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Tests of the AC Dipole Based Nonlinear Optics in the Tevatron - Part 1

This article presents the conditions, procedures, and measurements of the AC Dipole based nonlinear optics tests conducted in the Tevatron. The measurements include beta-function and betatron phase measurements, as well as the detection of sources for higher order modes.

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Tests of the AC Dipole Based Nonlinear Optics in the Tevatron - Part 1

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  1. Tests of the AC Dipole BasedNonlinear Optics Measurementsin the Tevatron Part 1 Ryoichi Miyamoto Univ. of Texas at Austin 5/2/2008

  2. Conditions and Procedures of the Tests • The tests were done 2/20/2008 • Usual 150 GeV uncoalesced proton beam was used. • Large oscillations are produced with a vertical AC dipole at E17. • First, linear optics (βy and ψy) was measured. • For the detuning measurement, an octupole family (OD) was • changed from 0 and 13 A (the nominal is 7 A). Measurements were • made for several different AC dipole currents. • For the sextupole driving term measurement, an skew sextupole • (S2A3A) was set between −30 and 45 A (The nominal is ~0 A). • Measurements were made for several different AC dipole currents.

  3. Beta-Function Measurement (Not New) δd = νd−ν −0.04 −0.02 +0.02 0 • An AC dipole produces the • artificial beta-beat. • The true beta-function can be • extrapolated by sweeping the • driving frequency.

  4. Betatron Phase Measurement (New) δd = νd−ν −0.04 −0.03 −0.02 +0.01 +0.02 +0.04 0 • An AC dipole also produces the • artificial phase-beat. • The true betatron phase advance • can be also extrapolated by • sweeping the driving frequency. • The fit function is a bit more complex:

  5. Detuning νd<ν When the tune depends on the action, the amplitude of the driven motion is determined by the following two Eqs: νd>ν

  6. An Example of Nonlinear System Equation of Motion: HO with a Cubic Potential • Consider 1st order of k • Expansion: x = x0 + x1 +… • Leading order: 1st order: A sextupole 1. shifts the central orbit. 2. produces a higher order mode (2ν).

  7. Shifts of the Closed Orbit by a Sextupole • A skew sextupole is changed • from 0 [A] to −30 [A]. • The closed orbit changes • more for a larger kick. The phase of the orbit flips if the sign of the sextupole current is flipped.

  8. 1-νd νd 2νd 1-2νd 1-ν ν 1-3νd 3νd Higher Order Modes • Higher frequency modes appear for a larger oscillation. • The frequencies of the higher order modes change • with the driving frequency.

  9. w What do we measure? The closed orbit formula: Behavior of worW s If we know βandψ, from information of two BPM’s 1 and 2,we can construct

  10. Generating Function of the 1st Order Mode • A fixed sextupole current, different AC dipole current. • The strength of the sextupole is determined from the step size. • The kick strength depends on the action. • The action is not factored out, here

  11. Generating Function of the 3rd Order Mode • A similar analysis is possible for the 3rd order mode (Two same condition). • Two generating functions for resonances of 2νd−ν and 2νd+ν.

  12. One way to Detect Sources

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