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Special Topic: Beta Beating and Lattice Correction

Special Topic: Beta Beating and Lattice Correction. We know that our general transfer matrix from one point to another in a lattice is. and that our beta function evolves according to. If we introduce a quadrupole of strength. at the initial point, this modifies our matrix as.

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Special Topic: Beta Beating and Lattice Correction

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  1. Special Topic: Beta Beating and Lattice Correction

  2. We know that our general transfer matrix from one point to another in a lattice is and that our beta function evolves according to. Special Topic: Beta Beating and Error Correction

  3. If we introduce a quadrupole of strength at the initial point, this modifies our matrix as These terms are the same Special Topic: Beta Beating and Error Correction

  4. Our distorted beta function becomes So the distortion will “beat” the periodicity of the cell with the periodicity of twice phase advance. Hence the term “beta beating”. Solving the problem in a closed ring is a bit more complicated, but the behavior from point to point is the same. Special Topic: Beta Beating and Error Correction

  5. Initial beta-beating in LHC Special Topic: Beta Beating and Error Correction

  6. Correcting Errors Generally, a dipole will produce a set of x deviations around the ring, and a quadrupole will produce a distorted beta function. So if I have a set of dipole and quadrupole corrector (“trim”) elements, I can write If I have a set of measured distortions, I can introduce a correction to cancel it. If N=M, then If N>M, then I must use a bit more sophisticated math to minimize the RMS deviation (there are several techniques). Special Topic: Beta Beating and Error Correction

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