Transverse Beam Transfer Functio ns via the Vlasov Equation
Transcript of Transverse Beam Transfer Functio ns via the Vlasov Equation
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Transverse Beam Transfer Functions via the Vlasov Equation
M. Blaskiewicz, V.H. Ranjbar C-AD BNL • Introduction • Calculating BTFs using particle simulations • Calculations with the Vlasov equation • Comparison
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Introduction • Stability and Response properties are encapsulated in Beam
Transfer Functions (BTFs). • Measurements are generally straightforward but can be
challenging. • The data can be quite rich. • Reliable predictions and improvements to the model can
result from an adequate theory. • The problem is important enough to warrant more than one
approach.
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Calculating BTFs using simulations Starter problem: Consider a kicker giving a kick f(τ) as
the bunch passes on turn 0. Let Df(k, τ) be the beam response (dipole moment) to this kick on turns k=0,1,…
Now take the forcing function Assuming linear response Hence, only one simulation is needed for all Q. NAPAC13 3
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Calculating BTFs using simulations In a real BTF with |ω1|< ω0/2=π/Trev Hence we need the impulse response to a sine kick and a cosine
kick to get the response for all frequencies. Also, BTF is one complex number and not a function of τ. Take the Fourier component of the beam response at the drive
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Calculations with the Vlasov equation Single particle equations
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BTF with f(τ) for n=2. Space charge and a Q=5 resonator wake with frτb=6. Solid lines are tracking results, crosses from Vlasov.
Zero chrom, sc and hi freq imped, comp3.plt
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Low frequency resolution BTF taken at frequency equal to inverse bunch length (n=2).
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low frequency BTF (n=0). Has a step function wake for various chromaticities.
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BTF with same wake and chrom but with f=1/4τb.(n=1) upper and lower sidebands differ.
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Like previous but now including space charge.
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Conclusions • Transverse beam transfer functions can be calculated reliably
using either simulations or the Vlasov equation. • Only 2 tracking runs are needed to get the full TBTF for
bunches short compared to the machine circumference. • The Vlasov approach requires more CPU power than tracking
for comparable results. This is especially true with large space charge.
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Discretize with linear interpolation The matrix element is left as a 2D integral. The driving terms for upper and lower sidebands are
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BTF with f(τ) for n=1. Solid lines are tracking results. Crosses are from Vlasov. Space charge is the collective effect.