# Spin-Dance Measurement

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

## General

Let the particle's momentum to rotate by an angle theta. Then its spin would turn with respect to the momentum by an angle: angl=(g-2)/2*gamma*theta, where gamma=E/m and (g-2)/2=0.00115965 for electrons. So, spin turns to the same direction as the momentum and by a larger angle.

## Spin Precession at CEBAF

In a note TJNAF-TN-97-021 C.Sinclair derived the expected spin turns for the halls A,B,C at CEBAF. The value of theta depends on the number of passes and on the angle in the hall's extraction arcs, taking into account the beam acceleration:
theta=(2n-1)*180o;
angl=EL/m*(g-2)/2*(2n2-n*(1-2a+b)-a*(1+b/2))*180o
where:
• EL 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.
Let us assume Hall A is running 5 passes at EL=0.812 GeV, so theta=4.5*360 degrees. Then spin makes an angle of about 22.167*360o to the momentum. Measuring the spin precession with an accuracy of 5o one can potentially measure the electron energy with an accuracy on about 0.06%. However the initial spin angle is not defined with a precision comparable to that and a better way of measuring the spin precession is to compare the synchronously measured beam polarizations from 2 halls running at different energies. This procedure is nicknamed "spin-dance". The direction of spin is changed on the injector and by convention the positive direction is clock-wise, as the direction of accelerated electrons is. It is convenient to run at a so-called "magic" energy when both halls should have a zero longitudinal polarization at the same angle at the injector. Since at the zero point the accuracy of the phase measurement is the highest, both halls should be able to measure this point at the same angle at the injector. More detailed information about spin precession at CEBAF can be founded in the paper .

## Spin Precession and Energy Fluctuation

There is one more important effect of spin precession. If the beam energy for the Moller measurements and a currently running experiment is different (as it is shown on the plot ) the beam polarization for the experiment and the beam polarization from the Moller measurements will be different. It means that the Moller measurements should be done at the same energy as the experiment in Hall A. Unfortunately, it is not possible always. Typical beam current for the Moller measurements is 0.2-1uAmp. Hall A energy lock is not available at this small current. Hall C energy lock can be used for Hall A Moller measurements if Hall C is running. If Hall C is OFF, ones can use ARC2 energy lock. There are three disadvantages of ARC2 enery lock:
• Energy in Hall A with Hall A energy lock and ARC2 energy lock is different (see the plot ).
• Minimal current for ARC2 energy lock is 1uAmp.
• Small energy fluctuation in ARC2 becomes large one at high passes (see the plot ).
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.

## April, 26 1998

"Spin-dance" measurement was performed on April,26 1998 by Moeller polarimeters in halls A and B, at a "magic" energy of about 0.8395 GeV per pass. Hall A ran at 4 passes while Hall B ran at 1(?) pass. The initial angle at the injector was 70.6 degrees.

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 -20o to the beam. At beam spin orientation of, say 80o the longitudinal and transverse components of asymmetry have different signs. Therefore at a spin angle of 90o the observed asymmetry should be non-zero and negative, if at 0o it is positive. The observed asymmetry curve is shifted by the transverse polarization by +3.0+/-0.4o 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+/-0.3o. Therefore the beam helicity should be zero at about -16+/-0.6o.

## May, 23 1998

The assumed energy per pass was 0.812 GeV. The initial angle at the injector was -61.6o.

Spin-dance results
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.

## May, 27 1999

Assumed linac was 0.550 GeV, 3 passes in hall A.

Spin-dance results
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%

## July, 13 1999

Assumed linac was 0.5437 GeV, 3 passes in hall A. The initial angle at the injector was 45.6o.

Spin-dance results
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+/-0.5o. That would mean that the optimal settings should be 39.2o (not 45.6o 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 40.7o. 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%

## July, 10-12 2000

The point at -81.6o Wien angle, at 163.5o 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 (23o) which was done right after the measurement at 163.5.

Comparison of the DC and RF measurements with Hall A Moller.
# 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 4o (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+/-1o (compare with -3.2+/-0.4o 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)

Spin angles, assumed linac energy of 0.5590GeV. The polarization for Hall A and C was evaluated in a Wien angle range of +/-85o. For Hall A Moller a correction for dead time of +1% was done. Hall A Compton result was projected to the DC mode using a correction factor of +1.9% obtained with Hall A Moller polarimeter. The errors shown are statistical.
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

#### Final results

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 1o 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.

Spin angles, assumed linac energy of 0.5590GeV. The final results from the polarimeters as well as the Wien filter angles are used. The Wien angle is assumed to have a random error of 1.0o. For Hall A Moller, corrections for the dead time of +1% and for the Levchuk effect of -0.4% were done. Hall A Compton result was projected to the DC mode using a correction factor of +1.9% obtained with Hall A Moller polarimeter. The errors shown are statistical. For Hall A, the average for two methods to measure the target angles (the dial angle and the measured angle) was used.
Polarimeter Predicted angle o no Wien errors Wien error of 1o
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 1o is shown on plot 6.

## December, 01 2000

Assumed linac was 0.565 GeV, 5 passes in hall A.

Spin-dance results
Wien angle Polarization
-36.16 69.2+/-0.15
24.7 53.7+/-0.15
-7.0 71.5+/-0.15
The spin-dance results are presented on the plot. The optimal Wien angle is -17.8o. The spin angle to the Z axis was calculated using two ways:

• 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 163o in this case).
The difference between the results of these two methods is presented on the same plot. The maximum difference of about 1.4o is indicating that the accuracy of the second method is reasonably good to make estimates of the spin angle while evaluating the accelerator tune.

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        0.00080
```
These results indicate a linac energy of 0.5653 and E(A)=4.586 GeV.

## September, 15 2001

Assumed linac was 0.5675 GeV, 5 passes in hall A.

Spin-dance results
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%.

## October, 17 2005

Assumed linac was 0.520 GeV, 3 passes in hall A.

Spin-dance results
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%.

## January, 13 2009

Assumed linac was 0.5835 GeV, 5 passes in hall A.

Spin-dance results
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 .

### Spin Dance Results Correction

The Spin Dance results were corrected after Mini Spin Dance in 02/11/2009 (see below). For the beam energy and current - see the Chart . Beam energy (HALLA:p) fluctuation was ~2.5MeV during the Moller measurements.
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 ~68o (see. plot ). From the Spin Dance this angle is ~58o. This discrepancy can be explained by the Linacs' energy disbalance ~0.8%.

The final corrected results:
Pz=-79.88 +/- 0.15(stat) +/- 2.0(syst-prelim) (dead time correction is not included)

## February, 11 2009

Assumed linac was 0.5835 GeV, 5 passes in hall A.

Spin-dance results
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 ~68o (see. plot ). From the Spin Dance this angle is ~58o. This discrepancy can be explained by the Linacs' energy disbalance ~0.8%.

The Wien filter angle 23o 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 19o.
The Wien filter angle 23o gives for the Hall A a bit lager polarization.

The final results for the Wien filter angle 23o:
Pz=74.50 +/- 0.15(stat) +/- 2.0(syst-prelim) (dead time correction is not included)

## August, 31 2009

The beam energy is 3483.9MeV(by M. Tiefenback), 549.4MeV(North Linac), 590.6MeV(South Linac), 64.125MeV(injector). 3 passes in hall A

Spin-dance results
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.