Calculations to find the rate on a bar as a function of the position of a beam element on target have been repeated for Juliette's up- and downstream collimators of April 26, labelled "11" and "10" on her collimator optimization page (http://www.phys.vt.edu/~jmammei/coll_opt.html).
The rate seen by a detector centred on the x-axis (the narrow dimension of the bar is parallel to the x-axis, the long dimension parallel to y) as a function of position (x, y) of a beam element on target is fitted by:
Rate(x, y) = a(1 + bx + cy2 + dxy2).
From this you can obtain:
1) Sensitivity to position modulation of the beam.
2) Estimate of how precisely bars need to be positioned and/or magnetic fields need to be matched in the QTOR sectors.
3) Estimate of the effect of beam size modulation.
The false asymmetry averaged over eight identical bars is:
ε = c x0 δx
when the beam moves by ±δx on spin flip (nominally ±20 nm) and x0 is the DC offset of the beam element from x = 0. The "b" term cancels.
If the Cerenkov bars are not positioned at the same radius, or the QTOR sectors produce different deflections of the electrons on their way to the Cerenkov bars, the b term no longer cancels when the false asymmetry is averaged over bars. The effect can be estimated by moving the position of the bar cut and fitting rate as a function of (x, y) on target once more. For a set of mismatched bars the residual b coefficient and false asymmetry can be found. Alternatively, the radial position of a bar can be treated as a random variable with standard deviation σr and the uncertainty in the average asymmetry calculated. The result is that the uncertainty in the false asymmetry averaged over all bars is:
δε = (1/4)(db/dr) σr δx
The false asymmetry from beam size modulation is:
ε = (1/6) c D δD
where D is the unrastered beam size (~100 µm) and ±δD is the change of beam size on helicity flip. The allowable change in size of the rastered beam scales inversely as the beam size, so is a factor of ~40 smaller in the rastered beam.
Following are results for Juliette's up- and downstream collimators, using the handlebar cut (xmin = 322 cm, xmax = 337 cm out to y = ±50 cm, a = 0.2), assuming that: ε = δε = 6 × 10-9, δx = 20 nm.
| Collimator | x0 (mm) | δD (µm, unrastered) | σr (mm) | QTOR field match (%) |
|---|---|---|---|---|
Upstream (#11) |
0.4 |
0.5 |
4.6 |
0.4 |
Downstream (#10) |
3.7 |
4.4 |
5.3 |
0.4 |
The sensitivities to position and size modulations are quite different for the up- and downstream collimators. As noted earlier, the downstream collimator should be better from this point of view, as the target more closely resembles a point. Put another way, the motion of the beam is smaller in relation to the scale of the defining aperture.
There is nothing to choose between the two collimators when it comes to matching Cerenkov bar positions or QTOR sector magnetic fields, however. The relative variation of rate with radial position of a bar is the same.
JB, 12 May 2005