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
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.
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 ≈ ₀₀₀₅ . This makes G E p /G M p close to zero in this Q ² range (see
⁰⁷
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.
90.
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