Tune Measurements

Tune Measurements as a Function of Beam Position
in the Superconducting Wigglers

Analysis of Data Taken 18-20 August 2003

21 August 2003
JAC

The measurements labelled "14EA" were taken with wiggler 14EA (WIG1_14E in CESRV, Serial Nr 2) excited to 21 kG. The measurements labelled "14EB" were taken with both wigglers 14EA and 14EB excited to 21 kG with the same polarity. Measurements labelled "19EB_HBUMP_A" were recorded with the two wigglers 14EB and 14EA at 21kG, but wiggler 19EB off. They serve as reference for the succeeding measurements with 19EB excited.

Description of the Analysis
On August 18, tune measurements vs vertical bumps 60 and 61 were recorded in data file W14EA3_VERT. Immediately thereafter, tune measurements versus horizontal bumps 72 and 73 were recorded in W14EA4_HBUMP, while recording phase measurements. Then tunes versus the vertical bumps were repeated (W14EA_VBUMP) in order to record phase measurements, which had been omitted earlier. These repeated tune measurements agree with the values previously obtained, as shown in the first two plots. The vertical and horizontal bump coefficients were stored in BIGGRP 91089. The lattice WIG1_14E21KG_T_OPT was loaded during these measurements. Vertical orbits with the vertical bumps at zero and 500cu were 92882 and 92883. The corresponding orbits for the horizontal bumps were 92890 and 92891.

In the following plots, the tune measurements are shown in red. They are compared to values calculated from the transfer functions calculated by the Vector Fields tracking algorithm applied to its own finite-element calculation of the wiggler field (w7-20_26.tracks). These calculations are shown in green. The vertical betatron frequency shift as a function of vertical position shows the cubic nonlinearity in the transfer function and is calculated as Del f = f_0 * betay/4pi * dthetay/dy. A fit to the transfer function thetay vs y shows very small quadratic and quartic terms, leaving only the cubic term to give a y dependence of the vertical tune. Similar calculations give the horizontal tune change and the horizontal dependences of the tune changes. The beta functions for this lattice and biggrp at the center of WIG14EA (WIG1_14E in CESRV) are betax=15.0m and betay=31.3m. According to CESRV, they vary by less than 5% for the bump amplitude ranges of the measurements. f_0 is 390 kHz.

CESRV calculations of the expected tune shift are shown in blue in these plots. These results were obtained for W14EA4_VBUMP with the following commands in CESRV:
*l:WIG1_14E_21KG_T_OPT
*read big 91089
*read o:92882
*run (fit model to reference orbit using all vertical steerings)
*cha sex 82 @0cu (repeat to turn off all sextupoles 11e:17e)
*q_t @216.05 242.5 (set the tunes to the measured zero-bump values)
*read o:92883 (read the orbit taken with vbumps 60,61 set to 500cu)
*veto vert all
*use gr vbump 60:61
*run (fit this orbit using only vbumps 60,61)
*show tune (get CESRV tune calculation for plot comparison)

Fitting to this orbit, CESRV finds bump settings for 60 and 61 of 347 and 442 cu. This is an indication that something isn't quite right, since the actual set values were 500cu. Also, setting CESRV to the values 500cu shows a bump which is not well closed.

The remaining points on the CESRV curve are obtained by scaling the values 347,442 according to the set values of the measurement.

The comparison shows that the measured tune shifts are much greater than those calculated from the transfer functions and CESRV, which agree with each other. This may be due to a mistake in the bump calibration arising from hysteretic effects during the calculation of coefficients. The same coefficients were used for the two vertical bump measurements. The measurements and calculation agree very well if the bump calibration is changed from 1.20cm/1000cu to 0.90cm/1000cu.

Another indication that the bump calibration may be wrong is obtained by comparison to the measurements of tune shift with both 7-pole wigglers on made by ST on 21 August. In this two-wiggler lattice, the vertical beta functions at the wigglers were 29m and 22m. The transfer functions for a single wiggler with beta=31m give a tune shift of 5.7 kHz. Scaling with the beta functions and adding the focussing effects of the two wigglers at Y=1cm, one would expect a shift of 9 kHz, similar to the measured value of 8 kHz. However, if one uses the single-wiggler value of 7.5 kHz obtained from measurement, one would get at value of 12kHz, differing substantially from the observed value. A tentative conclusion is that the vertical bump calibration was wrong by 25% on 18 August.
The third plot shows the comparisons for the horizontal bump. The transfer function cannot account for the vertical tune shift observed. However, since we know from magnetic measurement that the 7-pole magnets show a linear component (skew quad) which the Vector Fields calculation does not emulate, we cannot be surprised that there is also a discrepancy in second order.

Since the horizontal tune shift is not symmetric in the 7-pole wiggler, the biggest tune shift being on the side where the wiggle occurs, this measurement is sensitive to the polarity of the wiggler field. Measurement shows this to be the side of positive HBUMP, which is negative X (the inside of the CESR ring). This corresponds to the end-pole field pointing up in the 7-pole wiggler. (A positron entering the wiggler from the north makes a right turn toward negative x when entering.) The transfer function result for 1.8 GeV ELECTRONS shows the big tune shift to be on the +x side, in agreement with observation in CESR. (However, there appears to be a bump amplitude calibration problem here as well as for the vertical bump noted above.) The tentative conclusion is that the end-pole field is pointing up in both the Vector Fields model and in CESR (NB: A sign mistake was made during my presentation of 9/4, when I incorrectly claimed that the end-pole field points down in the VF calculation). However, this result is preliminary and still under discussion. It disagrees with the result using the two 7-pole wiggler cluster measurements of 21 August, which appear to show that CESRV wigglers have opposite polarity from the wigglers in CESRV using this same technique. (Note, however, that CESRV does not give conclusive results for this simpler single-wiggler case, as described below.) One possible resolution of this discrepancy would be a sign change during the generation of the Taylor map.

The CESRV calculation shown in this third plot is inconclusive, since it does not show the expected lack of symmetry. Possibly this is due to the order-3 Taylor map used in the tracking not being of sufficient accuracy to model this fairly small effect in the transfer function.
Analysis of the 8-pole wiggler measurements of 19 August has not yet been completed. However, it is already clear that the bump calibration will be unknown, since the bump coefficients were not stored in a biggrp and the calibration orbits were taken at bump settings which differed from those used during the scan. In particular, the values used during the scan were incorrect, since they were the same as those used before the wiggler was turned on and the bump coefficients recalculated.

Creation date: 09/05/03.