Magnet Displacement in the GEp-IIIExperiment at Jefferson Lab

Philip Carter

Christopher Newport University

July

Overview

• The GEp-III experiment measuredGE p/GM p for Q² from . to . (GeV/c)²

• GEp-I and GEp-II measured from. to . (GeV/c)²

• Scattered protons analyzed by amagnetic spectrometer

• Three quadrupole magnets and adipole

• Needed the absolute positionsof the quadrupole magnets towithin a few millimeters

Elastic electron scattering

γ∗

p

e

p′

e′• Reaction: ¹H(~e , e ′~p )

• We were only interested in elastic events

• Q²: four-momentum-transfer-squared

ω= Ee − E ′eq= pe −p

′e

Q2= |q|2−ω2

• Q² increases with higher beam energy

• Higher energy corresponds to shorter wave-lengths for the electron, probing deeperinto the proton

Proton form factors

• GE and GM :

– Among the simplest physical observables of the proton’s in-ternal structure

– Electric and magnet form factors

– Fourier transform of charge and current distributions

• Size of the proton (. fm)→Q²= . (GeV/c)²

• Low Q² data→ bulk charge and magnetization distribution:µpGE/GM =

• Asymptotically large Q² data (over – (GeV/c)²)→ pqcd

scaling

• Intermediate Q²: non-perturbative qcd, quark confinement,etc. This is the regime under study in GEp-III.

Proton form factors

• For Q²< (GeV/c)², GE and GM

follow the dipole form:

GD(Q2) =

1+Q2

Λ2

−2

Λ2= 0.71 (GeV/c)²

• The dipole form corresponds to charge and current densitiesfalling off exponentially

0 0

Cha

rge

or c

urre

nt d

ensi

ty

Distance from center

Rosenbluth separation

• Measure scattering cross section and compare it to Mott scat-tering

• Used for over years

• GM dominates for large Q², GE for small Q²

• Difficult to calculate GE for Q²> (GeV/c)²

dσ

dΩ=

dσ

dΩ

Mott

G 2E +τ

εG 2

M

1

1+τ

dσ

dΩ

Mott=α2E ′e cos2 θe

2

4E 3e sin4 θe

2

ε=

1+2 (1+τ) tan2 θe

2

−1

Existing Rosenbluth proton data

2 (GeV/c)2Q0.01 0.1 1 10

D /

GE

pG

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Andivahis et al, 1994Berger et al, 1971Borkowski et al, 1975Christy et al, 2004Janssens et al, 1966Price et al, 1971Qattan et al, 2005Simon et al, 1980Walker et al, 1994

2 (GeV/c)2Q0.1 1 10

DG pµ/

Mp

G

0.7

0.8

0.9

1

1.1

Andivahis et al, 1994Bartel et al, 1973Berger et al, 1971Borkowski et al, 1975Christy et al, 2004Janssens et al, 1966Price et al, 1971Qattan et al, 2005Sill et al, 1993Walker et al, 1994

• Good data for GE at Q²< (GeV/c)²

• Good data for GM up to Q²= (GeV/c)²

Recoil polarization

GE

GM=−

Pt

Pl

Ee + E ′e2M

tanθe

2

• The ratio of transverse to longitudinal polarization of the scat-tered proton directly measures the form factor ratio

• Measure polarization using a secondary analyzing reaction

• Measures the relative sign of GE and GM

• Better sensitivity to GE at high Q²

• Needs a highly polarized electron beam and high beam current

• Jefferson Lab: – µA, –% polarization

Existing recoil polarization data

2 (GeV/c)2Q0.2 0.3 0.4 1 2 3 4 5 6 7 8 9

p M/G

p E G pµ

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Andivahis et al, 1994Bartel et al, 1973Berger et al, 1971Christy et al, 2004Crawford et al, 2007Gayou et al, 2002Jones et al, 2000Jones et al, 2006Maclachlan et al, 2006Milbrath et al, 1998Qattan et al, 2005Ron et al, 2007

The experimental setup

• Conducted in Hall C of Jefferson Lab

• Detectors: BigCal and Focal Plane Polarimeter

• hms: quadrupole magnets, dipole magnet

• drift chambers, each with planes of signal wires

S0

DC1+DC2

S1X+S1Y

FPP1+FPP2

2CH

U V

′X, X

Y, Y′

cardsAmplifier-discriminator

Magnet position offsets

• Particles in hms focused by quadrupole magnets

• Deflected upwards by ° in dipole magnet

• Horizontal misalignment in quadrupole magnets→ horizontaldeflection

• Vertical misalignment→ vertical deflection

• Effects of rotation of quadrupole magnets difficult to separatefrom deflections

Importance of determining φbend

GE

GM=−

Pt

Pl

Ee + E ′e2M

tanθe

2

• Recoil polarization formula depends on spin at the target

• Pl at the target precesses in the dipole magnet to Pn at the focalplane

• Pt does not precess

• With a horizontal deflection φbend, Pl at the target would pre-cess slightly to Pt at the focal plane

• Pt approaches zero near Q²= (GeV/c)², so accurate knowledgeof φbend is essential

• Took data to determine the offsets of the quadrupole magnets

Beam optics

• Beam hits target, passes through central hole of sieve slit andthrough focal plane

• Particles detected in fpp

• Took data at several configurations of quadrupole magnets

• Solved for most likely magnet offsets

Magnet settings

q1 field strength q2 field strength q3 field strengthNominal 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

• Took several hundered thousand events at each settings

• Dipole field strength was always nominal

• q1, q2 and q3 settings were point-to-parallel focusing:dφfp

dφtgt=

Beam optics runs

Number ofRun 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

65963 Dipole only 1.65 143 959 After degaussing q3

65965 Dipole only 1.66 54 462 After degaussing q3 again65966 Dipole only 2.43 260 741

65967 Dipole only 5.29 500 000

65968 Dipole only −2.34 400 000

65969–65970 Dipole only 0.45 202 713

65971–65972 q1 plus dipole 0.45 408 086

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

Beam optics equations

φMCC=−arctan

∆xMCC

∆z MCC

= 0.033

ybeam= xMCC cos

12.01+φMCC

= 0.978xMCC

ytgt= y0 tgt− ybeam

φtgt=−arctan

ytgt+0.24

ytgt tan

12.01+φMCC

+1659.48

=−0.603ytgt−0.145

=−0.603y0 tgt+0.589xMCC−0.145

yfp= yPAW+ y0 fp

φfp=φPAW+φ0 fp

• Used knowledge of geometry and survey data to set up equa-tions

• Each magnet setting gave a pair of values for yPAW and φPAW

Measuring yPAW and φPAW

-0.02

-0.01

0

0.01

0.02

-20 -10 0 10 20

φ vs. y (no cuts)

-20

-10

0

10

20

-60 -40 -20 0 20

y vs. x (no cuts)

0

5000

10000

15000

-20 -10 0 10 20

y (no cuts)

0

5000

10000

15000

20000

-60 -40 -20 0 20

x (no cuts)

0

2000

4000

-20 -10 0 10 20

y (x cut)

0

500

1000

1500

2000

-60 -40 -20 0 20

x (y cut)

q1 reduced

-0.01

-0.005

0

0.005

0.01

-10 0 10

φ vs. y (no cuts)

-10

0

10

-60 -40 -20 0 20

y vs. x (no cuts)

0

5000

10000

15000

20000

-10 0 10

y (no cuts)

0

10000

20000

-60 -40 -20 0 20

x (no cuts)

0

5000

10000

15000

-10 0 10

y (x cut)

0

200

400

600

800

-60 -40 -20 0 20

x (y cut)

q1

Modeling the hms with cosy

• Propogation of particles through the magnets in the hms canbe modeled using cosy infinity

• The collaboration created a cosy script for the hms

• Script outputs coefficients relating target variables to focal planevariables

• Coefficients used: derivatives of yfp and φfp with respect to ytgt,

φtgt and δ=p fp−p tgt

p tgt

• I modified the script to find coefficients associated with quad-rupole magnet offsets at each magnet setting: derivatives of yfp

and φfp with respect to s¹, s² and s³.

Solving the optics equations

• Quantities to solve for: y0tgt, y0fp, φ0fp, s¹, s² and s³

•φbend can be derived from s¹, s² and s³

yfp

φfp

=

(yfp|ytgt) (yfp|φtgt)

(φfp|ytgt) (φfp|φtgt)

ytgt

φtgt

+

3∑

i=1

(yfp|si )+ (yfp|δi )δ

(φfp|si )+ (φfp|δi )δ

si

φbend=

3∑

i=1

(φfp|si )+ (φfp|δi )δ

si

• Each magnet setting gives a set of equations for yfp and φfp

• Results in an overdetermined set of equations

χ2=

N∑

i=1

f i (x1, x2, . . . , xm )− yi

σi

2

Minimizer program

• Written in c++ and root

• Set up system of equations and minimized

• Used a generic function minimizer (Migrad/Minuit)

• Minimized out of variables at a time

• Ran minimizer multiple times over a range of values of the fixedvariable

Results

• Held y0 fp fixed in the minimizer

• Holding y0 tgt or φ0 fp fixed gave the same results

y0 fp φ0 fp y0 tgt s¹ s² s³ χ²/Ndof φbend ytgt

−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

φ0 fp y0 tgt s¹ s² s³ φbend

Errors: 0.04 0.03 0.04 0.04 0.13 0.05

Results

Total sys- StatisticalQ² µpGE p/GM p µpγκp∆φbend tematic error error5.17 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

• Used a conservative limit of |∆φbend|< . mrad

• Uncertainty in φbend is responsible for % of the systematicerror in the experiment

• These results are included in the GEp-III paper recently pub-lished in Physical Review Letters