1.ABSTRACTTitle of Thesis: Magnet Displacement in the GEp-III Experiment at Jeerson LabDegree Candidate:Philip Charles CarterDegree and Year: Master of Science, Thesis Directed by:Edward Brash, Ph.D., Associate Professor, Department of Physics,Computer Science and EngineeringThe goal of the GEp-III experiment at Jeerson 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 , from . to . (GeV/c). In this experiment, high-energy electronsstruck a proton target, causing the electrons and protons to scatter. Elastically scatteredprotons 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, whicheach were roughly one meter in diameter, were needed to within a few millimeters. Inorder to measure these displacements, a series of measurements was taken of elasticallyscattered electrons traveling through the spectrometer. Using knowledge of the exper-imental geometry, together with this data, the most likely absolute positions of thesemagnets were determined.

2. Magnet Displacement in the GEp-IIIExperiment at Jeerson Labby Philip Charles CarterThesis submitted to the Graduate Faculty of Christopher Newport University in partialfulllment of the requirements for the degree of Master of Science Approved:Edward Brash, ChairDavid HeddleYelena ProkBrian Bradie 3. 2010 Philip Charles Carter 4. ii DEDICATIONDedicated to my parents Paul and Sandra, to my sister Angie, and to my brother-in-law Cale. Their supportin my seeking a masters degree and their loyalty throughall of the changes in my life have been invaluable. 5. iiiACKNOWLEDGMENTSFirst, 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 thesisresearch, was essential in bringing my degree and thesis to completion. I would alsolike to thank the other members of my thesis committee, David Heddle, Yelena Prok andBrian Bradie, for taking the time to review my thesis and to sit on the committee for mythesis defense. Lubomir Pentchev, the expert on beam optics for the G E p series of experiments atJeerson 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 thesisto Lubomir. He also provided the cosy script used to model particle motion through themagnets of the spectrometer, a core component of the set of programs I used in my thesisresearch. 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 gures used in this thesis. My time at Jeerson Lab and cnu would have been much less enjoyable had it notbeen for my friends in Newport News, especially Micah Veilleux, Jonathan Miller, SelinaMaley 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 contributedinformation 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 Jeerson Lab and from Our Lady of Mount CarmelChurch, has proved to be a valuable friend to me as well. Multiple members of the faculty and sta at cnu assisted me in the completion of 6. iv my degree in one way or another, especially Mary Lou Anderson and Pam Gaddis, theformer and current secretaries of the physics department, and Lyn Sawyer from the Of-ce of Graduate Studies. I would also like to thank all of the professors from whom I tookclasses. Of course, without the eorts of the entire GEp-III collaboration, this experiment couldnot 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. vTABLE OF CONTENTSSectionPage Dedication iiAcknowledgments iiiList of Tables viiList of FiguresviiiChapter 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 osets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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: Methodology17 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 PAW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 PAW and estimating errors . . . . . . . . . . . . . . . . . . . . 3.1.4 Details of tting data for each magnet setting . . . . . . . . . . . . . . . . . . 3.2 Measuring using paw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 3.3 Modeling magnetic elds 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 osets . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 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 Conclusion51 4.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A: Plots of the Data Runs 58Appendix B: Plots of the Fits72Bibliography 79 9. vii LIST OF TABLES NumberPage 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 PAW . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Fixed x PAW cuts used for each magnet setting and beam position x MCC , andcorresponding cuts on y PAW and PAW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 p G E p /G M p . . . . . . . . . . . . . . . . . . . . 4.4 Correlation coecients of the nal results at y fp = . . . . . . . . . . . . . . . . . . . 10. viii LIST OF FIGURES NumberPage 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, andtwo 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 p G E p /G M p data from the rst two recoil polarization experiments at Jeer-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 xed x PAW cut, and com-bined data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Same as Fig. 3.1, zoomed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Measured PAW data using a variable x PAW cut and a xed x PAW cut, and com-bined data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Histogram of from the nominal setting at the nal 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 ts of tgt vs. x MCC compared to the expected slope . . . . . . . . . . . . . . . . 3.9 Linear t of y PAW vs. x MCC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10 Linear t of PAW vs. x MCC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.11 2 /N dof when holding y fp , fp or y tgt xed while minimizing, accounting for terms in the analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. ix 3.12 2 /N dof when holding y fp , fp or y tgt xed while minimizing, not accountingfor terms in the analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.13 Selected minimizer results of each of the ve minimized variables at y 0 fp = ,compared to all results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 0 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 s for sets of equations that include the dipole and q1 settings . . . . . . . . . . . 4.4 s for sets of equations that include the dipole and q2 settings . . . . . . . . . . .4.5 s 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 1IntroductionThe GEp-III experiment was conducted at Jeerson Lab in Newport News, Virginia. Thegoal 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 , from. (GeV/c) to . (GeV/c). 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 theextraction of the electric form factor. In this experiment, high-energy electrons struck a proton target, in the form of liquidhydrogen, causing the electrons and protons to scatter. Scattered protons from elasticcollisions were detected after passing through a magnetic spectrometer, which consistedof three quadrupole magnets and a single dipole magnet. By measuring the positionand angle of the proton using a series of detectors located near the focal plane of thespectrometer, the momentum of the proton can be determined, as well as the degree towhich its spin precessed in the magnetic elements. In particular, a detailed knowledgeof this spin precession was a crucial component in the extraction of the form factor ratiofrom the data. Specically, the absolute positions of the quadrupole magnets, each of which wereapproximately one meter in diameter, were needed to within a few millimeters. In or-der to measure these displacements and rotations, a series of dedicated measurementswere taken of elastically scattered electrons traveling through the magnetic spectrometer,with various magnetic eld strengths in the magnetic elements. This is known as beamoptics data, because the scattered particles are deected as they pass through the seriesof 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 rst quadrupole magnet 14. of the spectrometer allowed only electrons incident at specic angles to pass through.Using the knowledge of the experimental geometry, together with these data, the mostlikely 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 Jeerson Lab. The rst experiment, which was pub-lished in February [], measured the form factor ratio for values of Q between be-tween . and . (GeV/c). GEp-II extended the measurement to . (GeV/c) and waspublished in February []. Data collection for GEp-III took place in and .The beam optics data, necessary to determine the absolute positions of the quadrupolemagnets, were taken in October . The nal results of the GEp-III experiment werepublished 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).1.1 History of nucleon structure studiesThe atomic nucleus was discovered in by Ernest Rutherford []. The nucleus wasshown to have internal structure in , when Rutherford discovered the proton []. Theneutron was discovered by James Chadwick in []. In , Otto Stern measured themagnetic moment of the proton []. In this experiment, Stern found that the protonsmagnetic moment was not that of a point particle of the protons charge and mass; thisdiscrepancy indicated that the proton had an internal structure. The magnetic moment of the neutron was measured in by Luis Alvarez and FelixBloch []. The electric and magnetic form factors of the proton were rst measured inthe 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 aboutone femtometer. For his ndings, Hofstadter won the Nobel Prize in physics in . Starting in the s, experiments revealed further evidence of composite nucleonstructure, with the rst direct evidence of quarks inside the proton published in [].Multiple experiments conducted from the s until the present have used Rosenbluthseparation 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 thanfor the neutron, and the protons magnetic form factor is more easily measured than itselectric form factor for high values of Q . Experiments to date have therefore provided arelative abundance of data for the magnetic form factor of the proton over a wide rangeof values of Q , in comparison to the other electric and magnetic form factors. Recent measurements of G E p and G E n at Jeerson Lab have made signicant contri-butions to the existing data for these form factors. As already discussed, the rst 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 Jeerson Lab, experimente-, will publish in the coming months [].1.2 The GEp-III experiment1.2.1 Experimental techniques for determining proton form factorsThe G E p experiments at Jeerson Lab, together with one other experiment carried outat 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 todetermine 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 values above (GeV/c).Rosenbluth separation has, however, been used to determine the magnetic form factorof the proton with good accuracy above (GeV/c). Recoil polarization can be used to de-termine the ratio G E p /G M p , and experiments using Rosenbluth separation have providedG M p , so the electric form factor G E p can be readily extracted. At present, Jeerson Lab is the only particle accelerator in the world that can pro-duce a beam with sucient intensity and duty factor such that the recoil polarizationtechnique can be used to determine G E p /G M p at high Q ; the technique requires a highlypolarized electron beam and high current in the energy range under study. Jeerson Labcan provide such a beam up to a beam energy of GeV at A and % polariza-tion. This allowed the GEp-III experiment to measure the form factor ratio up to a Q of 16. . (GeV/c). 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: H(e , e p ) The target was made of hydrogen (H), and a polarized electron beam (e ) was used. Thescattered 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 wasmeasured. A series of cuts was placed on the data in order to select only elastic collisions. 1.2.2 The experimental setupThe experiment was conducted in Hall C of Jeerson 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 lled with liquid hydrogen, cooled to K. Scattered elec-trons were detected using a lead-glass calorimeter, and scattered protons were detectedin the High Momentum Spectrometer, or hms. The hms consisted of three quadrupolemagnets and a dipole magnet that led to a detector hut (see Figs. 1.1 and 1.2). Inside thedetector 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 throughthe quadrupole and dipole magnets. The dipole eld strength was set at roughly . T.Given this eld strength and the physical layout of the dipole magnet, protons from thetarget were deected upwards by , allowing them to enter the detector hut. Electronswere detected in a calorimeter, and the time of detection was recorded for both protonsand electrons, in order to select elastic collisions. Other types of charged particles eitherwould not bend at the correct angle when reaching the dipole magnet, or would havea dierent time of ight through the spectrometer and would be excluded as inelasticevents. There was shielding along the direct line of sight between the target and thedetector 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 gure below.FPP1+FPP2 S1X+S1YDC1+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 CH plastic. 18. The three quadrupole magnets of the hms focused the beam of scattered protons. Thebeam was focused in order to allow a wider scattering cross section of protons to reachthe detector. The eld strengths of the quadrupole magnets were individually adjustable,with nominal eld strengths between . and . T. The hms was designed to accept a maximum central momentum of . GeV/c. Themomentum bite is about % and the solid angle acceptance is about msr. The nominalresolution 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 lesserresolution, the angle of any charged particle entering the detector. Data from both driftchambers 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, animaginary plane between the two drift chambers, yielding the focal plane coordinates x fp ,y fp , fp and fp . The x -axis points down and the y -axis points to the left when facing thefocal plane from the target. The angle is measured from the z -axis in the y z -plane and 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 detectedparticles over an area roughly cm tall and cm wide. The polarimeters determine the normal and transverse components of the spin ofincident particles which scatter in the blocks of analyzer material placed before eachpolarimeter. The normal component of the particles spin in the detector precesses inthe dipole magnet from the longitudinal component of the spin at the target, while torst order, the transverse component does not precess. For the beam optics study, thepolarization 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 . Betweeneach signal plane there are two planes of eld wires. The x and x wires are horizontaland measure position in the vertical (dispersive) direction. These two planes of x wiresare oset from each other to avoid a left-right ambiguity. The y and y wires are verti- 19. Y, YUV 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 amplier-discriminator cards are also shown. cal, measuring position in the horizontal direction, and are also oset from each other toavoid left-right ambiguity. The u and v planes are at angles to the x and x wires. Thisconcentration of near-horizontal wires gives the drift chambers better resolution in thedispersive direction, which allows for better reconstruction of the particle momentum.The drift chambers are diagrammed in Fig. 1.3. Each drift chamber was lled with a %/% argon-ethane mixture by weight. High-energy charged particles traveling through the drift chamber leave a trail of ionized gasparticles. These ionized particles drift to the nearest signal wire where they cause a pulsewhich is detected by an amplier-discriminator card. The particles trajectory can be re-constructed from the series of wires which sent a signal as the particle traveled throughthe drift chamber. Because there are six planes of wires in each drift chamber and onlyfour coordinates to determine (x , y , and ), the particles 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 inthe other drift chamber. This method of reconstructing each particles trajectory allows 20. for tracking multiple particle trajectories at once. 1.2.3 Magnet position osetsAs described in the previous section, the spectrometer arm of the experimental setupconsisted 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 upby , allowing the protons to enter the detector hut and be detected. The goal of thisresearch is to determine the absolute displacements of the three quadrupole magnets.Any horizontal displacement would defocus the proton beam, introducing a horizontalbend to the central trajectory through the magnets. This could strongly aect the ratio ofthe 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 ofthe 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 aect onlythe vertical bend angle, and this bend angle was measured and accounted for separately.A horizontal displacement in the dipole magnet would have almost no eect. Therefore,it was not necessary to investigate these osets.1.3 The physics behind GEp-IIIThe GEp-III experiment was designed to probe the interior of the proton by observing theresults from elastic collisions with polarized electrons. In general, when a high-energyelectron collides with a proton, any number of interactions can occur. The simplest ofthese 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 theproton, 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. eee e e e ppp p p p(a) One-photon exchange(b) Two-photon exchange (c) Two-photon exchangeFig. 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 interactionsare 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 thesubject of the GEp- experiment [], experiment number e-, the sister experimentto 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 , which has units of (GeV/c). Q iscalculated using Eqs. (.) to (.): = Ee Ee (.) q = pe pe(.) 2 22 Q = |q| (.) 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 correspond to higher beam energy and a shorterwavelength for the incident electrons, which allows the electron to probe deeper into theproton, revealing the protons internal structure. 22. 1.3.1 Proton form factorsThe physical property of the proton under investigation in this experiment is its Sachselectric 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. Theelectric and magnetic form factors are among the simplest physics observables of thenucleons internal structure. They correspond to the Fourier transforms of the nucleonscharge and current distributions, respectively. The electric and magnetic form factorsare related to the Dirac and Pauli form factors according to Eqs. (.) to (.):G E (Q 2 ) F(Q 2 ) F (Q 2 )(.)22 2G M (Q ) F(Q ) + F (Q ) (.) Q2 (.)4M 2 where F is the Dirac form factor, F is the Pauli form factor, is the anomalous mag-netic moment of the nucleon, and M is the mass of the proton. These form factors arefunctions of Q ; as indicated above, higher Q corresponds to probing deeper into theproton. Low values of Q correspond to bulk charge and magnetization distributions. AQ of . (GeV/c) corresponds roughly to . fm, the radius of the proton. At Q = ,F = F = , so G E p = and G M p = + p = p , the magnetic moment of the proton (approx-imately . nuclear magnetons). Previous experiments have found that both the electric and magnetic form factors ofthe proton can be described by the dipole form given in Eq. (.):2 Q2 G D (Q 2 ) = 1 +(.) 2This corresponds to the charge and current densities of the proton falling o exponen-tially for distances far from the protons center. The dipole form holds approximately forQ less than about one (GeV/c). In the equation, is a constant experimentally deter-mined to be . (GeV/c). The exponential distribution of charge corresponding to the dipole distribution of 23. Charge or current density00Distance 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 thistrend continues all the way to a radius of zero, because the derivative of exp(r ) = atr = . The derivative of the charge or current density of the proton would therefore bediscontinuous in the center of the proton, which is unphysical. As a result, the ratiosG E p /G D and G M p /p G D , which are approximately equal to unity at low Q , must deviatefrom unity at higher values of Q . Both of these ratios have been measured to sucientlyhigh Q in previous experiments to demonstrate that this is indeed the case.For asymptotically large Q (greater than to (GeV/c)), perturbative quantumchromodynamics (pqcd) predicts that the ratio G E p /G M p should become constant. In-termediate Q is the most theoretically challenging region, for which there are multipleconicting theoretical models. It is therefore necessary to measure the ratio G E p /G M p inthis range experimentally, to provide insight into which models may be correct. A por-tion of this intermediate region of Q , . (GeV/c) to . (GeV/c), is the reign of study inthe GEp-III experiment. 1.3.2 Rosenbluth separationOne 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. dd 2 21 =GE + GM(.)ddMott 1+ d2 E e cos2 e 2 = (.) d Mott 4E e sin4 e3 21 e = 1 + 2 (1 + ) tan2(.) 2 Eq. (.) gives the Mott cross section, which is the expected cross section of a pointlike,spin- particle. The portion of Eq. (.) in square brackets, along with the factor of 1/(1 +), is the adjustment to Mott scattering owing to the internal structure of the nucleon.Eq. (.) denes , the longitudinal polarization of the virtual photon. The factor is the ne-structure constant, approximately . E e is the beam energy,E e is the energy of the scattered electron, e is the electron scattering angle in the labora-tory frame, and was given in Eq. (.). Because is proportional to Q , G E dominates Eq. (.) for low Q and G M dominatesfor high Q . Rosenbluth separation has been used eectively to determine G E p up toQ ; 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 .Figs. 1.6 and 1.7 show a representative sample of measurements of G E p and G M p obtainedby Rosenbluth separation. Rosenbluth separation can also be used to determine the neutron form factors G E nand G M n . The neutron, however, is unstable when not in a nucleus, with a half-life ofabout minutes. Because free neutron targets would decay quickly, the neutron must bestudied indirectly using Rosenbluth separation on deuterium (H). Such studies are moredicult than studies of the proton form factors, since free protons are readily available inthe form of hydrogen. As a result, the neutron form factors are known much less preciselythan those of the proton. [][] The electric form factor of the neutron is dicult to separate from the magnetic formfactor when using Rosenbluth separation, in part because G E n is many times smaller thanG M n . Values of G E n measured using Rosenbluth separation have been indistinguishable 25. 1.61.41.2GEp / GD 10.80.6Andivahis et al, 1994Berger et al, 1971Borkowski et al, 1975 0.4Christy et al, 2004Janssens et al, 1966Price et al, 1971 0.2Qattan et al, 2005Simon et al, 1980Walker et al, 1994 0 0.01 0.1 110 Q2 (GeV/c)2Fig. 1.6: Rosenbluth separation data for G E p [] from zero and have had large error bars []. This is another limitation of the techniqueof Rosenbluth separation. 1.3.3 Recoil polarizationTo determine G E p accurately at values of Q greater than (GeV/c), the technique of recoilpolarization was developed, which was rst used in the rst G E p experiment at JeersonLab. In this technique, a polarized electron beam strikes an unpolarized target. Theincident electrons transfer some polarization to the scattered protons. The ratio G E p /G M pcan be determined by measuring the transverse and longitudinal components of the spinof the scattered proton, as described in Eqs. (.) to (.): 26. 1.11GMp/pGD 0.90.8 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 0.1 110 Q2 (GeV/c)2 Fig. 1.7: Rosenbluth separation data for G M p []e E e + E e 2 I 0 Pl = k he (1 + ) tan2 GM (.) 2M e I 0 Pt = 2 k he (1 + ) tan G E G M(.)2 I 0 Pn = 0 (.) 2 2 I0 GE + GM (.) GE Pt E e + E e e=tan (.)GM Pl 2M2 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 (GeV/c) and recoil polarizationcan accurately reveal G E p /G M p , the electric form factor of the proton can be readily ex-tracted. The term in Eq. (.) was given in Eq. (.). Eq. (.) follows immediately fromEqs. (.) 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. 1.81.61.41.2p GE/GMp 1p 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 00.2 0.3 0.4 1 2 345 6 7 89 Q2 (GeV/c)2Fig. 1.8: p G E p /G M p data from the rst two recoil polarization experiments at Jeerson Lab [][], compared to existing data from Rosenbluth separation [] with high current and high polarization. High current was necessary because at the beamenergies under investigation, only a small percentage of collisions in the target result inelastic collisions; data collection at low current would take years or decades. JeersonLab can provide a beam up to A and % polarization. The results of the twoG E p experiments at Jeerson Lab prior to GEp-III have been published and are shown inFig. 1.8. The data from Rosenbluth separation experiments are included for comparison.From this gure, recoil polarization clearly results in higher-quality data for G E p /G M p atvalues of Q 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 measurethe 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 scatteredparticles from the target are measured are called double polarization experiments. Likerecoil 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 protonform factor ratio G E p /G M p using a polarized target with a polarized beam. 29. CHAPTER 2Methodology2.1 The optics data taken2.1.1 Geometry of the experimental setupAs outlined in Chapter 1, to determine the horizontal displacements of the quadrupolemagnets 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, andthus the momentum of the scattered electron would be xed for a given scattering an-gle. The hms was positioned at a . angle to the electron beam. By adjusting theeld strengths of the magnets in the hms, it was congured to accept electrons with amomentum 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 specic angles. Scattered electrons were chosenrather than protons for the optics runs because protons can travel through the metal ofthe collimator. This experimental setup therefore ensured that electrons originating fromelastic collisions at a known location (that of the carbon target) and passing through oneof the holes in the sieve slit collimator were detected in the detector hut. A carbon target was used because such targets can be thinelectrons can scatteranywhere along the intersection of the beam and target, so a thinner target limits onesource of error in determining the location of the collision. Carbon also remains solidat high temperatures, allowing a higher beam current for more frequent collisions andbetter 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 slitfrom the spectrometer axis were measured in a survey. The gure also labels most of thequantities which will be discussed in the following sections. The position of the beamwas recorded from three beam position monitors or bpms along the beam line. Also seethe physical arrangement of the magnets in Fig. 1.1. The quantities y tgt and tgt are the y and positions of the scattered particle at thetarget, respectively, and y fp and fp are the corresponding positions at the focal plane.These four quantities are specied in the spectrometer coordinate system, the primarycoordinate system used in this research. The x -axis points down, the z -axis points fromthe 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 MCCand z MCC , so named because the electron beam was controlled by the Machine ControlCenter, or mcc. The mcc coordinate system is oset from the spectrometer coordinatesystem 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 spectrometercoordinate system is related to the beam coordinate system by the unknown but smalloset y 0 tgt . The quantity y beam is the same as x MCC , scaled to align with the spectrometercoordinate system. At the focal plane side, y PAW and PAW are in the coordinate systemused by the detector array. This coordinate system is supposed to be the same as thespectrometer coordinate system, but may be misaligned by a small amount, quantiedby the osets y 0 fp and 0 fp . The spectrometer coordinate system was considered to bendinside 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 MCCMCC = arctan= 0.033 (.)z MCCy beam = x MCC cos 12.01 + MCC = 0.978x MCC (.)y tgt = y 0 tgt y beam(.) y tgt + 0.24 tgt = arctan(.)y tgt tan 12.01 + MCC + 1659.48= 0.603y tgt 0.145= 0.603y 0 tgt + 0.589x MCC 0.145y fp = y PAW + y 0 fp (.)fp = PAW + 0 fp(.) Eq. (.) gives MCC (not shown in the diagram), the angle of the beam relative to the beamaxis where x MCC = . This quantity was calculated for each run using the bpm data, andfound to be roughly . to the left of the beam axis for each run. Distances are in unitsof millimeters and angles are in units of milliradians, except where explicitly designatedto be in degrees. 2.1.2 Primary goal of this researchThere are seven unknown quantities to be determined in the beam optics studies. Theprimary three quantities solved for were the misalignments in the y direction (using thespectrometer coordinate system in Fig. 2.1) of the three quadrupole magnets: s is the y 32. displacement of the rst quadrupole magnet (q1), and similarly for s and s . Anotherquantity to be solved for is the expected angular deection in the y z -plane of a particleentering the quadrupole magnets along the spectrometer axis (x = , y = ). If all threequadrupole magnets were aligned perfectly, such a proton would travel along the centralaxis of the spectrometer without deecting. Because of misalignments in the quadrupolemagnets, there can be a deection, bend (not shown in the diagram).The nal three unknown quantities are shown in Fig. 2.1: y 0 tgt , y 0 fp and 0 fp . Thequantity 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 0 fp aremeasures of any misalignment in the hms detector array relative to the spectrometer co-ordinate system. These three quantities could either be xed at zero (or another value)while solving for the quadrupole osets, or allowed to vary. In particular, y 0 fp and 0 fp areexpected to be very small, so it was possible either to allow these parameters to vary tosee if the solutions found give small numbers for the parameters, or to hold these valuesxed at zero to facilitate nding a solution for the quadrupole osets and for y 0 tgt .As shown in Eq. (.), the form factor ratio G E p /G M p is proportional to the ratio ofthe scattered protons spin components, Pt /Pl . At Q = 8.5, the highest energy setting ofthe experiment, Pt /Pl . This makes G E p /G M p close to zero in this Q range (see Fig. 1.8), and very sensitive to Pt . If all of the magnets of the hms were aligned perfectly,the longitudinal component of the protons spin at the target, Pl , would precess to thenormal component at the hms detector, while the transverse component Pt would notprecess at all. With horizontal magnet misalignments, particles experience a horizontaldeection bend and Pl at the target precesses slightly to the transverse direction at thedetector. This small precession can combine with the small component of Pt , changingthe measurement of Pt by a large percentage. As a result, an accurate knowledge of thehorizontal positioning of the quadrupole magnets is essential to accurately determine Ptand therefore G E p /G M p . 33. Tab. 2.1: Beam optics settings. All quadrupole eld strengths are relative to nom- inal. Dipole eld strength was always nominal. q1 eld strengthq2 eld strength q3 eld strengthNominal 1 11q1 0.700300q20 0.3959 0q30 0 0.5745q1 reduced0.7 11q2 reduced1 0.71q3 reduced1 10.7Dipole only 0 00 2.1.3 Description of data runs takenA series of beam optics runs were taken with the quadrupole magnets in various cong-urations. The magnet settings used are detailed in Tab. 2.1 and the runs taken are listedin Tab. 2.2. In every run, the dipole magnet eld strength was at its nominal setting. Forthe q1 reduced, q2 reduced and q3 reduced settings, one quadrupole magnet was set at% of its nominal eld strength, and all other magnets were at nominal eld strength.Data runs were also taken with all magnets at their nominal eld strengths, and with allquadrupole magnets were turned o. This list of congurations was based on LubomirPentchevs technical note [] on similar work done for GEp-II. The purpose of these runswas to investigate the way electrons traveled through the quadrupole magnets. This datacould then be used, in combination with knowledge of the physics involved and datafrom a survey of the equipment, to determine any osets from the expected positions ofthe quadrupole magnets. The rst runs taken used the so-called nominal setting, i.e. all three quadrupole mag-nets were set at their nominal eld strengths. Next, all of the quadrupole magnets wereturned o, leaving only the dipole magnet turned on. However, the q3 eld strength readback at G after turning o the quadrupole magnets, despite the current through allmagnets reading at zero. A procedure was followed which attempted to degauss q3, inorder to eliminate any residual magnetic eld. However, the degaussing was found tohave no eect on the readout for the q3 eld strength. Next, run was taken withthe current on q3 set to zero, despite a nonzero eld 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 65959Nominal1.66 539 183Before degaussing q3 65960Nominal1.66 927 239Before degaussing q3 659611.67 0Junk 659621.67 0Junk 65963Dipole only1.65 143 959After degaussing q3 659641.66 0Junk 65965Dipole only1.6654 462After degaussing q3 again 65966Dipole only2.43 260 741 65967Dipole only5.29 500 000 65968Dipole only 2.34 400 000 65969Dipole only0.45 6 952 65970Dipole only0.45 195 761 65971q1 plus dipole 0.4513 727 65972q1 plus dipole 0.45 394 359 65973q2 plus dipole 0.44 491 325 65974q3 plus dipole 0.45 285 053 65975Nominal0.46 377 604 65976q1 reduced 0.47 303 991 65977q2 reduced 0.50 201 150 65978q3 reduced 0.47 252 151 then taken, with a current on q3 chosen to make the eld strength read near zero. Af-ter this run, an alternate method of degaussing q3 was used. After turning o q3, theeld strength read near G, as before. After this, run was taken. Observingthat neither degaussing had any eect, it was concluded that the eld strength in q3 wasactually close to zero when q3 was turned o, and that the reading for the eld strengthwas incorrect. Next were runs through , a series of runs at varying beam x positions withonly the dipole magnet turned on. These runs were taken to nd the central hole ofthe sieve slit and set the beam x MCC position such that the central hole would appearnear y PAW = in the data (in its own coordinate systemsee Fig. 2.1). All of the holes ofthe sieve slit had the same diameter, . cm, except for the central hole which had adiameter of . cm. This allowed the central hole to be identied: the histograms ofy PAW showed a smaller peak for the central hole. See, for example, the histograms of y inFig. 2.2, also shown in Figs. a.4 and a.5. In Fig. .(a), the central hole the central hole was 35. 25000 12000 200001000015000 8000 60001000040005000200000-20 -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 wasnear y = , with larger peaks on each side, but y values with an absolute value greaterthan about cm were outside the acceptance of the spectrometer. As a result, it is onlypossible 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 beammoved. This is why these two runs have approximate numbers listed for the number ofgood events in Tab. 2.2, while all of the other runs have exact counts of events. Upon nding the central sieve slit hole and positioning the beam so that the centralhole 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 o and theeld 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 dfp plane, regardless of their initial angle: dtgt = (see Fig. 2.1). The q2 and q3 settings werealso set for point-to-parallel focusing. The eld strengths for these three settings werecalculated assuming no osets in the positions of the quadrupole magnets, using cosyinfinity [], a software package for modeling beam physics to arbitrary order using dif- 36. ferential algebra.2.2 Equations using the optics data2.2.1 Setting up the equationsWhile traveling through the quadrupole and dipole magnets, particles are deected in apredictable way. The magnetic eld 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 experiments conguration of magnets.For each magnet conguration, this script outputs the coecients used to project anelectrons position from the target side of the magnets to the detector side, or vice versa.The coecients for propagating the electron from the target side to the detector side arehere referred to as cosy coecients, and coecients propagating in the other directionas reverse cosy coecients. Equations using the cosy coecientsThe cosy coecients gave the derivatives of x , , y , and t at the focal plane with re-spect to x , , y , , t and at the target, where y and are in the directions shown onFig. 2.1, x is perpendicular to the page in that gure, is the out-of-page angle and t isp fp p tgt time. The quantity =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 reversecosy coecients 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 dyand d , written more compactly as (y |y ) and (y |); the second-orderderivatives are (y |y ), (y |y ) and so on. Six additional derivatives were needed for theanalysis: the derivatives of y fp and fp with respect to the three quadrupole shifts s, s and s . To nd (y fp |s) and (fp |s), The cosy script was modied by shifting q1 by + mmin 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 fp were taken. The derivatives withrespect to s and s were found in a similar way. The derivatives of y fp and fp with respect to y tgt , tgt and were used to determinethe horizontal osets (along the y axis) of the quadrupole magnets. The derivatives ofother focal plane variables were unnecessary, and the derivatives of y fp and fp with re-spect to the other target variables did not contribute strongly to the results because thesecoecients were small. Taking only the derivatives with respect to y tgt , tgt and , the general equation for y fpand fp using cosy coecients to rst order is given by: 3y fp (y fp |y tgt ) (y fp |tgt ) y tgt(y fp |s i ) + (y fp |i ) =+si(.)fp(fp |y tgt ) (fp |tgt ) tgt(fp |s i ) + (fp |i ) i =1 This equation provides the means to set up two equations for each set of data. FromEqs. (.) to (.), y tgt and tgt are functions of the beam position and angle, x MCC andMCC , and the oset y 0 tgt . The position and angle of the electrons at the detector canbe found by analyzing the data for each run. To account for any misalignment of thedetector, the y and values found in the analysis are designated y PAW and PAW , and arerelated to the values y fp and fp in Eq. (.) by Eqs. (.) and (.). The momentum term can be measured from the data as well. The error values for y fp and fp in the aboveequation were calculated using Eqs. (.) and (.):3 2y fp = (y PAW )2 +(y fp |i )()s i(.) i =13 2fp =(PAW )2 + (fp |i )()s i (.) i =1After nding the three quadrupole shifts, the horizontal bend angle bend can be de-termined using Eq. (.):3bend =(fp |s i ) + (fp |i ) s i(.)i =1This equation follows from Eq. (.) by choosing y tgt = tgt = ; i.e., it gives the horizon- 38. tal bend angle of a particle entering the spectrometer along the spectrometer axis. Thecosy coecients below are calculated for protons traveling through the spectrometer atenergies used in the experiment, with all magnets set to their nominal eld strengths.Eq. (.) gives bend , the error of bend , which is a function of the cosy coecients andthe calculated errors on s i : 3 23 bend2 bend = (s i )2 = (fp |s i ) + (fp |i ) (s i )2 (.) i =1 sii =1 Performing checks using reverse cosy coecients y tgt There are two ways of calculating the expected value of x MCC , the amount by which y tgtwould change if the beam were moved in the x MCC direction. As a consistency check onthe data, this quantity was calculated using both methods, comparing the results. Oney tgt solution follows immediately from Eqs. (.) and (.), which givex MCC= .. Thisratio can also be calculated using the reverse cosy coecients and a series of runs takenat various beam x positions, with the quadrupole magnets turned o. Eq. (.) gives y tgtfor given values of y fp and fp , and Eq. (.) gives tgt :y tgt = (y tgt |y fp )y fp + (y tgt |fp )fp (.) tgt = (tgt |y fp )y fp + (tgt |fp )fp (.) For a given dipole run, the quantities y fp , fp and the beam x are known to within con-stant osets, 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 nd x MCC. Similarly, tgt x MCCis found to equal . from Eq. (.), and can also be calculated using Eq. (.)and experimental data. 2.2.2 Solving the equationsEq. (.) 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 s, s and s in terms of 39. the beam position and angle, the y and positions for each run as measured from thedata, and three coordinate system osets. The unknown values in the resulting equationsare the three quadrupole shifts and the three other osets.Data were taken for each of the eight settings listed in Tab. 2.1 with the beam posi-tioned such that y PAW and PAW would be small (runs through in Tab. 2.2).The eight magnet settings give a total of equations by Eq. (.). These equationsand six unknowns form an overdetermined system of equations. In theory, if there wereno 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 wouldprovide redundant information, and the system of equations would reduce to six linearlyindependent equations, from which the six unknowns could be readily determined. Inpractice, there are unknowable measurement errors and the system of equations is in-consistent. However, it is still possible to nd the most likely values of the six unknowns.This is accomplished by attempting to quantify the error i associated with each func-tion f i and assigning a value to the system of equations, where the functions f i and are functions of the six unknowns. These unknowns are then varied until the value of is at a minimum. If the system of equations were consistent, the minimum would bezero; for an inconsistent system of equations, N dof is considered a good result, whereN dof is the number of degrees of freedom, equal to the number of equations minus thenumber of unknowns. For any overdetermined system of N equations and m unknownsof the form y i = f i (x 1 , x 2 , . . . , x m ), can be calculated as follows: N 2f i (x 1 , x 2 , . . . , x m ) y i2 = i =1 iFor the optics equations, the x i are the three quadrupole magnet shifts and three co-ordinate system osets. The y i are y fp and fp for each magnet setting, from Eq. (.). Theerrors i are estimated by determining the errors in measurement of y PAW and PAW . Themost likely values of the six unknowns can then be found by minimizing using a min-imization program. This program will return values of the unknowns along with errorvalues for each unknown, according to how strongly is changed when the value for 40. each unknown is varied. The i terms in the above equation are given by Eqs. (.) and (.). In these equa-tions, the measurement error on was very small in comparison with the measurementerrors y fp and fp . As a result, the error values i used in the minimization were deter-mined by the measurement errors on y fp and fp . Although there were other sources ofmeasurement error in the experiment, none were easily quantiable. Also, many othersources of error were indirectly accounted for in the errors in y fp and fp . For example, aninstability in the beam position would have resulted in wider peaks in the histograms ofy PAW and x PAW in the data and probably a larger error of the mean when tting a Gaussiancurve to the data. After nding the most likely values of the quadrupole osets, solving for bend isstraightforward. From Eq. (.), it is a function only of the quadrupole osets and ofthree cosy coecients. The error bend can be quantied using Eq. (.), where thevalues s i are the errors of s i found in the minimization. 41. CHAPTER 3AnalysisTo solve for the most likely osets of the quadrupole magnets, it was rst necessary todetermine the values of all measured data and calculated data. The values of y PAW , PAWand were found by analyzing the collected beam data using paw. The magnetic eldcoecients used in Eqs. (.) to (.) were calculated using cosy. The beam position atthe target was calculated using the position data reported by the beam position monitors(bpms). Finally, the most likely osets of the quadrupole magnets were determined usinga minimizer program.3.1 Measuring y PAW and PAWThe rst step after taking beam data was to analyze the raw data and then determine they position and angle at the focal plane of the hms. The raw data was analyzed usingthe Hall C engine, the standard analysis code used in Hall C at Jeerson Lab. This codeoutputs data les for each run which can be read and analyzed using paw. It is thenpossible to make one- and two-dimensional histograms of the data with various cuts. Byapplying appropriate cuts on the data, it was possible to determine the values of y PAW andPAW for each setting. These cuts were intended to select only the electrons that passedthrough the central hole of the sieve slit collimator, in order to measure the deection ofa 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 inthe horizontal direction. For most magnet settings, the central hole is readily apparentin histograms of y PAW vs. x PAW . See for example Figs. a.9 to a.11 and a.13, which each showa clear series of peaks in the y direction, corresponding to the columns of holes in thecollimator. 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 thesesettings were excluded from the analysissee 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 PAWaxis is vertical but only the horizontal particle deection was of primary interest. As aresult, data from particles that traveled through any of the sieve slit holes aligned withthe central hole in the y direction would be serviceable. However, for some settings thevalues of y PAW and PAW varied depending on x PAW . Also, in some settings a cut around ornear 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 directionThere were two possible methods of performing cuts on y PAW and x PAW . One was to plota two-dimensional histogram from the data and use a two-dimensional cut in a looparound 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 atable (see tables 3.1 and 3.2), and there was no apparent need to use a more complicatedcut for any of the magnet settings. Referring to the y vs. x histograms in Appendix a, itcan be seen that rectangular cuts on y vs. x are always sucient 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 PAWwas performed using a histogram of all y PAW data for a given setting. A histogram of x PAWwas then generated using this cut on y PAW and performed a cut on x PAW . Lastly, this cutwas 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 o the tails of the peak in the histogram, to assist in tting thedata. In this way, a cut of y PAW rened by a cut in x PAW was obtained that helped to isolatethe central hole. 3.1.2 Methods of performing a cut on x PAWIn 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 thedata collection. As a result, there was little variation in the central hole peak positionin x PAW . Compare for example the histograms of x PAW for the ve dipole runs (Figs. a.1to a.5), which have widely varying values of y PAW , but the peaks in x PAW remain essentiallyconstant. For the q1, q2 and q3 settings, the central hole in x PAW was not visible but therewas 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 rst was to imple-ment a cut around the central hole in x PAW when the central hole is visible. For the q1, q2and q3 settings, the cut was instead around the peak in x PAW . Using these cuts helped toisolate the central hole in y PAW , especially for the q1 setting. Compare the histograms ofy 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 cutson y PAW and PAW which were chosen after the x PAW cut was applied. For the q2 reducedsetting, the central hole in x PAW was visible but the central hole in y PAW was not. The x PAWcut 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 cutfor all settings, since it was not strictly necessary to isolate the central hole in x PAW andthere was little variation in the position of the central hole for the settings where it wasvisible. The limits of the cut were chosen by comparing the x PAW cuts used on settingswhere 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 tothe central hole, but it provided a simple means to select a cut in x PAW that was expectedto 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 positionx MCC , and corresponding cuts on y PAW and PAWx MCC (mm)x PAW (cm)y PAW (cm)PAW (rad) Dipole 1.65 0.8 to 60.5 to 3.60.002 to 0.0035 Dipole 2.43 1 to 6.5 0.7 to 5 0.0015 to 0.0045 Dipole 5.29 0 to 5.56.5 to 12 0.0015 to 0.0075 Dipole 2.342 to 67.2 to 2.80.0055 to 0 Dipole 0.45 1 to 62.8 to 1.8 0.0035 to 0.0025 q1 0.45 0 to 101.5 to 0.60.003 to 0.0025 q2 0.44 1 to 8.5 1.2 to 00.004 to 0.003 q3 0.45 1 to 8 1.6 to 0.20.0035 to 0.0025 q1 reduced 0.47 0.6 to 4.5 0.5 to 0.50.003 to 0.003 q2 reduced 0.50 2 to 3.5 q3 reduced 0.47 1.5 to 4 0.9 to 0.50.004 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 PAW x MCC (mm)x PAW (cm) y PAW (cm) PAW (rad)Dipole 1.65 1.6 to 4.51 to 4 0.004 to 0.005Dipole 2.43 1.6 to 4.5 0.4 to 5.5 0.003 to 0.006Dipole 5.29 1.6 to 4.56 to 12.50 to 0.009Dipole 2.341.6 to 4.57.5 to 20.007 to 0.001Dipole 0.45 1.6 to 4.52.5 to 1.8 0.003 to 0.003q1 0.45 1.6 to 4.51.5 to 0.6 0.005 to 0.0045q2 0.44 1.6 to 4.51.1 to 0 0.003 to 0.0025q3 0.45 1.6 to 4.51.6 to 0.10.003 to 0.002q1 reduced 0.47 1.6 to 4.50.5 to 0.5 0.003 to 0.003q3 reduced 0.47 1.6 to 4.50.8 to 0.4 0.005 to 0.005 Tab. 3.2 shows the y PAW and PAW cuts chosen after applying this cut on x PAW . Both methods of choosing a cut in x PAW were tried, nding values and error estimatesfor y PAW and PAW for each set of cuts. This resulted in two sets of mean values and errorsfor both y PAW and PAW . These two sets of results were combined by choosing a meanand error bar such that the new error bars spanned the error bars obtained by using thexed and variable x PAW cuts. The two sets of y PAW and PAW data and the combined set areplotted in Figs. 3.1 to 3.3. 3.1.3 Fitting y PAW and PAW and estimating errorsHaving applied a y PAW cut around the central hole of the sieve slit and either a xed orvariable cut on x PAW , a Gaussian curve was tted to the histogram of y PAW to nd its mean 45. 10864 yPAW (mm) 20-2-4-6Dipole Q1 Q2 Q3Q1rQ3r Dipole Dipole DipoleDipolex=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 xed x PAW cut(blue). Combined data is black. Settings are labeled by the magnet congura-tion and the beam x value. Fig. 3.2 provides a zoomed view of the six settings atthe central beam position. value. The cut on y PAW was tight enough to exclude the tails of the peak corresponding tothe central hole. This was done because the tails of the peaks in the data were not neces-sarily Gaussian, but a Gaussian curve t well to the area closer to the peak. The area tby the Gaussian curve extended to between and . away from the peak, dependingon the particular data being t. The mean of the Gaussian curve was taken as the meanvalue of y PAW , and the error on the mean returned by the tting command was taken asthe estimated error of the mean.After tting y PAW , a plot was made of PAW with the cuts on y PAW and x PAW applied.With these cuts, the histogram of PAW always had a single peak. Next, a cut was appliedon PAW to select the peak, cutting o the tails. The cuts extended to between . and. away from the peak, depending on the particular data set. With these three cuts, aGaussian curve was tted to PAW and the mean and error of the mean recorded. 46. 0-0.2yPAW (mm)-0.4-0.6-0.8-1 Dipole Q1 Q2 Q3 Q1rQ3r x=0.45 x=0.45 x=0.44 x=0.45x=0.47 x=0.47Fig. 3.2: Same as Fig. 3.1, zoomedFollowing 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 ts used foreach setting and beam position are shown in Appendix b. 3.1.4 Details of tting data for each magnet settingDipole settingThe runs taken at the dipole setting yielded some of the simplest data to analyze. Becauseall of the quadrupole magnets were turned o, there was no beam focusing and onlyparticles very close to the central trajectory of the hms reached the detector. Peaks iny PAW and x PAW were easily distinguished. Data was taken at the dipole setting at severalbeam positions until the central hole of the sieve slit was identied, the central hole beingsmaller than the others and so corresponding to a smaller peak on the histogram of y PAW .The methods described above of nding y PAW and PAW at the central hole worked without 47. 0.0050.0040.0030.002 PAW (mm) 0.001 0-0.001-0.002-0.003 Dipole Q1 Q2 Q3Q1rQ3r Dipole Dipole DipoleDipole 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=-2.34Fig. 3.3: Measured PAW data using a variable x PAW cut (red) and a xed x PAW cut (blue). Combined data is black. Settings are labeled by the magnet congura- tion and the beam x value. modication for the dipole setting. The dipole runs are plotted in Figs. a.1 to a.5. Nominal settingFor data taken at the nominal setting, the central sieve slit hole was not visible either iny PAW or x PAW . The nominal runs are plotted in Figs. a.6 and a.7. In the histograms of y PAWvs. x PAW , at least ve tails are visible for values of x PAW < which appear to correspond toseparate sieve slit holes in y PAW , but such tails were always excluded from the analysis ofthe other runs. Given its large negative values of x PAW , there was no reason to believe thatthe events in the central tail of the histogram came from a hole in x PAW near the centralhole of sieve slit. An attempt was made to nd the values of y PAW and PAW at the centralhole by doing a linear t of the central tail. Extrapolating the t line to x PAW wouldgive an approximate value for y PAW at the central hole, and the slope of the line wouldgive PAW . However, the linear t had large error bars and was not useful for nding 48. either y PAW or PAW . As a result, the nominal runs were excluded from the analysis. q1 settingIn 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, adjacentpeaks. One way to nd y PAW at the central hole is to t three Gaussian curves to the cen-tral peak and the two adjacent peaks. However, there is no apparent way to nd PAW atthe central hole using this method. A better method is to perform a cut on x PAW whichisolates the central hole in y . Compare the histograms of y PAW before and after applyingthe cut on x PAW in Fig. a.8. With such a cut, it is straightforward to perform Gaussian tson both y PAW and PAW at the central hole.There was a single peak in the histogram of x PAW , located near where the central holein x PAW was expected to be. One possible cut on x PAW was to choose limits centeredaround this peak; this is the x PAW cut listed in Tab. 3.1 and applied in the second histogramof y PAW in Fig. a.8. The other possible cut on x PAW was the xed cut listed in Tab. 3.2. Eitherof these cuts helps to isolate the central hole in y PAW . q2 and q3 settingsLike the q1 data, the q2 and q3 data each have a single peak in x PAW , so there were twooptions for choosing a cut on x PAW : tting around the peak or using the xed x PAW cut.The central hole in y PAW is easily distinguished in these settings, even without a cut onx PAW , so these settings presented no special challenges in nding the values of y PAW andPAW at the central hole. The q2 run is plotted in Fig. a.9, and the q3 run is plotted inFig. a.10. q1 reduced and q3 reduced settingsOut 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 eachhole 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 thehistograms of y PAW with no cuts on x PAW applied, the peak corresponding to the centralhole is larger than the adjacent peaks. After applying a cut on x PAW around the centralhole, 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 passingthrough other sieve slit holes that have the same y position as the central hole. The q1reduced run is plotted in Fig. a.11, and the q3 reduced run is plotted in Fig. a.13. q2 reduced settingIn the data for the q2 reduced setting, the central hole in x PAW is visible but the centralhole in y PAW is not. As can be seen in the second histogram of y PAW in Fig. a.12, applying acut on x PAW is ineective for nding the central hole in y PAW . Cuts on PAW likewise revealno distinct peaks in y PAW . As a result, the values of y PAW and PAW at the central hole couldnot be found, and this setting was excluded from the analysis.3.2Measuring using pawp fp p tgt To determine the eect of the (y fp |) and (fp |) cosy coecients, the value of =p tgtwas determined. This value was calculated per event by the analysis code engine usinga method that was accurate only for the nominal magnet setting. The value of didnot depend on the magnet setting, so the value taken was from the nominal setting atthe nal beam position. Fig. 3.4 shows a histogram of , with the elastic peak centeredaround .%. The mean and standard deviation of the elastic peak were determined bydoing a Gaussian t. Because > , it appears likely that the beam energy was slightlyhigher than what was expected during the data collection. 50. 6000 5000 4000 3000 2000 10000 -12 -10-8-6 -4 -202 (percent) Fig. 3.4: Histogram of from the nominal setting at the nal beam position. The elastic peak is centered around = .%. 3.3Modeling magnetic elds using a cosy scriptTo predict the motion of particles through the magnets, cosy infinity was used. A cosyscript was written that described the positions of the quadrupole and dipole magnets inthe hms, the spacing between them and their nominal eld strengths. Then, given a par-ticle type and momentum, the script outputs a table of coecients describing particlemotion through the magnets from the target to the focal plane, or vice versa. I modi-ed this script by adjusting the quadrupole magnet eld strengths to model each of themagnet congurations listed in Tab. 2.1. I also adjusted the target position in the scriptslightly to account for a misalignment revealed in a survey. I set the script to use elec-trons at . GeV/c, although the rst-order cosy coecients do not depend on beamenergy or particle mass, and only rst-order terms were used in the nal analysis. The other modication I made to the cosy script was to add a mm shift for each ofthe quadrupole magnets in order to nd the cosy coecients of y fp and fp with respectto each of the quadrupole shifts. Details of the coecients returned by the script aregiven in Sec. 2.2.1. Because the cosy script can run at only one magnet setting and conguration 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 outputwas stored in a separate le. These multiple scripts and output les, along with most ofthe rest of the components of my solving program, were coordinated using a makeleto reduce the possibility for human error, and to automate the process of nding cosycoecients.3.4 Determining the beam positionThe beam position was recorded using three bpms, which continuously read out x MCCand y MCC . Using the x MCC readings and the known z MCC position of each bpm, it waspossible to determine the value of x MCC at the target (where z MCC = ) using a t line. Thist line also determined MCC , the angle of the beam with respect to the beam axis. Thebeam position sometimes varied slightly even when a change in beam position was notrequested, so the values of x MCC and MCC at the target were calculated for each run. Theangle MCC was found to be approximately . for each run. See Fig. 3.5 for the t lineused for the q1 setting. Fit lines for the other settings looked similar.3.5 Using survey dataSurveys were taken of the area near the target to measure any mis-pointing between thebeam and spectrometer central axes, as well as any misalignment of the central hole ofthe sieve slit collimator. The survey results relevant to this optics study are shown inFig. 3.6. From this data, it was found that the central hole of the sieve slit was . mmout of position in the y direction, the target was . mm farther down the beamline thanintended, and the central beam axis was mis-pointed by . mm. The distance betweenthe z = axis and the sieve slit collimator was measured to be . mm. The sieve slit oset of . mm was accounted for in Eq. (.). The other survey resultswere used to calculate the distance between sieve slit collimator and the portion of thetarget 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 0Beam x (mm) -0.2C -0.4 -0.6B-0.8 -1 -1.2 A-1.4-3500 -3000 -2500 -2000 -1500 -1000-5000Beam z (mm) Fig. 3.5: Beam position readings projected to the target for the q1 data. Thepoint labeled a is the beam x reading from bpm a placed at its z position alongthe beamline, and similarly for bpms b and c. The red point gives the value ofx MCC used in the analysis. The distance of . mm was calculated from the geometry in Fig. 3.6 using the followingequation: 0.380.38sin(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 oset in the y direction ofthe 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 oset by . mm,using Eq. (.):0.38tan(12.01 ) 0.9= 0.20 (.)sin(12.01 ) cos(12.01 ) There was also a survey of the bpms, yielding osets in the x MCC direction between. and . mm for each bpm. However, although the bpm positions were needed for theoptics 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 t line is already very good. Introducing . mmosets in the x MCC direction would make it more dicult to t a straight line to the bpmdata.3.6 Performing checks on the data y tgt As discussed in the section on reverse cosy coecients on page , x MCCis expected totgt 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 |fp )(PAW + 0 fp )(.) tgt = (tgt |y fp )(y PAW + y 0 fp ) + (tgt |fp )(PAW + 0 fp ) (.)From these equations, y tgt and tgt are known to constant osets given experimentaldata for y PAW and PAW . 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 werecalculated using these equations: 54. 65(mm) 4 0 fp+25.55 32 0 fp1ytgt -1.30 y 0-1-2-3-2 -1012 3 4 5 6Beam 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 |fp )2 (PAW )2 tgt = (tgt |y fp )2 (y PAW )2 + (tgt |fp )2 (PAW )2Fig. 3.7 shows that the expected slope ts the data reasonably well, given the size ofthe error bars. The t lines on Fig. 3.8 are also close to the expected slope, although thecalculated error bars are much smaller and do not always reach the t lines. In Fig. 3.8,the data point at x = . appeared to be increasing the slope of the black t line, so asecond t line was drawn, excluding this point. This new t line is closer to the expectedslope, but still does not match. However, the error bars on this plot are probably some-what underestimated, and the t line slope is acceptably close to the slope expected.One possible reason that the t lines of tgt vs. x MCC do not have the expected slope isthat Eqs. (.) and (.) use only rst-order cosy coecients. There may be higher-ordereects for large values of x MCC , which are farther from the central beam position. Thedata 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 eects not accounted for in 55. 43(mrad)2 0 fp-1.19 1 0 fp 0tgt +0.02 y -1-2 -3 -2 -1 012 345 6Beam x (mm) Fig. 3.8: Two ts of tgt vs. x MCC (black tting all points, and blue tting fourpoints) compared to the expected slope (red) these equations. According to Eqs. (.) and (.), both y tgt and tgt should vary linearlywith changing x MCC , but this does not appear to be the case, at least in the plot of tgtvs. x MCC . Similarly, Eq. (.) predicts that y fp and fp vary linearly in x MCC , but Figs. 3.9and 3.10 suggest that the dependence of y fp and fp is non-linear far from the centralbeam position. To avoid any non-linear eects in the nal analysis, only data taken atthe central beam position (x MCC . mm) was used.3.7 Solving for the quadrupole osetsThe six variables solved for in the analysis were the three quadrupole magnet osets s,s and s and the three coordinate system osets y 0 fp , 0 fp and y 0 tgt . These variables arerelated to each other by Eqs. (.) to (.) and (.). Eq. (.) yields two equations foreach magnet setting. Six magnet settings were used in the nal analysis, so there were equations and six variables: an overdetermined system of equations. Among the datainput to these equations was the beam position x MCC and the measured position andangle of the beam at the focal plane, y PAW and PAW . The values of y PAW and PAW used were 56. 100 80 60(mm)40 0 fp20y -y fp 0 -20 -40 -60-3-2 -1012 34 56Beam x (mm) Fig. 3.9: Linear t 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. Onlydata taken at the central beam position was used. The terms in Eq. (.) were includedin the analysis, but they only had an eect on the value of compare Figs. 3.11 and 3.12,explained in the following section. 3.7.1 Considerations in minimizing the equationsIn principle, solving this system of equations is simply a matter of running a minimizerprogram to nd the most likely values of the variables. However, the minimizer used wasnot able to solve for all six variables at once. It was known that the coordinate systemosets would be small, so I tried putting limits in the minimizer, for example by holding < y 0 fp < . This resulted in the minimizer nding solutions only at the extremes of thespecied range, rather than nding a local minimum. The minimizer was able to ndreasonable solutions when any of the coordinate system osets were held to a xed value,allowing the minimizer to solve for the other ve variables. In the system of equations,y 0 fp , 0 fp and y 0 tgt are all confounded, because no data was taken that was intended toseparate them. This may explain why the minimizer could not solve for all six variables 57. 5 4 3(mrad)21 0 fp -0 fp-1 -2 -3 -3 -2 -1 0 12 345 6Beam x (mm)Fig. 3.10: Linear t of PAW vs. x MCC . The error bars on PAW were too small to draw. at once.The horizontal bend angle bend was calculated from the quadrupole magnet shiftsdetermined by the minimizer and some cosy coecients, using Eq. (.). The cosy co-ecients in this equation were calculated for protons, because the GEp-III experimentdetected protons in the hms. The cosy coecients for the optics runs were calculatedfor electrons. To rst order, these coecients are equal to each other, so no separatecosy script was needed. When the spectrometer arm of the experimental setup was re-congured to accept protons instead of electrons, the polarity on all four magnets wasreversed, so that a deection of an electron to the left in the electron conguration is stilla deection to the left for protons in the proton conguration. 3.7.2 Method used for minimizationMy 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 genericminimizer algorithm in root. Because the minimizer could not solve for all six variables 58. 7 6 5 2/Ndof 4 3 Holding y0 fp fixed 2Holding y0 tgt fixedHolding 0 fp fixedy0 fp quartic fit y0 tgt quartic fit 1 -2 -1.5 -1-0.500.5 1 1.5 2 y0 tgtFig. 3.11: 2 /N dof when holding y fp , fp or y tgt xed while minimizing, accountingfor terms in the analysis at once, I ran the minimizer multiple times, holding one of the three coordinate systemosets xed at various values. Six magnet settings were used, each giving two equations,and ve variables were minimized, so there are seven degrees of freedom.The solutions found were largely independent of which coordinate system oset wasxed. For example, in Tab. 4.1, the row of results corresponding to y 0 fp = was obtained byholding y 0 fp xed at , but nearly the same results could have been obtained by holding0 fp xed at ., or by holding y 0 tgt xed at .. All solution sets agreed with each otherto within the error values given by the minimizer.Fig. 3.11 shows the /N dof of the solutions found when holding each of the coordinatesystem osets xed. The x axis is the value of y 0 tgt found. The values are similar whenholding y 0 fp or y 0 tgt xed, but when holding 0 fp xed is often much higher. The solu-tion sets found when holding 0 fp also had larger error values. For the following analysis,the solutions found when holding y 0 fp xed were used, because was genera