Last updated: Tue, 15 September 2009 moller@jlab.org

theta=(2n-1)*180

angl=E

where:

- E
_{L}is the linac's energy - n is the number of passes
- m is the electron mass
- a=0.1125 is the ratio of the injector energy to the linac's energy
- b=-1/2.4 for Hall A, =0 for Hall B and 1/2.4 for Hall C; this factor comes from the extraction arc.

If the beam energy for the Moller measurement is nown, measured beam polarization can be corrected. This effect is small at low passes and if the beam polarization is maximal.

The angle at the injector was decreased which means turning the spin
anti-clockwise looking from the top, accordingly to Charles Sinclair.
In our notation it is the positive direction - from Z to X.
In the Hall A polarimeter the transverse polarization in the horizontal
plane changes the measurements while in Hall B it does not.
The supermendur "bottom" target was used (which is in fact the upper target).
The target plate is positioned at about -20^{o} to the beam.
At beam spin orientation of, say 80^{o} the longitudinal
and transverse components of asymmetry have different signs.
Therefore at a spin angle of 90^{o}
the observed asymmetry should be non-zero and negative, if at 0^{o}
it is positive. The observed asymmetry curve is shifted
by the transverse polarization by +3.0+/-0.4^{o} with respect
to a curve one would obtain if there were no transverse polarization
influence.

The observed
spin-dance curve
crosses zero at about -13+/-0.3^{o}. Therefore the beam helicity should
be zero at about -16+/-0.6^{o}.

hall | # passes | plot | angle at max (^{o}) | precession/360. | ang.shift(^{o}) | energy shift % |
---|---|---|---|---|---|---|

A | 5 | plot | -53.3+/-0.2 | 22.168 | -7.1+/-0.2 | -0.089+/- 0.003 |

B | 3 | . | -59.0+/-2.6 | 7.169 | -2.0+/-2.6 | -0.077+/- 0.010 |

A-B | . | . | . | . | . | -0.095+/- 0.048 |

Two ponts at the lowest injector angle fall out of the COS curve. Their errors are increased by hand. The reason has not been found. It may happen that at these angles the wien filter does not provide the spin turn expected.

Wien angle | Polarization |
---|---|

3.8 | 70.5+/-0.2 |

29.4 | 29.4+/-0.2 |

-94.8 | -2.9+/-0.2 |

The results presented on a plot. The maximal polarization in Hall A is 72.35+/-0.15%. The last Mott result is 70.3+/-0.5%

Wien angle | Polarization |
---|---|

-74.8 | 29.4+/-0.2 |

-47.6 | -4.3+/-0.2 |

-20.3 | -36.2+/-0.2 |

45.6 | -75.9+/-0.2 |

The results presented on a plot show the zero crossing at
about -50.8+/-0.5^{o}. That would mean that the optimal settings should be 39.2^{o} (not 45.6^{o}
as it had been set by the accelerator).
The angle predicted by my program (E.Ch) for the linac energy of 0.5437GeV is about 40.7^{o}.
We rounded the number to 40.^{o} and asked the MCC to tune to this angle for the future.
The maximal polarization in Hall A is 75.26+/-0.15%. The last Mott result is 70.3+/-0.%. The last Mott result is 73.8+/-0.8%

The point at -81.6^{o} Wien angle, at 163.5^{o} target angle has a high chi2 value of 21.
Apparently, the value of polarization changed just during the measurement. The first 3 points
are at about P=0.19, while the last 3 points are at 0.16. The latter matches the measurement
at the forward target angle (23^{o}) which was done right after the measurement
at 163.5.

# | Polarization with (DC) % | Polarization (RF) % | DC-RF, % relative |
---|---|---|---|

1 | 75.98+/-0.13 | 74.56+/-0.11 | 1.4+/-0.2 |

2 | 75.85+/-0.14 | 73.44+/-0.10 | 2.4+/-0.2 |

The results for all the measurements, presented on the plot 1
indicate a deviation of the Wien filter angles from the expected ones at large angles.
Removing the points at -85< theta_wien< 85 plot 2
does not change the results significantly. Hall A measures two "zero-crossings". Presumably because
of the Wien angle distortion the results from the left and right crossings differ considerably - by about 4^{o}
(see plot 3).

The transverse polarization of the beam has been taken into account for the analysis of Hall A Moller
data. The longitudinal polarization measured has not been affected by the transverse polarization,
since the forward and backward target angles are nearly the same and in such a case the transverse component
cancels while the average for two target position is taken (see plot 4).
The transverse polarization was evaluated taking a difference of the results at forward and
backward target angles (see the same plot). The predicted dependence was fit to these data yielding
the full polarization of 70.4+\-1.2 (compare with 76.2+/-0.12 for the longitudinal polarization) and
the angle of -5+/-1^{o} (compare with -3.2+/-0.4^{o} for the longitudinal polarization),
more or less consistent with the longitudinal polarization.

The final result for Hall A Moller: 76.90 +/- 0.13(stat) +/- 2.4(syst-prelim)

Polarimeter | Predicted angle ^{o} | Measured angle ^{o} | chi2/ndf | shift ^{o} | D(E)/E linac | Polarization % |
---|---|---|---|---|---|---|

Mott | 0. | 1.9+/-0.4 | 1.7 | +1.9+/-0.3 | - | 71.64+/-0.21 |

Compton A | -7.77 | -4.7+/-0.6 | 2.3 | +3.1+/-0.6 | 0.00029 | 73.12+/-0.45 |

Moller A | -7.77 | -3.2+/-0.3 | 8.7 | +4.6+/-0.3 | 0.00042 | 76.94+/-0.13 |

Moller B | -66.70 | -61.2+/-0.5 | 2.4 | +5.5+/-0.3 | 0.00049 | 71.70+/-0.48 |

Moller C | 54.37 | 57.8+/-0.4 | 4.7 | +3.4+/-0.3 | 0.00034 | 73.84+/-0.23 |

Fitting the curves with just the statistical errors for the polarization measurements
gives a high chi2, about 200 per degree of freedom. We assume that either there is a
considerable error on the Wien filter angle, or the polarimeters (Hall A Moller in particular)
has a systematic error showing up at large transverse polarizations of the beam. In both cases
the amplitude of the SIN curve is not affected by this error, though the phase might be
affected. We used the former assumption and attributed and error of 1^{o} to
all the angles. The fit was done in two steps. At the first step, the error on the angle
was ignored, while on the second step this error was translated to the polarization
error, using the already known slope of the curve, and added in quadratures.
The results of the fit for all the polarimeters, shown on
plot 5, are different by 1-2%
from the initial fit.

Polarimeter | Predicted angle ^{o} | no Wien errors | Wien error of 1^{o}
| ||||
---|---|---|---|---|---|---|---|

angle ^{o} | Polarization % | chi2/ndf | angle ^{o} | Polarization % | chi2/ndf | ||

Mott | 0. | 1.2±0.20 | 72.21±0.11 | 4.2 | 1.1±0.4 | 72.43±0.15 | 0.9 |

Compton | -7.77 | -4.2±0.30 | 72.67±0.43 | 4.7 | -4.6±0.6 | 72.40±0.52 | 3.4 |

Moller A | -7.77 | -3.9±0.04 | 77.07±0.07 | 230.0 | -3.7±0.3 | 75.21±0.14 | 3.4 |

Moller B | -66.70 | -60.4±0.40 | 69.71±0.47 | 1.8 | -60.4±0.6 | 69.80±0.55 | 1.2 |

Moller C | 54.37 | 57.0±0.10 | 73.24±0.08 | 42.4 | 57.5±0.4 | 73.60±0.14 | 4.0 |

Comparison of all polarimeters for the fit with the Wien errors of 1^{o} is shown on
plot 6.

Wien angle | Polarization |
---|---|

-36.16 | 69.2+/-0.15 |

24.7 | 53.7+/-0.15 |

-7.0 | 71.5+/-0.15 |

- as a difference of the Wien angle and the found optimal angle,
- reconstructed from the influence of the transverse polarization, using the measurements
at different target angles (23 and 163
^{o}in this case).

Comparison of the measured optimal angles in each hall with calculations for Elinac=0.5656:

Wien angle for max Pz, deg Hall measured calculated dE/d(thet)/E0 1/deg A -17.8+/-0.5 -21.5 0.00014 B -4.9+/-1.6 -10.7 0.00010 C -6.2+/-0.5 -6.3 0.00080These results indicate a linac energy of 0.5653 and E(A)=4.586 GeV.

Wien angle | Polarization |
---|---|

10.0 | -82.72+/-0.14 |

105.0 | 7.92+/-0.16 |

70.0 | -40.27+/-0.15 |

38.0 | -72.34+/-0.16 |

-60.0 | -25.34+/-0.14 |

-87.0 | 10.81+/-0.15 |

-30.0 | -63.24+/-0.18 |

The results presented on a plot. The maximal polarization in Hall A is 82.31+/-0.18%.

Wien angle | Polarization |
---|---|

-6.2 | 82.2+/-0.2 |

9.2 | 88.1+/-0.2 |

24.0 | 87.8+/-0.2 |

39.2 | 82.0+/-0.2 |

54.2 | 70.6+/-0.2 |

The results presented on a plot. The maximal polarization in Hall A is 88.82+/-0.18%.

Wien angle | Polarization |
---|---|

30.0 | -79.80+/-0.2 |

50.0 | -89.82+/-0.2 |

70.0 | -88.32+/-0.2 |

-30.0 | -4.94+/-0.2 |

54.0 | -90.23+/-0.2 |

In Hall A, the spin is parallel at a Wien angle of 53.3±1.°. From this, assuming symmetric linacs, we derive the energy per linac of 583.35 MeV, which gives the Hall A energy of 5.899 GeV. This is about 7 MeV higher than the measured energy. The difference might be caused by a linacs' disbalance. The spin would be parallel at 15.5° in Hall B (5 passes) and at 23° in Hall C (4 passes). The sum of squares of the longitudinal polarizations in all 3 halls would be maximized at about 30°.

A large chi2 of the spin dance fit (20/NDF) is likely caused by an energy instability, which affects the measurements at large spin angles (far from 54°)

The final results:

Pz= -81.58 +/- 0.23(stat)+/- 2.0(syst-prelim) (dead time correction is not included)

The plot is given
here .

After old on-line and off-line analysis (see plots a) and b) ) the Moller data was corrected on a difference between the beam energy readings from HALLA:p and the actual experiment energy 5892MeV (see plot c) ).

Spin precession for the Hall A and the beam energy 5892MeV gives maximum at ~68

The final corrected results:

Pz=-79.88 +/- 0.15(stat) +/- 2.0(syst-prelim) (dead time correction is not included)

Wien angle | Polarization |
---|---|

30.0 | -80.64+/-0.1 |

20.0 | -72.19+/-0.1 |

23.0 | -74.50+/-0.1 |

Mini Spin Dance results were analysed together with 01/13/09 Spin Dance results. After the Moller raw data analys (see plot b) )
we corrected the Moller data on a difference between the beam energy readings from HALLA:p and the actual experiment energy 5892MeV
(see plot c) ).

Spin precession for the Hall A and the beam energy 5892MeV gives maximum at ~68^{o} (see. plot ).
From the Spin Dance this angle is ~58^{o}. This discrepancy can be explained by the Linacs' energy disbalance ~0.8%.

The Wien filter angle 23^{o} was choosen as an optimal for the Hall A and the Hall C beam polarization values. After all corrections
from plot we can see that the optimal Wien filter angle for the Hall A and the Hall C is 19^{o}.

The Wien filter angle 23^{o} gives for the Hall A a bit lager polarization.

The final results for the Wien filter angle 23^{o}:

Pz=74.50 +/- 0.15(stat) +/- 2.0(syst-prelim) (dead time correction is not included)

Wien angle | Polarization |
---|---|

40.30 | -80.28+/-0.3 |

110.27 | 14.11+/-0.3 |

-70.22 | -5.37+/-0.3 |

14.20 | -89.61+/-0.3 |

The spin dance result is given here . Comparison of the spin dance and prediction for the beam polarization precession for linacs without dis-balance and with dis-balance 7.5% is given on plot1 for MCC beam energy (Eb=3.4812GeV) and on plot2 for Tiefenback beam energy (Eb=3.4839GeV). The linacs dis-balance by MCC definition corresponds (North Linac)/(South Linac). The Wien filter angle 14.2° was chosen as an optimal for the Hall A with the beam energy Eb=3.4839GeV (by M. Tiefenback) and linacs dis-balance=7.5%. The Wien filter angle 17.° is an optimal for the Hall A with the beam energy Eb=3.4812GeV(by MCC) and linacs dis-balance=7.5%. The beam polarization difference between 14.2° and 17° is 0.998682.

More detailed analysis of the Spin Dance results fit show much larger Chi2 for fit with three Wien filter angles 14.2, 44.3048 and 110.27deg (see plot3 ) than for three Wien filter angles -70.22, 14.2, 44.3048deg (see plot4 ). For large Wien filter angles,like 110deg, the beam trajectory is not strait. Because of that measurements with large Wien filter angles are less trusted for fit and end errors for such angles should be increased. The result of fit with increased error for Wien angle 110.27deg. is given on plot4 . From the more detailed Spin dance analysis ( plot4 and plot5 ) the optimal Wien filter angle for the Hall A is ~16.5deg.