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					      First Study of Three-body
Photodisintegration of He with Double
         Polarizations at HIgS
          Ph.D. Dissertation Defense

                   Xing Zong
             Committee members:
           Prof. Haiyan Gao (Advisor)
              Prof. Thomas Mehen
               Prof. John Thomas
               Prof. Henry Weller
                 Prof. Ying Wu
                 Feb 19th, 2010
• Introduction and Physics Motivation
• The Experiment
 –   Overview
 –   HIGS working principle
 –   Polarized 3He Target
 –   Neutron Detection
• Data Analysis
• Result and Discussion
• Summary & Outlook
•   Understanding Nuclear force has been
    a fundamental goal in nuclear physics:
    Hideki Yukawa: exchange of pion accounted
    for the force between two nucleons

•   Two nucleon (NN) system can be
    described by
    realistic NN potentials:
    –   long range one-pion exchange,
        intermediate attraction, short-range
    –   Modern NN potentials include: AV18[1] and
        CD Bonn[2].                                   [1] R.B.Wiringa et al. PRC 51, 38 (1995)
                                                      [2] R.Machleidt et al. PRC 53, R1483
    –   NN potentials reproduce NN scattering         (1996)
        database up to 350 MeV with high precision.
    –   They underbind triton
                   Three-nucleon system
•   Excellent testing ground of theory
     – simplest non-trivial nuclear system
     – sufficient complex to test the details of
     – small enough to allow exact calculations
     – Hamiltonian is written as
                     Pi 2
                H        Vij  Vijk
.                  i 2m     i j  i  j k

• Three-nucleon Force (3NF)                                          △
     (a) Fujita and Miyazawa first introduced 3NF in
     1957[1]; △ isobar yields an effective 3NF
     (b) Urbana IX is one of the most widely used
                                                       [1] J. Fujita and H. Miyazawa, Prog.
                                                       Theor. Phys. 17, 360 (1957)
                                                       [2] J. Carlson, et al. Nucl. Phys. A
                                                       401, 59 (1983)
Three-nucleon system: 3He

 ~90%              ~2%              ~8%

Polarized 3He is an effective neutron target
               Movitation I: Test 3-body calculations
   g  He  p  p  n

                   Deltuva            Golak
Calculation        AGS                Faddeev
NN                 CD Bonn            AV18
3 NF               △ isobar           Urbana IX

Nuclear            Siegert            Expliticit MEC:
EM current         theorem for        single nucleon
                   1body electric     current+ two (p
                   current, explict   r) body current
                   MEC for
                   multipoles and
                   h.o. terms E
Include            Yes                Only in bound
Coulomb?                              states
Relativistic       Yes                No                2005 Nagai data @10.2 and 16 MeV (green)
                      Motivation II: Test GDH sum rule
                 d                      2p 2 2
                                      
                                                                     Fundamental Interpretation: any particle
                   N     N   
                     P          A
                                               N                    with a nonzero anomalous magnetic
          thr
                                         MN 2
                                                                     moment has internal structure

                                              496b                   3He   GDH Sum Rule
     thr
             GDH 3 He
                                                   217  39b ???                       HIgS @ DUKE
     thr
             GDH 3 He                                              Extrapolated from low Q2 3He GDH (E94-010)
                                                                   measurement @ JLab, (E97-110 much lower Q2)
      2  3GeV                              247  38b
     p
                   GDH 3 He
                               31.9  9.6b
                                                                                                              

   2  3GeV
                  GDH 3 He                             23GeV
                                                                 GDH 3 He  Pn  
                                                                                  2 3GeV
                                                                                            GDHn  2  Pp  
                                                                                                                        GDH p
                                                        0.87  35  2  (0.027)  (26)

[1] M. Amarian, PRL 89, 242301(2002)   [2] J. L. Friar et al. PRC 42, 2310 (1990) [3] N. Bianchi, et al. PLB 450, 439 (1999)
Few-body calculations of GDH integral up to p

                                                                               Compare to our
                                                                               simple estimation:

                                                                              217  39b ???

                                                                              It is crucial to carry
                                                                              out 3-body
                                                                              measurement to
                                                                              provide stringent
                                                                              test of the theories!

Deltuva et al. PRC 72, 054004 (2005): Green (with △ isobar) and Blue (without △ isobar)
Golak et al. PRC 67, 054002 (2003) (Black curve)
• Introduction and Physics Motivation
• The Experiment
 –   Overview
 –   HIGS working principle
 –   Polarized 3He Target
 –   Neutron Detection
• Data Analysis
• Result and Discussion
• Summary & Outlook
                Experimental Overview

1. Beam: HIgS provides
   circularly polarized gray
   @ 11.4 MeV
2. Polarized 3He target: flip
   target spin to form
3. ONLY neutrons are
   detected! 7 detectors
   from 50 to 160 degs.
          High Intensity g-ray source

Progress in Particle and
Nuclear Physics 62 (2009) 257,
Henry R. Weller, et al.
             Experimental Setup

                          preserving mirror

22mm                                          target


Experiment Setup@ Duke FEL
 Spin Exchange Optical Pumping (SEOP)
  • Rb vapor in a weak B field is optically pumped

                                      N2 buffer gas

  • Spin exchange of hybrid alkali
Rb only:   Rb  3He  Rb  3He 

Hybrid:    Rb   K  Rb  K 
           K  3He  K  3He 
                                         Largest 3He cell ever made!
                          NMR Polarimetry
• A magnetic moment when placed in an
  external B-field
                       
             |lab  g M  B       g   gyromagnetic ratio
• Transform into a rotating frame rotates
  around the B field at frequency w
                        
         dM          dM             
             |lab          |rot w  M
          dt           dt
•   The motion of M in the rotating frame
                                       
             dM                  w
                    |rot  gM   B0  
               dt                     g 
• Apply oscillating RF field
                                                      
                        B coswt x  B1 sin wt y 
    B x  2 B1 coswt x   1                          
                            B1 coswt x  B1 sin wt  y

• Effective field in the rotating frame at
  frequency  wz
                 Beff  ( B0  ) z  B1 x
                                 ˆ      ˆ
NMR - Adiabatic Fast Passage (AFP)
• Ramp the holding field B0 z from below the
  resonance w / r to above it
                            Beff    ( B0  ) z  B1 x
                                              ˆ      ˆ

• AFP line shape

  B0    0
                        B0  w / g    
                                      2 1/ 2

      Amplitude of voltage at
      resonance, proportional to the
      sample polarization.
Water calibration to extract 3He Polarization
• The ratio of 3He signal to water signal
                0h wPn nV h           w                   rf frequency
           R p 
                0 wPn nV  p           Pn                  nuclear polarization
                                                        n     number density
• The definition of polarization                        V     volume of the cell
                         n  n                         
                     pn                                     magnetic moment
                         n  n
• The polarization of proton in water is given
   Pnp  tanh pw p / kTpg p    p w p / kTp g p
• The polarization of 3He is

          Pnh 
                Vw 2  2 p n pTh R                    p h pressure of 3 He cell

                  Vw h g pT p p h                    T h temperature of 3 He cell

W.Lorenzon et al. Phys. Rev. A, 47, 468 (1993)
K.Kramer et al. Nucl. Inst. Method A, 582, 318 (2007)
                   Spin up/down curves

The typical spin up/down curves:
measurement every 3 hrs. 17 mV
corresponds to ~40% polarization.

               During the run, the average
               polarization was 42%, and
               quite stable.
                Neutron Detection
                                             Important info from the signal:
                                             ADC: Pulse-height (energy)
                                             TAC: Pulse-shape
                                             discrimination (particle
                                             TDC: Time of flight (time)

                                        g                               n

                                        1D TAC: Proportional to the length of the
The traditional PSD working principle   trailing edge of the detector signal,
                                        therefore measure the particle type.
• Introduction and Physics Motivation
• The Experiment
 –   Overview
 –   HIGS working principle
 –   Polarized 3He Target
 –   Neutron Detection
• Data Analysis
• Result and Discussion
• Summary & Outlook
         11.4 MeV Run summary

Run Summary:                                Data Analysis Overview:
D2 run:               used as a             1. Calibration: relate ADC and
calibration of main experiment                 TDC to PH (energy) and
3He  run: took spin P and A alternatively      TOF (timing) information.
to form asymmetry and to reduce             2. Cuts: separate gammas and
systematic uncertainties.                      neutrons
Al run: determine gamma peak position       3. Integrated flux determination
to obtain timing information
                                            4. GEANT4 simulation to
N2 run: background subtraction                 determine the acceptance
                    Calibration I: ADC

Determine Pedestal (offset         Compare Cs source runs with simulation
resulting from electronics bias)   to find tune gains (energy per channel)
               Calibration II: TDC

Determine Gamma peak TDC   Compare D2 run and simulation to
position                   determine C (TDC Calibration constant)
     How to select neutrons?

  Gammas                   Neutron                                       Cuts Values
                                                                         PSD 6

                                                                         PH          0.2 MeVee

                                                                         TOF 1.1 MeV

Note: 1.  roughly corresponds to 30 TAC channels, 2. ee means electron equivalent
Cut effects

              En=1.1 MeV
              Monte Carlo Simulation
• Geant4 Simulation helps to determine
   – the ADC, TDC calibration constant
   – The back detector efficiency (for
      flux determination)
   – Main detector acceptance
• The acceptance is the convoluted
  effect of all the factors: the extended
  target effect and detecting efficiency of
  the detectors.
• G4 simulation was ran twice under the
  same conditions first with a point
  target (Run 1), then with 40cm long
  target (Run 2). Then divide the number
  of detected neutrons from Run 2, by
  the number of neutrons into the
  detectors from Run 1.
    Integrated flux determination
        g D  pn
       1257  36b
              A sin 2 
1. We use back detectors to monitor
   the gamma flux.
2. The info was used to extract DXS.
3. The D2 calibration run is based on
   the same principle.
Normalization Issue

                 Normalized Yields by
                    back detectors:
                 1. Spin P (black), spin
                    A (red)
                 2. A downward trend
                    is observed, which
                    gives rise to false
           Gamma peak normalization

                            Gamma peak method:
                            1. Only provide a relative
                               (not absolute) photon
                            2. Cell-dependent.
                            We use it to get relative
                              integrated flux between
                              spin P and A.

Compare run-to-
run stabilities
between back
detectors and
gamma peak
         Systematic uncertainty study
• Uncertainty from analysis cuts
 PSD cut: vary from 5 to 7
 PH cut: vary from 0.19 MeVee to 0.21 MeVee (5% change)
 TOF cut: vary the trailing edge by from 1.0 MeV to 1.2 MeV (+/- 3 ns).
• Uncertainty from HIgS beam
  Integrated Photon Flux: different methods for asymmetry and DXS
  Beam polarization: we assume 5% relative uncertainty
• Uncertainty from target
 Target polarization: 4% (NMR/EPR measurement)
 Target Thickness: 2%(uncertainty in the density measurement,
 temperature change)
• Introduction
• The Experiment
 –   Overview
 –   HIGS principle
 –   Polarized 3He Target
 –   Neutron Detector
• Data Analysis
• Result and Discussion
• Summary & Outlook
D2 differential cross section
                     Goal of this run is
                     consistency check:
                     • calibration
                     • data selection (cuts)
                     • simulation
                     • normalization

   A*sin2()         The two fits give us very
                     similar results. Using the
                     Bsin2()+C fit result, we
                     obtained total cross
                     section :1247+/-45 b
                     in agreement with world data:
                     0 = (1257+/-36 b)
                    I. Asymmetry Results

Systematic study includes PSD, PH,
TOF cuts variations, beam and target
polarization, and integrated photon

Top two curves are from Deltuva (CD
Bonn), bottom two from Golak (AV18).
En starts from 1.1 MeV. The fitted
average asymmetry agrees with
theory within 2.
      II. Unpolarized differential cross section
                                               CD Bonn

                                               CD Bonn+ △
   is the detector acceptance which
includes both detector efficiency and
the extended target effect.

Data is from En=1.1 MeV, corrected by
simulation (model-dependent) from 0.

 Statistical uncertainty:
                III: Total Cross Section
   Two methods:
   1. Fit the data with a
      constant times the AV-18
      curve (725b), the
      constant is about 1.053
   2. Expand the DXS:
            a P cos 
    d    l l

 Fit results: a  61 .7  2.9 b

Total cross section result:

776±18(stat.) ±32(sys.)±11(mod)b

Compared with 05 Nagai data at
10.2 MeV, our datum agrees with
theory much better!
• Introduction & Physics Motivation
• The Experiment
 –   Overview
 –   HIGS working principle
 –   Polarized 3He Target
 –   Neutron Detection
• Data Analysis
• Result and Discussion
• Summary & Outlook
• We carried out a first study of three-body
  photodisintegration of 3He at HIgS with 11.4 MeV
  circularly polarized photons.

• We have extracted three sets of results: asymmetry,
  unpol DXS and TXS.

• Results are compared to two sets of state-of-the-art
  three-body calculations from Deltuva and Golak using
  CD Bonn and AV18 potentials.

• Fitted average asymmetry is within 2 of the theoretical
  value, unpolarized DXS agrees reasonably with theory.

• Total cross section is obtained by two methods. The final
  result agrees with theoretical calculation much better
  than 2005 Nagai data.
• A new proposal was approved by HIGS
  physics advisory committee (PAC) in July
• PAC granted us 180 hrs to run
  measurements at three photon energies.
• Beam time could be as early as fall 2010.
3He   Three-body Photodisintegration
      Collaboration @ Duke HIgS
M. Ahmed, C. Arnold, M. Blackston, W. Chen, T.
    Clegg, D. Dutta, H. Gao (Spokesperson/
Contact Person), J. Kelley, K. Kramer, J. Li, R.
 Lu, B. Perdue, X. Qian, S. Stave, C. Sun, H.
 Weller (Co-Spokesperson), Y. Wu, Q.Ye, W.
            Zheng, X. Zhu, X. Zong

       Duke University, Durham, NC
• U.S. DOE contract number DE-FG02-
• Duke University School of Arts and
• TUNL MEP group, Capture group, TUNL
  staff & FEL staff
• Dr. Deltuva and Dr. Golak and their
          Historical Background
Nuclear interactions and Nobel Prizes
1949 Hideki Yukawa: exchange of pion
   accounted for the force between two
1969 Murray Gell-Mann:
   existence of more fundamental
   particles, ie. quarks

2004 Gross/Politzer/Wilczek: discovery
      of asymptotic freedom in QCD
                     Photos are from

Pseudoscaler Mesons: quark and   Vector Mesons: quark and
anti-quark spin antiparallel     anti-quark spin parallel

J=1/2             J=3/2
                   Nuclear force
• Range of nuclear force is ~ radius of alpha particle, 1.7 fm
• Intermediate attraction: nuclear binding
• Repulsive core. Proof: NN scattering data, energies above
  300 MeV, the s-wave phase shifts becomes negative
• Nuclear force has a tensor component. Proof: the presence
  of the a quadrupole moment for the deuteron ground state
• Nuclear force has a strong spin-orbit component. The triple
  p-waves from phase shift analysis at high energies can only
  be reproduced by adding a spin-orbit term to the central and
  tensor nuclear force component.
                       Yukawa potential
            g e  mr
   r                    Derived from Klein-Gordon eqn.
           4p r
With the increase of r, potential decreases very fast, which implies the short-
range characteristics of nuclear force.
• It contains an EM interaction & a phenomenological short
  and intermediate range

• NN interaction describe by V(r, p, 1, 2) where terms are
  the relative position, relative momentum, spin.

• It is based on AV14, 14 operators not related to charge,
 j
     i      , 
        i  j i  j   ,            i    j   , S , S 
                                                          ij   ij    i    j   , L  S, L  S    i    j   ...L  S 

• 4 charge operators
         Tij ,  i   j Tij , SijTij ,  zi   zj 
                                                            549

Exchange of mesons! It is                              770 782
based on field theoretical
perturbation theory.
Completely defined in terms                 -meson describes multiple-
of one-boson exchange!                      meson contributions in the single
                                            boson exchange.

Nonlocality: the potential acting at one point may depend on the the
value of the wavefunction at a different point.
It essentially describes the relativistic treatment.
All mesons with mass below nucleon mass are included.
Interaction between nucleons in the same spin-angular momentum is
identical for pp, np, and nn system. ----charge independence.
        Three-nucleon Force (3NF): UIX
Fujita-Miyazawa term


 Urbana IX potential:

Short range repulsive term:

  Multiple-pion exchange and repulsive contributions
                            Nuclear current operator
    • One-body current with Siegert operator

    • Meson Exchange Currents (MEC)
      include non-relativistic
In my calculations MEC means that meson exchange currents were included directly, i.e., (transversal) E and M multipoles were
calculated from spatial 1-body and 2-body currents. In my all other calculations (RCO) Siegert theorem was used, that assumes
current conservation and replaces dominant parts of electric multipoles by the Coulomb multipoles. In case of exact current
conservation MEC and Siegert would be identical (and I verified this practically with simple meson exchange model). However, it
is very hard to achieve exact current conservation with realistic NN models, especially if they are nonlocal like CD Bonn.
Therefore MEC and Siegert yield different results. Since the charge operator is theoretically known better than MEC's, it is
advantageous to use Siegert approach, where, in fact, dominant contribution of not well known MEC's are replaced by better
known 1N charge (you probably know all that). Therefore our standard calculation is Siegert. The first relativistic corrections are
of order (p/m)**2 and are charge corrections, i.e., they have entirely no effect on MEC results. The large difference between
MEC and RCO results means that the considered observable is very sensitive to relativistic corrections. I remember, that when I
did first calculations without RCO, MEC and Siegert were quite similar. Once again: my MEC does not includes 1N charge
relativistic corrections, but others two do so. One more remark: the calculations named "Siegert" have different meaning in my
and Golak calculations.
         Nuclear current operators
Siegert theorem:

 Electric multipoles = matrix elements of spatial current operator,
 Coulomb multipoles = matrix elements of charge operator.

electric multipoles = Coulomb multipoles + higher order terms.
The advantage of Siegert form is that Coulomb multipoles (charge), being
strongly dominated by 1-body operators, are known better than spatial
current operator. The uncertanties from spatial current operators enter
then only in higher order terms and magnetic multipoles.

Currents and nuclear forces are connected by continuity equation!!!
    More on nuclear current operators
•   The current is expanded in electric and magnetic multipoles.
•   The magnetic multipoles are calculated from the one- and two-baryon parts
    of the spatial current.
•   The electric multipoles use the Siegert form of the current without the long-
    wavelength approximation; assuming current conservation, the dominant
    parts of the one-baryon convection current and of the diagonal $\pi$- and
    $\rho$-exchange current are taken into account implicitly in the Siegert part
    of the electric multipoles by the Coulomb multipoles of the charge density;
    the remaining non-Siegert part of the electric multipoles not accounted for
    by the charge density is calculated using explicit one- and two-baryon
    spatial currents.

•   The potential and currents are related via continuity equation (charge
    conservation). Transversal MEC that do not contribute to the continuity
    equation are not constrained by this.
•   Potential and MEC's have to include the same meson exchanges. The
    same meson-nucleon coupling parameters have to be used for the potential
    and MEC's.
• Original Faddeev equations are for wave
  function components, their sum is the full
  wave function. In the differential form one
  needs to impose desired boundary
  conditions on the trial solutions. The
  scattering amplitudes can be extracted
  from the wave function, either from its
  asymptotic or from an additional integral
  with potentials.
• AGS equations are formulated not for observables, but for the
  transition operators
• If one wants to calculate the observables for a given reaction,
  one needs the amplitude for that reaction. On-shell elements
  of transition operators are those needed amplitudes, that's
  why transition operators are so important.
• They incorporate standard boundary conditions, so in some
  sense they are Faddeev equations in integral form. Particular
  (so-called on-shell) matrix elements of the transition operators
  are scattering amplitudes from which all observables can be
  calculated. On-shell means that initial and final state
  momenta and reaction energy satisfy energy conservation.
• On the other hand, from the half-shell matrix elements of the
  transition operators one can construct the full wave function.
  Half-shell means that only initial state momenta and reaction
  energy satisfy energy conservation.
       Deltuva’s RCO treatment
• one has to start with fully relativistic expression for one-
  nucleon e.m. current (4-dim. Dirac matrices and spinors)
  and then make an expansion in powers of (k/m_N).
  Nonrelativistic charge is of 0th order, nonrelativistic
  spatial current is of 1st order, whereas the leading
  relativistic corrections are of 2nd (charge) and 3rd
  (spatial) order. In the RCO calculations charge operator
  is a sum of those 0th and 2nd order terms.
                        GDH sum rule
• Anomalous magnetic moment: The difference between the observed
   gyromagnetic ratio of the electron and the value of exactly two predicted by Dirac's

   theory of the electron. The discrepancy is resolved using quantum electrodynamics.
• Low energy theorem: from Lorentz and gauge invariance.
• Unitarity means that the sum of probabilities of all possible
  outcomes is always 1. the S-matrix must be a unitarity operator,
            Im f (0)
• Unsubtracted dispersion relation: causality.
    GDH sum rule: A simple derivation
      Forward Compton scattering amplitude
                                                  
      f    f1  e *  e  f 2  i  e *  e  real photon energy
                                                        e transverse polarization vectors
                                                                             spin vector of nucleon
      Dispersion Relation for spin-amplitude, the dispersion relation for the spin-
      averaged amplitude f is the kramers-kronig relation from optics.

Re f 2 
          2
                   1
                                   A     P  d  2
                    4p   2
                              0             2  2

 Gell-Mann etc. related the zero energy limit of spin-flip         
                                                                         d                  2p 2 2
 amplitude to the square of the anomalous magnetic of
                                                                        
                                                                                  
                                                                                         N    
                                                                                              M  2
                                                                                                   N I
 the nucleon                                                      thr

 Low-energy theorem
    f 2 0  
                       / MP P
                           2  2
        GDH sum rule (more)
• The only assumption in deriving the equation:
  the scattering amplitude goes to zero in the limit,
  photo energy                         g 
• The weakest argument of the derivation, no-
  subtraction hypothesis
• Validity of no-subtraction hypothesis!!!!
• It is the only “QCD-input” to the sum-rule
• For fundamental charged particles with g=2 the
  no-subtraction hypothesis is applicable
FEL principle
      Selection Rule of Gamma transition

              0 or 1   2     3     4     5

p ip f
+1            M1(E2) E2      M3(E4) E4   M5(E6)

-1            E1       M2(E3) E3   M4(E5) E5
• Gyromagnetic ratio: ratio of the magnetic dipole
  moment to the mechanical angular
  momentum/spin. g = -2 B/hbar,
• AFP conditions
  the sweep rate slow enough

  rate fast enough:

  In this way, the magnetization of 3He follows the effective B field
  (“adiabatic”), while it is fast enough so the spin relaxation at resonance
  is minimal (“fast”). Here T1 and T2 are the longitudinal and transverse
  relaxation times.
• the Rb number density is ~10^15 cm-3 in
  the pumping cell and ~10^11 cm-3 in the
  target cell because of the different
  temperatures inside in each cell.
Neutron detection: PSD, MPD4 module

      D2 run Transition matrix element
       Deuteron ground state (spin is 1 and
       parity is positve):                          J p  1
L is the multipolarity of the incoming polarized gamma-ray. The mode of the
gamma-ray is P. p=0 magnetic multipoles, p=1 electric multipoles.
*** E1 radiation leads to   l 1

                                                   L=1, S=1, J=0,1,or 2
 Target Polarization measurement
                                              Two methods for cross calibration:
                                              1. Water calibration
                                              S_water=6.6 uV,
                                              S_NMR=15.1 mV -> P=41.1%
                                              2. EPR measurement
                                              So: two methods agree within 7%

   Hybrid cell “Linda” has a consistent
   performance during the run

For more, please refer to our targer paper:
K.Kramer et al, NIMA 582, Issue 2 (2007)

Comparison of NMR with/no detectors
Remove detectors, peak shift from 23.4G-> 25.4G
Expected change: 8.5% signal increase
Measurement: 15.13 mV->15.31 mV
Actual change: 1.2% increase, plus 0.8% measurement loss
So: mu metal results in 6.5% polarization loss
           Compton scattering
E  m0 c 2  E   E e
p  p ' p e
Ee  c 2 pe  m0 c 4
    2         22

E  cp
cp  cp  m0 c 2   2
                                                      
                          c 2 p 2  2 pp cos  p 2  m0 c 4

     1 1
m0 c    1  cos
     p p 
                                                          E
            h                            E 
    
                 1  cos                     1
                                                              1  cos 
            m0 c 
                                                           2
                                                       me c
            2.426  10 12 m
        m0 c
                  My Ph.D. work
       Set up the polarized 3He target experimental
       apparatus, started polarization measurement
2005   Sept: first HIgS run. Polarization was low.
       The problem was later traced to the mirror.
2006   Tested a hybrid 3He target, published a target paper.
       Simulation for projection. Changed the configuration
2007   to longitudinal setup and tested target
       Worked with Capture group to test detectors.
2008   May: first measurement at HIgS
       Analyzed experimental data and ran simulation
       May: Test run of the new target cell at HIgS
2010   Checked data analysis and finalized results
     III. Helicity-dependent diff. cross section difference

definition of the variables are the same as
It is calculated directly from the above
expression, but can be verified by
combining asymmetry and DXS results.
   The statistical uncertainty is

Systematic uncertainty is obtained in the
same way as before.
   Theory curves are the same as before

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