −0.7003 0 0 q2 0 0.3959 0 q3 0 0 −0.5745 q1 reduced 0.7 1 1 q2 reduced 1 0.7 1 q3 reduced 1 1 0.7 Dipole only 0 0 0 2.1.3 Description of data runs taken A series of beam optics runs were taken with the quadrupole magnets in various config- urations. The magnet settings used are detailed in Tab. 2.1 and the runs taken are listed in Tab. 2.2. In every run, the dipole magnet field strength was at its nominal setting. For the q1 reduced, q2 reduced and q3 reduced settings, one quadrupole magnet was set at % of its nominal field strength, and all other magnets were at nominal field strength. Data runs were also taken with all magnets at their nominal field strengths, and with all quadrupole magnets were turned off. This list of configurations was based on Lubomir Pentchev’s technical note [] on similar work done for GEp-II. The purpose of these runs was to investigate the way electrons traveled through the quadrupole magnets. This data could then be used, in combination with knowledge of the physics involved and data from a survey of the equipment, to determine any offsets from the expected positions of the quadrupole magnets. The first runs taken used the so-called nominal setting, i.e. all three quadrupole mag- nets were set at their nominal field strengths. Next, all of the quadrupole magnets were turned off, leaving only the dipole magnet turned on. However, the q3 field strength read back at − G after turning off the quadrupole magnets, despite the current through all magnets reading at zero. A procedure was followed which attempted to degauss q3, in order to eliminate any residual magnetic field. However, the degaussing was found to have no effect on the readout for the q3 field strength. Next, run  was taken with the current on q3 set to zero, despite a nonzero field strength reading. Run  was
  • 34.  Tab. 2.2: List of optics runs. The beam x values given were calculated in the analysis from the bpm values. Number of Run number Setting Beam x good events Notes 65959 Nominal 1.66 539 183 Before degaussing q3 65960 Nominal 1.66 927 239 Before degaussing q3 65961 — 1.67 0 Junk 65962 — 1.67 0 Junk 65963 Dipole only 1.65 143 959 After degaussing q3 65964 — 1.66 0 Junk 65965 Dipole only 1.66 54 462 After degaussing q3 again 65966 Dipole only 2.43 260 741 65967 Dipole only 5.29 500 000 65968 Dipole only −2.34 400 000 65969 Dipole only 0.45 6 952 65970 Dipole only 0.45 195 761 65971 q1 plus dipole 0.45 13 727 65972 q1 plus dipole 0.45 394 359 65973 q2 plus dipole 0.44 491 325 65974 q3 plus dipole 0.45 285 053 65975 Nominal 0.46 377 604 65976 q1 reduced 0.47 303 991 65977 q2 reduced 0.50 201 150 65978 q3 reduced 0.47 252 151 then taken, with a current on q3 chosen to make the field strength read near zero. Af- ter this run, an alternate method of degaussing q3 was used. After turning off q3, the field strength read near − G, as before. After this, run  was taken. Observing that neither degaussing had any effect, it was concluded that the field strength in q3 was actually close to zero when q3 was turned off, and that the reading for the field strength was incorrect. Next were runs  through , a series of runs at varying beam x positions with only the dipole magnet turned on. These runs were taken to find the central hole of the sieve slit and set the beam x MCC position such that the central hole would appear near y PAW =  in the data (in its own coordinate system—see Fig. 2.1). All of the holes of the sieve slit had the same diameter, . cm, except for the central hole which had a diameter of . cm. This allowed the central hole to be identified: the histograms of y PAW showed a smaller peak for the central hole. See, for example, the histograms of y in Fig. 2.2, also shown in Figs. a.4 and a.5. In Fig. .(a), the central hole the central hole was
  • 35.  25000 12000 20000 10000 15000 8000 6000 10000 4000 5000 2000 0 0 -20 -15 -10 -5 0 5 10 15 20 -25 -20 -15 -10 -5 0 5 10 15 20 25 (a) y , run , beam x = −. mm (b) y , runs –, beam x = . mm Fig. 2.2: Comparison of central sieve slit hole at two beam positions near y = − cm and an adjacent hole is near y =  cm. In Fig. .(b), the central hole was near y = , with larger peaks on each side, but y values with an absolute value greater than about  cm were outside the acceptance of the spectrometer. As a result, it is only possible to see the edges of the adjacent peaks in Fig. .(b). Runs  and  were mistakenly left running while the beam was being moved. The data was analyzed later to determine how many events were taken before the beam moved. This is why these two runs have approximate numbers listed for the number of good events in Tab. 2.2, while all of the other runs have exact counts of events. Upon finding the central sieve slit hole and positioning the beam so that the central hole was near y PAW = , data was taken using all of the magnet settings listed in Tab. 2.1. For the q1 setting, the second and third quadrupole magnets were turned off and the field strength of magnet q1 was chosen to give point-to-parallel focusing in the y direc- tion, so that all particle tracks passing through a given point will be parallel at the focal dφfp plane, regardless of their initial angle: dφtgt =  (see Fig. 2.1). The q2 and q3 settings were also set for point-to-parallel focusing. The field strengths for these three settings were calculated assuming no offsets in the positions of the quadrupole magnets, using cosy infinity [], a software package for modeling beam physics to arbitrary order using dif-
  • 36.  ferential algebra. 2.2 Equations using the optics data 2.2.1 Setting up the equations While traveling through the quadrupole and dipole magnets, particles are deflected in a predictable way. The magnetic field within and surrounding the magnets can be mod- eled and the particles’ motion can be predicted using cosy infinity. The GEp-III collab- oration has created a cosy script that models the experiment’s configuration of magnets. For each magnet configuration, this script outputs the coefficients used to project an electron’s position from the target side of the magnets to the detector side, or vice versa. The coefficients for propagating the electron from the target side to the detector side are here referred to as cosy coefficients, and coefficients propagating in the other direction as reverse cosy coefficients. Equations using the cosy coefficients The cosy coefficients gave the derivatives of x , θ , y , φ and t at the focal plane with re- spect to x , θ , y , φ , t and δ at the target, where y and φ are in the directions shown on Fig. 2.1, x is perpendicular to the page in that figure, θ is the out-of-page angle and t is p fp −p tgt time. The quantity δ = p tgt , where p tgt is the momentum of the scattered electron be- fore it enters the spectrometer and p fp is its momentum at the focal plane. The reverse cosy coefficients calculate the corresponding derivatives in the opposite direction, e.g. derivatives of target variables with respect to focal plane variables. The cosy script can output the above derivatives to arbitrary order. First-order deriva- dy dy tives include dy and dφ , written more compactly as (y |y ) and (y |φ); the second-order derivatives are (y |y ²), (y |y φ) and so on. Six additional derivatives were needed for the analysis: the derivatives of y fp and φfp with respect to the three quadrupole shifts s¹, s ² and s ³. To find (y fp |s¹) and (φfp |s¹), The cosy script was modified by shifting q1 by + mm in the y (horizontal) direction, leaving the other two quadrupole magnets in their nom-
  • 37.  inal positions, and the zero-order terms for y fp and φfp were taken. The derivatives with respect to s ² and s ³ were found in a similar way. The derivatives of y fp and φfp with respect to y tgt , φtgt and δ were used to determine the horizontal offsets (along the y axis) of the quadrupole magnets. The derivatives of other focal plane variables were unnecessary, and the derivatives of y fp and φfp with re- spect to the other target variables did not contribute strongly to the results because these coefficients were small. Taking only the derivatives with respect to y tgt , φtgt and δ, the general equation for y fp and φfp using cosy coefficients to first order is given by: 3 y fp (y fp |y tgt ) (y fp |φtgt ) y tgt (y fp |s i ) + (y fp |δi )δ = + si (.) φfp (φfp |y tgt ) (φfp |φtgt ) φtgt (φfp |s i ) + (φfp |δi )δ i =1 This equation provides the means to set up two equations for each set of data. From Eqs. (.) to (.), y tgt and φtgt are functions of the beam position and angle, x MCC and φMCC , and the offset y 0 tgt . The position and angle of the electrons at the detector can be found by analyzing the data for each run. To account for any misalignment of the detector, the y and φ values found in the analysis are designated y PAW and φPAW , and are related to the values y fp and φfp in Eq. (.) by Eqs. (.) and (.). The momentum term δ can be measured from the data as well. The error values for y fp and φfp in the above equation were calculated using Eqs. (.) and (.): 3 2 ∆y fp = (∆y PAW )2 + (y fp |δi )(∆δ)s i (.) i =1 3 2 ∆φfp = (∆φPAW )2 + (φfp |δi )(∆δ)s i (.) i =1 After finding the three quadrupole shifts, the horizontal bend angle φbend can be de- termined using Eq. (.): 3 φbend = (φfp |s i ) + (φfp |δi )δ s i (.) i =1 This equation follows from Eq. (.) by choosing y tgt = φtgt = ; i.e., it gives the horizon-
  • 38.  tal bend angle of a particle entering the spectrometer along the spectrometer axis. The cosy coefficients below are calculated for protons traveling through the spectrometer at energies used in the experiment, with all magnets set to their nominal field strengths. Eq. (.) gives ∆φbend , the error of φbend , which is a function of the cosy coefficients and the calculated errors on s i : 3 2 3 ∂ φbend 2 ∆φbend = (∆s i )2 = (φfp |s i ) + (φfp |δi )δ (∆s i )2 (.) i =1 ∂ si i =1 Performing checks using reverse cosy coefficients ∆y tgt There are two ways of calculating the expected value of ∆x MCC , the amount by which y tgt would change if the beam were moved in the x MCC direction. As a consistency check on the data, this quantity was calculated using both methods, comparing the results. One ∆y tgt solution follows immediately from Eqs. (.) and (.), which give ∆x MCC = −.. This ratio can also be calculated using the reverse cosy coefficients and a series of runs taken at various beam x positions, with the quadrupole magnets turned off. Eq. (.) gives y tgt for given values of y fp and φfp , and Eq. (.) gives φtgt : y tgt = (y tgt |y fp )y fp + (y tgt |φfp )φfp (.) φtgt = (φtgt |y fp )y fp + (φtgt |φfp )φfp (.) For a given dipole run, the quantities y fp , φfp and the beam x are known to within con- stant offsets, so Eq. (.) can be used to calculate y tgt for each run. It is then possible to ∆y tgt plot the calculated y tgt for each dipole run as a function of beam x to find ∆x MCC . Similarly, ∆φtgt ∆x MCC is found to equal . from Eq. (.), and can also be calculated using Eq. (.) and experimental data. 2.2.2 Solving the equations Eq. (.) gives two equations for each of the magnet settings listed in Tab. 2.1. These equa- tions combined with Eqs. (.) to (.) give the quadrupole shifts s¹, s ² and s ³ in terms of
  • 39.  the beam position and angle, the y and φ positions for each run as measured from the data, and three coordinate system offsets. The unknown values in the resulting equations are the three quadrupole shifts and the three other offsets. Data were taken for each of the eight settings listed in Tab. 2.1 with the beam posi- tioned such that y PAW and φPAW would be small (runs  through  in Tab. 2.2). The eight magnet settings give a total of  equations by Eq. (.). These  equations and six unknowns form an overdetermined system of equations. In theory, if there were no measurement errors and the equations were set up to account for all possible vari- ables, this system of  equations could be solved exactly; some of the equations would provide redundant information, and the system of equations would reduce to six linearly independent equations, from which the six unknowns could be readily determined. In practice, there are unknowable measurement errors and the system of  equations is in- consistent. However, it is still possible to find the most likely values of the six unknowns. This is accomplished by attempting to quantify the error σi associated with each func- tion f i and assigning a value χ ² to the system of equations, where the functions f i and χ ² are functions of the six unknowns. These unknowns are then varied until the value of χ ² is at a minimum. If the system of equations were consistent, the minimum χ ² would be zero; for an inconsistent system of equations, χ ² ≈ N dof is considered a good result, where N dof is the number of degrees of freedom, equal to the number of equations minus the number of unknowns. For any overdetermined system of N equations and m unknowns of the form y i = f i (x 1 , x 2 , . . . , x m ), χ ² can be calculated as follows: N 2 f i (x 1 , x 2 , . . . , x m ) − y i χ2 = i =1 σi For the optics equations, the x i are the three quadrupole magnet shifts and three co- ordinate system offsets. The y i are y fp and φfp for each magnet setting, from Eq. (.). The errors σi are estimated by determining the errors in measurement of y PAW and φPAW . The most likely values of the six unknowns can then be found by minimizing χ ² using a min- imization program. This program will return values of the unknowns along with error values for each unknown, according to how strongly χ ² is changed when the value for
  • 40.  each unknown is varied. The σi terms in the above equation are given by Eqs. (.) and (.). In these equa- tions, the measurement error on δ was very small in comparison with the measurement errors ∆y fp and ∆φfp . As a result, the error values σi used in the minimization were deter- mined by the measurement errors on y fp and φfp . Although there were other sources of measurement error in the experiment, none were easily quantifiable. Also, many other sources of error were indirectly accounted for in the errors in y fp and φfp . For example, an instability in the beam position would have resulted in wider peaks in the histograms of y PAW and x PAW in the data and probably a larger error of the mean when fitting a Gaussian curve to the data. After finding the most likely values of the quadrupole offsets, solving for φbend is straightforward. From Eq. (.), it is a function only of the quadrupole offsets and of three cosy coefficients. The error ∆φbend can be quantified using Eq. (.), where the values ∆s i are the errors of s i found in the minimization.
  • 41.  CHAPTER 3 Analysis To solve for the most likely offsets of the quadrupole magnets, it was first necessary to determine the values of all measured data and calculated data. The values of y PAW , φPAW and δ were found by analyzing the collected beam data using paw. The magnetic field coefficients used in Eqs. (.) to (.) were calculated using cosy. The beam position at the target was calculated using the position data reported by the beam position monitors (bpms). Finally, the most likely offsets of the quadrupole magnets were determined using a minimizer program. 3.1 Measuring y PAW and φPAW The first step after taking beam data was to analyze the raw data and then determine the y position and angle φ at the focal plane of the hms. The raw data was analyzed using the Hall C engine, the standard analysis code used in Hall C at Jefferson Lab. This code outputs data files for each run which can be read and analyzed using paw. It is then possible to make one- and two-dimensional histograms of the data with various cuts. By applying appropriate cuts on the data, it was possible to determine the values of y PAW and φPAW for each setting. These cuts were intended to select only the electrons that passed through the central hole of the sieve slit collimator, in order to measure the deflection of a single beam of electrons through the hms magnets. The most important cut in selecting the central hole was the cut on y PAW , which is in the horizontal direction. For most magnet settings, the central hole is readily apparent in histograms of y PAW vs. x PAW . See for example Figs. a.9 to a.11 and a.13, which each show a clear series of peaks in the y direction, corresponding to the columns of holes in the collimator. The peak near y = 0 in each of these plots corresponds to the central hole of
  • 42.  the sieve slit. For runs taken at the nominal magnet setting and the q2 reduced setting, it was not possible to isolate the central hole of the sieve slit in the y direction, so these settings were excluded from the analysis—see Figs. a.6, a.7 and a.12. Including a cut on x PAW was not essential to the analysis. This is because the x PAW axis is vertical but only the horizontal particle deflection was of primary interest. As a result, data from particles that traveled through any of the sieve slit holes aligned with the central hole in the y direction would be serviceable. However, for some settings the values of y PAW and φPAW varied depending on x PAW . Also, in some settings a cut around or near the central hole in x PAW made the central hole in y PAW more visible. For these reasons, cuts on x PAW were applied to the data for each setting. 3.1.1 Method of isolating the central sieve slit hole in the y PAW direction There were two possible methods of performing cuts on y PAW and x PAW . One was to plot a two-dimensional histogram from the data and use a two-dimensional cut in a loop around the area of interest. The other method was to use two one-dimensional his- tograms, specifying a high and low cut point for each. This method is equivalent to draw- ing a rectangle around the area of interest with a two-dimensional cut. In this analysis, the latter method was chosen, opting to use one-dimensional histograms for three rea- sons: the results are more easily reproducible, the cuts used are easily presented in a table (see tables 3.1 and 3.2), and there was no apparent need to use a more complicated cut for any of the magnet settings. Referring to the y vs. x histograms in Appendix a, it can be seen that rectangular cuts on y vs. x are always sufficient for isolating a given peak, except in cases where the peak cannot be isolated at all. For most settings, the central hole in y PAW is well separated from the adjacent holes, and the histogram of y PAW shows easily discernible peaks corresponding to each hole. But, for the reasons listed above, a cut on x PAW was also applied. An initial cut on y PAW was performed using a histogram of all y PAW data for a given setting. A histogram of x PAW was then generated using this cut on y PAW and performed a cut on x PAW . Lastly, this cut was applied on x PAW to a histogram of y PAW and did a tighter cut in y PAW . The cut on y PAW
  • 43.  was tight enough to cut off the tails of the peak in the histogram, to assist in fitting the data. In this way, a cut of y PAW refined by a cut in x PAW was obtained that helped to isolate the central hole. 3.1.2 Methods of performing a cut on x PAW In the x PAW (vertical) direction, the central hole is only visible for some magnet settings. However, the electron beam was held constant in the vertical direction throughout the data collection. As a result, there was little variation in the central hole peak position in x PAW . Compare for example the histograms of x PAW for the five dipole runs (Figs. a.1 to a.5), which have widely varying values of y PAW , but the peaks in x PAW remain essentially constant. For the q1, q2 and q3 settings, the central hole in x PAW was not visible but there was still a peak in x PAW near where the central hole should be (see Figs. a.8 to a.10). There were two apparent methods for choosing a cut on x PAW . The first was to imple- ment a cut around the central hole in x PAW when the central hole is visible. For the q1, q2 and q3 settings, the cut was instead around the peak in x PAW . Using these cuts helped to isolate the central hole in y PAW , especially for the q1 setting. Compare the histograms of y in Fig. a.8, where the central peak becomes more distinct after applying the cut on x . Tab. 3.1 shows the x PAW cuts selected for each setting at each beam position, and the cuts on y PAW and φPAW which were chosen after the x PAW cut was applied. For the q2 reduced setting, the central hole in x PAW was visible but the central hole in y PAW was not. The x PAW cut found is shown in the table, but this setting was excluded from the analysis. The second method for choosing a cut on x PAW was to use the same limits on the cut for all settings, since it was not strictly necessary to isolate the central hole in x PAW and there was little variation in the position of the central hole for the settings where it was visible. The limits of the cut were chosen by comparing the x PAW cuts used on settings where the central hole in x PAW was visible. The x PAW cut selected by this method was . mm < x PAW < . mm. This cut did not always include the entire peak corresponding to the central hole, but it provided a simple means to select a cut in x PAW that was expected to be close to the central hole for those settings where the central hole was not visible.
  • 44.  Tab. 3.1: Variable x PAW cuts used for each magnet setting and beam position x MCC , and corresponding cuts on y PAW and φPAW x MCC (mm) x PAW (cm) y PAW (cm) φPAW (rad) Dipole 1.65 −0.8 to 6 −0.5 to 3.6 −0.002 to 0.0035 Dipole 2.43 −1 to 6.5 0.7 to 5 −0.0015 to 0.0045 Dipole 5.29 0 to 5.5 6.5 to 12 0.0015 to 0.0075 Dipole −2.34 −2 to 6 −7.2 to −2.8 −0.0055 to 0 Dipole 0.45 −1 to 6 −2.8 to 1.8 −0.0035 to 0.0025 q1 0.45 0 to 10 −1.5 to 0.6 −0.003 to 0.0025 q2 0.44 1 to 8.5 −1.2 to 0 −0.004 to 0.003 q3 0.45 1 to 8 −1.6 to −0.2 −0.0035 to 0.0025 q1 reduced 0.47 0.6 to 4.5 −0.5 to 0.5 −0.003 to 0.003 q2 reduced 0.50 2 to 3.5 — — q3 reduced 0.47 1.5 to 4 −0.9 to 0.5 −0.004 to 0.004 Tab. 3.2: Fixed x PAW cuts used for each magnet setting and beam position x MCC , and corresponding cuts on y PAW and φPAW x MCC (mm) x PAW (cm) y PAW (cm) φPAW (rad) Dipole 1.65 1.6 to 4.5 −1 to 4 −0.004 to 0.005 Dipole 2.43 1.6 to 4.5 0.4 to 5.5 −0.003 to 0.006 Dipole 5.29 1.6 to 4.5 6 to 12.5 0 to 0.009 Dipole −2.34 1.6 to 4.5 −7.5 to −2 −0.007 to 0.001 Dipole 0.45 1.6 to 4.5 −2.5 to 1.8 −0.003 to 0.003 q1 0.45 1.6 to 4.5 −1.5 to 0.6 −0.005 to 0.0045 q2 0.44 1.6 to 4.5 −1.1 to 0 −0.003 to 0.0025 q3 0.45 1.6 to 4.5 −1.6 to −0.1 −0.003 to 0.002 q1 reduced 0.47 1.6 to 4.5 −0.5 to 0.5 −0.003 to 0.003 q3 reduced 0.47 1.6 to 4.5 −0.8 to 0.4 −0.005 to 0.005 Tab. 3.2 shows the y PAW and φPAW cuts chosen after applying this cut on x PAW . Both methods of choosing a cut in x PAW were tried, finding values and error estimates for y PAW and φPAW for each set of cuts. This resulted in two sets of mean values and errors for both y PAW and φPAW . These two sets of results were combined by choosing a mean and error bar such that the new error bars spanned the error bars obtained by using the fixed and variable x PAW cuts. The two sets of y PAW and φPAW data and the combined set are plotted in Figs. 3.1 to 3.3. 3.1.3 Fitting y PAW and φPAW and estimating errors Having applied a y PAW cut around the central hole of the sieve slit and either a fixed or variable cut on x PAW , a Gaussian curve was fitted to the histogram of y PAW to find its mean
  • 45.  10 8 6 4 yPAW (mm) 2 0 -2 -4 -6 Dipole Q1 Q2 Q3 Q1r Q3r Dipole Dipole Dipole Dipole x=0.45 x=0.45 x=0.44 x=0.45 x=0.47 x=0.47 x=1.66 x=2.43 x=5.29 x=-2.34 Fig. 3.1: Measured y PAW data using a variable x PAW cut (red) and a fixed x PAW cut (blue). Combined data is black. Settings are labeled by the magnet configura- tion and the beam x value. Fig. 3.2 provides a zoomed view of the six settings at the central beam position. value. The cut on y PAW was tight enough to exclude the tails of the peak corresponding to the central hole. This was done because the tails of the peaks in the data were not neces- sarily Gaussian, but a Gaussian curve fit well to the area closer to the peak. The area fit by the Gaussian curve extended to between σ and .σ away from the peak, depending on the particular data being fit. The mean of the Gaussian curve was taken as the mean value of y PAW , and the error on the mean returned by the fitting command was taken as the estimated error of the mean. After fitting y PAW , a plot was made of φPAW with the cuts on y PAW and x PAW applied. With these cuts, the histogram of φPAW always had a single peak. Next, a cut was applied on φPAW to select the peak, cutting off the tails. The cuts extended to between .σ and .σ away from the peak, depending on the particular data set. With these three cuts, a Gaussian curve was fitted to φPAW and the mean and error of the mean recorded.
  • 46.  0 -0.2 yPAW (mm) -0.4 -0.6 -0.8 -1 Dipole Q1 Q2 Q3 Q1r Q3r x=0.45 x=0.45 x=0.44 x=0.45 x=0.47 x=0.47 Fig. 3.2: Same as Fig. 3.1, zoomed Following the above procedure using both sets of y PAW and x PAW cuts (shown in ta- bles 3.1 and 3.2) resulted in the values plotted in Figs. 3.1 to 3.3. The Gaussian fits used for each setting and beam position are shown in Appendix b. 3.1.4 Details of fitting data for each magnet setting Dipole setting The runs taken at the dipole setting yielded some of the simplest data to analyze. Because all of the quadrupole magnets were turned off, there was no beam focusing and only particles very close to the central trajectory of the hms reached the detector. Peaks in y PAW and x PAW were easily distinguished. Data was taken at the dipole setting at several beam positions until the central hole of the sieve slit was identified, the central hole being smaller than the others and so corresponding to a smaller peak on the histogram of y PAW . The methods described above of finding y PAW and φPAW at the central hole worked without
  • 47.  0.005 0.004 0.003 0.002 φPAW (mm) 0.001 0 -0.001 -0.002 -0.003 Dipole Q1 Q2 Q3 Q1r Q3r Dipole Dipole Dipole Dipole x=0.45 x=0.45 x=0.44 x=0.45 x=0.47 x=0.47 x=1.66 x=2.43 x=5.29 x=-2.34 Fig. 3.3: Measured φPAW data using a variable x PAW cut (red) and a fixed x PAW cut (blue). Combined data is black. Settings are labeled by the magnet configura- tion and the beam x value. modification for the dipole setting. The dipole runs are plotted in Figs. a.1 to a.5. Nominal setting For data taken at the nominal setting, the central sieve slit hole was not visible either in y PAW or x PAW . The nominal runs are plotted in Figs. a.6 and a.7. In the histograms of y PAW vs. x PAW , at least five tails are visible for values of x PAW < − which appear to correspond to separate sieve slit holes in y PAW , but such tails were always excluded from the analysis of the other runs. Given its large negative values of x PAW , there was no reason to believe that the events in the central tail of the histogram came from a hole in x PAW near the central hole of sieve slit. An attempt was made to find the values of y PAW and φPAW at the central hole by doing a linear fit of the central tail. Extrapolating the fit line to x PAW ≈  would give an approximate value for y PAW at the central hole, and the slope of the line would give φPAW . However, the linear fit had large error bars and was not useful for finding
  • 48.  either y PAW or φPAW . As a result, the nominal runs were excluded from the analysis. q1 setting In the q1 data, if y PAW is plotted in a histogram with no cuts on x PAW , then the peak cor- responding to the central hole is indistinct, partially overlapped by the larger, adjacent peaks. One way to find y PAW at the central hole is to fit three Gaussian curves to the cen- tral peak and the two adjacent peaks. However, there is no apparent way to find φPAW at the central hole using this method. A better method is to perform a cut on x PAW which isolates the central hole in y . Compare the histograms of y PAW before and after applying the cut on x PAW in Fig. a.8. With such a cut, it is straightforward to perform Gaussian fits on both y PAW and φPAW at the central hole. There was a single peak in the histogram of x PAW , located near where the central hole in x PAW was expected to be. One possible cut on x PAW was to choose limits centered around this peak; this is the x PAW cut listed in Tab. 3.1 and applied in the second histogram of y PAW in Fig. a.8. The other possible cut on x PAW was the fixed cut listed in Tab. 3.2. Either of these cuts helps to isolate the central hole in y PAW . q2 and q3 settings Like the q1 data, the q2 and q3 data each have a single peak in x PAW , so there were two options for choosing a cut on x PAW : fitting around the peak or using the fixed x PAW cut. The central hole in y PAW is easily distinguished in these settings, even without a cut on x PAW , so these settings presented no special challenges in finding the values of y PAW and φPAW at the central hole. The q2 run is plotted in Fig. a.9, and the q3 run is plotted in Fig. a.10. q1 reduced and q3 reduced settings Out of all the magnet settings, the q1 reduced and q3 reduced settings provide the clear- est view of the sieve slit holes in the plot of y PAW vs. x PAW : peaks corresponding to each hole are visible, and the peak corresponding to the central hole appears smaller because
  • 49.  the central hole of the sieve slit is smaller than the others. The central hole is easily iso- lated in histograms of both y PAW and x PAW . The only consideration to note is that in the histograms of y PAW with no cuts on x PAW applied, the peak corresponding to the central hole is larger than the adjacent peaks. After applying a cut on x PAW around the central hole, the peak corresponding to the central hole in y PAW becomes smaller than the adja- cent peaks; the large peak before applying cuts is due to a large number of events passing through other sieve slit holes that have the same y position as the central hole. The q1 reduced run is plotted in Fig. a.11, and the q3 reduced run is plotted in Fig. a.13. q2 reduced setting In the data for the q2 reduced setting, the central hole in x PAW is visible but the central hole in y PAW is not. As can be seen in the second histogram of y PAW in Fig. a.12, applying a cut on x PAW is ineffective for finding the central hole in y PAW . Cuts on φPAW likewise reveal no distinct peaks in y PAW . As a result, the values of y PAW and φPAW at the central hole could not be found, and this setting was excluded from the analysis. 3.2 Measuring δ using paw p fp −p tgt To determine the effect of the (y fp |δ) and (φfp |δ) cosy coefficients, the value of δ = p tgt was determined. This value was calculated per event by the analysis code engine using a method that was accurate only for the nominal magnet setting. The value of δ did not depend on the magnet setting, so the value taken was from the nominal setting at the final beam position. Fig. 3.4 shows a histogram of δ, with the elastic peak centered around .%. The mean and standard deviation of the elastic peak were determined by doing a Gaussian fit. Because δ > , it appears likely that the beam energy was slightly higher than what was expected during the data collection.
  • 50.  6000 5000 4000 3000 2000 1000 0 -12 -10 -8 -6 -4 -2 0 2 δ (percent) Fig. 3.4: Histogram of δ from the nominal setting at the final beam position. The elastic peak is centered around δ = .%. 3.3 Modeling magnetic fields using a cosy script To predict the motion of particles through the magnets, cosy infinity was used. A cosy script was written that described the positions of the quadrupole and dipole magnets in the hms, the spacing between them and their nominal field strengths. Then, given a par- ticle type and momentum, the script outputs a table of coefficients describing particle motion through the magnets from the target to the focal plane, or vice versa. I modi- fied this script by adjusting the quadrupole magnet field strengths to model each of the magnet configurations listed in Tab. 2.1. I also adjusted the target position in the script slightly to account for a misalignment revealed in a survey. I set the script to use elec- trons at . GeV/c, although the first-order cosy coefficients do not depend on beam energy or particle mass, and only first-order terms were used in the final analysis. The other modification I made to the cosy script was to add a  mm shift for each of the quadrupole magnets in order to find the cosy coefficients of y fp and φfp with respect to each of the quadrupole shifts. Details of the coefficients returned by the script are given in Sec. 2.2.1. Because the cosy script can run at only one magnet setting and configuration of quad- rupole shifts at a time, I created multiple versions of the script corresponding to each
  • 51.  combination of magnet settings and shifts. After each iteration of the script, the output was stored in a separate file. These multiple scripts and output files, along with most of the rest of the components of my solving program, were coordinated using a makefile to reduce the possibility for human error, and to automate the process of finding cosy coefficients. 3.4 Determining the beam position The beam position was recorded using three bpms, which continuously read out x MCC and y MCC . Using the x MCC readings and the known z MCC position of each bpm, it was possible to determine the value of x MCC at the target (where z MCC = ) using a fit line. This fit line also determined φMCC , the angle of the beam with respect to the beam axis. The beam position sometimes varied slightly even when a change in beam position was not requested, so the values of x MCC and φMCC at the target were calculated for each run. The angle φMCC was found to be approximately .° for each run. See Fig. 3.5 for the fit line used for the q1 setting. Fit lines for the other settings looked similar. 3.5 Using survey data Surveys were taken of the area near the target to measure any mis-pointing between the beam and spectrometer central axes, as well as any misalignment of the central hole of the sieve slit collimator. The survey results relevant to this optics study are shown in Fig. 3.6. From this data, it was found that the central hole of the sieve slit was . mm out of position in the y direction, the target was . mm farther down the beamline than intended, and the central beam axis was mis-pointed by . mm. The distance between the z =  axis and the sieve slit collimator was measured to be . mm. The sieve slit offset of . mm was accounted for in Eq. (.). The other survey results were used to calculate the distance between sieve slit collimator and the portion of the target that intersects the spectrometer axis. This distance was found to be . mm, or . mm closer than the intended . mm. Eq. (.) also included this distance.
  • 52.  0.6 0.4 0.2 0 Beam x (mm) -0.2 C -0.4 -0.6 B -0.8 -1 -1.2 A -1.4 -3500 -3000 -2500 -2000 -1500 -1000 -500 0 Beam z (mm) Fig. 3.5: Beam position readings projected to the target for the q1 data. The point labeled a is the beam x reading from bpm a placed at its z position along the beamline, and similarly for bpms b and c. The red point gives the value of x MCC used in the analysis. The distance of . mm was calculated from the geometry in Fig. 3.6 using the following equation: 0.38 0.38 sin(12.01◦ ) − 0.9 − = 0.84 tan(12.01◦ ) cos(12.01◦ ) This data can also be used to predict a value for y 0 tgt , the offset in the y direction of the beam axis relative to the spectrometer axis. From Fig. 3.6, at the point where the tar- get intersects the spectrometer axis, the beam axis is expected to be offset by −. mm, using Eq. (.): 0.38 tan(12.01◦ ) − 0.9 = −0.20 (.) sin(12.01◦ ) cos(12.01◦ ) There was also a survey of the bpms, yielding offsets in the x MCC direction between . and . mm for each bpm. However, although the bpm positions were needed for the optics studies, this survey data did not prove useful. The bpm data shown in Fig. 3.5 did
  • 53.  Fig. 3.6: Diagram of relevant survey data. The distance of . mm is not to scale, but the angle and other distances are. The center of the central sieve slit hole is represented by the red circle. not use the bpm survey data, but the fit line is already very good. Introducing . mm offsets in the x MCC direction would make it more difficult to fit a straight line to the bpm data. 3.6 Performing checks on the data ∆y tgt As discussed in the section on reverse cosy coefficients on page , ∆x MCC is expected to ∆φtgt equal −. and ∆x MCC is expected to equal .. Combining Eqs. (.), (.), (.) and (.) yields the following equations: y tgt = (y tgt |y fp )(y PAW + y 0 fp ) + (y tgt |φfp )(φPAW + φ0 fp ) (.) φtgt = (φtgt |y fp )(y PAW + y 0 fp ) + (φtgt |φfp )(φPAW + φ0 fp ) (.) From these equations, y tgt and φtgt are known to constant offsets given experimental data for y PAW and φPAW . This data is plotted in Figs. 3.7 and 3.8 for each of the dipole runs, which were taken at varying beam positions. The error bars on the data points are were calculated using these equations:
  • 54.  6 5 (mm) 4 0 fp +25.55 φ 3 2 0 fp 1 ytgt -1.30 y 0 -1 -2 -3 -2 -1 0 1 2 3 4 5 6 Beam x (mm) Fig. 3.7: Fit of y tgt vs. x MCC (black) compared to the expected slope (red) ∆y tgt = (y tgt |y fp )2 (∆y PAW )2 + (y tgt |φfp )2 (∆φPAW )2 ∆φtgt = (φtgt |y fp )2 (∆y PAW )2 + (φtgt |φfp )2 (∆φPAW )2 Fig. 3.7 shows that the expected slope fits the data reasonably well, given the size of the error bars. The fit lines on Fig. 3.8 are also close to the expected slope, although the calculated error bars are much smaller and do not always reach the fit lines. In Fig. 3.8, the data point at x = . appeared to be increasing the slope of the black fit line, so a second fit line was drawn, excluding this point. This new fit line is closer to the expected slope, but still does not match. However, the error bars on this plot are probably some- what underestimated, and the fit line slope is acceptably close to the slope expected. One possible reason that the fit lines of φtgt vs. x MCC do not have the expected slope is that Eqs. (.) and (.) use only first-order cosy coefficients. There may be higher-order effects for large values of x MCC , which are farther from the central beam position. The data recorded did not contain all the values needed to do a full higher-order analysis, but the experimental data indicates that there are non-linear effects not accounted for in
  • 55.  4 3 (mrad) 2 0 fp -1.19 φ 1 0 fp 0 φtgt +0.02 y -1 -2 -3 -2 -1 0 1 2 3 4 5 6 Beam x (mm) Fig. 3.8: Two fits of φtgt vs. x MCC (black fitting all points, and blue fitting four points) compared to the expected slope (red) these equations. According to Eqs. (.) and (.), both y tgt and φtgt should vary linearly with changing x MCC , but this does not appear to be the case, at least in the plot of φtgt vs. x MCC . Similarly, Eq. (.) predicts that y fp and φfp vary linearly in x MCC , but Figs. 3.9 and 3.10 suggest that the dependence of y fp and φfp is non-linear far from the central beam position. To avoid any non-linear effects in the final analysis, only data taken at the central beam position (x MCC ≈ . mm) was used. 3.7 Solving for the quadrupole offsets The six variables solved for in the analysis were the three quadrupole magnet offsets s¹, s ² and s ³ and the three coordinate system offsets y 0 fp , φ0 fp and y 0 tgt . These variables are related to each other by Eqs. (.) to (.) and (.). Eq. (.) yields two equations for each magnet setting. Six magnet settings were used in the final analysis, so there were  equations and six variables: an overdetermined system of equations. Among the data input to these equations was the beam position x MCC and the measured position and angle of the beam at the focal plane, y PAW and φPAW . The values of y PAW and φPAW used were
  • 56.  100 80 60 (mm) 40 0 fp 20 y -y fp 0 -20 -40 -60 -3 -2 -1 0 1 2 3 4 5 6 Beam x (mm) Fig. 3.9: Linear fit of y PAW vs. x MCC . The error bars on y PAW were too small to draw. from the combined set of data as described in Sec. 3.1.2 and shown in Figs. 3.1 to 3.3. Only data taken at the central beam position was used. The δ terms in Eq. (.) were included in the analysis, but they only had an effect on the value of χ ² —compare Figs. 3.11 and 3.12, explained in the following section. 3.7.1 Considerations in minimizing the equations In principle, solving this system of equations is simply a matter of running a minimizer program to find the most likely values of the variables. However, the minimizer used was not able to solve for all six variables at once. It was known that the coordinate system offsets would be small, so I tried putting limits in the minimizer, for example by holding − < y 0 fp < . This resulted in the minimizer finding solutions only at the extremes of the specified range, rather than finding a local minimum. The minimizer was able to find reasonable solutions when any of the coordinate system offsets were held to a fixed value, allowing the minimizer to solve for the other five variables. In the system of equations, y 0 fp , φ0 fp and y 0 tgt are all confounded, because no data was taken that was intended to separate them. This may explain why the minimizer could not solve for all six variables
  • 57.  5 4 3 (mrad) 2 1 0 fp φ -φ 0 fp -1 -2 -3 -3 -2 -1 0 1 2 3 4 5 6 Beam x (mm) Fig. 3.10: Linear fit of φPAW vs. x MCC . The error bars on φPAW were too small to draw. at once. The horizontal bend angle φbend was calculated from the quadrupole magnet shifts determined by the minimizer and some cosy coefficients, using Eq. (.). The cosy co- efficients in this equation were calculated for protons, because the GEp-III experiment detected protons in the hms. The cosy coefficients for the optics runs were calculated for electrons. To first order, these coefficients are equal to each other, so no separate cosy script was needed. When the spectrometer arm of the experimental setup was re- configured to accept protons instead of electrons, the polarity on all four magnets was reversed, so that a deflection of an electron to the left in the electron configuration is still a deflection to the left for protons in the proton configuration. 3.7.2 Method used for minimization My minimizer program was written in c++ with the root libraries for data analysis [], using the Migrad minimization algorithm from the Minuit library. This is the generic minimizer algorithm in root. Because the minimizer could not solve for all six variables
  • 58.  7 6 5 χ2/Ndof 4 3 Holding y0 fp fixed 2 Holding y0 tgt fixed Holding φ0 fp fixed y0 fp quartic fit y0 tgt quartic fit 1 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 y0 tgt Fig. 3.11: χ 2 /N dof when holding y fp , φfp or y tgt fixed while minimizing, accounting for δ terms in the analysis at once, I ran the minimizer multiple times, holding one of the three coordinate system offsets fixed at various values. Six magnet settings were used, each giving two equations, and five variables were minimized, so there are seven degrees of freedom. The solutions found were largely independent of which coordinate system offset was fixed. For example, in Tab. 4.1, the row of results corresponding to y 0 fp =  was obtained by holding y 0 fp fixed at , but nearly the same results could have been obtained by holding φ0 fp fixed at ., or by holding y 0 tgt fixed at .. All solution sets agreed with each other to within the error values given by the minimizer. Fig. 3.11 shows the χ ²/N dof of the solutions found when holding each of the coordinate system offsets fixed. The x axis is the value of y 0 tgt found. The χ ² values are similar when holding y 0 fp or y 0 tgt fixed, but χ ² when holding φ0 fp fixed is often much higher. The solu- tion sets found when holding φ0 fp also had larger error values. For the following analysis, the solutions found when holding y 0 fp fixed were used, because χ ² was generally lowest
  • 59.  7 Holding y0 fp fixed Holding y0 tgt fixed Holding φ0 fp fixed y0 fp quadratic fit 6 5 χ2/Ndof 4 3 2 -3 -2 -1 0 1 2 3 y0 tgt Fig. 3.12: χ 2 /N dof when holding y fp , φfp or y tgt fixed while minimizing, not ac- counting for δ terms in the analysis for these solutions. To investigate the dependence of the results on the value chosen for y 0 fp , I ran the minimizer with y 0 fp fixed for values between − and  mm. The values of χ ² and the horizontal bend angle φbend for each value of y 0 fp were then calculated. Fig. 3.12 shows χ ² when running the minimizer without accounting for the δ terms in Eq. (.). Here, χ ² was quadratic rather than quartic, but again χ ² is lowest when holding y 0 fp fixed. The δ terms are included in the following analysis. For each fixed value of y 0 fp , the minimizer ran using multiple combinations of equa- tions to determine the dependence of the solution on the specific set of equations used. According to Eq. (.), each magnet setting gives two equations to be minimized. These pairs of equations were never split up; either both equations from a given setting were included, or neither. The equations corresponding to the dipole setting were always in- cluded in the minimization because the dipole setting served as a baseline of the particle motion in the hms, without the influence of the quadrupole magnets. The nominal and
  • 60.  q2 reduced settings were always excluded, because there was no usable data from these settings. When calculating the quadrupole magnet offset s¹, the equations from the q1 setting were used in every minimization. In like manner, the q2 equations were always included when solving for s ², and the q3 equations when solving for s ³. This was done because these settings provided data on how particles traveled through each quadrupole magnet without the effects of the other quadrupole magnets. The program then cycled through each combination of equations that contained the required settings and that had at least as many equations as unknowns. The minimizer ran for each of these sets of equations and recorded the solutions found. Only solutions which had χ ²/N dof <  were kept, as well as sets of equations having the same number of equations as unknowns, for which χ ² = N dof = . Results from the minimizer were therefore selected based on which magnet settings were included in the minimization and on the value of χ ² for the solution found. The histograms in Fig. 3.13 show all results found in black and the selected results in red for y 0 fp = . Distributions of results found at other values of y 0 fp were similar. The displayed width of each histogram is five times the error value found by the minimizer when using the equations from all magnet settings. Some points fell outside this range, but they would not have formed any significant peaks in the histograms. After cycling through each combination of equations, the values found for the five minimized variables were averaged. See Figs. 4.1 to 4.5 for the results found with y 0 fp fixed at zero. Tab. 4.1 shows the solutions found for various values of y 0 fp between − and  mm, along with the average (solid horizontal line) and standard deviation (dashed lines). 3.7.3 Estimating errors The minimization algorithm outputs error values for each minimized variable. Because y 0 fp is always held fixed, there is no error value for that variable. After running the min- imizer for multiple sets of equations, the standard deviation of the solutions found can
  • 61.  12000 2500 10000 2000 8000 1500 6000 1000 4000 500 2000 0 0 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.4 0.45 0.5 0.55 0.6 0.65 φ (mrad) y (mm) 0 fp 0 tgt 8000 1000 7000 6000 800 5000 600 4000 3000 400 2000 200 1000 0 0 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 1.2 1.25 1.3 1.35 1.4 1.45 1.5 1.55 s1 (mm) s2 (mm) 400 350 300 250 200 150 100 50 0 3.4 3.6 3.8 4 4.2 4.4 4.6 s3 (mm) Fig. 3.13: Selected minimizer results (red) of each of the five minimized vari- ables at y 0 fp = , compared to all results (black). All results are weighted by the inverse square of the error returned by the minimizer.
  • 62.  be used as an estimate of the error for that variable. However, these solutions are not independent of each other, since they are all generated from subsets of the same system of equations. Another way of estimating error values is by choosing one of the error val- ues returned by the minimizer, for example the error value returned when the minimizer solved all equations at once, corresponding to the rightmost points in Figs. 4.1 to 4.5. To determine an error value for y 0 fp , the minimizer can be run holding y 0 tgt fixed in- stead of y 0 fp . As discussed previously, the solutions found are nearly the same holding either of these offsets fixed. The error values found when holding y 0 tgt fixed were also similar to those found when holding y 0 fp fixed. The five minimized equations were functions of the fixed parameter y 0 fp , and thus there was no way of determining which value of y 0 fp was correct. The error values previ- ously discussed estimate the uncertainty for each variable at a given value of y 0 fp . Vari- ation in the solutions found for different values of y 0 fp is another way of estimating the error on each variable. For example, if y 0 fp is assumed to be between − and  mm, the range of values for each variable can be taken to quantify the error for that variable.
  • 63.  CHAPTER 4 Results and Conclusion Figs. 4.1 to 4.5 show the results of minimization using multiple subsets of equations, hold- ing y 0 fp fixed at zero. The average values of the variables are designated by the solid hori- zontal line in each figure, and the standard deviation is shown in dotted lines. Tab. 4.1 shows the results of this method of minimization for values of y 0 fp between − and  mm. The numbers given in the table for φ0 fp , y 0 tgt and the three quadrupole offsets s¹, s ² and s ³ were calculated from the averages of multiple minimizations as in Figs. 4.1 to 4.5. The χ ²/N dof given for each row is calculated from the five minimized offsets and the offset y 0 fp . There were seven degrees of freedom. The horizontal bend angle φbend is calculated from the three quadrupole offset shifts by Eq. (.). For reference, y tgt is also listed, which is equal to y 0 tgt − . by Eq. (.). The error values in Tab. 4.2 correspond to the data in Tab. 4.1 where y 0 fp was held fixed. They are the same as the error bars shown on the rightmost points in Figs. 4.1 to 4.5, where the minimizer used all equations all once. The error value for y 0 fp came from a minimization where y 0 tgt was held fixed instead. The errors did not strongly depend on Tab. 4.1: Final results for |y 0 fp | ≤  mm. In this data, y 0 fp was held fixed, and the minimizer solved for φ0 fp , y 0 tgt , s¹, s ² and s ³. The χ ² data corresponds to blue quartic fit function in 3.11. All units are millimeters and milliradians. y 0 fp φ0 fp y 0 tgt s¹ s² s³ χ ²/N dof φbend y tgt −10.0 −0.41 1.23 1.02 3.31 7.25 2.83 −0.43 0.79 −7.5 −0.28 1.06 0.86 2.83 6.44 2.18 −0.30 0.62 −5.0 −0.15 0.88 0.70 2.34 5.63 1.86 −0.17 0.44 −2.5 −0.01 0.70 0.55 1.86 4.81 1.74 −0.04 0.26 0.0 0.12 0.53 0.39 1.37 4.00 1.76 0.09 0.09 2.5 0.25 0.35 0.23 0.88 3.18 1.82 0.22 −0.09 5.0 0.39 0.18 0.08 0.38 2.37 1.91 0.36 −0.26 7.5 0.52 −0.00 −0.08 −0.11 1.55 1.93 0.50 −0.44 10.0 0.66 −0.18 −0.24 −0.61 0.73 1.88 0.63 −0.62
  • 64.  Tab. 4.2: Final error values for each variable at y fp = . All units are millimeters and milliradians. y 0 fp 0.14 φ0 fp 0.04 y 0 tgt 0.03 s¹ 0.04 s² 0.04 s³ 0.13 φbend 0.05 the value of y 0 fp , so they were not included in Tab. 4.1. The value of χ ²/N dof was less than . for values of y 0 tgt between −. and . mm, tak- ing the green fit line of χ ² in Fig. 3.11, which was obtained by holding y 0 tgt fixed. This fit line gives a slightly narrower range of acceptable values of y 0 tgt than the fit line obtained when holding y 0 fp fixed. The range −. < y 0 tgt < . corresponds to −. < y 0 fp < .. Us- ing instead the blue quadratic χ ² function in Fig. 3.12, χ ² is less than . for in the range −. < y 0 tgt < ., or −. < y 0 fp < .. All three coordinate system offsets were assumed to be small, with |y 0 fp | <  mm and |φ0 fp | <  mrad. In the plots in this chapter and the last chapter, y 0 fp =  was used because it varied more strongly than φ0 fp or y 0 tgt , and a small value of y 0 fp results in small values for the other two offsets. In contrast, setting y 0 tgt = − mm, which would otherwise seem reasonable, gives y 0 fp =  mm. In Eq. (.), the expected value of y 0 tgt was calculated to be −. mm using the survey data, which corresponds to y 0 fp = . mm. From the data in Tab. 4.1, the angles φbend and φ0 fp are strongly correlated and nearly equal to each other, and near y 0 fp = , φbend and all of the coordinate system offsets are close to zero. This suggests that the true values for these variables are all close to zero, unless all of the coordinate system and quadrupole magnets are misaligned in such a way that the overall apparent misalignment is zero. The following equations give each variable as a function of y 0 tgt :
  • 65.  y 0 fp = −14.178y 0 tgt + 7.492 φ0 fp = −0.758y 0 tgt + 0.521 s¹ = 0.888y 0 tgt − 0.079 s ² = 2.778y 0 tgt − 0.108 s ³ = 4.619y 0 tgt + 1.555 φbend = −0.753y 0 tgt + 0.493 2 4 3 2 χ /N dof = 0.884y 0 tgt + 0.024y 0 tgt − 1.098y 0 tgt + 0.347y 0 tgt + 1.910 (with δ) χ 2 /N dof = 0.101y 0 tgt + 0.050y 0 tgt + 2.031 2 (without δ) These are linear fits for the range − < y 0 fp < , with the minimizer accounting for the δ terms in Eq. (.). The δ term actually made some of the equations quadratic, but a linear fit described the data well in this range. The blue fit functions of χ ²/N dof from Figs. 3.11 and 3.12 are also given. The horizontal bend angle φbend is calculated by Eq. (.), reproduced below with the δ terms dropped. (The δ terms did not have a strong effect on the result.) The following row in the equation inserts the numeric values of the cosy coefficients. The last two rows solve for φbend using the quadrupole magnet shifts calculated for y 0 fp = . 3 φbend = (φfp |s i )s i i =1 = 0.25s¹ − 0.80s ² + 0.27s ³ = 0.25 · 0.39 − 0.80 · 1.37 + 0.27 · 4.00 = 0.09 From this calculation, it becomes apparent that the horizontal defocusing in q2 par- tially cancels the defocusing in q3, resulting in a smaller overall horizontal bend angle. This partial cancellation holds for other small values of y 0 fp , so that despite a relatively large calculated offset for q3, the bend angle φbend remains small. The final published data used |φbend | < . mrad from the range of values of φbend found in Tab. 4.1. For com- parison, the nominal resolution of the hms is  mrad. The value of φbend contributed a systematic error γκp ∆φbend on Pt /Pl , where γ is the
  • 66.  Tab. 4.3: Final results and error estimations for µp G E p /G M p . Q ² is in (GeV/c)²; all other numbers are in units of nuclear magnetons. Q² µp G E p /G M p µp γκp ∆φbend Total systematic error Statistical error 5.17 0.443 0.016 0.018 0.066 6.70 0.327 0.020 0.022 0.105 8.49 0.138 0.037 0.043 0.179 Tab. 4.4: Correlation coefficients of the final results at y fp =  φ0 fp y 0 tgt s¹ s² s³ φ0 fp 1 y 0 tgt −0.067 1 s¹ −0.025 −0.731 1 s² −0.000 −0.715 0.615 1 s³ 0.017 −0.605 0.423 0.948 1 proton’s boost factor and κp is its anomalous magnetic moment. Taking ∆φbend < . gives the error contributions shown in Tab. 4.3. The uncertainty due to ∆φbend consti- tutes roughly % of the systematic error for the GEp-III experiment. Final results for µp G E p /G M p are also given in Tab. 4.3. Tab. 4.4 shows the correlation coefficients for the five minimized variables when hold- ing y 0 fp fixed at zero. These coefficients were returned by the root’s minimizer algorithm when solving all equations at once. The correlation matrices for solutions found at other values of y 0 fp are similar. 4.1 Conclusion This research resulted in the most likely values for the horizontal offsets of the three quad- rupole magnets in the hms, as well as the expected horizontal bend angle φbend . The angle φbend was of particular importance to the GEp-III experiment because it directly in- fluenced the spin precession of particles traveling through the hms. This precession, in turn, affected the value of Pt /Pl which is directly proportional to the ratio G E p /G M p . If there were a horizontal deflection φbend through the hms, the longitudinal component of the spin of the scattered proton would precess slightly to the transverse direction at the hms detector. It was determined that the most likely value for φbend was zero, with ∆φbend = . mrad.
  • 67.  0.2 0.15 (mrad) 0.1 0 fp 0.05 φ 0 -0.05 D+Q1 D D+Q2 D+Q3 D+Q1+Q2 D+Q1+Q3 D+Q2+Q3 D+Q1+Q2+Q3 D+Q1+Q2+Q3+Q1r+Q3r D+Q1+Q2+Q1r+Q3r D+Q1+Q3+Q1r+Q3r D+Q2+Q3+Q1r+Q3r D+Q1+Q1r D+Q2+Q1r D+Q3+Q1r D+Q1+Q3r D+Q2+Q3r D+Q3+Q3r D+Q1+Q2+Q1r D+Q1+Q3+Q1r D+Q2+Q3+Q1r D+Q1+Q2+Q3r D+Q1+Q3+Q3r D+Q2+Q3+Q3r D+Q1+Q2+Q3+Q1r D+Q1+Q2+Q3+Q3r D+Q1r+Q3r D+Q1+Q1r+Q3r D+Q2+Q1r+Q3r D+Q3+Q1r+Q3r Fig. 4.1: φ0 fp for sets of equations that include the dipole setting This uncertainty in φbend results in an error γκp ∆φbend on Pt /Pl and on G E p /G M p . This is approximately % of the systematic error in the GEp-III experiment, and is therefore an essential contribution to the estimation of systematic error in the experiment. These findings have been included in the GEp-III paper published in June  [].
  • 68. s1 (mm) y0 tgt (mm) 0.4 0.5 0.32 0.34 0.36 0.38 0.42 0.44 0.48 0.52 0.54 0.56 0.58 D D+Q1 D+Q1 D+Q1+Q2 D+Q2 D+Q1+Q2 D+Q1+Q3 D+Q3 D+Q1+Q3 D+Q1+Q2+Q3 D+Q2+Q3 D+Q1+Q2+Q3 D+Q1+Q1r D+Q1+Q1r D+Q2+Q1r D+Q1+Q2+Q1r D+Q1+Q2+Q1r D+Q3+Q1r D+Q1+Q3+Q1r D+Q1+Q3+Q1r D+Q2+Q3+Q1r D+Q1+Q2+Q3+Q1r D+Q1+Q2+Q3+Q1r D+Q1+Q3r D+Q1+Q3r D+Q2+Q3r D+Q1+Q2+Q3r D+Q1+Q2+Q3r D+Q3+Q3r D+Q1+Q3+Q3r D+Q1+Q3+Q3r D+Q2+Q3+Q3r D+Q1+Q2+Q3+Q3r D+Q1+Q2+Q3+Q3r D+Q1r+Q3r D+Q1+Q1r+Q3r D+Q1+Q1r+Q3r D+Q2+Q1r+Q3r D+Q1+Q2+Q1r+Q3r D+Q1+Q2+Q1r+Q3r D+Q3+Q1r+Q3r Fig. 4.2: y 0 tgt for sets of equations that include the dipole setting D+Q1+Q3+Q1r+Q3r D+Q1+Q3+Q1r+Q3r Fig. 4.3: s¹ for sets of equations that include the dipole and q1 settings D+Q2+Q3+Q1r+Q3r D+Q1+Q2+Q3+Q1r+Q3r D+Q1+Q2+Q3+Q1r+Q3r 
  • 69. s3 (mm) s2 (mm) 4 3.8 3.9 4.2 1.3 1.4 1.5 1.25 1.35 1.45 4.1 D+Q3 D+Q2 D+Q1+Q3 D+Q1+Q2 D+Q2+Q3 D+Q2+Q3 D+Q1+Q2+Q3 D+Q1+Q2+Q3 D+Q3+Q1r D+Q2+Q1r D+Q1+Q3+Q1r D+Q1+Q2+Q1r D+Q2+Q3+Q1r D+Q2+Q3+Q1r D+Q1+Q2+Q3+Q1r D+Q1+Q2+Q3+Q1r D+Q3+Q3r D+Q2+Q3r D+Q1+Q3+Q3r D+Q1+Q2+Q3r D+Q2+Q3+Q3r D+Q2+Q3+Q3r D+Q1+Q2+Q3+Q3r D+Q1+Q2+Q3+Q3r D+Q3+Q1r+Q3r D+Q2+Q1r+Q3r D+Q1+Q3+Q1r+Q3r D+Q1+Q2+Q1r+Q3r D+Q2+Q3+Q1r+Q3r D+Q2+Q3+Q1r+Q3r Fig. 4.5: s ³ for sets of equations that include the dipole and q3 settings Fig. 4.4: s ² for sets of equations that include the dipole and q2 settings D+Q1+Q2+Q3+Q1r+Q3r D+Q1+Q2+Q3+Q1r+Q3r 
  • 70.  APPENDIX A Plots of the Data Runs The following plots show several histograms for each of the optics runs taken: two-dimen- sional histograms of y PAW vs. x PAW and φPAW vs. y PAW , and one-dimensional histograms of x PAW and y PAW . The quantities y PAW and φPAW are described in the text—see page  for a description and Fig. 2.1 for a diagram of the quantities. The quantity x PAW is described in Sec. 3.1.2. For most runs, there are four one-dimensional histograms. The first two were gen- erated with no cuts on the data. The second histogram of y PAW has the corresponding variable cut on x PAW from Tab. 3.1. The second x PAW histogram has a cut on y PAW , also cor- responding to the variable x cut in Tab. 3.1. For runs where it was not possible to isolate the central hole of the sieve slit collimator in either the x or y directions, the correspond- ing histograms were not generated. This was the case for both the x PAW and y PAW cuts in the nominal magnet setting and for the y PAW cut in the q2 reduced magnet setting; these settings were therefore excluded from the analysis. The units on all of the following plots are centimeters and radians. Beam x values in the captions are given in millimeters. The captions designate which data run or runs are plotted (refer to Tab. 2.2). Where two runs are listed, the data from the runs was combined to create the plot.
  • 71.  0.02 20 0.01 0 0 -0.01 -20 -0.02 -20 0 20 -60 -40 -20 0 20 φ vs. y (no cuts) y vs. x (no cuts) 10000 10000 7500 5000 5000 2500 0 0 -20 0 20 -60 -40 -20 0 20 y (no cuts) x (no cuts) 2000 4000 1500 1000 2000 500 0 0 -20 0 20 -60 -40 -20 0 20 y (x cut) x (y cut) Fig. a.1: Dipole only, beam x = . (runs  and )
  • 72.  0.02 20 0.01 0 0 -0.01 -20 -0.02 -20 0 20 -60 -40 -20 0 20 φ vs. y (no cuts) y vs. x (no cuts) 15000 10000 10000 5000 5000 0 0 -20 0 20 -60 -40 -20 0 20 y (no cuts) x (no cuts) 3000 6000 2000 4000 1000 2000 0 0 -20 0 20 -60 -40 -20 0 20 y (x cut) x (y cut) Fig. a.2: Dipole only, beam x = . (run )
  • 73.  0.02 20 0.01 10 0 0 -0.01 -10 -0.02 -20 -20 -10 0 10 20 -60 -40 -20 0 20 φ vs. y (no cuts) y vs. x (no cuts) 20000 15000 15000 10000 10000 5000 5000 0 0 -20 -10 0 10 20 -60 -40 -20 0 20 y (no cuts) x (no cuts) 6000 2000 4000 1000 2000 0 0 -20 -10 0 10 20 -60 -40 -20 0 20 y (x cut) x (y cut) Fig. a.3: Dipole only, beam x = . (run )
  • 74.  0.02 20 0.01 10 0 0 -0.01 -10 -0.02 -20 -20 -10 0 10 20 -60 -40 -20 0 20 φ vs. y (no cuts) y vs. x (no cuts) 20000 20000 15000 10000 10000 5000 0 0 -20 -10 0 10 20 -60 -40 -20 0 20 y (no cuts) x (no cuts) 8000 6000 6000 4000 4000 2000 2000 0 0 -20 -10 0 10 20 -60 -40 -20 0 20 y (x cut) x (y cut) Fig. a.4: Dipole only, beam x = −. (run )
  • 75.  0.02 20 0.01 0 0 -0.01 -20 -0.02 -20 0 20 -60 -40 -20 0 20 φ vs. y (no cuts) y vs. x (no cuts) 10000 10000 5000 5000 0 0 -20 0 20 -60 -40 -20 0 20 y (no cuts) x (no cuts) 2000 6000 4000 1000 2000 0 0 -20 0 20 -60 -40 -20 0 20 y (x cut) x (y cut) Fig. a.5: Dipole only, beam x = . (runs  and )
  • 76.  0.02 20 0.015 15 0.01 10 0.005 5 0 0 -0.005 -5 -0.01 -10 -0.015 -15 -0.02 -20 -20 -10 0 10 20 -60 -40 -20 0 20 φ vs. y (no cuts) x 10 2 y vs. x (no cuts) 2 x 10 1800 1800 1600 1600 1400 1400 1200 1200 1000 1000 800 800 600 600 400 400 200 200 0 0 -20 -10 0 10 20 -60 -40 -20 0 20 y (no cuts) x (no cuts) Fig. a.6: Nominal, beam x = . (run  and )
  • 77.  0.02 20 0.015 15 0.01 10 0.005 5 0 0 -0.005 -5 -0.01 -10 -0.015 -15 -0.02 -20 -20 -10 0 10 20 -60 -40 -20 0 20 φ vs. y (no cuts) y vs. x (no cuts) 60000 50000 50000 40000 40000 30000 30000 20000 20000 10000 10000 0 0 -20 -10 0 10 20 -60 -40 -20 0 20 y (no cuts) x (no cuts) Fig. a.7: Nominal, beam x = . (run )
  • 78.  0.01 10 0.005 0 0 -0.005 -10 -0.01 -10 0 10 -60 -40 -20 0 20 φ vs. y (no cuts) y vs. x (no cuts) 20000 15000 20000 10000 10000 5000 0 0 -10 0 10 -60 -40 -20 0 20 y (no cuts) x (no cuts) 800 15000 600 10000 400 5000 200 0 0 -10 0 10 -60 -40 -20 0 20 y (x cut) x (y cut) Fig. a.8: Dipole plus q1, beam x = . (runs  and )
  • 79.  0.01 20 0.005 10 0 0 -0.005 -10 -0.01 -20 -20 -10 0 10 20 -60 -40 -20 0 20 φ vs. y (no cuts) y vs. x (no cuts) 30000 30000 20000 20000 10000 10000 0 0 -20 -10 0 10 20 -60 -40 -20 0 20 y (no cuts) x (no cuts) 20000 1000 15000 10000 500 5000 0 0 -20 -10 0 10 20 -60 -40 -20 0 20 y (x cut) x (y cut) Fig. a.9: Dipole plus q2, beam x = . (run )
  • 80.  0.01 20 0.005 10 0 0 -0.005 -10 -0.01 -20 -20 -10 0 10 20 -60 -40 -20 0 20 φ vs. y (no cuts) y vs. x (no cuts) 15000 15000 10000 10000 5000 5000 0 0 -20 -10 0 10 20 -60 -40 -20 0 20 y (no cuts) x (no cuts) 1000 8000 6000 750 4000 500 2000 250 0 0 -20 -10 0 10 20 -60 -40 -20 0 20 y (x cut) x (y cut) Fig. a.10: Dipole plus q3, beam x = . (run )
  • 81.  0.02 20 0.01 10 0 0 -0.01 -10 -0.02 -20 -20 -10 0 10 20 -60 -40 -20 0 20 φ vs. y (no cuts) y vs. x (no cuts) 15000 20000 15000 10000 10000 5000 5000 0 0 -20 -10 0 10 20 -60 -40 -20 0 20 y (no cuts) x (no cuts) 2000 4000 1500 1000 2000 500 0 0 -20 -10 0 10 20 -60 -40 -20 0 20 y (x cut) x (y cut) Fig. a.11: q1 reduced, beam x = . (run )
  • 82.  0.02 10 0.01 0 0 -0.01 -10 -0.02 -10 0 10 -60 -40 -20 0 20 φ vs. y (no cuts) y vs. x (no cuts) 10000 10000 7500 7500 5000 5000 2500 2500 0 0 -10 0 10 -60 -40 -20 0 20 y (no cuts) x (no cuts) 400 200 0 -10 0 10 y (x cut) Fig. a.12: q2 reduced, beam x = . (run )
  • 83.  0.02 20 0.01 10 0 0 -0.01 -10 -0.02 -20 -20 -10 0 10 20 -60 -40 -20 0 20 φ vs. y (no cuts) y vs. x (no cuts) 10000 10000 5000 5000 0 0 -20 -10 0 10 20 -60 -40 -20 0 20 y (no cuts) x (no cuts) 2000 1500 1500 1000 1000 500 500 0 0 -20 -10 0 10 20 -60 -40 -20 0 20 y (x cut) x (y cut) Fig. a.13: q3 reduced, beam x = . (run )
  • 84.  APPENDIX B Plots of the Fits The following plots show the Gaussian fits on y PAW and φPAW at the central hole of the sieve slit collimator for each setting and beam position. The central hole was isolated in each of the settings by taking cuts on x PAW and y PAW , and then a Gaussian fit was performed to find the mean value and error of the mean. These numbers were used in the analysis discussed in Chapter 3. In the nominal and q2 reduced settings, it was not possible to isolate the central hole in the y PAW direction, so these runs were not used in the analysis and there are no fits of those data. The black histograms and fit lines correspond to the variable x cuts discussed in Sec. 3.1.2 and given in Tab. 3.1. The histograms and fit lines is plotted in red correspond to the fixed x cuts given in Tab. 3.2. As in the previous appendix, the units on all of the following plots are centimeters and radians. Beam x values in the captions are given in millimeters. The captions designate which data run or runs are plotted (refer to Tab. 2.2). Where two runs are listed, the data from the runs were combined to create the plot.
  • 85.  140 200 120 175 150 100 125 80 100 60 75 40 50 25 20 0 0 0 1 2 3 -0.002 0 0.002 y fits φ fits Fig. b.1: Dipole only, beam x = . (runs  and ) 250 160 200 140 120 150 100 80 100 60 40 50 20 0 0 1 2 3 4 5 0 0.002 0.004 y fits φ fits Fig. b.2: Dipole only, beam x = . (run )
  • 86.  250 200 175 200 150 150 125 100 100 75 50 50 25 0 0 8 10 12 0.002 0.004 0.006 y fits φ fits Fig. b.3: Dipole only, beam x = . (run ) 225 350 200 300 175 250 150 200 125 150 100 75 100 50 50 25 0 0 -7 -6 -5 -4 -3 -0.004 -0.002 0 y fits φ fits Fig. b.4: Dipole only, beam x = −. (run )
  • 87.  250 180 160 200 140 120 150 100 100 80 60 50 40 20 0 0 -2 -1 0 1 -0.002 0 0.002 y fits φ fits Fig. b.5: Dipole only, beam x = . (runs  and ) 90 60 80 70 50 60 40 50 40 30 30 20 20 10 10 0 0 -1.5 -1 -0.5 0 0.5 -0.002 0 0.002 y fits φ fits Fig. b.6: Dipole plus q1, beam x = . (runs  and )
  • 88.  100 90 80 80 70 60 60 50 40 40 30 20 20 10 0 0 -1 -0.75 -0.5 -0.25 0 -0.004 -0.002 0 0.002 y fits φ fits Fig. b.7: Dipole plus q2, beam x = . (run ) 90 70 80 60 70 50 60 50 40 40 30 30 20 20 10 10 0 0 -1.5 -1 -0.5 -0.002 0 0.002 y fits φ fits Fig. b.8: Dipole plus q3, beam x = . (run )
  • 89.  40 30 35 25 30 20 25 20 15 15 10 10 5 5 0 0 -0.4 -0.2 0 0.2 0.4 -0.002 0 0.002 y fits φ fits Fig. b.9: q1 reduced, beam x = . (run ) 30 30 25 25 20 20 15 15 10 10 5 5 0 0 -0.5 0 0.5 -0.004 -0.002 0 0.002 0.004 y fits φ fits Fig. b.10: q3 reduced, beam x = . (run )
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