Many analytically established error tolerances have been confirmed using numerical methods. Simulation indicates that a baseline array of beam position/profile profile monitors and correctors placed roughly every quarter betatron wavelength in phase advance will allow machine operation in the presence of all anticipated errors and will not lead to intolerable beam quality degradation.

A DIMAD model of the wiggler to reinjection transport was constructed and used to study effects of error levels consistent with the analytically derived error budget. An "ensemble" of ten randomly distributed sets of alignment and powering errors was generated, and the beamline performance modeled when the errors were imposed. Correction of on-momentum orbit errors only was then implemented, and the performance analysis repeated. In this analysis, four criteria for machine and beam behavior were examined:

1. central orbit behavior
2. chromatic behavior - of the orbit (off-momentum orbits, dispersion, and momentum compaction), and of beam properties about the orbit (beam envelopes, phase advance)
3. large amplitude behavior - geometric aberrations and phase space distortion
4. horizontal/vertical coupling

Conclusions and comments are summarized at the end of this section.

The figure presented below displays horizontal and vertical central orbits for the sampled ten random error sets ("ten seeds") before and after correction. All orbits remain within the 2 inch vacuum chamber before correction. All orbits are significantly improved by the correction process.

We have found that the rms residual orbit tracks the rms BPM misalignment down to submillimeter levels (see the following plot, which displays corrected orbits with an rms BPM misalignment of 1 mm rather than the 0.5 mm used in the preceding figures). We plan to rigidly fixture BPMs to adjacent quadrupoles (which will be aligned to 500 microns, rms) and calibrate the BPM by beam centering in the quad. This will provide BPM alignment to well below 1 mm rms, which in turn will allow orbit correction to submillimeter rms residuals, adequate for machine operation.

Off Momentum Orbits/Dispersions/Momentum Compaction - The figures below show plots of off-momentum orbits at ⁺/_-3% for ten random error sets before and after orbit correction. In this case, the correction is done for the on-momentum central orbit only. We find that simple correction of just the on-momentum central orbit leads to generally adequate machine performance; the off-momentum orbits track the corresponding on-momentum orbits, indicating that the correction is more or less local. If ongoing studies indicate the need to do so, the off-momentum orbits can be separately constrained during the orbit correction process as well, and/or dispersion correction can be activated during or after orbit correction.

Histograms of dispersions and momentum compaction (specifically, M₁₆, M₃₆, and M₅₆ from wiggler to reinjection point) for each of the ten random seeds studied, before and after correction of the on-momentum central orbit, are given below. Before correction, horizontal dispersion and momentum compaction are fairly near design values, but spurious vertical dispersion can become rather large (up to ~0.3 m). This is probably due to the fact that this type of lattice tends to have large T₃₃₆, T₃₄₆, T₄₃₆, and/or T₄₄₆ values. Vertical quadrupole misalignments will cause angular kicks that couple to these aberrations and thereby generate large effective vertical dispersion. Localized correction of the on-momentum orbit will compensate such effects at their source; this is seen in the figure, where we note that all dispersive matrix elements return to near-design values following orbit correction.

Momentum Scans of Orbit and Betatron Properties - The next graphs present the results of momentum scans of beam position, angle, and lattice functions at the reinjection point, and path length variation and phase advance from wiggler to reinjection point, before and after correction of the on-momentum central orbit. The orbit correction process significantly reduces the final position and angular offsets and their variation with momentum. This is consistent with the above discussion. The seed-dependent scatter in path length is well within the range of compensation available in the machine; the momentum dependence is nearly ideal. Beam envelope functions exhibit only moderate dependence on momentum errors both on and off momentum, even before correction. They are nearly ideal after correction.

The figure below displays the relative emittance distortion vs. momentum for each of 10 random seeds, before and after correction of the on- momentum central orbit. A beam with 10 times the design geometric emittance of 0.16 mm-mrad (a 13 mm-mrad normalized emittance at 42 MeV) was used in the DIMAD "line geometric aberrations" operation to evaluate the phase space distortion. Performance is acceptable in all cases, and approaches the ideal machine behavior after correction. Results are similar if unequal emittances (horizontal twice vertical or vice-versa) are used.

Horizontal/vertical coupling about a specified orbit can be computed using the DIMAD "rmatrix" operation, which evaluates the linear transfer matrix about the orbit of interest and evaluates the determinants of the various transverse sub- blocks. The following tables present the results of this process for each random distribution studied, before and after correction of the central orbit. Before correction, the H/V coupling parameters DET B and DET C are small; after correction they are vanishingly small. The transport system is therefore not expected to experience inherent coupling problems. Any such difficulties will be due to magnetic field errors and/or coupling in the RF cavities, both of which will be evaluated in future simulations.

The results detailed above indicate that machine performance is acceptable for errors consistent with the error budget. The simulation effort is ongoing. Over the next months, the model will be extended to include magnetic inhomogeneities, perturbations of pi-bend alignment and excitation, nonlinearities in the wiggler, and a more accurate cavity transformation (including H/V coupling and chromatic effects). Machine parameter sensitivities to individual errors, as well as global response to multiple errors, will be investigated. Activities will then shift to modeling of commissioning and operational processes such as alternate orbit correction algorithms, dispersion/momentum compaction adjustment, lattice/beam phase space matching, error resolution and correction of other lattice properties. A schedule of projected activities is given elsewhere.

Last modified: 25 March 1997
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