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.
Magnet Displacement in the GEp-III
Experiment at Jeﬀerson Lab
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
Edward Brash, Chair
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.
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
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
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
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
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.
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
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.
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
. (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-
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.
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.
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.
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-
U V Amplifier-discriminator
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
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
e e e e e e
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-
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.
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 ) (.)
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. (.):
G D (Q 2 ) = 1 + (.)
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
Charge or current density
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. (.)
dσ dσ 2 τ 2 1
= GE + GM (.)
dΩ dΩ Mott ε 1+τ
dσ α2 E e cos2 θe
dΩ Mott 4E e sin4 θe
ε = 1 + 2 (1 + τ) tan2 (.)
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
GEp / GD
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
0.01 0.1 1 10
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 (.):
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
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 (.)
I 0 Pt = −2 k · he τ (1 + τ) tan G E G M (.)
I 0 Pn = 0 (.)
2 τ 2
I0 ≡ GE + GM (.)
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
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
0.2 0.3 0.4 1 2 3 4 5 6 7 89
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
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.
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
Fig. 2.1 diagrams the distances and angles of interest in the optics runs. The distance
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
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:
φMCC = − arctan = 0.033◦ (.)
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
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
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 .
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
Tab. 2.2: List of optics runs. The beam x values given were calculated in the
analysis from the bpm values.
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
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
-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
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-
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-
tives include dy
, 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-
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:
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
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 (.):
∆y fp = (∆y PAW )2 + (y fp |δi )(∆δ)s i (.)
∆φfp = (∆φPAW )2 + (φfp |δi )(∆δ)s i (.)
After ﬁnding the three quadrupole shifts, the horizontal bend angle φbend can be de-
termined using Eq. (.):
φbend = (φfp |s i ) + (φfp |δi )δ s i (.)
This equation follows from Eq. (.) by choosing y tgt = φtgt = ; i.e., it gives the horizon-
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 (.)
∂ si i =1
Performing checks using reverse cosy coeﬃcients
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
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
plot the calculated y tgt for each dipole run as a function of beam x to ﬁnd ∆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
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:
f i (x 1 , x 2 , . . . , x m ) − y i
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
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.
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
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
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.
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
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.
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
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